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the recent proliferation of smartphones and tablets has been seen as a key enabler for anywhere , anytime wireless communications . the rise of online services , such as facebook and youtube , significantly increases the frequency of users online activities . due to this continuously increasing demand for wireless access , a tremendous amount of data is circulating over today s wireless networks . this increase in demand is straining current cellular systems , thus requiring novel approaches for network design . in order to cope with this wireless capacity crunch , device - to - device ( d2d ) communication underlaid on cellular systems , has recently emerged as a promising technique that can significantly boost the performance of wireless networks @xcite . in d2d communication , user equipments ( ues ) transmit data signals to each other over a direct link instead of through the wireless infrastructure , i.e. , the cellular network s evolved node bs ( enbs ) . the key idea is to allow direct d2d communication over the licensed band and under the control of the cellular system s operator @xcite . recent studies have shown that the majority of traffic in cellular systems consists of downloading contents such as videos or mobile applications . usually , popular contents , such as certain youtube videos , are requested more frequently than others . as a result , enbs often end up serving different mobile users with the same contents using multiple duplicate transmissions . in this case , following the enb s first transmission of the content , such content is now locally accessible to others in the same area , if ues resource blocks ( rbs ) can be shared with others . newly arriving users that are within the transmission distance can receive the old " contents directly from those users through d2d communication . here , the enb only serves users that request new " content , which has never been downloaded . through this d2d communication , we can reduce considerable redundant requests to enb , so that the traffic burden of enb can be released . our main contribution is to propose a novel approach to d2d communication , which allows to exploit the social network characteristics so as to reduce the load on the cellular system . to achieve this goal , first , we propose an approach to establish a d2d subnetwork to maintain the data transmission successfully . as a d2d subnetwork is composed by individual users , the connectivity among users can be intermittent . however , the social relations in real world tend to be stable over time . such social ties can be utilized to achieve efficient data transmission in the d2d subnetwork . we name this social relation assisted data transmission wireless network by offline social network ( offsn ) . second , we assess the amount of traffic that can be offloaded , i.e. , with which probability can the requested contents be served locally . to analyze this problem , we study the probability that a certain content is selected . this probability is affected by both external ( influence from media or friends ) and internal ( user s own interests ) factors . while users interests are difficult to predict , the external influence which is based on users selections can be easily estimated . to this end , we define an online social network ( onsn ) that encompasses users within the offsn , which reflect users online social ties and influence to each other . in this paper , we adopt practical metrics to establish offsn , and the indian buffet process to model the influence in onsn . then we can get solutions for the previous two problems . latter we will integrate offsn and onsn to carry out the traffic offloading algorithm . further more , in order to evaluate the performance of our algorithm , we will set up the chernoff bound of the number of old contents user selects . to make the analysis more accurate , we also derive the approximated probability mass function ( pmf ) and cumulative distribution function ( cdf ) of it . from the numerical results , we show that under certain circumstances , our algorithm can reduce a considerable amount of of enb s traffic . our simulations based on the real traces also proved our analysis for the traffic offloading performance . consider a cellular network with one enb and multiple users . in this system , two network layers exist over which information is disseminated . the first layer is the onsn , the platform over which users acquire the links of contents from other users . once a link is accessed , the data package of contents must be transmitted to the ue through the actual physical network . taking advantage of the social ties , the offsn represents the physical layer network in which the requested contents of links to transmit . an illustration of this proposed model is shown in fig.[fig : onsn and offsn ] . each active user in the onsn corresponds to the ue in the offsn . in the offsn , the first request of the content is served by the enb . subsequent users can thus be served by previous users who hold the content , if they are within the d2d communication distance . information dissemination in both onsn and offsn.,scaledwidth=20.0% ] in the area covered by an enb , the density of users in public areas such as office buildings and commercial sites , is much higher than that in other locations such as sideways and open fields . indeed , the majority of the data transmissions occurs in those fixed places . in such high density locations , forming d2d networks as an offsn becomes a natural process . thus , we can distinguish two types of areas : highly dense areas such as office buildings , and white " areas such as open fields . in the former , we assume that d2d networks are formed based on users social relations . while in the latter , due to the low density , the users are served directly by the enb . the offsn is a reflection of the local users social ties . proper metrics need to be adopted to depict the degree of the connections among users . in @xcite , the authors identify that human mobility shows a very high degree of temporal and spatial regularity , and that each individual returns to a few highly frequented locations with a significant probability . thus , such social ties lead to higher probabilities to transmit data among users . the contact duration distribution between two users is assumed to be a continuous distribution , which has a positive value for all real values greater than zero . in addition , users encounter duration usually centers around a mean value . so we can adopt a @xmath0 distribution which is widely used in modeling the call durations @xcite to model the call duration between two users . to find the value for the two parameter @xmath1 and @xmath2 , we need the mean and variance of the contact duration . as shown in fig . [ fig : encounter history ] , given the contact duration @xmath3 and the number of encounters @xmath4 between ue @xmath5 and ue @xmath6 corresponding to user @xmath5 and user @xmath6 in the onsn , an estimate of the expected contact duration length @xmath7 and variance @xmath8 can be given by : @xmath9 @xmath10 contact history between ue i and ue j.,scaledwidth=11.6% ] given the mean and variance of the contact duration , we can derive the contact duration distribution : @xmath11 . thus , the probability density function ( pdf ) of contact duration will be given by : @xmath12 where @xmath13 . then , we can calculate the probability of the contact durations that are qualified for data transmission . if the contact duration is not sufficient to complete a data package transmission , the communication session can not be carried out successfully . we adopt a closeness metric , @xmath14 to represent the probability of establishing a successful communication period between ue @xmath5 and ue @xmath6 , which ranges from @xmath15 to @xmath16 . the qualified contact duration is the complementary of the disqualified communication duration probability , so @xmath14 can be given by : @xmath17 where @xmath18 is the minimal contact duration required to successfully transmit one content data package , @xmath19 is the lower incomplete gamma function . hence , we can use the closeness metric @xmath14 to describe the communication probability between two ues , which can also be seen as the weight of the link between ue @xmath5 and ue @xmath6 . then , a threshold @xmath20 can be defined to filter the boundary between different offsns and white " areas . to cluster users based on metrics such as closeness , we can adopt algorithms such as those used in social networks as in @xcite . then , with a properly chosen @xmath20 , each pair in the offsn will have a strong direct neighboring relationship . onsn is the platform for content links to disseminate . we define the number of users in the onsn as @xmath21 which , in turn , corresponds to @xmath21 ues in the offsn . the total number of available contents in the onsn is denoted by @xmath22 , @xmath23 . given the large volume of content available online , @xmath24 . @xmath25 represents the set of contents that have viewing histories and @xmath26 is the set of contents that do not have any . we adopt the indian buffet process ( ibp ) @xcite model which serves as a powerful tool for getting the content popularity distribution and predicting users selections . the ibp is a stochastic process in which each diner samples from an infinite selection of dishes on offer at a buffet . the first customer will select its preferred dishes according to a poisson distribution with parameter @xmath27 . since all dishes are new to this customer , no external information exists so as to influence the selection . however , once the first customer completes the selection , the following customers will have some prior information about those dishes based on the first customer s feedback . customers learn from the previous selections to update their beliefs on the dishes and the probabilities with which they will choose the dishes . the behavior of content selection behavior in onsn is analogous to the dish selection in an ibp . if we view our onsn as an indian buffet , the online contents as dishes , and the users as customers , we can interpret the contents spreading process online by an ibp . so the probability distribution can be implemented from the ibp directly . one realization of indian buffet process.,scaledwidth=19.0% ] in fig . [ fig : ibp ] , we show one realization of an ibp . customers are labeled by ascending numbers in a sequence . the shaded block represents the @xmath28th user selected dish @xmath1 . in ibp , the first customer selects each dish with equal probability of @xmath29 , and ends up with the number of dishes following @xmath30 distribution . for subsequent customers @xmath31 , the probability of also having dish @xmath1 already belonging to previous customers is @xmath32 , where @xmath33 is the number of customers prior to @xmath28 with dish @xmath1 @xcite . repeating the same argument as the first customer , customer @xmath28 will also have @xmath34 new dishes not tasted by the previous customers following a @xmath35 distribution . the probabilities of selecting certain dishes act as the prior information @xmath36 . for old " dishes which have been tasted before , @xmath37 . for new" dishes which have not been sampled before , @xmath38 . after user @xmath28 completes its selection , @xmath39 will be updated to @xmath40 . this learning process is also illustrated in fig . [ fig : ibp ] . @xmath41 is the number of dishes that have not been sampled before user @xmath28 s selection session . in the previous section , we have introduced the basic model to formulate the subnetwork for d2d communication and predict users selection . in this section , we can integrate the two layer networks together and propose the traffic offloading algorithm based on the offsn and onsn models . as the inter - offsn interference of d2d communication can be restricted by methods such as power control @xcite@xcite , we can ignore the interference among different offsns . thus , we place an emphasis on the intra - offsn interference due to resource sharing between d2d and cellular communication . during the downlink period of d2d communication , ues will experience interference from other cellular and d2d communications as they share the same subchannels . thus , we can define the transmission rate of users served by the enb as @xmath42 , and by d2d communication with co - channel interference as @xmath43 , @xcite : @xmath44 @xmath45 where @xmath46 , @xmath47 and @xmath48 are the transmit power of enb , d2d transmitter @xmath49 and @xmath50 , respectively , @xmath51 is the channel response of the link between ue or enb @xmath5 and ue @xmath6 , @xmath52 is the additive white gaussian noise ( awgn ) at the receivers , and @xmath53 represents the presence of interference from d2d to cellular communication , satisfying @xmath54 , otherwise @xmath55 . here , @xmath56 , so @xmath57 represents the interference from the other d2d pairs that share spectrum resources with pair @xmath49 . the transmission rate of the users that are only served by the enb without underlying d2d , thus also without co - channel interference is given by : @xmath58 in the proposed model , even though enb can offload traffic by d2d communication , controlling the switching over cellular and d2d communication causes extra data transmission . thus , there exists a certain cost such as control signals transmission and information feedback during the access process @xcite . therefore , for the enb that is serving a certain user @xmath28 , we propose the following utility function : @xmath59r_d+m_n^0 r_c - m_n c_c,\ ] ] where @xmath60 is the overhead cost for controlling the resource allocation process . we propose a novel and robust algorithm that can offload the traffic of enb without any sacrifice on the quality of service . the algorithm consists of multiple stages . in the first stage , the enb collects the encounter history between users to compute the closeness @xmath14 . then , based on specific situation such as time and locations , enb chooses @xmath20 dynamically . by checking if @xmath61 , i.e. , if ue @xmath5 and ue @xmath6 satisfy the predefined closeness threshold , the enb can decide on whether to add this user into the offsn or not . by choosing a proper @xmath20 and power control , the interference among different offsns can be avoided . this process will continue until no more users can be further added to the enb s list . then , the users in the established offsn can construct a communication session with only intra - offsn interference . for websites that provide a portal to access content , such as facebook and youtube , the enb will assign a special tag . once a user visits such tagged websites , the enb will inspect whether the user is located in an offsn or a white " area . if the user is in a white " area , any requests of users will be served by the enb directly . if the user is located in an offsn , the enb will wait the user s future activities . by browsing online , current user can have the prior information @xmath62 of the content distribution in the onsn based on previous user s requests . as soon as the user requests data , the enb detects if there are any resources in the offsn , and then choose to set up d2d communication or not based on the feedback . for old content , the enb will send control signal to the ue @xmath5 that has the highest closeness @xmath63 with user @xmath28 . then , ue @xmath28 and ue @xmath5 establish a d2d communication link . even if the d2d communication is setup successfully , the enb still waits until the data transmitting process finishes . if the d2d communication fails , the enb will revert back to serve the user directly . for new contents , the enb serves the user directly . after the selection is complete , the prior updates to the posterior @xmath64 . the proposed d2d communication algorithm is summarized in algorithm @xmath16 . * offsn and onsn generation * enb collects encounter information in cellular network find the closeness @xmath14 between two ues forming onsn by the users of corresponding ues * 2 . user activity detection * * 3 . service based on onsn activities * to evaluate the traffic offloading performance of our algorithm , we derive a bound on the amount of traffic that can be offloaded . we show that this problem is equivalent to the amount of contents that have been downloaded . while those locally accessible contents is related to the number of total contents and new contents selected by the users . before we start to derive a closed - form expression on the number of old content , we first try to find its bound . here , we adopt the chernoff bound for analysis . let @xmath65 be a sequence of independent trials with @xmath66 , @xmath67 , and @xmath68 $ ] . then : for any @xmath69 , there is a bound when @xmath70 $ ] @xmath71^\mu,\ ] ] and a bound when @xmath72 @xmath73^\mu.\ ] ] for the case of our model , the total number of contents user selects is @xmath74 , the number of new contents @xmath75 . then , the number of old contents @xmath76 . hence , define the expected number of old contents @xmath77 , @xmath78 , with a chernoff bound of @xmath79^{\frac{n-1}{n}\alpha},\\ \end{split}\ ] ] when @xmath80 $ ] , and @xmath81^{\frac{n-1}{n}\alpha},\\ \end{split}\ ] ] when @xmath82 . as we can see , the number of old contents is the difference of two possion distribution which follows the skellam distribution . we can approximate pmf and cdf for the skellam distribution using the saddlepoint approximation . then , we have the approximated pmf and cdf for the number of old contents as follows : @xmath83 @xmath84 , \end{split}\ ] ] where @xmath85 and @xmath86 . with the cdf function we can get the approximate number of old contents that each user select . thus , we can estimate the traffic that can be offloaded . to evaluate the performance of our algorithm , we exploit a data set of sensor mote encounter records and corresponding social network data of a group of participants at university of st andrews by the crawdad team @xcite . in the first data set , they deployed @xmath87 t - mote invent devices over a period of @xmath88 days among @xmath87 users in the department of computer science building . this data set helps us to establish our physical layer offsn . in the second data set , they collected the participants facebook friend lists to generate a social network topology . with those information we can generate the corresponding onsn . then , we adopt the ibp to generate users selections online under the assumption that the size of content library is unbounded . we assume that the content selection process has already been performed for a number of times . thus , the enb can obtain the prior information of the content distribution . 0.3 0.3 0.3 as the d2d communication distance increases , the enb will have more possibilities for detecting available contents providers . as a result , the performance of traffic offloading will be better with a larger maximum distance . this assumption is shown in fig . [ fig : traffic and distance ] . in this figure , we can see that , increasing the maximum communication distance , yields a decrease in the enb s data rate and an increase in the amount of offloaded traffic . however , we note that , with the increase of the transmission distance , the associated ue costs ( e.g. , power consumption ) will also increase . thus , the increase of d2d communication distance will provide additional benefits to the enb , but not for users . in fig . [ fig : traffic and enb cost ] , we show the variation of the sum - rate at the enb as the cost for control signal varies . as the enb has to arrange the inter change process between cellular and d2d communication , necessary control information is needed . moreover , additional feedback signals are required for monitoring the d2d communication and checking its status . those costs will affect the traffic offloading performance of the system . in our simulation , we define the cost as the counteract to the gain in data rate from 5% to 50% . as we can see , increasing the cost on control signal , the offload traffic amount is decreased . in order to know the amount of traffic that can be offloaded , we specified a special case when @xmath89 , then plot the chernoff bound and saddlepoint approximation of the cdf of the @xmath90th user s number of old contents in fig . [ fig : bound ] . as we can see , the approximated cdf lies between the chernoff bound . we have mentioned in the previous section , the mean value of the number of old contents is @xmath91 . so there is a gap between the number of @xmath92 and @xmath93 . then we simulate the @xmath90th user s selection and plot the empirical cdf . the simulated empirical cdf also lies between the upper and lower chernoff bound as we expected . in addition , the approximated cdf line is quite close to the simulated empirical cdf line , which proves our analysis in the previous section . in this paper , we have proposed a novel approach for improving the performance of d2d communication underlaid over a cellular system , by exploiting the social ties and influence among individuals . we formed the offsn to divide the cellular network into several subnetworks for carrying out d2d communication with only intra - offsn interference . also we established the onsn to analyse the offsn users online activities . by modeling the influence among users on contents selection online using the indian buffet process , we have obtained the distribution of contents requests , and thus can get the probabilities of each contents to be requested . using our proposed algorithm , the traffic of enb has been reduced . simulation results based on real traces have shown that different parameters for enb and users will lead to different traffic offloading performances . c. xu , l. song , z. han , d. li , and b. jiao , resource allocation using a reverse iterative combinatorial auction for device - to - device underlay cellular networks , " _ ieee globe communication conference ( globecom ) _ , anaheim , ca , dec . 3 - 7 , 2012 . n. golrezaei , a. f. molisch , and a. g. dimakis , base - station assisted device - to - device communication for high - throughput wireless video networks , " _ ieee international conference on communications ( icc ) _ , pp . 7077 - 7081 , ottawa , canada , jun . 10 - 15 , 2012 . f. li and j. wu , localcom : a community - based epidemic forwarding scheme in disruption - tolerant networks , " _ proc . ieee conference sensor , mesh and ad hoc communications and networks ( secon ) _ , pp . 574 - 582 , rome , italy , jun . 22 - 26 , 2009 . j. guo , f. liu , and z. zhu estimate the call duration distribution parameters in gsm system based on k - l divergence method , " _ ieee international conference on wireless communications , networking and mobile computing ( wicom ) _ , pp . 2988 - 2991 , shanghai , china , sep . 21 - 25 , 2007 . g. bigwood , d. rehunathan ma . bateman , t. henderson , and s. bhatti , exploiting self - reported social networks for routing in ubiquitous computing environments , " _ proceedings of ieee international conference on wireless and mobile computing , networking and communication ( wimob 08 ) _ , pp . 484 - 489 , avignon , france , oct . 12 - 14 , 2008 .
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device - to - device ( d2d ) communication has seen as a major technology to overcome the imminent wireless capacity crunch and to enable new application services . in this paper
, we propose a social - aware approach for optimizing d2d communication by exploiting two layers : the social network and the physical wireless layers .
first we formulate the physical layer d2d network according to users encounter histories .
subsequently , we propose an approach , based on the so - called indian buffet process , so as to model the distribution of contents in users online social networks . given the social relations collected by the evolved node b ( enb )
, we jointly optimize the traffic offloading process in d2d communication .
in addition , we give the chernoff bound and approximated cumulative distribution function ( cdf ) of the offloaded traffic . in the simulation , we proved the effectiveness of the bound and cdf .
the numerical results based on real traces show that the proposed approach offload the traffic of enb s successfully .
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elliptic curves underpin some of the most advanced mathematics currently being pursued , most notably ( probably ) the proof of fermat s last theorem by sir andrew wiles . recreational number theory is enjoyed by many people with * no * professional mathematics qualifications . the purpose of the present survey is to show how , even a small knowledge of elliptic curves and a computer , can help solve many problems . these problems are all reasonably simple to understand but can lead to some very interesting mathematics . we start with a simple * problem : * if possible , find rational numbers @xmath0 with @xmath1 where @xmath2 is a specified non - zero integer . the first stage is to analyze the problem . we will find that solutions are related to rational points on the elliptic curve @xmath3 the rational points form a group , often denoted @xmath4 , which is finitely generated . this means that every element @xmath5 can be written @xmath6 where @xmath7 are points of infinite order , called generators , and @xmath8 is a point of finite order , called a torsion point . the subset of points of finite order is small and easily determined . for example , the above curve has @xmath9 as elements of @xmath8 , together with the point at infinity which is the group identity . the torsion points often correspond to trivial solutions of the problem , and only rarely give a solution we seek . the quantity @xmath10 is called the * rank * of the elliptic curve . it can be zero , so that there are only torsion points , which usually means no non - trivial solutions . we can estimate the rank using the famous birch and swinnerton - dyer ( bsd ) conjecture and a moderate amount of computing . if the rank is computed to be greater than one , it is often reasonably easy to find at least one generator , though there are exceptions to this rule . the real computational problems are with curves where the rank is estimated to be exactly @xmath11 . a by - product of the bsd calculations is an estimate of the height of the point - the higher the height , the more digits in the rational coordinates and so the point is harder to find ( in general ) . there are two possible height normalizations available and i use the one described in silverman @xcite , which gives values half that of pari s current * ellheight * command . let me state at the outset that my primary interest is in computing * actual * numerical values for solutions of the many problems discussed rather than just proving a solution exists . to get anywhere in this subject , you need a computer and some software . to help with the algebra in the analysis of the problems , some form of symbolic algebra package would be very useful . i tend to use an old ms - dos version of derive but this falls down sometimes , so i have been known to use maxima ( free ) or even mathematica ( not - free ) . more importantly , you need a package which does the number theory calculations on elliptic curves . the most powerful is magma , but this costs money . pari is a free alternative but does not have so many built - in capabilities . the vast bulk of my code is written using pari , since i only recently purchased magma as a retirement present to myself ! if you have an old 32-bit machine , i would recommend ubasic for development work as it is very fast . i have tried to give the correct credit where appropriate . if you think i have failed in this task , please contact me . i have also tried to make each problem description self - contained so there is quite a lot of repetition between sub - sections . any mistakes are , of course , purely my fault . i would be extremely pleased to hear from readers with problems i have not covered . what is an * elliptic curve * ? since i am aiming to make most of this report understandable to non - professionals , i will take a very simplistic approach , which will certainly appal some professionals . but you can get quite far in using elliptic curves , without worrying about topics such as cohomology or galois representations . there are several books that can be recommended on elliptic curves . at an introductory level silverman and tate @xcite is excellent . more advanced are the books by husemller @xcite , knapp @xcite and cassels @xcite . the two volumes by silverman @xcite @xcite have become the standard mathematical introduction . consider the simple cubic @xmath13 which has zeros at @xmath14 . these are the only zeros , so the curve is non - negative for @xmath15 and @xmath16 , but strictly negative for @xmath17 and @xmath18 . make the very simple change of @xmath19 to @xmath20 giving @xmath21 which gives the curve in the following graph . firstly , this curve is symmetric about the x - axis , because if @xmath22 lies on the curve so does @xmath23 . secondly , there are no points for @xmath17 or @xmath18 , since @xmath24 in these intervals . this means that the curve has two disconnected components . the first for @xmath16 is a closed shape , usually called the `` egg '' - for obvious pictorial reasons . the second for @xmath15 is usually called the infinite component . this is all very nice and pretty , but it does not help us solve our problems . the problems we consider look for integer or rational solutions , and we can link these to finding rational points on the elliptic curve . a rational point is one where both coordinates are rational . there are @xmath25 obvious rational points on this curve , namely @xmath26 , @xmath27 , and @xmath28 . a little simple search shows that @xmath29 gives @xmath30 , so @xmath31 and @xmath32 are also rational points on the curve . let @xmath33 and @xmath34 . the line joining @xmath35 to @xmath36 has equation @xmath37 , and meets the curve where @xmath38 which gives the cubic equation @xmath39 this has at most @xmath25 real solutions , and we already know two of them , @xmath40 and @xmath29 . we also know from basic algebra that the product of the three roots is @xmath41 . thus the third point of intersection must be @xmath42 which gives @xmath43 . so from two rational points we get a third rational point ! thinking about this procudure , if @xmath35 and @xmath36 are any rational points , the line joining them will have rational coefficients . so the intersection with the curve will be a cubic with rational coefficients . the product of the three roots will be the negative of the constant in the cubic and so rational . since we know two of the roots are rational , the third must be rational . but the line has rational coefficients , so the y - coordinate must also be rational . so from two rational points we get a third rational point . this operation is thus closed on the set of rational points @xmath4 . a closed binary operation is the first requirement for a group . with some further fiddling , we can define an `` addition '' operation on @xmath4 , which gives a commutative group . this is discussed further in the appendices . the crucial point is that this group is finitely generated . this means that there are points @xmath44 such that @xmath45 where @xmath46 is any rational point , @xmath47 are integers and @xmath8 is an element of a small subset of @xmath4 which is called the torsion subgroup and which can be easily computed . thus , all we need to do to find rational points is to find @xmath48 and hence use these to find a solution to our original problem . the value of @xmath10 is called the * rank * of the elliptic curve , and it can be @xmath49 . for example , @xmath50 has rank zero . the challenge is to compute these generator points . it can be very difficult , but that is what provides the challenge . let @xmath42 , @xmath51 and @xmath52 . then @xmath53 and @xmath54 . it is a natural question to ask if this happens for other integers apart from @xmath55 . @xmath56 the following lemma shows , however , that we are doomed to failure . * lemma . * the only integer solution is the one above . * we can assume , without loss of generality , that @xmath57 and so , either all of @xmath58 are positive or two of them are negative and one positive . suppose , firstly , that all the variables are positive and we can assume @xmath59 . we have @xmath60 and @xmath61 if the right - hand - side is negative then @xmath62 , contradicting @xmath63 being an integer . if @xmath64 , the only value of @xmath19 giving a positive rhs is @xmath51 , but this gives @xmath65 . @xmath66 gives @xmath67 and @xmath68 . if @xmath42 , @xmath69 , which only gives integer values when @xmath70 , again for @xmath68 . if we now assume two of @xmath58 are negative , let them be @xmath71 and @xmath19 , with @xmath72 . but implies @xmath73 , a contradiction . thus , the only purely integer solution is the one at the start of the section . we now consider @xmath74 . we have @xmath75 , and , substituting , we have that @xmath76 must satisfy the quadratic equation @xmath77 for fixed @xmath2 , we can generate rational values of @xmath71 , and test if this quadratic factors into linear factors . for @xmath78 , the following results were obtained in a simple search .equal sum and product results [ cols= " < , > , > , > " , ] ( 8,8 ) ( 0,0)(1,0)12 ( 0,0)(0,1)8 ( 0,8)(1,0)12 ( 12,0)(0,1)8 ( 0,0)(2,1)8 ( 8,4)(1,-1)4 ( 0,8)(2,-1)8 ( 8,4)(1,1)4 finally , for the @xmath79 configuration , consider the rectangle shown . we have @xmath80 with a solution to the @xmath79 configuration when @xmath81 and @xmath82 . from @xmath83 and @xmath84 we have the quadratic equation @xmath85 which must have a discriminant being a rational square to give rational @xmath86 . thus , there exists @xmath87 , such that @xmath88 defining @xmath89 and @xmath90 gives the quartic @xmath91 using mordell s method , this can be transformed to the equivalent elliptic curve @xmath92 with @xmath93 there are , thus , two free parameters @xmath94 and @xmath95 . for each possible pair , we try to find curves with rank greater than zero , and possible generators . then generate points on the curve , and values of @xmath96 and solve the quadratic for @xmath86 . from these find @xmath97 and test if they are all positive and if @xmath82 . no solution has been found , and , in fact , bremner and guy conjecture that a solution does not exist . i must admit that i agree with them . an interesting question is how small can @xmath98 be ? it is easy to adapt the code for this . consider the @xmath99 magic square @xmath100 where we assume @xmath101 . can we find such a square where the nine entries are distinct squares ? so far , only a single square with @xmath102 square entries has been found , by andrew bremner @xcite @xcite . it is @xmath103 on the principle that one solution usually implies more than one , i have spent a lot of computer time looking for more solutions . they are all based on the elliptic curve approach described by bremner , starting from making @xmath55 entries square . to illustrate the approach , suppose we take the first and third rows to all be squares , so @xmath104 and we can , without loss of generality , consider all these variables to be rational . we have @xmath105 and @xmath106 if we think in terms of vectors , the first relation says that the lengths of @xmath107 are the same . thus each vector must just be a rotation of another , so @xmath108 and @xmath109 for some angles @xmath110 and @xmath111 . these relations imply that the sines and cosines are rational , so @xmath112 and @xmath113 where @xmath114 . these give @xmath115 in terms of @xmath116 . substituting into the second relation gives a quadratic @xmath117 where @xmath118 is a fairly complicated rational function in @xmath119 . if this quadratic is to give rational values for @xmath120 , the discriminant @xmath121 must be a rational square . computing this , and clearing as many square terms as possible , we eventually arrive at the quartic in @xmath95 , assuming @xmath96 has been selected , @xmath122 @xmath123 this quartic has a rational solution when @xmath124 and @xmath125 . thus the quartic is birationally equivalent to an elliptic curve . after a reasonable amount of computer algebra , we find the relevant curve to be @xmath126 where @xmath127 with @xmath128 and @xmath129 and @xmath130 . the inverse transformation is @xmath131 the elliptic curve has @xmath25 finite points of order @xmath132 , at @xmath133 , @xmath134 and @xmath135 . there are also points of order @xmath136 when @xmath137 . the torsion subgroup is thus @xmath138 or @xmath139 . numerical testing suggests the former is always the case . if we take the denominator of the f - transformation , solve for @xmath140 , and substitute into the elliptic curve we find a point @xmath141 which gives a rational point on the curve . adding to @xmath26 gives @xmath142 which has @xmath143 , which suggests the curves have rank at least one . as a numerical example , @xmath144 , gives @xmath145 . the positive value gives @xmath146 , but the negative value gives @xmath147 . this gives @xmath148 and @xmath149 as possible values . from these , we can compute @xmath0 with @xmath150 @xmath151 with @xmath152 @xmath153 @xmath154 sadly , none of the other @xmath25 elements of the magic square are squares . we can easily perform all these calculations in pari , as well as considering the @xmath155 other distinct configurations of six squares . bremner s solution is found quickly in several configuration calculations , but no second solution has yet been discovered . i would like to express my sincere thanks to andrew bremner for all his help , and his courtesy in answering lots of questions . the standard elliptic curve over @xmath156 can be written @xmath157 with @xmath158 . firstly , we transform the equation of the curve to a simpler form . from @xmath159 we have @xmath160 so @xmath161 define @xmath162 , so @xmath163 giving @xmath164 so that @xmath165 now , define @xmath166 , giving @xmath167 and @xmath168 finally , set @xmath169 and @xmath170 leading to @xmath171 which we write as @xmath172 with @xmath101 . this is the form we use for all the elliptic curves in this survey . it has the great advantage of symmetry about the u - axis , so , if @xmath173 is a point on the curve , so is @xmath174 . we define addition of rational points in the usual chord - and - tangent way , see silverman and tate s @xcite excellent introduction to elliptic curves . if @xmath175 the line joining the points has equation @xmath176 and so substituting this into the equation gives a cubic in @xmath177 for the points where the line and curve meet . the constant of the cubic is @xmath178 which is rational , and also is @xmath179 where @xmath180 is the u - coordinate of the third intersection of the line and the curve . thus @xmath181 and the corresponding v - coordinate ( call it @xmath140 ) is also rational . thus combining two rational points in this way gives a third rational point . this gives a closed binary operation on the set of rational points , which is often denoted by @xmath4 . this is the first requirement for defining a group . to get the associative property needed for a group , we actually need to take the result of the operation to be the point @xmath182 and not the actual point of intersection @xmath183 . we define the `` addition '' of rational points as @xmath184 with this binary operation , the set of rational points can be shown to be an abelian group . the identity ( denoted by @xmath185 ) is the point at infinity , which is best approached by projectivising the curve , setting @xmath186 and @xmath187 to give @xmath188 and the point at infinity is @xmath189 . this gives the inverse of @xmath173 as @xmath174 . @xmath4 is a finitely generated group , and is isomorphic to @xmath190 , where @xmath10 is called the rank of the curve and @xmath191 is called the * torsion subgroup * and is one of * @xmath192 , * @xmath193 . as proven by mazur @xcite , though first conjectured by beppo levi much earlier in the 20th century . this means that there exist @xmath10 rational points @xmath7 such that all rational points can be written @xmath194 where @xmath195 and @xmath196 . the hard part in all the problems is finding @xmath197 . if @xmath46 is a point of order @xmath132 then @xmath198 , so @xmath199 . the only way this can happen is if the v - coordinate of @xmath46 is zero , so that the u - coordinate is a zero of @xmath200 . we want this root to be rational , but , since the coefficient of @xmath201 is @xmath11 , any rational root will actually be an integer root . thus point of order @xmath132 in @xmath4 can only come from integer roots of @xmath202 . there is a theorem of nagell and lutz , which states that for curves of the form , all rational torsion points have integer coordinates . there are two important points to note . this result does not always hold for curves of the form , and points of infinite order can have integer coordinates . if @xmath4 has a point of order @xmath132 , at @xmath203 say , if we define @xmath204 the elliptic curve reduces to the very simple but important form @xmath205 with @xmath206 . ten of the fifteen different torsion subgroup types have points of order @xmath132 so this form is a basic structure . suppose @xmath207 for a curve of this form . the point @xmath208 comes from the intersection of the tangent at @xmath46 with the curve . some simple algebra shows that the w - coordinate of @xmath209 is thus @xmath210 so must be a rational square and , by nagell - lutz , must be an integer square . this is an incredibly useful result . we now show how to transform the quartic @xmath211 with @xmath212 , into an equivalent elliptic curve . we first consider the case when @xmath213 , which is very common . if @xmath214 for some rational @xmath215 , we substitute @xmath216 and @xmath217 , giving @xmath218 thus , suppose @xmath219 we describe the method given by mordell on page @xmath220 of @xcite , with some minor modifications . we first get rid of the cubic term by making the standard substitution @xmath221 giving @xmath222 where @xmath223 we now get rid off the quartic term by defining @xmath224 , where @xmath177 is a new variable and @xmath225 is a constant to be determined . this gives the quadratic in @xmath63 @xmath226 if @xmath71 were rational , then @xmath63 would be rational , so the discriminant of this quadratic would have to be a rational square . the discriminant is a cubic in @xmath177 . we do not get a term in @xmath227 if we make @xmath228 , giving @xmath229 and , if we substitute the formulae for @xmath230 , and clear denominators we have @xmath231 with @xmath232 and @xmath233 if we set the denominator of to zero , we have @xmath234 which gives @xmath235 automatically giving a point on the curve . often , it will just be a torsion point , but , occasionally , we get the coordinates of a point of infinite order . note also , that if @xmath236 , then @xmath237 so we always have a point of order @xmath132 from @xmath238 . in this latter case , if we transfer the origin to this point of order @xmath132 and simplify we get @xmath239 with @xmath240 if @xmath241 , we need a rational point @xmath173 lying on the quartic . let @xmath242 , so that @xmath243 giving @xmath244 @xmath245 where the coefficient of @xmath246 is , in fact , @xmath247 . define , @xmath248 , and then @xmath249 giving @xmath250 @xmath251 we now , essentially , complete the square . we can write @xmath252 if we set @xmath253 and @xmath254 this gives @xmath255 and if we define @xmath256 we have @xmath257 multiply both sides by @xmath258 , giving @xmath259 which , on defining @xmath260 , gives @xmath261 define , @xmath262 and @xmath263 giving @xmath264 which is an elliptic curve . if we define @xmath265 , we transform to the form @xmath266 all of the above are easy to program in any symbolic package . a variant of the above problem , that occurs often enough to be important , is when @xmath236 , so the quartic is @xmath267 applying the above formulae gives an elliptic curve where the cubic right - hand - side has a linear factor , so the curve has a point of order @xmath132 . transforming to make this point the origin , and doing a variable scaling gives @xmath268 in this section , we describe a very simple method for searching for rational points on @xmath270 where @xmath271 . suppose @xmath272 where we set @xmath273 , with @xmath274 and @xmath275 , and @xmath276 . then @xmath277 it is clear that @xmath278 iff @xmath279 , so suppose both @xmath280 . now @xmath281 can not divide @xmath282 since , if it did , that would give @xmath283 against @xmath274 . thus @xmath284 . similarly , we must have @xmath285 , so @xmath286 which implies there must exist an integer @xmath287 with @xmath288 and @xmath289 . thus @xmath290 and we can always write @xmath2 as @xmath291 where @xmath292 is squarefree . putting @xmath293 and @xmath294 into gives @xmath295 so that @xmath296 and @xmath297 . thus @xmath298 and , since @xmath292 is squarefree by definition , @xmath299 . thus there must exist an integer @xmath140 such that @xmath300 . thus @xmath301 so that @xmath302 , and , since @xmath303 we must have @xmath304 . so we can determine possible values for @xmath292 from the squarefree factors of @xmath305 . if @xmath306 , there will be no negative @xmath292 values . now , @xmath307 and we can easily write @xmath308 where @xmath215 is squarefree . define @xmath309 , @xmath310 and @xmath311 , so that @xmath312 and @xmath313 thus we can look for solutions @xmath314 of and substitute into the right - hand - side of to see if we get a rational square . quite often @xmath315 and we can parameterize @xmath316 as @xmath317 , @xmath318 and @xmath319 , which makes for a very simple program . if @xmath320 , we can find an initial non - trivial solution @xmath321 of and use standard quadric parameterisation to give @xmath322 where @xmath323 . this gives @xmath324 this simple method provides lots of points of small heights . we can develop further by taking the numerator and denominator of this equation separately as @xmath325 @xmath326 where @xmath225 is squarefree . we try to satisfy both simultaneously . values of @xmath225 must divide the resultant of the two quadratics on the left - hand - sides of these two relations , so @xmath327 so there are only a computable finite set of possible values . suppose we find an initial solution @xmath328 to the first quadratic , then the line @xmath329 meets it again where @xmath330 substituting into the second quadratic , we must have @xmath331 with @xmath332 @xmath333 @xmath334 @xmath335 @xmath336 finding rational solutions to is a non - trivial problem . there is not enough space to describe the many tricks of the trade that should be employed to do this efficiently . this entire method has been the workhorse for finding hundreds of solutions . isogenies are a crucial tool in finding points with large heights by allowing us to consider curves which * might * have rational points with much smaller heights . the classic reference is the paper of vlu @xcite . in this case , points of order @xmath132 have coordinates @xmath338 , and other torsion points occur in pairs @xmath339 . if @xmath8 denotes the torsion subgroup , let @xmath340 be the subset consisting of all points of order @xmath132 together with * one * point from each of the pairs @xmath341 . consider , first , the curve @xmath351 with just the torsion point @xmath26 . then @xmath352 , @xmath353 and @xmath354 . thus equation is @xmath355 and setting @xmath356 gives the standard 2-isogenous curve @xmath357 suppose we find a point on the 2-isogenous curve @xmath359 . solving the left - hand equation for @xmath360 will usually give a non - rational value for @xmath71 . if , however , we form @xmath209 we find it has h - coordinate @xmath361 putting this into the left - hand equation we find @xmath362 . for a curve of the form @xmath370 , a point of inflexion would occur when @xmath371 . this gives @xmath372 but there is not a point of inflexion at @xmath373 since @xmath95 is negative at this point . we can , however , get a simpler form for the 3-isogenous curve by using this point .
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several problems which could be thought of as belonging to recreational mathematics are described .
they are all such that solutions to the problem depend on finding rational points on elliptic curves .
many of the problems considered lead to the search for points of very large height on the curves , which ( as yet ) have not been found .
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in the current studies of ultracold quantum gases , a great deal of interest has been drawn to the study of bose - einstein condensates ( becs ) loaded into optical lattices ( ols ) , i.e. , spatially periodic potentials induced by the interference between counterpropagating laser beams book1,book2,morsch06 . besides playing a crucial role in effectively tuning the interaction strength in the condensate , i.e. , the ratio between the kinetic and interaction energies @xcite , ols offer an extremely useful tool for studies of the transition between the superfluid and mott - insulator states @xcite , and for the investigation of effects in the matter - wave dynamics due to the interplay between nonlinearity and the quasi - discreteness , which is induced by a deep lattice potential @xcite . the mean - field dynamics of the bec loaded into the ol is described by the cubic gross - pitaevskii equation ( gpe ) with the periodic potential book1,book2,morsch06 . the respective bogoliubov s excitation spectrum features a band structure , similar to electronic bloch bands in solid state . if the ol potential is deep enough , the lowest - band dynamics may be approximated by the discrete nonlinear schrdinger ( nls ) equation trombettoni01 . using this correspondence , the bec dynamics was studied in the framework of the nonlinear - lattice theory , see works trombettoni01,abdullaev01,alfimov02 and short review @xcite . the presence of the ol gives raise to energetic and dynamical instabilities , which have been predicted theoretically wu01,smerzi02,konotop02,wu03,menotti03,taylor03,kramer03,kramer05 and studied experimentally @xcite . an important application of the ols is their use for the creation and stabilization of matter - wave solitons . in particular , the periodic potential gives rise to localized gap solitons in the case of _ repulsive _ two - body interactions , as was predicted theoretically @xcite and demonstrated experimentally @xcite ( with the attractive interactions , bright matter - wave solitons were created and observed in condensates of @xmath0li @xcite and @xmath1rb weiman atoms ) . more generally , the use of time- and space - modulated fields acting on atoms is a powerful tool for the control of soliton properties @xcite ; for instance , while the gpe without external potentials admits stable soliton solutions only in the 1d geometry sulem99,ablowitz04 , ol potentials can stabilize solitons in any higher dimension @xcite . unlike 1d solitons , a necessary existence condition for their multidimensional counterparts , stabilized by means of ols , is that the soliton s norm must exceed a certain threshold value . another very useful tool frequently used in experiments with ultracold atomic gases is the control of the strength and sign of two - body interactions by means of an external magnetic field near the feshbach resonance @xcite . further , recent works proposed to exploit the possibility to control the strength of _ three - body _ interactions between atoms , independently from the control of the two - body collisions @xcite . one motivation for such studies is related to the possibility of creating new exotic strongly correlated phases in ultracold gases . indeed , quantum phases , such as topological ones or spin liquids , turn out to be ground - states of the hamiltonian including three- or multi - body - interaction terms , an example being fractional quantum - hall states described by pfaffian wave functions @xcite . in a recent work @xcite , a 1d bose gas with @xmath2-body attractive interactions was studied in the mean - field approximation , with the objective to create highly degenerate ground - states of hamiltonians including many - body terms . for the three - body interactions ( @xmath3 ) , the system is described by a _ gpe , i.e. , the respective term in the energy density is proportional to @xmath4 , where @xmath5 is the single - atom mean - field wave function ( in the general case , a similar term is proportional to @xmath6 ) . soliton solutions can be found for each @xmath2 , but they represent the stable ground - state , with negative energy ( which is defined as per eqs . ( [ e - functional ] ) and ( [ h0 ] ) , see below ) , only for @xmath7 , being unstable excited states with positive energy at @xmath8 . for @xmath3 , soliton solutions are 1d counterparts of the well - known townes solitons @xcite , which play the role of the separatrix between collapsing and decaying localized states . the townes - like solitons with fixed norm ( which is @xmath9 , in the notation adopted below ) exist only at a single critical value of the interaction strength , at which they feature the infinite degeneracy fersino08 : _ all _ the solitonic wave functions , @xmath10 ^{-1/2}$ ] , with arbitrary width @xmath11 ( see eqs . ( [ critical - value ] ) and ( quintic-1 ) below ) , have _ zero energy _ but different values of the chemical potential , @xmath12 . a relevant issue is how this infinite degeneracy is lifted by an external potential , especially by a periodic one corresponding to the ol @xcite . when the two - body interaction is present , the mean - field equation is the gpe with the cubic - quintic ( cq ) nonlinearity @xcite . as said above , it has been shown @xcite that it is possible to tune the strength of the two - body interactions independently from the three - body ones . in addition to that , in the framework of the effective gpe for the bec loaded into a nearly 1d ( cigar - shaped " ) trap with tight transverse confinement , an effective attractive quintic term appears , in the absence of any three - atom interactions , as a manifestation of the residual deviation from the one - dimensionality @xcite . in any case , if the two - body interaction is repulsive while its three - body counterpart is attractive , soliton solutions to the cq gpe can be found in an exact analytical form ( in the free space ) , but they feature an unstable eigenvalue in the bogoliubov - de gennes spectrum of small perturbations around them @xcite , while the instability of the townes - like solitons in the quintic equation is subexponential , being accounted for by a zero eigenvalue . the issue we address in this paper is the possibility to stabilize such solitons by means of the ol potential . previously , the stabilization of originally unstable solitons by means of the ol was considered , in the 2d @xcite and 1d @xcite settings alike , only for localized states of the townes type ( recently , the stabilization of 2d solitons against the _ supercritical collapse _ by the ol was also demonstrated in the cq model in 2d , with both cubic and quintic terms being attractive @xcite ) . it was found that the ol with any value of its strength ( i.e. , with zero threshold ) opens a _ stability window _ around the critical point corresponding to the townes solitons . in this work , we demonstrate that the ol opens a stability window for solitons in the cq model ( with the repulsive cubic and attractive quintic terms ) too , but only if the lattice strength exceeds a _ finite _ threshold value . apart from the context of bec , where the nonlinearity degree is related to the number of atoms simultaneously involved in the contact interaction , nls equations with the power - law and cq nonlinearities are also known as spatial - domain models of the light propagation in self - focusing media kivshar03 ( for a brief overview of optical models based on the cq - nls equation , including references to experimental realizations , see recent works @xcite ) . in the case of the cubic nonlinearity ( the kerr medium ) , effects of imprinted lattices on the transmission of light beams have been investigated both in local @xcite and nonlocal @xcite models . the paper is structured as follows . in section ii , we introduce the cq gpe corresponding to the mean - field description of the 1d bose gas with two - body repulsive and three - body attractive interactions . properties of the ( unstable ) soliton solutions to this equation are also recapitulated in section ii . in section iii , we use the variational approximation ( va ) ( see ref . @xcite for a review ) to discuss effects of the ol on the solitons . we introduce an appropriate ansatz and compute the corresponding energy . the limit of the vanishing two - body interaction is considered too and compared to previous results @xcite . in section iv , the stability region for the soliton solution in the presence of the repulsive two - body interaction and ol is determined and compared with numerical findings . the effect of an additional harmonic - trap potential is studied in section v , showing that the stability region depends on the matching between minima of the periodic potential and the location of the minimum of the harmonic trap . in section vi we present our conclusions . the quantum many - body hamiltonian for the 1d bose gas with @xmath2-body contact attractive interactions is @xmath13 ^{% \mathcal{n}}\left [ \hat{\psi}(x)\right ] ^{\mathcal{n}}\right\ } , \label{ham - contact}\]]where @xmath14 is the bosonic - field operator , @xmath15 is the nonlinearity strength and @xmath16is the single - particle hamiltonian , @xmath17 being the external potential . the case of @xmath7 in the homogeneous limit ( @xmath18 ) corresponds to the integrable lieb - liniger model @xcite . for attractive interactions ( @xmath15 ) , its analytical solution was obtained by means of the bethe ansatz @xcite and for a large number of particles , @xmath19 , the energy of the exact ground - state solution coincides with that obtained in the mean - field approximation calogero75 . in the attractive lieb - liniger model , a finite ground - state energy per particle is provided by fixing product @xmath20 to a constant value @xcite , while for @xmath21 one has to set @xmath22 @xcite . in the heisenberg representation , the equation of motion for field @xmath23 is @xmath24 = \hat{h}_{0}\hat{\psi}-c\left ( \hat{\psi}^{\dag } \right ) ^{\mathcal{n% } -1}\left ( \hat{\psi}\right ) ^{\mathcal{n}-1}\hat{\psi}. \label{dyn}\]]the mean - field approximation reduces eq . ( [ dyn ] ) to the corresponding gpe with the power - law nonlinearity , @xmath25where the macroscopic wave function @xmath26 is normalized to the total number of atoms , @xmath19 , and the nonlinearity degree is related to the order of the multi - body interactions , @xmath2 : @xmath27thus , the usual two - body interaction ( @xmath7 ) corresponds to @xmath28 , and the three - body interaction ( @xmath3 ) to @xmath29 . equation ( [ gnls ] ) conserves the energy , @xmath30 \psi ( x ) , \label{e - functional}\]]which is the classical counterpart of quantum hamiltonian ( [ ham - contact ] ) . in eq . ( [ h0 ] ) , @xmath17 is the external trapping potential , which typically includes a superposition of an harmonic magnetic trap and periodic ol potential , @xmath31 , where the harmonic confining term is @xmath32 . we take the periodic potential as @xmath33 , where @xmath34 is proportional to the power of the laser beams which build the ol , and @xmath35 , with @xmath36 ; here , @xmath37 is the wavelength of the beams , and @xmath38 the angle between them ( the period of the lattice is @xmath39 ) . parameter @xmath40 measures a mismatch between the minimum of the parabolic potential ( at @xmath41 ) and the closest local minimum of the lattice potential : when @xmath42 ( @xmath43 ) a minimum ( maximum ) of @xmath44 coincides with the minimum of @xmath45 . in fact , except for section v , we consider the situation without the parabolic trap ( i.e. , @xmath46 ) , therefore we set @xmath42 in this case . the time - independent power - law gpe corresponding to eq . ( [ gnls ] ) is ( from now on , we use normalized units , with @xmath47 and @xmath48 ) @xmath49 \psi ( x)=\mu \psi ( x ) , \label{gnls - mu}\]]where @xmath50 is the chemical potential , @xmath51 and the norm of the wave function is @xmath9 . in the free - space cubic model ( @xmath28 and @xmath52 ) , eq . ( [ gnls - mu ] ) is the integrable nls equation , whose multi - soliton solutions can be obtained by means of the inverse scattering method @xcite . the commonly known single - soliton nls solution is @xmath53where @xmath54 and @xmath55 is a real amplitude , the respective value of the chemical potential being @xmath56 . for a general value of @xmath57 , the integrability is lost even in the absence of the external potential @xcite ; nevertheless , the respective single - soliton solutions can be found in an explicit form ablowitz81,polyanin04 . for the attractive three - body interactions ( @xmath29 ) , eq . ( [ gnls ] ) is the self - focusing quintic gpe , whose stationary version is @xmath58 \psi ( x)=\mu \psi ( x ) . \label{quintic - mu}\]]for @xmath59 , if one fixes coefficient @xmath60 in front of the interaction term , the townes - like solitons exist for a particular value of the norm of the wave function @xcite . on the other hand , fixing the normalization of the wave function ( recall that the norm is @xmath9 in our units ) amounts , for @xmath61 , to fixing a relation among the chemical potential and the interaction strength @xcite , so that for each @xmath60 it is possible to obtain a single soliton solution ( although , as mentioned above , these solutions provide the ground - state in the infinite system only for @xmath62 , i.e. , for @xmath63 ) . however , for @xmath29 ( i.e. , @xmath3 ) chemical potential @xmath50 remains indefinite , assuming arbitrary negative values , while the soliton solution of the form @xmath64satisfies the unitary normalization condition at a single ( critical ) value of the interaction strength @xcite , @xmath65at @xmath66 , all solutions ( [ quintic-1 ] ) share a common value of the energy , which is simply @xmath67 @xcite , as follows from eqs . ( [ e - functional ] ) and ( [ critical - value ] ) . if the two - body interaction is added to the three - body attraction , the mean - field equation is the gpe with the cq nonlinearity , @xmath68 \psi ( x)=\mu \psi ( x ) . \label{cubic - quintic - mu}\]]as said above , we chiefly focus on the case of the _ repulsive _ two - body interactions , i.e. , @xmath69 . a family of exact soliton solutions to eq . ( [ cubic - quintic - mu ] ) with @xmath59 can be obtained in the exact form @xcite , which , for @xmath69 , is @xmath70where @xmath71 , and the maximum value of the density , at the soliton s center , is @xmath72a simple derivation of eq . ( [ cubic - quintic-1 ] ) is presented in appendix a. obviously , for @xmath73 solution ( [ cubic - quintic-1 ] ) reduces to townes - like soliton ( [ quintic-1 ] ) . imposing the above - mentioned normalization,@xmath74on solution ( [ cubic - quintic-1 ] ) , one arrives at relation@xmath75from where it follows that , for @xmath76 , soliton solutions with @xmath77 satisfying normalization condition ( [ 1 ] ) exist for @xmath78 . however , these solutions are unstable @xcite ( in particular , because they do not satisfy the vakhitov - kolokolov stability criterion kolokolov73 ) . in the following section we discuss how the ol can stabilize such localized solutions . both for @xmath79 and @xmath80 ( @xmath7 and @xmath81 ) , and for the gpe with the mixed cq nonlinearity , the presence of the periodic potential makes it necessary to resort to approximate methods for finding solitons . to this end , we use the va ( variational approximation ) @xcite based on the _ ansatz _ which yields exact soliton solution ( quintic-1 ) of the quintic nls equation in the absence of the external potential : @xmath82here , width @xmath11 is the variational parameter to be determined by the minimization of the energy , while amplitude @xmath55 will be found from normalization condition ( [ 1 ] ) . we expect that ansatz ( [ var ] ) , which does not explicitly include the modulation of the wave function induced by the ol , may give a reasonable estimate of the soliton s energy for sufficiently small values of ol strength @xmath34 in eq . ( [ ol ] ) , cf . the known result for the 2d equation with the cubic nonlinearity ( @xmath79 ) and ol potential @xcite . in the case of the 3d gpe which includes the cubic term and harmonic trap , this approach leads to an estimate for the critical value of the number of atoms above which the condensate collapses , that was found to be in a reasonable agreement with results produced by the numerical solution of the gpe fetter95,ruprecht95 . in 1d , the va based on the gaussian ansatz also provides for quite an accurate approximation to exact soliton solution ( [ cubic-1 ] @xcite . similar analyses carried out in the 1d model including the cubic term and ol @xcite have demonstrated that ( unlike the 2d and 3d cases ) the 1d soliton trapped in the ol potential does not have an existence threshold in terms of its norm ( number of atoms ) . the energy to be minimized in the framework of the va is obtained by inserting ansatz ( [ var ] ) in the gpe energy functional given by eq . ( e - functional ) . the kinetic and quintic - interaction energy terms in the functional both scale as @xmath83 ; then , the energy per particle computed from expression ( [ e - functional ] ) is @xmath84 , \label{e3}\]]@xmath85where @xmath86 is defined in eq . ( [ critical - value ] ) . for @xmath52 ( without the ol ) , the scenario discussed in the previous section for the uniform cq gpe with the attractive three - body and repulsive two - body interactions is recovered , as energy ( [ e3 ] ) reduces in that case to @xmath87for @xmath73 , the energy is positive when @xmath88 ( i.e. , @xmath89 , see eq . ( [ beta ] ) ) and vanishes at @xmath90 ; for @xmath91 ( i.e. , @xmath92 ) one obtains @xmath67 , in agreement with the above - mentioned exact result showing the infinite degeneracy of soliton family ( [ quintic-1 ] ) , while for @xmath93 the energy is negative and diverges ( to @xmath94 ) at @xmath95 , signaling , in terms of the va , the onset of the collapse . with @xmath76 , expression ( [ e3-homog ] ) does not give rise to any minimum of the energy , which agrees with the known fact of the instability of all the solitons in this case @xcite . a detailed study of minima of variational energy ( [ e3 ] ) is presented in appendix b. in the following subsection , we consider the case of the self - focusing quintic gpe in the presence of the ol ( @xmath96 ) , while the discussion of the general case ( @xmath97 ) is given in section iv . here we address the stability of localized variational mode ( [ var ] ) , for different values the ol parameters , strength @xmath34 and wavenumber @xmath98 , keeping @xmath73 . the results of the analysis of minima of the variational energy ( [ e3 ] ) , presented in appendix b , can be summarized as follows ( see also fig . [ fig1 ] ) : for @xmath99 , the infinitely deep minimum of the energy is obtained at @xmath95 , which corresponds to the collapse , as shown in fig . [ fig1](a ) . for @xmath100 , the collapse may be avoided , and three possibilities arise : there exists another special value , @xmath101 , such that for every @xmath60 between @xmath102 and @xmath86 the energy has a minimum at @xmath103 and a maximum at @xmath104 , while for @xmath105 the energy does not have a minimum at any finite value of @xmath11 , see fig . [ fig1](d ) . actually , two different situations should be distinguished for @xmath106 : there exists a specific value ( refer to appendix b ) , @xmath107(with @xmath108 ) such that , for @xmath109 , the energy has a _ global _ minimum at @xmath110 ( which , thus , represents the _ ground - state _ of the boson gas in this situation ) , while , for @xmath111 , the energy minimum at @xmath110 is a _ local _ one . in other words , taking into regard the fact that , as shown by eq . ( [ e3 ] ) , the energy - per - particle approaches value @xmath112 at large @xmath11 , we conclude that , for @xmath113 ( @xmath111 ) , the energy satisfies inequality @xmath114 ( @xmath115 ) , as showed in figs . [ fig1](b , c ) . obtained in the framework of the quintic gpe versus @xmath116 ( in units of @xmath117 ) for @xmath118 ( a ) ; @xmath119 ( b ) ; @xmath120 ( c ) ; @xmath121 ( d ) . in ( a ) the solid ( dotted ) line is the energy for @xmath93 ( @xmath66 ) ; in ( b)-(c ) , points of the energy minimum and maximum , @xmath122 and @xmath123 , are indicated . ] from the above analysis , we infer that for @xmath124 the ground - state is a delocalized one ( although the metastable state , corresponding to the above - mentioned local energy minimum , exists for @xmath111 ) , for @xmath119 the ground - state is represented by a finite - size soliton configuration ( in agreement with ref . @xcite ) and for @xmath93 it is collapsing . equation ( [ c_2_star ] ) shows that the width of the stability region depends on ratio @xmath125 : keeping fixed all other parameters , the decrease of the lattice spacing ( i.e. , the increase of @xmath98 ) leads to a reduction of the stability region . equation ( [ c_2_star ] ) also shows that for @xmath126 the va formally predicts @xmath127 : however , for @xmath128 , the ground - state is delocalized and the variational ansatz ( [ var ] ) can not be used , as it does not take into account the modulation induced by the deep ol potential . in fig . [ deloc-1 ] , we plot the numerically found ground - state of the quintic gpe in a 1d box ( @xmath129 ) . it is seen that , with the increase of @xmath130 , the configuration becomes broader , until a critical value is reached , as discussed in @xcite . in the inset of fig . [ deloc-1 ] we plot the squared width @xmath131 of the numerically found ground - state @xmath132 versus @xmath60 , which makes the delocalization transition evident : for @xmath124 the width @xmath133 is @xmath134 , while around @xmath135 the width suddenly decreases . variational estimate ( [ c_2_star ] ) for the critical value @xmath136 , as predicted by the va ( see eq . ( [ c_2_star ] ) ) , is displayed in fig . [ comparison_quintic ] , together with numerical results . one observes a reasonable agreement between them , especially for small @xmath34 , which is due both to the use of the more adequate ansatz ( [ var ] ) , rather than a gaussian , and also because @xmath137 is found as the value at which the global ( rather than local ) minimum disappears . . solid lines , starting from the narrowest configuration , refer to @xmath138 ( recall that @xmath139 ) , and the dashed line refers to @xmath140 . parameters are @xmath141 , @xmath142 and @xmath143 . inset : squared width @xmath144 of the ground - state as a function of @xmath60 ( the dot - dashed line is a guide to the eye ) . critical value @xmath145 obtained from the numerical analysis is @xmath146 , which should be compared with the corresponding value ( [ c_2_star ] ) predicted by the variational approximation , @xmath147 . ] as a function of @xmath148 , according eq . ( [ c_2_star ] ) ( for the quintic gpe ) , the dashed line corresponding to @xmath149 . discrete symbols represent results obtained from the numerical solution of the quintic gpe . they designate the transition form the localized ground - state to the extended one ( parameters are the same as in fig . ( [ deloc-1 ] ) . according to the variational approximation , the ground - state is delocalized ( @xmath150 ) below the dotted line , and it collapses ( @xmath151 ) for @xmath60 above the dashed line . ] the most interesting situation occurs when the two - body repulsive interaction ( @xmath76 ) competes with the attractive three - body collisions ( @xmath15 ) . as said above , all solitons in the free space ( @xmath52 ) are strongly unstable in this situation @xcite , and the possibility of their stabilization by the ol was not studied before . the analysis of variational energy ( [ e3 ] ) , presented in appendix b , yields the following results for this case . for @xmath93 , the energy does not have a minimum at finite @xmath11 , hence the ol can not stabilize the solitons in this case . if @xmath66 , the energy has a global minimum at a finite value of @xmath11 , when @xmath152for @xmath88 , the energy features a global minimum at finite @xmath11 for @xmath153 , where the modified critical value is @xmath154cf . definition ( [ c_2_star ] ) for @xmath155 . the value @xmath137 depends upon @xmath156 , vanishing for @xmath156 larger than the critical value @xmath157 . this means that , to balance the destabilizing effect of the repulsive two - body interactions , the strength of the periodic potential , @xmath34 , must _ exceed _ its own critical value , @xmath158otherwise , eq . ( [ c_2_star_g ] ) yields @xmath159 , i.e. , the ol can not stabilize the solitons . in fig . [ deloc-2 ] we plot the numerically found ground - state of cq gpe ( [ cubic - quintic - mu ] ) for several values of @xmath34 . it is seen that , at small @xmath34 , the wave function @xmath5 remains delocalized , until a critical value is reached . in the inset of fig . [ deloc-2 ] the squared width of the numerically generated ground - state is plotted versus @xmath34 . in fig . [ comparison_epsilon ] , we compare critical value @xmath160 , as given by eq . ( [ epsilon_critical ] ) , with numerical results : for small @xmath161 , the predicted linear dependence of @xmath160 on @xmath161 is well corroborated by the numerical results , the relative error in the slope being @xmath162 . in principle , the comparison between variational estimate ( [ epsilon_critical ] ) and numerical results might be further improved by choosing a variational wave function which , in the limit of @xmath52 ( uniform space ) would reproduce exact cq soliton ( [ cubic - quintic-1 ] ) . however , the calculations with such an ansatz are extremely cumbersome . . solid lines , starting from the narrowest wave function , refer to @xmath163 , and the dashed line refers to @xmath164 . parameters are @xmath165 , @xmath166 , @xmath142 , @xmath167 . inset : the squared width of the ground - state versus @xmath168 ( the dot - dashed line is a guide to the eye ) . critical value @xmath169 obtained from the numerical data is @xmath170 , which should be compared to the variational prediction given by eq . ( [ epsilon_critical ] ) , which is @xmath171 . ] versus @xmath172 as given by eq . ( [ epsilon_critical ] ) . symbols refer to results obtained from the numerical solution for the ground - state of the cubic - quintic gpe . they represent the delocalization transition . the parameters are the same as in fig . ( [ deloc-2 ] ) . ] in this section we aim to use the variational approximation based on ansatz ( [ var ] ) for examining the combined effect of the parabolic trapping potential acting along with an ol , i.e. , we take eq . ( [ quintic - mu ] ) with external potential@xmath173cf . ( [ ol ] ) , and disregard binary collisions ( @xmath73 ) . value @xmath174 ( @xmath175 ) corresponds to the matching ( largest mismatch ) between the minimum of the harmonic potential and a local minimum of the lattice potential . the respective variational energy is obtained from ( e - functional ) with potential ( [ full - v ] ) : @xmath176 . \label{e3-tot}\ ] ] with @xmath177 , the soliton is stable for @xmath88 , and it collapses otherwise . with @xmath178 , a richer behavior is predicted by the va . the system does stabilize for @xmath88 , while , for @xmath93 , the presence of the mismatched harmonic trap gives rise to a metastability region . since @xmath179 as @xmath95 and @xmath180 as @xmath90 , one can encounter two possibilities : either @xmath181 is positive for all @xmath11 ( and there are no energy minima ) , or equation @xmath182 has two roots , corresponding to a local minimum and a maximum . the equation for the value of @xmath11 at which energy ( [ e3-tot ] ) reaches the local minimum is @xmath183where @xmath184 , and @xmath185@xmath186one can see that , for @xmath66 ( i.e. , @xmath92 ) , eq . ( [ cond - trap ] ) does not have a nonvanishing solution if @xmath98 is smaller than a critical value , @xmath187while it has a nonvanishing solution for @xmath188 . actually , for @xmath93 ( i.e. , @xmath189 ) , eq . ( [ cond - trap ] ) with @xmath190 has two nonvanishing roots , one of which is a local minimum , while such roots do not exist for @xmath191 . for @xmath188 , the right - hand side of eq . ( [ cond - trap ] ) has a maximum value , which fixes the maximum value of @xmath192 , i.e. , the maximum value of @xmath60 , which we refer to as @xmath193 . then , for @xmath194 , the variational energy does not have a local minimum . for @xmath195 there appears a finite metastability region , in terms of wavenumber @xmath98 , as illustrated by fig . [ q ] . in other words , for fixed @xmath60 , metastable states appear at large values of @xmath34 . plane of the model ( including the parabolic trap ) the metastable region from the unstable one . the parameters are @xmath196 , @xmath197 , and @xmath198 . ] in this work we have studied the effect of the ol ( optical lattice ) on the 1d bose gas with attractive three - body and repulsive two - body interactions , described by the gpe ( gross - pitaevskii equation ) with the cq ( cubic - quintic ) nonlinearity . actually , the effective quintic attractive term in the gpe may be induced by the residual deviation of the condensate , tightly trapped in a cigar - shaped confining potential , from the one - dimensionality ( when the three body losses are negligible ) @xcite or by three - body interaction terms between atoms according to recent proposals @xcite . in the absence of an external potential , soliton solutions to this equation with the cq nonlinearity are known in the exact form , but they all are strongly unstable . we have demonstrated that the ol opens a stability window for the solitons , provided that the ol strength , @xmath34 , exceeds a finite minimum value . the size of the stability window depends on @xmath199 , where @xmath98 is the ol s wavenumber . we have also considered effects of the additional harmonic trap , finding that , if the quintic nonlinearity is strong enough ( @xmath99 ) , a metastability region may arise , depending on the mismatch between minima of the periodic potential and harmonic trap . assuming that @xmath200 is real , we look for localized solutions to the cq nls equation , @xmath201with @xmath15 and @xmath69 . interpreting @xmath202 as a formal time variable and @xmath203 as the coordinate of a particle , eq . ( [ cubic - quintic - mu - app ] ) formally corresponds to the newton s equation of motion of this particle , @xmath204where the effective mass is @xmath205 , and the potential is @xmath206with an arbitrary additive constant chosen so as to have @xmath207 . potential ( [ potential - app ] ) for @xmath77 , which corresponds to normalizable solutions , is plotted in fig . [ potential ] . condition @xmath208 yields expression ( [ a - cubic - quintic ] ) for the soliton s amplitude . for @xmath209 . ] further , we make use of the conservation of the corresponding hamiltonian , @xmath210the boundary conditions for localized solutions , @xmath211 , @xmath212 , select @xmath213 in eq . ( [ energy - app ] ) . taking into regard the fact that @xmath214 for @xmath215 , and looking for solutions with @xmath216 at @xmath217 , one obtains from here the soliton solution in an implicit form , @xmath218it further follows from eq . ( [ quadr - app-1 ] ) that@xmath219with @xmath220 . in eq . ( [ quadr - app-2 ] ) , we use notation @xmath221 and @xmath222 . thus , from eq . ( quadr - app-2 ) one obtains @xmath223 ^{2}+4a^{2}a^{4}% \mathcal{e}^{2}}. \label{quadr - app-3}\]]one can easily check that this expression yields @xmath224 for @xmath73 , and that @xmath225 , as it must be . finally , using relation @xmath226 , one obtains eq . ( cubic - quintic-1 ) from eq . ( [ quadr - app-3 ] ) , after a straightforward algebra . in this appendix we aim to study minima of variational energy ( [ e3 ] ) . when @xmath73 , one sees that , for @xmath93 , the energy per particle tends to @xmath94 at @xmath95 , and to @xmath112 at @xmath227 . then , with regard to @xmath228 , no local ( metastable ) minima exist , and variational wave function ( [ var ] ) is not the ground - state for any finite width . for @xmath66 , one obtains the global minimum at @xmath229 , which implies the collapse . for @xmath230 , the situation is different : @xmath231 as @xmath232 ( because @xmath89 ) , and @xmath233 for @xmath234 . then , it is necessary to find the value of @xmath192 at which derivative @xmath181 has two real zeros . introducing the parameter @xmath235with @xmath236 defined as per eq . ( [ beta ] ) , one can write condition @xmath182 as @xmath237where @xmath238 , as defined above . equation ( [ t - cond ] ) can be satisfied if @xmath239 is smaller than a maximum value , @xmath240 , and it then has two roots , @xmath241 and @xmath242 , which correspond , respectively to the minimum at @xmath110 , and maximum at @xmath104 ( see fig . [ fig1 ] ) . for @xmath243 , eq . ( [ t ] ) has no roots , hence the variational energy has no minima at finite values of the soliton s width , @xmath11 . a plot of @xmath241 as a function of @xmath239 is presented in fig . [ fig_theta ] , where the maximum value of @xmath241 is @xmath244 . the energy minimum at @xmath241 is a global one if @xmath245 ; using eq . ( [ e3 ] ) , this condition reads @xmath246as one can see from fig . [ fig_theta ] , condition ( [ condition - app ] ) is satisfied for @xmath247 , where @xmath248 ; then , a global minimum exists only for @xmath249 , while for @xmath250 the minimum is local , corresponding to a metastable state . using the value of @xmath137 and definition ( [ t ] ) , one arrives at eq . ( [ c_2_star ] ) . as a function of parameter @xmath239 ( defined in eq . ( [ t - cond ] ) ) for @xmath73 . the dashed line is the plot of function @xmath251 versus @xmath239 . the maximum value of @xmath252 at @xmath253 is indicated . ] for @xmath76 ( recall it corresponds to the two - body repulsion ) , variational energy ( [ e3 ] ) for @xmath93 does not have a minimum at finite values of @xmath11 . however , for @xmath66 a finite minimum is possible . indeed , with definition of @xmath156 as per eq . ( [ g - cond - text ] ) , condition @xmath182 can be written as @xmath254for @xmath255 , eq . ( [ g - cond ] ) has two roots . by imposing the condition that the value of the energy at @xmath110 be smaller than @xmath112 , one gets @xmath256 . then , similar to the situation considered above , a global minimum exists only for @xmath257 , while for @xmath258 the minimum is local . for @xmath88 , condition @xmath259 reads @xmath260one can see that condition ( [ t - g - cond ] ) is satisfied for @xmath261 , with @xmath262 . then , for @xmath263 , i.e. , for @xmath34 small enough , the variational energy does not have a minimum . imposing the condition that the minimum is global leads to @xmath247 , with @xmath264 . then , for @xmath265 , i.e. for @xmath34 smaller than a critical value , the variational energy can not have a _ global _ minimum at a finite value of @xmath11 , i.e. , localized states can not realize a global minimum . functions @xmath266 and @xmath267 are plotted in fig . [ t - vs - g ] ; in fig . [ theta - max ] , we plot maximum value @xmath268 of @xmath241 for @xmath269 , as a function of @xmath156 . _ acknowledgement : _ we thank l. salasnich for useful comments and discussions . this work was partially supported by esf project instans and by miur projects quantum field theory and statistical mechanics in low dimensions " , and quantum noise in mesoscopic systems " . abdullaev , b. b. baizakov , s. a. darmanyan , v. v. konotop , and m. salerno , phys . a * 64 * , 043606 ( 2001 ) ; i. carusotto , d. embriaco , and g. c. la rocca , _ ibid_. * 65 * , ( 2002 ) 053611 ; b. b. baizakov , v. v. konotop , and m. salerno , j. phys . b * 35 * , ( 2002 ) 5105 ; e. a. ostrovskaya and y. s. kivshar , phys . * 90 * , ( 2003 ) 160407 ; opt . exp . * 12 * , ( 2004 ) 19
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we study the effect of an optical lattice ( ol ) on the ground - state properties of one - dimensional ultracold bosons with three - body attraction and two - body repulsion , which are described by a cubic - quintic gross - pitaevskii equation with a periodic potential . without the ol and with a vanishing two - body interaction term , soliton solutions of the townes type are possible only at a critical value of the three - body interaction strength , at which an infinite degeneracy of the ground - state occurs ; a repulsive two - body interaction makes such localized solutions unstable .
we show that the ol opens a stability window around the critical point when the strength of the periodic potential is above a critical threshold .
we also consider the effect of an external parabolic trap , studying how the stability of the solitons depends on matching between minima of the periodic potential and the minimum of the parabolic trap .
| 12,035 | 241 |
the methods used to calculate the energy and the lifetime of a resonance state are numerous @xcite and , in some cases , has been put forward over strong foundations @xcite . however , the analysis of the numerical results of a particular method when applied to a given problem is far from direct . the complex scaling ( complex dilatation ) method @xcite , when the hamiltonian @xmath5 allows its use , reveals a resonance state by the appearance of an isolated complex eigenvalue on the spectrum of the non - hermitian complex scaled hamiltonian , @xmath6 @xcite . of course in an actual implementation the rotation angle @xmath7 must be large enough to rotate the continuum part of the spectrum beyond the resonance s complex eigenvalue . moreover , since most calculations are performed using finite variational expansions it is necessary to study the numerical data to decide which result is the most accurate . to worsen things the variational basis sets usually depend on one ( or more ) non - linear parameter . for bound states the non - linear parameter is chosen in order to obtain the lowest variational eigenvalue . for resonance states things are not so simple since they are embedded in the continuum . the complex virial theorem together with some graphical methods @xcite allows to pick the best numerical solution of a given problem , which corresponds to the stabilized points in the @xmath7 trajectories @xcite . other methods to calculate the energy and lifetime of the resonance , based on the numerical solution of complex hamiltonians , also have to deal with the problem of which solutions ( complex eigenvalues ) are physically acceptable . for example , the popular complex absorbing potential method , which in many cases is easier to implement than the complex scaling method , produces the appearance of nonphysical complex energy stabilized points that must be removed in order to obtain only the physical resonances @xcite . the aforementioned issues explain , at some extent , why the methods based only in the use of real @xmath2 variational functions are often preferred to analyze resonance states . these techniques reduce the problem to the calculation of eigenvalues of real symmetric matrices @xcite . of course , these methods also have its own drawbacks . one of the main problems was recognized very early on ( see , for example , the work by hol@xmath8ien and midtdal @xcite ) : if the energy of an autoionizing state is obtained as an eigenvalue of a finite hamiltonian matrix , which are the convergence properties of these eigenvalues that lie in the continuum when the size of the hamiltonian matrix changes ? but in order to obtain resonance - state energies it is possible to focus the analysis in a global property of the variational spectrum : the density of states ( dos)@xcite , being unnecessary to answer this question . the availability of the dos allows to obtain the energy and lifetime of the resonance in a simple way , both quantities are obtained as least square fitting parameters , see for example @xcite . despite its simplicity , the determination of the resonance s energy and width based in the dos is far from complete . there is no a single procedure to asses both , the accuracy of the numerical findings and its convergence properties , or which values to pick between the several `` candidates '' that the method offers @xcite . recently , pont _ et al _ @xcite have used _ finite size scaling _ arguments @xcite to analyze the properties of the dos when the size of the hamiltonian changes . they presented numerical evidence about the critical behavior of the density of states in the region where a given hamiltonian has resonances . the critical behavior was signaled by a strong dependence of some features of the density of states with the basis - set size used to calculate it . the resonance energy and lifetime were obtained using the scaling properties of the density of states . however , the feasibility of the method to calculate the resonance lifetime laid on the availability of a known value of the lifetime , making the whole method dependent on results not provided by itself . the dos method relies on the possibility to calculate the ritz - variational eigenfunctions and eigenvalues for many different values of the non - linear parameter @xmath9 ( see kar and ho @xcite ) . for each basis - set size , @xmath3 , used , there are @xmath3 variational eigenvalues @xmath10 . each one of these eigenvalues can be used , at least in principle , to compute a dos , @xmath11 , resulting , each one of these dos in an approximate value for the energy , @xmath12 , and width , @xmath13 , of the resonance state of the problem . if the variational problem is solved for many different basis - set sizes , there is not a clear cut criterion to pick the `` better '' result from the plethora of possible values obtained . this issue will be addressed in section [ model ] . in this work , in order to obtain resonance energies and lifetimes , we calculate all the eigenvalues for different basis - set sizes , and present a recipe to select adequately certain values of @xmath3 , and one eigenvalue for each @xmath3 elected , that is , we get a series of variational eigenvalues @xmath14 . the recipe is based on some properties of the variational spectrum which are discussed in section [ some - properties ] . the properties seem to be fairly general , making the implementation of the recipe feasible for problems with several particles . actually , because we use scaling properties for large values of @xmath3 , the applicability of the method for systems with more than three particles could be restricted because the difficulties to handle large basis sets . the set of approximate resonance energies , obtained from the density of states of a series of eigenvalues selected following the recipe , shows a very regular behaviour with the basis set size . this regular behaviour facilitates the use of finite size scaling arguments to analyze the results obtained , in particular the extrapolation of the data when @xmath15 . the extrapolated values are the most accurate approximation for the parameters of the resonance state that we obtain with our method . this is the subject of section [ recipe ] , where we present results for models of one and two particles . following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths in section [ golden - rule ] . using the scaling function that characterizes the behaviour of the approximate energies as a guide , it is possible to find a very good approximation to the resonance width since , again , the data generated using our prescription seems to converge when @xmath4 . finally , in section [ discusion ] we summarize and discuss our results . when one is dealing with the variational spectrum in the continuum region , some of its properties are not exploited to obtain more information about the presence of resonances , usually the focus of interest is the stabilization of the individual eigenvalues . the stabilization is achieved varying some non - linear variational parameter . if @xmath9 is the inverse characteristic decaying length of the variational basis functions , then the spectrum of the kinetic energy scales as @xmath16 , moreover , for potentials that decay fast enough , the spectrum of the whole hamiltonian _ also _ scales as @xmath16 for large ( or small ) enough values of @xmath9 ( see appendix ) . this is so , since the variational eigenfunctions are @xmath2 approximations to plane waves _ except _ when @xmath9 belongs to the stabilization region . when @xmath9 belong to the stabilization region of a given variational eigenvalue , say @xmath17 , then @xmath18 ( where @xmath0 is the resonance energy ) and the variational eigenfunction @xmath19 has the localization length of the potential well . we intend to take advantage of the changes of the spectrum when @xmath9 goes from small to large enough values . the variational spectrum satisfies the hylleras - undheim theorem or variational theorem : if @xmath3 is the basis set size , and @xmath20 is the @xmath21 eigenvalue obtained with a variational basis set of size @xmath3 , then @xmath22 actually , since the threshold of the continuum is an accumulation point , then for small enough values of @xmath9 and a given @xmath23 there is always a @xmath24 such that @xmath25 for the kinetic energy variational eigenvalues , and for fixed @xmath26 , and @xmath27 , if the ordering given by equation ( [ ordering ] ) holds for some @xmath9 then it is true for all @xmath9 . of course this is not true for a hamiltonian with a non zero potential that support resonance states . so , we will take advantage of the variational eigenvalues such that for @xmath9 small enough satisfies equation ( [ ordering ] ) but , for @xmath9 large enough @xmath28 despite its simplicity , the arguments above give a complete prescription to pick a set of eigenstates that are particularly affected by the presence of a resonance . choose @xmath3 and @xmath29 arbitrary , and then look for the smaller values of @xmath30 and @xmath27 such that the two inequalities , equations . ( [ ordering ] ) and ( [ reverse ] ) are fulfilled . so far , all the examples analyzed by us show that if the inequalities are satisfied for some @xmath30 and @xmath27 then they are satisfied too by the eigenvalues @xmath31 , for @xmath32 . to illustrate how our prescription works we used two different model hamiltonians . the first model , due to hellmann @xcite , is a one particle hamiltonian that models a @xmath33-electron atom . the second one is a two particle model that has been used to study the low energy and resonance states of two electrons confined in a semiconductor quantum dot @xcite . the details of the variational treatment of both models will be kept as concise as possible . the one particle model has been used before for the determination of critical nuclear charges for @xmath33-electron atoms @xcite , it also gives reasonable results for resonance states in atomic anions @xcite and continuum states @xcite . the interaction of a valence electron with the atomic core is modeled by a one - particle potential with two asymptotic behaviours . the potential behaves correctly in the regions where electron is far from the atomic core ( @xmath34 electrons and the nucleus of charge @xmath35 ) and when it is near the nucleus . the hamiltonian , in scaled coordinates @xmath36 , is @xmath37 where @xmath38 and @xmath39 is a range parameter that determines the transition between the asymptotic regimes , for distances near the nucleus @xmath40 and in the case @xmath41 the nucleus charge is screened by the @xmath34 localized electrons and @xmath42 . another advantage of the potential comes from its analytical properties . in particular this potential is well behaved and the energy of the resonance states can be calculated using complex scaling methods . so , besides its simplicity , the model potential allows us to obtain the energy of the resonance by two independent methods and check our results . the two particle model that we considered describes two electrons interacting via the coulomb repulsion and confined by an external potential with spherical symmetry . we use a short - range potential suitable to apply the complex scaling method . the hamiltonian @xmath5 for the system is given by @xmath43 where @xmath44 , @xmath45 the position operator of electron @xmath46 ; @xmath47 and @xmath48 determine the range and depth of the dot potential . after re - scaling with @xmath47 , in atomic units , the hamiltonian of equation ( [ hamiltoniano ] ) can be written as @xmath49 where @xmath50 . the variational spectrum of the two particle model , equation ( [ hamil2part ] ) , and all the necessary algebraic details to obtain it , has been studied with great detail in reference @xcite so , until the end of this section , we discuss the variational solution of the one particle model given by equation ( [ hamil ] ) . the discrete spectrum and the resonance states of the model given by equation ( [ hamil ] ) can be obtained approximately using a real @xmath51 truncated basis set @xmath52 to construct a @xmath53 hamiltonian matrix @xmath54 . we use the rayleigh - ritz variational method to obtain the approximations @xmath55 @xmath56 for bound states this functions are variationally optimal . the functions @xmath57 are @xmath58 and @xmath59 are the associated laguerre polynomials of @xmath60 order and degree @xmath61 . the non - linear parameter @xmath9 is used for eigenvalue stabilization in resonance analysis @xcite . note that @xmath9 plays a similar role that the finite size of the box in spherical box stabilization procedures @xcite , as stated by kar _ et . @xcite . resonance states are characterized by isolated complex eigenvalues , @xmath62 , whose eigenfunctions are not square - integrable . these states are considered as quasi - bound states of energy @xmath0 and inverse lifetime @xmath1 . for the hamiltonian equation ( [ hamil ] ) , the resonance energies belong to the positive energy range @xcite . using the approximate solutions of hamiltonian ( [ hamil ] ) we analyze the dos method @xcite that has been used extensively to calculate the energy and lifetime of resonance states , in particular we intend to show that 1 ) the dos method provides a host of approximate values whose accuracy is hard to assess , and 2 ) if the dos method is supplemented by a new optimization rule , it results in a convergent series of approximate values for the energy and lifetime of resonance states . the dos method relies on the possibility to calculate the ritz - variational eigenfunctions and eigenvalues for many different values of the non - linear parameter @xmath9 ( see kar and ho @xcite ) . the localized dos @xmath63 can be expressed as @xcite @xmath64 since we are dealing with a numerical approximation , we calculate the energies in a discretization @xmath65 of the continuous parameter @xmath9 . in this approximation , equation ( [ densidad_sin_suma ] ) can be written as @xmath66 where @xmath67 is the @xmath30-th eigenvalue of the @xmath53 matrix hamiltonian with @xmath68 and @xmath39 fixed . in complex scaling methods the hamiltonian is dilated by a complex factor @xmath69 . as was pointed out long time ago by moiseyev and coworkers @xcite , the role played by @xmath9 and @xmath70 are equivalent , in fact , our parameter @xmath9 corresponds to @xmath71 . besides , the dos attains its maximum at optimal values of @xmath9 and @xmath0 that could be obtained with a self - adjoint hamiltonian without using complex scaling methods @xcite . so , locating the position of the resonance using the maximum of the dos is equivalent to the stabilization criterion used in complex dilation methods that requires the approximate fulfillment of the complex virial theorem @xcite . the values of @xmath72 and @xmath73 are obtained performing a nonlinear fitting of @xmath63 , with a lorentzian function , @xmath74}.\ ] ] one of the drawbacks of this method results evident : for each pair @xmath75 there are several @xmath11 , and since each @xmath11 provides a value for @xmath76 and @xmath77 one has to choose which one is the best . kar and ho @xcite solve this problem fitting all the @xmath11 and keeping as the best values for @xmath0 and @xmath1 the fitting parameters with the smaller @xmath78 value . at least for their data the best fitting ( the smaller @xmath78 ) usually corresponds to the larger @xmath30 . this fact has a clear interpretation , if the numerical method approximates @xmath0 with some @xmath79 , where @xmath3 is the basis set size of the variational method , a large @xmath30 means that the numerical method is able to provide a large number of approximate levels , and so the continuum of positive - energy states is `` better '' approximated . in a previous work @xcite we have shown that a very good approximation to the energy of the resonance state is obtained considering just the energy value where @xmath11 attains its maximum . we denote this value as @xmath76 . figure [ prefig1 ] shows the approximate resonance energy @xmath76 for different basis set size @xmath3 , where @xmath29 is the index of the variational eigenvalue used to calculate the dos . we used the values @xmath80 and @xmath81 corresponding to the ones used before @xcite in the analysis of @xmath82 resonances . the figure [ prefig1 ] also shows the value calculated using complex scaling . it is clear that the accuracy of all the values shown is rather good ( all the values shown differ in less than 6@xmath83 ) , and that larger values on @xmath84 provide better values for the resonance energy . these facts are well known from previous works , _ i.e. _ almost all methods to calculate the energy of the resonance give rather stable and accurate results for @xmath0 . however , the practical importance of this fact is reduced : these are uncontrolled methods , so the accuracy of the values obtained from the dos can not be assessed ( without a value independently obtained ) and these values do not seem to converge to the value obtained using complex scaling when @xmath3 is increased and @xmath84 is kept fixed . there is another fact that potentially could render the whole method useless : for small or even moderate @xmath29 , the values @xmath76 become _ unstable _ ( see figure [ prefig1 ] ) when @xmath3 is large enough . this last point has been pointed previously @xcite . in the problem that we are considering is rather easy to obtain a large number of variational eigenvalues in the interval where the resonances are located , allowing us to calculate @xmath76 up to @xmath85 , but this situation is far from common see , for example , references @xcite . so far we have presented only results about the behaviour of the one particle hamiltonian , from now on we will discuss both models , equations ( [ hamil ] ) and [ hamil2part ] . it is known that the variational eigenvalues @xmath17 do not present crossings when they are calculated for some fixed values of @xmath3 , _ i.e _ the variational spectrum is non - degenerate for any finite hamiltonian matrix . as a matter of fact the avoided crossings between successive eigenvalues in the variational spectrum are the watermark of a resonance . an interesting feature emerges when the variational spectrum for many different basis set sizes @xmath3 are plotted together versus the parameter @xmath9 . besides the places where @xmath86 attains its minimum value , which correspond to the stabilization points , there are some gaps which correspond to crossings between eigenvalues obtained with different basis set sizes , see figure [ prefig2 ] . moreover , the crossings corresponds to eigenvalues with different index @xmath29 , and are the states that satisfy the inequalities equations ( [ ordering]),and ( [ reverse ] ) . it is worth to remark that the main features shown by figure [ prefig2 ] are independent of the number of particles of the hamiltonian and the particular values of the threshold of the continuum . figure [ bundle2p ] shows the behaviour of the variational eigenvalues obtained for the two particle hamiltonian equation ( [ hamil2part ] ) . in this case the ionization threshold is not the asymptotic value of the potential , but it is given by the energy of the one particle ground state . the resonance state came from the two - particle ground state that becomes unstable and enters into the continuum of states when the quantum dots becomes `` too small '' to accommodate two electrons . for more details about the model , see reference @xcite . the left panel of figure [ prefig3 ] shows the behaviour of the maximum value of the dos , @xmath87 , for the one particle hamiltonian , obtained for different basis - set sizes and fixed @xmath29 ( in this case @xmath85 ) , and the @xmath88 obtained choosing a `` bundle '' of states that are linked by a crossing , these states have @xmath89 and @xmath90 respectively . from our numerical data , the maximum value of the dos scales with the basis - set size following two different prescriptions . for @xmath29 fixed , @xmath91 , with @xmath92 , while when the pair @xmath93 is chosen from the set of pairs that label a bundle of states @xmath94 , with @xmath95 . in particular , for @xmath85 we get that @xmath96 , and @xmath97 when @xmath98 . of course we can pick sets of states that are not related by a crossing . for instance , we also picked sets with a simple prescription as follows : choose a given initial pair @xmath99 and form a set of states with the states labeled by @xmath100 and so on . figure [ prefig3](a ) shows two examples obtained choosing @xmath101 , @xmath102 and @xmath103 and @xmath104 both with @xmath105 . quite interestingly , the data in figure [ prefig3 ] show that the scaled maxima of the dos for a bundle and two different sets seem to converge to the _ same _ value when @xmath106 , but only for the bundle the scaling function is @xmath107 . the advantage obtained from picking those eigenvalues @xmath17 that belong to a given bundle is still more evident when the corresponding dos and @xmath0 are calculated . the right panel of figure [ prefig3 ] shows the @xmath0 obtained from the dos whose maxima are shown in the left panel . it is rather evident that these values now seem to converge , besides , the extrapolation to @xmath4 results in a more accurate approximate value for @xmath0 . in contradistinction , the values for @xmath0 corresponding to a fixed index @xmath29 ( the values shown in the figure [ prefig3 ] correspond to @xmath85 ) do not seem to converge anywhere close to the value obtained using complex rotation . figure [ er2par ] shows the resonance energies obtained from the bundles of states shown in figure [ bundle2p ] for the two - particle model . since the numerical solution of this model is more complicated than the solution of the one - particle model the number of approximate values is rather reduced . however , it seems that the data also supports a linear scaling of @xmath76 with @xmath108 . many real algebra methods to calculate resonance energies use a golden rule - like formula to calculate the resonance width . in this section we will use the formula and stabilization procedure proposed by tucker and truhlar @xcite that we will describe briefly for completeness . this projection formula seems to work better for one - particle models . for two - particle models its utility has been questioned @xcite , so to analyze the width of the resonance states of the quantum dot model we fitted the corresponding dos using equation ( [ lorentz ] ) . the method of tucker and truhlar @xcite is implemented by the following steps . choose a basis @xmath109 where @xmath9 is a non - linear parameter . diagonalize the hamiltonian using up to @xmath3 functions of the basis . look for the stabilization value @xmath110 and its corresponding eigenfunction @xmath111 which are founded for some value @xmath112 . define the projector @xmath113 where @xmath114 is the normalized projection of @xmath111 onto the basis @xmath115 for any other @xmath9 . diagonalize the hamiltonian @xmath116 in the basis @xmath115 , again as a function of @xmath9 , and find a value @xmath117 of @xmath9 such that @xmath118 where @xmath119 denotes eigenvalue @xmath84 of the projected hamiltonian for the scale factor @xmath117 , and @xmath120 is the corresponding eigenfunction . with the previous definitions and quantities , the resonance width @xmath1 is given by @xmath121 where @xmath122.\ ] ] despite some useful insights , the procedure sketched above does not determine all the intervening quantities , for instance there are many solutions to equation ( [ seg - estabili ] ) and , of course , the stabilization method provides several good candidates for @xmath111 and @xmath112 . we are able to avoid some of the indeterminacies associated to the tucker and truhlar procedure using a bundle of states associated to a crossing , so @xmath111 and @xmath112 are given by any of the eigenfunctions associated to a bundle and @xmath123 comes from the stabilization procedure . then we construct projectors @xmath124 where @xmath19 is one of the variational eigenstates that belong to a bundle of states . with the projectors @xmath125 we construct hamiltonians @xmath126 , and find the solutions to the problem @xmath127 since there is not an a priori criteria to choose one particular solution of equation ( [ tercera - estabili ] ) we show our numerical findings for several values of @xmath84 . figure [ prefig4 ] shows the behaviour of the resonance width calculated with equation ( [ golden2 ] ) , where we have used @xmath128 as @xmath111 and @xmath129 , where @xmath130 and @xmath131 . despite that the different sets corresponding to different values of @xmath84 do not converge to any definite value , for @xmath3 large enough all the sets scale as @xmath132 , with @xmath133 . since the resonance energy scales as @xmath107 , at least when a bundle of states with a crossing is chosen to calculate approximations ( see figure [ prefig3 ] ) , we suggest that the right scaling for @xmath1 is given by @xmath134 . of course for a given basis size , particular variational functions , stabilization procedures and so on , we can hardly expect to find a proper set of @xmath1 whose scaling law would be @xmath107 . instead of this we propose that the data in the right panel can be fitted by @xmath135 then the best approximation for the resonance width is obtained fitting the curve and selecting the @xmath136 as @xmath137 for @xmath138 the closest value to one . as pointed in reference @xcite , the projection technique to calculate the width of a resonance can be implemented if a suitable form of the projection operator can be found . as this procedure is marred by several issues we used the dos method to obtain the approximate widths of a resonance state of the two particle model . figure [ gama2par ] shows the widths calculated associated to the energies shown in figure [ er2par ] , the parameters of the hamiltonian are exactly the same . there is no obvious scaling function that allows the extrapolation of the data but , even for moderate values of @xmath3 , it seems as the data converge to the value obtained using complex scaling . in this work we analyzed the convergence properties of real @xmath51 basis - set methods to obtain resonance energies and lifetimes . the convergence of the energy with the basis - set size for bound states is well understood , the larger the basis set the better the results and these methods converge to the exact values for the basis - set size going to infinite ( complete basis set ) . this idea is frequently applied to resonance states . the increase of the basis - set size in some commonly used methods does not improve the accuracy of the value obtained for the resonance energy @xmath0 , as showed in figure [ prefig1 ] . this undesirable behavior comes from the fact that the procedure is not variational as in the case of bound states . moreover , the exact resonance eigenfunction does not belong to the hilbert space expanded by the complete basis set . in this work we presented a prescription to pick a set or bundle of states that has linear convergence properties for small width resonances . this procedure is robust because the choice of different bundles results in very similar convergence curves and energy values . in fact , in the method described here , the pairs @xmath93 of the bundles play the role of a second stabilization parameter together with the variational parameter @xmath9 . of a second we tested the method in others one and two particle systems and the general behavior of them is the same . the results are very good in all cases leading to an improvement in the calculation of the resonance energies . nevertheless we have to note that the method could no be applied in cases where two or more resonance energies lie very close because the overlapping bundles . the lifetime calculation is more subtle . the use of golden - rule - like formulas , as we applied here , always give several possible outcomes for the width @xmath139 , corresponding to different pseudo - continuum states @xmath140 . the projection technique , equation ( [ golden2 ] ) , is not the exception and it is not possible to select _ a priori _ which value of @xmath141 is the most accurate . the linear convergence of the dos with basis - set size suggests that the scaling in the lifetime value , in accordance with the energy scaling , should be linear . regrettably , the projection method gives discrete sets of values which can not be tuned to obtain an exact linear convergence . our recipe is to choose the set @xmath139 whose scaling is closest to the linear one , then the best estimation for the resonance width is obtained from extrapolation . many open questions remain on the analysis of the different convergence properties of resonance energy and lifetime . the method presented here to obtain the resonance energy from convergence properties works very well , but the appearance of bundles in the spectrum is not completely understood . even there is not a rigorous proof , the numerical evidence supports the idea that the behaviour of the systems studied here is quite general . in this appendix we give arguments that support our assumptions on the scaling of the eigenenergies with the basis - set parameter @xmath9 . we present our argument for one body hamiltonians , but it is straightforward to generalize to more particles with pair interactions decaying fast enough at large distances . 1 . let @xmath142 be an @xmath143 matrix with all its matrix elements having the form @xmath144 , where @xmath145 , then if @xmath146 the eigenvalues of @xmath142 scales with @xmath147 : @xmath148=0\;\rightarrow det[a(1)-\frac{\lambda(\eta)}{f(\eta)}\,i]=0 \rightarrow \lambda(1)=\frac{\lambda(\eta)}{f(\eta ) } \,.\ ] ] 2 . let @xmath149 @xmath143 be symmetric matrices with @xmath150 , and @xmath151 the eigenvalues of @xmath149 and @xmath152 respectively , in nondecreasing order , then , by the minimax principle @xcite consider a spherical one - particle potential with compact support , @xmath154 if @xmath155 , and finite , @xmath156 ( both conditions could be relaxed , but we adopt them for simplicity ) . let the basis - set functions be of the form @xmath157 with @xmath158 and @xmath159 , then the coefficients take the form @xmath160 . and then , by equation ( [ asa ] ) , all the eigenvalues of the kinetic energy have the same scaling with @xmath16 . we have to show that , in both limits , @xmath162 and @xmath163 , for all the potential matrix elements hold @xmath164 , and then , by the wielandt - hoffman theorem @xcite , the eigenenergies are a perturbation of the eigenvalues of the kinetic energy . a u hazi and h s taylor , phys . rev . a * 14 * , 2071 ( 1976 ) . a ferrn , o osenda and p serra , phys . a * 79 * , 032509 ( 2009 ) . h feschbach , ann . ny * 19 * , 627 ( 1962 ) n moiseyev , c corcoran , phys . rev . a * 20 * , 814 ( 1979 ) . m e fisher , in _ critical phenomena _ , proceedings of the 51st enrico fermi summer school , varenna , italy , m. s. green , ed . ( academic press , new york 1971 ) . s e tucker and d. g. truhlar , j.chem . * 86 * , 6251 ( 1987 ) . f r manby and g doggetty , j. phys . b : at . mol . . phys . * 30 * , 3343 ( 1997 ) j. h. wilkinson , _ the algebraic eigenvalue problem _ , oxford university press , london , ( 1965 ) .
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the resonance states of one- and two - particle hamiltonians are studied using variational expansions with real basis - set functions .
the resonance energies , @xmath0 , and widths , @xmath1 , are calculated using the density of states and an @xmath2 golden rule - like formula .
we present a recipe to select adequately some solutions of the variational problem .
the set of approximate energies obtained shows a very regular behaviour with the basis - set size , @xmath3 .
indeed , these particular variational eigenvalues show a quite simple scaling behaviour and convergence when @xmath4 . following the same prescription to choose particular solutions of the variational problem
we obtain a set of approximate widths . using the scaling function that characterizes the behaviour of the approximate energies as a guide
, it is possible to find a very good approximation to the actual value of the resonance width .
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several diagnostic protocols are usually adopted by dermatologists for analyzing and classifying skin lesions , such as the so - called _ abcd - rule _ of dermoscopy @xcite . due to the subjective nature of examination , the accuracy of diagnosis is highly dependent upon human vision and dermatologist s expertise . computerized dermoscopic image analysis systems , based on a consistent extraction and analysis of image features , do not have the limitation of this subjectivity . these systems involve the use of a computer as a second independent and objective diagnostic method , which can potentially be used for the pre - screening of patients performed by non - experienced operators . although computerized analysis techniques can not provide a definitive diagnosis , they can improve biopsy decision - making , which some observers feel is the most important use for dermoscopy @xcite . recently , numerous researches on this topic propose systems for the automated detection of malignant melanoma in skin lesions ( e.g. , @xcite ) . in our previous study on dermoscopic images @xcite , the segmentation of the skin area and the lesion area was achieved by a semi - automatic process based on otsu algorithm @xcite , supervised by a human operator . here , we propose a full automatic segmentation method consisting of three main steps : selection of the image roi , selection of the segmentation band , and segmentation . the paper is organized as follows . in section [ proposedapproach ] we describe the proposed algorithm , providing details of its main steps . in section [ expres ] we provide a thorough analysis of experimental results on the isic 217 dataset @xcite . conclusions are drawn in section [ conclusioni ] . the block diagram of the segmentation algorithm proposed for dermoscopic images , named sdi algorithm , is shown in fig . [ fig : overall ] . the three main steps are described in the following . in order to achieve an easier and more accurate segmentation of the skin lesion , it is advisable to select the region of interest ( roi ) , i.e. , the subset if image pixels that belong to either the lesion or the skin . this region excludes image pixels belonging to ( usually dark ) areas of the image border and/or corners , as well as those belonging to hair , that will not be taken into account in the subsequent steps of the sdi algorithm . in the proposed approach , the value band of the image in the hsv color space is chosen in order to select dark image pixels ; these are excluded from the roi if they cover most of the border or the angle regions of the image . concerning hair , many highly accurate methods have been proposed in the literature @xcite . here , we adopted a bottom - hat filter in the red band of the rgb image . an example of the roi selection process is reported in fig . [ fig : roiselection ] for the isic 2017 test image no . 15544 . here , we observe that the wide dark border on the left of the image , as well as the dark hair over the lesion , have properly been excluded from the roi mask . [ cols="^,^ " , ] we proposed the sdi algorithm for dermoscopic image segmentation , consisting of three main steps : selection of the image roi , selection of the segmentation band , and segmentation . the reported analysis of experimental results achieved by the sdi algorithm on the isic 2017 dataset allowed us to highlight its pro s and con s . this leads us to conclude that , although some accurate results can be achieved , there is room for improvements in different directions , that we will go through in future investigations . this research was supported by lab gtp project , funded by miur . w. stolz , a. riemann , a. b. cognetta , l. pillet , w. abmayr , d. holzel , p. bilek , f. nachbar , m. landthaler , and o. braun - falco , `` abcd rule of dermoscopy : a new practical method for early recognition of malignant melanoma , '' _ european journal of dermatology _ , vol . 4 , pp . 521527 , 1994 . m. burroni , r. corona , g. delleva , f. sera , r. bono , p. puddu , r. perotti , f. nobile , l. andreassi , and p. rubegni , `` melanoma computer aided diagnosis : reliability and feasibility study , '' _ clinical cancer research _ , vol . 10 , pp . 18811886 , 2004 . m. e. celebi , h. a. kingravi , b. uddin , h. iyatomi , y. a. aslandogan , w. v. stoecker , and r. h. moss , `` a methodological approach to the classification of dermoscopy images . '' _ computerized medical imaging and graphics _ , vol . 31 , no . 6 , pp . 362373 , september 2007 . i. maglogiannis and c. n. doukas , `` overview of advanced computer vision systems for skin lesions characterization , '' _ ieee transactions on information technology in biomedicine _ , vol . 13 , no . 5 , pp . 721733 , 2009 . v. cozza , m. r. guarracino , l. maddalena , and a. baroni , `` dynamic clustering detection through multi - valued descriptors of dermoscopic images , '' _ statistics in medicine _ , 30 , no . 20 , pp . 25362550 , 2011 . [ online ] . available : http://dx.doi.org/10.1002/sim.4285 m. e. celebi , q. wen , h. iyatomi , k. shimizu , h. zhou , and g. schaefer , `` a state - of - the - art survey on lesion border detection in dermoscopy images , '' in _ dermoscopy image analysis _ , m. e. celebi , t. mendonca , and j. s. marques , eds.1em plus 0.5em minus 0.4emcrc press , 2015 , pp .
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we propose an automatic algorithm , named sdi , for the segmentation of skin lesions in dermoscopic images , articulated into three main steps : selection of the image roi , selection of the segmentation band , and segmentation .
we present extensive experimental results achieved by the sdi algorithm on the lesion segmentation dataset made available for the isic 2017 challenge on skin lesion analysis towards melanoma detection , highlighting its advantages and disadvantages .
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evolutionary algorithms ( ea ) are a group of computational techniques , which employ the theory of natural selection to a population of individuals to generate better individuals . genetic programming ( gp ) is a paradigm of ea which uses hierarchical , tree structure , variable length representation to code solutions of a problem . gp can be used to intelligently search the solution space for finding the optimal solution of a problem . there are many gp tools ( lil - gp , genexprotools , gplab , ecj , open beagle ) @xcite developed by gp practitioners . however , none of these address the following demands of end user before applying them to solve symbolic regression problems : ( i ) ease of use and ( ii ) small learning curve . many of these tools are open source and available freely ( lil - gp , ecj , open beagle , gplab for matlab ) @xcite whereas the rest are available commercially ( genexprotools ) @xcite . many of these tools necessitate modification in source code in order to generate required experimental environment . determining the final solution , produced by these tools , demands translation of the output or digging the log files . due to these reasons , the interest of researchers and engineers in gp may get reduced . theses reasons motivated us to develop our own gp framework @xcite which uses the postfix notation for an individual representation . we have considered the following features which need to be supported by postfix - gp framework : ( i ) easy to extend , ( ii ) simple and quick procedure for the configuration and running of gp , ( iii ) a set of algorithm implementation for : ( a ) generating the initial population , ( b ) selection mechanisms , and ( c ) genetic operators , ( iv ) visualization of : ( a ) postfix - gp run analysis and ( b ) evolved solution with statistical measures , ( v ) one - step and multi - step prediction support , ( vi ) visualization of results for one - step and multi - step predictions , and ( vii ) storage and retrieval of evolved solutions to and from file . postfix - gp has been used in experimental work @xcite , @xcite , @xcite , @xcite . this paper presents the design and implementation of postfix - gp , an object oriented software framework for genetic programming . section [ sec : introduction to gp ] gives introduction to gp . section [ sec : postfix - gp ] presents the design of postfix - gp . moreover , the section also gives the implementation details and main features of postfix - gp . section [ sec : case study ] presents postfix - gp as a solution modeling tool by solving the benchmark symbolic regression problem . section [ sec : featurecomparison ] compares the features of postfix - gp with lil - gp @xcite , ecj @xcite , and jclec @xcite frameworks . this is followed by conclusions in section [ sec : conclusions ] . standard gp @xcite employs a variable length , tree structure scheme for an individual representation . the tree can be used to represent logical expressions ( if - then - else ) , boolean expressions ( and , or , not ) or algebraic expressions . the symbolic regression aims to find the functional relationship ( mathematical expression ) between given instances of inputs - outputs . gp can be used to perform symbolic regression ( sr ) . when using gp for solving symbolic regression problems , the user need to specify the following items : ( i ) gp configuration parameters , ( ii ) terminal set and function set , ( iii ) fitness function . the main steps of standard gp @xcite are as follows : 1 . random generation of an initial population of candidate solutions in tree form using the elements of function set and terminal set , selected by the modeler ( user ) . 2 . calculating fitness value of every individual of the population on the given training dataset ( fitness cases ) 3 . selecting parents for mating based on the calculated fitness values , determined in previous step 4 . applying sub - tree crossover and mutation ( genetic operators ) on selected parents for generating a new population of individuals . the process is repeated until the termination condition is fulfilled . the important features of the proposed postfix - gp are categorized into : ( i ) training dataset , function set , and terminal set related , ( ii ) gp parameters related , ( iii ) test dataset prediction related , ( iv ) gp run analysis related , and ( v ) serialization and de - serialization of gp experiments and results . training dataset , function set , and terminal set related features : * loading of training dataset * loading of binary and unary functions * loading of constants gp parameters related : * selection of method to generate initial population * configuration of gp parameters like population size , number of generations , crossover rate , mutation rate * selection of crossover and mutation type * selection of selection scheme test dataset prediction related : * loading out - of - sample test dataset for one - step predictions * visualization of results of one - step predictions with statistical measures * loading out - of - sample test dataset for multi - step predictions * visualization of results of multi - step predictions with statistical measures gp run analysis related : * plotting best adjusted fitness vs number of generations * plotting average adjusted fitness vs number of generations * plotting solution size vs number of generations serialization and de - serialization of gp experiments and results : * serialization of gp parameters , function set , terminal set , and obtained solutions to a file * de - serialization of gp parameters , function set , terminal set , and obtained solutions from a file this section presents the implementation details of postfix - gp . this will be useful to the reader in understanding and customizing the proposed postfix - gp framework . for details related to postfix - gp solution representation scheme , refer @xcite , @xcite . postfix - gp framework is developed using microsoft .net framework @xcite on windows xp operating system . the zedgraph @xcite class library is used for plotting the charts . zedgraph is an open source graph library for .net platform . + the class diagram for postfix - gp is depicted in [ fig : classdiagram ] . the classes can be grouped into following categories : ( i ) representation ( , ) , ( ii ) population ( ) , ( iii ) crossover operator ( , , , ) , ( iv ) mutation operator ( , , ) , ( v ) selection schemes ( , , , ) , ( vi ) gp parameters ( ) , and ( vii ) statistical analysis of results ( ) . [ cols="^ " , ] * lil - gp * : lil - gp @xcite is a gp toolkit implemented in c programming language . the toolkit is efficient and fast , as it is implemented in c. however , it is difficult to extend the toolkit compare to other object - oriented implementations of gp systems . lil - gp uses only tree structure for an individual representation and provides limited fitness measures . moreover , the toolkit does not provide graphical user interface to read an input ( training ) data file . the toolkit uses a parameter file to load gp parameters . the toolkit produces six reporting files ( .sys,.gen,.prg,.bst,.his and .stt ) that provide the statistical information of the gp run . there are many patches developed by different researchers to fix the bugs and to improve the functionality of the basic lil - gp . * ecj * : ecj @xcite is a java based framework for evolutionary computation and genetic programming . ecj is designed using the object - oriented concepts . classes of ecj framework are divided into four layers @xcite : ( i ) utility layer , ( ii ) basic and custom evolutionary computation layer , ( iii ) basic and custom genetic programming layer , and ( iv ) problem layer . as the framework is implemented in java , it is slower in speed . moreover , ecj uses a tree ( and not an arrays ) to represent an individual , which requires dynamic memory allocation . thus , the framework consumes large memory . ecj determines gp parameters from a parameter file . ecj determines classes to be loaded , the type of problem to be solved , the type of technique to use to solve the problem , and the way to report the statistical results of the run from the parameter file at the run time @xcite . this provides easy to extend functionality to the ecj . ecj stores statistical information of gp run in a text file . moreover , it provides flexibility to produce the extra output files through class customization but does not provide a gui to visualize this information . * jclec * : jclec @xcite is a java based framework for evolutionary computation and genetic programming . jclec @xcite is designed using the object - oriented concepts . the classes of jclec framework are divided in three layers : ( i ) system core , ( ii ) experiments runner ( reads an ea script file , execute all indicated algorithms and produce report files ) , and ( iii ) genlab ( a gui on the top of experiments runner and system core layers , provides functionality to edit the experiment files and to view the gp run results ) @xcite . gp parameters can be set by the user either through genlab gui or through an xml ( configuration ) file . however , the structure of configuration files is not user friendly . the framework provides the gui to visualize statistical information of gp run . the jclec framework is easy to extend . this paper presented the design and implementation of postfix - gp framework . the implementation details of postfix - gp , including an individual representation , different crossover operators , mutations , and selection mechanism were also presented . postfix - gp provides user interactive gui for performing different activities . the user can load training dataset , function set , and constants . the user can set the gp parameters through gui . the evolved solutions with their statistical measures can be visualized through gui . moreover , the user can also perform one - step and multi - step predictions using gui . the evolved solutions can be stored in binary format and can be retrieved later on . the user can also analyze postfix - gp run through gui . postfix - gp as a solution modeling tool was presented by solving symbolic regression problem . postfix - gp addresses the requirements of ease of use and small learning curve before utilizing it to solve the problems . it was developed to minimize the user s time required to set up and run gp experiments . c. gagn and m. parizeau , `` open beagle : a new c++ evolutionary computation framework , '' in _ gecco 2002 : proceedings of the genetic and evolutionary computation conference_.1em plus 0.5em minus 0.4em new york : morgan kaufmann publishers , 9 - 13 july 2002 , p. 888 . [ online ] . available : http://www.cs.ucl.ac.uk/staff/w.langdon/ftp/papers/gecco2002/gecco-2002-15.pdf v. dabhi and s. chaudhary , `` semantic sub - tree crossover operator for postfix genetic programming , '' in _ proceedings of seventh international conference on bio - inspired computing : theories and applications ( bic - ta 2012 ) _ , ser . advances in intelligent systems and computing , j. c. bansal , p. k. singh , k. deep , m. pant , and a. k. nagar , eds . 201.1em plus 0.5em minus 0.4emspringer india , 2013 , pp . 391402 . , `` time series modeling and prediction using postfix genetic programming , '' in _ advanced computing communication technologies ( acct ) , 2014 fourth international conference on _ , feb 2014 , pp . 307314 . x. li , c. zhou , w. xiao , and p. c. nelson , `` prefix gene expression programming , '' in _ late breaking paper at genetic and evolutionary computation conference ( gecco2005 ) _ , washington , d.c . , usa , 25 - 29 jun . 2005 , pp . 2531 . j. h. holland , _ adaptation in natural and artificial systems : an introductory analysis with applications to biology , control and artificial intelligence_.1em plus 0.5em minus 0.4emcambridge , usa : mit press , 1992 .
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this paper describes postfix - gp system , postfix notation based genetic programming ( gp ) , for solving symbolic regression problems .
it presents an object - oriented architecture of postfix - gp framework .
it assists the user in understanding of the implementation details of various components of postfix - gp .
postfix - gp provides graphical user interface which allows user to configure the experiment , to visualize evolved solutions , to analyze gp run , and to perform out - of - sample predictions .
the use of postfix - gp is demonstrated by solving the benchmark symbolic regression problem .
finally , features of postfix - gp framework are compared with that of other gp systems .
+ * _ keywords- _ * postfix genetic programming ; postfix - gp framework ; object oriented design ; gp software tool ; symbolic regression
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let @xmath0 be a standard brownian motion . it @xcite defined the multiple stochastic integral of a function @xmath1 , @xmath2 taking care to ensure that the diagonal sets , like @xmath3 , do not contribute at all . for this reason the integral has very good properties and is easy to work with . however , for a function of the form @xmath4 we have that , in general , @xmath5 that means the it multiple integral does not behave like the integral with respect to a product measure . many years later , hu and meyer @xcite introduced ( although they believed that this integral was already known @xcite , page 75 ) a multiple integral , @xmath6 , which followed the ordinary rules of multiple integration . they called it the multiple stratonovich integral . furthermore , hu and meyer stated the relationship between the it and stratonovich integrals , the celebrated hu meyer formula , adding the contribution of the diagonals to the it integral : for a function @xmath7 symmetric with good properties , @xmath8}\frac{n!}{(n-2j)!j!2^j } i_{n-2j } \biggl(\int_{{\mathbb{r}}_+^{j}}f(\bolds\cdot , t_1,t_1,t_2,t_2,\ldots , t_{j},t_{j } ) \,d t_{1}\cdots d t_{j } \biggr).\ ] ] this formula is simple because the quadratic variation of the brownian motion is @xmath9 , and the integral over coincidences of order three or superior are zero . following their ideas , sol and utzet @xcite proved a hu meyer formula for the poisson process . again , in that case , the formula is relatively simple because the variations of any order of the process can always be written in terms of the poisson process and @xmath9 . from another point of view , engel @xcite , working with a general process with independent increments , related the ( it ) multiple stochastic integral with the theory of vector valued measures , and masani @xcite , using also vector valued measures and starting from the wiener s original ideas , developed both the it and stratonovich integrals ( with respect to the brownian motion ) and proved many profound results . the vector measures approach is no simple matter ; engel s work covers 82 pages , and masani s covers 160 . an important and clarifying contribution was made by rota and wallstrom @xcite who used combinatorial techniques to show the features of the multiple stochastic integration . they did not really work with integrals , but with products of vector measures . however , the path towards a general theory of multiple stochastic integration had been laid . see also prez abreu @xcite for an interesting generalization to hilbert space valued random measures . further , vershik and tsilevich @xcite , in a more algebraic context , constructed a fock factorization for a lvy process , and some important subspaces can be described through rota and wallstrom concepts . we should also mention the very complete survey by peccati and taqqu @xcite in which a unified study of multiple integrals , moments , cumulants and diagram formulas , as well as applications to some new central limit theorems , is presented . it is worth remarking that rota and walstrom s @xcite combinatorial approach to multiple integration has been extended to the context of free probability in a very interesting and fertile field of research , started by anshelevich ( see @xcite and the references therein ) . in fact , rota and walstrom s ideas fit very well with the combinatorics of free probability ( see nica and speicher @xcite ) and noncommutative lvy processes . our renewed interest in rota and walstrom s paper @xcite was motivated by anshelevich s work . in the present paper we use the powerful rota and wallstrom s @xcite combinatorial machinery to study the stratonovich integral ( the integral with respect to the product random measure ) with respect to a lvy processes with finite moments up to a convenient order . the key point is to understand how the product of stochastic measures works on the diagonal sets , and that leads to the diagonal measures defined by rota and wallstrom @xcite . for a lvy process those measures are related to the powers of the jumps of the process , and hence to a family of martingales introduced by nualart and schoutens @xcite , called teugels martingales , which offer excellent properties . specifically , these martingales have deterministic predictable quadratic variation and this makes it possible to easily construct an it multiple stochastic integral with respect to different integrators , which can be interpreted as an integral with respect to a random measure that gives zero mass to the diagonal sets . with all these ingredients we prove a general hu meyer formula . the paper uses arduous combinatorics because of our need to work with stochastic multiple integrals with respect to the different powers of the jumps of the process , and such integrals can be conveniently handled through the lattice of the partitions of a finite set . as in the brownian case ( see , e.g. , @xcite ) , there are alternative methods to construct a multiple stratonovich integral based on approximation procedures , and it is possible to relax the conditions on the integrator process by assuming more regularity on the integrand function . such regularity is usually expressed in terms of the existence of _ traces _ of the function in a convenient sense . the advantage of using lvy processes with finite moments lies in the fact that simple @xmath10 estimates for the multiple stochastic integral of simple functions can be obtained , and then the multiple stratonovich integral can be defined in an @xmath11 space with respect to a measure that controls the behavior of the functions on the diagonal sets . in this way , the problem of providing a manageable definition of the traces is avoided . we would like to comment that an impressive body of work on multiple stochastic integrals with respect to lvy processes has been done by kallenberg , kwapien , krakowiak , rosinski , szulga , woyczinski and many others ( see @xcite and the references therein ) . however , their approach is very different from ours , and assumes different settings to those used in this work . for this reason , we have only used a few results by those authors . the paper is organized as follows . in section [ sec2 ] we review some combinatorics concepts and the basics of the stochastic measures as vector valued measures . in section [ random ] we introduce the random measures induced by a lvy process , and we identify the diagonal measures in such a case . in section [ measure ] we study the relationship between the product and it measures of a set , and we obtain a hu meyer formula for measures . in section [ integral ] we define the multiple it stochastic integral and the multiple stratonovich integral and also prove the general hu meyer formula for integrals . in section [ sec6 ] , as particular cases , we deduce the classical hu meyer formulas for the brownian motion and for the poisson process . we also study the case where the lvy process is a subordinator , and prove that both the multiple it stochastic integral and the multiple stratonovich integral can be computed in a pathwise sense . finally , in order to make the paper lighter , some of the combinatorial results are included as an . we need some notation of the combinatorics of the partitions of a finite set ; for details we refer to stanley @xcite , chapter 3 , or rota and wallstrom @xcite . let @xmath12 be a finite set . a partition of @xmath12 is a family @xmath13 of nonvoid subsets of @xmath12 , pairwise disjoint , such that @xmath14 . the elements @xmath15 are called the _ blocks _ of the partition . denote by @xmath16 the set of all partitions of @xmath12 , and write @xmath17 for @xmath18 . given @xmath19 , we write @xmath20 if each block of @xmath21 is contained in some block of @xmath22 ; we then say that @xmath21 is a refinement of @xmath22 . this relationship defines a partial order that is called the _ reversed refinement order _ , and it makes @xmath16 a lattice . we write @xmath23 , which is the minimal element , and @xmath24 the maximal one . we say that a partition @xmath25 is of type @xmath26 if @xmath22 has exactly @xmath27 blocks with 1 element , exactly @xmath28 blocks with 2 elements , and so on . in the same way , for @xmath29 and @xmath30 , we say that the segment @xmath31 $ ] is of type @xmath32 if there are exactly @xmath27 blocks of @xmath22 in @xmath21 ; there are exactly @xmath28 blocks of @xmath22 that each one gives rise to 2 blocks of @xmath21 , etc . necessarily , @xmath33 in that situation , the mbius function of @xmath31 $ ] is @xmath34 we use the mbius inversion formula , that in the context of the lattice of the partitions of a finite set , says that for two functions @xmath35 , @xmath36 if and only if @xmath37 ( see @xcite , proposition 3.7.2 ) . as we commented in the , we will introduce two random measures on a @xmath38-dimensional space , and the diagonal sets will play an essential role . diagonal sets can be conveniently described through the partitions of the set @xmath39 . we use the notation introduced by rota and wallstorm @xcite . let @xmath40 be an arbitrary set , and consider @xmath41 . given @xmath42 , we write @xmath43 if @xmath44 and @xmath45 belong to the same block of @xmath22 . put @xmath46 and @xmath47 the sets @xmath48 are called _ diagonal sets_. note that @xmath49 and @xmath50 . for example , for @xmath51 and @xmath52 , we have @xmath53 and @xmath54 the sets corresponding to the minimal and maximal partitions are specially important @xmath55 and @xmath56 if @xmath57 , then @xmath58 the above notation @xmath59 is coherent with the reversed refinement order @xmath60 in particular , @xmath61 . let @xmath62 be a complete probability space . in this paper , a random measure @xmath63 on a measurable space @xmath64 is an @xmath10-valued @xmath21-additive vector measure , that means , a map @xmath65 such that for every sequence @xmath66 , such that @xmath67 , @xmath68 the @xmath21-additive vector measures defined on a @xmath21-field inherit some basic properties of the ordinary measures , but not all . so , for a sake of easy reference , we write here a uniqueness property translated to our setting . the proof is the same as the one for ordinary measures . [ monotona ] let @xmath63 and @xmath69 be two random measures on @xmath64 , and consider a family of sets @xmath70 closed under finite intersection and such that @xmath71 . then @xmath72 assume that the measurable space @xmath64 satisfies that for every set @xmath73 and every @xmath42 , we have @xmath74 . as rota and wallstrom @xcite point out , this condition is satisfied if @xmath40 is a polish space and @xmath75 its borel @xmath21-algebra . we extend the definition of _ good random measure _ introduced by rota and wallstorm @xcite to a family of measures ; specifically , we say that the random measures @xmath76 over a measurable space @xmath64 are _ jointly good random measures _ if the finite additive product vector _ measure _ @xmath77 defined on the product sets by @xmath78 can be extended to a ( unique ) @xmath21-additive random measure on @xmath79 . this extension , obvious for ordinary measures , is in general not transferred to arbitrary vector measures ( see engel @xcite , masani @xcite and kwapien and woyczynski @xcite ) . given a good random measure @xmath63 ( in the sense that the @xmath38-fold product @xmath80 satisfies the above condition ) , the starting point of rota and wallstrom ( @xcite , definition 1 ) is to consider new random measures given by the restriction over the diagonal sets ; specifically , for @xmath42 they define @xmath81}_\pi(c):=\phi^{\otimes n}(c_{\pi})\qquad \mbox{for } c\in\mathcal{s}^{\otimes n}.\ ] ] the following definitions are the extension of these concepts to a family of random measures . let @xmath82 be jointly good random measures on @xmath64 . for a partition @xmath42 , define @xmath83 and @xmath84 in agreement with the notation in rota and wallstrom @xcite , when @xmath85 , we simply write @xmath86 for @xmath87 and @xmath88}_\pi$ ] for the corresponding measure given in ( [ def - ito ] ) . since @xmath89 , then @xmath90 , that is the product measure . the measure @xmath91 is called _ the it multiple stochastic measure _ relative to @xmath82 . as the ordinary multiple it integral , the it multiple stochastic measure gives zero mass to every diagonal set different from @xmath92 : [ 0diagonals ] let @xmath42 such that @xmath93 . for every @xmath73 , we have @xmath94 from ( [ incompatibilitat ] ) we have @xmath95 . the basic result of rota and wallstrom @xcite , proposition 1 , is transferred to this situation : [ strato - ito1 ] @xmath96 and @xmath97 where @xmath98 is the mbius function defined in section [ partitions ] . the equality ( [ mesura - prod ] ) is deduced from ( [ union ] ) and the definitions ( [ def - producte ] ) and ( [ def - ito ] ) . the equality ( [ mesura - ito ] ) follows from ( [ mesura - prod ] ) and the mebius inversion formula ( [ moebius ] ) . let @xmath99 \}$ ] be a lvy process , that is , @xmath100 has stationary and independent increments , is continuous in probability , is cadlag and @xmath101 . in all the paper we assume that @xmath100 has moments of all orders ; however , if the interest is restricted to multiple integral up to order @xmath102 , then it is enough to assume that the process has moments up to order @xmath103 . denote the lvy measure of @xmath100 by @xmath104 , and by @xmath105 the variance of its gaussian part . the existence of moments of @xmath106 of all orders implies that @xmath107 and @xmath108 . write @xmath109,\nonumber\\[-8pt]\\[-8pt ] k_2&= & \sigma^2+\int_{\mathbb{r } } x ^2 \nu(dx ) \quad\mbox{and}\quad k_n=\int_{\mathbb{r } } x ^n \nu(dx)<\infty,\qquad n\ge3.\nonumber\end{aligned}\ ] ] from now on , take @xmath110 $ ] and @xmath111)$ ] . the basic random measure @xmath112 that we consider is the measure induced by the process @xmath100 itself , defined on the intervals by @xmath113s , t])=x_t - x_s,\qquad 0\le s\le t\le t,\ ] ] and extended to @xmath114)$ ] . the measure @xmath112 is an independently scattered random measure , that is , if @xmath115)$ ] are pairwise disjoint , then @xmath116 are independent . the random measures induced by the powers of the jumps of the process , @xmath117 , are also used . consider the _ variations _ of the process @xmath100 ( see meyer @xcite , page 319 ) @xmath118_t=\sum_{0<s\le t } ( \delta x_s ) ^2+\sigma^2t,\\ x ^{(n)}_t&=&\sum_{0<s\le t } ( \delta x_s ) ^n,\qquad n \ge3 . \nonumber\end{aligned}\ ] ] the processes @xmath119 are lvy processes such that @xmath120=k_n t\qquad \forall n\ge1.\ ] ] so , the centered processes , @xmath121 are square integrable martingales , called _ teugels martingales _ ( see nualart and schoutens @xcite ) , with predictable quadratic covariation @xmath122 we denote by @xmath123 the random measure induced by @xmath124 , and for @xmath125 , @xmath126 ( we indistinctly use both @xmath127 and @xmath112 ) . every @xmath123 is a independently scattered random measure . for @xmath128)$ ] , @xmath129=k_{n+m } \int_{a\cap b}dt+ k_nk_m \int_{a } dt\int _ { b}dt.\ ] ] we stress the following property , which is the basis of all the paper , and is a consequence of theorem 10.1.1 by kwapien and woyczynski @xcite . for every @xmath130 , the random measures @xmath131 are jointly good random measures on @xmath132^n,\mathcal{b } ( [ 0,t]^n ) ) $ ] . rota and wallstrom @xcite define the diagonal measure of order @xmath38 of @xmath112 as the random measure on @xmath133 $ ] given by @xmath134).\ ] ] to identify the diagonal measures is a necessary step to study the stochastic multiple integral . in the case of a random measure generated by a lvy process we show that the diagonal measures are the measures generated by the variations of the process . [ diagonal - simple ] for every @xmath135)$ ] and @xmath136 , @xmath137 where @xmath123 is the random measure induced by @xmath124 . since both @xmath138 and @xmath123 are random measures , by proposition [ monotona ] it is enough to check the equality for @xmath139 $ ] . consider an increasing sequence of equidistributed partitions of @xmath140 $ ] with the mesh going to 0 ; for example , take @xmath141 and let @xmath142 to shorten the notation , write @xmath143 instead of @xmath144 . consider the sets @xmath145^n\cup(t_1,t_2 ] ^n\cup\cdots\cup(t_{2^m-1},t ] ^n.\ ] ] random measures are sequentially continuous and @xmath146^n_{\widehat1}$ ] , when @xmath147 , so we have that @xmath148)=\lim_m \sum_{k=0}^{2^m-1 } ( \phi ( ( t_k , t_{k+1 } ] ) ) ^n= \lim_m \sum_{k=0}^{2^m-1 } ( x_{t_{k+1}}-x_{t_{k } } ) ^n\ ] ] in @xmath10 . for @xmath149 , @xmath150_t=\phi_2 ( ( 0,t ] ) \qquad \mbox{in probability},\ ] ] so the proposition is true in this case . for @xmath151 , by it s formula , @xmath152 \nonumber\\ \label{equa } & & \qquad = n\int_0^t \biggl(\sum_{k=0}^{2^m-1 } ( x_{s-}- x_{t_{k } } ) ^{n-1}\mathbf{1 } _ { ( t_{k},t_{k+1}]}(s ) \biggr ) \,dx_s\\ \label{equb } & & \qquad\quad{}+\pmatrix{n\cr2}\int_0^t \biggl(\sum_{k=0}^{2^m-1 } ( x_{s-}- x_{t_{k } } ) ^{n-2}\mathbf{1}_{(t_{k},t_{k+1}]}(s ) \biggr ) \,d[x , x]_s\\ \label{equc } & & \qquad\quad{}+\sum_{j=3}^n\sum_{k=0}^{2^m-1}\sum_{t_{k}<s\le t_{k+1 } } \pmatrix{n\cr j } ( x_{s-}- x_{t_{k } } ) ^{n - j } ( \delta x_s ) ^j.\end{aligned}\ ] ] for @xmath153 , the corresponding term in ( [ equc ] ) is @xmath154}(s ) \biggr ) \,dx^{(j)}_s.\ ] ] hence , ( [ equa ] ) , ( [ equb ] ) and ( [ equd ] ) have the same structure @xmath155 where @xmath156}(s)$ ] is a predictable process and @xmath157 is a semimartingale . since @xmath158 is left continuous , @xmath159 moreover , @xmath160 and the process @xmath161 \}$ ] is cadlag and adapted , and as a consequence , it is prelocally bounded ( see pages 336 and 340 in dellacherie and meyer @xcite ) . by the dominated convergence theorem for stochastic integrals ( dellacherie and meyer @xcite , theorem 14 , page 338 ) , @xmath162 finally , for @xmath163 , the term in ( [ equc ] ) is @xmath164 , and the proposition is proved . diagonal measures associated to a random measure of the form @xmath165 are needed . this is an extension of the previous proposition , and it is a key result for the sequel . [ diagonal - diversos ] let @xmath130 , @xmath102 , and @xmath135)$ ] . then @xmath166 as in the proof of the last proposition and with the same notation , it suffices to prove that for all @xmath167 @xmath168 ) \ ] ] in probability . this convergence follows from proposition [ diagonal - simple ] by polarization . the hu meyer formula gives the relationship between the product measure @xmath169 and the it stochastic measures @xmath170 . in this section we obtain this formula for measures and in the next one we extend it to the corresponding integrals . the idea of hu meyer formula is the following . given @xmath171^n)$ ] , we can decompose @xmath172 so @xmath173 next step is to express each @xmath174 as a multiple it stochastic measure . for example , take @xmath175 and @xmath176 . then , @xmath177 and we will prove that @xmath178 that is , both the product measure and the product set on the last two variables collapse to produce a diagonal measure , and since @xmath179 , we get an it measure . to handle in general this property , we need some notation . given a partition @xmath180 with blocks @xmath15 , we can order the blocks in agreement with the minimum element of each block . when necessary , we assume that the blocks have been ordered with that procedure , and we simply say that @xmath15 are ordered . in that situation , we write @xmath181 we start considering a set @xmath182 , with @xmath135)$ ] , and later we extend the hu meyer formula to an arbitrary set @xmath183^n)$ ] . [ hu - meyer1 ] let @xmath135)$ ] . then @xmath184 to prove this theorem we need two lemmas the first one is an invariance - type property of product measures under permutations . we remember some standard notation . we denote by @xmath185 the set of permutations of @xmath186 . consider @xmath187 . 1 . for a partition @xmath180 with blocks @xmath15 , we write @xmath188 for the partition with blocks @xmath189 . note that in general the blocks @xmath190 are not ordered , even when @xmath191 are . 2 . for a vector @xmath192 , we write @xmath193 given @xmath194 , we put @xmath195 [ permutacio ] let @xmath187 and @xmath130 . then for every @xmath196^n)$ ] , @xmath197 and @xmath198 define the vector measure @xmath199 for @xmath200 , we have that @xmath201 and it is clear that @xmath202 then , equality ( [ permut1 ] ) follows from proposition [ monotona ] . to prove ( [ permut2 ] ) , first note that , by definition , the it stochastic measure satisfies @xmath203 moreover @xmath204 . so it suffices to prove ( [ permut2 ] ) for a set @xmath205 . denote by @xmath206 the @xmath207-algebra trace of @xmath114^n)$ ] with @xmath133^n_{\widehat0}$ ] , which is composed by all sets @xmath92 , with @xmath208^n)$ ] . this @xmath21-algebra is generated ( on @xmath133^n_{\widehat0}$ ] ) by the family of rectangles @xmath209 , with @xmath210 pairwise disjoint . by proposition [ monotona ] , we only need to check ( [ permut2 ] ) for this type of rectangle , and the property reduces to ( [ permut1 ] ) . the next lemma is an important step in proving theorem [ hu - meyer1 ] . to have an insight into its meaning , consider the following example : let @xmath51 and @xmath211 . with a slight abuse of notation , we can write @xmath212 by theorem [ diagonal - diversos ] , @xmath213 however , if you consider @xmath214 , even though @xmath215 and @xmath21 have the same number of blocks with 1 element and the same number of blocks with 2 elements ( they have the same type ) , the computation of @xmath216 is not so straightforward . the lemma gives such computation . its proof demands some combinatorial results and it is transferred to appendix [ prova - lema ] . [ lema - basic ] let @xmath130 , @xmath180 with blocks @xmath15 ( ordered ) , and @xmath217)$ ] . then @xmath218 proof of theorem [ hu - meyer1 ] by proposition [ strato - ito1 ] , @xmath219}(a^{n}).\ ] ] so it suffices to prove that @xmath220}(a^{n})={\mathrm{st}}_{\widehat0}^{\overline\sigma}(a^{\ # \sigma}).\ ] ] by the second statement in proposition [ strato - ito1 ] we have @xmath221}(a^{n})=\sum_{\pi\in[\sigma,\widehat1]}\mu ( \sigma , \pi)\phi ^{\otimes n}_\pi(a^n)=\sum_{\pi\in[\sigma,\widehat1]}\mu(\sigma , \pi ) \phi^{\otimes n } ( a^n_{\ge\pi } ) .\ ] ] by lemma [ lema - basic ] , @xmath222 let @xmath223 be the blocks of @xmath180 ( ordered ) and write @xmath224 the partition @xmath225 $ ] , with blocks @xmath226 , induces a unique partition of @xmath227 , with blocks @xmath228 such that @xmath229 ( see proposition [ bijeccio ] in the ) . hence , for @xmath230 , @xmath231 thus , from ( [ hu - form3 ] ) and lemma [ lema - basic ] , @xmath232 by ( [ hu - form2 ] ) and ( [ hu - form4 ] ) using again the bijection between @xmath233 $ ] and @xmath234 stated in proposition [ bijeccio ] in the , and proposition [ strato - ito1 ] , we obtain @xmath235}(a^{n})&=&\sum_{\pi\in[\sigma,\widehat1]}\mu ( \sigma , \pi ) ( \phi_{s_1}\otimes\cdots\otimes\phi_{s_m } ) ( a^m_{\ge\pi^*})\\ & = & \sum_{\rho\in\pi_{m}}\mu(\widehat0,\rho ) ( \phi_{s_1}\otimes\cdots\otimes\phi_{s_m } ) ( a^m_{\ge\rho } ) = { \mathrm{st}}_{\widehat 0}^{\overline\sigma}(a^{\#\sigma}).\end{aligned}\ ] ] in order to extend the hu meyer formula for a general set in @xmath236^n)$ ] , we use a set function to express for an arbitrary set the contraction from @xmath237 to @xmath238 . that is , given a partition @xmath180 , with blocks @xmath223 ordered , we want to contract a set @xmath208^n)$ ] into a set of @xmath114^{\ # \sigma})$ ] according to the structure of the @xmath21-diagonal sets . with this purpose , define the function @xmath239^{\#\sigma } & \longrightarrow & [ 0,t]^n,\nonumber\\[-8pt]\\[-8pt ] ( x_1,\ldots , x_m ) & \longrightarrow & ( y_{1},\ldots , y_{n } ) , \nonumber\end{aligned}\ ] ] where @xmath240 , if @xmath241 . for example , if @xmath51 and @xmath242 , @xmath243 note that @xmath244 see appendix [ q - sigma ] for more details . [ hu - meyer2 ] let @xmath208^n)$ ] . then @xmath245 we separate the proof in two steps . in the first one , we show that it is enough to prove the theorem for a rectangle of the form @xmath246 where @xmath247 are pairwise disjoint . in the second step we check formula ( [ hu - meyer2-form ] ) for those rectangles . _ first step . _ by proposition [ monotona ] , it suffices to prove the theorem for a rectangle @xmath209 . since every rectangle can be written as a disjoint union of rectangles such that every two components are either equal or disjoint , we consider one of this rectangles , @xmath200 , where for every @xmath248 , @xmath249 or @xmath250 . now we show that the formula ( [ hu - meyer2-form ] ) applied to @xmath251 is invariant by permutations : specifically , we see that for any permutation @xmath187 @xmath252 the first equality is deduced from ( [ permut1 ] ) . for the second one , applying proposition [ perm - delta](i ) , we have @xmath253 where @xmath254 is the permutation that gives the correct order of the blocks of @xmath188 ( see the lines before proposition [ perm - delta ] ) . by lemma [ permutacio ] @xmath255 where the last equality is due to the fact that @xmath256 by the definition of @xmath257 [ see ( [ barra2 ] ) ] . finally , @xmath258 because we are adding over all the set @xmath259 . _ second step . _ consider @xmath260 with @xmath261 pairwise disjoint and @xmath262 . by theorem [ hu - meyer1 ] , @xmath263 where the last equality is due to the fact that @xmath264 and the definition of the it measure @xmath265 . let @xmath266 be the partition with blocks @xmath267 there is a bijection between the elements @xmath180 , with @xmath268 , and @xmath269 such that @xmath270 where we use equality ( [ projector2 ] ) in the . then , @xmath271 where the last equality is due to the fact that if @xmath272 , then @xmath273 [ see ( [ projector2 ] ) ] . we extend theorem [ hu - meyer2 ] to integrals with respect to the random measures involved . we first define an it - type multiple integral and an integral with respect to the product measure . we generalize the multiple it integral with respect to the brownian motion ( it @xcite ; see also @xcite ) to a multiple integral with respect to the lvy processes @xmath274 . as we will prove , that integral can be interpreted as the integral with respect to the it stochastic measure . the ideas used to construct this integral are mainly it s ; however , the fact that these processes ( in general ) are not centered obstructs the classical isometry property , being substituted by an inequality . write @xmath275^n,\mathcal{b}([0,t]^n ) , ( dt)^{\otimes n})$ ] . denote by @xmath276 the set of the so - called it - elementary functions , having the form @xmath277 where @xmath278)$ ] are pairwise disjoint , and @xmath279 is zero if two indices are equal . it is well known ( see it @xcite ) that @xmath276 is dense in @xmath280 . consider @xmath281 and define the multiple it integral of @xmath282 with respect to @xmath274 by @xmath283 let @xmath281 and @xmath284 . then @xmath285\le\alpha_{\mathbf{r}}\int _ { [ 0,t]^n}f^2(t_1,\ldots , t_n ) \,dt_1\cdots dt_n,\ ] ] where @xmath286 is a constant that depends on @xmath287 but not on @xmath282 . the proof follows exactly the same steps as that of theorem 4.1 in engel @xcite . the key point is that the measures @xmath288 can be written as @xmath289 where @xmath290 is the centered and independently scattered random measure corresponding to @xmath291 . the extension of the multiple it stochastic integral to @xmath280 , stated below , is proved as in the brownian case ( see it @xcite ) . [ ito - ext - th ] the map @xmath292 can be extended to a unique linear continuous map from @xmath280 to @xmath10 . in particular , @xmath293 satisfies the inequality @xmath294\le\alpha_{\mathbf{r}}\int _ { [ 0,t]^n}f^2(t_{1},\ldots , t_{n } ) \,dt_1\cdots dt_n.\ ] ] as in the brownian case , it is useful to express the multiple integral in terms of iterated integrals of the form @xmath295 where @xmath296 is a permutation of @xmath186 . this integral is properly defined for @xmath297 . this can be checked using the decomposition of @xmath298 as a special semimartingale @xmath299 , where , as we said in section [ random ] , @xmath291 is a square integrable martingale with predictable quadratic variation @xmath300 . the previous iterated integral then reduces to a linear combination of iterated integrals of type @xmath301 being @xmath302 either @xmath9 or @xmath303 . hence , at each iteration , the integrability condition @xmath304<\infty\ ] ] of a predictable process @xmath305 with respect to @xmath306 can be easily verified . next proposition gives the precise expression of the multiple integral as a sum of iterated integrals . since we are integrating with respect to different processes , we need to separate the space @xmath133^n$ ] into simplexs . let @xmath297 . then @xmath307 where @xmath308 , and the integrals on the right - hand side are interpreted as iterated integrals . by linearity and density arguments , it suffices to consider a function @xmath309 where @xmath310 $ ] are pairwise disjoint , and a computation gives the result . when @xmath311 , we write @xmath312 instead of @xmath313 ; in that case , the multiple it integral enjoys nicer properties . [ nicer ] 1 . let @xmath297 . then @xmath314 where @xmath315 is the symmetrization of @xmath282 @xmath316 2 . assume @xmath317=0 $ ] . for @xmath318 , @xmath319=\delta_{n , m } k_2^n n!\int _ { [ 0,t]^n}\widetilde{f } \widetilde g \,d\mathbf{t},\ ] ] where @xmath320 , and 0 otherwise . 3 . let @xmath297 be a symmetric function . then @xmath321 we now state the relationship between the it stochastic measure @xmath170 and the it multiple integral @xmath322 . [ stochastic - int ] let @xmath208^n)$ ] and @xmath323 . then @xmath324 by ( [ desigualtat - integral ] ) the map @xmath325 defines a vector measure on @xmath114^n)$ ] . on the left - hand side of ( [ ito - int - mes ] ) , the it measure satisfies @xmath326 now , look at right - hand side of ( [ ito - int - mes ] ) . for @xmath327 we have that @xmath328^n_\pi$ ] . for all @xmath93 , @xmath329\le \alpha_n\int_{[0,t]^n } \mathbf{1}_{c_\pi}\,dt_1\cdots dt_n\le\alpha _ n\int _ { [ 0,t]^n } \mathbf{1}_{[0,t]^n_\pi } \,dt_1\cdots dt_n=0.\ ] ] hence , @xmath330 from ( [ f-0 ] ) and ( [ f-00 ] ) , it suffices to prove ( [ ito - int - mes ] ) for a set @xmath205 . as in the proof of the second part of lemma [ permutacio ] , this can be reduced to check that equality for a rectangle @xmath331\times\cdots\times(s_n , t_n]$ ] , with the intervals pairwise disjoint . this follows from the fact that both sides of ( [ ito - int - mes ] ) are equal to @xmath332)\cdots\phi_{r_n}((s_n , t_n])$ ] . the property @xmath333 is lost when the integrators are different . however , from proposition [ stochastic - int ] and ( [ permut2 ] ) we can deduce the following useful property : [ perm - int ] let @xmath297 , and @xmath323 , where @xmath334 . consider @xmath187 . then @xmath335 given a map @xmath336^n\to{\mathbb{r}}$ ] , the integral with respect to the product measure @xmath169 is called the multiple stratonovich integral , and denoted by @xmath6 . its basic property is that the integral of a product function factorizes @xmath337 where @xmath338 in order to construct this integral , we consider ordinary simple functions of the measurable space @xmath339^n,\mathcal{b}([0,t]^n))$ ] . specifically , denote by @xmath340 the set of functions with the form @xmath341 where @xmath342^n ) , i=1,\ldots , k$ ] . for such @xmath282 , define the multiple stratonovich integral by @xmath343 the integral of a simple function does not depend on its representation , and it is linear . moreover , [ hu - meyer - funct - prop ] let @xmath344 . then we have the hu meyer formula @xmath345 where the function @xmath346^{\#\sigma } \to[0,t]^n$ ] is introduced in ( [ qsigma ] ) , @xmath347 is the vector whose components are the sizes of the ordered blocks of @xmath21 , and @xmath348 is the multiple it integral of order @xmath349 with respect to the measures @xmath350 . by linearity , it suffices to consider @xmath351 , where @xmath208^n)$ ] . a generic term on the right - hand side of ( [ hu - meyer - funct1 ] ) is @xmath352 and @xmath353 hence , by proposition [ stochastic - int ] , @xmath354 and ( [ hu - meyer - funct1 ] ) follows from theorem [ hu - meyer2 ] . let @xmath180 , with @xmath355 , and denote by @xmath356 the image measure of the lebesgue measure @xmath357 by the function @xmath346^{m } \to[0,t]^n$ ] . the image measure theorem implies that for @xmath358^n\to{\mathbb{r}}$ ] measurable , positive or @xmath359-integrable , @xmath360^n } f(t_1,\ldots , t_n ) \,d\lambda_\sigma(t_1,\ldots , t_n)= \int_{[0,t]^m } f(q_\sigma(t_1,\ldots , t_m ) ) \,dt_1 \cdots dt_m.\hspace*{-28pt}\ ] ] define on @xmath114^n)$ ] the measure @xmath361 and write @xmath362 for @xmath363^n,\mathcal { b}([0,t]^n),\lambda_n)$ ] . in order to extend the multiple stratonovich integral we need the following inequality of norms : let @xmath344 . then @xmath364\le c\int_{[0,t]^n}f^2\ , d\lambda_n,\ ] ] where @xmath251 is a constant . by ( [ hu - meyer - funct1 ] ) , ( [ desigualtat - integral ] ) and ( [ image ] ) , @xmath365 & \le & c \sum_{\sigma\in\pi_n } { \mathbb{e}}\bigl [ \bigl(i_{\#\sigma}^{\overline\sigma}(f\comp q_\sigma ) \bigr)^2 \bigr]\\ & \le & c \sum_{\sigma\in\pi_n}\int_{[0,t]^{\#\sigma}}(f\comp q_\sigma)^2\ , dt_1\cdots dt_{\#\sigma}\\ & = & c \sum_{\sigma\in\pi_n}\int_{[0,t]^{n}}f^2 \,d\lambda_\sigma\\ & = & c\int _ { [ 0,t]^{n}}f^2 \,d\lambda_n.\end{aligned}\ ] ] the main result of the paper is the following theorem : the map @xmath366 can be extended to a unique linear continuous map from @xmath362 to @xmath10 , and we have the hu meyer formula @xmath367 the extension of @xmath368 to a continuous map on @xmath362 is proved using a density argument and inequality ( [ desigualtat - strato ] ) . to prove the hu meyer formula , let @xmath369 and @xmath370 such that @xmath371 in @xmath362 . for every @xmath180 , we have @xmath372 in @xmath373 ; hence , from theorem [ ito - ext - th ] the it integrals on the right - hand side of ( [ hu - meyer - funct ] ) converge , and the formula follows from proposition [ hu - meyer - funct - prop ] . 1 . let @xmath374,dt)$ ] . then @xmath375 and @xmath376 this result is easily checked for simple functions @xmath377 and extended to the general case by a density argument . 2 . in order to prove the hu meyer formula for @xmath368 it is enough to assume that the process @xmath100 has moments up to order @xmath103 . 3 . for @xmath378 , the measure @xmath359 is singular with respect to the lebesgue measure on @xmath133^n$ ] . for example , for @xmath149 and @xmath379 , let @xmath380\}$ ] be the diagonal of @xmath133 ^ 2 $ ] . then @xmath381 is concentrated in @xmath382 , that has zero lebesgue measure , but @xmath381 is nonzero @xmath383 ^ 2}\mathbf{1}_{d}(s , t ) \,d\lambda _ { \widehat 1}(s , t)=\int _ { [ 0,t]}\mathbf{1}_{d}(t , t ) \,dt = t.\ ] ] 4 . as in the brownian case ( see @xcite and the references therein ) , there are other procedures to construct the multiple stratonovich integral . the main difficulty in every approach is that the usual condition @xmath297 in it s theory is not sufficient to guarantee the multiple stratonovich integrability of @xmath282 . the reason is that one needs to control the behavior of @xmath282 on the diagonal sets @xmath133_\sigma^n$ ] that have zero lebesgue measure when @xmath384 . we solve this difficulty using the norm induced by the measure @xmath385 , which seems to be appropriate for dealing with the diagonal sets , avoiding in this way the difficulty of a manageable definition of the _ traces_. when the function @xmath369 is symmetric , the hu meyer formula can be considerably simplified . we show that we can assume that symmetry on @xmath282 without loss of generality . let @xmath369 . then @xmath386 , where @xmath315 is the symmetrization of @xmath282 [ see ( [ simetrizacio ] ) ] . the proof is straightforward for @xmath387 , @xmath171^n)$ ] , using lemma [ permutacio ] . by linearity the equality @xmath388 is extended to @xmath389 , and by density to @xmath362 . next we show the hu meyer formula for a symmetric function @xmath282 . in general ( for @xmath282 symmetric ) , the function @xmath390 is nonsymmetric , but as we will see in the proof of the next theorem , its multiple it integral depends only on the block structure of @xmath21 ( the type of @xmath21 ) . for example , with @xmath391 , @xmath392 and @xmath393 , we have that @xmath394 that is nonsymmetric . its integral is @xmath395 take @xmath396 . then @xmath397 and @xmath398 where the last equality is due to proposition [ perm - int ] . we use the following notation : given nonnegative integers @xmath399 such that @xmath400 , we write @xmath401=(\underbrace{1,\ldots,1}_{r_1},\underbrace { 2,\ldots , 2}_{r_2},\ldots).\ ] ] note that this corresponds to @xmath402 when @xmath403 we also write @xmath404 for @xmath405 , with @xmath21 the above partition . let @xmath406 be a symmetric function . then @xmath407 } ( f\comp q_{r_1,\ldots , r_k } ) , \ ] ] where the sum is extended over all nonnegative integers @xmath408 such that @xmath400 , for @xmath409 . let @xmath369 symmetric . for every @xmath180 and @xmath187 , @xmath410 where ( [ equa ] ) is due to proposition [ perm - delta](ii ) , the equality ( [ equb ] ) follows from the symmetry of @xmath282 and ( [ equc ] ) from proposition [ perm - int ] and the fact that @xmath257 gives the correct order of @xmath188 [ see ( [ barra2 ] ) ] . this implies that all the partitions that have the same number of blocks of 1 element , the same number with two elements , etc . ( i.e. , they have the same type ) give the same it multiple integral in the hu meyer formula . to obtain ( [ hu - meyer - simetric ] ) it suffices to count the number of partitions of @xmath39 with @xmath27 blocks with 1 element , @xmath28 blocks with 2 elements@xmath411 blocks with @xmath412 elements , which is @xmath413 _ final remark . _ one may expect that by decomposing the lvy process into a sum of two independent processes , one with the small jumps and the other with the large ones , the assumption of the existence of moments could be avoided . however , this decomposition introduces dramatic changes to the context of the work , and such an extension is beyond the scope and purposes of the present paper . when @xmath414 is a standard brownian motion , @xmath415)=t \quad\mbox{and}\quad \phi_n=0,\qquad n\ge3.\vadjust{\goodbreak}\ ] ] it follows that in the hu meyer formula only the partitions with all blocks of cardinality 1 or 2 give a contribution , and all the it integrals are a mixture of multiple stochastic brownian integrals and lebesgue integrals . we can organize the sum according the number of blocks of two elements . for a partition having @xmath45 blocks of @xmath416 elements , and @xmath406 symmetric , the multiple it integral is @xmath417}_{n - j}(f)\\ & & \qquad=\int_{[0,t]^{n - j}}f(s_1,\ldots , s_{n-2j},\\ & & \hspace*{84.4pt}t_1,t_1,\ldots , t_{j},t_{j } ) \,dw_{s_1}\cdots dw_{s_{n-2j } } \,d t_{1}\cdots d t_{j}\\ & & \qquad = i_{n-2j } \biggl(\int_{[0,t]^{j}}f(\bolds\cdot , t_1,t_1,\ldots , t_{j},t_{j } ) \,d t_{1}\cdots d t_{j } \biggr),\end{aligned}\ ] ] where the last equality is due to a fubini - type theorem . therefore , @xmath418}\frac{n!}{(n-2j)!j!2^j}\nonumber\\[-8pt]\\[-8pt ] & & \hspace*{17.6pt}{}\times i_{n-2j } \biggl(\int_{[0,t]^{j}}f(\bolds\cdot , t_1,t_1,t_2,t_2,\ldots , t_{j},t_{j})\ , d t_{1}\cdots d t_{j } \biggr),\nonumber\end{aligned}\ ] ] which is the classical hu meyer formula ( see @xcite ) . on the other hand , in the measure @xmath385 only participate the measures @xmath359 corresponding to the partitions above mentioned . consider the measure @xmath419 on @xmath133 ^ 2 $ ] , that is , for a positive or @xmath420 integrable function @xmath421 , @xmath422 ^ 2}h(s , t ) \,d\ell_2(s , t)=\int_{[0,t]}h(t , t)\,dt.\ ] ] given the partition @xmath180 , @xmath423 we have @xmath424 let @xmath425 be a standard poisson process with intensity 1 , and consider the process @xmath426 . for every @xmath102 , @xmath427 and hence , a multiple it integral can be reduced to a linear combination of multiple integrals where all the integrators are @xmath428 or @xmath429 . for @xmath406 symmetric , each integral @xmath430 } ( f\comp q_{r_1,\ldots , r_k})$ ] in ( [ hu - meyer - simetric ] ) can be expressed in terms of the number of lebesgue integrals that appear @xmath431 } ( f\comp q_{r_1,\ldots , r_k})\\ \hspace*{-5pt}&&\qquad=\sum_{j=0}^{r_2+\cdots+r_k}\hspace*{-0.9pt}i_{r_1+\cdots+r_k - j } \biggl ( \int_{[0,t]^j } \biggl(\mathop{\sum_{l_1,\ldots , l_j = r_1 + 1}}_{\mathrm { different}\mbox { } } ^{r_1+\cdots+r_k}\hspace*{-0.9pt } ( f\comp q_{r_1,\ldots , r_k})\\ \hspace*{-5pt}&&\hspace*{204.5pt}{}\times ( t_1,\ldots , t_{r_1+\cdots+r_k } ) \biggr)\,dt_{l_1}\cdots dt_{l_j } \biggr),\end{aligned}\ ] ] and the hu meyer formula of sol and utzet @xcite can be deduced from this expression . a subordinator is a lvy process with increasing paths . an important example of a subordinator with moments of all orders is the gamma process , denoted by @xmath432 , which is the lvy process corresponding to an exponential law of parameter 1 . its lvy measure is @xmath433 the law of @xmath434 is gamma with mean @xmath9 and scale parameter equal to one . a gamma process can be represented as the sum of its jumps , that are all positive , @xmath435 the lvy measure of @xmath436 is ( see schoutens @xcite ) @xmath437 and the teugels martingales are @xmath438 in this case , unlike the brownian motion and the poisson process , the hu meyer formula does not simplify , due to the fact that the diagonal measures can not be expressed in a simple way in terms of , say , the process and a deterministic measure . however , for a gamma process , and in general , for a subordinator without drift ( see below for the definition ) with moments of all orders , both the multiple it and stratonovich integrals can be computed pathwise integrating with respect to an ordinary measure . this is a multivariate extension of the property that states that the stochastic integral and the pathwise lebesgue stieljes integral with respect to a semimartingale of bounded variation are equal ; such property was proved for the integral with respect to a lvy process of bounded variation by millar @xcite under weak conditions , and part of our proof follows his scheme . let @xmath439 be a subordinator . the lvy it representation of @xmath100 takes the form @xmath440 with @xmath441 ( see sato @xcite , theorems 21.5 and 19.3 ) . the number @xmath442 is called the _ drift _ of the subordinator , and we will assume that @xmath443 . consider the sequence of stopping times @xmath444 with disjoint graphs that exhaust the jumps of @xmath445 , and @xmath100 only has jumps on these times ( see , e.g. , dellacherie and meyer @xcite , theorem b , page xiii , for a construction of this sequence ) . denote by @xmath446 the set of @xmath38-tuples @xmath447 , with @xmath448 , and all entries different . for @xmath130 , define a measure on @xmath133^n$ ] by @xmath449 where @xmath450 is a dirac measure at point @xmath451 , with the convention that the sum is 0 if @xmath452 . we have the following property : let @xmath439 be a subordinator without drift and with moments of all orders . with the preceding notation , for every @xmath297 , @xmath453^n}f \,d m_{r_1,\ldots , r_n}\qquad \mbox{a.s.}\ ] ] first , note two facts : \(a ) @xmath454 is a finite measure @xmath455^n)&=&\sum_{(t_{i_1},\ldots , t_{i_n})\in j_n } ( \delta x_{t_{i_1 } } ) ^{r_1}\cdots(\delta x_{t_{i_n } } ) ^{r_n}\\ & \le&\sum_{t_{i_1}\le t,\ldots , t_{i_n}\le t } ( \delta x_{t_{i_1 } } ) ^{r_1}\cdots(\delta x_{t_{i_n } } ) ^{r_n}\\ & = & x_t^{(r_1)}\cdots x_t^{(r_n)}<\infty.\end{aligned}\ ] ] \(b ) if the intervals @xmath331,\ldots,(s_n , t_n]$ ] are pairwise disjoint , then ( [ path ] ) is true for @xmath456\times\cdots \times ( s_n , t_n]}$ ] . the proof is straightforward . we separate the proof of the proposition in two steps . _ step _ 1 . formula ( [ path ] ) is true for every map @xmath457^n\to { \mathbb{r}}$ ] @xmath458-measurable and bounded , where @xmath458 is the @xmath21-field on @xmath133^n$ ] generated by the rectangles @xmath331\times\cdots\times(s_n , t_n]$ ] , with @xmath331,\ldots,(s_n , t_n]$ ] pairwise disjoint . to prove this claim we use a convenient monotone class theorem . denote by @xmath459 the family of functions that satisfy ( [ path ] ) ; it is a vector space such that : @xmath460 . if @xmath461 , @xmath462 for some constant @xmath463 , and @xmath464 , then @xmath465 . to see ( i ) , consider the dyadic partition of @xmath133 $ ] with mesh @xmath466 , write @xmath467,\qquad j=1,\ldots , 2^k,\ ] ] and define @xmath468 by the remark ( b ) at the beginning of the proof , @xmath469^n}f_k \,d m_{r_1,\ldots , r_n}.\ ] ] moreover , @xmath470 out off the diagonal sets @xmath133^n_\sigma$ ] , with @xmath471 , and then @xmath470 a.e . with respect to the lebesgue measure , and in @xmath280 . therefore , @xmath472 on the other hand , for every @xmath473 , the measure @xmath454 does not charge on any of the above mentioned diagonal sets . thus , the convergence @xmath470 is also @xmath454-a.e . by the monotone convergence theorem , @xmath474^n}f_k \,d m_{r_1,\ldots , r_n}=\int_{[0,t]^n}1\ , d m_{r_1,\ldots , r_n},\ ] ] and ( i ) follows . point ( ii ) is deduced directly from the monotone convergence theorem and taking into account that under the conditions in ( ii ) we have @xmath475 in @xmath280 . again by remark ( b ) above , the indicator of a set @xmath331\times \cdots\times(s_n , t_n]$ ] , with @xmath331,\ldots,(s_n , t_n]$ ] pairwise disjoint , is in @xmath459 , and this family of sets is closed by intersection . by the monotone class theorem , it follows that all bounded @xmath476-measurable functions are in @xmath459 . _ step _ 2 . extension of ( [ path ] ) to all @xmath297 . first , note that @xmath458 is the @xmath21-field generated by the borelian sets @xmath477^n)$ ] such that @xmath478^n_{\widehat0}$ ] . then , given @xmath477^n)$ ] , the indicator @xmath479^n_{\widehat0}}$ ] is @xmath458 measurable . let @xmath480 , and assume @xmath481 . there is a sequence of simple ( and then bounded ) functions such that @xmath482 . define @xmath483^n_{\widehat0}}$ ] , which is @xmath458 measurable , and @xmath484 a.e . with respect to the lebesgue measure . the convergence is also in @xmath280 , and then @xmath485 . on the other hand , @xmath484 , @xmath454-a.e . so @xmath486^n}f_m^0 \,d m_{r_1,\ldots , r_n}= \int_{[0,t]^n}f \,d m_{r_1,\ldots , r_n}.\ ] ] by step 1 , we get the result . for a general @xmath297 , decompose @xmath487 . finally , for a subordinator without drift and with moments of all orders , the multiple stratonovich measure can be identified with the @xmath38-fold product measure of @xmath488 . so for @xmath489 , by definition , @xmath490^n}f \,d\phi^{\otimes n}.\ ] ] using similar arguments as in the previous proposition , but easier , it is proved that @xmath490^n}f \,d\phi^{\otimes n}\qquad \forall f\in l^2(\lambda_n).\ ] ] then , the hu meyer formula can be transferred to a pathwise context . fix a partition @xmath180 , with blocks @xmath15 . let @xmath491 , with blocks @xmath492 ; each block @xmath493 is the union of some of the blocks @xmath15 . hence , we can consider the partition @xmath227 that gives the relationship between the @xmath493 s and the @xmath494 s , that is , @xmath495 has blocks @xmath228 defined by @xmath496 [ bijeccio ] let @xmath180 with @xmath355 . with the above notation , the map @xmath497&\longrightarrow & \pi_m,\\ \pi&\mapsto&\pi^*\end{aligned}\ ] ] is a bijection and , for @xmath498 $ ] , @xmath499 moreover , @xmath500 where @xmath501 is the mbius function on @xmath502 . \1 . for a subset @xmath505 we denote by @xmath506 the image of @xmath507 by @xmath508 @xmath509 given a partition @xmath180 , with blocks @xmath223 , let @xmath188 be the partition with blocks @xmath510 defined by @xmath511 . note that in general the blocks @xmath190 are not ordered . the application @xmath512 is a bijection and for @xmath513 , @xmath514 this last property is clear , because if @xmath515 , and @xmath516 , then @xmath517 \2 . for a vector @xmath192 , we write @xmath519 and the application @xmath520 determines a bijection on @xmath504 , that we also denote by @xmath508 . for a set @xmath194 , we write @xmath521 in particular , for @xmath522 , @xmath523 notice that if we look for the position of a particular set , say @xmath524 , in @xmath525 , we find it at place @xmath526 @xmath527 this last observation gives some light to the next property : \(i ) let @xmath532 . write @xmath533 therefore , @xmath534 the condition on the right is equivalent to @xmath535 [ see ( [ equivalencia ] ) ] . so , returning to the @xmath536 s , @xmath537 call @xmath538 and @xmath539 . we have @xmath540 hence , @xmath541 . consider a partition @xmath180 with blocks @xmath15 ( ordered ) . if the elements of each block are consecutive numbers , then , @xmath543 when @xmath21 does not fulfill the previous condition , the expression ( [ descomposicio ] ) is not valid . however , since we are interested in computing @xmath544 , thanks to lemma [ permutacio ] , we fortunately can permute both the set and the product measure to make things work . the next proposition is essential for this purpose . let @xmath223 be the blocks of @xmath21 ordered . if @xmath21 is such that @xmath553 , by theorem [ diagonal - diversos ] , @xmath554 for the general case , let @xmath508 be the permutation given by proposition [ particio ] and write @xmath555 . by proposition [ particio ] ( first ) , and @xmath556 ( second ) , we have @xmath557 by lemma [ permutacio ] and the first part of the proof , @xmath558 where @xmath559 . for every @xmath560 , @xmath561 given a partition @xmath180 , with blocks @xmath15 ( ordered ) , the function @xmath405 [ see ( [ qsigma ] ) ] is defined by @xmath562^m & \longrightarrow & [ 0,t]^n,\\ ( x_1,\ldots , x_m ) & \to & ( y_{1},\ldots , y_{n}),\end{aligned}\ ] ] where @xmath240 , if @xmath241 . this function is a bijection between @xmath133^m$ ] and @xmath133^n_{\ge\sigma}$ ] , and it is borel measurable because @xmath563 given a partition @xmath180 , with blocks ( ordered ) @xmath564 and a permutation @xmath187 , as we commented , the blocks of @xmath188 in general are not ordered . it is convenient to consider the permutation @xmath565 that gives the correct order of the blocks of @xmath188 , that means , @xmath566 if @xmath567 is the first block of @xmath188 , @xmath568 if @xmath569 is the second block , and so on ; in other words , @xmath570 are the blocks of @xmath188 ordered . remember that we defined [ see ( [ barra ] ) ] the @xmath349-dimensional vector @xmath571 .
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in the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by rota and wallstrom [ _ ann .
probab . _ * 25 * ( 1997 ) 12571283 ] , we present an it multiple integral and a stratonovich multiple integral with respect to a lvy process with finite moments up to a convenient order . in such a framework ,
the stratonovich multiple integral is an integral with respect to a product random measure whereas the it multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets .
a general hu meyer formula that gives the relationship between both integrals is proved .
as particular cases , the classical hu meyer formulas for the brownian motion and for the poisson process are deduced .
furthermore , a pathwise interpretation for the multiple integrals with respect to a subordinator is given .
, and . .
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recent experiments and observations have already opened up what one can actually call the era of precision cosmology . it is now hoped that next years will see rather impressive advances leading not just to the determination of key cosmological parameters with an accuracy not even dreamed five years ago , but also to unprecedented scrutinies on fundamental aspects of particle and field theories . in particular , the connection between recent boomerang ( de bernardis et al . 2000 ) and maxima ( hanany et al . 2000 ) experiments with theory is twofold . they are linked through predictions from either inflationary ( and perhaps cosmic string ) models by using fundamental particle physics arguments ( hu and white 1996 ) or from general more or less standard cosmological scenarios which may or may not include some new concepts such as quintessence ( cadwell , dave and steinhardt 1998 ) . the aim of this paper will concentrate at a particular aspect of cmb anisotropy measurements : the dependence of the position of the first doppler peak on the values of the relevant cosmological parameters within the realm of a quintessence model . kamionkowski , spergel and sugiyama ( 1994 ) originally derived the simple relation that the position of the first doppler peak @xmath2 , where @xmath3 , which has been the subject of some controversy . while frampton , ng and rohm ( 1998 ) have recently derived a similar dependence using a simple quintessential model , weinberg ( 2000 ) has argued that this formula is not even a crude approximation when @xmath4 is smaller than @xmath5 . for a very recent discussion on quntessential model of cmb anisotropies see bond et al . ( 2000 ) . assuming the holding of the cosmic triangle condition , we obtain in this paper that @xmath6 only depends on the topology of the universe , so confirming the original proposal by kamionkowski , spergel and sugiyama ( 1994 ) . in order to derive an expression relating the position of the first doppler peak with the scalar field potential of quintessence models , we choose a general tracking model with time - dependent parameter @xmath7 for the state equation ( zlatev , wang and steinhardt 1999 ) . this kind of models can be related to particle physics and may solve the so - called cosmic coincidence problem ( steinhardt 1997 ) . one typically considers tracking quintessence fields @xmath8 for a ratra - peebles potential @xmath9 with a given constant parameter @xmath10 ( ratra and peebles 1988 ) . during its cosmic evolution , the equation of state parameter @xmath11 passes through several distinct regimes ( brax , martin and riazuelo 2000 ) , including first a kinetic regime lasting until @xmath12 with @xmath13 , then the transition and potential regimes chracterized by @xmath14 , to finally reach the proper tracking regime at @xmath15 , where @xmath16 ( with @xmath17 for radiation dominated universe and @xmath18 for matter dominated universe ) . on this regime , there exists a particular solution for the scalar field , @xmath19 ( in which @xmath20 is the conformal time , with @xmath21 the scale factor ) which is an attractor able to solve the cosmic coincidence problem . for @xmath22 a change of @xmath11 should then be expected when the universe becomes matter dominated once the the surface of last scattering is overcome . since the tracking regime is characterized by @xmath23 ( @xmath24 being the sound velocity defined as @xmath25 ( brax , martin and riazuelo 2000 ) ) , parameter @xmath11 becomes a constant given by @xmath26 when matter dominates . it has been recently argued that @xmath27 ( balbi et al . 2001 ) , so that if we choose for definiteness @xmath28 , then @xmath29 for the vacuum quintessence field . in this case , the ratra - peebles potential becomes @xmath30 since the corresponding tracking solution , @xmath31 , would then correspond to a constant parameter @xmath32 for the quintessence field , it should satisfy the constraint equation derived from the corresponding conservation laws and cosmological field equations ( di pietro and demaret 1999 , gonzlez - daz 2000 ) . in terms of the cosmological parameters @xmath33 , @xmath34 , and the quintessence field potential @xmath35 , this constraint equation can be written as ( gonzlez - daz 2000 ) @xmath36 @xmath37 } + \omega_{\lambda}\left(\frac{v}{v_0}\right)^{(3\omega+4)/[3(\omega+1)]},\ ] ] where @xmath38 , the subscript 0 means current value and @xmath39 . ( 2 ) corresponds to the generalized quintessence model with negative constant parameter for the state equation recently suggested ( gonzlez - daz 2000 ) . besides the contributions from the topological curvature , @xmath40 , and gravitationally observable mass , @xmath41 , this model distinguishes two essentially distinct contributions from vacuum energy : a varying cosmological term with positive energy density @xmath5 assumed to satisfy the conservation law @xmath42 at sufficiently small redshifts , and a quintessence negative energy density such that while @xmath43 ( @xmath44 being the current value of the hubble constant ) is always negative , @xmath45 is always positive , so satisfying the ford - roman s quantum interest conjecture ( ford and roman 1999 ) . adapting then the relation between the position of the first doppler peak @xmath6 and the cosmological parameters @xmath33 first derived by frampton , ng and rohm ( 1998)to our generalized model , we can finally obtain a general relation between @xmath6 and the quintessence potential @xmath35 of the form : @xmath46 @xmath47}\right]\right|_{z = z_r } s\left\{\frac{\sqrt{|\omega_k|}v_0 ' } { 3(\omega+1)v_0 } \int_{\phi(z=0)}^{\phi(z = z_r ) } \frac{d\phi}{\left(\frac{v}{v_0}\right)^{(3\omega+1)/[6(\omega+1)]}}\right\ } , \ ] ] where @xmath48 is the redshift at recombination , we have used the relation ( di pietro and demaret 1999 ) @xmath49 and @xmath50 for @xmath51 , @xmath52 for @xmath53 and @xmath54 for @xmath55 . in the case of interest @xmath29 , where we obtain both topological and dynamical acceleration , eqn . ( 3 ) reduces to @xmath56\right|_{z = z_r } s\left\{\frac{\sqrt{|\omega_k|}v_0 ' } { v_0 } \int_{\phi(z=0)}^{\phi(z = z_r)}d\phi\left(\frac{v}{v_0}\right)^{1/2}\right\ } , \ ] ] and there is a solution to the constraint equation ( 2 ) given by @xmath57+\xi_0\right\}^{-1},\ ] ] where @xmath58 @xmath59 in this case , the cosmological parameters must satisfy the relations @xmath60\geq 0\ ] ] @xmath61 > 0 .\ ] ] together with the triangle equation @xmath62 , eqns . ( 9 ) and ( 10 ) will restrict the values that the cosmological parameters may take on . on the other hand , we note from eqns . ( 4 ) and ( 6 ) that for large @xmath63 and @xmath64 one obtains @xmath65 and @xmath66 ; that is , while the potential ( 6 ) can be consistently interpreted as a generalization for smaller values of @xmath63 from the ratra - peebles potential , the attractor solution @xmath67 must necessarily change into @xmath68 as the universe becomes matter dominated . performing the derivative and integration in eqn . ( 5 ) after inserting solution ( 6 ) , we finally obtain for the position of the first doppler peak when @xmath29 @xmath69 } \left.t\left\{\sqrt{\frac{|\omega_k|}{\omega_v\varphi_0 } } f\left[\varphi(z ) , r_0\right]\right|_{0}^{z_r}\right\ } , \ ] ] where @xmath70 $ ] is the elliptic integral of the first kind ( abramowitz and stegun 1965 ) , @xmath71 @xmath72 @xmath73 and @xmath74 for @xmath51 and , unlike the function @xmath75 , @xmath76 both for the closed and open cases . @xmath0 can run between two extreme values , such that @xmath77 a plot for the position of the first doppler peak , @xmath6 , against the parameter @xmath0 is given in fig . it can be seen that the maximum value for @xmath6 is reached when @xmath6 approaches the value 190 at the flat case @xmath51 , decreasing slowly therefrom as @xmath78 increases , a little more steeply for @xmath79 than for @xmath53 . this seems to quite reasonably reproduce the results provided by boomerang , maxima and previous ( netterfield et al . 1997 , vollaek et al . 1997 ) experiments . although uncertainties about our results for @xmath29 must come from the possibility of having values of @xmath10 other than just unity ( provided they are on the interval ( 0,2 ) , or small deviation from time - independence of @xmath11 during cosmic evolution after recombination , they seems to strongly suggest a nearly flat topology for the universe . we note that cbm anisotropies and cosmic acceleration can be easily related in our model . to see this , let us consider solution ( 6 ) for the the flat case , i.e. @xmath80\right\}^{-1 } , \ ] ] with @xmath81 this allows us to also construct a plot for the luminosity distance versus redshift by using the expression ( gonzlez - daz 2000 ) @xmath82\right|_{0}^{z}\right\ } .\ ] ] one can see readily that our flat solution in fact gives rise to a @xmath83 plot with a nearly straight line between @xmath84 and @xmath85 which appears to slightly accelerate thereafter , the full @xmath86 that corresponds to the @xmath63-interval of presently available type ia supernova observations , @xmath87 $ ] , being around 12 . this accelerating behaviour for the universe conforms quite well the data obtained from distant supernova ia ( perlmutter et al . 1999 , reiss et al . it is worth noticing that though our universe model is dynamically accelerating it still is topologically uniform as @xmath88 exactly vanishes in the flat case . as one separates from flatness one model produces topologically decelerating scenarios . consistent solutions to the constraint equation for the quintessence potential have been obtained which all correspond to particular fixed sets of values for @xmath33 , @xmath89 , in such a way that the resulting value of the position of the first doppler peak @xmath6 becomes automatically fixed once just one of the parameters @xmath0 , @xmath4 or @xmath90 is fixed , but does not depend on the ratio @xmath91 . one can also conclude that @xmath6 acquires a maximum value which fits fairly well experimental results at @xmath51 ( i.e. @xmath92 ) . as the topology of the universe separates from flatness ( @xmath93 if @xmath94 , or @xmath95 if @xmath96 ) , @xmath6 always decreases from its maximum value . since @xmath97 , from the condition that @xmath98 be real it follows that the quintessence field @xmath8 is pure imaginary . this amounts to the interpretation that the classical solution is axionic , and hence the quintessence field can be taken as a genuine component of dark matter in such a way that one can define for the gravitationally observable mass of the universe the quantity @xmath99 , with @xmath100 . thus , one can always adjust our results to the currently favoured combination @xmath101 , @xmath102 for any @xmath0 . we finally remark that the results obtained in this paper refer only to the case of a quintessence parameter for state equation @xmath29 ( i.e. @xmath28 ) . it is expected that results even more adjusted to observation ( i.e. closer to 200 ) can also be obtained within the model of this paper by slightly shifting @xmath11 towards more realistic smaller values approaching -0.8 . abramowitz , m. , and stegun , i.a . , 1965 , handbook of mathematical functions : dover , 1995 balbi , a. , baccigalupi , c. , matarrese , s. , perrotta , f. , and vittorio , n. 2001 , , 547 , l89 de bernardis , p. et al . 2000 , nature , 404 , 955 bond , j.r . 2000 , astro - ph/0011379 brax , p. , martin , j. , and riazuelo , a. 2000 , phys . d62 , 103505 caldwell , r.r . , dave , r. , and steinhardt , p.j . 1998 , phys . 80 , 1582 di pietro , e. , and demaret , j. 1999 , gr - qc/9908071 ford , l.h . , and roman , t.a . 1999 , d60 , 104018 frampton , p.h . , ng , y.j . , rohm , r.m . 1998 , mod . a13 , 2541 gonzlez - daz , p.f . 2000 , phys . d62 , 023513 hanany , s. et al . 2000 , , 545 , 1 hu , w. , and white , m. 1996 , , 471 , 30 kamionkowski , k , spergel , d.n . , and sugiyama , n. 1994 , , 426 , 57 netterfield , c.b . 1997 , , 474 , 47 perlmutter , s. et al . 1999 , , 517 , 565 ratra , b. , and peebles , p.j.e . 1988 , phys . d37 , 3406 steinhardt , p.j . 1997 , critical problems in physics , v.l . fitch and d.r . marlow , princeton : princeton university press weinberg , s. 2000 , phys . d62 , 127302 zlatev , l. , wang , l. , and steinhardt , p.j . 1999 , lett . 82 , 896
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by using a tracking quintessence model we obtain that the position of the first doppler peak in the spectrum of cmb anisotropies only depends on the topology of the universe , @xmath0 , for any value of the ratio @xmath1 , so that such a dependence is perfectly valid in the range suggested by supernova observations .
| 4,215 | 84 |
at the end of the last century , the astronomical observations of high redshift type ia supernovae ( snia ) indicated that our universe is not only expanding , but also accelerating , which conflicts with our deepest intuition of gravity . with some other observations , such as cosmic microwave background radiation ( cmbr ) , baryon acoustic oscillations ( bao ) and large - scale structure ( lss ) , physicists proposed a new standard cosmology model , @xmath0cdm , which introduces the cosmological constant back again . although this unknown energy component accounts for 73% of the energy density of the universe , the measured value is too small to be explained by any current fundamental theories.@xcite-@xcite if one tries to solve this trouble phenomenologically by setting the cosmological constant to a particular value , the so - called fine - tuning problem would be brought up , which is considered as a basic problem almost any cosmological model would encounter . a good model should restrict the fine - tuning as much as possible . in order to alleviate this problem , various alternative theories have been proposed and developed these years , such as dynamical dark energy , modified gravity theories and even inhomogeneous universes . recently , a new attempt , called torsion cosmology , has attracted researchers attention , which introduces dynamical torsion to mimic the contribution of the cosmological constant . it seems more natural to use a pure geometric quantity to account for the cosmic acceleration than to introduce an exotic energy component . torsion cosmology could be traced back to the 1970s , and the early work mainly focused on issues of early universe , such as avoiding singularity and the origin of inflation . in some recent work , researchers attempted to extend the investigation to the current evolution and found it might account for the cosmic acceleration . among these models , poincar gauge theory ( pgt ) cosmology is the one that has been investigated most widely . this model is based on pgt , which is inspired by the einstein special relativity and the localization of global poincar symmetry@xcite . et al_. made a comprehensive survey of torsion cosmology and developed the equations for all the pgt cases.@xcite based on goenner s work , nester and his collaborators@xcite found that the dynamical scalar torsion could be a possible reason for the accelerating expansion . et al_.@xcite extended the investigation to the late time evolution , which shows us the fate of our universe . besides pgt cosmology , there is another torsion cosmology , de sitter gauge theory ( dsgt ) cosmology , which can also be a possible explanation to the accelerating expansion . this cosmological model is based on the de sitter gauge theory , in which gravity is introduced as a gauge field from de sitter invariant special relativity ( dssr ) , via the localization of de sitter symmetry.@xcite dssr is a special relativity theory of the de sitter space rather than the conventional minkowski spacetime , which is another maximally symmetric spacetime with an uniform scalar curvature @xmath1 . and the full symmetry group of this space is de sitter group , which unifies the lorentz group and the translation group , putting the spacetime symmetry in an alternatively interesting way . but in the limit of @xmath2 , the de sitter group could also degenerate to the poincar group . to localize such a global symmetry , de sitter symmetry , requires us to introduce certain gauge potentials which are found to represent the gravitational interaction . the gauge potential for de sitter gauge theory is the de sitter connecion , which combines lorentz connection and orthonormal tetrad , valued in @xmath3(1,4 ) algebra . the gravitational action of dsgt takes the form of yang - mills gauge theory . via variation of the action with repect to the the lorentz connection and orthonormal tetrad , one could attain the einstein - like equations and gauge - like equations , respectively . these equations comprise a set of complicated non - linear equations , which are difficult to tackle . nevertheless , if we apply them to the homogeneous and isotropic universe , these equations would be much more simpler and tractable . based on these equations , one could construct an alternative cosmological model with torsion . analogous to pgt , dsgt has also been applied to the cosmology recently to explain the accelerating expansion.@xcite our main motivation of this paper is to investigate ( i)whether the cosmological model based on de sitter gauge theory could explain the cosmic acceleration ; ( ii)where we are going , i.e. , what is the fate of our universe ; ( iii ) the constraints of the parameters of model imposed by means of the comparison of observational data . by some analytical and numerical calculations , we found that , with a wide range of initial values , this model could account for the current status of the universe , an accelerating expanding , and the universe would enter an exponential expansion phase in the end . this paper is organized as follows : first , we summarize the de sitter gauge theory briefly in sec . [ sec : de - sitter - gauge ] , and then show the cosmological model based on de sitter gauge theory in sec . [ sec : cosm - evol - equat ] . second , we rewrite these dynamical equations as an autonomous system and do some dynamical analysis and numerical discussions on this system in the sec . [ sec : autonomous - system ] and [ sec : numer - demonstr ] . next in the [ sec : supern - data - fitt]th section , we compare the cosmological solutions to the snia data and constrain the parameters . last of all , we discuss and summarize the implications of our findings in section [ sec : summary - conclusion ] . [ supernovae data fitting]in dsgt , the de sitter connection is introduced as the gauge potential , which takes the form as @xmath4 -r^{-1}e^b_\mu & 0 \end{array } \right ) \in \mathfrak{so}(1,4),\ ] ] where @xmath5 , @xmath6 and @xmath7 , which combines the lorentz connection and the orthonormal tetrad are 4d coordinate indices , whereas the capital latin indices @xmath8 and the lowercase latin indices , @xmath9 denote 5d and 4d orthonormal tetrad indices , respectively . ] . the associated field strength is the curvature of this connection , which is defined as @xmath10 -r^{-1}t^b_{~\mu\nu } & 0 \end{array } \right ) \in \mathfrak{so}(1,4),\end{aligned}\ ] ] where @xmath11 , @xmath12 is the de sitter radius , and @xmath13 and @xmath14 are the curvature and torsion of lorentz - connection , @xmath15 which also satisfy the respective bianchi identities . the gauge - like action of gravitational fields in dsgt takes the form , @xcite @xmath16.\label{gym}\end{aligned}\ ] ] here , @xmath17 , @xmath18 is a dimensionless constant describing the self - interaction of the gauge field , @xmath19 is a dimensional coupling constant related to @xmath18 and @xmath12 , and @xmath20 is the scalar curvature of the cartan connection . in order to be consistent with einstein - cartan theory , we take @xmath21 and @xmath22 , where @xmath23 . assuming that the matter is minimally coupled to gravitational fields , the total action of dsgt could be written as : @xmath24 where @xmath25 denotes the action of matter , namely the gravitational source . now we can obtain the field equations via variational principle with respect to @xmath26 , @xmath27 \label{feq2}% & & \nabla_{\nu}f_{ab}^{~~\mu\nu}-r^{-2}\left(y^\mu_{~\,\lambda\nu } e_{ab}^{~~\lambda\nu}+y ^\nu_{~\ , \lambda\nu } e_{ab}^{~~\mu\lambda } + 2t_{[a}^{~\mu\lambda } e_{b]\lambda}\right ) = 16\pi gr^{-2}s^{\quad \mu}_{{\rm m}ab},%\end{aligned}\ ] ] where@xmath28 represent the effective energy - momentum density and spin density of the source , respectively , and @xmath29 is the contorsion . it is worth noticing that the nabla operator in eq . ( [ feq1 ] ) and ( [ feq2 ] ) is the covariant derivative compatible with christoffel symbols \{@xmath30 } for coordinate indices , and lorentz connection @xmath31 for orthonormal tetrad indices . readers can be referred to ref.@xcite for more details on dsgt . since current observations favor a homogeneous , isotropic universe , we here work on a robertson - walker ( rw ) cosmological metric @xmath32.\ ] ] for robertson - walker metric , the nonvanishing torsion tensor components are of the form , @xmath33 where @xmath34 denotes the vector piece of torsion , namely , in components , the trace of the torsion , and @xmath35 indicates the axial - vector piece of torsion , which corresponds in components to the totally antisymmetric part of torsion . @xmath34 and @xmath35 are both functions of time @xmath36 , and their subscripts , + and - , denote the even and odd parities , respectively . the nonvanishing torsion 2-forms in this case are @xmath37 where @xmath38 . according to the rw metric eq . and the torsion eq . , the field equations could be reduced to @xmath39 \label{el-11}% & & \frac{\ddot a^2 } { a^2 } + \left(\dot t_+ + 2\frac{\dot a } a t_+ - 2\frac{\ddot a } a + \frac{6}{r^2}\right)\dot t_+ -\frac 1 4 \left(\dot t_- + 2 \frac { \dot a } a t_-\right)\dot t_- - t_+^4 + \frac 3 2 t_+^2 t_-^2 - \frac1 { 16 } t_-^4 \nonumber\\ & & \quad+ \frac { \dot a } a(4 t_+^2 - 3 t_-^2)t_+ - \left(5\frac{\dot a^2 } { a^2 } + 2 \frac k { a^2 } + \frac3 { r^2}\right ) t_+^2 + \frac 1 2 \left(\frac 5 2\frac{\dot a^2 } { a^2 } + \frac k { a^2 } + \frac 3 { r^2}\right ) t_-^2- 2\frac{\dot a } a \left(\frac{\ddot a } { a}- 2\frac{\dot a^2 } { a^2 } \right.\nonumber \\[0.2 cm ] & & \quad \left . - 2 \frac k { a^2}- \frac6 { r^2}\right)t_+ - \frac 4 { r^2 } \frac{\ddot a } a -\frac{\dot a^2 } { a^2 } \left(\frac{\dot a^2}{a^2 } + 2\frac k { a^2}\right ) + \frac2 { r^2 } -\frac{k^2}{a^4 } - \frac2 { r^2}\frac k { a^2 } + \frac6 { r^4 } = -\frac{16\pi g p}{r^2 } , \\[0.3 cm ] \label{yang1 } % & & \ddot t_- + 3 \frac{\dot a } a \dot t_- + \left ( \frac 1 2 t_-^2 - 6 t_+^2 + 12 \frac { \dot a } a t_+ + \frac{\ddot a } a - 5\frac{\dot a^2}{a^2 } - 2\frac k { a^2}+ \frac 6 { r^2}\right ) t_-=0 , \\[0.3 cm ] \label{yang2}% & & \ddot t_+ + 3 \frac{\dot a } a \dot t_+ -\left ( 2 t_+^2 -\frac 3 2 t_-^2 - 6\frac{\dot a } a t_+ -\frac { \ddot a } a + 5 \frac { \dot a^2 } { a^2 } + 2 \frac k { a^2}- \frac 3 { r^2}\right ) t_+ - \frac 3 2 \frac{\dot a } a t_-^2-\frac{\dddot a } a - \frac{\dot a\ddot a}{a^2 } \nonumber\\ & & \quad+ 2\frac { \dot a^3 } { a^3 } + 2\frac{\dot a } a \frac k { a^2 } = 0 , % \end{aligned}\ ] ] where eqs . and are the @xmath40 and @xmath41 component of einstein - like equations , respectively ; and eqs . and are 2 independent yang - like equations , which is derived from the @xmath42 and @xmath43 components of lorentz connection . the spin density of present time is generally thought be very small which could be neglected . therefore , we here assumed the spin density is zero . the bianchi identities ensure that the energy momentum tensor is conserved , which leads to the continuity equation : @xmath44 equation can also be derived from eqs . - , which means only four of eqs . - are independent . with the equation of state ( eos ) of matter content , these four equations comprise a complete system of equations for five variables , @xmath45 . by some algebra and differential calculations , we could simplify these 5 equations as : @xmath46t_+,\\ \ddot{t}_- & = & -3h\dot{t}_- -\left[-\frac{15}{2}t_{+}^{2}+\frac{33h t_{+}}{2}-6h^{2}-\frac{3k}{a^2}+\frac{8}{r^{2}}+\frac{5}{4}t^{2}_{-}+\frac{3}{2}\dot{t}_+ \right.\nonumber\\ & & \left . + \frac{4\pi g}{3}(\rho+3p)\right]t_{-},\\ \dot{\rho}&=&-3h(\rho+p),\\ \label{eq : eos } w&=&\frac{p}{\rho } , \end{aligned}\ ] ] where @xmath47 is the hubble parameter . if we rescale the variables and parameters as @xmath48 where @xmath49 is the hubble radius in natural units , these variables and parameters would be dimensionless . under this transformation , - remain unchanged expect for the terms including @xmath50 and @xmath51 , which change into @xmath52 and @xmath53 respectively . the contribution of radiation and spatial curvature in current universe are so small that it could be neglected , so we here just consider the dust universe with spatial flatness , whose eos is equal to zero . by some further calculations , these equations could be transformed to a set of six one - order ordinary derivative equations , which forms a six - dimensional autonomous system , as follows , @xmath54t_+ + 6h\rho , \\[0.2 cm ] \dot{t_{+}}&=&p,\\ \dot q&=&-3h q -\left(-\frac{15}{2}t_{+}^{2}+\frac{33h t_{+}}{2}-6h^{2}+\frac{8}{r^{2}}+\frac{5}{4}t^{2}_{-}+\frac{3}{2}p + \rho \right)t_{-},\\ \dot{t_{-}}&=&q,\\ \dot{\rho}&=&-3h\rho . \label{rho}\end{aligned}\ ] ] for such an autonomous system , we can use the dynamics analysis to investigate its qualitative properties . critical points are some exact constant solutions in the autonomous system , which indicate the asymptotic behavior of evolution . for example , some solutions , such as heteroclinic orbits , connect two different critical points , and some others , such as homoclinic orbits , are a closed loop starting from and coming back to the same critical point . in the dynamics analysis of cosmology , the heteroclinic orbit is more interesting.@xcite thus , critical points could be treated as the basic tool in dynamics analysis , form which one could know the qualitative properties of the autonomous system . by some algebra calculation , we find all the 9 critical points ( @xmath55 ) of this system , as shown in table 1 . . [ tab : critical - points]the critical points and their corresponding eigenvalues . the point 9 is not physically acceptable , for its negative energy density . [ cols="<,<,<",options="header " , ] based on the current observations , the present density of torsion in our universe is very small , so it is reasonable to assume that the initial values of all torsions and their first order derivative are zero at @xmath56 . but their second order derivatives does not vanish yet , which would have a significant impact on the history and future of the evolution of our universe . in this case , the number of parameter and initial value is reduced , and the rest parameters and initial values are just @xmath12 and @xmath57 . it is easy to find the best - fit of these 2 parameters , which are shown in table [ tab : best - fit ] . and the minimal @xmath58 is @xmath59 , whereas the value for @xmath0cdm is @xmath60 , with @xmath61 . in fig . [ fig : chi2-distribution ] we show the @xmath58 distribution with respect to @xmath12 and @xmath52 compared to @xmath0cdm model , where the plane @xmath62 corresponds to the value of @xmath0cdm . furthermore , we plot the contours of some particular confidence levels , as shown in fig . [ fig : contour ] . from these figures , we could find that the evolution of our universe is insensitive to the initial value , which alleviate the fine - tuning problem . the @xmath58 distribution with respect to @xmath12 and @xmath52 , compared to the @xmath0cdm , the plane @xmath63 . here we assume that all the torsions and their first order derivatives vanish at present time.,width=642,height=340 ] the 68.3% , 95.4% and 99.7% @xmath58 confidence contours of dsgt with respect to @xmath12 and @xmath52 , using the union 2 dataset . here we also assume that the current - time values for all the torsions and their first order derivatives are zero . the yellow point is the best - fit point . , height=264 ] the astronomical observations imply that our universe is accelerating to a de sitter spacetime . this gives us a strong motive to consider the cosmic evolution based on the de sitter gauge theory instead of other gravity theories . the localization of de sitter symmetry requires us to introduce curvature and torsion . so in de sitter gauge theory , the torsion is an indispensable quantity , by which people tried to include the effect of spin density in gravity theory at first . but now this essential quantity might account for the acceleration of our universe , if we apply dsgt to cosmology . we found the cosmological equations for dust universe in dsgt could form an autonomous system by some transformations , where the evolution of the universe is described in terms of the orbits in phase space . therefore , by dynamics analysis to the dsgt cosmology , one could study the qualitative properties of this phase space . we found all 9 critical points , as shown in table [ tab : critical - points ] . we also analyzed the stabilities of these critical points , and found among these critical points there is only one positive attractor , which is stable . the positive attractor alleviates the fine - tuning problem and implies that the universe will expand exponentially in the end , whereas all other physical quantities will turn out to vanish . in this sense , dsgt cosmology looks more like the @xmath0cdm , than pgt cosmology . and we conducted some concrete numerical calculations of this the destiny of our model of the universe , which confirms conclusions from dynamics analysis . finally , in order to find the best - fit values and constraints of model parameters and initial conditions , we fitted them to the union 2 snia dataset . the maximum likelihood estimator here we used is the @xmath58 estimate . by minimizing the @xmath58 , we found the best - fit parameters @xmath64 and the corresponding @xmath65 , while the value for @xmath0cdm is @xmath60 , with @xmath66 . note that we here set all the initial values of torsions and their first - order derivatives to zero at @xmath67 , since the contribution of torsion to the current universe is almost negligible . we also plotted the confidence contour fig . [ fig : contour ] with respect to @xmath12 and @xmath52 , from which it is easy to see that the fine - tuning problem is alleviated and the evolution is not so sensitive to the initial values and model parameters . if we want to go deeper into cosmology based on de sitter gauge theory , there are a lot of work need to be done . we should fit this model to some other observations , like bao and lss etc , to constrain the parameters better . we also could study the perturbations in the early universe , and compare the results to cmbr data . these issues will considered in some upcoming papers . this work is supported by srfdp under grant no . 200931271104 and shanghai natural science foundation , china , under grant no . 10zr1422000 . t. padmanabhan , _ cosmological constant : the weight of the vacuum _ , _ phys . * 380 * ( 2003 ) 235 - 320 , [ hep - th/0212290 ] ; x .- z . li and j .- g . o(n ) phantom , a way to implement w @xmath68 -1 _ _ phys . * d69 * ( 2004 ) 107303 , [ hep - th/0303093 ] . hao and x .- z . li , _ phantom cosmic dynamics : tracking attractor and cosmic doomsday _ , _ phys . _ * d70 * ( 2004 ) 043529 , [ astro - th/0309746 ] ; + c .- j . feng and x .- z . li , _ cardassian universe constrained by latest observations _ _ phys . lett . _ * b692 * ( 2010 ) 152 - 156 , [ arxiv:0912.4793 ] . f. w. hehl , _ four lectures on poincar gauge field theory _ in _ proc . of the 6th course of the school of cosmology and gravitation on spin , torsion , rotation , and supergravity _ , eds . p. g. bergmann and v. de sabbata ( new york : plenum ) p5 + m. blagojevi , _ gravitation and gauge symmetries _ , iop publishing , bristol , 2002 . h. chen , f .- h . ho and j. m. nester , c .- h . wang , and h .- j . yo , _ cosmological dynamics with propagating lorentz connection modes of spin zero _ , _ jcap _ * 0910 * ( 2009 ) 027 , [ arxiv:0908.3323 ] ; + p. baekler , f. w. hehl , and j. m. nester , _ poincare gauge theory of gravity : friedman cosmology with even and odd parity modes . analytic part _ rev . _ * d83 * ( 2011 ) 024001 , [ arxiv:1009.5112 ] ; + f .- h . ho and j. m. nester , _ poincar gauge theory with coupled even and odd parity dynamic spin-0 modes : dynamic equations for isotropic bianchi cosmologies _ , [ arxiv:1106.0711 ] ; + f .- h . ho and j. m. nester , _ poincar gauge theory with even and odd parity dynamic connection modes : isotropic bianchi cosmological models _ , [ arxiv:1105.5001 ] . li , c .- b . sun , and p. xi , _ torsion cosmolgical dynamics _ , _ phys . _ * d79 * ( 2009 ) 027301 , [ arxiv:0903.3088 ] ; + x .- z . li , c .- b . sun , and p. xi , _ statefinder diagnostic in a torsion cosmology _ , _ jcap _ * 0904 * ( 2009 ) 015 , [ arxiv:0903.4724 ] ; + x .- c . ao , x .- z . li , and p. xi _ analytical approach of late - time evolution in a torsion cosmology _ lett . _ * b694 * ( 2010 ) 186 - 190 , [ arxiv:101.4117 ] . guo , c .- g . huang , z. xu , and b. zhou , _ on beltrami model of de sitter spacetime _ _ * a19 * ( 2004 ) 1701 - 1710 , [ hep - th/0311156 ] ; + h .- y . guo , c .- g . huang , z. xu , and b. zhou , _ on special relativity with cosmological constant _ , _ phys . lett .. _ * a331 * ( 2004 ) 1 - 7,[hep - th/043171 ] ; + h .- y . guo , c .- g . huang , z. xu , and b. zhou , _ three kinds of special relativity via inverse wick rotation _ * 22 * ( 2005 ) 2477 - 2480 , [ hep - th/0508094 ] ; + h .- y . guo , c .- g . huang , b. zhou , _ temperature at horizon in de sitter spacetime _ , _ europhys . * 72 * ( 2005 ) 1045 - 1051 . li , y .- b . zhao , and c .- b . sun , _ heteroclinic orbit and tracking attractor in cosmological model with a double exponential potential _ , _ class . * 22 * ( 2005 ) 3759 - 3766 , [ astro - ph/050819 ] ; + j .- hao and x .- z . li , _ an attractor solution of phantom field _ , _ phys . _ * d67 * ( 2003 ) 107303 , [ gr - qc/0302100 ] .
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a new cosmological model based on the de sitter gauge theory ( dsgt ) is studied in this paper . by some transformations ,
we find , in the dust universe , the cosmological equations of dsgt could form an autonomous system .
we conduct dynamics analysis to this system , and find 9 critical points , among which there exist one positive attractor and one negative attractor .
the positive attractor shows us that our universe will enter a exponential expansion phase in the end , which is similar to the conclusion of @xmath0cdm .
we also carry out some numerical calculations , which confirms the conclusion of dynamics analysis .
finally , we fit the model parameter and initial values to the union 2 snia dataset , present the confidence contour of parameters and obtain the best - fit values of parameters of dsgt .
| 7,943 | 202 |
the _ kissing number _ @xmath6 is the highest number of equal non - overlapping spheres in @xmath7 that touch another sphere of the same size . in three dimensions the kissing number problem asks how many white billiard balls can _ kiss _ ( i.e. touch ) a black ball . the most symmetrical configuration , 12 balls around another , is achieved if the 12 balls are placed at positions corresponding to the vertices of a regular icosahedron concentric with the central ball . however , these 12 outer balls do not kiss one another and each may be moved freely . this space between the balls prompts the question : _ if you moved all of them to one side , would a 13th ball fit ? _ this problem was the subject of the famous discussion between isaac newton and david gregory in 1694 . most reports say that newton believed the answer was 12 balls , while gregory thought that 13 might be possible . this problem is often called the _ thirteen spheres problem_. the problem was finally solved by schtte and van der waerden in 1953 @xcite . a subsequent two - page sketch of an elegant proof was given by leech @xcite in 1956 . leech s proof was presented in the first edition of the well - known book by aigner and ziegler @xcite ; the authors removed this chapter from the second edition because a complete proof would have to include too much spherical trigonometry . the thirteen spheres problem continues to be of interest , and new proofs have been published in the last several years by hsiang @xcite , maehara @xcite ( this proof is based on leech s proof ) , brczky @xcite , anstreicher @xcite , and musin @xcite . note that for @xmath8 , the kissing number problem is currently solved only for @xmath9 @xcite , and for @xmath10 @xcite ( see @xcite for a beautiful exposition of this problem ) . if @xmath0 unit spheres kiss the unit sphere in @xmath7 , then the set of kissing points is an arrangement on the central sphere such that the ( euclidean ) distance between any two points is at least 1 . this observation allows us to state the kissing number problem in another way : _ how many points can be placed on the surface of @xmath11 so that the angular separation between any two points be at least @xmath12 ? _ it leads to an important generalization . a finite subset @xmath13 of @xmath11 is called a _ spherical @xmath14-code _ if for every pair @xmath15 of @xmath13 with @xmath16 its angular distance @xmath17 is at least @xmath14 . let @xmath13 be a finite subset of @xmath18 . denote @xmath19 the set @xmath13 is then a spherical @xmath20-code . denote by @xmath21 the largest angular separation @xmath20 with @xmath22 that can be attained in @xmath18 , i.e. @xmath23 in other words , _ how are @xmath0 congruent , non - overlapping circles distributed on the sphere when the common radius of the circles has to be as large as possible ? _ this question is also known as the problem of the `` inimical dictators '' , namely _ where should @xmath0 dictators build their palaces on a planet so as to be as far away from each other as possible ? _ the problem was first asked by the dutch botanist tammes @xcite ( see ( * ? ? ? * section 1.6 : problem 6 ) ) , while examining the distribution of openings on the pollen grains of different flowers . the tammes problem is presently solved for several values of @xmath0 , namely for @xmath1 by l. fejes tth @xcite ; for @xmath2 by schtte and van der waerden @xcite ; for @xmath3 by danzer @xcite ( for @xmath24 see also brczky @xcite ) ; and for @xmath4 by robinson @xcite . the tammes problem for @xmath5 is of particular interest due to its relation to both the kissing problem and the kepler conjecture @xcite . actually , this problem is equivalent to _ the strong thirteen spheres problem _ , which asks to find the maximum radius of and an arrangement for 13 equal size non - overlapping spheres in @xmath25 touching the unit sphere . it is clear that the equality @xmath26 implies @xmath27 . brczky and szab @xcite proved that @xmath28 . bachoc and vallentin @xcite have shown that @xmath29 . we solved the tammes problem for @xmath5 in 2012 @xcite . we proved that + _ the arrangement @xmath30 of 13 points in @xmath31 is the best possible , the maximal arrangement is unique up to isometry , and @xmath32 . _ in this paper , using very similar method we present a solution to the tammes problem for @xmath33 . we note that there is an arrangement of 14 points on @xmath18 such that the distance between any two points of the arrangement is at least @xmath34 ( see ( * ? ? ? 4 ) and http://neilsloane.com/packings/dim3/pack.3.14.txt ) . this arrangement is shown in fig . [ fig1 ] . and its contact graph @xmath35 . @xmath36.,title="fig : " ] and its contact graph @xmath35 . @xmath36.,title="fig : " ] the first upper bound @xmath37 was found in @xcite . actually , this value is the famous fejes tth bound @xmath38 for @xmath33 . brczky and szab @xcite improved the fejes tth bound and proved that @xmath39 . bachoc and vallentin @xcite using the sdp method have shown that @xmath40 . the arrangement @xmath41 of @xmath42 points in @xmath31 gives a solution of the tammes problem , moreover the maximal arrangement for @xmath33 is unique up to isometry and @xmath43 . * contact graphs . * let @xmath13 be a finite set in @xmath44 . the _ contact graph _ @xmath45 is the graph with vertices in @xmath13 and edges @xmath46 such that @xmath47 . * shift of a single vertex . * we say that a vertex @xmath48 _ can be shifted _ , if , in any open neighbourhood of @xmath49 there is a point @xmath50 such that @xmath51 where for a point @xmath52 and a finite set @xmath53 by @xmath54 we denote the minimum distance between @xmath55 and points in @xmath56 . * danzer s flip . * danzer ( * ? ? ? 1 ) defined the following flip . let @xmath57 be vertices of @xmath45 with @xmath58 . we say that @xmath49 is flipped over @xmath59 if @xmath49 is replaced by its mirror image @xmath60 relative to the great circle @xmath59 ( see fig . [ fig3 ] ) . we say that this flip is _ danzer s flip _ if @xmath61 . * irreducible contact graphs . * we say that a graph @xmath45 is _ irreducible _ provided it does not allow danzer s flip and no vertex in @xmath13 can be shifted . the concept of irreducible contact graphs was invented by schtte - van der waerden @xcite , fejes tth @xcite , and danzer @xcite . actually , in these papers as well as in our paper @xcite this concept has been used for solutions of the tammes problem with @xmath62 . recently , we enumerated all of the irreducible contact graphs ( with and without danzer s flip ) for @xmath63 @xcite . * @xmath41 and @xmath35 . * denote by @xmath41 the arrangement of 14 points in fig . 1 . let @xmath64 . it is not hard to see that the graph @xmath35 is irreducible . * maximal graphs @xmath65 . * let @xmath13 be a subset of @xmath31 with @xmath66 and @xmath67 . denote by @xmath65 the graph @xmath45 . actually , this definition does not assume that @xmath65 is unique . we use this designation for some @xmath45 with @xmath67 . * graphs @xmath68 . * let us define five planar graphs @xmath69 ( see fig . [ eliminated ] ) , where @xmath70 , and @xmath71 . note that @xmath72 , is obtained from @xmath35 by removing certain edges . .,title="fig : " ] .,title="fig : " ] .,title="fig : " ] .,title="fig : " ] .,title="fig : " ] @xmath65 is isomorphic to @xmath68 with @xmath73 or @xmath74 . @xmath65 is isomorphic to @xmath75 and @xmath76 . it is clear that lemma 2 yields theorem 1 . now our goal is to prove these lemmas . here we give a sketch of our computer assisted proof . for more details see http://dcs.isa.ru/taras/tammes14/ . the combinatorial properties of @xmath77 have previously been considered [ @xcite , ( * ? ? ? * chap . vi ) , @xcite and @xcite ] . in particular , for @xmath33 we have : 1 . @xmath65 is a planar graph with 14 vertices ; 2 . any vertex of @xmath65 is of degree @xmath78 or @xmath79 ; 3 . any face of @xmath65 is a polygon with @xmath80 or @xmath81 vertices ; 4 . if @xmath65 contains an isolated vertex @xmath82 , then @xmath82 lies in a hexagonal face . moreover , a hexagonal face of @xmath65 can not contain two or more isolated vertices . [ cor1 ] in our papers @xcite the main relations between these parameters were considered ( ( * ? ? ? * propositions 3.63.11 ) , ( * ? ? ? * proposition 4.11 ) ) . let us list those results here . let @xmath83 be an irreducible contact graph in @xmath31 with faces @xmath84 . let @xmath85 . denote by @xmath86 , @xmath87 the set of its angles . here @xmath88 denotes the number of vertices of @xmath84 . 1 . @xmath89 for all @xmath90 and @xmath91 . @xmath92 for all @xmath90 and @xmath91 , where @xmath93 is the angle of the equilateral spherical triangle with side length @xmath94 . @xmath95 for all vertices @xmath82 of @xmath96 . here @xmath97 denotes the set of angles with the vertex in @xmath82 . 4 . if @xmath98 then @xmath84 is an equilateral triangle with angles @xmath99 5 . in the case @xmath100 , @xmath84 is a spherical rhombus and @xmath101 , @xmath102 . moreover , we have the equality : @xmath103 6 . in the case @xmath104 , @xmath84 is a convex equilateral spherical polygon with angles @xmath105 . denote by @xmath106 vertices of @xmath84 . the polygon @xmath84 is uniquely defined ( up to isometry ) by its @xmath107 angles and @xmath94 . then functions @xmath108 and @xmath109 , where @xmath110 and @xmath111 are also uniquely defined . it follows that + ( a ) @xmath110 for @xmath112 ; and + ( b ) @xmath113 for all @xmath114 . 7 . now consider the case when there is an isolated vertex inside @xmath84 . ( it is only if @xmath115 . ) define @xmath116 then @xmath117 our proof of lemma 1 consists of two parts : + ( i ) create the list @xmath118 of all graphs with 14 vertices that satisfy proposition 3.1 ; + ( ii ) using linear approximations and linear programming remove from the list @xmath118 all graphs that do not satisfy the known geometric properties of @xmath65 ( proposition 3.2 ) . to create @xmath118 we use the program _ plantri _ ( see @xcite ) . this program is the isomorph - free generator of planar graphs , including triangulations , quadrangulations , and convex polytopes . ( brinkmann and mckay s paper @xcite describes plantri s principles of operation , the basis for its efficiency , and recursive algorithms behind many of its capabilities . ) the program generates about @xmath119 billion graphs in @xmath118 , i.e. graphs that satisfy proposition 3.1 . namely , @xmath118 contains @xmath120 graphs with triangular and quadrilateral faces ; @xmath121 with at least one pentagonal face and with triangular and quadrilaterals ; @xmath122 with at least one hexagonal face which do not contain isolated vertices . the list of graphs with one and more isolated vertices relies on graphs in @xmath123 with @xmath124 that contain at least @xmath125 hexagons . for instance , the list of graphs in @xmath118 with exactly one isolated vertex consists of @xmath126 graphs . however , this list may contain isomorphic graphs . let @xmath127 be a finite point set such that its contact graph @xmath45 is irreducible . properties ( i)-(iv ) are combinatorial properties of @xmath45 . there are several geometric properties . note that all faces of @xmath45 are convex . since all edges of @xmath45 have the same length , @xmath20 , all its faces are spherical equilateral convex polygon with number of vertices at most @xmath128 . consider now a planar graph @xmath96 with given faces @xmath129 that satisfy corollary 2.1 . we are going consider embeddings of this graph into @xmath31 as an irreducible contact graph @xmath45 for some @xmath127 . any embedding of @xmath96 in @xmath31 is uniquely defined by the following list of parameters ( variables ) : + ( i ) the edge length @xmath94 ; + ( ii ) the set of all angles @xmath86 , @xmath87 of faces @xmath84 . here by @xmath88 we denote the number of vertices of @xmath84 . let us consider a graph @xmath96 from @xmath118 . we start from the level of approximation @xmath130 . now using proposition 3.2 we write the linear equalities and inequalities below . \(a ) from proposition 3.2(3 ) we have 14 linear equalities @xmath131 \(b ) since @xmath132 , from proposition 3.2(2 ) we have @xmath133 where @xmath134 . \(c ) for a quadrilateral @xmath135 with angles @xmath136 we have the equalities @xmath137 and inequalities @xmath138 \(d ) for a quadrilateral @xmath135 we also have the linear inequalities @xmath139 these inequalities follow from proposition 3.2(5 ) . we have @xmath140 if we consider the maximum and minimum of @xmath141 with @xmath142 $ ] and @xmath143 , then we obtain these inequalities . so from these linear equalities and inequalities we can obtain maximum and minimum values for each variable . it gives us a domain @xmath144 which contains all solutions of this system if they there exist . if @xmath144 is empty , then we can remove @xmath96 from the list @xmath118 . the first step , @xmath130 , `` kills '' almost all graphs . after this first step all that remained were @xmath145 graphs without isolated vertices , @xmath146 graphs with one isolated vertex , and no graphs with two and more isolated vertices . next we consider @xmath147 . in this step @xmath144 is divided into two domains and for both we can add the same linear constraints as we did when @xmath130 . moreover , for this step we add new linear constraints for polygons with five and higher vertices . in this level we obtain the parameterdomain @xmath148 . if this domain is empty , then @xmath96 can not be embedded to @xmath31 and it can be removed from @xmath118 . actually , for @xmath149 we can repeat the previous step , divide @xmath148 into two domains and obtain additional constraints as we did when @xmath147 for both parts independently . we can repeat this procedure again and again . in fact , by increasing @xmath150 we increase the number of sub - cases . however , in practically every step some sub - cases vanish . we repeat this process for @xmath151 and obtain a chain of embedded domains : @xmath152 if this chain is ended by the empty set , then @xmath96 can be removed from @xmath118 . in the case that a graph @xmath96 after @xmath153 steps still `` survives '' , i. e. @xmath154 , then it is checked by numerical methods , namely by the so called nonlinear `` solvers '' . ( we used , in particular , ipopt . ) if a solution there exists , then @xmath96 is declared as a graph that can be embedded , and if not , then @xmath96 is then removed from @xmath118 . in @xcite are given some numerical details for this algorithm . in this section we present a proof of lemma 2 . actually , two approaches are considered here - geometric and analytic . both methods are rely on the geometric properties of @xmath65 . the first method we already applied to prove lemma 2 in our solution of the tammes problem for @xmath5 @xcite . in this method by using the symmetries of a graph @xmath155 we find certain relations between the variables and using them we prove that @xmath156 . the geometric method is elementary , but it is not trivial and is relatively tricky . for @xmath157 , we found a proof that is based on the geometric approach . however , for the cases @xmath158 , we could not find a simple geometric proof . for those cases we apply the analytic approach . the idea of the analytic approach is very similar to the connelly s `` stress matrix '' method @xcite . perhaps , this method is not as elementary and explicit as the geometric approach , however it works for all cases and can be applied with a computer assistant . * geometric approach : the case @xmath166 . * let @xmath155 . proposition 3.2(3,4,5 ) allows us to prove some equalities for variables @xmath167 . namely , assume that two vertices , a and b , of @xmath96 are adjacent to two triangles and two quadrilaterals . then @xmath168 . we denote the correspondent angles by @xmath169 and @xmath170 respectively . ( here , as above , @xmath171 denote the angle of the equilateral triangle . ) we have @xmath172 if additionally , @xmath173 and @xmath174 are the opposite vertices of a quadrilateral @xmath135 in @xmath96 , then the equality @xmath175 ( see proposition 3.2.5 ) implies the equality @xmath176 . it is not hard to prove that if @xmath186 $ ] , where @xmath187 is sufficiently small , then @xmath94 is uniquelly defined , i.e. there is a continuous function @xmath188 on @xmath189 $ ] such that @xmath190 . since @xmath191 , if @xmath192 is a solution of ( 4.1 ) , then @xmath193 is also a solution . it implies that the function @xmath188 is even , i. e. @xmath194 . we present this function in fig . 4 . note that @xmath200 . from fig . [ dbyx ] we can see that @xmath201 for @xmath202 . therefore , @xmath203 . it can be rigorously proved . indeed , @xmath204 which means that the function @xmath205 is monotonically decreasing . since all @xmath206 , we obtain @xmath207 . * analytic approach : the case @xmath214 . * suppose that @xmath215 is a configuration of @xmath0 points in the sphere @xmath31 such that @xmath45 is the maximal graph . then points in @xmath13 can not to get closer together . so any slight motion of points in @xmath13 can not increase the minimal distance @xmath20 . therefore , @xmath13 is an infinitesimally rigid configuration @xcite . we say that an @xmath216 symmetric matrix @xmath217 is the _ equilibrium stress _ matrix if to each pair of distinct vertices @xmath218 of @xmath219 we have @xmath220 , @xmath221 when @xmath218 is not an edge of @xmath222 , and for each @xmath90 , the equilibrium equation @xmath223 holds . here @xmath224 is the unit tangent vector at the point @xmath225 to the great circle that passes through the points @xmath225 and @xmath226 . actually , these conditions for the equilibrium stress matrix can be derived from the karush - kuhn - tucker conditions @xcite , where stresses @xmath227 correspond to lagrange multipliers and the equilibrium equation corresponds to the stationary condition . note that , the inequality @xmath220 holds because @xmath228 can not be increased . in fact , after computations we have approximation intervals for all parameters of @xmath233 therefore , we can compute intervals also for all @xmath234 and @xmath235 , where @xmath236 let @xmath237 $ ] and @xmath238 $ ] . here @xmath239 and @xmath240 are sufficiently small numbers . thus , ( 4.3 ) implies @xmath241 j. leech , the problem of the thirteen spheres , math . gazette * 41 * ( 1956 ) , 22 - 23 . levenshtein , on bounds for packing in @xmath247-dimensional euclidean space , sov . dokl . * 20*(2 ) , 1979 , 417 - 421 .
|
the tammes problem is to find the arrangement of @xmath0 points on a unit sphere which maximizes the minimum distance between any two points .
this problem is presently solved for several values of @xmath0 , namely for @xmath1 by l. fejes tth ( 1943 ) ; for @xmath2 by schtte and van der waerden ( 1951 ) ; for @xmath3 by danzer ( 1963 ) and for @xmath4 by robinson ( 1961 ) .
recently , we solved the tammes problem for @xmath5 .
the optimal configuration of 14 points was conjectured more than 60 years ago . in the paper
, we give a solution of this long - standing open problem in geometry .
our computer - assisted proof relies on an enumeration of the irreducible contact graphs .
| 6,283 | 220 |
in 2003 , belle reported the discovery of a charmoiniumlike neutral states x(3872 ) with mass=@xmath21 mev with width @xmath22 mev @xcite and latter confirmed by do @xcite , cdf @xcite and barbar @xcite . this discovery fed excitement in the charmonium spectroscopy because of unconventional properties of the state . x(3872 ) could not be explained through ordinary meson(@xmath23 ) and baryon ( qqq ) scheme . the conventional theories predicts complicated color neutral structures and search of such exotic structures are as old as quark model @xcite . after the discovery of x(3872 ) , the large number of charge , neutral and vector states have been detected in various experiments , famous as the xyz states . recently , the charge bottomoniumlike resonances @xmath24 and @xmath25 have been reported by belle collaboration in the process @xmath26 and @xmath27 @xcite . moreover , a state reported by besiii collaboration in @xcite as @xmath28 in the @xmath29 reaction , again the besiii collaboration reported a state @xmath30 from invariant mass @xmath31 in the @xmath32 reaction @xcite , whereas the belle @xcite and cleo @xcite reconfirmed the status of the state . the sub structure of the all these states are still a open question , they might driven exotic structure like tetraquark , molecular or hybrid , expected as per theory of qcd , needs theoretical attention . + in the present study , we focus on the molecular structure , as meson - antimeson bound state , just like deuteron . the multiquark structures have been studied since long time @xcite . t@xmath33rnqvist , in @xcite predicted mesonic molecular structures , introduced as @xmath34 by using one pion exchange potential . with heavy flavour mesons , various authors predicted the bound state of @xmath0 and @xmath12 as a possible mesonic molecular structures as well as studied the possibilities of the @xmath4 and @xmath15 as vector - vector molecule @xcite , also it have been studied in various theoretical approaches like potential model @xcite , effective field theory @xcite , local hidden gauge approach @xcite etc .. + in the variational scheme , we have used the potential model approach to study the meson - antimeson bound system . for that , we have used the hellmann potential @xcite ( superposition of the coulomb + yukawa potential ) with one pion exchange potential ( opep ) . we assume that the colour neutral states experience residual force due to confined gluon exchange between quarks of the hadrons ( generally known as residual strong force ) , skyrme - like interaction . as mentioned by greenberg in ref . @xcite and also noted by shimizu in ref @xcite that this dispersive force ( also called london force ) or the attraction between colour singlet hadron comes from the virtual excitation of the colour octet dipole state of each hadron @xcite . indeed , long ago skyrme @xcite in 1959 and then guichon @xcite , in 2004 had remarked that the nucleon internal structure to the nuclear medium does play a crucial role in such strong effective force of the n - n interaction . in the study of the s - wave n - n scattering phase shift , in ref.@xcite , khadkikar and vijayakumar used the colour magnetic part of the fermi - breit unconfined one - gluon - exchange potential responsible for short range repulsion and sigma and pion are used for bulk n - n attraction . in this way , with such assumption of the interaction , the mass spectra of the dimesonic bound states are calculated . + for molecular binding , the ref.@xcite found that the quark exchange alone could not bind the system , led to include one pion exchange . the ref.@xcite mention some additional potential strength required with one pion exchange . whereas , the dynamics at very short distance led to complicated heavy boson exchange models as studied in @xcite . in all these studies @xcite , one common conclusion was extracted that the highly sensitive dependence of the results on the regularisation parameter . to avoid these dependency and complicated heavy boson exchange in this phenomenological study , we used the hellmann potential in accordance to delicate calculation of attraction and repulsion at short distance . the overall hellmann potential represents the residual strong interaction at short distance in flavour of the virtual excitation of the colour octet dipole state of each colour neutral states . the opep is included for long range behaviour of the strong force . the ope potential could be split into two parts ( i ) central term with sipn - isospin factor ( ii ) tensor part . we have analyse the effect of these two parts . by calculating the spin - isospin factor as in @xcite , we have found the symmetry braking in our results which was also discussed by t@xmath33rnqvist in @xcite , whereas , the tensor term to be found play a very crucial role implicit the necessity of it . in that way , bound state of @xmath0 is compared with the state x(3872 ) which also have been predicted as mesonic molecule by the authors of ref.@xcite whereas states @xmath24 and @xmath24 which are close to the @xmath12 and @xmath15 threshold . + to test the internal structure of the state , in general , one have to look for the decay pattern of the state . in ref . @xcite , the hadronic decays of the x(3872 ) have been studied in accordance to its decay mode sensitive to the short or long distance structure of the state . to test the compared states as dimesonic system , we have used the binding energy as input for decay calculation . we have adopt the formula developed by authors of ref . @xcite for the partial width sensitive to the long distance structure of the state , whereas , the formula for the decay mode sensitive to short distance structure of the state is taken from @xcite . in ref . @xcite , authors predicted existence of the neutral spin-2 ( @xmath35=@xmath36 ) partner of x(3872 ) , would be @xmath4 bound state , and in same way expected spin-2 partner of @xmath12 , would be @xmath15 bound state , on the basis of heavy quark spin symmetry and calculated the hadronic and radiative decays . we have used formula of @xcite for radiative decay for predicated states . with calculated binding energy , the decay properties are in good agreement with @xcite . + the article is organize as follows , after the brief introduction we have discussed the hellmann potential and ope potential in sec - ii then in sec - iii we presents our theoretical approach for the calculation of the mass spectra as well as for decay width , in sec - iv we present our results for deuteron as per our model and generalize our approach . in sec - v and vi we presents our results of dimesonic states and finally we summarise our present work . two color neutral states formed the bound states , just like deuteron is the bound state of proton and neutron . in the case of the dimesonic states , the meson and antimeson are takeover as a constituents , forming a bound state . the interaction potential between two color neutral constituents taken as a phenomenological hellmann potential @xcite with one pion exchange potential(opep ) @xcite . the colour neutral states experience residual force due to confined gluon exchange between quarks of the hadrons . the attraction between colour singlet hadron comes from the virtual excitation of the colour octet dipole state of each hadron . when two colour neutral states are come enough close to each other so the quark of the one colour neutral states can feel to quark of the other state creates manifestation of the energy . once the interaction become strong enough then this manifestation of the energy being the source for the creation for the quark anti quark pair . this residual force is analogous to the residual interaction between two electrical neutral atoms , experienced the van der waals force . the overall hellmann potential represents the residual strong interaction at short distance , where , these pseudopotential has the form , namely @xmath37 here , the constant @xmath38 is the residual strength of the strong running coupling constant between the two colour neutral states and b are the strength of the yukawa potential whereas @xmath39 is the relative separation between constituents . the value of the @xmath38 could be determined through the model , such as @xmath40 where @xmath41 and @xmath42 are constituent masses , m=2@xmath41 @xmath42/(@xmath41+@xmath42 ) , @xmath43=1 gev , @xmath44=0.250 gev and @xmath45 is number of flavour @xcite . so , the coulomb interaction is constrained by the value of @xmath38 . the constant b and c appeared in the yukawa potential of eq.(1 ) play very trivial role on overall characteristic of the hellmann potential . in the fig(3 ) ( see appendix - a ) , one can observe that once the coulombic interaction is fixed , the variation of the constant b and c are inversely proportional to each other . as the value of the b increases , it increases the repulsive nature of the potential while the constant c smoothing the curve as well as it increases the strength of the yukawa part - inversely to b. to show such nature of the hellmann potential , in the fig(3 ) , the graph have been plotted for various set of values of the b and c. for the bound state , the value of b can take both positive and negative values . with the negative values of the b the overall potential become attractive . for more detailed discussion on the hellmann potential , we suggest to the readers for ref.@xcite . in our case of the dimesonic bound states calculations , we assume that the hellmann potential at very short distance cares for the delicate cancellation of attraction and repulsion respectively which is mainly taken care by heavy boson exchange in the one boson exchange model . + whereas , the long range behaviour of the interaction part is accomplished by one pion exchange . the model carried the net potential as the hellmann potential plus ope potential plus relativistic correction . we introduced and discussed the relativistic correction after this section . for instance , to fix the model ( or the potential ) for deuteron which is widely believed to be have molecular structure and well studied in obe potential model , we fit the values of the constant @xmath46 and @xmath47 to get approximate binding energy of deuteron . then , to generalize the model to dimesonic system , we arrive on the relation between b and c in accordance of the mass of the system , + if @xmath48 , where @xmath49 is the threshold mass of dimesonic state and @xmath41 mass of deuteron and n is integer number ( n=1,2,3 .. ) , then @xmath50 here , @xmath46 and @xmath47 are the constant , fitted for deuteron binding energy . + the one pion exchange potential ( opep ) taken for long range interaction which is well studied for nn - interaction . the ope potential for nn - interaction takes the form @xcite @xmath51\end{aligned}\ ] ] where @xmath52 is the nucleon - pion coupling constant , @xmath53 and @xmath54 are spin and isospin factors respectively while the @xmath55 and @xmath56 are defined as @xmath57 @xmath58 whereas the @xmath59 is the usual tensor operator expressed as @xmath60 , is mainly responsible for long range tail of the potential and play very crucial role in the nn - interaction . the expression of opep in the eq.(4 ) is for the point like pion . while , in a more realistic picture where the pion itself has its own internal structure , it is natural to introduced the usual form factor due to the dressing of the quarks . by introducing this finite size effect @xcite , we have \(a ) ( b ) ( c ) ( d ) @xmath61 @xmath58 thus , the function @xmath62 and @xmath63 with the finite size effect take the form @xmath64\ ] ] @xmath65\ ] ] now , one pion exchange potential for dimesonic system could be written as @xmath66 @xmath67\end{aligned}\ ] ] such that , the opep becomes for dimesonic systems , @xmath68 where the @xmath69=0.69 is the meson - pion coupling constant , @xmath49 and @xmath70 are the average mass of two constituent of the dimesonic state and pion mass respectively , while @xmath71 is the pion form factor . the mesons and quark masses with their quantum numbers are taken from the listing of particle data group@xcite . the constituent meson - quark coupling constant may derive by using the goldberger - treiman relation on suitable estimates of known @xmath72 coupling constant . the relation between quark - boson and nucleon - boson coupling could expressed as @xcite @xmath73 the effective ope potential can be split into a central and tensor term proportional to @xmath74 @xmath75 and @xmath59 @xmath75 respectively . in ref.@xcite , thomas et.al . mentioned and discussed sign convention and detail calculation of spin - isospin factor . in ref.@xcite , authors showed the inconsistent sign convention adopted by ref.@xcite in the calculation of spin - isospin factor of @xmath76(@xmath36 ) and then derive and explain overall sign for determination of @xmath74 @xmath75 . we are agreed with ref.@xcite and adopt the same . for total spin s , total isospin state i and charge conjugation parity c , the spin - isospin factor for central term are given by @xcite @xmath77 the matrix element of the tensor oerator for different spin state are real numbers and it is well discussed by t@xmath33rnqvist in ref.@xcite . for example , the matrix element of @xmath59 in the case of deuteron @xmath78 with such interaction the opep potential shown in eq(11 ) , one need to solve the coupled channel schr@xmath33dinger equation . in the present study , we have focus only on the s - wave spectra of dimesonic state . as such , we have taken only the s - wave tensor contribution of particular spin state . the matrix element of such tensor operator of dimesonic states are given by @xmath79 the value of the form factor @xmath80 in opep affects the strength of the potential very drastically . the sensitivity of the @xmath80 have discussed by authors of ref.@xcite . in the fig(1 ) , the nature of the opep in different spin - isospin state are shown for different values of @xmath80 ( noted that only the s - wave contribution of the tensor interaction is considered ) . one can analysed from the plot , as the value of the @xmath80 increases the strength of the potential is also increases drastically . in ref.@xcite , t@xmath33rnqvist noted the value of @xmath80 fall in range 0.8 - 1.5 gev for fit the nn scattering data where as in case of mesonic molecule , specially , the heavy meson which are very small compared to the size of the nucleon , the larger value of @xmath80 is expected and the large value of @xmath80 increases the binding energy . ref.@xcite shows the results are very sensitive to @xmath80 and the binding energy no longer monotonically increases with @xmath80 with opep model . for instant , we fixed @xmath80=1.5 gev consistent with nn - scattering data and increase the dominance of the hellmann potential , such that we arrive at relation of the eq(2 ) . + the factors @xmath74 @xmath75 and @xmath59 @xmath75 makes the opep spin and isospin state dependent . as per the literature @xcite , the most attractive channel is ( i , s)=(0,0 ) while the channel ( i , s)=(2,0 ) is the most repulsive . but , as in fig(1 ) , we observed the channel ( i , s)=(1,1 ) is most attractive while ( i , s)=(0,1 ) noted as most repulsive one . the next order of the attractive channels are ( i , s)=(0,0 ) and ( i , s)=(1,2 ) . whereas , the channels ( i , s)=(0,1),(i , s)=(2,0 ) and ( i , s)=(1,0 ) expected to be unbound . the observed change in the nature of the these channels may be due to only the s - wave contribution of the tensor interaction . it clearly indicates the dominant effect of tensor operator at long range part of the potential . the discussed effect of opep is also reflect in our results , tabulated in table - ii and table - iii . .masses of the mesons ( in mev)@xcite [ cols="^,^,^,^,^,^,^,^ " , ] there are large possibilities of dimesonic states with heavy light flavours , also shown in our calculated mass spectrum(see table - iii ) . we have predicted the mass and root mean square radius . for instant with @xmath76 dimesonic states , for the decay properties , we focus on these two states @xmath81 and @xmath82 as spin-2 partner of @xmath0 and @xmath12 molecules respectively and also expected as per heavy quark spin symmetry(hqss ) . the study of decay properties with mass spectra is very important to investigate their sub structure . we have calculated the hadronic and radiative decay of these states . the hadronic decay modes of @xmath81 @xmath83 by using the eq(20)(with @xmath84=0.993 ) , the calculated partial decay width , we have @xmath85 similarly the decay width for @xmath82 with the same formula ( with @xmath84=0.998 ) to be found @xmath86 to understand the electromagnetic branching fraction we need to understand the interaction with photon and s - wave mesons and their contribution due to light and heavy quarks @xcite . in addition to , radiative decay is more sensitive to long distance molecular structure . the radiative decay mode of @xmath81 expressed as @xmath87 thus the decay width for the state calculated as per eq(23 ) ( with @xmath88 = 1 gev ) , we have @xmath89 the calculated partial decay widths are in good agreement with the results calculated in ref@xcite . the radiative decay for the state @xmath82 could also calculated similar to @xmath81 . @xmath90 for this decay mode of @xmath82 , in our model the radiative decay of @xmath82 to be found forbidden with suggestive value of @xmath88 = 1 gev . with very large value of @xmath88 ( @xmath918 ) , we have get the comparable results with ref . moreover , the radiative decay width for @xmath82 calculated in @xcite is @xmath91 10 ev , which is very small . with very large value of @xmath88 , we have found @xmath91 14 ev . ( see the table - iv , for the results of calculated partial widths with comparisons with others ) . in summary , we have calculated the mass spectra of dimesonic states with heavy - light flavour mesons by using the hellmann and one pion exchange potential . the calculated binding energy of dimesonic states are found overestimated which is also expected in variational approach . we have analysed the change in the binding energy due to charge conjugation parity and isospin , agreed with ref.@xcite suggested isospin symmetry braking . we have also discussed the effect of tensor term in one pion exchange . we includes only the s - wave contribution of tensor term and observed shuffling of channels which is different from expected channels as in @xcite . it is remarkably pointing out the effect of the tensor term and its contribution in different processes , led us to conclude that it can not be ignore . whereas , the contribution from the relativistic correction in the potential to the binding energy is @xmath92 which is almost one third to our previous work @xcite . + in addition , we have calculated the decay properties of @xmath0 , @xmath12 , @xmath4 and @xmath15 ( the formalism adopted from ref.@xcite ) , using our calculated binding energy as input where formalism mainly dependent on the masses . to enforced the our predictions , we have attempted the decay calculations as well to test the molecular structure of compared states with dimesonic states on the basis of decay modes which may have responsible for their long and short distance structure , as studied in the literature @xcite . we have found comparable results of decay properties ( with our mass spectra ) @xcite , support the prediction of @xmath0 , @xmath12 , @xmath4 and @xmath15 bound states as mesonic molecules . we support the molecular structure of the x(3872 ) as @xmath0 bound state with @xmath93 dominant decay mode . the long distance radiative decay may gives information of the mixing of charge to neutral channel of the bound state which lack in this study . we have predicted the mass spectra and decay properties of the possible spin-2 partner ( @xmath94 ) of x(3872 ) in charm sector as well as spin-2 partner of @xmath12 in the bottom sector , as miguel et.al studied in @xcite , also expected in heavy quark spin - flavour symmetry . + apart form these states , we have suggest mass spectra of possible dimesonic states as mesonic molecules with @xmath95 , @xmath96 , @xmath97 , @xmath98 and @xmath99 mesons . wei chen et.al in @xcite suggested open flavour tetraquark structure with having strange quark , with mass range 6.9 - 7.3 gev , with @xmath100 or @xmath101 . in ref.@xcite , gui - jun ding suggest @xmath5 molecule . the masses of @xmath102 and @xmath76 dimesonic states with @xmath95 , @xmath96 , @xmath97 and @xmath98 mesons are fall in the same range of 7.1 - 7.3 gev . moreover , we have also predicted the masses of the dimesonic states having @xmath99 mesons as a constituent . we look forward to the experimental facilities and working groups to take more attention for searches of the possible dimesonic states as well as for the confirmations of our theoretical predictions . + * acknowledgements * d. p. rathaud would like to thanks prof . r. c. johnson for the useful discussion . a. k. rai acknowledge the financial support extended by d.s.t . , government of india under serb fast track scheme sr / ftp /ps-152/2012 . \(a ) ( b ) \(c ) ( d ) \(e ) ( f ) \(g ) ( h ) \(i ) ( j ) \(i ) for hadronic decay @xcite : one of the constituent meson decay into two mesons : @xmath103 + here , m is the mesonic molecule ( m = d+b , where d and b are constituents ) decaying into product mesons a , d , e in which one of the meson is the constituent of the dimesonic state , for example , @xmath104 , here , @xmath105 or @xmath106 + from ref @xcite , on the bassis of the tree level approximation the amplitude is given by @xcite @xmath107 where g=0.69 the quarks meson coupling constant , @xmath108= 92.2 mev pion decay constant where as @xmath109=0.35 @xmath110 is the coupling constant to the dimesonic state to the charge or neutral channels @xcite . thus , decay formula reads @xmath111 where @xmath112 is the three momentum of the pion . while @xmath113 and @xmath114 are the the four momenta of ad and ac systems . since the amplitude is dependent on the invariant masses we have @xmath115=@xmath116 and @xmath117=@xmath118 of the final state ea and ad pairs respectively @xcite . here , @xmath119 is the k@xmath120llen function . for the known value of @xmath115 the range of the @xmath117 could be determined by the values of momentum @xmath121 is parallel or antiparallel to @xmath122 as here , @xmath129 is the dimesonic coupling constant , @xmath130 is the fine structure constant , @xmath131gev is the charm quark mass , @xmath132 @xmath133 ( @xmath134 is the light quark mass ) , @xmath70 and @xmath135 is the pion and kaon masses , respectively . whereas , @xmath108 and @xmath136 are the pion and kaon decay constant , respectively . thus the partial radiative decay width given as for the known value of the @xmath138 the range of @xmath139 could be determined by the values of momentum @xmath122 is parallel or antiparallel to @xmath140 as @xmath141 here , @xmath142 @xmath143 and @xmath140 are the photon energy and momentum in the @xmath138 cm frame , respectively . whereas , the amplitude for the @xmath15 read as with @xmath145=@xmath146 for @xmath15 system and could be calculated according to eq.(b8 ) . @xmath42=4.6 gev is mass of the bottom meson . for more detailed of the calculations and formalism of decay calculation we refer to ref.@xcite . al.(particle data group ) , chin . c * 38 * , 090001 ( 2014 ) . choi et al . 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in this work , we have calculated the mass spectra and decay properties of dimesonic ( meson - antimeson ) bound state in the variational scheme . the intermesonic interaction considered as the hellmann potential and one pion exchange potential .
the mass spectra of the @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 , @xmath5 , @xmath6 , @xmath7 , @xmath8 , @xmath9 , @xmath10,@xmath11 , @xmath12 , @xmath13 , @xmath14 , @xmath15 , @xmath16 , @xmath17 etc .. are calculated .
the states x(3872 ) , @xmath18 , @xmath19 and @xmath20 are compared with @xmath0 , @xmath4 , @xmath12 and @xmath15 dimesonic bound states . to probe the molecular structure of the compared states
, we have calculated the decay properties sensitive to their long and short distance structure of hadorinc molecule , our results suggested the compared states driven the molecular structure . apart from these the other calculated mass spectra of dimesonoic states are predicted and for such bound states , the experimental search are suggested .
pacs numbers : : 12.39.jh , 12.39.pn , 13.25.jx,14.40.rt
| 9,384 | 381 |
a natural and very general extension of the einstein - maxwell lagrangian yielding a general system of equations for a non - minimal coupling between the gravitational and electromagnetic fields , with non - linear terms , was set up and studied in @xcite . within this general theory , a special theory , worth of discussion , arises when one restricts the general lagrangian to a lagrangian that is einstein - hilbert in the gravity term , quadratic in the maxwell tensor , and the couplings between the electromagnetism and the metric are linear in the curvature terms . the motivations for setting up such a theory are phenomenological , see , e.g. , @xcite for reviews and references.this theory has three coupling constants @xmath0 , @xmath1 and @xmath2 , which characterize the cross - terms in the lagrangian between the maxwell field @xmath10 and terms linear in the ricci scalar @xmath11 , ricci tensor @xmath12 , and riemann tensor @xmath13 , respectively . the coupling constants @xmath0 , @xmath1 and @xmath2 have units of area , and are a priori free parameters , which can acquire specific values in certain effective field theories . more specifically , the action functional of the non - minimal theory linear in the curvature is @xmath14 where @xmath15 is the determinant of the spacetime metric @xmath16 , and the lagrangian of the theory is @xmath17 here @xmath18 , @xmath19 being the gravitational constant and we are putting the velocity of light @xmath20 equal to one , @xmath21 is the maxwell tensor , with @xmath22 being the electromagnetic vector potential , and latin indexes are spacetime indexes , running from 0 to 3 . the tensor @xmath23 is the non - minimal susceptibility tensor given by @xmath24 where @xmath0 , @xmath1 , and @xmath2 are the mentioned phenomenological parameters . the action and lagrangian ( [ action1])-([susceptibility2 ] ) describe thus a three - parameter class of models , non - minimally coupled , and linear in the curvature @xcite . lagrangians of this type have been used and studied by several authors . the first and important example of a calculation of the three couplings was based on one - loop corrections to quantum electrodynamics in curved spacetime , a direct and non - phenomenological approach considered by drummond and hathrell @xcite . this model is effectively one - parameter since the coupling constants are connected by the relations @xmath25 , @xmath26 , @xmath27 . the positive parameter @xmath3 appears naturally in the theory , and is constructed by using the fine structure constant @xmath28 , and the compton wavelength of the electron @xmath29 , @xmath30 . in these models it is useful to define a radius @xmath31 , an effective radius related to the non - minimal interaction , through @xmath32 . thus , the corresponding effective radius for the non - minimal interaction in this case , is the drummond - hathrell radius @xmath33 , given by @xmath34 . in @xcite one also finds a quantum electrodynamics motivation for the use of generalized einstein - maxwell equations . phenomenological models , i.e. , models based on external considerations to obtain the couplings , or parameters , @xmath0 , @xmath1 , and @xmath2 , have also been considered . prasanna @xcite wanting to understand how the strong equivalence principle can be weakly violated in the context of a non - minimal modification of maxwell electrodynamics , has shown that @xmath35 , @xmath3 a free parameter , is a good phenomenologically model . another type of requirement , one with mathematical and physical motivations , is to impose that the differential equations forming the non - minimal einstein - maxwell system are of second order ( see , e.g. , @xcite ) . for instance , in @xcite , by imposing a kaluza - klein reduction to four dimensions from a gauss - bonnet model in five dimensions , thus guaranteeing second order equations for the electric field potential @xmath36 , and metric @xmath16 , it was discussed a model in which @xmath37 and @xmath38 , i.e. , with @xmath5 , @xmath6 and @xmath7 . so the extra non - minimal term is a kind of gauss - bonnet term , and the model is called the gauss - bonnet model . yet another type of requirement , this time purely mathematical , was suggested in @xcite . the idea is connected with the symmetries of the non - minimal susceptibility tensor @xmath23 ( see eq . ( [ susceptibility2 ] ) ) . for instance , one can recover the relations @xmath37 and @xmath38 , used in @xcite , by the ansatz that the non - minimal susceptibility tensor @xmath39 is proportional to the double dual riemann tensor @xmath40 , i.e. , @xmath41 , for some @xmath42 ( see , @xcite for details and motivations ) . analogously , one can use the weyl tensor @xmath43 in the relation @xmath44 , for some @xmath45 , or the difference @xmath46 instead of @xmath40 , to introduce some new linear relations between @xmath47 , namely @xmath48 and @xmath49 . yet another type of requirement is to choose the parameters so that one obtains exact solutions . as we will see this will lead to a model with @xmath37 and @xmath9 , i.e. , @xmath5 , @xmath8 , @xmath9 . since this model is integrable we call it the integrable model . a subcase of this has additional interest and is called the fibonacci soliton . up to now we have a theory defined through eqs . ( [ action1])-([susceptibility2 ] ) , with each chosen set of values for the parameters @xmath0 , @xmath1 , and @xmath2 , giving a model . we have seen that the reduction from three - parameter models to one - parameter models , specified by the one parameter @xmath3 and the relations between @xmath0 , @xmath1 and @xmath2 , happens in several instances , either through direct calculation , as in @xcite , or through phenomenological and other considerations , as in @xcite-@xcite or @xcite and here . this certainly simplifies the analysis , and we will consider this one - parameter type of models , in which @xmath0 , @xmath1 , and @xmath2 , have a specified relation to the parameter @xmath3 . for all these models , one can pick an effective radius @xmath50 , as in the drummond - hathrell case , which gives the range of the non - minimal interaction between the gravitational and electric fields . of course , @xmath31 can be set to zero , in the case the world is pure einstein - maxwell , or otherwise can have a given specified value . the radius @xmath33 defined above is a candidate but in principle not the unique choice . thus , possible estimations of the parameter @xmath3 , and so of @xmath31 , from , for instance , astrophysical observations , are undoubtedly of interest ( see , e.g. , @xcite ) . now , after choosing a model , specified by @xmath3 and by the relations between @xmath0 , @xmath1 , and @xmath2 , it is important to study exact solutions . exact solutions of the equations of non - minimal electrodynamics in non - linear gravitational wave backgrounds were obtained in @xcite@xcite , and a non - minimal bianchi - i cosmological solution was discussed in @xcite in this context . here we want to study charged black hole and other charged solutions of non - minimal models . the reissner - nordstrm solution is a standard solution in pure einstein - maxwell theory , with two horizons , an event and a cauchy horizon , and a timelike singularity at the center ( see , e.g. , @xcite ) . paradigmatic charged black hole solutions also appear in the framework of einstein theory minimally coupled to non - linear electromagnetic fields , as well as other matter fields . such solutions were found by bardeen and others @xcite-@xcite and the main feature is that they are regular , without singularities inside the horizon . within quartic gravity non - singular charged black hole solutions have also been found @xcite . since the non - minimal theory we are considering possesses new degrees of freedom , namely , the phenomenological parameter @xmath3 and its relations to @xmath0 , @xmath1 , and @xmath2 , we believe that these allow to introduce new aspects to the problem of finding black hole and other solutions of each chosen model . three aspects can be mentioned . first , one wants to have a gauge in order to compare the new solutions . thus , we study the reissner - nordstrm solution , the trivial solution in this context , where @xmath51 with @xmath52 , in order to understand the novel features , such as causal and singularity structure , of the new solutions . second , one should try to search for models exactly or quasi - exactly soluble . this requirement will take us , among the models cited above and the many other possible models , to two interesting models . they are , the gauss - bonnet type model where @xmath5 , @xmath6 and @xmath7 , and the integrable model where @xmath5 , @xmath8 , @xmath9 . in both models we perform a detailed analysis . third , we vary @xmath3 within each of the two nontrivial models and in trying to consider non - minimal extensions of the reissner - nordstrm solution , we search for features that are similar or distinct from the two paradigmatic solutions , the reissner - nordstrm solution itself and the bardeen solutions . in our search for charged black hole solutions in non - minimal models we work in schwarzschild coordinates and impose certain requirements . the first requirement is connected with the electric field @xmath53 . we demand it is a regular function on the interval @xmath54 , the origin being also regular ( i.e. , the value @xmath55 is finite ) , and for large values of @xmath56 the electric field is coulombian , @xmath57 , where @xmath58 is the electric charge . the second requirement is concerned with the metric functions @xmath16 . these should take finite values at the center ( @xmath59 ) . horizons at @xmath60 are not excluded , and far away the solutions should be asymptotically flat . upon these conditions and solving the non - minimal equations in certain cases we will find electric charged solutions with one horizon only , thus causally distinct from the reissner - nordstrm . however , like reissner - nordstrm , the solutions have a singularity at the center , although here the singularity is spacelike instead , as the schwarzschild case , but conical , thus much milder . we have also found in one model a gravitational charged soliton , without horizons , where the fields are well behaved , apart from a mild conical singularity at the center . although this and other solutions with horizons are almost regular at the center , we have not obtained strictly non - singular black hole of the type found by bardeen and others @xcite-@xcite . the difference is based on two aspects . firstly , we consider spherically symmetric static solutions with @xmath61 , in contrast to the schwarzschild , reissner - nordstrm , and minimal regular bardeen solutions . secondly , we assume that the values @xmath62 and @xmath63 are finite , but can differ from one . moreover , we admit , that @xmath62 can , in principle , be equal to zero . this means that the scalar curvature invariants for such a metric can take infinite values at the center , and the solution of the non - minimal einstein equations is not regular at the center in this general sense . however , in many cases the singularity is a conical one , so much milder than the nasty ones of schwarzschild and reissner - nordstrm . in summary , we find charged black hole solutions different in horizon structure from the reissner - nordstrm but similar to bardeen black holes , and although singular , in certain cases can be considered quasi - regular conical singularities , in - between schwarzschild and reissner - nordstrm types of singularities and the no singularities of bardeen . this paper is organized as follows . in section [ genericalities ] , in particular in subsection [ genericalities2 ] , using the lagrangian formalism of the introduction , we establish a three - parameter non - minimal einstein - maxwell model . in subection [ static ] , we set up static equations for studying black holes and reduce the three - parameter model to a one - parameter model . in section [ solutions ] we study specific spherically symmetric one - parameter solutions . in subsection [ basics ] we define the basic quantitities and basic variables . in subsection [ rnmodel ] we display the reissner - nordstrm solution as a preparation , where @xmath51 with @xmath52 . in subsection [ gbmodel ] we analyze in detail a one - parameter model , the gauss - bonnet model , with @xmath5 , @xmath6 , @xmath7 ( i.e. , @xmath37 and @xmath38 ) , using the known solution of the abel equation , the key equation of the model , and the dynamical system associated with this model . we display the charged black hole solutions , and we focus on a specific exact solution and its critical properties . in subsection [ nonamemodel ] we consider in detail an exactly integrable one - parameter model , the integrable model , with @xmath5 , @xmath8 , @xmath9 ( i.e. , @xmath37 and @xmath9 ) . we examine a special sub - model , the fibonacci soliton , of this one - parameter model and present the corresponding exact solution . in subsection [ table ] we present , by means of a table , the summary of the results of the models studied . in section [ conc ] we conclude . the variation of the lagrangian ( [ action1])-([susceptibility2 ] ) with respect to the metric yields ( see , @xcite for details ) , @xmath64 \ , . \label{standardform}\ ] ] the energy - momentum tensor of the pure electromagnetic field , @xmath65 , is @xmath66 the definitions for the other three parts of the stress - energy tensor , @xmath67 , @xmath68 and @xmath69 , are @xmath70 @xmath71 - f^{ln}(r_{il}f_{kn } + r_{kl}f_{in } ) - \nonumber\\ & & { - } r^{mn } f_{im } f_{kn } { - } \frac{1}{2 } \nabla^l \nabla_l ( f_{in}f_{k}^{\ n } ) { + } \frac{1}{2}\nabla_l \left [ \nabla_i(f_{kn}f^{ln } ) { + } \nabla_k(f_{in}f^{ln } ) \right ] \ , , \label{part2}\end{aligned}\ ] ] @xmath72 in addition , the non - minimal electrodynamics associated with the lagrangian ( [ action1])-([susceptibility2 ] ) obeys the equation @xmath73 = 0 \ , , \quad \nabla_k f^{*ik } = 0 \ , , \label{m1}\ ] ] where @xmath74 is the maxwell tensor and @xmath75 is dual to it . consider now these master equations in the case of a static spherically symmetric spacetime . using schwarzschild coordinates , the line element for a static spherically symmetric system can be put in the form @xmath76 where @xmath77 and @xmath78 are metric potentials that depend on the radial coordinate @xmath56 only . this form of the line element is useful as when @xmath77 and @xmath79 are simultaneously zero it signals the presence of an event horizon . assume also that the electromagnetic field inherits the static and spherical symmetries . then the potential four - vector of the electric field @xmath36 has the form @xmath80 from ( [ potentiala ] ) the maxwell tensor is equal to @xmath81 , where a prime denotes the derivative with respect to @xmath56 . to characterize the electric field , it is useful to introduce a new scalar quantity @xmath53 as @xmath82 . then the electric field squared is @xmath83 from which one obtains in turn @xmath84 . since the expressions @xmath79 and @xmath85 enter frequently in the master equations , it is sometimes convenient to use the functions @xmath86 and @xmath87 defined as @xmath88 and @xmath89 . in summary , the functions @xmath90 and @xmath91 are alternatives to the functions @xmath92 , @xmath93 , and @xmath94 . the maxwell equations ( [ m1 ] ) with ( [ potentiala ] ) give only one non - trivial equation , namely , @xmath95^{\prime } = 0 \ , , \label{eequ}\ ] ] which can be integrated immediately to give @xmath96 \right.\ ] ] @xmath97 where @xmath58 is a constant , to be associated with the central electrical charge of the solution . this equation gives the electric field of a central charge , corrected by the radial component of the dielectric permeability tensor , @xmath98 . this component describes the vacuum screening effect on the central charge , due to the interaction of the vacuum with curvature , analogously to the screening of a charge by a non - homogeneous medium in a spherical cavity . supposing the spacetime to be asymptotically flat , i.e. , @xmath99 , one can see that ( [ 1e ] ) yields asymptotically the coulomb law @xmath100 , and the constant @xmath58 indeed coincides with the total electric charge of the object . the equations for the gravitational field ( [ standardform ] ) with ( [ t0])-([part3 ] ) and the metric potentials ( [ metric1 ] ) redefined as in ( [ sigman ] ) can be rewritten as a pair of equations for @xmath86 and @xmath87 , respectively , @xmath101^{\prime}}{\kappa r^2 } = - \left ( e^2\right)^{\prime \prime } n ( q_1+q_2+q_3)\nonumber\\ + \left ( e^2\right)^{\prime } \left[- \frac{1}{2}(q_1+q_2+q_3)\left ( n^{\prime } + \frac{8n}{r}\right)+ \frac{n}{r}(2q_1+q_2 ) \right]\nonumber\\ + e^2 \left [ \frac{1}{2 } + ( q_1+q_2+q_3)\left ( n^{\prime \prime } + 3 n^{\prime } \ \frac{\sigma^{\prime}}{\sigma } + 2n \frac{\sigma^{\prime \prime}}{\sigma } - \frac{n^{\prime}}{r } - 2 \frac{n}{r^2 } \right ) \right.\nonumber\\ \left . + ( 2q_1+q_2)\left ( 2\frac{n^{\prime}}{r } + 2\frac{n}{r } \frac{\sigma^{\prime}}{\sigma } + \frac{n}{r^2 } \right ) + q_1 \frac{(n-1)}{r^2 } \right]\ , , \label{2e}\end{aligned}\ ] ] @xmath102\nonumber\\ + e^2 \left[(q_1+q_2+q_3 ) \frac{2\sigma^{\prime } } { r \sigma } - \frac{2q_3}{r^2 } \right ] \,.\label{3e}\end{aligned}\ ] ] the first equation can be reduced to an equation for @xmath53 and @xmath86 , by extracting the term @xmath103 from the second one . the second equation contains the unknown functions @xmath53 and @xmath87 only . thus , eqs . ( [ 1e])-([3e ] ) form the key system of equations for the non - minimal einstein - maxwell model of a static spherically symmetric object . it is a system of three ordinary differential equations of second order for the three unknown functions , @xmath53 , @xmath86 and @xmath87 . the form of equations is not canonical . in principle , the electric field @xmath53 can be extracted explicitly from ( [ 1e ] ) as a function of @xmath104 , @xmath105 , @xmath106 , @xmath107 , @xmath103 and @xmath56 . inserting such @xmath53 into the eqs . ( [ 2e])-([3e ] ) , we obtain equations for @xmath86 and @xmath87 of fourth order in their derivatives . below we focus on models admitting solutions to eqs . ( [ 1e])-([3e ] ) , such that they can be represented by a series expansion regular at @xmath108 , i.e. , @xmath109 where @xmath110 symbolizes generically the functions @xmath53 , @xmath86 , and @xmath87 . our purpose is to find solutions satisfying three conditions : first , the electric field @xmath53 should be a continuous function regular at @xmath108 ( @xmath111 ) and also should be of coulombian form at @xmath112 . second , the metric functions @xmath86 and @xmath113 should be regular at @xmath108 . third , in terms of the functions @xmath93 and @xmath94 the asymptotic flatness requires that @xmath114 , and @xmath115 , @xmath116 . so , essentially , one can put , @xmath117 and @xmath118 . note that the regularity of the functions @xmath53 , @xmath86 , and @xmath87 at @xmath108 does not guarantee that the solution of the einstein - maxwell model is characterized by regular curvature invariants . for instance , when @xmath119 is finite but @xmath120 , the model displays a conical singularity and the scalar invariants of the curvature tend to infinity as @xmath121 . in considering solutions such that the fields are finite we try to be as close as possible to bardeen s idea of having black hole solutions without singularities , by finding a regular @xmath53 and putting the metric coefficients in the form @xmath122 and @xmath123 , for some @xmath124 @xcite . as we will see it will turn out that this is not achieved , since although the electric and metric potentials are regular , the black hole solutions found here are singular at the center , where the curvature invariants blow up . notwithstanding , these solutions are very interesting . using the ansatz ( [ regular ] ) and the eqs . ( [ 1e])-([3e ] ) one can couple the values @xmath55 , @xmath119 , @xmath125 , and @xmath0 , @xmath1 , @xmath2 . the relations are different for the cases @xmath126 and @xmath127 , which we now analyze . _ ( i ) @xmath126 : _ when @xmath126 , but @xmath128 , one obtains from the system ( [ 1e])-([3e ] ) that @xmath129 , and the decomposition ( [ regular ] ) is valid at @xmath121 , when the following conditions are satisfied , @xmath130= 2q_1 + q_2\ , , \label{regularity2}\end{aligned}\ ] ] for generic @xmath0 , @xmath1 , and @xmath2 . there are two specific cases . when @xmath37 , but both @xmath131 and @xmath132 , then @xmath133 and @xmath134 . when @xmath37 and @xmath135 , simultaneously , then @xmath136 , providing @xmath0 is negative , and @xmath119 is fixed by the relation @xmath137 . as in the case @xmath127 , see below , here the ricci scalar @xmath138 takes an infinite value at @xmath108 . _ ( ii ) @xmath127 : _ when all three functions , @xmath53 , @xmath86 , and @xmath87 are regular at @xmath108 , and @xmath87 , appearing in the denominator of eqs . ( [ 1e])-([3e ] ) , does not vanish at @xmath108 , one obtains from the system ( [ 1e])-([3e ] ) the following set of equations @xmath139 = q \ , , \quad e^2(0 ) \ 2 q_3 = 0 \ , , \nonumber\\ n(0 ) \left [ 1 + \kappa e^2(0 ) ( q_1 - q_2 - 2 q_3 ) \right ] = 1 + q_1 \kappa e^2(0 ) \ , , \label{regularity1}\end{aligned}\ ] ] for generic @xmath0 , @xmath1 , and @xmath2 . since the charge of the object , @xmath58 , is considered to be non - vanishing , one obtains immediately from the first equation of the set that @xmath140 and @xmath120 . thus , we infer , @xmath141 } \ , , \quad q_2 = \frac{2 [ n(0)-1 ] + \kappa e(0 ) q}{2 n(0 ) \kappa e^2(0 ) } \ , , \quad q_3=0\ , . \label{r2}\ ] ] the relations ( [ regularity1 ] ) give that the curvature invariants are infinite in the center @xmath108 . for instance , when @xmath86 and @xmath87 are regular in the center and @xmath142 , then the ricci scalar @xmath143 tends to infinity at @xmath121 , since @xmath120 , as well as generally @xmath144 . now we want analyze the simplest cases of the system of eqs . ( [ 1e])-([3e ] ) . one sees that there is an immediate simplification when @xmath37 , since second order derivatives and products of first order derivatives disappear from the equations . in such a case the system ( [ 1e])-([3e ] ) reduces to @xmath145 @xmath146 @xmath101^{\prime}}{\kappa } = r \left ( e^2\right)^{\prime } ( 2q_1+q_2 ) n + \nonumber\\ + e^2 \left [ \frac{r^2}{2 } - q_1 + 2 r ( 2q_1+q_2)\left ( n^{\prime } + n \frac{\sigma^{\prime}}{\sigma } \right ) + n ( 3q_1 + q_2 ) \right]\ , . \label{13e}\end{aligned}\ ] ] moreover , the three - dimensional matrix , composed of the coefficients in front of the first derivatives @xmath147 , @xmath105 and @xmath148 , has rank two . this means that for @xmath37 the system ( [ 11e])-([13e ] ) can be reduced to one algebraic equation and two differential equations of the first order . the corresponding algebraic equation is @xmath149 e - q = 0 \ , , \label{cubic}\ ] ] which in turn links two functions , @xmath53 and @xmath86 . now from equation ( [ cubic ] ) , one can consider three subcases , that emerge from the case @xmath37 . first we consider briefly the trivial case in this context , @xmath150 i.e. , the reissner - nordstrm limit , see subsection [ rnmodel ] . then we consider two interesting non - trivial cases : first , @xmath151 , second , @xmath152 , when @xmath151 it is easy to express @xmath86 in terms of @xmath53 . in the subsection [ gbmodel ] we consider a specific model in this class , characterized by the supplementary condition @xmath153 this is the gauss - bonnet type model , which has been considered as an important model in @xcite , and for which we present an extended analysis with relevant new details . for the second case @xmath154 @xmath53 decouples from @xmath86 and we deal with a cubic equation for the determination of the electric field . we will consider such a model , the integrable model , in subsection [ nonamemodel ] . one should first define three quantities , @xmath155 , @xmath156 , @xmath157 , as follows @xmath158 now , the models we are going to discuss here are essentially one parametric , with @xmath0 , @xmath1 , and @xmath2 being a multiple of some parameter @xmath3 . it is useful to introduce first a quantity @xmath31 , given through @xmath159 with @xmath31 being a radius . from @xmath155 , @xmath156 , and @xmath31 , one can then construct two independent dimensionless quantities , namely @xmath160 and @xmath161 the @xmath162 quantity gives the deviation from the standard reissner - nordstrm case , and @xmath163 fixes the ratio between the total mass to the total charge of the object . in addition , for what follows below , it is useful to write the equations of motion by defining two dimensionless variables , a normalized radius @xmath164 and a normalized electric field @xmath165 , defined as follows @xmath166 also , the function @xmath167 can be defined in terms of another useful function @xmath168 , i.e. , @xmath169 the physical interpretation of @xmath168 is connected with the effective mass of the object , i.e. , @xmath170 , as we will see below . with these quantities defined we now discuss the reissner - nordstrm limit and the two new models . when the non - minimal parameters @xmath0 , @xmath1 , and @xmath2 are set to zero , i.e. , @xmath171 and so @xmath172 as well , we can integrate immediately eqs . ( [ 1e])-([3e ] ) . in terms of the above functions , the system of key equations can be rewritten as @xmath173 @xmath174 @xmath175 the solutions to these equations are @xmath176 @xmath177 @xmath178 the function @xmath168 in ( [ nrn ] ) is here given by @xmath179 where @xmath163 is defined in ( [ k0 ] ) , and one can see through eq . ( [ 2eqdimrn ] ) that @xmath180 obeys the equation @xmath181 of course one can transform to the original fields @xmath53 , @xmath93 , and @xmath94 , giving @xmath182 @xmath183 @xmath184 which are the usual reissner - nordstrm functions . note , then , that the physical interpretation of @xmath168 given in ( [ yrn ] ) is connected with the effective mass of the object , i.e. , @xmath170 , which in turn is given by the definition @xmath185 thus , @xmath168 is related to the dimensionless effective mass , since @xmath186 , with @xmath187 , @xmath188 being the asymptotic mass of the object . we study in this context the usual static spacetimes with @xmath172 , i.e. , the schwarzschild and reissner - nordstrm spacetimes , which are special solutions of the full system of equations . these two solutions are heavily singular , both the metric and the kretschmann scalar diverge at @xmath108 . so these solutions are outside the spirit of the solutions we want to find . they do not serve as models . however , they are of interest to set the nomenclature , and to have a gauge with which we can compare the solutions we find in the two models studied below . note that for these solutions @xmath189 . in these cases the problem of searching for horizons is equivalent to finding their radial position @xmath190 through the solutions of @xmath191 , where @xmath170 is the effective mass . in terms of dimensionless mass @xmath168 , ( [ nyxrn ] ) , or eq . ( [ yrn ] ) , this condition can be written as @xmath192 . for the schwarzschild metric @xmath193 , where @xmath194 is defined in ( [ k0 ] ) . then , one obtains only one horizon at @xmath195 , which is just the schwarzschild radius , @xmath196 . for the reissner - nordstrm metric one has @xmath197 , and the equation @xmath198 gives three different cases : ( i ) @xmath199 ( i.e. , @xmath200 , or @xmath201 in the standard notation ) : there are two solutions @xmath202 , and @xmath203 , corresponding to the outer and inner horizons , respectively , of a usual reissner - nordstrm black hole . ( ii ) @xmath204 ( i.e. , @xmath205 , or @xmath206 ) : there is one horizon only , given by @xmath207 , corresponding to an extremal black hole . ( iii ) @xmath208 ( i.e. , @xmath209 , or @xmath210 ) : there is no solution to the equation @xmath192 . the object is a naked singularity . consider now , in eqs . ( [ action1])-([susceptibility2 ] ) , the following specific one - parameter model @xmath211 for some parameter @xmath3 . in this model the susceptibility tensor is proportional to the double - dual riemann tensor and is divergence - free @xcite , i.e. , @xmath212 , and @xmath213 moreover , the coupled einstein and electromagnetic equations are second order in the derivatives , which is the reason why this model is called a gauss - bonnet model . gauss - bonnet gravity in five and higher dimensions has the property that its equations are of second order , lovelock gravity being a generalization of it . actually , one can show that the model specified by ( [ modeladopted ] ) comes from kaluza - klein reduction to four dimensions from five a dimensional gauss - bonnet theory , i.e. , einstein gravity plus a gauss - bonnet term @xcite . using ( [ modeladopted ] ) , eqs . ( [ 11e])-([13e ] ) convert , respectively , into @xmath214 @xmath215 @xmath216^{\prime } = \frac{1}{2}\kappa e^2 \left [ r^2 + 2 q(1 - n ) \right]\ , . \label{13e1}\ ] ] after appropriate redefinitions these equations agree with the ones discussed in @xcite . the correspondingly modified algebraic eq . ( [ cubic ] ) coincides with ( [ 11e1 ] ) . we now consider two ways of analyzing this system of equations : first , we use a power series expansion , and second , we apply the formalism of dynamical systems . \(i ) the abel equation 0.1 cm using from eq . ( [ 11e1 ] ) that @xmath217\ , , \label{equane}\ ] ] one obtains the abel equation ( see , e.g. , @xcite ) for @xmath53 from ( [ 13e1 ] ) , @xmath218 similarly , eliminating @xmath53 from eq . ( [ 11e1 ] ) one can transform eq . ( [ 13e1 ] ) into the abel equation for a function @xmath219 , here defined as @xmath220 thus , eq . ( [ 13e1 ] ) is given by @xmath221 \left [ r^2 + 2q \theta(r ) \right ] = \frac{\kappa q^2}{2 } \ , . \label{equat}\ ] ] clearly , the values @xmath55 and @xmath119 are related through @xmath222 . searching for solutions @xmath86 regular at @xmath108 , we have to consider @xmath55 to be non - vanishing , @xmath140 . in such a case eq . ( [ equae ] ) yields @xmath223 . this means that @xmath3 has to be positive in these solutions . thus , although we analyze all cases , we tend to focus in models with @xmath224 . finally , a possible solution , regular at @xmath108 , should be characterized by @xmath225 . it is convenient to use the auxiliary quantities @xmath226 , @xmath227 , and @xmath162 used before ( see eqs . ( [ rqpropersaid])-([aratio ] ) ) , to write @xmath228 using eqs . ( [ 11e1])-([13e1 ] ) , plus the asymptotic conditions @xmath117 , @xmath118 , as well as the condition that the electric field is asymptotically coulombian @xmath100 , one can obtain the following formula for the asymptotic mass @xmath188 , @xmath229 \right\ } = \lim_{r\to\infty } r(1-n ) \ , . \label{equam2}\ ] ] 2.7 cm ( ii ) the solution of the abel equation for small @xmath56 0.1 cm in the vicinity of the point @xmath108 the solutions for @xmath53 , @xmath86 and @xmath87 are assumed to have a polynomial form of the type given in eq . ( [ regular ] ) . the decomposition of a regular solution @xmath53 with non - vanishing @xmath55 is @xmath230 \ , , \label{small1}\ ] ] and the corresponding @xmath86 with finite @xmath119 is given by @xmath231 for this solution the effective mass @xmath170 becomes equal to zero at @xmath108 . moreover , @xmath232 , when electric and non - minimal radii coincide , i.e. , @xmath233 , @xmath234 . 0.4 cm ( iii ) power series expansion with respect to @xmath235 0.1 cm the decomposition of the electric field yields @xmath236 \ , , \label{18}\ ] ] where the @xmath237 are defined below . infinity is a regular point for @xmath86 , thus , taking into account ( [ equam2 ] ) one obtains the following decomposition of @xmath86 @xmath238 where again , the @xmath237 are given below . the function @xmath239 is equal to one in the schwarzschild and the reissner - nordstrm cases , but not in general . when @xmath240 the logarithm of this function can be represented by the decomposition @xmath241\ , . \label{expansionofab}\end{aligned}\ ] ] the @xmath237 coefficients can be taken from eq . ( [ 17 ] ) , and starting from @xmath242 can be found by the recurrence formula @xmath243 with @xmath244 the decompositions ( [ 18])-([expansionofab ] ) are regular at @xmath245 and absolutely converge in the interval @xmath246 , where @xmath247 . note that the terms @xmath248 and @xmath249 are the schwarzschild and reissner - nordstrm terms , respectively , and that the @xmath237 for @xmath250 are the terms that give the post reissner - nordstrm behavior . note also that for @xmath52 , i.e. , the reissner - nordstrm case ( or the schwarzschild case when , further , @xmath251 ) , the function @xmath252 in ( [ expansionofab ] ) is equal to one , as it should . for @xmath253 , eqs . ( [ 18])-([expansionofab ] ) give useful asymptotic formulas , showing that , for @xmath53 and @xmath86 , the first post - reissner - nordstrm terms are of fifth order in @xmath254 , and that the decomposition for @xmath255 starts with a term of fourth order . when necessary one should convert from @xmath106 and @xmath252 to @xmath78 and @xmath77 . numerical calculations , see figs . [ k=2sqrt2gaussbonnetmodel]-[k=1gaussbonnetmodel ] , confirm that the corresponding curves tend to the corresponding horizontal asymptotes , when @xmath56 goes to infinity , i.e. , @xmath53 tends to zero , and @xmath256 and @xmath94 tend to 1 . it follows from ( [ 18 ] ) that , for positive @xmath3 , @xmath257 , the curvature coupling effects on the electric field are analogous to a dielectric medium , since the asymptotic electric field effectively decreases . for negative @xmath3 , @xmath258 , there are no solutions with @xmath86 regular at @xmath108 , and , since we are mostly interested in regular or quasi - regular solutions , we do not fully discuss this case . , @xmath6 and @xmath7 , of gravitational electrically charged objects characterized by @xmath259 , for @xmath260 ( @xmath199 ) plots ( a ) , ( b ) , ( c ) , and ( d ) depict the metric potentials @xmath261 and @xmath94 , the function @xmath262 , and the electric field @xmath263 , respectively , as functions of @xmath264 , for solutions with different values of the non - minimal quantity @xmath162 . the reissner - nordstrm black hole has @xmath172 , the curves of which are clearly shown in the plots . in this case the curves for @xmath261 and @xmath94 have two zeros representing the inner and outer horizons , and for @xmath253 they go to one , while the curve @xmath263 tends to zero , respectively , and when @xmath265 these curves tend to @xmath266 ( the electric field is in a logarithmic scale ) . for @xmath267 and @xmath268 the black holes behave quite similarly as the case @xmath172 , with two horizons , and the function @xmath263 tends to finite values as @xmath265 . for @xmath269 the curve for @xmath261 tends to a finite negative value when @xmath121 , and takes the value zero only once . on the other hand @xmath94 has two zeros one at the same point as @xmath261 , the event horizon , and the other at @xmath108 , signaling the presence of a singularity there . the function @xmath263 tends to finite values as @xmath265 . for @xmath270 the curves @xmath256 and @xmath94 have one zero , and thus one horizon only , at the same @xmath56 , and then tend to infinity as @xmath265 , an analogous behavior to the schwarzschild black hole . the function @xmath271 tends to finite values as @xmath265 . see text for more details.,width=648,height=480 ] , @xmath6 and @xmath7 , of gravitational electrically charged objects characterized by @xmath259 , for @xmath204 plots ( a ) , ( b ) , ( c ) , and ( d ) depict the metric potentials @xmath261 and @xmath94 , the function @xmath262 , and the electric field @xmath263 , respectively , as functions of @xmath264 , for solutions with different values of the non - minimal quantity @xmath162 . the extremal reissner - nordstrm black hole has @xmath172 , the curves of which are clearly shown in the plots . in this case the curves for @xmath261 and @xmath94 have one double zero representing an extremal horizon , and for @xmath253 they go to one , while the curve @xmath263 tends to zero , respectively , and when @xmath265 these curves tend to @xmath266 ( the electric field is in a logarithmic scale ) . for @xmath267 and @xmath268 the black holes behave quite similarly as the case @xmath172 , with one horizon , and the function @xmath263 tends to finite values as @xmath265 . for @xmath269 the curve for @xmath261 tends to a finite negative value when @xmath121 , and takes the value zero only once . @xmath94 has two zeros one at the same point as @xmath261 , signaling there is only one horizon , and the other at @xmath108 , signaling the presence of a singularity there . the function @xmath263 tends to finite values as @xmath265 . for @xmath270 the curves @xmath256 and @xmath94 have at the same @xmath56 , one zero , and thus one horizon only , and then tend to infinity as @xmath265 , an analogous behavior to the schwarzschild black hole . the function @xmath263 tends to finite values as @xmath265 . see text for more details.,width=648,height=475 ] , @xmath6 and @xmath7 , of gravitational electrically charged objects characterized by @xmath259 , for @xmath272 ( @xmath208 ) plots ( a ) , ( b ) , ( c ) , and ( d ) depict the metric potentials @xmath261 and @xmath94 , the function @xmath262 , and the electric field @xmath263 , respectively , as functions of @xmath264 , for solutions with different values of the non - minimal quantity @xmath162 . the reissner - nordstrm naked singularity has @xmath172 , the curves of which are clearly shown in the plots . in this case the curves for @xmath261 and @xmath94 have no zeros , and for @xmath253 they go to one , while the curve @xmath263 tends to zero , respectively , and when @xmath265 these curves tend to @xmath266 ( the electric field is in a logarithmic scale ) . for @xmath267 and @xmath268 the naked singularity behaves quite similarly as the case @xmath172 , and the function and @xmath263 tends to finite values as @xmath265 . for @xmath269 the curve for @xmath261 tends to a finite positive value as @xmath121 , while @xmath94 has a zero at @xmath108 , signaling the presence of a singularity there . the function @xmath263 tends to finite values as @xmath265 . for @xmath270 the curves @xmath256 and @xmath94 have one zero at the same @xmath56 , and thus one horizon only , and then tend to infinity as @xmath265 , an analogous behavior to the schwarzschild black hole . thus , by tuning the non - minimal quantity @xmath162 one can turn a reissner - nordstrm naked singularity , which has @xmath172 , into a black hole , when @xmath270 . the function @xmath263 tends to finite values as @xmath265 . see text for more details . , width=648,height=489 ] we now study this model , specified through eq . ( [ modeladopted ] ) , using a dynamical system analysis . 0.5 cm ( i ) first analysis : the plots and numerics 0.3 cm _ ( a ) key dynamic equation : _ in order to find the regular solutions @xmath53 , @xmath86 and @xmath87 in the whole interval @xmath273 let us transform the master eqs . ( [ 11e1 ] ) , ( [ 12e11 ] ) and ( [ 13e1 ] ) to the independent variable @xmath274 , a dimensionless radius , and to the unknown dimensionless function @xmath168 given in eqs . ( [ nrnnew ] ) and ( [ yrn ] ) , i.e. , @xmath275\ , . \label{ydef}\ ] ] the physical interpretation of @xmath168 is connected with the so - called effective mass of the object , @xmath170 , see eq . ( [ nyxrn ] ) . putting these definitions into eq . ( [ 13e1 ] ) , we obtain the following key equation @xmath276 this equation is indeed a key one , since using its solution , @xmath168 , we can represent explicitly the electric field by @xmath277 the metric function @xmath167 by eq . ( [ nyxrn ] ) , and @xmath278 by the integral form @xmath279 moreover , @xmath78 follows from @xmath280 and @xmath77 from @xmath281 , i.e. , @xmath282 \exp{\left\{2a \int^x_{\infty } \frac{d x'}{x'}\,z^2(x')\right\ } } \ , . \label{newb}\ ] ] note that the function @xmath168 is also a function of the quantity @xmath162 , so in general should be written as @xmath283 . 0.5 cm _ ( b ) three typical cases : _ in this problem there are two independent dimensionless quantities , constructed from @xmath155 , @xmath156 , and @xmath31 , namely @xmath162 and @xmath163 , see eqs . ( [ aratio ] ) and ( [ k0 ] ) . note that @xmath163 , besides fixing the ratio between the total mass and the charge of the object , gives the value of the dimensionless mass @xmath168 at @xmath245 , since @xmath284 . taking into account the quantity @xmath163 , and in conformity with the reissner - nordstrm solution , let us distinguish three different situations , ( i ) @xmath199 , ( ii ) @xmath204 , ( iii ) @xmath208 , within each situation the quantity @xmath162 can vary from zero to infinity [ k=2sqrt2gaussbonnetmodel]-[k=1gaussbonnetmodel ] display typical cases in each situation , and fig . [ yingausbonnetmodel ] shows the behavior of @xmath168 . in slightly more detail : ( i ) @xmath199 ( i.e. , @xmath200 ) : for @xmath199 , we use @xmath285 as a typical value for the numerical analysis , see the plots in fig . [ k=2sqrt2gaussbonnetmodel ] , ( see fig . [ k=2sqrt2gaussbonnetmodel]a for @xmath256 , fig . [ k=2sqrt2gaussbonnetmodel]b for @xmath94 , fig . [ k=2sqrt2gaussbonnetmodel]c for @xmath286 , and fig . [ k=2sqrt2gaussbonnetmodel]d for @xmath53 ) . when @xmath172 this case gives the usual reissner - nordstrm black hole with two horizons . for other @xmath162s there are also black holes , some with different properties . ( ii ) @xmath204 ( i.e. , @xmath205 ) : for @xmath287 see the plots in fig . [ k=2gaussbonnetmodel ] , ( see fig . [ k=2gaussbonnetmodel]a for @xmath256 , fig . [ k=2gaussbonnetmodel]b for @xmath94 , fig . [ k=2gaussbonnetmodel]c for @xmath286 , and fig . [ k=2gaussbonnetmodel]d for @xmath53 ) . when @xmath172 this case gives the extreme reissner - nordstrm black hole with one horizon . for other @xmath162s there are also interesting solutions with black holes . ( iii ) @xmath208 ( i.e. , @xmath209 ) : for @xmath208 , we use @xmath272 as a typical value for the numerical analysis , see the plots in fig . [ k=1gaussbonnetmodel ] , ( see fig . [ k=1gaussbonnetmodel]a for @xmath256 , fig . [ k=1gaussbonnetmodel]b for @xmath94 , fig . [ k=1gaussbonnetmodel]c for @xmath286 , and fig . [ k=1gaussbonnetmodel]d for @xmath53 ) . when @xmath172 this case gives a reissner - nordstrm naked singularity , a solution without horizons . for other @xmath162s there are also solutions . and @xmath283 for the non - minimal gauss - bonnet model , with @xmath288 , @xmath6 and @xmath7 , of gravitational electrically charged objects characterized by @xmath259 plots of the curves @xmath289 and @xmath283 are shown , with @xmath290 . for @xmath291 ( with @xmath292 ) the characteristic curve @xmath293 is not intercepted by the function @xmath283 . for @xmath269 the intersection takes place at @xmath294 . for @xmath268 the point of crossing floats along the characteristic curve , and the integral curve has two branches . finally , for @xmath267 there is no intersection of @xmath283 with the characteristic curve . in the plot @xmath289 it is shown explicitly the existence of a point obeying @xmath295 with positive @xmath296 . , width=648,height=453 ] 0.5 cm _ ( c ) scaling of the key equation : _ the key eq . ( [ key ] ) remains invariant after the following scale transformations : @xmath297 thus , the critical values of @xmath162 that one may eventually encounter when @xmath272 , are also critical values that one can easily find for arbitrary @xmath163 using the formula @xmath298 . 0.5 cm ( ii ) second analysis : critical properties of the family of the solutions 0.3 cm _ ( a ) about the mass function @xmath283 when @xmath294 , @xmath289 - the critical value of the quantity @xmath162 , @xmath296 : _ we now write explicitly that @xmath180 is a function of both @xmath164 and @xmath162 , @xmath299 since this is important to our analysis . plots of this dimensionless mass function @xmath283 are displayed in fig . [ yingausbonnetmodel ] . in fig . [ yingausbonnetmodel]a , @xmath283 is shown for several values of @xmath162 , and in fig . [ yingausbonnetmodel]b , a plot for @xmath289 as a function of @xmath162 is shown . a simple qualitative analysis shows that the mass function @xmath283 at the central point @xmath294 , i.e. @xmath289 , as a function of the quantity @xmath162 , has to possess a zero . indeed , when @xmath172 , @xmath300 , ( where , recall , @xmath301 ) , corresponding thus to the reissner - nordstrm solution . when @xmath302 , @xmath303 , corresponding thus to the schwarzschild solution . note as well that when @xmath304 , @xmath305 . in addition , when @xmath306 , @xmath307 , a condition at infinity that holds for arbitrary @xmath162 . in order to prove our assertion , that @xmath289 as a function of @xmath162 possesses a zero , consider then @xmath289 as a function of the quantity @xmath162 in the interval @xmath308 . one can see , that @xmath309 , and @xmath310 . supposing that @xmath289 is continuous in such an interval , one can conclude that there exists at least one specific value of the @xmath162 quantity for which @xmath311 , where @xmath296 is the value of @xmath162 for which @xmath311 . [ yingausbonnetmodel]b shows that , for @xmath272 the zero of the function @xmath312 happens , when @xmath313 . but the most interesting fact is that the curve @xmath314 displays a discontinuity in the first derivative with respect to @xmath162 just at @xmath269 . one can see explicitly a finite jump of the derivative at this point , @xmath269 . nevertheless , the function @xmath315 itself is continuous at this point . that is why @xmath296 is a critical value of the quantity @xmath162 . for other @xmath163s the critical values can be found using the scaling properties ( [ scaletransformations ] ) , yielding @xmath316 . for instance @xmath317 and , for the typical case we study , @xmath318 . for negative @xmath162 , @xmath319 asymptotically when @xmath320 . 0.5 cm _ ( b ) the critical points of the associated autonomous two - dimensional dynamical system : _ the key eq . ( [ key ] ) can be put as an autonomous dynamical system @xmath321 where @xmath322 and @xmath323 is an auxiliary parameter . in eq . ( [ autonomoussystem ] ) there is one critical point at @xmath324 in the vicinity of this critical point the variables @xmath164 and @xmath180 are connected by the relation @xmath325 , which means this point is a saddle point when @xmath162 is positive , and a center when @xmath162 is negative . if @xmath326 there are two separatrices @xmath327 . the equation for @xmath328 can also be written as dynamical , @xmath329 \ , , \label{21}\ ] ] which is much more complicated than eq . ( [ autonomoussystem ] ) for @xmath168 . nevertheless , if @xmath3 is positive ( i.e. , @xmath162 is positive ) , one can find the critical points immediately , @xmath330 in order to present the integral curves for the total interval of the auxiliary parameter @xmath323 , we resort to numerical calculations . the results are presented in figs . [ k=2sqrt2gaussbonnetmodel]-[k=1gaussbonnetmodel ] . it is clear that for the critical @xmath269 the curve for @xmath331 tends to one when @xmath112 , and takes a finite value @xmath332 at the center of the object , @xmath108 . this critical curve is a separatrix between the curves having @xmath270 and those having @xmath333 . the same type of behavior happens with the curves for @xmath94 . one also has that for @xmath334 there exists a unique integral curve for @xmath168 with asymptotic value given by @xmath335 , and for @xmath265 one has @xmath295 . again , this curve behaves as a separatrix . similar reasoning goes to the curve @xmath53 . in other words , the critical point @xmath336 , given in eq . ( [ criticalpointsy1/a ] ) , is just a saddle point , corresponding to the critical value @xmath296 , obtained numerically , and it is the final point of the unique integral curve , which coincides with the separatrix of the function @xmath337 at small @xmath164 . when @xmath258 and , thus , @xmath162 is negative , there are no regular or quasi - regular solutions to the master equations for the whole interval @xmath338 . 0.5 cm _ ( c ) vertical asymptotes and the electric barrier : _ from eq . ( [ key ] ) one sees that , when @xmath339 , the derivative @xmath340 becomes infinite and vertical asymptotes appear in the graph of @xmath168 versus @xmath164 . at the point @xmath341 , for which @xmath342 , the electric field ( [ newe ] ) becomes infinite , and so these vertical asymptotes can be interpreted in terms of an electric barrier . when @xmath267 , figs . [ k=2sqrt2gaussbonnetmodel]d , [ k=2gaussbonnetmodel]d , and [ k=1gaussbonnetmodel]d show that for all three values of @xmath163 the curve for the electric field has a form of a finite barrier . indeed for @xmath253 the electric field @xmath53 tends to zero , then at some radius it reaches a maximum , the barrier height , and finally goes to a finite positive value when @xmath265 . a charged test particle with sufficiently high energy can overcome this barrier , be trapped in the potential well and then oscillate inside . when @xmath172 , more specifically , when @xmath343 , the barrier height increases and tends towards the center @xmath265 . for @xmath172 ( see again figs . [ k=2sqrt2gaussbonnetmodel]d , [ k=2gaussbonnetmodel]d , and [ k=1gaussbonnetmodel]d ) one obtains the standard , reissner - nordstrm , behavior , @xmath344 . this means that the electric barrier has become infinite , taking its maximum value ( an infinite value ) at the center @xmath108 . in other words , the vertical asymptote for the electric field appears at @xmath108 . finally , when @xmath326 this vertical asymptote and the position of the infinite electric barrier shift from the center towards positive @xmath56 values , then stop at some value of the quantity @xmath162 , drift back again to smaller values of @xmath56 and , finally , the infinite electric barrier disappears at @xmath345 , ( with @xmath292 ) . when @xmath269 the curve tends to the horizontal asymptote at @xmath121 , there is yet no trap . when @xmath346 , one can see a finite electric barrier with the corresponding traps , for all values of @xmath163 . since the integrals in ( [ newab ] ) and ( [ newb ] ) diverge when @xmath53 is discontinuous , then in searching for solutions with regular functions @xmath53 , @xmath256 , and @xmath94 , we should reject all the cases where vertical asymptotes appear . for @xmath347 , numerical calculations show that vertical asymptotes do not appear . so we will mainly consider solutions for this range of the quantity @xmath162 , i.e. , @xmath345 . ( d ) horizons : _ to analyze the @xmath345 case , with a non - singular electric field and possible non - singular metric potentials @xmath79 and @xmath77 , we have studied previously , for comparison , the usual static spacetimes with @xmath172 , i.e. , the schwarzschild and reissner - nordstrm spacetimes , which are special solutions of eq . ( [ metric1 ] ) . one has for these that @xmath348 . for @xmath345 one sees from ( [ expansionofab ] ) that @xmath349 , in contrast to the @xmath172 case . nevertheless , as in the @xmath172 case , horizons are still given by the condition @xmath350 , or @xmath192 . we have analyzed this numerically . for @xmath270 the results are the following : ( i ) @xmath199 ( i.e. , @xmath200 ) ) : a typical case is @xmath351 , see fig . [ k=2sqrt2gaussbonnetmodel ] . one finds that for @xmath352 , there is only one horizon . ( ii ) @xmath204 ( i.e. , @xmath205 ) : for @xmath287 , see fig . [ k=2gaussbonnetmodel ] . one finds that for @xmath353 , there is one horizon also . ( iii ) @xmath208 ( i.e. , @xmath209 ) ) : a typical case is @xmath354 , see fig . [ k=1gaussbonnetmodel ] . one finds that for @xmath355 , there is only one horizon also . note that figs . [ k=2sqrt2gaussbonnetmodel]-[k=1gaussbonnetmodel ] show that when @xmath270 , for all formal possibilities ( @xmath199 , @xmath204 , @xmath208 ) the curves @xmath79 tend monotonically to minus infinity and cross the line @xmath356 only once . thus , for arbitrary @xmath270 the plots of @xmath79 are continuous , irregular at the center and characterized by one horizon . these solutions have thus an analogous behavior to the schwarzschild black hole . moreover , by tuning the non - minimal quantity @xmath162 one can turn a reissner - nordstrm naked singularity , with @xmath208 and @xmath172 , into a black hole , when @xmath270 . now , since the @xmath269 is a very special case , we discuss it in particular . 0.5 cm _ ( e ) the solution with @xmath269 : _ in the framework of the model under discussion , i.e. , when @xmath288 , @xmath6 , and @xmath7 , all the three functions , @xmath53 , @xmath256 , and @xmath94 , are regular in the interval @xmath357 if and only if @xmath358 ( recall @xmath359 ) . this means that we deal with a one - parameter family of exact solutions , the arbitrary quantity being @xmath360 , and all the curvature coupling constants , @xmath0 , @xmath1 and @xmath2 , being expressed explicitly via @xmath163 . the critical @xmath3 corresponding to @xmath296 is thus , with the help of eqs . ( [ rq])-([k0 ] ) , given by , @xmath361 . we now consider these solutions in more detail : ( i ) there are two different solutions , corresponding to the separatrices @xmath362 at small @xmath164 . the physical solution , the solution that gives the appropriate limit when @xmath363 and has no jumps on the derivative of the characteristic curve @xmath364 , is @xmath365 . the plots of @xmath256 , @xmath94 and @xmath53 are displayed in figs . [ k=2sqrt2gaussbonnetmodel]-[k=1gaussbonnetmodel ] . ( ii ) the solutions , which we discuss , are characterized by finite values at the center , and @xmath140 , @xmath366 , and @xmath367 , but @xmath225 and @xmath368 . ( iii ) for this model @xmath369 , and there are two distinct critical radii ; first , the radius for which @xmath370 ; second , the radius for which @xmath371 which signals an infinite redshift surface , and an event horizon in the case of static spacetimes , such as the ones we are treating here . ( iv ) for @xmath372 , one finds that such a solution exists when @xmath373 , i.e. , @xmath374 ; this is a necessary condition ( see , e.g. , ( [ small2 ] ) ) . moreover , within this case it is possible to have such a zero when @xmath375 , i.e. , @xmath376 or @xmath377 . for @xmath378 , one has , @xmath379 is positive and without zeros . ( v ) for the infinite redshift surface and event horizon , @xmath371 , one finds that when @xmath269 , for the critical quantity , the function @xmath94 is zero both at @xmath108 and at the radius for which @xmath372 . ( vi ) the kretschmann scalar diverges at @xmath108 , so although the metric functions are regular , spacetime is not , time stops . 0.5 cm _ ( f ) remark : _ some important aspects derived from qualitative and numerical analyses of this non - minimal model , with @xmath288 , @xmath6 and @xmath7 , can be found in @xcite . our results are in thorough concordance with this initial analysis . we have supplemented those aspects on several grounds , of which we stress briefly three novel details obtained here : ( i ) we have found a complete converging decomposition of the solution of the abel equation based on the recurrence formula ( [ 191 ] ) . this gives us not only the asymptotic decompositions for @xmath112 , but also the possibility to link the limiting formulas for @xmath265 and @xmath112 ( see ( [ 18])-([expansionofab ] ) ) . ( ii ) we have formulated and discussed the problem of the infinite electric barrier , associated with the vertical asymptote appearing when @xmath380 , thus completing physically and mathematically the analysis given in @xcite . ( iii ) we have found , first , the critical value of the non - minimal parameter @xmath3 , namely , @xmath361 , second , the scaling law of the critical parameter @xmath381 for different values of the asymptotic quantity @xmath382 , namely , @xmath383 , third , the significance of the choice for @xmath163 , namely , @xmath199 , @xmath204 , and @xmath208 , in the qualitative analysis . now , we consider the model with @xmath384 which , as we have seen , has the property that @xmath53 decouples from @xmath86 and we deal with a cubic equation for the determination of the electric field . since the basic feature of the model is that it is integrable , we call it the integrable model . for such a model the susceptibility tensor @xmath23 can be written in terms of the einstein tensor @xmath385 as follows , @xmath386 , % \label{chie } $ ] where we have put @xmath387 . thus , the model becomes one - parametric and we can introduce the dimensionfull quantities defined above , @xmath156 , @xmath157 , @xmath31 , and the dimensionless quantities @xmath162 and @xmath163 . we assume that @xmath157 inherits the sign of the charge @xmath58 and the quantity @xmath162 can be positive or negative depending on the sign of @xmath3 . then , we introduce the two dimensionless variables , the normalized radius @xmath164 and the normalized electric field @xmath165 , defined in ( [ dimssrn ] ) . in terms of these , the system of key equations can be rewritten as @xmath388 @xmath389 = 1 - 2 z + ( x^2+a ) z^2 \ , , \label{2eqdim}\ ] ] @xmath390 clearly , eq . ( [ 1eqdim ] ) for @xmath165 is the key equation for finding @xmath167 and @xmath278 from ( [ 2eqdim ] ) and ( [ 3eqdim ] ) , respectively . if instead of @xmath167 one uses @xmath168 then eq . ( [ 2eqdim ] ) gives an equation for @xmath391 of the type of eq . ( [ ydashrn ] ) or eq . ( [ key ] ) , but more complicated , which for this analysis is not very illuminating . for this model it is better to start analyzing the electric field @xmath53 , or its redefinition @xmath165 . consider now eq . ( [ 1eqdim ] ) in detail . ( [ 1eqdim ] ) is a one - parameter algebraic equation of third order for the dimensionless electric field . below we denote its solution as @xmath392 . the function @xmath392 can be generally presented by the well - known cardano formula , nevertheless we prefer to analyze qualitatively this one - parameter family of solutions . depending on the value of the quantity @xmath162 the solution @xmath392 can possess one or three real branches . the corresponding plots are presented in fig . [ electricfieldinnonamemodel ] . when @xmath172 , one obtains , as it should , the coulombian solution @xmath393 . the curves displaying @xmath392 for non - vanishing values of the quantity @xmath162 are more sophisticated . and @xmath9 , the rescaled electric field @xmath392 , @xmath394 , as a function of the scaled radius @xmath164 @xmath395 of gravitational electrically charged objects characterized by @xmath396 ( a ) displays the solution @xmath392 when the non - minimal quantity @xmath162 satisfies the inequality @xmath397 . there are three real branches of the solution @xmath392 . nevertheless , only one of them is defined on the whole interval @xmath398 . only one branch tends to the horizontal asymptote @xmath399 at @xmath363 . ( b ) displays the solution @xmath392 in the case @xmath400 . there are three real starting points @xmath401 , @xmath402 , @xmath403 for the three corresponding branches of the solution @xmath392 . nevertheless , two branches are not continuous , only the third being defined on the whole interval @xmath404 . ( c ) displays the solution @xmath392 for the important case @xmath405 , the fibonacci soliton . ( d ) displays the solution @xmath392 in the case @xmath406 . ( e ) displays the solution @xmath392 in the case @xmath172 which is a coulombian electric field . the curve is not continuous . ( f ) gives an example of the solution for negative @xmath162 with a vertical asymptote . when @xmath162 tends to zero remaining negative , the vertical asymptote shifts towards the line @xmath294 , and the solution @xmath407 converts , finally , into the coulombian solution . at large values of @xmath164 the plots of the function @xmath408 tend to the coulombian curve @xmath409 for all values of the quantity @xmath162 . , width=648,height=648 ] 0.2 cm _ ( i ) @xmath326 : _ when @xmath162 is positive , the functions @xmath392 take finite values for all values of the quantity @xmath162 , see the curves a , b , c , d in fig . [ electricfieldinnonamemodel ] . the initial values @xmath410 satisfy the cubic equation @xmath411 . there is only one real solution of this cubic equation for @xmath326 , if the discriminant @xmath412 is positive , i.e. , when @xmath413 , see box @xmath414 of fig . [ electricfieldinnonamemodel ] for details . this plot displays three real branches of the solution @xmath392 , nevertheless , only one of them is defined on the whole interval @xmath398 . two other branches are defined for @xmath415 only and contact at the point @xmath416 . only one branch tends to the horizontal asymptote @xmath399 at @xmath363 . when @xmath417 , the discriminant @xmath418 is negative or equal to zero , which guarantees that there are three real starting points , @xmath401 , @xmath402 , @xmath403 for the three corresponding branches of the solution @xmath392 , see the curves in boxes b , c , d of fig . [ electricfieldinnonamemodel ] . nevertheless , when @xmath419 , two branches of the solution @xmath392 are not continuous , only the third being defined on the whole interval @xmath420 , see box b of fig . [ electricfieldinnonamemodel ] . when @xmath421 all three branches are continuous and are defined on the whole interval @xmath422 , one of them is asymptotically coulombian . there are three horizontal lines @xmath423 , @xmath399 and @xmath424 , which yield distinct ranges for the functions @xmath425 , @xmath426 , and @xmath427 . clearly , the curve of coulombian type is in - between the separatrices @xmath399 and @xmath428 . the model with critical @xmath162 , call it @xmath296 again , is the one that has @xmath234 , i.e. , @xmath405.this model can be solved analytically , and we consider this case in detail below . finally , when @xmath162 tends to zero remaining positive , the starting points @xmath401 tend to minus infinity , and @xmath402 , @xmath403 grow infinitely . clearly , at @xmath429 the branch @xmath426 is the only branch that remains visible at finite values of @xmath164 , and is coulombian . 0.2 cm _ ( ii ) @xmath172 : _ when @xmath172 , one obtains the coulombian solution @xmath393 , see the curve in box @xmath430 of fig . [ electricfieldinnonamemodel ] . 0.2 cm _ ( iii ) @xmath267 : _ when @xmath162 is negative , the coefficient @xmath431 in the first term of eq . ( [ 1eqdim ] ) vanishes at @xmath432 and the line @xmath433 is the vertical asymptote of the graph @xmath392 , see the example of the curve for @xmath434 in box @xmath435 of fig . [ electricfieldinnonamemodel ] . when @xmath162 tends to zero remaining negative , the vertical asymptote shifts towards the line @xmath294 , and the solution @xmath407 converts , finally , into the coulombian solution . at large values of @xmath164 the plots of the function @xmath408 tend to the coulombian curve @xmath409 for all values of the quantity @xmath162 . thus , for @xmath436 the solutions @xmath392 are not regular at all in the range @xmath398 . to study the gravitational part of the solution , we observe that the solution of ( [ 2eqdim ] ) for @xmath106 can be presented in quadratures @xmath437 \right\ } \ , , \label{31eqdim}\end{aligned}\ ] ] where @xmath392 is supposed to be already found . the constant of integration , @xmath163 , can clearly be related to the asymptotic mass of the object @xmath188 . when @xmath438 and @xmath439 , eq . ( [ 31eqdim ] ) yields that @xmath440 , as defined in ( [ k0 ] ) . on the other hand , searching for a solution @xmath441 , which is finite at @xmath294 , and taking into account that @xmath410 is finite , we should require that @xmath442 \ , . \label{kzreq}\ ] ] in this case the formula ( [ 31eqdim ] ) transforms into @xmath443 \ , , \label{431eqdim}\end{aligned}\ ] ] providing @xmath444 since the electric field at the center satisfies the condition @xmath445 ( see ( [ 1eqdim ] ) ) , then @xmath446 can be rewritten as @xmath447 in accordance with the first relation from ( [ regularity1 ] ) . thus , for the family of solutions with regular functions @xmath448 and @xmath449 the quantity @xmath163 is a function of the value @xmath410 , i.e. , depends on the quantity @xmath162 according to the formula ( [ kzreq ] ) , @xmath450 . also , from ( [ 3eqdim ] ) for @xmath252 one finds @xmath451 \right\ } \ , , \label{21eqdim}\ ] ] with @xmath392 being found from ( [ 1eqdim ] ) . for the asymptotically coulombian branch we have to set @xmath452 and thus @xmath453 . then , the function @xmath79 is taken from @xmath454 and the function @xmath77 is taken from @xmath78 and @xmath252 , @xmath281 . , @xmath8 and @xmath9 , of gravitational electrically charged objects characterized by @xmath396 plots ( a ) , ( b ) , ( c ) , and ( d ) depict the functions @xmath455 , @xmath77 , @xmath456 and @xmath457 , respectively , as functions of @xmath264 , for three typical values of the quantity @xmath162 , when @xmath458 . these functions are regular and take finite values at @xmath108 . the solution with @xmath459 is a solution without horizons , since @xmath455 and @xmath77 are positive everywhere . indeed the @xmath405 solution is a soliton of the model , the fibonacci soliton . it has a mild conical singularity at the center , and is a solution without horizons.,width=648,height=486 ] in the reissner - nordstrm solution and in the previous discussed model , i.e. , the gauss - bonnet model ( see sections [ rnmodel ] and [ gbmodel ] , respectively ) , we divided the solutions according to @xmath199 , @xmath204 , and @xmath208 . here it is no more convenient to make such a division . the reason is that @xmath163 is not a free quantity here , rather @xmath450 . since it is not a free quantity , we can not classify the models with respect to it , we can only calculate this quantity ( numerically ) after solving the problem as a whole . in the gauss - bonnet model of section [ gbmodel ] we could classify through @xmath163 because the equations for @xmath165 and @xmath167 can not be decoupled , thus , @xmath410 and @xmath446 are connected . this means that we can choose @xmath163 as a convenient quantity for the classification , the dependent quantity being @xmath410 . here , in this model , the solution for @xmath165 satisfies a decoupled equation , the latter does not depend on @xmath167 . thus solving the decoupled equation for @xmath392 , we can classify the quantity @xmath410 as the independent one . then , we obtain @xmath441 and , as we see , @xmath446 depends on @xmath460 , an so @xmath163 is a dependent quantity , @xmath450 . an explicit example where @xmath461 is calculated is given below for the case @xmath405 . so , as when discussing the electric field , we again divide the analysis into three cases , here with subcases . 0.2 cm _ ( i ) @xmath326 : _ we should further divide into three subcases : @xmath462 : the electric field is discontinuous or irregular at the center . since in this work we focus on regular electric fields everywhere , although interesting , we do not discuss these models here . @xmath234 , i.e , @xmath405 : we plot in fig . [ metricinnonnamemodel ] , boxes ( a ) , ( b ) , ( c ) , ( d ) , the functions @xmath79 , @xmath77 , @xmath463 , and @xmath457 , respectively . from the figure it is clear that @xmath464 is characterized by the absence of horizons . for @xmath405 the solution is regular , or better , quasi - regular , and is a soliton of the theory , the fibonacci soliton . due to its interest , the case @xmath465 will be solved next explicitly . @xmath466 : this case is interesting . in this case we plot in fig . [ metricinnonnamemodel ] , boxes ( a ) , ( b ) , ( c ) , ( d ) , the functions @xmath79 , @xmath77 , @xmath463 , and @xmath457 , respectively , for two positive values of @xmath162 within this range , namely , @xmath467 . from the figure it is clear that positive @xmath162 in this range gives one horizon , and the solution possesses a quasi - regular center . since the singularity at the center is a conical one , these black hole solutions can be considered as quasi - regular solutions , and thus are of great interest . note that extremal black holes have only one zero , which in turn is a double zero . so from the figure above , the non - minimal black holes do not characterize as extremal . rather , they are of the schwarzschild type , with one horizon , a spacelike singularity , but here , different from schwarzschild , the singularity is mild , it is a conical singularity . 0.2 cm _ ( ii ) @xmath468 : _ it is the reissner - nordstrm case . for this case the electric field is irregular at the center . 0.2 cm _ ( iii ) @xmath469 : _ for these cases the electric field is discontinuous or irregular at the center , like the reissner - nordstrm case , and we do not discuss these models here . let @xmath405 . then the cubic equation ( [ 1eqdim ] ) takes the form @xmath470 = 0 \ , . \label{3to2}\ ] ] one sees that eq . ( [ 3to2 ] ) splits into one linear equation and one quadratic equation . one branch of solutions of ( [ 3to2 ] ) , the linear one , describes a constant electric field @xmath471 , or , equivalently , @xmath472 . this branch is of no great interest . another branch @xmath473 is given by the function @xmath474 $ ] , which is bounded . the graph of this function starts from @xmath475 and tends asymptotically to the line @xmath476 . the behavior of such electric field is not of coulombian type , and will be not discussed further . yet , there is a third branch . the branch @xmath477 given by the formula @xmath478 \ , , \label{z3}\ ] ] describes a coulombian type electric field . at @xmath363 , one has @xmath479 , or equivalently , @xmath57 . the graph of this function starts from @xmath480 . interesting to note that the starting points @xmath481 are associated to the well - known fibonacci series and the `` golden section '' @xmath482 . for the coulombian type solution , the function @xmath483 , which we write simply as @xmath167 when suitable , is regular at the center only if the constant @xmath163 satisfies ( [ kzreq ] ) . the corresponding quadrature for @xmath167 is @xmath484 \ , , \label{nsi}\ ] ] where @xmath278 is given by ( [ si ] ) . clearly , @xmath485 and @xmath486 , so that @xmath487 , and the relations ( [ regularity1 ] ) are satisfied . the plot of the function @xmath167 for @xmath405 is shown in fig . [ metricinnonnamemodel ] . clearly , the function @xmath167 is positive in the interval @xmath488 . for the coulombian type solution ( [ z3 ] ) the function @xmath489 , which we write simply as @xmath278 when suitable , is given by @xmath490 with @xmath491 being equal to @xmath492 . then one finds @xmath79 from @xmath454 and @xmath493 . the function @xmath494 is also positive , and @xmath495 . thus , this solution is a solution without horizons and is regular . although the curvature scalars diverge , the singularity at the center is a mild one , it is a conical singularity . the asymptotic mass @xmath188 of the object defined as @xmath496 \right\ } \ , , \label{mass1}\ ] ] is represented in this case by the integral @xmath497 \ , . \label{mass2}\ ] ] numerical calculations give the value @xmath498 which yields in addition , @xmath499 this is thus a very interesting solution . it is a soliton , in the sense that is made of the very own fields of the theory , the gravitational and electric fields , it is a solution without horizons , and it is quasi - regular , with a conical singularity at the center . note also that the value @xmath405 can be regarded as a critical one . there are two reasons for this . first , it is clear from fig . [ electricfieldinnonamemodel ] , that for the unique case @xmath405 there exists a bifurcation point , in which the two branches of the curve @xmath392 , or @xmath449 , intersect . when @xmath500 , the coulombian branch of the electric field curve is discontinuous . when @xmath501 , three continuous regular branches exist . the second reason is that at @xmath405 the corresponding curve on fig . [ metricinnonnamemodel ] plays a role of a separatrix ; when @xmath502 , desirable curves , continuous and regular at the center , exist , otherwise they do not appear . in fig . [ table ] we present a table in which the results of the various models studied are summarized . we have shown that the original non - minimal einstein - maxwell theory with three parameters @xmath0 , @xmath1 , and @xmath2 , is reducible in natural different ways to a theory with one parameter @xmath3 only , in which the three parameters obey two relations between themselves . we have then studied two special models for static spherically symmetric solutions obeying the following requirements : the electric field @xmath53 is regular everywhere in the interval @xmath503 , being coulombian far from the center . from the solutions of this class we extract the ones , for which the metric coefficients @xmath256 and @xmath94 are regular at the center @xmath108 and tend to one asymptotically as @xmath504 . the first non - minimal model , the gauss - bonnet model ( with @xmath5 , @xmath6 , @xmath7 , @xmath3 free ) , displays charged black hole solutions with one horizon only , when the dimensionless non - minimal quantity @xmath162 , with @xmath259 naturally appearing in the model , exceeds a critical value @xmath296 , @xmath270 . although the black hole is electrically charged the solutions found have one horizon only , and are similar in this connection to the schwarzschild solution . when @xmath333 , the solutions are discontinuous in the interval @xmath503 , or irregular at the center @xmath108 . another main result in this model is that there exists a unique solution , the solution for the critical value @xmath505 , which does not possess horizons and is characterized by regular fields @xmath53 , @xmath261 and @xmath94 , with @xmath506 and @xmath507 , although the curvature invariants blow at the origin . the second model , the integrable model ( with @xmath508 , @xmath8 , @xmath9 , @xmath3 free ) , is also characterized by one critical value @xmath509 of the non - minimal quantity @xmath162 . when @xmath267 or @xmath500 , the solutions are irregular . when @xmath502 one obtains black holes with electric field regular everywhere and with only one horizon , like the schwarzschild solution . finally , when @xmath405 , i.e. , at the critical value of the quantity @xmath162 , there exists a solitonic solution with a conical singularity at the center , but otherwise well behaved this solution can be called the fibonacci soliton , since the well - known @xmath510 number ( @xmath511 ) , associated with the golden section , appears naturally in the expressions for the central values of the electric field @xmath512 and the metric coefficients are also related to @xmath510 , namely @xmath513 and @xmath514 $ ] . summing up , we can say that the non - minimal curvature induced interaction between the gravitational and electromagnetic fields provides an electric field , of static spherically symmetric charged objects , which is regular everywhere for different relations between the coupling constants @xmath0 , @xmath1 and @xmath2 . as for additional regularity of the metric coefficients @xmath94 and @xmath93 , the non - minimal interaction can provide models which have very specific , critical , values for the coupling constants , in which the geometry has at most a conical , and thus mild , singularity . this is in line with problem posed by bardeen @xcite , where one should look for theories with regular black hole solutions . we have partially solved it within these models . ab thanks the hospitality of centra / ist in lisbon , and a special grant from fct to invited scientists . this work was partially funded by fundao para a cincia e tecnologia ( fct ) - portugal , through project poci / fp/63943/2005 , and by russian foundation for basic research through project rfbr 08 - 02 - 00325-a .
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using a lagrangian formalism , a three - parameter non - minimal einstein - maxwell theory is established .
the three parameters , @xmath0 , @xmath1 and @xmath2 , characterize the cross - terms in the lagrangian , between the maxwell field and terms linear in the ricci scalar , ricci tensor , and riemann tensor , respectively .
static spherically symmetric equations are set up , and the three parameters are interrelated and chosen so that effectively the system reduces to a one parameter only , @xmath3 . specific black hole and other type of one - parameter solutions are studied .
first , as a preparation , the reissner - nordstrm solution , with @xmath4 , is displayed .
then , we seek for solutions in which the electric field is regular everywhere as well as asymptotically coulombian , and the metric potentials are regular at the center as well as asymptotically flat . in this context , the one - parameter model with @xmath5 , @xmath6 , @xmath7 , called the gauss - bonnet model , is analyzed in detail .
the study is done through the solution of the abel equation ( the key equation ) , and the dynamical system associated with the model .
there is extra focus on an exact solution of the model and its critical properties . finally ,
an exactly integrable one - parameter model , with @xmath5 , @xmath8 , @xmath9 , is considered also in detail .
a special sub - model , in which the fibonacci number appears naturally , of this one - parameter model is shown , and the corresponding exact solution is presented .
interestingly enough , it is a soliton of the theory , the fibonacci soliton , without horizons and with a mild conical singularity at the center .
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coherent control of atomic and molecular dynamics using optical fields has attracted much attention , both theoretically and experimentally @xcite . thus far , most theoretical work has focused on the idealized case of isolated systems , where loss of quantum phase information due to decoherence , i.e. coupling to the environment , is ignored . such effects are , however , crucial to control in realistic systems , since loss of phase information results in loss of control . for this reason efforts to understand control in external environments @xcite @xcite and to compensate for the resultant decoherence ( e.g. , @xcite@xcite ) are of great interest . there exist a number of basic interference schemes@xcite that embody the essence of coherent control . one is the @xmath0 vs. @xmath1 photon scenario where control results from interference between state excitation using @xmath0 and @xmath1 photons simultaneously . in this letter we provide an analytic solution for control in the two - level @xmath0 vs. @xmath1 photon control scenario in the presence of decoherence . for simplicity , we examine the 1 vs. 3 photon case , although the solutions obtained below apply equally well to the @xmath0 vs. @xmath1 photon case , with obvious changes in the input rabi frequencies and relative laser phases . in 1 vs. 3 photon control@xcite a continuous wave electromagnetic field composed of a superposition of a fundamental and third harmonic wave is incident on a system . by varying the relative phase and amplitude of the fundamental and the third harmonic one can alter the population of the state excited by the incident field . clearly , decoherence can be expected to diminish the 1 vs. 3 photon induced interference , and hence the control over excitation . although extensive theoretical @xcite - @xcite and experimental @xcite - @xcite studies have been carried out on the 1 vs. 3 photon coherent control scenario , there has been no serious examination of the stability of this control scheme in an external environment , barring a derivation of a simple analytical expression for the autoionization of a two - level atomic system for weak laser intensities , using the rate approximation @xcite . amongst the various possible influences of an environment on a system we focus on the loss of phase coherence , that is , dephasing . dephasing is expected to occur on a time scale more relevant to control , since the duration of control field can be on the order of a picosecond or less , wheras the typical time scale for energy transfer is considerably longer @xcite . in this paper we show that the 1 vs. 3 photon phase control scenario ( which controls the population ) in a two - level system , when coupled to an environment , reduces to the analytically soluble monochromatic field case , but with an effective rabi frequency that is determined by the relative phase and amplitudes of the two fields . sample results for control as a function of relative laser phase in the presence of dephasing are then provided . the possiblity of solving the off - resonance case is also noted . consider a two - level bound system interacting with an continuous wave ( cw ) electromagnetic field and assume that the energy levels undergo random stark shifts without a change of state during collisions with an external bath , e.g. , elastic collisions between atoms in a gas . the cw field @xmath2 is treated classically , and the ground and the excited energy eigenstates states , of energy @xmath3 and @xmath4 are denoted @xmath5 and @xmath6 , respectively . in general , the system density operator @xmath7 obeys the liouville equation , @xmath8-{\mr}\rho . \label{liouville12l}\end{aligned}\ ] ] here @xmath9 , where the free atomic hamiltonian term is @xmath10 and the atom - field interaction term within the dipole approximation is @xmath11\ ] ] with electric dipole operator @xmath12 . the second term in eq . ( [ liouville12l ] ) , @xmath13 , is a dissipative term that can have a variety of nonequivalent forms associated with various master equations . below we assume simple exponential dephasing of the off - diagonal @xmath14 . in the simplest 1 vs. 3 control scenario , a two - level system is subject to the linearly polarized laser field : @xmath15 , \label{efield2l } \end{aligned}\ ] ] where @xmath16 is the real time - independent amplitude and @xmath17 is the phase of the corresponding field , with @xmath18 . here the subscripts @xmath19 denotes the fundamental and its third harmonic , and @xmath20 denotes the complex conjugate of the terms that precede it . the fields have frequencies @xmath21 and @xmath22 , chosen so that the third - harmonic and the three fundamental photons are on resonance with the transition from the ground state @xmath5 to the excited state @xmath6 . in the standard scenario @xcite , control is obtained by changing the relative phase and amplitudes of two fields , which results in the alteration of the degree of interference between the two pathways to the excited state . within the rotating - wave approximation , the slowly varying density - matrix elements of the states @xmath5 and @xmath6 , @xmath23 , ( @xmath24 ) and @xmath25 obey the following set of equations : @xmath26\nonumber\\ \pd { \sigma_{22}}{t}&= & { \mrm{im } } [ ( { \mu}_{12}^{(3)}{\mathcal{e}}_f ^3 /\hbar+ { \mu}_{12}{\mathcal{e}}_h e^{i\phi } /\hbar)\sigma_{21 } ] , \label{density_matrix112l}\nonumber\\ \pd { \sigma_{21}}{t}&=&-\gamma_{{p } } \sigma_{21 } + \frac{i}{2 } ( { \mu}_{21}^{(3)}{\mathcal{e}}_f ^3 /\hbar+ { \mu}_{21}{\mathcal{e}}_h e^{-i\phi } /\hbar)(\sigma_{11}-\sigma_{22 } ) , \label{density_matrix212l}\end{aligned}\ ] ] with @xmath27 here @xmath28 is the dephasing rate , @xmath29 is the frequency difference between levels @xmath30 and @xmath31 and @xmath32 . the quantities @xmath33 and @xmath34 denote the one - photon matrix element for the harmonic field and the effective three - photon matrix element for the fundamental field for the @xmath35 transition . below , we use @xmath36 and @xmath37 , omitting the subscripts for simplicity . the controllable relative phase is @xmath38 . it is convenient to define the one- and three - photon rabi frequencies by @xmath39 and @xmath40 given in terms of their amplitudes and phases , by @xmath41 and @xmath42 . note that , although @xmath43 and @xmath44 are real for a bound system , we derive all the equations under the assumption that they can be complex so that the analysis can be extended to complex matrix elements arising in transitions to the continuum . since @xmath45 and @xmath46 are real and positive , @xmath47 and @xmath48 are determined by @xmath43 and @xmath44 : @xmath49 to amalgamate these rabi frequencies and the relative laser phase of @xmath50 , we define the effective rabi frequency @xmath51 : @xmath52 where @xmath51 is real and positive . here @xmath51 and @xmath53 are related to @xmath54 and @xmath55 as @xmath56/ [ { \cos{(\phi+\theta_h)}+\frac{|\omega_f|}{|\omega_h|}\cos{\theta_f}}],\end{aligned}\ ] ] where @xmath57 it is worth noting some features of @xmath51 that are evident from eq . ( [ omegaeff3 ] ) . first , the total excitation probability obtained in lowest order perturbation theory for 1 vs. 3 photon phase control in a two - level system @xcite is proportional to @xmath51 . hence , @xmath51 can be used to predict the controlled population , and its dependence on @xmath50 , when the fields are weak . further ( see below ) , @xmath51 plays a major role in determining the transient behavior of the excited state population for any field intensity in the absence or presence of dephasing . second , the interference term in eq . ( [ omegaeff3 ] ) can be controlled by varying @xmath58 , that is , by manipulating the relative phase @xmath50 of the two fields . since @xmath43 and @xmath44 are real in a bound system , possible values of @xmath59 are 0 and @xmath60 . when @xmath59=0 , , @xmath61 , @xmath62 . on the other hand , when @xmath63 , , @xmath64 , @xmath65 . thus , opposite interference effects are observed depending on the signs of @xmath43 and @xmath44 . third , @xmath66 . if @xmath67 , the smallest @xmath51 is not zero , and thus complete destructive interference , that is , zero excitation from the ground to the excited state , does not occur . rewriting eq . ( [ density_matrix212l ] ) in terms of @xmath51 and @xmath53 , @xmath68 \\ \pd { \sigma_{22}}{t}&= & { \mrm{im } } [ \omega_{\rm{eff } } e^{i\theta } \sigma_{21 } ] , \label{redensity_matrix112l}\\ \pd { \sigma_{21}}{t}&=&-\gamma_{{p } } \sigma_{21 } + \frac{i}{2 } [ { \omega_{\rm{eff}}}e^{-i\theta } ] ( \sigma_{11}-\sigma_{22 } ) . \label{redensity_matrix212l}\end{aligned}\ ] ] and introducing @xmath69 , @xmath70 , and @xmath71 , gives @xmath72 the resultant equations are now of standard form@xcite , but with @xmath51 replacing the rabi frequency of the single field case discussed in ref @xcite . note that the longtime steady - state solution to eqs . ( [ cut ] ) to ( [ cwt ] ) is found by setting @xmath730 , giving @xmath740 . this implies that , regardless of initial conditions and for sufficiently large time , pure dephasing leads to an equilibrium state with equal populations in the ground and the excited states and with no remaining coherence . substituting eq . ( [ cvt ] ) into eq . ( [ cwt ] ) gives a simple equation for @xmath75 : @xmath76 in the important case where initially the ground state is populated and the coherence is zero [ i.e. , @xmath77 , @xmath78 , the excited state population @xmath79 is given by @xmath80+\frac{1}{2}\,\ , & { \mrm{for } } & \gamma_{\mrm{p } } < 2\omega_{\rm{eff } } , \label{oscillation } \\ \rho_{22}=\frac{[-\lambda_2 e^{\lambda_1 t}+\lambda_1 e^{\lambda_2 t } ] } { 2(\lambda_2-\lambda_1)}+\frac{1}{2}\,\,\,\,\,\,\,\,\,\,\,\ , & { \mrm{for } } & \gamma_{{p } } > 2\omega_{\rm{eff } } , \label{mono}\\ \rho_{22}=-\frac{e^{-\frac{\gamma_{{p } } t}{2}}}{2}(1+\frac{\gamma_{{p}}t}{2})+\frac{1}{2}\,\,\,\,\,\,\,\,\,\,\,\ , & { \mrm{for } } & \gamma_{{p}}= 2\omega_{\rm{eff } } , \label{expo}\end{aligned}\ ] ] where @xmath81 , and @xmath82 $ ] . the general behavior of the solution is seen to be determined by relative size of the dephasing time and the period of the rabi oscillation . analogous analytic results can be obtained for @xmath83 which decays with rate @xmath84 . if the external field is intense enough so that @xmath85 , then @xmath86 shows oscillations that are exponentially damped with time . on the other hand , if dephasing dominates over the rabi oscillation , so that @xmath87 or @xmath88 , @xmath89 increases monotonically . however , in all cases @xmath86 reaches a stationary value of 0.5 at long times and @xmath90 for short times . the behavior of the excited state population for several values of @xmath91 for a given value of @xmath51 ( here chosen as 2@xmath92 ) is sketched in fig . [ 2lcwanal ] . for @xmath93 , the introduction of dephasing increases the period of the oscillation and causes the amplitudes to decay as @xmath94 . although this is a cw laser field case , we can extract the result for the field being switched off at a specific time , i.e. , a square pulsed laser which is on from @xmath95 to @xmath96 , by examining the population at time @xmath97 . ( this assumes that there are no additional energy levels excited by the frequency breadth of the truncated cw source ) . significantly , one can end up with an increased @xmath89 even for a larger dephasing , depending on the pulse duration . for example , assume that we turn off the field at @xmath98=1 . if there is no dephasing , then the excited state population at @xmath98 = 1 is 0 and is thus less than that of any of the other cases with dephasing . on the other hand , for @xmath99 ( here @xmath100 ) , there is no oscillation ; @xmath89 just increases monotonically towards 0.5 , where the system reaches the steady - state slower with increasing dephasing . if we were to consider a pulse rather than a cw laser field for this relatively strong dephasing case , the excited state population would be expected to increase up to 0.5 with the increase in the pulse duration . typical behavior of @xmath89 and of the 1 vs. 3 photon phase control profile ( i.e. , @xmath89 as a function of generic phase control variable @xmath58 ) for several values of rabi frequencies and @xmath91 are shown in fig . [ control2lt ] . here we assume that the fields are abruptly turned off at the times indicated in the figure captions to produce a square pulse and the intensities are chosen so that @xmath101 , to enhance the interference effects . the effective rabi frequency is then @xmath102 . while @xmath103 leads to a complete constructive interference of the two transition amplitudes , @xmath104 leads to a complete destructive interference , , no excitation from the ground to the excited state . the typical control behavior seen in fig . [ control2lt ] depends upon the pulse duration , as well as upon @xmath51 and @xmath105 . for example , when the field is weak and @xmath106 , then @xmath89 is given by [ eq ( [ oscillation])-([expo ] ) ] @xmath107~~{\mrm{for}}~~\gamma_{{p } } = 0 \label{precos}\ ] ] @xmath108 hence , the system shows a @xmath109 rule " . [ a similar rule obtains from eq . ( [ oscillation ] ) - eq . ( [ expo ] ) when @xmath110 and @xmath111 , when @xmath112 and @xmath113 and @xmath114 are much less than one , and for @xmath115 when @xmath116 . ] note also that eq . ( [ precos ] ) predicts oscillatory behavior of @xmath89 as a function of @xmath51 at fixed @xmath98 , as observed later below . the control profiles for small @xmath54 and @xmath117 ( thin dashed lines in fig . [ control2lt ] ) are then seen to be monotonically decreasing from the maximum excitation at @xmath103 , to zero excitation at @xmath104 , i.e. they follow the @xmath109 rule " . by contrast , for strong intensity ( thin solid lines in fig . [ control2lt ] ) in which there are many rabi cycles during the pulse , the control curve is not necessarily monotonic since the final excited populations are determined by the time at which the fields are turned off . introducing dephasing is seen to lead to a decreased range of control whose magnitude depends on the relative strength of the dephasing and on the effective rabi frequencies , according to eqs . ( [ oscillation ] ) - ( [ expo ] ) . figure [ control2lt ] demonstrates that phase control profiles are strongly dependent on the pulse duration . for weak intensities , as the pulse duration increases , the degree of control improves and the control curve continues to approximately follow a @xmath118 law ( e.g. , eq . ( [ cos ] ) ) . this behavior is seen both in the absence and in the presence of dephasing , although dephasing reduces the yield for a given pulse duration . in the strong field case the control profile varies strongly with pulse duration . in particular , with @xmath119 , if the pulse duration is smaller than the oscillation period ( = 1/2 ) of the @xmath120 case , then @xmath89 decreases with increasing @xmath58 , as shown in fig . [ control2lt](a ) . for the pulse duration greater than that period , the control profile no longer follows @xmath118 and the maximal yields start to appear at @xmath121 , as shown in figs . [ control2lt](b ) to ( d ) . in all strong intensity cases , the addition of dephasing results in a decay of @xmath89 with a rate of @xmath122 for a given @xmath51 . thus the degree of the control worsens in the presence of dephasing as the pulse duration increases . note that the introduction of dephasing leads to a degree of control @xmath123 that converges to 0.5 , where @xmath123 is defined as the difference between the maximum and minimum excited state populations . finally , we note that this treatment can be extended in two directions . the most obvious is to extend it to the general two - level @xmath0 photon + @xmath1 photon interference scenario@xcite where the structure of the problem is exactly the same as that of the 1 vs. 3 photon case . the equations above therefore hold , but with the one and three photon rabi frequencies and phases replaced by the @xmath0 and @xmath1 photon rabi frequencies and phases . the second is to consider the more general case that includes the equal detuning of both fields from the @xmath124 transition , i.e. @xmath125 , and where the populations of levels @xmath6 and @xmath5 decay with the same rate @xmath126 . ( [ redensity_matrix112l ] ) to ( [ redensity_matrix212l ] ) become : @xmath127 , \label{gredensity_matrix112l}\\ \pd { \sigma_{22}}{t}&= & -\gamma_d(\sigma_{22}-\sigma_{2e } ) + { \mrm{im } } [ \omega_{\rm{eff}}e^{i\theta } \sigma_{21 } ] , \label{gredensity_matrix222l}\\ \pd { \sigma_{21}}{t}&= & -(\gamma_d+\gamma_{{p}}+i\delta)\sigma_{21 } + \frac{i}{2 } [ { \omega_{\rm{eff}}}e^{-i\theta } ] ( \sigma_{11}-\sigma_{22 } ) . \label{gredensity_matrix212l}\end{aligned}\ ] ] here @xmath128 and @xmath129 are the steady - state values of @xmath130 and @xmath131 , respectively , when @xmath132 , and are introduced to allow for relaxation to equilibrium . in terms of @xmath133 , @xmath134 , and @xmath75 , the above equations lead to @xmath135 where @xmath136 , @xmath137 and @xmath138 . note that these are then of the same form as the usual bloch equations for a monochromatic field . torrey gave detailed analytical solutions for these equations in the monochromatic field case @xcite and the same analytical solutions for the 1 vs. 3 photon phase control case can be used , where the single field @xmath139 considered by torrey is replaced by @xmath51 . we do not pursue this direction in this letter . in summary , we have obtained an analytic solution for @xmath0 vs. @xmath1 photon phase control of a two - level system in an environment described by a @xmath140 dephasing time , with @xmath141 and @xmath142 as a specific example . the results should serve as a prototype for understanding the results of @xmath0 vs. @xmath1 photon phase control in more complicated systems , such as controlled xenon ionization and ibr photodissociation@xcite . 100 p. brumer and m. shapiro , _ principles of the quantum control of molecular processes _ ( wiley , new york , 2003 ) . s. a. rice and m. zhao , _ optical control of molecular dynamics _ ( wiley , new york , 2000 ) . h. rabitz , r. de vivie - riedle , m. motzkus , and k. kompa , science * 288 * , 824 ( 2000 ) . j. cao , c. j. bardeen , and k. r. wilson , j. chem . phys . * 113 * , 1898 ( 2000 ) . m. demirplak and s. a. rice , j. chem . * 116 * , 8028 ( 2002 ) . j. gong and s. a. rice , j. chem . phys . * 120 * , 3777 ( 2004 ) . b. d. fainberg and v. a. gorbunov , j. chem . phys . * 117 * , 7222 ( 2002 ) . m. a. nielsen and i. l. chuang , _ quantum computation and quantum information _ ( cambridge university press , cambridge , 2000 ) . m. shapiro and p. brumer , , 052308 ( 2002 ) . p. zanardi and m. rasetti , , 3306 ( 1997 ) . l. duan and g. guo , , 737 ( 1998 ) . d. a. lidar , d. bacon , and k. b. whaley , , 4556 ( 1999 ) . j. i. cirac and p. zoller , , 4091 ( 1995 ) . c. j. myatt , b. e. king , q. a. turchette , c. a. sackett , d. kielpinski , w. m. itano , c. monroe , and d. j. wineland , nature * 403 * , 269 ( 2000 ) . d. j. tannor , r. kosloff , and a. bartana , faraday discuss . * 113 * , 365 ( 1999 ) and references are therein . m. shapiro , j. w. hepburn , p. brumer , chem . . lett . * 149 * , 451 ( 1988 ) c. k. chan , p. brumer , and m. shapiro , j. chem . phys . * 94 * , 2688 ( 1991 ) . d. petrosyan and p. lambropoulos , phys . lett . * 85 * , 1843 ( 2000 ) . d. petrosyan and p. lambropoulos , phys . a * 63 * , 043417 ( 2001 ) . j. c. camparo and p. lambropoulos , , 552 ( 1997 ) . c. chen , y - y . yin and d.s . elliott , phys . . lett . * 64 * , 507 ( 1990 ) . c. chen , y - y . yin and d.s . elliott , phys . lett . * 65 * 1737 , ( 1990 ) . x. wang , r. bersohn , k. takahashi , m. kawasaki , and h.l . kim , j. chem . * 105 * , 2992 ( 1996 ) . karapanagioti , d. xenakis , d. charalambidis and c. fotakis , j. phys . b * 29 * , 3599 ( 1996 ) . for a review see : r. j. gordon , l. zhu , and t. seideman , acc . 32 * , 1007 ( 1999 ) . l. k. iwaki and d. l. dlott , j. phys . a. * 104 * , 9101 ( 2000 ) . h. graener , r. zrl and m. hofmann , j. phys . b. * 101 * , 1745 ( 1997 ) . h. c. torrey , phys . rev . * 76 * , 1059 ( 1949 ) . l. allen and j. h. eberly , _ optical resonance and two - level atoms _ ( wiley , new york , 1975 ) . h. han and p. brumer ( manuscript in preparation ) . h. han , ph.d . dissertation , university of toronto , 2004 . excited state population as a function of time for various dephasing rates , @xmath91 shown inside the box for @xmath143 . all the variables are in dimensionless units . data points are connected by straight lines as a guide . excited state population versus relative phase for two different @xmath144 : solid lines and dashed lines denote the case at @xmath145 and @xmath146 , respectively . thin lines and thick lines denote the case at @xmath147 and @xmath148 , respectively . fields are turned off at ( a ) @xmath149 , ( b ) @xmath150 , ( c ) @xmath151 , and ( d ) @xmath152 . note that the data in panel ( d ) is too widely spaced to produce the last @xmath153 maxima , whose exact locations can be predicted from eq . ( [ cos ] ) . all the variables are in dimensionless units . data points are connected by straight lines as a guide .
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decoherence effects on the traditional @xmath0 vs. @xmath1 photon coherent control of a two - level system are investigated , with 1 vs. 3 used as a specific example .
the problem reduces to that of a two - level system interacting with a single mode field , but with an effective rabi frequency that depends upon the fundamental and third harmonic fields .
the resultant analytic control solution is explored for a variety of parameters , with emphasis on the dependence of control on the relative phase of the lasers .
the generalization to off - resonant cases is noted .
| 7,803 | 148 |
neural networks ( nn s ) have become in the last years a very effective instrument for solving many difficult problems in the field of signal processing due to their properties like non - linear dynamics , adaptability , self - organization and high speed computational capability ( see for example @xcite and the papers therein quoted ) . aim of this paper is to show the feasibility of the use of nn s to solve difficult problems of signal processing regarding the so called virgo project . gravitational waves ( gw s ) are travelling perturbations of the space - time predicted by the theory of general relativity , emitted when massive systems are accelerated . up to now , there is only an indirect evidence of their existence , obtained by the observations of the binary pulsar system psr 1913 + 16 . moreover , the direct detection of gw s is not only a relevant test of general relativity , but the start of a new picture of the universe . in fact , gw s carry complementary information with respect to electromagnetic and optical waves , since the gw s are practically not absorbed by the matter . the aim of the virgo experiment is the direct detection of gravitational waves and , in joint operation with other similar detectors , to perform gravitational waves astronomical observations . in particular , the virgo project is designed for broadband detection from @xmath0 to @xmath1 . the principle of the detector is shown in figure 1 . a @xmath2 @xmath3 arm - length michelson interferometer with suspended mirrors ( test masses ) is used . the phase difference @xmath4 between the two arms is amplified using fabry - perot cavities of finesse @xmath5 in each arm . aiming for detection sensitivity of @xmath6 , virgo is a very delicate experimental challenge because of the competition between various sources of noise and the very small expected signal . in fact , the interferometer will be tuned on the dark fringe , and then the signal to noise ratio will be mainly limited , in the above defined range of sensitivity , by residual seismic noise , thermal noise of the suspensions photon counting noise ( shot noise ) . in figure 2 the overall sensitivity of the apparatus is shown . in this figure it is easy to see the contribution of the different noise sources to the global noise . in this context we use a multi - layer perceptron ( mlp ) nn with the back - propagation learning algorithm to model and identify the noise in the system , because we experimentally found that fir nn s and elman nn s did not work in a satisfying manner . both the fir @xcite and elman @xcite models proved to be very sensible to overfitting and were not stable . furthermore the elman network required a great number of hidden units , while the fir network required a great number of delay terms . instead , the mlp proved succesfull and easy to train because we used the bayesian learning paradigm . nn s are massively parallel , distributed processing systems . they are composed of a large number of processing elements ( called nodes , units or neurons ) which operate in parallel . scalars ( called weights ) are associated to the connections between units and determine the strength of the connections themselves . computational capability is due to the connections between the units and to their collective behaviour . furthermore , information representation is distributed , i.e. no single unit is charged with specific information . nn s are well - known for their universal approximation capability @xcite . system identification consists in finding the input - output mapping of a dynamic system . a discrete - time multi - input multi - output ( mimo ) system ( or a continuous - time sampled - data system ) can be described with a set of non - linear difference equations of the form ( _ input - state - output _ representation ) : @xmath7 where @xmath8 is the state vector of the system , @xmath9 is the input vector and @xmath10 is the output vector . since we can not always access the state vector of the system , therefore we can use an input - output representation of the form : @xmath11 \label{eq : io}\end{aligned}\ ] ] where @xmath12 and @xmath13 are the maximum lags of the input and output vectors , respectively , and @xmath14 is the pure time - delay ( or dead - time ) in the system . this is called an arx model ( autoregressive with exogenous inputs , ) and it can be shown that a wide class of discrete - time systems can be put in this form @xcite . to build a model of the form ( [ eq : io ] ) , we must therefore obtain an estimation of the functional @xmath15 $ ] , which generally is nonlinear . given a set of input - output pairs , a neural network can be built @xcite which approximates the desired functional @xmath16 $ ] . such a network has @xmath17 inputs and @xmath18 outputs ( see figure 3 ) . a difficulty in this approach arises from the fact that generally we do not have information about the model order ( i.e. the maximum lags to take into account ) unless we have some insight into the system physics . furthermore , the system is non - linear . recently @xcite a method has been proposed for determining the so - called _ embedding dimension _ of nonlinear dynamical systems , when the input - output pairs are affected by very low noise . furthermore , the lags can be determined by evaluating the _ average mutual information _ ( ami)@xcite . such methodologies , although not always successful , can be nevertheless used as a starting point in model design . in the virgo data analysis , the most difficult problem is the gravitational signal extraction from the noise due to the intrinsic weakness of the gravitational waves , to the very poor signal - to - noise ratio and to their not well known expected templates . furthermore , the virgo detector is not yet operational , and the noise sources analyzed are purely theoretical models ( often stationary noises ) , not based on experimental data . therefore , we expect a great difference between the theoretical noise models and the experimental ones . as a consequence , it is very important to study and to test algorithms for signal extraction that are not only very good in signal extraction from the theoretical noise , but also very adaptable to the real operational conditions of virgo . for this reason , we decided the following strategy for the study , the definition and the tests of algorithms for gravitational data analysis . the strategy consists of the following independent research lines . the first line starts from the definition of the expected theoretical noise models . then a signal is added to the virgo noise generated and the algorithm is used for the extraction of the signal of known and unknown shape from this noise at different levels of signal - to - noise ratio . this will allow us to make a number of data analysis controlled experiments to characterize the algorithms . the second line starts from the real measured environmental noise ( acoustic , electromagnetic , ... ) and tries to identify the noise added to a theoretical signal . in this way we can test the same algorithms in a real case when the noise is not under control . using this strategy , at the end , when in a couple of years virgo will be ready for the first test of data analysis , the procedure will be moved to the real system , being sure to find small differences from theory and reality after having acquired a large experience in the field . as we have seen in the introduction , the virgo interferometer can be characterized by a sensitivity curve , which expresses the capability of the system to filter undesired influences from the environment , and which could spoil the detection of gravitational waves ( such a noise is generally called seismic noise ) . the sensitivity curve has the following expression : @xmath19 + s_{\nu } & \quad f\geq f_{\textrm{min } } \\ s(f_{\textrm{min } } ) & \quad f < f_{\textrm{min } } \end{array } \right . \label{eq : exprsenscurve}\ ] ] where : * @xmath20 * @xmath21 is the shot noise cut - off frequency * @xmath22 is the pendulum mode * @xmath23 is the mirror mode * @xmath24 is the shot noise the contribution @xmath25 of violin resonances is given by : @xmath26 ^{2}+\phi _ { i}^{2}}+\frac{1}{i^{4}}\frac{f_{i}^{(f)}}{f}\frac{c_{f}\phi _ { i}^{2}}{\left [ \left ( \frac{f}{if_{i}^{(f)}}\right ) ^{2}-1\right ] ^{2}+\phi _ { i}^{2 } } \label{eq : violinres}\ ] ] where the different masses of close and far mirrors are taken into account : * @xmath27 * @xmath28 * @xmath29 * @xmath30 * @xmath31 note that we used a simplified curve for our simulations , in which we neglected the resonances ( see figure 4 ) . samples of the sensitivity curve @xmath32 can be obtained by evaluating the expression ( [ eq : exprsenscurve ] ) at a set of frequencies @xmath33 , @xmath34 $ ] . the samples of the sensitivity curve allow us to obtain the system transfer function ( in the frequency domain ) , @xmath35 , such that : @xmath36 > from this , by means of an inverse discrete fourier transform , samples of the system transfer function ( in the time domain ) can be obtained . our aim is to build a model of the system transfer function ( [ eq : tranfun ] ) . assuming that the interferometer input noise is a zero mean gaussian process , by feeding it to the system ( i.e. filtering it through the system transfer function ) we obtain a coloured noise . the so obtained white noise - colored noise pairs can then be used to train an mlp , as shown in figure 3 . the first step in building an arx model is the model order determination . to determine suitable lags which describe the system dynamics , we used the ami criterion @xcite . this can be seen as a generalization of the autocorrelation function , used to determine lags in linear systems . a strong property of the ami statistic is that it takes into account the non - linearities in the system . usually , the lag is chosen as the first minimum of the ami function . the result is reported in figure 5 , in which the first minimum is at 1 . to find how many samples are necessary to unfold the ( unknown ) state - space of the model ( the so called _ embedding dimension _ @xcite ) we used the method of @xcite , the lipschitz decomposition . the result of the search is reported in figure 6 . from the figure we can see that , starting from lag three , the order index decreases very slowly , and so we can derive that the width of the input window is at least three . in order to test the nn s capability in solving the problem , we chose a width of 5 , both for input and output ( i.e. @xmath37 ) . in this way , we obtained a nn with a simple structure . furthermore , some preliminary experiments showed that the system dead - time is @xmath38 ; this gives the best description of the system dynamics . another fundamental issue is the nn complexity , i.e. the number of units in the hidden layers of the nn . usually the determination of the network complexity is critical because of the risks of overfitting . since the nn was trained following a bayesian framework , then overfitting was of no concern ; so we directed our search for a model with the minimum possible complexity . in our case , we found a hidden layer with 6 @xmath39 units is optimal . the bayesian learning framework ( see @xcite and @xcite ) allows the use of a _ distribution _ of nn s , that is , the model is a realization of a random vector whose components are the nn weights . the so obtained nn is the _ most probable _ given the data used to train it . this approach avoids the bias of the cross - validatory techniques commonly used in practice to reduce model overfitting @xcite . to allow for a smooth mapping which does not produce overfitting , several regularization parameters ( also called _ hyperparameters _ ) embedded in the error criterion have been used : * one for each set of connections out of each input node , * one for the set of connections from hidden to output units , * one for each of the bias connections , * one for the data - contribution to the error criterion . usually , the hyperparameters of the first three kinds are called _ alphas _ , while the last is called a _ beta_. the approach followed in the application of the bayesian framework is the `` exact integration '' scheme , where we sample from the analytical form of the distribution of the network weights . this can be done if we assume an analytic form for the prior hyperparameters distribution , in particular a distribution uniform on a logarithmic scale : @xmath40 this distribution is _ non - informative _ , i.e. it embodies our complete lack of knowledge about the values the hyperparameters should take . the chosen model was trained using a sequence of little less than a million of patterns ( we sampled the system at 4096hz for 240s ) normalized to zero mean and unity variance with a _ whitening _ process . note that the input - output pairs were processed through discrete integrators to obtain pattern - target pairs , as shown in figure 3 . the nn was then tested on a 120s long sequence . the nn was trained for @xmath41 epochs , with the hyperparameters being updated every @xmath42 epochs . a close look at the @xmath43 hyperparameters shows that all the inputs are relevant for the model ( they are of the same magnitude ; note that this further confirms the pre - processing analysis ) . the @xmath44 hyperparameter shows that the data contribution to the error is very small ( as we would expect , since the data are synthetic ) . the simulations were made using the matlab@xmath45 language , the netlab toolbox @xcite and other software designed by us . in figure 7 , the psds of the target and the predicted time series are shown ; in the lower part of the figure is reported the psds of the prediction residuals . in figure 8 , the psd of a 100hz sine wine added to the noise is shown , with the signal extracted by the network . as can be seen , the network recognizes the frequency of the sine wave with the maximum precision allowed by the residuals . in this paper we have shown some preliminary tests on the use of nn s for signal processing in the virgo project . some observations can be elicited from the experimental results : * in evaluating the power spectral densities ( psds ) , we made the hypothesis that the system is sampled at @xmath46 . it is only a work hypothesis , but it shows how the network reproduces the system dynamics up to @xmath47 . note that the psds are nearly the same also if we were near the nyquist frequency . * the psd of the residuals shows a nearly - white spectrum , which is index of the model goodness ( see @xcite ) . - : : to increase the system model order and to test if there are significant differences in prediction ; - : : to test the models with a greater number of samples to obtain a better estimate of the system dynamics ; - : : to model the noise inside the system model to improve the system performance and to allow a multi - step ahead prediction ( i.e. an output - error model )
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in this paper a neural networks based approach is presented to identify the noise in the virgo context .
virgo is an experiment to detect gravitational waves by means of a laser interferometer .
preliminary results appear to be very promising for data analysis of realistic interferometer outputs .
| 4,083 | 67 |
the degree of linear polarization of sunlight scattered by an asteroid toward an observer depends on the phase - angle , namely the angle between the asteroid - sun and the asteroid - observer directions . the phase - polarization curves of all atmosphereless bodies of the solar system exhibit qualitatively similar trends , but their detailed features vary according to the specific properties ( including primarily the geometric albedo ) of individual surfaces . in the phase - angle range @xmath1 , asteroids exhibit the so - called branch of _ negative polarization _ , in which , in contrast to what is expected from simple single rayleigh - scattering or fresnel - reflection mechanisms , the plane of linear polarization turns out to be parallel to the plane of scattering ( the plane including the sun , the target and the observer ) . the plane of linear polarization becomes perpendicular to the scattering plane , a situation commonly described as _ positive polarization _ , at phase angle values larger than the so - called _ inversion angle _ , which is generally around @xmath2 . a few years ago , @xcite discovered a class of asteroids exhibiting peculiar phase - polarization curves , characterized by a very unusual extent of the negative polarization branch , with an inversion angle around @xmath3 , much larger than the values commonly displayed by most objects . since the prototype of this class is the asteroid ( 234 ) barbara , these objects have been since then commonly known as _ barbarians_. only half a dozen barbarians are known today : ( 234 ) barbara , ( 172 ) baucis , ( 236 ) honoria , ( 387 ) aquitania , ( 679 ) pax , and ( 980 ) anacostia @xcite . the polarimetric properties of the barbarians are fairly unexpected . the observed large negative polarization branch is not predicted by theoretical models of light scattering , but in fairly special situations , including surfaces composed of very regularly - shaped particles ( spheres , crystals ) or surfaces having considerable microscopic optical inhomogeneity @xcite . although barbarians are certainly uncommon , they do exist , and the interpretation of their polarization features may lead to important advances in our understanding of both light - scattering phenomena , and of the origin and evolution of these objects . potential explanations range from peculiar surface composition and/or texture , to the possible presence of anomalous properties at macroscopic scales due the presence of large concavities associated with big impact craters @xcite . for instance , ( 234 ) barbara has a very long rotation period , which might be the effect of a big collision . @xcite suggested that ( 234 ) barbara could have a surface characterised by large - scale craters . this is confirmed by an analysis of still unpublished occultation data by one of us ( pt ) . in terms of taxonomy based on spectro - photometric data , all known barbarians are classified as members of a few unusual classes , including @xmath4 , @xmath5 , and ( in only one case ) @xmath6 . ( 234 ) barbara itself is an @xmath5 asteroid ( here we use the taxonomic classification of * ? ? ? . however , there are @xmath4-class asteroids which are normal " objects not exhibiting the barbarian properties . this fact seems to rule out a direct relationship between taxonomic class ( based on the reflectance spectrum ) and polarimetric properties . on the other hand , @xmath4 , @xmath5 and @xmath6 classes are located , in a principal component analysis plane , along adjacent locations , which although non - overlapping , seem to represent some kind of continuous spectral alteration surrounding the most common @xmath7 class complex . the fact that the six known barbarians identified so far belong all to one of these three classes suggests that surface composition could be responsible for their polarimetric properties . even more important , two @xmath4-class barbarians , ( 387 ) aquitania and ( 980 ) anacostia , exhibit very similar reflectance spectra , both sharing the rare property of displaying the spectral signature of the spinel mineral @xcite . actually , it was exactly the fact that ( 980 ) anacostia was found to be a barbarian that led @xcite to observe polarimetrically ( 387 ) aquitania , and to discover that also this object shares the same polarimetric behaviour . spinel ( [ fe , mg]al@xmath8o@xmath9 ) is a mineral characterized by indistinct cleavage and conchoidal , or uneven fracture properties . in terms of optical properties , the mgal@xmath8o@xmath9 form of spinel has a fairly high refractive index ( @xmath10 ) , which becomes even higher in the spinel variety having a high iron content ( hercynite ) ( @xmath11 , i.e. , much above the values characterizing the most common silicates present on asteroid surfaces , * ? ? ? spinel is an important component of calcium aluminum - rich inclusions ( cai ) found in all kinds of chondritic meteorites . cais are refractory compounds which are thought to be among the first minerals to have condensed in the proto - solar nebula . they are the oldest samples of solid matter known in our solar system , and they are used to establish the epoch of its formation @xcite . in terms of spectroscopic properties , spinel is characterized by the absence ( or extreme weakness ) of absorption bands around 1@xmath12 m , and by the presence of a strong absorption band at 2@xmath12 m . @xcite concluded that , to model the available near - ir spectra of spinel - rich asteroids , it is necessary to assume abundances of the order of up to 30% of cai material on the surface . this extreme abundance , which causes a high refractive index , might also be responsible for the anomalous polarization properties . such high cai abundances have never been found in meteorite on earth ( so far , the richest cai abundance , found on cv3 meteorites , is about 10% ) . therefore , @xcite conclude that spinel - rich asteroids might be more ancient than any known sample in our meteorite collection , making them prime candidates for sample return " missions . many interesting questions are certainly open . which processes are involved in the onset of physical mechanisms which produce the barbarian polarimetric behaviour ? are barbarians really among the oldest objects accreted in our solar system ? if so , why are they the only one class of primitive objects being characterized by an anomalous polarimetric behaviour ? why are they so rare ? are they unusually weak against collisions and fragmentation ? do space - weathering phenomena affect their polarimetric properties ? are the taxonomic classifications of some @xmath4 , @xmath5 , @xmath6 objects possibly wrong ( in such a way that all barbarians might be member of a unique class and not spread among three of them ) ? it is clear that an important pre - requisite to improve our understanding of these objects , and to draw from them some possible inferences about the origin , composition and subsequent evolution of the planetesimals orbiting the sun at the epoch of planetary formation , is to find new members of the barbarian class , but where to look for them ? an aid to a barbarian search comes from the fact , recently confirmed by @xcite , that the spinel - bearing barbarian ( 980 ) anacostia belongs to a family of high - inclination asteroids . this family is named watsonia from its lowest - numbered member ( 729 ) watsonia . ( 980 ) anacostia belongs to a small grouping which is included in the watsonia family and merges with it at larger values of mutual distances between the members . in other words , ( 980 ) anacostia belongs to a distinct sub - group of the watsonia family , but the possible independence of this sub - group from the rest of the family is highly uncertain and questionable @xcite . another known barbarian , ( 387 ) aquitania , though not being a member of the watsonia family , is also located in close vicinity in the space of proper orbital elements ( see section[results ] ) . finally , a member of the watsonia family , asteroid ( 599 ) luisa , is not a known barbarian ( no published polarimetric measurements are available for it ) , but it is known to be one of the few spinel - rich asteroids identified so far ( thus sharing some common properties with both anacostia and aquitania ) . one should also note that the distribution of albedos of the members of the watsonia family observed at thermal ir wavelengths by the wise satellite @xcite is strongly peaked around values between @xmath13 and @xmath14 with only a few likely interlopers , suggesting that the family is not a statistical fluke web facility available at url http://mp3c.oca.eu/ ] . a possible common collisional origin of these asteroids opens new perspectives for the search of new barbarians and for the interpretation of their properties . asteroid families are the outcomes of fragmentation of single parent bodies disrupted by catastrophic collisions . therefore , if ( 980 ) anacostia , and possibly also ( 387 ) aquitania , were issued from the disruption of a common parent body exhibiting the rare properties which produce the barbarian polarization phenomenon , it would be natural to expect that also the other , still not observed members of the watsonia family should be found to be barbarians . moreover , among the watsonia family members at least one , ( 599 ) luisa , is a known spinel - rich asteroid , like anacostia and aquitania . finally , wise albedo values in the same range of those of watsonia family members have also been derived for most barbarian asteroids known so far , namely 234 , 172 , 236 , 679 and 980 . ccrrcr@@xmath15lr@@xmath15lc date & time ( ut ) & & & phase angle & & & wise + & & ( sec ) & & & & & albedo + & & & & & + 2013 07 05 & 23:41 & 480 & 5492 & 18.79 & @xmath161.14 & 0.10 & 0.05 & 0.10 & @xmath17 + 2013 07 29 & 01:44 & 960 & & 18.31 & @xmath161.01 & 0.09 & 0.00 & 0.09 + 2013 06 03 & 09:40 & 2000 & 42365 & 23.30 & @xmath160.83 & 0.15 & @xmath160.06 & 0.15 & @xmath18 + 2013 08 03 & 09:08 & 960 & & 18.55 & @xmath161.73 & 0.12 & @xmath160.06 & 0.12 + 2013 07 12 & 23:50 & 960 & 56233 & 17.83 & @xmath161.07 & 0.16 & @xmath160.09 & 0.16 & @xmath19 + 2013 08 05 & 01:03 & 2200 & & 19.31 & @xmath161.09 & 0.12 & @xmath160.07 & 0.12 + 2013 07 30 & 00:38 & 2000 & 106059 & 18.30 & @xmath161.06 & 0.30 & @xmath160.15 & 0.31 + 2013 08 04 & 01:09 & 4000 & & 18.82 & @xmath160.94 & 0.14 & 0.00 & 0.14 + 2013 08 28 & 01:24 & 4800 & & 19.64 & @xmath160.84 & 0.20 & @xmath160.09 & 0.20 + 2013 07 06 & 01:33 & 1400 & 106061 & 20.21 & @xmath160.94 & 0.11 & 0.06 & 0.11 + 2013 08 09 & 02:46 & 4000 & & 23.97 & @xmath160.57 & 0.12 & @xmath160.05 & 0.12 + 2013 07 06 & 02:05 & 960 & 144854 & 21.30 & @xmath160.81 & 0.12 & 0.13 & 0.12 + 2013 08 05 & 02:06 & 4000 & & 24.19 & @xmath160.15 & 0.12 & @xmath160.13 & 0.13 + 2013 08 13 & 06:28 & 4000 & 236408 & 18.31 & @xmath160.97 & 0.15 & 0.22 & 0.15 & @xmath20 + 2013 07 07 & 02:40 & 3440 & 247356 & 19.97 & 0.10 & 0.15 & @xmath160.33 & 0.15 + 2013 04 17 & 08:58 & 4800 & 320971 & 23.78 & 0.11 & 0.31 & @xmath160.06 & 0.31 + 2013 06 03 & 07:08 & 3440 & & 20.53 & @xmath160.03 & 0.22 & @xmath160.06 & 0.21 + light ) for the targets of the present investigation , compared with the polarization curves ( in @xmath21 light ) of the ( 234 ) barbara , and of ( 12 ) victoria , a large @xmath4-class asteroid which does not exhibit the barbarian behaviour . black symbols : the seven targets exhibiting the barbarian polarimetric behaviour ; red symbols : our two targets that display a `` normal '' polarimetric behaviour ; small blue symbols and blue curve : available data for ( 234 ) barbara @xcite , and the corresponding best - fit curve using the linear - exponential relation @xmath22 + c \cdot \alpha$ ] , where @xmath23 is the phase angle in degrees ; dashed , green curve : the best - fit curve for the @xmath4-class asteroid ( 12 ) victoria ( for the sake of clearness , individual observations of this asteroid are not shown).,width=283 ] our target list includes nine objects that are members of the watsonia family ( not limited to the anacostia sub - group ) according to @xcite . these objects are listed in table [ tab_observations ] . we note that a more recent family search whose results are publicly available at the astdys web site also identified a watsonia family , but using a more conservative value of the critical level of mutual distances between family members to be adopted to define the family . as a consequence , the astdys version of the watsonia family has a smaller membership list , and does not include our target objects 42365 , 144854 , 247356 and 320971 . we observed our targets using the vlt fors2 instrument @xcite in imaging polarimetric mode , and obtained @xmath24 broadband linear polarization measurements in the @xmath25 special filter from april to september 2013 . the choice of an @xmath25 filter ( instead of the standard @xmath21 filter traditionally used in many asteroid polarimetric observations ) was dictated by the need of improving the s / n ratio . polarimetric measurements were performed with the retarder wave - plate at all positions between 0 and 157.5 , at 22.5 steps . for each observation , the exposure time cumulated over all exposures varied from 8min ( for 5492 ) to 1h and 20min ( for 320971 ) . data were then treated as explained in @xcite , and our measurements are reported adopting as a reference direction the perpendicular to the great circle passing through the object and the sun . this way , @xmath26 represents the flux perpendicular to the plane sun - object - earth ( the scattering plane ) minus the flux parallel to that plane , divided by the sum of these fluxes . it is therefore identical to the parameter indicated as @xmath27 in many asteroid polarimetric studies . in these conditions , for symmetry reasons , @xmath28 values are expected to be zero , and inspection of their values confirms that this is the case for our observations . at small phase angles ( @xmath29 ) , all asteroids exhibit negative polarization , whereas at phase angles @xmath30 , nearly all asteroids exhibit positive polarization . by contrast , at phase angle @xmath31 , barbarians still exhibit a relatively strong negative polarization ( @xmath32% ) . therefore , to identify barbarians , we decided to measure the polarization at phase angles @xmath33 , and establish whether the measured polarization plane is found to be parallel or perpendicular to the scattering plane . the results of our polarimetric measurements are given in table [ tab_observations ] and shown in figure [ figure1 ] , in which a comparison is also made with the phase - polarization curve of ( 234 ) barbara and ( 12 ) victoria , a big , non - barbarian @xmath4-class asteroid . in the figure each target is represented by a different symbol , and some targets have more than one measurement . in the observed phase - angle range , seven of our targets show a polarization value @xmath32% , consistent with the value exhibited by ( 234 ) barbara . in fact , ( 234 ) barbara exhibits marginally higher polarization values ( in absolute value ) than our targets , possibly due to the different filter in which the observations were performed . the striking result of our observing campaign is that seven out of nine asteroids of our target list are barbarians . the two exceptions are 247356 and 320971 . however , we know _ a priori _ that all family member lists are expected to include some fraction of random interlopers @xcite , then we conclude that both non - barbarian objects in our target list may be family interlopers . for instance , asteroid 320971 is listed as a watsonian member by @xcite , but it is not included among the watsonia members identified in the new astdys list of asteroid families . on the other hand , three other targets not included in the astdys member list , but present in the @xcite member list , are found to be barbarians . we conclude therefore that , apart from some details concerning family membership , the watsonia family is an important repository of barbarian asteroids . this is the immediate and most important result of our investigation . we also note that the number of watsonia barbarians identified in our observing campaign is larger than the whole sample of barbarians previously known . we also note that , except in case of 236408 and 247356 ( only a single measurement each ) the resulting @xmath34 values are always well consistent with zero . our results confirm once again that asteroid polarimetry can provide an important contribution to asteroid taxonomy , as noted in the past by several authors ( see , for instance , * ? ? ? several problems , however , are now open , and deserve further theoretical and observational efforts . the first issue to be addressed is the relationship between the barbarian polarimetric features and the unusual amount of spinel measured via spectroscopy in some of the known barbarians . in other words , the new barbarians that we have found in our investigation must be spectroscopically observed in the visible and near - ir wavelengths in order to check whether they exhibit the spinel features . we do believe that this will be the case , since we already know three objects that are both spinel - rich and barbarians ( 234 barbara , 387 aquitania and 980 anacostia ) , ad we know also that the watsonia family includes at least one other spinel - rich asteroid ( 599 luisa ) @xcite , but we clearly need confirmation from new observations . similarly , we need to search for the barbarian polarimetric feature in other spinel - rich asteroids . in particular , we are interested not only in 599 luisa , but also in the henan family , which is known to include at least three spinel - rich asteroids @xcite . next , we need to understand why the barbarian and the anomalous spinel abundance phenomena are so rare . apparently , only a handful of barbarian parent bodies existed , and only their disruption into many fragments made it possible to identify today larger numbers of these objects . in principle , one might wonder whether the unusual properties displayed by these objects might also be in some ( still obscure ) way a consequence of the collisional events themselves . another open problem is the origin of ( 387 ) aquitania . this asteroid is not included in any family list , including both the most recent classification available at the astdys site , and the @xcite classification of high-@xmath35 families . yet , this object is quite close , in terms of orbital elements , to ( 980 ) anacostia and the rest of the watsonia family . the values of proper elements for ( 387 ) aquitania and ( 980 ) anacostia ( available in the astdys database ) are @xmath36 , @xmath37 , @xmath38 , for ( 387 ) aquitania , and @xmath39 , @xmath40 , @xmath41 for ( 980 ) anacostia , where @xmath42 is the proper semi - major axis , @xmath43 the proper eccentricity , and @xmath35 is the proper inclination . the only one relevant difference , which prevents any family search to include them in an acceptable group , is the proper eccentricity . one should conclude that there is no relation between ( 387 ) aquitania and ( 980 ) anacostia , and that the similarity of their orbital semi - major axes and inclinations are just a coincidence . moreover , both asteroids display stable orbits in terms of characteristic lyapunov exponents , and any possible evolution due to non - gravitational forces , ( e.g. the yarkovsky effect ) seem also very unlikely , because both asteroids are fairly large , with sizes of the order of @xmath44 km according to thermal radiometry data . finally , the collision which could have produced a family including two fragments of this size had to be extremely energetic , and should have produced a huge family , of which there is no evidence today . on the other hand , aquitania and anacostia exhibit too many relevant similarities in their physical properties , which suggests that they may come from a unique parent body fragmented by a collision . a very tentative scenario is that , in a very early epoch , a violent collision destroyed a big parent body which was the progenitor of the barbarians we see today in this zone . most of the first - generation fragments produced by the event are now gone due to gravitational perturbations and yarkovsky - driven drift in semi - major axis . the current watsonia family could be the product of a more recent disruption of one of the original fragments of the first event . this region could still include several first - generation fragments , preferentially the biggest ones which , due to their size , experienced no or extremely weak yarkovsky evolution , like ( 387 ) aquitania and ( 980 ) anacostia . some gravitational perturbations could have also played some role . if one looks at the @xmath42 - @xmath43 or @xmath42 - @xmath45 plots making use of the facility available at the astdys site , it is easy to see that the watsonia family is crossed , in semi - major axis , by a couple of thin resonances , one of which is located close to the value of @xmath42 of both ( 387 ) aquitania and ( 980 ) anacostia . so , we can not rule out entirely the idea that some limited evolution in eccentricity of these objects could be due to these perturbation effects and the two objects might have been originally closer in the proper element space . we stress again that this kind of scenario is only eminently speculative and deserves further analysis before it can be accepted as a plausible one . we point out that a so large relative fraction of observed members of the watsonia family turning out to be barbarians has a further implications to be taken into account . if the parent body was differentiated , or had some gradient of composition as a function of depth , with the presence of cai material limited to some thin shell , then the spinel signature would be present only in a minority of the family members . this is in disagreement with what we observe , indicating that the anomalous composition of the parent body was not limited to some thin region . in conclusion , we are not looking simply at some kind of surface process ( like alteration by space weathering or impacts ) , but at the properties displayed by asteroids originated from different depths inside a single parent body which was anomalously spinel - rich in most of its volume ( this can be even more true if the watsonia family is a secondary event derived by a first - generation destruction of a larger body . ) the issues of section [ intro ] are still open and deserve careful attention . the immediate result of our investigation is that we have now a much larger sample of barbarians than before at disposal for future physical investigations ( including photometric , spectroscopic , and polarimetric campaigns . ) at the same time , our results definitively confirm the family of watsonia as a sizeable group of barbarians having a common collisional origin , and open new perspectives for the development of models in order to try and sort out in a coherent scenario several pieces of observational evidence which appear now to form a complicated puzzle . part of this work was supported by the cost action mp1104 _ polarization as a tool to study the solar system and beyond_. in particular , funding for short term scientific mission cost- stsm - mp1104 - 12750 of ac at the armagh observatory is kindly acknowledged . the work of b.n . has been supported by the ministry of education and science of serbia , under the project 176011 . a careful review by k. muinonen helped to improve the paper .
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barbarians , so named after the prototype of this class ( 234 ) barbara , are a rare class of asteroids exhibiting anomalous polarimetric properties .
their very distinctive feature is that they show negative polarization at relatively large phase - angles , where all `` normal '' asteroids show positive polarization .
the origin of the barbarian phenomenon is unclear , but it seems to be correlated with the presence of anomalous abundances of spinel , a mineral usually associated with the so - called calcium aluminum - rich inclusions ( cais ) on meteorites .
since cais are samples of the oldest solid matter identified in our solar system , barbarians are very interesting targets for investigations .
inspired by the fact that some of the few known barbarians are members of , or very close to the dynamical family of watsonia , we have checked whether this family is a major repository of barbarians , in order to obtain some hints about their possible collisional origin .
we have measured the linear polarization of a sample of nine asteroids which are members of the watsonia family within the phase - angle range @xmath0 .
we found that seven of them exhibit the peculiar barbarian polarization signature , and we conclude that the watsonia family is a repository of barbarian asteroids .
the new barbarians identified in our analysis will be important to confirm the possible link between the barbarian phenomenon and the presence of spinel on the surface .
[ firstpage ] asteroids : general polarization .
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the introduction of reflective optical elements revolutionized the design and imaging capabilities of telescopes @xcite . while the possibility of creating large diameter objectives was one of the primary aims , a significant advantage is that reflective optical elements provide a means of reducing or eliminating the spherical and chromatic aberrations that are inherent in lens based systems . here we explore the opposite extreme of microfabricated optics . microfabricated optical elements are key components for the development and integration of optics into a range of research and commercial areas @xcite . to date , the majority of the work in microphotonics has been in refractive elements , i.e. microlenses . however , in this regime microlenses typically have significant numerical apertures and surface curvatures , which introduce large aberrations . a number of groups have in recent years discussed the design and fabrication of concave micromirrors @xcite . these examinations are largely driven by two purposes ; for optical tweezing , and for integration into atom optics . spherical mirrors have been demonstrated to collect light from single ions @xcite . parabolic mirrors , similar to those described here but with larger length scales , have been also been used as highly efficient collectors of light from single ions @xcite , atoms @xcite , and point sources @xcite . in a similar manner they may be used to tightly focus light onto atomic samples , which has to date been shown with refractive optics @xcite . in this work we consider the use of reflective micro optical components for focusing light . we present the construction and optical characterization of parabolic reflectors with an open aperture of radius 10 @xmath0 m and measured focal lengths that range from 24 @xmath0 m to 36 @xmath0 m . detailed mapping of the focused intensity field is made possible by the development of a previously unreported adaptation of a confocal microscope that allows the illumination of the reflector with collimated light , while still maintaining the highly - desirable large - numerical - aperture confocal collection . using this device we obtain 3d data about the focal plane demonstrating diffraction limited focussing . we also discuss the application of the parabolic mirror for use in atomic physics and tweezing experiments . the details of the fabrication of concave paraboloid structures through ion - beam milling are covered in @xcite . briefly , a focused ion - beam ( gallium ions with typical currents of 50 - 300 pa and accelerating voltages of 30 kv ) is used to precisely sculpt a silicon substrate with the required mirror profile , which is subsequently coated with gold to provide a highly - reflective coating . in focused ion - beam milling , controlling the dose of ions to which an area is exposed allows a region of the surface to be sputtered to a known depth , due to a linear relationship between depth and dose in silicon . the applied dose is a function of the beam current , the dwell time , and number of passes the beam makes over an area . by tracing a number of concentric discs of increasing radius , whilst linearly increasing the dose , a parabolic depression can be milled into the substrate . in principle this would create a stepped contour , however due to edge - effects of the milling process , as well as redistribution of etched material , a larger number of passes creates a smooth contour of the parabolic dish . further details of the construction and characterization can be found in @xcite , where an rms roughness of 4.0 nm was measured by afm over the range of the concave parabolic surface . due to the identical manufacturing process similar values are expected in this work . the propagation of light using microscopic optical elements is a well represented topic in the literature . here we restrict our discussion to the behavior of light fields after wavelength scale apertures and curved surfaces . in their work , goldwin and hinds @xcite derive analytic results for spherical mirrors , which they further compare with numerical integration of maxwell s equations . meanwhile , bandi _ et . _ consider the propagation of light after a wavelength - scale aperture using a fresnel representation of the fields @xcite , which offers the possibility of adding focusing to the formalism . here , however , the reflected field was modeled using the angular spectrum method @xcite , which provides a mapping of an electric field from a particular plane into a secondary plane ; @xmath2 where @xmath3 are the cartesian spatial directions , @xmath4 is the wavevector in the @xmath5-direction , @xmath6 is the electric field , and @xmath7 is the 2d fourier spectrum of the electric field in the plane @xmath8 . the result of this compact equation can be understood by first noting that the spatial spectrum of an electric field , @xmath9 can be translated from a plane @xmath10 to another plane @xmath8 using the helmholtz propagator , @xmath11 where the helmholtz propagator in reciprocal space is @xmath12 , @xcite . we then note that the electric field can be calculated from its spatial spectrum in a plane are by the inverse fourier transform , @xmath13 , @xmath14 these relations , eq . ( 2 ) and eq . ( 3 ) then clearly show the result of eq . ( 1 ) , and can be used to calculate the electric field in an arbitrary plane , given that is is known in one plane . the essential details of calculating the final intensity profile using the angular spectrum method can be clearly seen in fig . [ fig : theory_schematic](c ) ; the initial electric field is fourier transformed , the helmholtz propagator is then applied , before the inverse fourier spectrum is taken . finally , the resulting intensity field is found from the modulus - squared of the electric field . example codes are available on request from the corresponding author . for this work , in the plane of @xmath10 a curvature is numerically added to the phase - fronts within the area of the aperture of the mirror . the curvature is expressed as @xmath15 , where @xmath16 is the total wavevector , @xmath17 , @xmath18 is the radial distance from the center of the mirror , @xmath19 is the mirror aperture radius , and @xmath20 is the focal length . this modification of the optical wavefront represents the position dependent optical path - length difference , due to the spatially varying propagation distance to the metal surface , fig . [ fig : theory_schematic ] . care must be taken to include all modes of the field , including the evanescent ones , in order to return the full field simulation . all simulations were performed in matlab with the focal length , @xmath20 and the position of the surface as free parameters for fitting to the measured data . although transmission confocal imaging provides a powerful means of examining microlenses @xcite , reflection images of focusing elements require post analysis to interpret the data . in traditional reflection confocal microscopy the illuminating laser is focused onto the imaging plane at a point that is confocal with a pinhole in the imaging axis of the microscope . while this allows accurate probing of the surface it precludes imaging the focal plane of our concave mirrors directly . instead one may directly image about the center of curvature of the mirror and then infer the transfer function of the mirror assuming the point spread functions of the other optical elements are known . in this work we devise a scheme that allows interrogation of the focus while illuminating a mirror with a collimated laser beam , fig . [ fig : confocal_schematic ] . this arrangement means that the intensity distribution that would be formed in an optical tweezing experiment @xcite , for example , can be faithfully interrogated . probing the focal plane of the parabolic mirrors requires illumination with collimated light co axial with the mirror axis ( orthogonal to the substrate surface ) . the working distance of the objective lens used ( olympus uplansapo 40x/0.9@xmath21 ) is only 180 @xmath0 m , which severely limits introducing further optical elements before the object . to circumvent these restrictions a non - polarizing beam splitter ( npbs ) was introduced between the objective lens and the scanning column of a commercial confocal microscope ( olympus fv1000 scan head with olympus ix81 inverted microscope ) . this enabled an external probe beam to be introduced through the back aperture of the objective lens using the npbs to combine the beam into the imaging axis , see fig . [ fig : confocal_schematic ] . the probe , a commercial solid - state 589 nm laser ( based on sum - frequency generation using two lasing lines of an nd : yag laser ) , was focused , after fiber - coupling to clean up the spatial mode , at the back - focal plane of the objective lens . the specific probe wavelength was chosen for convenience ( good detector efficiency and high optical power in a single transverse mode ) - the numerical simulations were also conducted for this wavelength to account for wavelength - dependent diffraction effects . the collimation of the beam emitted was optimized by varying the axial position of the coupling lens . the npbs , optical fiber , and coupling lens were mounted in a holder with three axis adjustment , which allowed for overlapping of the optical axes . the emitted beam had a diameter of 1 mm , justifying the assumption of a spatially uniform intensity profile across the micromirror aperture ( 20 @xmath0 m diameter ) . the magnification of the combined optical system was calibrated by measuring the aperture diameter of the mirrors . in order to visually verify the surface quality of the mirrors , as well as to measure the aperture diameter of the mirrors the surface of the mirrors was also probed using conventional confocal microscopy @xcite , without removal of the additional optical elements . no aberrations were observed to have been introduced . m parabolic micromirror . a ) shows data collected from the surface of the mirror substrate ; b ) taken 25 @xmath0 m before the focus ( @xmath22 m in fig [ fig : data_vs_simulation ] ) , c ) taken at the position of the focus , and d ) taken 25 @xmath0 m after the focus ( @xmath23 m in fig [ fig : data_vs_simulation ] ) . the black scale bars in a ) , b ) , d ) indicate a length of 10 @xmath0 m , which is the width of image c ) . the linear colormap for each image is normalized from white to black for minimum to maximum value respectively.,width=321 ] optical measurements were taken for four mirrors with different focal lengths , all with 10 @xmath0 m radii apertures . two dimensional scans were taken , with 77 nm pixel size , of the reflected intensity field in defined planes parallel to the surface of the substrate , fig . [ fig : raw_data ] . a series of such scans , separated by 1 @xmath0 m , then return a high - resolution , 3d map of the intensity field . the raw data were then radially averaged about the optical axis of the beam . the optical axis was independently found by fitting a 2d gaussian to individual scans near the focal plane and then linearly extrapolating the center of mass fits to regions where a simple 2d - gaussian fit was not valid , _ e.g. _ , such as in fig . [ fig : raw_data](b ) . ( note , however , that about the focus , fig . [ fig : raw_data](c ) , a 2d - gaussian gives an excellent fit to the data . ) this step was taken to ensure any residual tilts in the optical system were accounted for . the averaged data were then combined , as shown in fig . [ fig : data_vs_simulation](a ) , to visualize the cylindrically symmetric intensity field . the focal length was extracted from the distance between the surface and the peak of the intensity , as well as by the best fit of the numerical model , fig . [ fig : data_vs_simulation](b ) . these data show excellent qualitative agreement as the detailed structure of the field away from the focus , which is emphasized by the logarithmic scale on the color axis . to further illustrate the agreement between experiment and theory the on - axis intensity profile of the data and the simulation are shown in fig . [ fig : data_vs_simulation](c ) . m mirror , using a log - scale to highlight the detail in the intensity field away from the focus . the data in each figure are each normalized , with the same color - axis for both . the color - axis , [ -4 0 ] , corresponds to a range of four orders of magnitude in intensity . the measured ( solid blue line ) and simulated ( dashed line ) on - axis intensity profiles , c ) , and radial intensity profiles at the focus , d ) , show agreement between experiment and model . also included in d ) , as a red dotted line , is a gaussian fit to the simulated data.,width=453 ] quantitative results from the measurements are in table [ table ] , below . the measured waists given are the minimum e@xmath24 radii returned from gaussian fits about the waist . the waists in the orthogonal directions were found to be equal and to occur in the same place , within measurement error . it can be seen in fig . [ fig : data_vs_simulation](d ) that a gaussian is an excellent fit to both simulation and experimental data near the focus . the diffraction limit was calculated as @xmath25 , where @xmath26 is the numerical aperture , @xmath27 which was derived using the measured focal length @xcite . it is clear that there is good agreement between the measured minimum beam waists and the prediction from diffraction theory . .measured beam waists ( e@xmath24 radius ) and focal lengths for a range of parabolic mirrors with 10 @xmath0 m radius aperture . errors , where quoted , are one standard deviation . [ cols="^,^,^,^,^ " , ] the intensity patterns observed in these data suggest that the these mirrors would be favorable for trapping of small clouds of ultra - cold atoms , or even single atoms @xcite . for a total power of 1 w in the illuminating beam we estimate a peak intensity at the focus of @xmath28 w / m@xmath29 ; with a dipole trapping wavelength of @xmath30 nm this corresponds to a trap depth of 370 @xmath0k for rubidium atoms @xcite . the trap depth would scale at least with the square of the aperture of the mirrors ( the power captured scales as the aperture diameter squared , while the peak intensity increases with the numerical aperture ) meaning that deeper traps could readily be designed . two dimensional arrays of single trapped atoms could be formed from one laser beam illuminating a congruent 2d array of mirrors . a feature of this array is that the illumination of individual mirrors would then provide single atom addressability , but on the length scale of the mirror aperture , which is inherently larger than the single atom focus . m. the blue dotted line corresponds to the position of peak intensity , the spot of arago , after an open aperture of diameter equal to that of the simulated mirror.,width=377 ] although we have shown , in fig . [ fig : data_vs_simulation ] , that the angular spectrum method yields agreement with the measured data in the regime probed here , further simulations show that focusing power of parabolic micro - mirrors is limited at longer focal lengths . in fig . [ fig : focal_vs_simulation ] the position of peak intensity as a function of mirror design focal length shows that diffraction from the aperture edge begins to dominate at focal lengths significantly longer than the aperture radius . in the limit of large focal length the location of the peak intensity tends towards that predicted for diffraction from the edges ; this is shown in fig . [ fig : focal_vs_simulation ] by the dotted line which is found from simulating diffraction from an aperture of radius equal the that of the mirrors . for the shortest focal length measured in this work , @xmath31 m , we derive a numerical aperture of @xmath32 for operation in water . as shown in @xcite by merenda _ , stable 3d trapping is to be expected in this regime . further work will be conducted to demonstrate optical tweezing using these mirrors . construction and diffraction limited focusing from parabolic micromirrors with radii of @xmath33 m ( @xmath34 ) has been demonstrated . simulations performed using the angular spectrum method show clear agreement with data and reproduce detailed features evident in the measured 3d intensity field . the results offer promise for use of these micromirrors for experiments in optical trapping in both biological and atomic - physics experiments . the authors are extremely grateful to alastair sinclair for initial project design and for discussions , to jonathan pritchard for help with simulations , and to john harris for assistance . pfg acknowledges the generous support of the royal society of edinburgh .
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we report on the fabrication and diffraction - limited characterization of parabolic focusing micromirrors . sub micron
beam waists are measured for mirrors with 10-@xmath0 m radius aperture and measured fixed focal lengths in the range from 24 @xmath0 m to @xmath1 m .
optical characterization of the 3d intensity in the near field produced when the device is illuminated with collimated light is performed using a modified confocal microscope .
results are compared directly with angular spectrum simulations , yielding strong agreement between experiment and theory , and identifying the competition between diffraction and focusing in the regime probed .
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the internet has become a near indispensable tool with both private individuals and organizations becoming increasingly dependent on internet - based software services , downloadable resources like books and movies , online shopping and banking , and even social networking sites . the issue of network security has become significant due to the prevalence of software with malicious or fraudulent intent . malware is the general term given to a broad range of software including viruses and worms designed to infiltrate a computer system without the owner s permission @xcite@xcite . cohen s conclusion in his 1987 paper that computer viruses are potentially a severe threat to computer systems @xcite is still valid in real networks today @xcite@xcite@xcite . current security systems do little to control the spread of malicious content throughout an entire network @xcite@xcite . most security systems are designed to protect a single computer unit . these properly protected units make up only a fraction of online computers . these highlight the necessity of examining the dynamics of the spread of malware in order to be able to develop proper control strategies . studies on the spread of malware in computer networks date back to the late 1980s @xcite and are generally based on the mathematical approach to the spread of diseases in biological populations . math models developed for spread of malware within a computer network such as the kephart - white model and other models adapted from it are based on the kermack - mckendrick model . these models have an implicit assumption that all nodes in the network are always available for `` contact '' @xcite@xcite . however , it is a basic limitation of malware that it can only be passed on to another computer if there is a path through which information can be passed @xcite , so the states of the nodes of the network whether they are online or offline have an effect on the dynamics of the spread . in this work , we model the spread of malware utilizing an ising system to represent an isolated computer network . the state of each node is a composite of its connection status and health . the spin state of a node defines its connection status to be either online or offline . connections are established with the premise that autonomous networks configure themselves @xcite . the health status describes whether a node has been infected or not , and infection can propagate only among online nodes . the ising model was originally intended for simulating the magnetic domains of ferromagnetic materials . its versatility has allowed it to be applied to other systems wherein the behavior of individuals are affected by their neighbors @xcite@xcite@xcite . it has been applied to networks and network - like systems @xcite such as neural networks @xcite@xcite , cooperation in social networks , and analysing trust in a peer - to - peer computer network @xcite . a computer network is modeled by an @xmath0 ising spin system . associated with each node is a spin @xmath1 corresponding to two possible states : @xmath2 for online and @xmath3 for offline . the local interaction energy is given by @xmath4 the interaction parameter , @xmath5 , determines the degree and type of dependence of @xmath1 on its neighbors . the nearest neighbors or local neighborhood are defined according to the network topology and are usually von neumann or moore neighborhoods @xcite@xcite . summing up all local energies gives the total energy , @xmath6 , of the system . global energy , @xmath6 , is associated with network efficiency and more efficient networks are characterized by lower energies . note that while interaction energies are explicitly dependent on the nearest neighbors , the state of each node is implicitly dependent on the state of the entire system . a node will change its configuration provided that the new energy of the system is lower than the previous . if the resulting energy is higher , the new configuration is accepted with probability @xmath7 in the standard ising procedure , @xmath8 is the change in energy , @xmath9 is temperature , and @xmath10 is the boltzmann constant . here , @xmath9 relates to network traffic . to model the spread of infection , each node is assigned a health status separate from its spin . the health status is either infected or susceptible . every online susceptible has a probability @xmath11 of becoming infected , where @xmath12 offline nodes do not transmit or receive data . hence , they do not participate in the infection part . [ [ program - specifics ] ] program specifics + + + + + + + + + + + + + + + + + the computer network is a @xmath13 lattice . nearest neighbors are defined to be the four adjacent nodes . the interaction parameters are all set to @xmath14 . eq.[generalising ] becomes @xmath15 for the interaction energy calculations , circular boundary conditions are imposed . parameters are scaled such that @xmath16 . initially , all nodes are offline ( @xmath17 ) . every time step , the entire system is swept in a left - to - right top - to - bottom fashion , evaluating each node for a possible change in state . the mean energy per node @xmath18 of each configuration is stored and averaged at the end of the run . the spread of infection begins with a single infective . at @xmath19 , one node is selected at random and infected . as the infection spreads , the number of susceptibles , @xmath20 , and infectives , @xmath21 , for each time step are stored . because no means for removal of infection is provided , all nodes eventually become infected . it is at this time that the program is terminated . the model was tested for @xmath9-values ranging from @xmath22 to @xmath23 . the infection curves of five trials were averaged for each @xmath9 . the average infection curve was normalized by dividing it by the total number of nodes to get the fraction of infectives @xmath24 . because it can no longer be assumed that nodes are always available for connection , a regular decay equation is used to model the fraction of infectives curve . a system with @xmath25 nodes has @xmath20 susceptibles and @xmath21 infectives at time @xmath26 . within the time - frame @xmath27 , the number of susceptibles being converted to infectives is @xmath28 . as time passes , @xmath28 decreases as the population of susceptibles is exhausted . thus , the probability of conversion , given by @xmath29 decreases with time . in equation form , this is @xmath30 where @xmath31 is the decay constant . the solution to eq.[decayeq ] is @xmath32 where @xmath33 , the initial number of susceptibles , is just the total number of units in the system . using these , the expression for the number of infectives , @xmath21 may be written as @xmath34 this may be normalized to @xmath35 note that the actual rate of spread varies with time , and @xmath31 provides a measure of the average rate of spread . the fits were made using the unweighted levenberg - marquardt algorithm of gnuplot ver.4.2 @xcite initialized with @xmath36 . for consistency , because some runs terminate very rapidly , we consider only the first 50 time - steps . during the first 50 iterations _ : the rate of spread of the infection increases with @xmath9 . for the above graphs , the resulting decay constants are:@xmath37 , @xmath38 , @xmath39 , @xmath40 , @xmath41 , and @xmath42 , width=240 ] from fig.[allcompare ] , it appears that the spread of infection becomes faster as @xmath9 increases . for @xmath43 and @xmath44 , the rates of spread are very slow , neither reaching @xmath45-infected at the last iteration . particularly , for @xmath43 , no new infectives were produced . these low - traffic systems are not dynamic as nodes have a low probability of coming online from their initial offline state . the network is also very efficient , @xmath46 and @xmath47 , which may be interpreted as information exchange being limited to necessary transactions . for this reason , there is little information exchange and hence a slow spread . for very high @xmath9 , as in @xmath48 and @xmath23 , the spread is rapid and nearly @xmath49 infection is reached . this suggests that very high traffic means a large volume of information exchange that leads to a faster spread of infection . the system is also inefficient at very high @xmath9 , with @xmath50 . it is worth mentioning that the average infection curves of @xmath51 and @xmath52 nearly coincide indicating rates of spread that are very similar . -dependence of rates _ : the increase in the rate of infection corresponds with the decrease in efficiency in the network . note that @xmath6-values are negative.,width=240 ] the observations are supported by the calculated decay constants . the calculated @xmath31 initially increases with traffic but is capped off at very high @xmath9 where it becomes constant . this behavior is similar to the saturation region in a traffic network where flux saturates at high densities . the saturation region indicates that information exchange is no longer freely flowing and that some kind of congestion has occurred @xcite . in fig . [ exponents ] , there is an evident transition that occurs in both the average rate of spread and the efficiency of the network . at the `` congested '' region , the efficiency of the network is very low ; while at the `` free flow '' region , the efficiency of the network is comparatively high . congestion occurs because networks can only handle a limited amount of traffic in the form of data packets . when there is too much traffic , the network is forced to store or drop packets making it inefficient @xcite@xcite . an increase in packet loss with increasing data traffic is reflected by the decrease in efficiency at the congestion region . the congestion is most likely a result of the limited size of the network and the `` finite - size effect '' may be confirmed by testing a larger network@xcite . our ising model approach accounts for the connection status of nodes in an infected network . unlike most epidemic models where all nodes are assumed to be always connected , the model allows malware propagation only among online nodes . we found that the rate of infection becomes faster in less efficient networks with higher data traffic and saturates as the network becomes congested . deepayan chakrabarti , yang wang , chenxi wang , jurij leskovec , and christos faloutsos . , `` epidemic thresholds in real networks '' , _ acm transactions on information and systems security _ , vol . 10 , no . 4 , january 2008 , article 13 jeffrey o. kephart , et al . `` biologically inspired defenses against computer viruses '' _ proceedings of the 14th international joint conference on artificial intelligence _ vol . 1 , montreal , quebec , canada , 1995 , pp.985 - 996
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we introduce an ising approach to study the spread of malware .
the ising spins up and down are used to represent two states online and offline of the nodes in the network .
malware is allowed to propagate amongst online nodes and the rate of propagation was found to increase with data traffic . for a more efficient network ,
the spread of infection is much slower ; while for a congested network , infection spreads quickly .
computer networks ; computer viruses ; epidemiology
| 2,797 | 111 |
hera deeply inelastic scattering ( dis ) results on structure functions demonstrate a rapid bremsstrahlung growth of the gluon density at small x. when interpreted in the same framework as the parton model , this growth is predicted to saturate because the gluon occupation number in hadron wave functions saturate at a value maximally of order @xmath1 ; dynamically , nonlinear effects such as gluon recombination and screening by other gluons deplete the growth of the gluon distribution@xcite . gluon modes with @xmath2 are maximally occupied , where @xmath3 is a dynamically generated semi - hard scale called the saturation scale . for small @xmath4 , @xmath5 is large enough that high occupancy states can be described by weak coupling classical effective theory@xcite . this color glass condensate description of high energy hadrons and nuclei is universal and has been tested in both dis and hadronic collisions . in particular , saturation based phenomenological predictions successfully describe recent lhc p+p data @xcite and predict possible geometrical scaling of transverse momentum distribution@xcite similar to the geometrical scaling observed previously in dis . the object common to dis and hadronic collisions is the dipole cross section @xmath6 . in the cgc framework , the dipole cross section can be expressed in terms of expectation values of correlators of wilson lines representing the color fields of the target . the energy dependence of this quantity comes from renormalization group evolution but to get the realistic impact parameter dependence one has to rely on models involving parametrizations constrained by experimental data . in the large @xmath7 limit , the dipole cross section is related to the un - integrated gluon distribution inside hadron / nucleus as @xmath8^{2}. \label{eq : unint - gluon}\ ] ] for hadron - hadron collisions , the inclusive gluon distribution which is @xmath9-factorizable into the products of un - integrated gluon distributions in the target and projectile is expressed as @xmath10 two models of the dipole cross - section that have been extensively compared to hera data are the ip - sat @xcite and the b - cgc @xcite models . in the former the impact parameter dependence is introduced through a normalized gaussian profile function @xmath11 and in the latter through a scale @xmath12 . for a detailed discussion of the parameters involved in these models and their values from fits to hera data , see ref . @xcite . the saturation scale in the fundamental representation for both the models can be calculated self consistently solving @xmath13=2(1-e^{-1/2})$ ] . the corresponding adjoint saturation scale @xmath14 , relevant for hadronic collisions , is obtained by multiplying @xmath15 by 9/4 . in the range @xmath16-@xmath17 , the behaviour of @xmath14 ( see fig.[fig : satscale ] left ) at @xmath18 can be approximated by a function of the form @xmath19 with @xmath20 for the b - cgc model and @xmath21 for the ip - sat model . [ fig : multdist ] multiparticle production in high energy hadronic collisions can be treated self consistently in the cgc approach . the glasma flux tube picture @xcite predicts @xcite that the n - particle correlation is generated by the negative binomial distribution @xmath22 . it is characterized by two parameters , the mean multiplicity @xmath23 and @xmath24 . at a given impact parameter of the collision , the mean multiplicity @xmath25 is obtained by integrating eq . [ eq : ktfact1 ] over @xmath26 . in the glasma picture , the parameter @xmath27 with @xmath28 @xcite . the quantity @xmath29 shown in fig.[fig : satscale ] ( right ) is the number of flux tubes in the overlap area @xmath30 of two hadrons . convolving @xmath31 with the probability distribution @xmath32 for an inelastic collision at @xmath33-fig . [ fig : multdist ] ( left)-one obtains @xcite the n - particle inclusive multiplicity distribution as shown in fig . [ fig : multdist ] ( right ) . various kinematic variables exhibit scaling with the saturation scale@xcite . the mid - rapidity multiplicity density scales with functional forms like @xmath34 and @xmath35 whereas a linear functional form seem to provide very good fit to the energy dependence of @xmath36 as shown in fig.[fig : scaling][left ] . these results are suggestive that @xmath37 is the only scale that controls the bulk particle multiplicity . in ref . @xcite it has been shown that @xmath26 spectra in @xmath38 collisions exhibit geometric scaling assuming a simple form of @xmath37 . in our case we use a scaling variable @xmath39 , where @xmath37 is directly calculated in the ip - sat model . as shown in fig.[fig : scaling][right ] , an approximate scaling below @xmath40 is observed for transverse momentum distribution in @xmath38 collision energy @xmath41 gev . going to lower energies we observe systematic deviations from the universal curve . + in summary , our description of multiplicity distribution successfully describes bulk lhc p+p data . in particular , we observe that the dominant contribution to multiplicity fluctuations is due to the intrinsic fluctuations of gluon produced from multiple glasma flux tubes rather than from the fluctuations in the sizes and distributions of hotspots . the @xmath26-spectra in p+p at high energies exhibits universal scaling as a function of @xmath39 . the observed scaling indicates that particle production in this regime is dominantly from saturated gluonic matter characterized by one universal scale @xmath37 . ridge like two particle correlation structures in @xmath42 in high multiplicity p+p collisions may provide more detailed insight into its properties @xcite . v. khachatryan _ et al . _ [ cms collaboration ] , phys . lett . * 105 * , 022002 ( 2010 ) . k. aamodt _ et al . _ [ alice collaboration ] , eur . j. c * 68 * , 345 ( 2010 ) . a. dumitru , k. dusling , f. gelis , j. jalilian - marian , t. lappi , r. venugopalan , arxiv:1009.5295 [ hep - ph ] .
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dipole models based on various saturation scenarios provide reasonable fits to small - x dis inclusive , diffractive and exclusive data from hera .
proton un - integrated gluon distributions extracted from such fits are employed in a @xmath0-factorization framework to calculate inclusive gluon distributions at various energies .
the n - particle multiplicity distribution predicted in the glasma flux tube approach shows good agreement with data over a wide range of energies .
hadron inclusive transverse momentum distributions expressed in terms of the saturation scale demonstrate universal behavior over a wider kinematic range systematically with increasing center of mass energies .
saturation ; lhc p + p collision ; cgc ; deep inelastic scattering
| 1,757 | 177 |
it is widely believed that there are supermassive black holes at the centers of galaxies , and these are hypothesized to be the central engines for active galactic nuclei ( agns ) and gamma ray bursts ( grbs ) . two main possibilities are considered as the energy source . one is the gravitational energy of accreting matter and the other is the rotational energy of the black hole or the accretion disk surrounding it . however , the details of the energy extraction process are not clear . it is also not understood well how the energy is converted into that of agns or grbs . blandford and znajek showed that the rotational energy of a rotating black hole can be extracted in the form of poynting flux along magnetic field lines penetrating the event horizon @xcite , which is known as the blandford - znajek ( bz ) mechanism . its efficiency depends on the rotational velocity of the black hole and the configuration of the magnetic field : the extraction of the rotational energy becomes more efficient the more magnetic field lines penetrates the event horizon and the more rapidly the black hole rotates . in the bz mechanism , poloidal magnetic fields which penetrate the event horizon play a crucial role for the energy extraction as well as for the formation of jets associated with agns . in fact , some numerical studies reported that poynting - dominated jets were produced @xcite . bick and janis showed that a magnetic field without an electric current is expelled from the event horizon of a maximally rotating black hole @xcite . this is analogous to the meissner effect in a superconductor . this effect for a rapidly rotating black hole would decrease the efficiency of the bz mechanism , though the larger rotational velocity of the black hole would increase the efficiency . in realistic astrophysical cases , however , there would be plasma around the black hole . how the meissner - like effect is affected by the existence of plasma is the main subject of this paper . we clarify the effect of an electric current on the meissner - like effect of an extreme black hole . komissarov and mckinney studied numerically the meissner - like effect of a kerr black hole @xcite . they carried out numerical simulations for a highly conductive magnetosphere until it almost reaches steady state , and there was no sign of the meissner - like effect in their numerical results . in this paper , we study how an electric current affects the meissner - like effect by solving a stationary problem analytically . since realistic situations are , in general , very complicated , it is difficult to model them . in order to reveal the essence of the plasma effect , we consider a very simple toy model : ( i ) we consider a stationary , axisymmetric force - free system of the electromagnetic field and plasma ; ( ii ) we consider a static spherically symmetric black hole spacetime with a degenerate horizon as a background spacetime rather than a rotating black hole . the degenerate horizon is the origin of the meissner - like effect in a vacuum black hole spacetime @xcite , and hence , by studying the electromagnetic field in this spacetime , we can see whether the meissner - like effect remains even in the case with an electric current . the spacetime considered in this paper is known as the reissner - nordstrm ( rn ) spacetime . by these assumptions , the basic equations reduce to only one quasi - linear elliptic equation for the magnetic flux function called the grad - shafranov ( gs ) equation @xcite . for the black hole spacetime , the gs equation has three regular singular points : one is at the event horizon , and the other two are at the inner and outer light surfaces on which the velocities of the magnetic field lines agree with the speed of light . for non - extreme cases , one boundary condition is imposed at each regular singular point so that the magnetic field is smooth everywhere . however , for a given electric current function , the obtained solution for the magnetic flux need not be @xmath0 but at most @xmath1 @xcite . although numerical @xmath0 solutions have been obtained by iteratively changing the functional form of the electric current @xcite , a mathematically rigorous proof for the existence of a @xmath0 solution has not yet been presented . furthermore , in the extreme case , two kinds of boundary condition must be imposed at once on the event horizon . we shall mention all these difficulties in solving the gs equation in iv . as will be shown in v , the monopole component is a unique configuration of the magnetic field on the event horizon if there is not an electric current . since there is no magnetic monopole in nature , this result implies the meissner - like effect of the extreme rn black hole . in order to study the electromagnetic field coupled to an electric current around an rn black hole , we use a perturbative method which includes two expansion parameters . one of these parameters corresponds to the rotational angular velocity of the magnetic fields . namely , we consider slow - rotating magnetic fields as was first considered by blandford and znajek @xcite . the other parameter is the ratio of the distance from the event horizon to the horizon radius , since we consider only the vicinity of the event horizon , which includes the inner light surface . although we can not take into account the outer light surface in our perturbative method , we can obtain approximate solutions sufficient to study the meissner - like effect with an electric current . this paper is organized as follows . in ii , we introduce the rn black hole as a background geometry . then we show the gs equation for the rn spacetime in iii ; the detailed derivation of the gs equation is given in appendices a and b. the regularity conditions for the gs equation and difficulties in solving this equation are described in detail in iv . using perturbative analyses , we study the cases with and without an electric current in v and vi , respectively . vii is devoted to summary and discussion . in appendix c , we show the relation between the kerr - schild coordinate system and the standard static coordinate system of the rn spacetime . in appendix d , we give a proof of a theorem on the magnetic field obtained by the present perturbative method . in this paper , we adopt the geometrized units , in which the newton s gravitational constant and the speed of light are unity , and the abstract index notation : small latin indices , excluding @xmath2 and @xmath3 , indicate the type of tensor , whereas small greek indices , excluding @xmath4 and @xmath5 , represent components with respect to the coordinate basis . the exceptional indices @xmath2 , @xmath3 , @xmath4 , and @xmath5 denote the components of time , and the radial and azimuthal coordinates in the spherical polar coordinate system . the signature of the metric is diag@xmath6 $ ] . we consider a static and spherically symmetric spacetime of the following metric : @xmath7 with @xmath8 where we assume @xmath9 . this spacetime is known as the rn spacetime . there are two horizons , which are determined by @xmath10 : @xmath11 and @xmath12 represent the radius of the event and cauchy horizons , respectively . the case of @xmath13 is called the extreme case . maxwell s equations are given by @xmath14 } & = & 0 , \label{eq : maxwell - eq-1}\\ \nabla_bf^{ab } & = & 4\pi j^{a } , \label{eq : maxwell - eq}\end{aligned}\ ] ] where @xmath15 is the field strength tensor of the electromagnetic field , @xmath16 is a current density , and @xmath17 is the covariant derivative . if the field strength tensor is expressed by using a 4-vector potential @xmath18 as @xmath19 then eq . ( [ eq : maxwell - eq-1 ] ) is trivially satisfied , where @xmath20 is the ordinary derivative . as mentioned in @xmath21i , hereafter , we consider the axisymmetric and stationary electromagnetic field in the rn spacetime . in order to make the problem simple , we assume that the system field satisfies the force - free condition @xmath22 hereafter , we focus on the system of only eqs . ( [ eq : maxwell - eq ] ) and ( [ eq : ff - condition ] ) . the formulation of a force - free electrodynamics field in the black hole spacetime was given by macdonald and thorne @xcite . our formulation is based on their work . in the case of a stationary , axisymmetric electromagnetic field , we can define the `` angular velocity '' of the magnetic field as @xmath23 the reason why @xmath24 can be regarded as the angular velocity of the magnetic field is described in appendix [ derivation ] . from eqs . ( [ eq : maxwell - eq ] ) and ( [ eq : ff - condition ] ) , the gs equation is obtained as @xmath25 where a prime @xmath26 represents a derivative with respect to @xmath3 , @xmath27 @xmath28 where @xmath29 and @xmath30 are the electric current and the magnetic flux through an axisymmetric polar cap , which are defined by eqs . ( [ eq : i - def ] ) and ( [ eq : psi - def ] ) , respectively , and @xmath31 is defined by @xmath32 the derivation of the gs equation ( [ eq : gs - eq ] ) is given in appendices [ derivation ] and [ i - psi ] . in general , the gs equation for a black hole magnetosphere has three regular singular points : one is at the event horizon @xmath10 , and the other two are at the light surfaces defined by @xmath33 . a light surface is a timelike hypersurface on which the rotational speed of magnetic field lines are equal to the speed of light . the inner light surface given by a function @xmath34 has a spacelike section with spherical topology , whereas the outer one , @xmath35 , has that of cylindrical topology . in the case of the kerr spacetime , the kerr - schild coordinate system is often adopted , for example , in numerical simulations ( e.g. , @xcite ) , since there is no coordinate singularity on the event horizon . therefore , we give the kerr - schild coordinate system for the rn spacetime in appendix [ ks ] . in appendix [ ks ] , we show that the singular point on the event horizon in the gs equation appears even if we adopt kerr - schild coordinates . this is also true in the case of the kerr spacetime , though it is not shown in this paper ( we will show it elsewhere ) . as long as a stationary magnetic field is considered , the singular point of the basic equation will appear on the event horizon , since there is no timelike killing vector field on or inside the event horizon , or in other words , the stationary configuration can not be realized inside the black hole . in this section , we consider only the case of @xmath36 . then , by virtue of the symmetry of the background spacetime , without loss of generality , we may assume @xmath37 . the case of @xmath38 will be treated as specific cases later . on the symmetry axis @xmath39 , both @xmath30 and @xmath40 should vanish by their definitions . @xmath40 is a function of @xmath30 , and thus , @xmath41 should vanish . in appendix [ ks ] , we give the components of @xmath15 in the kerr - schild coordinate system @xmath42 . from eq . ( [ eq : rt - comp ] ) , we have @xmath43 where @xmath44 since the kerr - schild coordinate system is non - singular on the event horizon , @xmath45 must be finite there . thus , we have @xmath46 . this leads to @xmath47 the above condition corresponds to the horizon boundary condition derived by znajek for the kerr black hole @xcite . it is seen from the gs equation ( [ eq : gs - eq ] ) that , in order that @xmath30 , @xmath48 and @xmath49 are finite on the event horizon @xmath50 , the following condition should be satisfied @xmath51 however , this condition is satisfied if @xmath30 satisfies the regularity condition ( [ eq : h - regularity ] ) , and thus no additional constraint is imposed by this equation . in the extreme case , since the equation @xmath10 has a double root @xmath52 , @xmath53 should vanish as well as @xmath54 so that @xmath45 is finite on the event horizon . these conditions imply @xmath55 on the event horizon @xmath56 . we can see from the gs equation ( [ eq : gs - eq ] ) that , in order that @xmath30 , @xmath48 , and @xmath49 are finite on the event horizon for the extreme case , not only eq . ( [ horizon ] ) but also the following condition should be satisfied , @xmath57 this condition is satisfied if @xmath30 satisfies the regularity conditions ( [ eq : h - regularity-1 ] ) and ( [ eq : h - regularity-2 ] ) , and thus no additional constraint is imposed by this equation . as mentioned , the light surfaces are singular points of the gs equation ( [ eq : gs - eq ] ) . in the extreme case , the radial coordinates of the light surfaces , which are the roots of the equation @xmath33 in the domain @xmath58 , are given by @xmath59 in order that @xmath49 and @xmath60 are finite on the light surfaces , @xmath61 must vanish there . this requirement leads to the following regularity conditions on the light surfaces : @xmath62 where @xmath63 , where @xmath64 and @xmath65 . , scaledwidth=80.0% ] for simplicity , in this subsection , we assume that @xmath24 is constant . by virtue of this assumption , the conditions ( [ eq : neumann ] ) become neumann boundary conditions on the light surfaces . here , we consider the non - extreme case @xmath66 . let us assume that the functional form of @xmath67 has already been determined before solving the gs equation . then , by solving the horizon regularity condition ( [ eq : h - regularity ] ) , we obtain @xmath30 on the event horizon . since @xmath30 is the magnetic flux through the polar cap ( see appendix b ) , @xmath30 should vanish at @xmath39 . thus , there seems to be no freedom for setting a boundary condition for @xmath30 in solving eq . ( [ eq : h - regularity ] ) , but this is not true . since @xmath39 is a regular singular point of eq . ( [ eq : h - regularity ] ) , there still remains one degree of freedom for choosing a boundary value of the second - order derivative of @xmath30 . hence , a dirichlet boundary condition for the gs equation is determined on the event horizon by the regularity condition ( [ eq : h - regularity ] ) . by imposing this dirichlet boundary condition at @xmath50 and further neumann boundary conditions at @xmath68 ( [ eq : neumann ] ) and on the equatorial plane ( e.g. , the reflection - symmetric boundary condition @xmath69 ) , a solution for the gs equation ( [ eq : gs - eq ] ) is uniquely determined in the domain @xmath70 . by imposing two neumann boundary conditions ( [ eq : neumann ] ) on the two light surfaces @xmath71 and the equatorial plane @xmath72 , and a further dirichlet boundary condition @xmath73 on the symmetry axis @xmath39 , a solution for the gs equation is uniquely determined in the domain @xmath74 . for the domain @xmath75 , if we impose a boundary condition at @xmath30 for @xmath76 , then we can obtain a solution for the gs equation . the gs equation can be solved for these three domains , @xmath77 , @xmath74 , and @xmath78 , independently by the above procedure ( see fig . [ fg : non - extreme ] ) . thus , for an arbitrary electric current @xmath67 , the obtained solution for @xmath30 is , in general , not @xmath0 but at most @xmath1 at the boundaries @xmath71 . it was first reported by contopoulos , kazanas , and fendt ( ckf ) that the continuity of the first - order derivative of @xmath30 at the light surface as well as the continuity of @xmath30 itself determines the functional form of the electric current @xmath67 in the case of the pulsar magnetosphere @xcite . they numerically obtained @xmath0 solutions for @xmath30 by an iterative method in which both @xmath30 and the functional form of @xmath67 are determined simultaneously . the ckf method was used by several authors to study the pulsar magnetosphere and they showed that this method also suitable for their case of interest @xcite . however , it should be noted that a mathematically rigorous proof for the existence of the @xmath0 solution for the gs equation has not yet been given . although there is at most one light surface in the case of the pulsar magnetosphere , there can be two light surfaces if @xmath24 is a non - vanishing constant in the case of the black hole magnetosphere ( see fig . [ fg : non - extreme ] ) . thus , if we chose the functional form of @xmath67 such that @xmath30 is @xmath0 in the domain @xmath79 , imposing an asymptotic boundary condition for @xmath76 does not guarantee the continuity of the derivative of @xmath30 at the outer light surface @xmath80 . thus , in order to obtain a solution which is @xmath0 in the domain @xmath81 , we need to solve the gs equation for the outermost domain @xmath82 as a cauchy problem with a boundary data for @xmath30 and the derivatives of @xmath30 on @xmath80 . this implies that we can not impose the asymptotic boundary condition for @xmath83 . in general , it is difficult to solve an elliptic - type differential equation numerically , such as the gs equation , as a cauchy problem due to the numerical instability . thus , in the case with two light surfaces , it is difficult to numerically obtain a solution for the gs equation in the outermost domain @xmath84 . however , this might not be a serious problem , since we may understand whether the blandford - znajek mechanism works by studying only the domain @xmath85 . uzdensky applied the ckf method to the magnetospheres of the schwarzschild black hole @xcite and of the kerr black hole @xcite , but he focused on only the cases in which there is only one light surface by virtue of a particular assumption on @xmath24 : uzdensky assumed that @xmath24 asymptotically decreases , and hence there is no outer light surface . thus , uzdensky succeeded in numerically obtaining global solutions without solving cauchy problems for the gs equation . , but for an extreme rn black hole @xmath86 . , scaledwidth=80.0% ] here , we consider the extreme case @xmath86 , which is the main case of interest in this paper . in this subsection , we also assume that @xmath24 is constant . the horizon regularity conditions ( [ eq : h - regularity-1 ] ) and ( [ eq : h - regularity-2 ] ) give boundary values of @xmath30 and the derivative of @xmath30 . thus , in the extreme case , we must solve the gs equation ( [ eq : gs - eq ] ) as a cauchy problem even for the domain @xmath87 ( see fig . [ fg : extreme ] ) . as mentioned , it is difficult to numerically solve the gs equation as a cauchy problem , and hence , it seems to be difficult to numerically obtain a solution for the physically important domain @xmath88 in the extreme case . further , even if we find a procedure for numerically solving the gs equation as a cauchy problem , the regularity condition on the inner light surface @xmath68 may not be satisfied for an arbitrary functional form of electric current @xmath67 : we must assume the functional form of @xmath67 to solve the gs equation as a cauchy problem , but , in general , the assumed electric current @xmath67 does not satisfy the regularity condition on the inner light surface . as a result , it seems to be impossible to obtain a solution for the gs equation numerically , which is finite on the inner light surface , in the extreme case . in this sense , the perturbative analytic approach discussed in vi is very important . we should note that even if we find analytically the electric current @xmath40 which guarantees the finiteness of @xmath30 and its derivative on the inner light surface , such a electric current @xmath40 might not guarantee the finiteness of both @xmath30 and its derivative on the outer light surface . this implies that either the force - free condition should break down near the outer light surface or the rotational velocity should decay far from the black hole so that the outer light surface does not exist , as in the situation studied by uzdensky . in this section , we consider the vacuum case @xmath89 . here , we consider the case of @xmath38 on the event horizon , @xmath90 . even if @xmath24 does not vanish except on the event horizon , it satisfies @xmath91 thus , on the event horizon , @xmath92 or @xmath93 should hold . in the former case , @xmath30 vanishes on the event horizon , implying that the magnetic flux does not penetrate the event horizon . in the latter case , from eq . ( [ eq : gs - eq ] ) , in order that @xmath30 , @xmath48 , and @xmath49 are finite on the event horizon , the following equation should be satisfied : @xmath94 in the extreme case , since @xmath95 also holds on the event horizon @xmath56 , we have @xmath96 the solution of the above equation which satisfies the regularity condition on the symmetry axis @xmath39 is @xmath97 where @xmath98 is an integration constant . the above solution implies that the magnetic field which can penetrate the event horizon is the only monopole component . by contrast to the extreme case , non - monopole components can penetrate the event horizon in the non - extreme case , since , in this case , eq . ( [ eq : h - regularity-0 ] ) does not necessarily imply @xmath99 . as a result , we can conclude that the meissner - like effect of the extreme black hole appears in the vacuum case . in the case that @xmath24 vanishes everywhere , we can obtain global solutions . the solutions which satisfy the regularity condition on the symmetry axis @xmath39 are written in the form @xmath100 where @xmath101 is the associated legendre function of the first kind with @xmath102 . then , the gs equation ( [ eq : gs - eq ] ) becomes @xmath103 in the extreme case , in order that @xmath104 , @xmath105 , and @xmath106 are finite on the event horizon , @xmath104 of @xmath107 vanishes , and the only monopole component @xmath108 may remain . in this case , from the regularity condition on the event horizon ( [ eq : h - regularity ] ) , we have @xmath109 for both extreme and non - extreme cases . thus , @xmath73 is a solution which satisfies the regularity condition on the symmetry axis @xmath39 . the magnetic field does not penetrate the event horizon at all . in this section , we focus on the extreme case @xmath52 and assume that the rotational velocity of the magnetic field @xmath24 is constant . in the case of @xmath38 , it is seen from eq . ( [ eq : gs - eq ] ) that since @xmath110 , the following condition should be satisfied on the event horizon : @xmath111 the above condition allows @xmath112 on the event horizon . however , since @xmath40 should vanish on the symmetry axis @xmath39 , the allowed constant is zero . thus , in this case , the same argument as was used in the vacuum case discussed in the previous section is also true . the allowed configuration of the magnetic field on the horizon is only the monopole component ( [ eq : monopole ] ) . hence , hereafter , we focus on the case of @xmath36 . we are interested in the configuration of the magnetic field near the event horizon . in order to analyze the gs equation , we introduce the following dimensionless quantities : @xmath113 using these quantities , the gs equation ( [ eq : gs - eq ] ) becomes @xmath114 where @xmath115 , \\ { \cal u}&=&2(y-1-\varepsilon^2y^4\sin^2\theta)\partial_y\psi , \\ { \cal w}&=&-\varepsilon^2 y^5\left[\sin2\theta \partial_\theta\psi -{\cal s}\left(\psi\right)\right].\end{aligned}\ ] ] the regularity conditions on the event horizon ( [ eq : h - regularity-1 ] ) and ( [ eq : h - regularity-2 ] ) become @xmath116 the regularity conditions on the light surfaces ( [ eq : neumann ] ) become @xmath117 where @xmath118 here , we assume that @xmath119 can be written in the form of taylor series around the event horizon , @xmath120 , as @xmath121 the coefficients @xmath122 are , in principle , determined by the gs equation ( [ gs ] ) with the regularity conditions ( [ hbo ] ) , ( [ hbdo ] ) , and ( [ ls ] ) . using the expression ( [ taylor ] ) , eqs . ( [ hbo ] ) and ( [ hbdo ] ) can be rewritten in the forms @xmath123 if we fix the functional form of @xmath124 , we obtain @xmath125 and @xmath126 from the above equations , and further , we obtain @xmath122 of @xmath127 from eq . ( [ gs ] ) ; in order to get @xmath122 for @xmath128 , we use an equation obtained by @xmath129-times differentiation of eq . ( [ gs ] ) with respect to @xmath130 . for example , for @xmath131 , by evaluating the gs equation ( [ gs ] ) on the event horizon @xmath120 , we have @xmath132 \nonumber \\ & & = -2\cot\theta \left(\frac{d\psi^{(0)}}{d\theta}-\frac{{\cal s}(\psi^{(0)})}{\sin2\theta}\right ) \left(\frac{1}{\sin^2\theta}+12\varepsilon^2\right ) -\varepsilon^2\psi^{(1)}\left[2 -\frac{1}{2\sin^2\theta}\frac{d^2{\cal s}}{d\psi^2}(\psi^{(0)}){\psi^{(1)}}\right ] . \label{gs-2 } \nonumber \\\end{aligned}\ ] ] here , we should again note that @xmath124 can not be freely specified . the functional form of @xmath124 must be chosen such that the regularity condition ( [ ls ] ) on the inner light surface is satisfied . we consider slowly rotating magnetic fields , or in other words , we assume @xmath133 . we rewrite the basic equations in the form of the power series with respect to @xmath134 , and then we construct a solution of @xmath119 on the horizon , i.e. , @xmath125 , by perturbative procedures with respect to @xmath134 . although , as mentioned , it seems to be impossible to determine the functional form of @xmath135 numerically , we can find it by this method . in order to construct a perturbative solution for @xmath122 , we write @xmath136 further , we assume @xmath137 from eq . ( [ rls ] ) , the location of the inner light surface is written as @xmath138 because @xmath139 , we can express the quantities on the inner light surface by using the quantities on the event horizon . for example , @xmath140 at @xmath141 is written as @xmath142 since the main purpose of this study is to see the effect of an electric current on the configuration of the magnetic field on the event horizon , we focus on @xmath125 . for this purpose , we rewrite eqs . ( [ hbd ] ) and ( [ gs-2 ] ) in more appropriate forms as follows . by differentiating eq . ( [ hb ] ) with respect to @xmath4 , we have @xmath143 from eq . ( [ hbd ] ) , we have @xmath144 by subtracting eq . ( [ hb - diff ] ) from the above equation , we obtain @xmath145 it is easy to integrate the above equation , and we have @xmath146 where @xmath147 is an integration constant . in order to obtain @xmath126 , we use eq . ( [ eq : psi1 ] ) rather than eq . ( [ hbd ] ) . by substituting eq . ( [ hb ] ) into eq . ( [ hb - diff ] ) and using eqs . ( [ cali - def ] ) and ( [ cals - def ] ) , we have @xmath148 the above equation is equivalent to eq . ( [ horizon ] ) . by substituting eq . ( [ horizon-2 ] ) into the first term on the right - hand side of eq . ( [ gs-2 ] ) , we obtain @xmath149 . \label{eq : psi0}\end{aligned}\ ] ] we shall use the above equation rather than eq . ( [ gs-2 ] ) . here , we obtain the zeroth - order solutions for @xmath125 . hereafter , the arguments of @xmath150 and @xmath151 are @xmath152 as long as we do not specify them . first of all , we write down the equations to obtain the zeroth - order solutions for @xmath125 . from the lowest order of eqs . ( [ ls ] ) , ( [ hb ] ) , and ( [ eq : psi0 ] ) , we have @xmath153 where from eqs . ( [ cali - def ] ) , ( [ cals - def ] ) , and ( [ assumptions ] ) , @xmath154 we can easily integrate eq . ( [ eq : psi00 ] ) and obtain @xmath155 where @xmath156 is an integration constant . the above result implies that @xmath152 has only the monopole component . then , substituting eq . ( [ 00-sol ] ) into eq . ( [ hb0 ] ) , we have @xmath157 where @xmath158 it is non - trivial whether the lowest order of the inner light surface regularity condition ( [ ls00 ] ) is satisfied by @xmath152 and @xmath159 obtained above . from eqs . ( [ i1 ] ) and ( [ s2 ] ) , we have @xmath160 it is easy to check that eqs . ( [ 00-sol ] ) and ( [ s0-sol ] ) satisfy eq . ( [ ls00 ] ) . namely , we have obtained a small electric current which satisfies the lowest order of the inner light surface regularity condition . it is worthwhile to notice the meaning of the zeroth - order solutions for @xmath125 , i.e. , @xmath152 . in the limit @xmath161 , @xmath24 and @xmath40 become zero , whereas @xmath125 becomes @xmath152 . since the case @xmath162 corresponds to the vacuum case , i.e. , the case without an electric current , @xmath152 corresponds to the vacuum solution . as we showed in v , the vacuum solution has only the monopole component on the event horizon . ( [ 00-sol ] ) is consistent with this fact . the small electric current @xmath163 can be regarded as a result of the slowly rotating monopole field @xmath152 . next , we consider the correction of @xmath164 to the zeroth - order solution for @xmath125 . hereafter , we assume @xmath165 . in our perturbative method , the case of @xmath166 is quite different from the case @xmath167 . if we choose @xmath166 , we can obtain only the trivial solution @xmath168 using our perturbative method . we prove this statement in appendix [ proof ] . the equations determining @xmath169 are derived from eqs . ( [ ls ] ) , ( [ hb ] ) , and ( [ eq : psi0 ] ) of @xmath164 : @xmath170 where @xmath171 we can see from eq . ( [ eq : psi01 ] ) that @xmath169 is also the monopole solution . thus , the first - order correction merely adds a constant of @xmath164 to the integration constant of the zeroth - order solution . as a result , without loss of generality , we may assume for the first - order solutions that @xmath172 from eqs . ( [ hb1 ] ) , ( [ s3 ] ) , and the above equation , we have @xmath173 we should check the inner light surface regularity condition ( [ ls ] ) . since @xmath174 , eq . ( [ ls1 ] ) becomes @xmath175 in order to estimate the above equation , we need @xmath176 , which is the zeroth - order solution for @xmath126 . we obtain @xmath176 from eq . ( [ eq : psi1 ] ) of @xmath177 as @xmath178 where we assume that @xmath147 is the order of unity . substituting eq . ( [ 00-sol ] ) into the above equation , we obtain @xmath179 by substituting eq . ( [ 10-sol ] ) into eq . ( [ ls10 ] ) and using the functional form of @xmath180 given by eq . ( [ s0-sol ] ) , we can see that eq . ( [ ls10 ] ) is satisfied . here it is worthwhile to notice that @xmath176 , as well as @xmath152 , necessarily corresponds to a vacuum solution . it is easy to check that eq . ( [ 10-sol ] ) is consistent with eq . ( [ eq : non - monopole ] ) . hereafter , we will make frequent use of eqs . ( [ ls00 ] ) and ( [ ls10 ] ) without giving an explicit reference . now , we consider the correction of @xmath181 to the zeroth - order solution for @xmath125 . the equations determining @xmath182 can be obtained from eqs . ( [ ls ] ) , ( [ hb ] ) , and ( [ eq : psi0 ] ) of @xmath181 : where @xmath184 in order to derive the above equations , we have used @xmath185 and @xmath186 obtained by substituting @xmath187 into eq . ( [ eq : psi1 ] ) . it should be noted that @xmath188 appears in eqs . ( [ ls02 ] ) and ( [ eq : psi02 ] ) . in order to determine @xmath188 , we use eq . ( [ horizon-2 ] ) of @xmath181 : @xmath189=0.\ ] ] by substituting eq . ( [ ls02 ] ) into the above equation , we have @xmath190.\ ] ] by subtracting the above equation from eq . ( [ eq : psi02 ] ) , we obtain the equation for @xmath188 as @xmath191 it is easily seen from eq . ( [ 10-sol ] ) that @xmath192 is a particular solution for the above equation . thus , the general solution of the above equation is expressed by a linear combination of this particular solution and general solutions of the following homogeneous equation : @xmath193 the general solution of eq . ( [ homo - eq ] ) is given by @xmath194,\ ] ] where @xmath195 and @xmath196 are arbitrary constants , and @xmath197 and @xmath198 are the associated legendre functions of the first and second kinds with @xmath199 and @xmath102 , respectively . from the boundary condition at @xmath39 , @xmath196 must vanish . hence , the most general solution of eq . ( [ 20-eq ] ) , which satisfies the boundary condition at @xmath39 , is given by @xmath200 where we have used @xmath201 , and @xmath202 is an arbitrary constant . here it is worthwhile to notice that the above result also corresponds to a vacuum solution . it is easy to check that eq . ( [ 20-sol ] ) , as well as @xmath152 and @xmath176 , is consistent with eq . ( [ eq : non - monopole ] ) . by using eq . ( [ s0-sol ] ) and substituting eq . ( [ 10-sol ] ) into eq . ( [ eq : psi02 ] ) , we have @xmath203\sin^4\theta . \label{02-eq}\ ] ] the above equation implies that , in general , the magnetic field on the event horizon includes non - monopole components of @xmath181 . we can easily integrate this equation and obtain @xmath204 , \label{02-sol}\ ] ] where we have chosen the integration constant so that @xmath205 , i.e. , this correction consists of only non - monopole components . the functional form of @xmath206 is determined by using eq . ( [ hb2 ] ) as @xmath207,\end{aligned}\ ] ] where @xmath208 is defined by eq . ( [ xhat ] ) . the solution with the corrections up to @xmath181 behaves near the inner light surface as @xmath209 although the solutions for @xmath210 for @xmath211 have been derived in the previous sections , we again show them with a slightly different parameterization : @xmath212\sin^2\theta , \\ \psi^{(1)}_0&=&c^{(0)}_0c'\sin^2\theta , \\ \psi^{(1)}_1&=&0 , \\ \psi^{(2)}_0&=&\frac{1}{2}\left[3\left(c^{(0)}_0{c'}^2 -10c^{(0)}_2\right)\cos\theta+c^{(0)}_0c'\right]\sin^2\theta , \end{aligned}\ ] ] where @xmath202 is given in this parameterization as @xmath213 by using the same parameterization as the above , the electric current is given by @xmath214,\end{aligned}\ ] ] where @xmath208 is defined by eq . ( [ xhat ] ) . we see that the arbitrary constants are only @xmath215 ( @xmath216 ) and @xmath147 . the reason this result takes this form is because if we choose @xmath119 and @xmath217 on the event horizon such that the regularity conditions ( [ hb ] ) and ( [ hbd ] ) are satisfied , then @xmath119 and @xmath135 are completely determined . we studied a force - free magnetosphere in a static spherically symmetric black hole spacetime with a degenerate event horizon . we have found that if an electric current exists , higher multipole components of the magnetic field can be superposed upon the monopole component on the event horizon even if the two horizons degenerate into one horizon . this result is consistent with the numerical result given by komissarov and mckinney : they showed that the magnetic field lines of higher multipole components can penetrate an extreme kerr black hole if conductivity exists . the detailed geometrical structures of the extreme kerr black hole and the extreme reissner - nordstrm black hole are different from each other . however , since the degenerate structures of the horizons of these black holes are similar , the present results may be applicable to a certain extent for the extreme kerr black hole . if we require that there is no monopole component in the lowest - order configuration on the horizon , or equivalently , @xmath218 , we obtain the trivial solution @xmath168 , even though we take all - order corrections into account ( see appendix [ proof ] ) . thus , the proposition in appendix d seems to imply that there is no non - trivial configuration without a monopole component on the event horizon of the extreme reissner - nordstrm black hole , even if an electric current exists . but this is not necessarily true . in order to see this fact , note that there is an exact monopole solution for the grad - shafranov equation ( [ eq : gs - eq ] ) : @xmath219 with the electric current @xmath220 where @xmath98 is an arbitrary constant . by contrast , the proposition in appendix d implies that , if @xmath218 , there is no higher - order correction by which the configuration of the perturbative solution on the event horizon approaches to the monopole configuration in our perturbation scheme . in other words , our perturbative solution with vanishing @xmath152 can not approach to the above exact solution , even if we take into account all - order corrections . this fact suggests that even if an exact solution with a non - monopole configuration on the event horizon exists , the perturbative solution with vanishing @xmath152 can not approach to such a solution in our perturbation scheme . this possibility may arise from the assumption for the electric current ( [ assumptions ] ) , which may be too strong , though the present analytic perturbation studies are impossible without this assumption . we would like to stress again that it is very difficult to obtain a stationary force - free magnetosphere by solving the grad - shafranov equation for the extreme black hole spacetime numerically . thus , we need to invoke analytic methods , as in the present study , or numerical techniques to follow the dynamical evolution of a force - free maxwell field until a stationary configuration is realized , as komissarov and mckinney used . as discussed in this paper , in the case that there are two light surfaces in the extreme reissner - nordstrm black hole spacetime , even though the magnetic field is regular , both on the event horizon and inner light surface , it will be singular on the outer light surface . if the angular velocity of the magnetic field is constant , two light surfaces necessarily exist . thus , if the dynamical evolution of a force - free maxwell field can be followed until it becomes stationary , then it is expected that the angular velocity decays far from the black hole so that the outer light surface does not exist . the extremity of charge or angular momentum changes the structure of boundary conditions for the grad - shafranov equation and seems to strongly affect global structures of the black hole magnetosphere . finally , we would like to suggest that analytic solutions obtained by this perturbation scheme becomes a benchmark for a numerical scheme to obtain solutions for stationary configurations of astrophysical magnetospheres , since our perturbation scheme is also suitable for non - extreme black hole cases . the authors would like to thank m. takahashi for useful lectures and discussions on black hole magnetospheres . is supported by a jsps grant - in - aid for creative scientific research no . a stationary , axisymmetric electromagnetic field implies @xmath221 . then , from eq . ( [ eq : f - def ] ) and the force - free condition ( [ eq : ff - condition ] ) , we have @xmath222 . using these equations , the components of @xmath15 in the static coordinate system ( [ metric ] ) are written in the form @xmath223 where @xmath24 is defined by eq . ( [ eq : omegaf - def ] ) , and @xmath224 is the determinant of the intrinsic metric of the spacelike hypersurface labeled by @xmath2 . note that @xmath24 can be regarded as the angular velocity of the magnetic field . we consider an observer with an angular velocity @xmath225 . his or her 4-velocity is given by @xmath226 , where @xmath227 is a normalization factor . the electric field for this observer is given by @xmath228 , and we can easily see from eq . ( [ eq : f - comp ] ) that @xmath229 vanishes . thus we may say that this observer is co - moving with the magnetic field , and the angular velocity of the magnetic field is @xmath24 . substituting eq . ( [ eq : f - comp ] ) into the jacobi identity @xmath230}=0 $ ] , we have @xmath231 the above equation implies that @xmath232 , or equivalently , the equi-@xmath24 surface agrees with the equi-@xmath233 surface . thus we have @xmath234 using eq . ( [ eq : f - comp ] ) , the maxwell equations imply the following equations : the @xmath2-component implies @xmath235 the @xmath5-component implies @xmath236 the @xmath3-component implies @xmath237 and the @xmath4-component implies @xmath238 where @xmath239 using eq . ( [ eq : f - comp ] ) , the force - free condition implies @xmath240 substituting eq . ( [ eq : m - theta - comp ] ) to eq . ( [ eq : f - r - comp ] ) , and substituting eq . ( [ eq : m - r - comp ] ) to eq . ( [ eq : f - theta - comp ] ) , we have @xmath241 from the above equations , we have @xmath242 the above equations imply @xmath243 using the above equation , eqs . ( [ eq : f - r - comp - dash ] ) and ( [ eq : f - theta - comp - dash ] ) imply @xmath244 from eqs . ( [ eq : m - t - comp ] ) and ( [ eq : m - phi - comp ] ) , we have @xmath245 \nonumber \\ & & = -4\pi\alpha\sqrt{\gamma } \left(j^\varphi - j^t\omega_{\rm f}\right ) , \label{eq : m - tphi - comp}\end{aligned}\ ] ] where @xmath31 is defined by eq . ( [ eq : d - def ] ) . noting that @xmath24 is a function of @xmath233 and substituting eq . ( [ eq : current ] ) into the right hand side of eq . ( [ eq : m - tphi - comp ] ) , we have @xmath246 where @xmath247 and @xmath248 + \frac{r^2}{2}\frac{d{\cal b}^2}{da_\varphi}.\ ] ] the above equation is called the grad - shafranov equation . measured from the symmetry axis @xmath39 along the polar cap . , scaledwidth=50.0% ] here we introduce electric current @xmath40 and magnetic flux @xmath30 on a spacelike hypersurface labeled by @xmath2 which penetrate downward and upward an axisymmetric polar cap , respectively . these quantities were first introduced by macdonald and thorne @xcite and are related to @xmath249 and @xmath233 as follows . the polar cap is parameterized by @xmath250 and @xmath5 , where @xmath250 is the proper length on the polar cap from @xmath39 . the coordinates @xmath3 and @xmath4 on the polar cap are given as functions of @xmath250 , i.e. , @xmath251 and @xmath252 : by definition , @xmath253 , and we assume that @xmath254 ( see fig . [ fg : polar - cap ] ) . the orthonormal tangent vectors of the polar cap are @xmath255 then , the upward unit normal to the polar cap is @xmath256 we assume that the edge of the polar cap is @xmath257 and @xmath258 . then , denoting the proper length @xmath250 at the edge by @xmath259 , we have @xmath260d\ell \nonumber \\ & = & -\frac{1}{2}\int_0^{\ell_{\rm e}}\frac{d ( \alpha b_\varphi)}{d\ell } d\ell = -\frac{1}{2}\alpha b_\varphi\biggl|_{(r,\theta)=(r_{\rm e},\theta_{\rm e } ) } , \label{eq : i - def}\end{aligned}\ ] ] where we have used eqs . ( [ eq : m - r - comp ] ) and ( [ eq : m - theta - comp ] ) in the second equality and assumed that @xmath261 from the regularity requirement . we can easily see from the above equation that the electric current through the polar cap is a function of the coordinate values of the edge , @xmath262 . by a similar consideration to that for the electric current @xmath40 , the magnetic flux @xmath30 can be written in the form @xmath263 d\ell = 2\pi\int_0^{\ell_{\rm e}}\frac{d a_\varphi}{d\ell } d\ell = 2\pi a_\varphi |_{(r,\theta)=(r_{\rm e},\theta_{\rm e } ) } , \label{eq : psi - def}\end{aligned}\ ] ] where @xmath264 ( @xmath265 ) is the components of the skew tensor in the spacelike hypersurface labeled by @xmath2 , and we have assumed @xmath266 from a regularity requirement . thus we have @xmath267 rewriting eq . ( [ eq : gs - equation ] ) using @xmath40 and @xmath30 , the grad - shafranov equation eq . ( [ eq : gs - eq ] ) is obtained . the line element of the reisner - nordstrm spacetime with the kerr - schild coordinate system @xmath42 is given by @xmath268 where @xmath269 and @xmath270 . the relation between the kerr - schild coordinate system and the static one is given by @xmath271 from the above relation , we have @xmath272 , @xmath273 , and @xmath274 , and @xmath275 using the above relations , we have @xmath276 by virtue of the stationary , axisymmetric nature of the electromagnetic field , we can easily see that the components of @xmath15 in the kerr - schild coordinate system are given as @xmath277 . \label{eq : rt - comp } \\\end{aligned}\ ] ] it should be noted that all the ordinary derivatives of the kerr - schild coordinates are equivalent to those of the static coordinates for the stationary axisymmetric field @xmath278 , @xmath279 thus , even if the kerr - schild coordinate system is adopted , the equation for @xmath280 takes exactly the same form as eq . ( [ eq : gs - equation ] ) . * proposition * : within the perturbation scheme developed in this paper , the solution for @xmath125 with vanishing lowest - order solution @xmath152 is the only trivial solution @xmath168 . 0.3 cm _ proof . _ we prove this by induction . we have already shown that , if we impose the non - existence of a monopole component for the lowest order of the perturbative solution , we obtain @xmath281 here , we assume that @xmath282 for @xmath283 , or equivalently , @xmath284 then , we have @xmath285 where @xmath286 and we have used @xmath287 , which is required from the regularity condition at @xmath39 . substituting the above equation into eq . ( [ hb ] ) , we obtain an equation of @xmath288 , @xmath289 integrating the above equation , we have @xmath290^{-2\pi { \cal i}_0{}'},\ ] ] where @xmath291 is an integration constant . the regularity condition implies @xmath292 . hence , if @xmath291 does not vanish , @xmath293 must be satisfied , where @xmath294 is an arbitrary constant . in the neighborhood of @xmath295 , @xmath296 and @xmath297 . hence , if @xmath291 does not vanish , we have @xmath298 if we require the finiteness of @xmath299 at @xmath295 , then @xmath291 must vanish , and , as a result , @xmath300 is obtained . q.e.d . r. d. blandford and r. l. znajek , mon . not . r. astron * 179 * , 433 ( 1977 ) . j. c. mckinney and c. f. gammie , astrophys . j. * 611 * , 977 ( 2004 ) . j. c. mckinney , astrophys . j. * 630 * , l5 ( 2005 ) . j. f. hawley and j. h. krolik , astrophys . j. * 641 * , 103 ( 2006 ) . j. bick and v. janis , mon . not . r. astron * 212 * , 899 ( 1985 ) . s. s. komissarov and j. c. mckinney , mon . not . r. astron . lett . * 377 * , l49 ( 2007 ) . j. bick , v. karas and t. ledvinka , astro - ph/0610841 . d. macdonald and k. s. thorne , mon . not . r. astron . soc . * 198 * , 345 ( 1982 ) . i. contopoulos , d. kazanas and c. fendt , astrophys . j. * 511 * , 351 ( 1999 ) . j. ogura and y. kojima , prog . . phys . * 109 * , 619 ( 2003 ) . d. a. uzdensky , astrophys . j. * 603 * , 652 ( 2004 ) . d. a. uzdensky , astrophys . j. * 620 * , 889 ( 2005 ) . a. gruzinov , phys . . lett . * 94 * , 021101 ( 2005 ) . a. n. timokhin , mon . not . r. astron . soc . * 368 * , 1055 ( 2006 ) . j. bick and l. dvork , phys . d * 22 * , 12 ( 1980 ) . s. s. komissarov , mon . not . r. astron . soc . * 326 * , 41 ( 2001 ) . s. nagataki , astrophys . j. * 704 * , 937 ( 2009 ) . r. l. znajek , mon . not . r. astron . soc . * 179 * , 457 ( 1977 ) .
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it is known that the meissner - like effect is seen in a magnetosphere without an electric current in black hole spacetime : no non - monopole component of magnetic flux penetrates the event horizon if the black hole is extreme . in this paper , in order to see how an electric current affects the meissner - like effect , we study a force - free electromagnetic system in a static and spherically symmetric extreme black hole spacetime . by assuming that the rotational angular velocity of the magnetic field is very small , we construct a perturbative solution for the grad - shafranov equation , which is the basic equation to determine a stationary , axisymmetric electromagnetic field with a force - free electric current .
our perturbation analysis reveals that , if an electric current exists , higher multipole components may be superposed upon the monopole component on the event horizon , even if the black hole is extreme .
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parametric resonance is a fundamental physical phenomenon that is encountered eventually in every area of science . in different disciplines , however , different facets of this rich phenomenon play a major role and are highlighted . parametric instability and multistable regimes in nonlinear dynamics @xcite , noise driven transitions among stable states in statistical physics @xcite , wave mixing and frequency conversion in wave dynamics @xcite are topics of primary interest . in electrical and optical engineering the low - noise properties of parametric amplifiers attract attentions , as well as non - classical statistical properties of the electromagnetic field generated by parametric devices @xcite . in superconducting electronics , the idea of using josephson junctions for quantum limited parametric amplification is under attention and development since the 1980s @xcite . during the last years the field revived by challenges of quantum information technology . the circuit - qed design , initially proposed for qubit manipulation and measurement @xcite , was employed for developing a variety of parametric devices @xcite . the circuit - qed approach is based on a combination of extended linear electromagnetic elements ( transmission lines and resonators ) with josephson junctions as nonlinear lumped elements . the design is flexible , allowing for diverse methods of parametric pumping , phase preserving and phase sensitive amplification schemes , different numbers or input and output ports , distributed josephson nonlinearities @xcite . the most of developed amplifiers are engineered in such a way that the dominant pump tone is sent through the same port as the signal , and parametric resonance is achieved by mixing them in nonlinear josephson elements . a different method is available for tunable superconducting cavities @xcite . the device consists of a resonator terminated with one ( or more ) dc - squid(s ) that determines the reflection condition at the cavity edge and hence the cavity resonance spectrum . parametric resonance is achieved by rapid modulation of a magnetic flux through the squid with an appropriate frequency . a number of interesting parametric effects have been observed with such a device : phase sensitive amplification @xcite , frequency conversion @xcite , radiation and multistability regimes above the parametric threshold @xcite , quantum entanglement of output photons @xcite , generation of photons out of vacuum noise @xcite - an analog of the dynamical casimir effect @xcite . in this paper we formulate a consistent theory of parametric resonance in a tunable superconducting cavity . we aim at a unified picture of the phenomenon below and above the parametric threshold . to this end we include into consideration the squid nonlinearity , and damping due to connection to a transmission line . the latter provides a stage for studying the parametric amplification . we develop a full nonlinear description of the cavity resonance dynamics and the amplification effect in the classical limit , and study small quantum fluctuations of amplified and radiative fields . for certainty we consider parametric excitation of the main cavity mode @xmath0 by pumping with a frequency @xmath1 close to twice the cavity resonance , @xmath2 . the overall picture of nonlinear parametric resonance in the tunable cavity is rather rich and complicated . at very small pump strength the cavity intrinsic dynamics resembles the one of the duffing oscillator @xcite showing a bifurcation of the cavity response and bistability . however , the scattering of an external incidental wave is qualitatively different from the duffing case : the scattering is inelastic , the reflected wave undergoes amplification or deamplification depending on the phase shift between the input tone and the pump ( phase sensitive amplification ) . with increasing pump strength , the amplification effect increases , and at the same time the resonance narrows such that the bifurcation occurs at ever smaller input amplitudes . eventually , while approaching the parametric threshold , the cavity response becomes nonlinear at any small input amplitude . further increase of the pump strength leads to an instability of the cavity zero - amplitude state and the formation of finite - amplitude states accompanied by stationary parametric radiation at the half frequency of the pump . the radiative states are bistable in a certain window of detuning of the pump frequency from the cavity resonance . outside of this interval at red detuning the radiative states coexist with the stable zero - amplitude state ( tristability ) , and the latter one becomes dominant at far red detuning . remarkably all these multistable regimes have been observed in experiment with a high quality tunable cavity @xcite . the multistability regimes are accompanied by random jumps among the stable states induced by thermal or quantum noise . these large amplitude fluctuations have small probability away from the bifurcation points and the parametric threshold , but become significant in the vicinity of these critical points ( cf . ref . and references therein ) . these effects are out of the scope of this paper , here we restrict to small quantum fluctuations around well defined classical states outside of the critical regions , both below and above the parametric threshold . the bifurcation of the duffing oscillator response is employed in josephson bifurcation amplifiers ( jba ) for dispersive qubit readout @xcite . this method also applies to the parametric regime below the threshold ( josephson parametric bifurcation amplifier , jpba ) . the novel feature here is the possibility to measure _ amplitude _ of the amplified probing tone , which exhibits strong dispersion with respect to the detuning near the threshold , and can be advantageous for high fidelity qubit readout . the parametric radiation above the threshold offers yet another strategy for the qubit readout based on the significant contrast between the strengths of the output radiation above the threshold and the amplified noise below the threshold . the paper is organized as follows . sections [ sec : device ] , [ sec : ftcavity ] and [ sec : losses ] are devoted to the development of the theoretical framework for describing parametric resonance in a high quality tunable cavity . in sec . [ sec : classical ] we consider the nonlinear cavity response to a classical input signal below and above the parametric threshold , and in sec . [ sec : application ] apply the results for the analysis of parametric amplification and methods of dispersive qubit readout . section [ sec : quantum ] is devoted to the analysis of quantum fluctuations . -cavity is terminated by a dc squid at the right end , and is capacitively coupled to a transmission line at the left end ; the squid is flux biased ( phase @xmath3 ) via inductive coupling to a flux line imposing a driving phase @xmath4 ; @xmath5 and @xmath6 are the phase values at the right and left ends of the cavity , respectively . an incidental signal fed in from the transmission line is reflected , separated from the input , and then analyzed . ] the device we study is sketched in fig . [ fig : device ] . its main part is a tunable superconducting strip line cavity terminated with a squid @xcite . the cavity is weakly coupled to a transmission line that feeds an external microwave signal in and provides means for probing the field inside the cavity . the cavity is a spatially extended system of length @xmath7 with inductance @xmath8 and capacitance @xmath9 per unit length , and the cavity state is characterized by the superconducting phase field @xmath10 . we use the lagrangian formalism @xcite to describe the nonstationary dynamics of @xmath10 . the lagrangian of the entire device consists of the sum of the lagrangians of the cavity , transmission line , and the coupling , @xmath11 = \mathcal{l}_{\text{cav } } + \mathcal{l}_{tl } + \mathcal{l}_{c } \,.\ ] ] the lagrangian of the cavity in its turn consists of the lagrangian of the bare cavity @xmath12 , and the lagrangian of the squid @xmath13 $ ] , @xmath14 + \mathcal{l}_{s}[\phi_d ] \nonumber \\ \label{eq : lcav_x_cos_orig0 } & = & { \left(\frac{\hbar}{2 e}\right)^2}\frac{c_0}{2 } \int_0^d d x \left(\dot \phi^2 - v^2 \phi'^2 \right ) \\ & + & \left [ { \left(\frac{\hbar}{2 e}\right)^2}\frac{2 c_j}{2 } \dot \phi_d^2 + 2 e_j \cos{f(t ) } \cos\phi_d \right ] \nonumber \,.\end{aligned}\ ] ] here @xmath15 is the field propagation velocity , @xmath16 is the boundary value of the cavity field at the squid , and @xmath17 is the phase across the squid controlled by external magnetic flux , see fig . [ fig : device ] . the squid is assumed symmetric for simplicity , with two identical josephson junctions , each having a josephson energy @xmath18 and a capacitance @xmath19 . the phase @xmath17 appears in eq . as an external time - dependent parameter that is able to excite parametric resonance . in fact it is a dynamical variable that describes , together with the variable @xmath20 , the dynamics of two coupled josephson oscillators of the squid driven by the external electromagnetic field @xmath21 . in appendix [ sec : squid ] we show that in the limit of small @xmath22 the @xmath23-oscillator decouples from the @xmath20-oscillator . moreover , for experimentally relevant circuit parameters , the @xmath23-oscillator follows the drive field adiabatically because the resonance frequency of the @xmath23-oscillator is large compared to a typical resonance frequency of the cavity . a detailed derivation of eq . and the connection of @xmath17 to the external field @xmath21 is provided in appendix [ sec : squid ] . we assume here that the controlling field @xmath17 is composed of a constant biasing part @xmath24 and a small harmonic oscillation with amplitude @xmath25 , @xmath26 it is worth mentioning that the constraint @xmath22 is essential , otherwise the two josephson oscillators become coupled and exhibit complex , even chaotic behavior under external drive @xcite . proceeding to the other components of the device , we suppose the transmission line to have the same characteristic parameters @xmath9 and @xmath8 as the cavity , @xmath27 = { \left(\frac{\hbar}{2 e}\right)^2}\frac{c_0}{2 } \int_{-\infty}^0 d x \left(\dot \phi_{tl}^2 - v^2 \phi_{tl}'^2 \right ) \,.\ ] ] the capacitive coupling is described with the lagrangian @xmath28 where @xmath6 and @xmath29 are the field values at the different sides of the coupling capacitor @xmath30 . we first consider the cavity decoupled from the input line , @xmath31 . the goal will be to identify the cavity frequency spectrum and investigate the parametric resonance . the lagrangian @xmath32 , eq . , explicitly contains two dynamical variables , the phase field @xmath10 , and its boundary value @xmath33 . variation of the associated action with respect to @xmath10 leads to the wave equation , @xmath34 supplemented by the boundary condition @xmath35 at the open end of the cavity . variation with respect to the boundary value @xmath33 yields the boundary condition , @xmath36 where @xmath37 and @xmath38 . under static biasing , @xmath39 , the linearized boundary condition of eq . determines the set of cavity eigen modes @xcite , @xmath40 the frequency spectrum @xmath41 is non - equidistant , and can be tuned by varying the bias @xmath24 . although the first term at the rhs of eq . can in principle be tuned to zero , at @xmath42 , in practice it dominates over the second term , at least for the lowest cavity modes , by virtue of the large parameter @xmath43 , where @xmath44 is the josephson plasma frequency . indeed , given typical experimental values , @xmath45 and @xmath46 , the plasma frequency is @xmath47 , while the cavity fundamental frequency is @xmath48 , i.e. , by one order of magnitude smaller ( for typical cavity parameters@xcite @xmath49 , @xmath50 , and @xmath51 ) . furthermore , the cavity inductive energy is typically small , @xmath52 , compared to the josephson energy @xmath53 . taking advantage of this relation , and neglecting the capacitive term in eq . , we get the approximate solutions @xmath54 the solutions of the spectral equation are graphically illustrated in fig . [ fig : spectrum](a ) , while fig . [ fig : spectrum](b ) shows the cavity spectrum as a function of the parameter @xmath55 . , and @xmath56 ( solid ) and @xmath57 ( dashed ) ; ( b ) cavity spectrum @xmath58 vs. @xmath55 according to eq . , the vertical line indicates the value @xmath59 used in ( a ) . ] the lagrangian formalism is sufficient for analyzing the classical parametric resonance . to describe the quantum dynamics the hamiltonian approach is more convenient . we derive the cavity hamiltonian by expanding the cavity field over the complete set of cavity eigen modes , @xmath60 where @xmath61 are time - dependent coefficients , and @xmath62 obey eq . . using expansion and noticing that the set of functions @xmath63 is non - orthogonal , we present the lagrangian ( [ eq : lcav_x_cos_orig0 ] ) after some algebra in the form , @xmath64 - v(q_n , t ) \,.\end{aligned}\ ] ] here the `` masses '' of the mode oscillators are given by the expressions , @xmath65 and @xmath66\ ] ] is a nonstationary nonlinear potential that mixes the eigen modes ( see appendix [ sec : lcav_n ] for details of the derivation ) . it is convenient to absorb the factors @xmath67 and @xmath41 into the rescaled coordinate , @xmath68 and redefine the mode expansion in eq . accordingly . then introducing the conjugated momenta , @xmath69 , we arrive at the cavity hamiltonian , @xmath70 for small pumping amplitudes and weak non - linearity , the potential @xmath71 in eq . could be considered perturbatively . however , the perturbative approach does not apply to the case of parametric resonance , when the pumping frequency matches an algebraic sum of the cavity eigen frequencies , @xmath72 . in this case the corresponding cavity modes are strongly mixed and undergo complex time evolution . a particular case is the degenerate parametric resonance for @xmath73 . in this paper we consider for certainty the degenerate parametric resonance of the fundamental mode , @xmath74 . the method outlined below is straightforwardly extended to a non - degenerate parametric resonance . first we perform a canonical transformation corresponding to a transition to the rotating frame with frequency @xmath75 . this is conveniently done in terms of a complex variable , @xmath76 for which the transformation reads @xmath77 . the equations of motion for the amplitudes @xmath78 read , @xmath79 at this point we take advantage of small values of the pumping amplitude , @xmath25 , and the field amplitude , @xmath22 , and expand the potential @xmath71 in powers of these small parameters , keeping only the first non - vanishing terms , @xmath80 close to the resonance , @xmath81 , the variable @xmath82 depends slowly on time while all the other variables contain rapid time oscillations . after averaging over these oscillations we arrive at the shortened equation of motion for @xmath82 ( we skip the mode index @xmath83 below ) , @xmath84 with the parameters @xmath85 when applying the canonical transformation to coordinate and momentum , @xmath86 , and averaging over fast oscillations , the cavity hamiltonian is cast into the form , @xmath87 this hamiltonian corresponds to the metapotential of the parametric lumped element oscillator @xcite , i.e. the degenerate parametric resonance in the cavity is mapped on the one in a lumped element oscillator . the mapping is defined by eqs . and ( [ eq : def_alphan ] ) , where the effective pump strength @xmath88 , and the nonlinearity coefficient @xmath89 are expressed through generic cavity parameters . according to the experimental values discussed in sec . [ subsec : cavmodes ] , parameter @xmath90 in eq . is estimated as @xmath91 . for such a small value of @xmath90 , the parameters @xmath88 and @xmath89 are approximated , using the spectral equation ( [ eq : dispersion ] ) , @xmath92 where @xmath93 is the cavity impedance and @xmath94 is the quantum resistance . it follows from these estimates that the effective pump strength @xmath88 is substantially reduced compared to the amplitude of the phase modulation in the squid , and the effective nonlinearity of the cavity oscillator is significantly smaller than the underlying bare nonlinearity of the squid oscillator ( @xmath95 for the josephson potential ) . these remarkable properties result from the fact that the cavity is almost shortcut to the ground at the edge @xmath96 by virtue of large josephson energy in eq . ( @xmath97 ) , hence the boundary value of the field amplitude @xmath20 is small . the small values of the effective oscillator parameters are essential for the validity of the resonance approximation . the latter requires the evolution of @xmath98 to take place on a time scale much larger than the period of the cavity fundamental mode , @xmath99 , over which the initial hamiltonian is averaged . it is instructive to express the constraints earlier imposed on the phases , @xmath100 , in terms of the amplitude @xmath98 and the pump strength @xmath88 , @xmath101 or equivalently , @xmath102 . in other words , the constraints ( [ eq : constraint_a ] ) are more stringent than the ones required for the resonance approximation , @xmath103 . on the other hand , these constraints provide sufficient room for the pumping strength to be increased above the parametric threshold beyond the resonance width @xmath104 ( see eq . in the next section ) , @xmath105 , for a high quality cavity . in most of our calculations we restrict to the lowest order @xmath106-dependence in eq . , however , in some cases it is useful to keep higher order terms . in particular , the second order term @xmath107 will introduce , after averaging over time , a nonlinear shift of the resonator frequency , proportional to @xmath108 . this shift is evaluated in eq . in appendix [ sec : squid ] , and in terms of the effective pump strength it reads , @xmath109 this shift could be used in practice for evaluating the actual magnitude of the pump power acting upon the squid , which is usually not known . also , it causes quenching of the parametric instability at large pump strength , as will be shown in sec . [ sec : paramresonance ] . the parametric effect in the closed cavity is an idealization . the connection to the external transmission line gives rise to the qualitatively important new features : firstly , the cavity field is allowed to leak out of the cavity , giving rise to the cavity damping , and secondly , an external electromagnetic signal can be fed into the cavity and amplified . our aim in this section will be to include these features into eq . , and derive the relation between the input and output fields , thus preparing the framework for the further investigation of parametric amplification . our derivation closely follows the input - output theory @xcite , ( see also illuminative derivations in refs . ) . aiming at the analysis of the quantum dynamics of the open cavity , we describe the field in the transmission line in terms of spatial modes , similar to eq . for the cavity , @xmath110 with @xmath111 . opening of the cavity invokes also an additional set of modes @xmath112 , however , in the weak coupling limit these modes do not contribute in the main approximation and are neglected here . focusing on the effect of cavity damping at weak coupling , we will only keep the cross term in the coupling lagrangian ( [ eq : lcavtl ] ) , @xmath113 and neglect the quadratic terms , thus neglecting small corrections to the kinetic energies . with this simplification , and retaining only the fundamental mode field in the cavity lagrangian , we write the total lagrangian in the form , @xmath114 the corresponding hamiltonian reads , to first order of the weak coupling ( @xmath115 ) , @xmath116 repeating the derivation of the previous section we derive coupled equations of motion for the cavity amplitude @xmath117 and the spectral amplitudes of the transmission line , @xmath118 , @xmath119 here we introduced the cavity damping rate , @xmath120 near the parametric resonance the equations of motion for the slow variables , @xmath121 , and @xmath122 , take the form , after averaging over rapid time oscillations , @xmath123 with @xmath124 . we eliminate the transmission line modes from eq . , invoking the solutions of eq . , @xmath125 with initial conditions @xmath126 at time @xmath127 , and substituting it into eq . . within the resonance approximation , the factor @xmath128 in the integrand is to be replaced with @xmath129 , and the integration over the wave vector @xmath130 to be extended to the entire axis . after making these approximations we arrive at the langevin equation for the cavity amplitude , @xmath131 with the input flux amplitude @xmath132 the amplitude @xmath133 is associated , as shown in appendix [ sec : langevin ] , with the incident ( right - going ) wave in the transmission line , @xmath134 , taken at the boundary @xmath135 . the solution of eq . can be equivalently expressed in terms of the amplitude at a future time , @xmath136 , @xmath137 , which defines the output flux amplitude @xmath138 via a relation similar to eq . with @xmath139 substituting for @xmath140 . this output amplitude is associated with the reflected ( left - going ) wave in the transmission line , @xmath141 , taken at @xmath135 ( appendix [ sec : langevin ] ) . the relation between the output and input amplitudes reads , @xmath142 the parametric pumping couples the cavity field amplitude @xmath98 and its complex conjugate , and it is convenient to rewrite eq . in the matrix form , @xmath143 where @xmath144 the conservative part of the dynamics in eqs . , is determined by the effective hamiltonian @xmath145 with @xmath146 from eq . , and @xmath147 being the phase shift between the input amplitude @xmath148 and the pump . besides the damping @xmath149 associated with the opening of the cavity , there might also be internal losses in the cavity , e.g. caused by the cavity resistance . a way to account for these losses is a model with a fictitious transmission line coupled to the cavity , that acts as a scattering channel with a noisy input amplitude @xmath150 and an associated damping rate @xmath151 . this would lead to an enhanced damping rate , @xmath152 , at the lhs of eq . , and also introduce an additional input term , @xmath153 , at the rhs of this equation . the damping effect results in the broadening of the resonance , and if the resonance becomes sufficiently broad , higher cavity modes might also be excited , despite the non - equidistant property of the cavity spectrum . in this case , the isolated mode dynamics of eq . would be replaced by a more complex dynamics of parametrically excited coupled modes . to ensure the validity of the single - mode approximation , the condition @xmath154 must be met , where @xmath155 is the resonance width of the first cavity mode . for @xmath156 , and the cavity spectrum given by eq . and parameters of sec . [ subsec : cavmodes ] , the anharmonicity is of the order , @xmath157 . this implies that the cavity quality factor @xmath158 should not be less than @xmath159 . this corresponds to a small coupling capacitance in eq . , @xmath160 , assuming that the internal losses are not dominant , @xmath161 . in this section we analyze the cavity response to a noiseless classical input signal . we consider harmonic inputs , which have the form @xmath162 , where @xmath163 is the detuning of the input signal from the half frequency of the pump . for the input frequency @xmath164 the cavity response is stationary , and it can be fully analyzed in the nonlinear regime . for detuned inputs , we restrict to small input amplitudes ; at large amplitudes the nonlinear response becomes complex and exhibits a transition to a chaotic regime . we start with the analysis of the intrinsic parametric resonance in the cavity in the absence of input signals , @xmath165 . due to the damping , any initial cavity state evolves towards one of the steady states that define the picture of the parametric resonance . these steady states depend crucially on the pump strength @xmath88 , and also on the detuning of the pump frequency from the cavity resonance , @xmath166 . if @xmath167 , only the trivial steady state , @xmath168 , exists for all values of the detuning @xmath169 . if @xmath170 , the trivial state turns unstable within the interval @xmath171 , and instead two non - trivial stable steady states , @xmath172 , emerge at the threshold @xmath173 , and persist for all @xmath174 , see fig . [ fig : homogeneous](a ) . these states have identical amplitudes , @xmath175 and are @xmath176-shifted in phase , with @xmath177 . in the further red detuned region , @xmath178 , the trivial steady state solution , @xmath168 , becomes stable again , such that the three stable states coexist there . simultaneously , two new unstable states emerge having the same amplitude , @xmath179 . in the limit of @xmath180 , @xmath181 ( the undamped duffing oscillator ) , the nontrivial stable and unstable states merge , forming a manifold of marginally stable states with indefinite phase @xmath182 and amplitude @xmath183 . the steady states of the damped cavity at @xmath170 originate from the fixed points of the cavity hamiltonian @xmath184 , eq . , which are illustrated in the insets of fig . [ fig : homogeneous](a ) for the mono- , bi- , and tristable regions . the damping @xmath104 introduces the threshold for the emerging nontrivial states , and shifts the positions of the steady states in phase space away from the fixed points . for @xmath185 , stable ( solid ) and unstable ( dotted ) ; dashed vertical lines separate mono- , bi- and tristable regions ; insets show phase portraits of corresponding regions . ( b ) boundary of parametric instability without ( solid ) and with ( dotted ) account of nonlinear frequency shift , eq . , for @xmath186 and @xmath187 , cf . eq . ; yellow region corresponds to bistable high - amplitude state , blue region indicates coexistence of stable high - amplitude and zero - amplitude states . ( @xmath186 ) . ] the pump parameters where new steady states occur are determined by the stability properties of the underlying linear system , characterized by the matrix @xmath188 . its determinant , @xmath189 , causes divergence at the parametric instability threshold , @xmath190 , where the fixed point @xmath168 turns unstable . in a linear system this would lead to exponentially growing solutions in the parameter regime @xmath191 and @xmath192 , with a rate @xmath193 . in the nonlinear system this global instability is lifted by the bifurcation of the fixed point @xmath168 into the two new stable steady states . the cavity field , as it leaks into the transmission line , generates an outgoing field with the amplitude @xmath194 according to eq . . for the steady state , eq . , the flux radiated into the transmission line amounts to @xmath195 the nonlinear effect of the cavity resonance shift induced by the pump , mentioned in sec . [ sec : parametric ] , eq . , leads to the quenching of the parametric instability at strong pumping as observed in experiment @xcite . by taking into account this shift , the actual pump detuning becomes @xmath196 , and the parametric instability condition modifies accordingly , @xmath197 where @xmath198 . the modified boundary of parametric instability in the @xmath199-plane is depicted in fig . [ fig : homogeneous](b ) : the instability region is bounded by the maximum blue detuning , @xmath200 , and it is also bounded by a maximum pump strength at given detuning , e.g. @xmath201 at @xmath202 . . insets show the basins of attraction of high - amplitude ( red and green ) and zero - amplitude ( blue ) states , for @xmath203 ( left ) and @xmath204 ( right ) ( @xmath205 , @xmath186 ) . ] in the experiment @xcite , all the described states of the parametrically pumped cavity have been observed : the subthreshold monostable regime at blue detuning , as well as the above - threshold bistable and tristable regimes at red detuning . the visibility of particular stable states in the multistable regime is defined by the probabilities of their occupation , which are determined by the relative areas of the respective basins of attraction , i.e. the phase space regions from which trajectories asymptotically approach the respective state . examples of the attractor basins in the red - detuned region , @xmath206 , are shown in the insets of fig . [ fig : attr_basins ] where the blue basin belongs to the zero - amplitude state , and the red and green attractor basins are those of the high - amplitude states . the relative areas of the latter rapidly decrease and become very small in the far red - detuned region , as shown on the main panel in fig . [ fig : attr_basins ] , implying that these states are much less populated . a similar conclusion is drawn from the calculation of the probability to escape from the high - amplitude states @xcite , which is much larger than the one for the trivial state , @xmath168 , at far red detuning . these arguments explain why in the experiment @xcite the boundary of parametric resonance is washed out at red detuning , in contrast to the sharp boundary at blue detuning , which is determined by the threshold for the nontrivial steady states . now we turn to the discussion of the cavity response to a weak signal with zero detuning , @xmath164 , and complex amplitude @xmath207 . it is instructive to first review the response of the driven duffing oscillator @xcite , which corresponds to the limit @xmath208 in eq . . in this case the detuning @xmath169 refers to the deviation of the input frequency from the cavity resonance . the cavity response is given by the equation , @xmath209 the maximum response is achieved at @xmath210 , along the tilted line @xmath211 , and amounts to @xmath212 , independent of @xmath89 . as a consequence of the tilted resonance line , the cavity response can display bistability , with two coexisting stable states , as shown in fig . [ fig : asqrsq_delta](a ) . the bistability emerges above the critical value of the driving amplitude , @xmath213 , and at the detunings , @xmath214 . the bistability region is confined by the bifurcation lines , @xmath215 \,,\ ] ] forming a wedge in the ( @xmath169-@xmath216 ) plane , as illustrated in fig . [ fig : dpdbifurc_ain ] with black lines . an ideal duffing cavity fully reflects the input signal , so the amplitude of the output , @xmath217 , carries no information about the resonance , @xmath218 . such information is only available for a lossy cavity , where @xmath219 on the other hand , the phase @xmath220 of the output signal is sensitive to the position of the resonance . this is the working principle of the josephson bifurcation amplifiers @xcite , where the variation of @xmath220 under sweeping the input power through the bistability region is exploited for the qubit readout . according to eq . vs. pump detuning @xmath169 for different values of the pump strength @xmath88 , below threshold , @xmath221 ( a , b ) , and above threshold , @xmath222 ( c , d ) . solid and dashed lines mark stable and instable states , respectively . ( @xmath223 , @xmath224 , @xmath225 , @xmath226 ) . ] , for different subthreshold values of the pump strength , @xmath227 ( @xmath224 , @xmath225 , @xmath226 ) . ] switching on the parametric pumping , @xmath228 , qualitatively changes the cavity response . now the amplitude of the cavity field is determined by the equation , @xmath229 \right ) , \nonumber\\ \label{eq : detd } d & = & \det(\mathcal{a } ) = \zeta^2 + \gamma^2 - \epsilon^2 \,.\end{aligned}\ ] ] in the subthreshold regime @xmath167 , the cavity response remains qualitatively similar to the duffing oscillator , see figs . [ fig : asqrsq_delta](a)-(b ) . the role of the parametric pumping in this regime is to effectively reduce the damping term , @xmath230 . this makes the resonance more narrow and , at the same time , strongly increases the cavity amplitude along the tilted resonance line @xmath210 . another important feature is an explicit dependence of the cavity field on the phase shift @xmath147 of the input with respect to the parametric pump . the maximum value of the cavity field is , @xmath231 similar to the duffing limit , this value is independent of the nonlinearity coefficient @xmath89 . the maximum response diverges at @xmath232 , which can be compared to the resonance catastrophe of a linear parametric oscillator . while in the linear case the divergence occurs at @xmath233 , the nonlinearity here shifts the divergence towards an infinite red detuning . as a consequence of the resonance narrowing , the critical bifurcation point moves towards the origin , @xmath234 when @xmath235 , as illustrated in fig . [ fig : dpdbifurc_ain ] . above the threshold , @xmath170 , the resonance splits into two branches , as shown in figs . [ fig : asqrsq_delta](c)-(d ) , each branch consisting of two non - degenerate steady states , one pair being stable and the other unstable . these states originate from the degenerate nontrivial states in the absence of an input signal , cf . [ fig : homogeneous ] , the degeneracy being now lifted by the input . the distance between the branches increases with @xmath88 . the scattering by the parametrically pumped cavity is always inelastic , in contrast to the duffing cavity , and the output signal in general differs significantly from the input signal , not only in phase but also in the absolute value , @xmath236 . using the input - output relation in eq . , and the steady state solution @xmath237 in eq . , the output amplitude can be expressed as a function of the input amplitude , @xmath238 with the parameters @xmath239 the relation in eq . maps the points of the unit circle , @xmath240 , onto the phase - dependent curve , @xmath241 , and determines the phase - dependent gain @xmath242 the @xmath147-dependence of the gain @xmath243 and the quadratures of @xmath244 are illustrated in fig . [ fig : rsq_phin ] . in the monostable ( subthreshold ) regime the output amplitude @xmath194 is amplified ( @xmath245 ) or deamplified ( @xmath246 ) depending on the input phase . for @xmath247 the points @xmath241 form a strongly elongated curve in phase space , centered at @xmath248 . in the quasilinear regime , where the parameters @xmath249 and @xmath250 in eq . are approximately independent of @xmath147 , this curve approaches an ellipse with the half axes @xmath251 giving the maximum / minimum gain factor along those quadratures . for negligible internal losses , @xmath252 , the amplified and deamplified quadratures are related according to @xmath253 . in the limit @xmath254 the gain factors become equal and reduce to the reflection coefficient of the duffing oscillator , eq . . in the bistable regime above the threshold the corresponding output amplitudes @xmath241 are mapped on two distinct closed curves in phase space , with a @xmath176-phase shift between them , as shown in fig . [ fig : rsq_phin ] . the offset from the origin is due to the parametric radiation generated by the cavity . -plane , in the subthreshold regime ( @xmath255 [ blue , red ] ) and above the threshold ( @xmath256 [ yellow ] ) ; insets show the dispersion of the gain with the input phase @xmath147 below threshold ( left ) and above ( right ) . ( @xmath257 , @xmath258 , @xmath225 , @xmath226 ) . ] the cavity response has a simple stationary form only when the frequency @xmath259 of the input signal strictly matches the half - frequency of the pump , @xmath75 . if the input is time - dependent in the rotating frame , e.g. @xmath260 with @xmath163 , the combination of the time - periodic force with the nonlinearity leads to the formation of a region in phase space where the cavity amplitude @xmath98 evolves chaotically , as illustrated in fig . [ fig : phasespace ] for the bistable regime above the parametric threshold . with increasing input amplitude and detuning a chaotic layer forms around the instable fixed points of the hamiltonian , and its area grows with @xmath261 and @xmath262 , see fig . [ fig : phasespace ] . however , as long as the stable fixed points persist in the presence of the time - dependent drive , the amplitude of the damped cavity evolves into a time - periodic limit cycle around them , and then the time - average of @xmath263 gives only small corrections to the stationary result . driven by detuned input amplitude @xmath260 . the trajectories @xmath264 are evaluated in the conservative limit , neglecting the damping term on the lhs of eq . , for ( a ) @xmath265 , ( b ) @xmath266 , and ( c ) @xmath267 . in ( b)-(c ) trajectories are represented stroboscopically , at times @xmath268 , ( @xmath269 ) . ( @xmath270 , @xmath202 , @xmath271 , @xmath272 , @xmath186 , @xmath226 ) . ] in this section we evaluate the response of an ideal cavity @xmath273 to a detuned signal in the monostable regime , @xmath274 . we restrict to a linear response assuming @xmath275 . suppose the input signal in eq . consists of two conjugated harmonics , @xmath276 ( signal and idler in the terminology of non - degenerate parametric amplification ) . then the output field , as well as the field in the cavity , will also consist of the combination of the same harmonics . the output amplitudes are related to the input via the equation generalizing eqs . -([eq : def_q ] ) , @xmath277 where @xmath278 the coupling between the conjugated harmonics is a fingerprint of parametric amplification : an input at frequency @xmath279 generates outputs at frequencies @xmath279 and @xmath280 , and conversely an output at frequency @xmath279 consists of the contributions of inputs at frequencies @xmath279 and @xmath280 . in particular , for @xmath281 , eq . yields , @xmath282 amplification of the detuned signal is characterized by two gain factors , direct gain @xmath283 , and interconversion gain , @xmath284 . these two gains are fundamentally related , @xmath285 , which is the consequence of the fundamental property of the matrix elements in eq . , @xmath286 . for the quantum fields , this property guarantees the unitary relation between the input and output quantum states ( see later in sec . [ sec : quantum ] ) . for small detuning @xmath287 ( single resonance ) , and large detuning @xmath288 ( split resonance ) , at fixed pump strength @xmath289 ; ( b ) @xmath290 for fixed detuning @xmath291 and increasing pump strengths , @xmath292 ( from bottom to top ) . ( @xmath226 ) . ] the amplification of detuned signals possesses another interesting property - the appearance of resonance features , as illustrated in fig . [ fig : gain_dw__dpd](a ) . the resonance structure of the gain is determined by the determinant @xmath293 , eq . . it has a single minimum , at @xmath265 , within the interval of relatively small detuning , @xmath294 , and the gain factor @xmath290 is accordingly single peaked at @xmath164 . however , at larger detunings , @xmath295 two resonance peaks emerge , situated symmetrically with respect to @xmath164 at @xmath296 the origin of these resonances can be understood from the behavior of the response function of a conventional damped linear oscillator , @xmath297 . at small damping , @xmath298 , the resonance is close to the eigen frequency @xmath0 , @xmath299 . with increasing damping the resonance is pulled towards the zero frequency , and stays at the zero frequency as soon as @xmath300 . similarly , the resonances in the response of the linearized parametric oscillator , eq . with @xmath301 , are at small @xmath104 close to the oscillator eigen frequencies , @xmath302 , as in eq . , but are pulled towards @xmath164 with increasing @xmath104 , and eventually merge when @xmath303 , eq . . in this section we discuss the application of the parametrically pumped cavity for signal amplification , and for dispersive qubit readout . in what follows we shall neglect internal losses in the cavity and assume @xmath304 . the amplification characteristics of the nonlinear parametric cavity depend on many parameters : pump and input strengths and detunings from the cavity resonance , relative phase shift , nonlinearity and damping , which makes the overall picture pretty intricate . the output power @xmath305 as a function of the input power @xmath216 for on - resonance input , @xmath164 , is depicted in fig . [ fig : drsqdbsq_ain ] for various values of pump strengths and pump detunings . the major phenomenon here is the appearance of multistable regimes . the bistable regime establishes already below the threshold , @xmath306 , in the red detuning region , @xmath307 , as shown on fig . [ fig : drsqdbsq_ain](b ) . above the threshold , the mono- , bi- , and tristable regimes exist at different detunings , as shown on fig . [ fig : drsqdbsq_ain](c ) . moreover , in the latter regime , the output power does not approach zero value at @xmath308 due to the effect of parametric radiation . for the amplification purpose the monostable regime in fig . [ fig : drsqdbsq_ain](a ) is the most suitable . the output power in this regime depends monotonically on the input power , but exhibits pronounced nonlinearity with increasing pump strength at input power levels @xmath309 . the maximum differential gain is achieved at small input power , and for phase shift @xmath310 . the gain is controlled by the quantity @xmath249 in eq . , and at large @xmath311 , @xmath312 the gain increases while approaching the threshold ( cf . fig . [ fig : asqrsq_delta](b ) ) , @xmath313 , in the quasilinear approximation , and then it is limited by the nonlinearity . let us evaluate this upper bound for the gain at @xmath232 and @xmath202 . in this case , @xmath314 . extracting the amplitude @xmath237 from eq . , with @xmath315 , @xmath316 we get @xmath317 in a similar way we can evaluate the absolute minimum of deamplification . this is achieved at @xmath318 , where @xmath319 , and @xmath320 leading to the equation for minimum gain , @xmath321 we note that the nonlinear deamplification is more efficient than the amplification : the product of the maximum and minimum nonlinear gains significantly deviates from unity , in contrast to the linear case , @xmath322 with these results we conclude that the maximum amplification ( deamplification ) efficiency is controlled by the parameter @xmath323 , and therefore a relatively small nonlinearity coefficient is required to achieve a large parametric effect . as we will see later , the same conclusion is also valid for the nonclassical properties of the fluctuations . at this point it is appropriate to estimate the output signal - to - noise ratio for parametric amplification , referring to the results of the noise analysis in sec . [ sec : noisec ] . according to eqs . and the amplified noise increases in the vicinity of the threshold , however , the noise amplification is less efficient than the signal amplification , giving the ratio ( for the quasilinear limit ) , @xmath324 this ratio is large as soon as @xmath325 . vs. input power @xmath216 , below threshold , @xmath167 ( a , b ) , and above threshold , @xmath326 ( c ) . ( a ) @xmath327 and @xmath328 ( from bottom to top ) ; ( b ) @xmath329 and @xmath330 ( from bottom to top ) ; the dotted line refers to the duffing limit , @xmath180 . ( c ) @xmath331 and @xmath332 ( from top to bottom ) for each of the parameters instable branches are indicated by dashed lines . ( @xmath272 , @xmath186 ) . ] amplification of a detuned signal , @xmath333 , has qualitatively similar properties in the vicinity of the parametric threshold , @xmath334 . here the gain factor @xmath290 has a quasi - lorentzian shape , peaked at @xmath164 , as shown in fig . [ fig : gain_dw__dpd](b ) , the maximum gain increases while approaching the parametric threshold , while the bandwidth shrinks to zero . however , the bandwidth can be considerably increased , maintaining rather high gain , by working away from the parametric threshold in the region where the gain peak splits , @xmath335 , eq . . here a wide frequency plateau emerges around @xmath265 , see fig . [ fig : gain_dw__dpd](b ) , where the gain factor is nearly constant over a frequency interval given by the distance between the resonances , @xmath336 . vs. input power @xmath216 for @xmath337 and @xmath338 . inset : maximum gain factor g vs. @xmath216 and detuning @xmath169 ; black lines indicate the boundaries of the bistable region ( wedge ) cf . [ fig : dpdbifurc_ain ] ; white vertical lines indicate the parameter traces used in the main figure . the average output power @xmath339 is obtained from eq . in the presence of white gaussian noise in the input . ( @xmath272 , @xmath186 ) . ] the bifurcation regime of the cavity nonlinear response in the absence of parametric pumping is employed for dispersive qubit readout using jba @xcite , for a review see ref . and references therein . with this method , the phase shift of a reflected ( or transmitted ) probing signal is measured while ramping the signal amplitude . the result is sensitive to the detuning of the signal tone from the cavity resonance , which is pulled by the qubit by @xmath340 , depending on the qubit state . one may take advantage of the high parametric gain for probing a qubit state by measuring the _ amplitude _ of the output signal instead of the phase shift . the amplified signal exhibits significant dispersion over the cavity - pump detuning thus providing high contrast for the qubit readout . the basis of the method can be understood from fig . [ fig : meanrsq_ain ] ; here the average output power is plotted against the input power for different detunings below the threshold , @xmath341 . the bistability wedge for this pump strength is illustrated in the inset , compare also fig . [ fig : dpdbifurc_ain ] . the lowest three curves in fig . [ fig : meanrsq_ain ] correspond to values of the detuning within the monostable regions , either to the right or to the left of the critical bifurcation point , as indicated by white cuts in the inset ( in the latter case , @xmath342 , the ramped input signal should not cross the bifurcation line ) . the other two curves correspond to crossing through the bistability wedge or very close to the critical bifurcation point ( here the average output power in the presence of classical noise is plotted , which then exhibits a gradual transition from the low- to the high - amplitude branch of the bifurcation curve ) . the output contrast is extremely sensitive to the detuning : it is up to factor of 10 for detunings differing by a linewidth @xmath104 already at rather small input power , @xmath343 . in practice , a cavity frequency pull exerted by the qubit may be of the order @xcite @xmath344 , i. e. comparable to the linewidth , @xmath345 . the output contrast can be further enhanced by increasing the pump strength towards the threshold . it is also possible to ramp the pump strength rather than input power . the possibility to operate with several parameters gives room for further optimization . vs. detuning @xmath169 and pump strength @xmath88 for @xmath346 . the black line separates bistable and tristable regimes , cf . [ fig : homogeneous ] ; the white line spans between two qubit - state dependent , effective detunings @xmath347 ( @xmath348 , @xmath349 , @xmath350 , @xmath186 ) . inset : noise photon number @xmath351 , eq . , vs. @xmath169 , @xmath88 as in main figure . ] an alternative strategy for the dispersive qubit readout is provided by parametric _ radiation _ above threshold . this method , illustrated in fig . [ fig : rsq_delta_epsilon ] , is based on the fact that in the absence of an input signal , @xmath352 , the output signal is zero in the monostable region below the threshold ( at blue detunings ) , @xmath353 , while it is finite above the threshold , @xmath354 , where it equals , @xmath355 , according to eq . . the maximum contrast is achieved by choosing the pumping strength , @xmath356 , and the optimum biasing detuning , @xmath357 , as illustrated in fig . [ fig : rsq_delta_epsilon ] . such a choice guarantees that the blue shifted point , @xmath358 , lies in the monostable region close to the threshold , while the red shifted point , @xmath359 , lies in the bistable region and not in the tristable region where the trivial cavity state , @xmath360 , dominates . then the output radiation power does not depend on @xmath88 , @xmath361 this value is to be compared to the noise value in the monostable region below the threshold . the amplified vacuum noise is given by eq . in sec . [ sec : quantum ] and illustrated in the inset of fig . [ fig : rsq_delta_epsilon ] , @xmath362 since the noise diverges at the threshold , the point @xmath358 is to be chosen not too close to the threshold . it is sufficient to depart from the threshold by @xmath363 to have the noise level , @xmath364 . then for @xmath365 , the radiation to noise contrast becomes , @xmath366 so far we discussed the classical regime of parametric resonance in the tunable cavity . in this section , we extend the formalism to the quantum regime , and investigate the quantum properties of the field inside the cavity , and of the output field . the hamiltonian description of the cavity parametric dynamics is a convenient starting point for the extension to the quantum regime . to this end we revisit eq . of sec . [ subsec : hamiltonian ] and impose canonical commutation relations , @xmath367 = { \text{i}}\hbar$ ] , on the conjugated variables of the eigen modes of the closed cavity . these commutation relations obviously translate to the commutation relations for the resonant variables , @xmath368 = { \text{i}}\hbar$ ] , because of the canonical nature of the transformations made in sec . [ sec : parametric ] . the unitary operator , which explicitly defines the corresponding quantum canonical transformation is@xmath369 averaging over rapid oscillations leads to the quantum hamiltonian coinciding with the one in eq . with quantum operators replacing respective classical variables . the quantization of the fundamental mode oscillator implies the quantization of the variable @xmath263 in terms of the conventional commutation relation for the annihilation operator , @xmath370=1 $ ] . due to the linear coupling of the cavity to the transmission line , eqs . -([eq : coupl_h ] ) , the input - output formalism outlined in sec . [ sec : losses ] straightforwardly extends to the quantum regime . to this end , the classical amplitudes of the transmission line modes are to be replaced with the bosonic annihilation and creation operators , with @xmath371 = \delta(k - k')$ ] . from these commutation relations follows the commutation relation for the incoming field operator , @xmath372 = \delta(t - t')$ ] , and similarly for the outgoing field operator @xmath138 . the scattering relation , eq . , has the same form in the quantum regime , @xmath373 while the quantum langevin equation for the cavity operator @xmath374 becomes , @xmath375 this quantum langevin equation , together with eq . preserves the commutation relation @xmath376 = 1 $ ] for the cavity mode , as shown in appendix [ sec : commutation ] . the conservative part of eq . is a dynamical equation associated with the hamiltonian , @xmath377 the full analytical solution to the nonlinear quantum equation eq . is unknown . in what follows we restrict to the limit of small quantum fluctuations around the classical stationary states . such a restriction is valid far from the bifurcation points and the parametric threshold . some exact results for the critical fluctuations at such points can be found in literature @xcite , also quantum jumps in multistable regimes have been investigated @xcite . to study quantum fluctuations within the framework of a linearized quantum langevin equation , we assume the cavity field operators to be of the form , @xmath378 , where @xmath82 is a steady state solution of the classical nonlinear equation , eq . , and @xmath379 describes small quantum fluctuations , @xmath380 similarly , we separate the classical amplitude and quantum fluctuations of the input field in the transmission line , @xmath381 , @xmath382 . then we expand eq . around @xmath82 up to linear order in the quantum fluctuation @xmath379 to obtain , @xmath383 herein we introduced the effective detuning @xmath384 and the ( complex ) pump strength @xmath385 by adding the terms proportional to the classical amplitude @xmath82 . we note that @xmath82 itself depends on the bare parameters @xmath169 and @xmath88 . quantitatively , the parameter regions where this approximation is valid are identified in appendix [ sec : validity ] . the analysis of eq . goes along the lines of sec . [ subsec : detuned ] , where the response to a classical detuned signal was evaluated . by introducing fourier harmonics of the quantum fluctuations in the transmission line , @xmath386 and similarly in the cavity , the solution of the linear eq . is cast into the form , @xmath387 where @xmath388 cf . it follows from this equation , that modes with frequencies @xmath389 and @xmath390 are coupled pairwise by virtue of the parametric pumping . this property underlines the generation of correlated pairs of photons with frequencies @xmath391 , which is analogous to the photon generation under non - degenerate parametric resonance . the denominator in eq . turns to zero at @xmath392 , if the relation @xmath393 holds , leading to the divergence of fluctuations at the corresponding parameter values . this happens at the parametric threshold , and at the bifurcation points , and indicates the enhancement of critical fluctuations . the full power spectrum of the field in the cavity consists of the sharp line of the amplified ( or generated ) classical signal , @xmath394 , together with the noise power spectrum , @xmath395 , @xmath396 solving eq . and assuming thermal noise in the input field , @xmath397 , where @xmath398 , we calculate for the noise power spectrum @xmath399 at zero temperature , the noise power spectrum reduces to @xmath400 which can be interpreted as the amplified vacuum noise of the input , manifesting itself as real photons in the cavity . the noise power spectrum in eq . has a resonance structure equivalent to the resonances in the classical response to a detuned signal discussed in sec . [ subsec : detuned ] . the only difference is that now the effective pump parameters enter eqs . instead of the bare pump parameters , since we allow here for a finite classical amplitude @xmath82 . accordingly , a single resonance at @xmath392 is observed under the condition @xmath401 and otherwise two resonances are found at @xmath402 cf . eqs . and . in fig . [ fig : nafluc_w ] the noise power spectrum @xmath403 is presented as a function of the pump detuning @xmath169 for @xmath404 and @xmath405 . in the monostable regime , @xmath406 , where @xmath407 , the effective pump parameters in eq . are identical to the bare parameters , while in the bistable regime , @xmath174 , with @xmath82 given by eq . , they are @xmath408 and @xmath409 . the condition identifies the interval @xmath410 around the parametric threshold , where the resonance lies at @xmath411 . outside that interval , once the resonance is split , the separation grows with the parameter distance from the threshold , both below and above the threshold . at the parametric threshold itself , @xmath173 , the noise power diverges . the total number of photons in the cavity at zero temperature is @xmath412 , with the noise power @xmath413 this quantity enters the validity criterium for the linearized langevin equation , eq . , which is analyzed in appendix [ sec : validity ] . of the cavity field , eq . , vs. pump detuning @xmath169 for @xmath406 the classical cavity amplitude is @xmath414 , while @xmath415 for @xmath174 according to eq . . the resonances , eq . , are indicated by the grey lines . ( @xmath404 , @xmath416 , @xmath186 ) . ] similar to the in - cavity field , the full power spectrum of the output field consists of the sharp line of the amplified ( generated ) classical signal , @xmath417 , and the noise power spectrum @xmath418 , @xmath419 the relation between the input and output field operators is similar to the one for a detuned classical signal in sec . [ subsec : detuned ] , eqs . , @xmath420 with matrix elements now dependent on the effective pump parameters , @xmath421 and @xmath422 is given by eq . . the matrix elements obey the fundamental relation , @xmath423 which provides the correct commutation relation for the output operators , @xmath424 = \delta({\delta_k}- { \delta_k}')$ ] . equation describes an input - output relation for a linear non - degenerate amplifier @xcite with signal and idler modes having frequencies @xmath389 and @xmath390 , respectively , while the input classical tone at @xmath392 plays the role of an additional pump . indeed , the renormalization of the generic pump parameters in eq . is an effect of this additional pump that increases the overall pump strength by @xmath425 , and also affects the detuning @xmath169 similar to eq . . we note that eq . is valid both below and above the threshold , and in the latter case it includes the classical parametric radiation acting as an additional pump signal even in the absence of the classical input . with the corresponding renormalization of the quantity @xmath426 that characterizes the amplifier gain , eq . , we cast the input - output relation , eq . into the form , @xmath427 where we introduced the standard notation for a non - degenerate parametric amplifier , @xmath428 the mapping in eq . is provided by a unitary two - mode squeezing operator @xcite , @xmath429 \hat b({\delta_k } ) s^\dag[\xi],\\ & & s[\xi ] = \exp\left(\int_{0}^\infty \!\ ! d{\delta_k}\left(\xi({\delta_k } ) \hat b^\dag({\delta_k } ) \hat b^\dag(-{\delta_k } ) - \text{h.c . } \right)\right ) \ , , \nonumber\end{aligned}\ ] ] where @xmath430 . this implies that the stationary state of the output field is a pure state provided the input is a pure state . this is true in spite of because the evolution of the total system , including the cavity variable , is formally non - unitary due to the presence of the dissipative term in the langevin equation ( [ eq : eom_a_linearized ] ) . the noise power spectrum of the output field can be computed from eq . , and for thermal noise input it reads , @xmath431 .\end{aligned}\ ] ] at zero temperature this equation reduces to @xmath432 and describes the generation of real photons from the vacuum under parametric resonance . this phenomenon is closely related to the dynamical casimir effect - the creation of real photons from vacuum fluctuations by an accelerated mirror @xcite . here the role of the moving mirror is played by the time - dependent boundary condition , driven by the modulated magnetic flux through the squid . the output noise , being proportional to @xmath395 , inherits the resonant behavior of the noise power spectrum in the cavity , as discussed in sec . [ sec : noisea ] and shown on fig . [ fig : nafluc_w ] . in the deep subthreshold regime , for very weak pump strength , @xmath433 , and in absence of an input signal , @xmath405 , eq . takes the form , @xmath434 \left[\gamma^2 + ( { \delta_k}- \delta)^2\right ] } \,.\ ] ] in this limit the resonances move towards @xmath435 , and the resonant structure of @xmath436 , approaches the one computed in @xcite and observed in @xcite . the total photon flux in the output field is @xmath437 , with the noise photon flux , @xmath438 at zero temperature . below the parametric threshold , @xmath439 , the effective parameters in eq . are identical to the bare ones , and eq . reduces to @xmath440 . above the threshold , @xmath441 , with @xmath442 given by eq . , eq . becomes @xmath443 the output noise level is illustrated in fig . [ fig : nctot_vac_dpd_eps ] as a function of @xmath88 and @xmath169 for @xmath416 . the right panel demonstrates the effect of back - bending of the threshold line due to the pump - induced frequency shift , eq . [ fig : homogeneous](b ) in sec . [ sec : paramresonance ] ) . the noise is enhanced at the parametric threshold and decreases while moving away from the threshold , there it is estimated as @xmath364 for @xmath444 . , eq . , vs. @xmath169 and @xmath88 , assuming ( a ) bare pump detuning , and ( b ) taking into account the pump - induced frequency shift , eq . . ( @xmath405 , @xmath186 ) . ] since the noise near the parametric threshold becomes strong , it is useful to evaluate the conditions for the output coherent signal dominating over the noise , @xmath445 . above the parametric threshold , the signal - to - noise ratios are identical for the output field and the field inside the cavity ( for @xmath416 ) , @xmath446 therefore the limitation established by eq . for the field in the cavity applies as well to the output field , @xmath447 below the threshold , the maximum amplified signal is , according to eq . , @xmath448 for @xmath202 and @xmath449 . comparing this with eq . , we arrive at the constraint on the input signal , @xmath450 this bound is of order @xmath104 for @xmath451 , and decreases both at weak pumping and close to the threshold . this is explained , at small @xmath452 , by the fact that the amplification of vacuum noise is small , while the classical signal remains finite , and , close to the threshold , by the fact that amplification of the signal is more efficient than the amplification of the noise . the constraint in eq . is qualitatively similar to the one for the field inside the cavity given by eq . . a homodyne detection scheme allows for measurement of the quadratures of the output signal , and characterization of quadrature fluctuations @xcite . with this method , the output field is mixed with a strong classical field of a local oscillator , @xmath453 , and the intensity of the mixed signal is measured . this intensity is proportional to the output field quadrature , @xmath454 , @xmath455 the phase @xmath456 refers to the phase shift of the local oscillator with respect to the parametric pump ; variation of @xmath456 allows accessing all the quadratures individually . the mean quadrature is determined by the classical output signal @xmath457 separating the classical and quantum components , @xmath458 , @xmath459 , and using the spectral representation of the noise quadratures , @xmath460 , we present the corresponding power spectrum in the form @xmath461 , where @xmath462 is the squeezing power spectrum @xcite . note that by virtue of the stationary state of the cavity , @xmath463 , hence only symmetric correlations between the sidebands contribute to the integral , i.e. the squeezing power characterizes the two - mode squeezing . we calculate the squeezing power assuming vacuum fluctuations of the input , using eq . for the output field operators . the result reads , @xmath464 \,.\end{aligned}\ ] ] equation @xmath465 corresponds to pure vacuum fluctuations . the noise squeezing power varies with the phase @xmath456 , the maximum and minimum values reached at @xmath466 and @xmath467 , respectively , with @xmath468 { \text{re}}({\tilde \epsilon } ) } { 2{\tilde \zeta}\gamma { \text{re}}({\tilde \epsilon } ) + [ \gamma^2-{\tilde \zeta}^2+|{\tilde \epsilon}|^2+{\delta_k}^2 ] { \text{im}}({\tilde \epsilon } ) } .\end{aligned}\ ] ] the corresponding extreme values are determined by the quantity @xmath469 , and have the form , @xmath470 which is similar to the classical gain of the ideal amplifier , eq . , including the relation , @xmath471 . however , the maximum squeezing and maximum quadrature gain do not generally correspond to the same value of mixing phase @xmath456 . moreover , the amplified classical signal , eq . , contains an additional phase , @xmath220 , which is controlled by the input signal phase @xmath147 . by varying the latter one may control the signal - to - noise ratio for the quadratures . in fig . [ fig : squeeze_w_theta](a - b ) the squeezing power @xmath472 is shown for @xmath405 , @xmath202 , and two different values of @xmath170 . the @xmath456-values of maximum and minimum squeezing power are indicated by black lines . , eq . ( a - b ) , and in - cavity noise @xmath473 , eq . , ( c - d ) for @xmath474 ( left column ) and @xmath475 ( right column ) . the quadrature phases for maximum and minimum squeezing power , eq . are indicated by the black lines . ( @xmath202 , @xmath416 , @xmath186 ) . ] it is useful to also quantify the in - cavity squeezing , by calculating the squeezing power for the quadrature operator @xmath476 in analogy to eq . . although the phase @xmath456 in this case is not related to any externally tunable phase , it might be relevant for a quadrature - dependent coupling to a qubit placed in the cavity , or to another transmission line . assuming vacuum fluctuations in the input field we calculate the internal squeezing power using eq . , @xmath477 \,.\end{aligned}\ ] ] further evaluation of the minimum uncertainty of the cavity quadrature , @xmath478 , results in the value @xmath479 , as in the case of linear parametric amplifiers @xcite , i.e. a factor @xmath479 below the vacuum limit . in figs . [ fig : squeeze_w_theta](c - d ) @xmath480 is illustrated for @xmath416 , @xmath202 and @xmath191 , in comparison to the external squeezing @xmath481 of figs . [ fig : squeeze_w_theta](a - b ) . as a consequence of the effective detuning @xmath384 in eq . , @xmath482 is not symmetric around @xmath392 , as is the case for @xmath481 . , eq . , vs. @xmath88 for fixed @xmath483 and for @xmath484 ( from bottom to top , the corresponding parameter values are marked with crosses on panel ( b ) ) . ( b ) @xmath485 and ( c ) gain @xmath486 vs. @xmath169 and @xmath147 , for fixed @xmath487 ( indicated by the vertical line in ( a ) ) . ( @xmath488 , @xmath186 ) . ] the two - mode squeezing is a nonclassical property of the amplified noise that originates from the production of noise photons in entangled pairs . further information about the nonclassical properties of the correlated output photons is provided by a two - photon correlation function , and characteristics of two - photon entanglement . we start with evaluating the second - order correlation function @xcite , @xmath489 in the presence of the classical output component , this equation takes the form , @xmath490 explicitly , using eq . , we obtain for @xmath491 and input vacuum noise , @xmath492 \bigr ] \ , . \nonumber\end{aligned}\ ] ] in fig . [ fig : g2](a ) the normalized correlation function , @xmath493 , is presented as a function of the pumping strength @xmath88 for several values of the pump detuning @xmath169 . in the duffing limit , @xmath494 , all the terms in eq . vanish except of the first one , yielding the coherent state limit , @xmath495 . the same is also true for large pumping strength above the threshold , @xmath496 . this is explained by the rapid growth of classical radiation power that dominates over the fluctuations , @xmath497 ( kinks on the curves at @xmath498 ) . at the intermediate pump strengths both bunching ( @xmath499 ) and antibunching ( @xmath500 ) are possible . for pure output noise in the absence of classical output , @xmath501 ( i.e. for @xmath416 below the threshold ) , only bunching occurs , @xmath502 , where the degree of bunching exceeds that of classical chaotic radiation , @xmath503 this can be interpreted as a consequence of the pair production of noise photons . when @xmath504 , also antibunching is possible @xcite due to the interplay between the classical and the quantum contribution to the correlation , last line in eq . . it occurs within a relatively narrow window of parameters , @xmath167 , @xmath505 , @xmath506 , for which the phase dependence in the last term in eq . can introduce a sign change . the dependence of @xmath485 as a function of the input phase @xmath147 and the pump detuning @xmath169 is illustrated in fig . [ fig : g2](b ) . pronounced antibunching ( blue regions ) is observed for @xmath507 , and for those values of @xmath147 where the gain approaches unity , @xmath508 , compare fig . [ fig : g2](c ) . ( solid ) and logarithmic negativity @xmath509 ( dashed ) vs. pump strength @xmath88 for @xmath510 ( from top to bottom ) . the vertical lines mark the values of @xmath88 at which the resonance , eq . , is encountered at the chosen value of @xmath389 . ( @xmath202 , @xmath416 , @xmath186 ) . ] the degree of entanglement between the two modes with frequencies @xmath389 and @xmath390 can be quantified with the entanglement entropy @xcite , @xmath511 where @xmath512 is the reduced density matrix of one of the involved modes . if these modes are entangled , the entropy takes a positive value , @xmath513 . we compute the entanglement entropy for the amplified vacuum noise , using the two - photon wave function of the squeezed state , @xmath514 which is obtained by applying the squeezing operator , eq . , to the vacuum input , @xmath515\,|0\rangle$ ] , and using the decomposition equation @xcite . the reduced density matrix has the form , @xmath516 giving the entanglement entropy @xcite , @xmath517 the entropy is nonzero for all @xmath228 , and follows closely the squeezing parameter @xmath518 , asymptotically approaching the linear dependence , @xmath519 , for @xmath520 . the entanglement entropy @xmath521 is shown as function of @xmath88 in fig . [ fig : logneg ] ( solid lines ) , for several values of the detuning @xmath389 , and for @xmath405 and @xmath202 . for small detuning , @xmath522 , the entropy reaches the maximum at the threshold , @xmath232 , at which @xmath523 [ @xmath524 diverges . with increasing value of the detuning @xmath389 this maximum shifts towards the value of @xmath525 , at which @xmath526 exhibits the resonance , eq . . the entropy rapidly decreases above the threshold , analogous to the behaviour of @xmath485 , due to the emergence of the classical radiative state , @xmath527 , that suppresses @xmath526 . a convenient measure of entanglement for gaussian states is provided by the logarithmic negativity @xcite related to the covariance matrix for the two entangled modes . the covariance matrix @xmath528 is defined through a 4-vector composed of the quadratures , @xmath529 , @xmath530 then splitting the covariance matrix into @xmath531 submatrices , @xmath532 , the logarithmic negativity is defined as @xmath533 where @xmath534 and @xmath535 . for entangled states the logarithmic negativity takes positive values . for amplified vacuum noise we obtain a simple result , using eq . , @xmath536 i.e. the logarithmic negativity is equal to twice the squeezing parameter @xmath518 . the logarithmic negativity @xmath537 is shown in fig . [ fig : logneg ] with dashed lines . its functional behavior is basically equivalent to that of the entropy @xmath521 . our calculation shows that the degree of the two - mode entanglement is significantly enhanced in the presence of the parametric resonance . to evaluate the exact maximum entanglement value one needs to go beyond the quasilinear approximation and include the nonlinear effect . we make a qualitative estimate by taking the function @xmath538 at the threshold , @xmath232 , @xmath202 , and at @xmath392 , and for the cavity field given by eqs . and assuming input power , @xmath539 , corresponding to one photon per bandwidth . this yields an estimate , @xmath540 with a numerical constant of order one . this crude estimate seems to agree with more accurate evaluation of the critical fluctuations @xcite . for values @xmath541 achievable in tunable cavities , the entanglement entropy can accordingly reach the values @xmath542 . this is significantly larger than the values calculated @xcite for a non - resonant open transmission line with modulated boundary , and also exceeds the values reported for experimental parametric josephson devices @xcite . we have developed a consistent theory of parametric resonance in a high quality tunable superconducting cavity . we considered the nonlinear classical dynamics of the cavity both below and above the parametric threshold , and analyzed amplification of external signals , and parametric radiation . we also studied quantum properties of the amplified and radiative fields . the non - equidistance of the cavity frequency spectrum enabled us to formulate the theory of the degenerate parametric resonance in terms of the one encountered in a nonlinear parametric oscillator . we identified the parameters of this effective oscillator as functions of the cavity generic characteristics , and investigated the multistable cavity dynamics in a relevant range of the effective parameters . the operation of the device in the monostable regime as a nonlinear parametric amplifier is characterized with a phase - dependent differential gain , which increases at small input power and reaches the maximum value at the parametric threshold . we found that this maximum value scales with the ratio of the damping coefficient and the nonlinearity coefficient , @xmath323 . we also found that the relation between the maximum and minimum gain for an ideal linear amplifier is violated in the nonlinear regime , @xmath543 . extremely small values of @xmath89 available in tunable cavities allows for very large gain and strong amplification vs. deamplification contrast . amplification of detuned signals was found to exhibit sideband resonances within a specific region of the cavity parameters . this effect can be used for enhancing the amplification bandwidth while maintaining high gain . the application of the device as a parametric bifurcation amplifier was discussed in regard to dispersive qubit readout . the advantage of the parametric regime compared to the conventional jba is a high sensitivity of the strength of the output signal to the variation of the cavity frequency . this , together with a high amplification gain , provides a potential for improving the fidelity of qubit single shot readout . yet another suggested method for qubit readout is based on a high contrast between the strengths of parametric radiation above the threshold and amplified noise below the threshold . small - amplitude quantum fluctuations around the classical signal were investigated for the in - cavity field and the output field . the limit of small fluctuations is appropriate in a wide range of the device parameters except of small regions of critically enhanced fluctuations close to the bifurcation points and the parametric threshold . the theory is analogous to the one for a quantum linear amplifier . the strength of the amplified noise increases in the vicinity of the threshold in accord with the classical gain . the same is also true for the two - mode squeezing and the entanglement quantified with the entanglement entropy and the logarithmic negativity . at the threshold , the estimated magnitude of the squeezing parameter may reach the values of a few units , exceeding that achievable e.g. in non - resonant josephson mixers . the second order coherence is dominated by strong bunching for small classical inputs , resulting from the production of noise photons in pairs . however , for classical inputs with strength comparable to the vacuum noise , significant antibunching is predicted resulting from the interference of the classical and quantum field components . to conclude , we note that the developed theory straightforwardly extends to the regime of non - degenerate parametric resonance , when the pumping frequency is commensurate with a combination of cavity resonances . similarly , in this case , strongly enhanced amplification gain is to occur near the parametric threshold , as well as strongly enhanced two - mode squeezing and entanglement of the cavity modes selected by the resonance . yet another extension of the theory is readily done for a two - sided cavity parametrically pumped by two squids , attached to both sides of the cavity @xcite . the dynamics of this device is equivalent to the single - sided parametric cavity , provided the squids are operated at the same pump frequency . the parametric resonance is then controlled by an effective pump strength , which depends on the phase shift between the actual pumps . for equal pump amplitudes the parametric effect is maximum for the out - of - phase pumping ( `` breathing '' mode ) , while for the in - phase pumping ( `` translational '' mode ) the parametric instability is completely suppressed . + _ acknowledgement _ we acknowledge useful discussions with chris wilson , per delsing , gran johansson , konrad lehnert , and tim duty . support from fp-7 ip solid is gratefully acknowledged . in this appendix we derive the lagrangian of the flux - tunable cavity , eq . , and give arguments for its validity . we start with a description of the squid establishing the connection between the cavity and the pump line . the generalized coordinates of the squid are the superconducting phase @xmath544 at the cavity edge @xmath96 , the phase @xmath3 dropping over the inductance @xmath545 of the squid loop , and the phase @xmath546 dropping over the coupling inductance @xmath547 of the pump line , see fig . [ fig : device ] . the squid is modelled as symmetric , with two identical josephson junctions , each having a josephson energy @xmath18 and a capacitance @xmath19 . to simplify notation we assume that the squid is grounded in such a way that its geometric inductance @xmath545 is divided into two equal parts @xmath548 , with a phase drop of @xmath23 over each part . thus , the phase difference on one of the josephson junctions is @xmath549 and @xmath550 on the other . the coupling to the flux line is inductive , with a mutual inductance @xmath551 . the full squid lagrangian is @xmath552 or , written with the capacitive energy of a josephson junction @xmath37 and the inductive energy of the squid loop @xmath553 @xmath554 separating the @xmath20-dependent terms ( first line ) from the purely @xmath23-dependent ones ( second line ) , @xmath555 + \mathcal{l}_{s}[f]$ ] , the former can be combined with the bare cavity lagrangian @xmath556 with the inductive energy of the cavity @xmath557 . together , these form the lagrangian @xmath32 of the flux - tunable cavity , eq . . for typical cavity and junction dimensions the orders of the three inductive energies in the lagrangian , eqs . - , are distinctly different . the dominant energy , @xmath558 , determined by the small geometric inductance of the squid loop ( @xmath559 ) , is larger than the josephson energy of the squid , @xmath560 , and that dominates over the inductive energy of the cavity , @xmath52 ( for @xmath561 ) . furthermore , the josephson plasma frequency @xmath562 is high compared to the fundamental cavity resonance , @xmath48 ( compare sec . [ subsec : cavmodes ] ) . the equations of motion for @xmath20 and @xmath23 , according to the full lagrangian @xmath563 $ ] , @xmath564 describe two coupled nonlinear oscillators . for @xmath565 the equilibrium is @xmath566 . in general , the coupled dynamics of nonlinear , driven oscillators features chaotic behaviour . we restrict our analysis to the case @xmath22 and assume that this is fulfilled even in the presence of a resonant excitation by the external field @xmath21 . under this condition the equation of motion for @xmath23 , eq . , decouples from the other oscillator , @xmath567 and the dynamical equation for @xmath20 , eq . , then depends only parametrically on @xmath17 , cf . . we suppose the external force of the form @xmath568 , @xmath569 , and separate the squid phase response @xmath570 into a constant equilibrium shift @xmath24 , governed by the equation , @xmath571 and a small harmonic oscillation , @xmath572 , driven by @xmath573 , @xmath574 assuming @xmath575 , we write the stationary solution in the form , @xmath576 , @xmath577 where @xmath578 is the frequency of the @xmath23-oscillator , which is much larger than the frequency of the pump , @xmath579 . linearized around the equilibrium shift @xmath24 , eq . becomes @xmath580 \sin\phi_d \nonumber\\ & + & { e_{l,\text{cav}}}d \phi'_d = 0 \,.\end{aligned}\ ] ] for @xmath581 this boundary condition determines the cavity mode spectrum , eqs . and . further expanding eq . to the second order with respect to @xmath106 leads to the pump induced shift of the cavity frequencies . indeed , averaging over time , we get a correction to the josephson energy , @xmath582 , which will modify eq . accordingly . in particular , for the fundamental mode we get from eq . , @xmath583 one could also expand eq . to the second order with respect to @xmath106 , which would lead , after the time averaging , to a shift of the static bias @xmath24 , and eventually to an additional shift of the cavity frequencies . however , this effect is small , by virtue of the parameter @xmath584 , compared to the shift ( [ eq : pump_frshift ] ) . in this appendix , we express the lagrangian of the flux - tunable cavity , eq . , in the mode representation , eq . , based on the expansion of the cavity field . firstly , making use of eq . , the overlap integrals of the non - orthogonal modes are @xmath585 where we have defined the coefficients @xmath67 , eq . . with these , the bulk contribution to the cavity lagrangian becomes @xmath586 \nonumber\\ & + & \frac{1}{2 } \sum_{n , m } \bigl [ - { 2c_j \over c_0 } \cos{k_n d } \cos{k_m d } \dot q_n \dot q_m \bigr.\nonumber\\ & & \bigl . + v^2 \cos{k_n d } \left({2c_j \over c_0 } k_m^2 \cos{k_m d } + k_m \sin{k_m d } \right ) q_n q_m \bigr ] \nonumber \,.\end{aligned}\ ] ] in the remaining boundary contribution of eq . , we firstly separate a time dependent , nonlinear potential term @xmath587 from the harmonic contribution , @xmath588 the mode - representation , eq . , of the harmonic part becomes , using @xmath589 , @xmath590 the first term of this cancels directly with a term in the bulk contribution , eq . . further , using the definition of the modes in eq . , we note that @xmath591 leading to further cancellation of terms between the bulk and the boundary contribution . the remaining terms are @xmath592 - v(q_n , t ) \,,\end{aligned}\ ] ] with @xmath593 . this is the mode representation of the cavity lagrangian in eq . . in this appendix we show the relation of the flux amplitude @xmath594 introduced in eq . to the incoming field , and similarly for the flux amplitude @xmath194 of the outgoing field , as well as their mutual relation given in eq . . the incoming and outgoing fields in the transmission line are defined , respectively , as @xmath595 \\ \label{eq : phiout } \!\!\!\!\!\ ! \phi_{out}(t ) \!&=&\ ! { e\over \hbar } \sqrt { \!\frac{\hbar}{\pi c_0 } } \!\int_0^{\infty } \!\!\!\ ! { d k \over \sqrt{\omega_k } } \!\!\left [ a_k(t_1 ) e^{-{\text{i}}\omega_k ( t - t_1 ) } \!+\ ! \text{h.c . } \right ] .\end{aligned}\ ] ] these are based on the solutions of eq . , which is expressed in terms of initial amplitudes @xmath596 at a time @xmath127 in the past , or in terms of final amplitudes @xmath597 at a time @xmath598 in the future , @xmath599 such a definition is justified , as will be shown below , by attributing different propagation directions along the transmission line for the incoming and outgoing field components , which can be separated by circulators and hence have physical meaning . we firstly use the solution to evaluate the field in the transmission line , eq . with @xmath600 , @xmath601 } \\ & & + { 4e c_c \over \pi \hbar c_0 d}\sqrt{2 d \omega_0 \over c_0 m_0 } \int_{t_0}^t \!\!d t ' p(t ' ) \int_0^\infty \!\!\!d k \cos(\omega_k ( t - t ' ) ) \cos{kx } \nonumber .\end{aligned}\ ] ] the contribution from the first line can be straightforwardly identified with @xmath602 from eq . , and equals @xmath603 . the integral in the second line is evaluated ( for @xmath604 ) @xmath605 \nonumber \\ & = & \frac{d}{d t } \int_0^\infty \frac{d k}{kv } \left [ \sin(k v(t - t ' ) - kx ) + \sin(k v(t - t ' ) + kx ) \right ] \nonumber\\ & = & \frac{\pi}{2v } \frac{d}{d t } \left [ \text{sgn}(t - t ' - x / v ) + \text{sgn}(t - t ' + x / v ) \right ] \nonumber \\ & = & \frac{\pi}{v } \left [ \delta(t - t ' - x / v ) + \delta(t - t ' + x / v ) \right ] .\end{aligned}\ ] ] the second @xmath169-function gives a contribution at @xmath606 , whereas the first , for @xmath604 , is not included in the integration limits , @xmath607 , and therefore @xmath608 taken together , the field in the transmission line reads @xmath609 alternatively , the transmission line field , eq . , can be evaluated from the second solution for the @xmath610 , eq . , yielding @xmath611 by subtracting eqs . and at @xmath135 we can establish a relation between the incoming and the outgoing field components , @xmath612 note that the last term can also be expressed by the derivate of the cavity field at @xmath135 , @xmath613 , using @xmath614 in the weak coupling approximation . we can now evaluate eq . at @xmath615 , and insert in eq . , such that the transmission line field is expressed as a linear combination of @xmath602 and @xmath616 alone , @xmath617 demonstrating the role of @xmath602 and @xmath616 as incoming and outgoing field components . finally , we want to relate the general input - output relation , eq . , with the slow varying amplitudes of the resonant approximation . to that end we separate the fast time oscillation with frequency @xmath75 in eqs . and ( [ eq : phiout ] ) , @xmath618 where @xmath619 within the resonant approximation , @xmath620 , @xmath621 , these quantities coincide with the ones defined in sec . [ sec : losses ] , cf . the cavity momentum is expressed in the rotating frame as well , @xmath622 with the slowly time - dependent cavity amplitude @xmath263 . by setting these expressions into eq . , multiplying with @xmath623 and averaging over fast oscillation , the corresponding input - output relation is obtained in the rotating frame , cf . in this appendix we show that the quantum langevin equation , eq . , preserves the commutation relation of the cavity amplitude . to this end we express the solution of eq . in terms of the propagator @xmath625 with the hamiltonian that governs the dynamics of the isolated cavity , @xmath626 ( cf . the operator @xmath98 refers to the schrdinger picture and coincides initially with the heisenberg operator @xmath627 . at time @xmath628 the solution is @xmath629 where we have used the fact that @xmath184 and @xmath630 commute since @xmath98 is uncorrelated with the operators @xmath596 of the incoming transmission line modes of which @xmath133 is composed . using this solution we are able to evaluate the equal time commutator , @xmath631 = e^{-2\gamma ( t - t_0 ) } \bigl[\ , u^{-1}(t , t_0 ) a(t_0 ) u(t , t_0)\,,\bigr . \\ & & \hspace*{2 cm } \bigl . u^{-1}(t , t_0 ) a^\dag(t_0 ) u(t , t_0 ) \,\bigr ] + { \text{i}}\sqrt{2\gamma } e^{-\gamma ( t - t_0)}\nonumber\\ & & \hspace*{0.5cm}\times \int_{t_0}^t \!\!d t ' e^{-\gamma t ' } \bigl ( \left[u^{-1}(t , t_0 ) a(t_0 ) u(t , t_0),\ , b^\dag(t ' ) \right ] \bigr . \nonumber\\ & & \phantom{\hspace*{0.5cm}\times \!\int_{t_0}^t \!\!d t ' e^{-\gamma t ' } } - \left[b(t'),\ , u^{-1}(t , t_0 ) a^\dag(t_0 ) u(t , t_0 ) \right ] \bigr ) \nonumber\\ & & \hspace*{0.5cm}+ 2\gamma \iint_{t_0}^t d t ' d t '' e^{-\gamma(t'+t '' ) } \left [ b(t'),\ , b^\dag(t '' ) \right ] \,.\end{aligned}\ ] ] the mixed commutators vanish , again with the argument of initially uncorrelated cavity and transmission line operators , leaving @xmath632 = } \nonumber\\ & & \hspace*{0.5 cm } e^{-2\gamma ( t - t_0 ) } u^{-1}(t , t_0 ) \left [ a(t_0),\ , a^\dag(t_0 ) \right ] u(t , t_0 ) \nonumber\\ & & \hspace*{0.5 cm } + 2\gamma \iint_{t_0}^t d t ' d t '' e^{-\gamma(t'+t '' ) } \left [ b(t'),\ , b^\dag(t '' ) \right ] \,.\end{aligned}\ ] ] finally , using @xmath633 = 1 $ ] and @xmath634 = \delta(t'-t'')$ ] , we arrive at the desired result , @xmath635 = e^{-2\gamma ( t - t_0 ) } - \left(e^{-2\gamma ( t - t_0 ) } - 1 \right ) = 1 \,.\ ] ] the invariance of the commutation relation under the langevin evolution , eq . , follows from the correct combination of the damping term , @xmath636 , and the fluctuations in the amplitude @xmath594 . averaging over fluctuations would violate the exact unitary evolution and break the commutation relation . having evaluated the magnitude of the quantum fluctuations , we are able to discuss the region of validity of the linearized equation , eq . . two assumptions have been made for the derivation : the amplified signal at frequency @xmath637 has been treated as a classical field , @xmath638 , and its magnitude to exceed the amplified external noise , @xmath639 . together these conditions are ( cf . eq . ) , @xmath640 we analyze these conditions separately above and below the threshold , at zero temperature , and at @xmath202 for simplicity . above the threshold , @xmath191 , the parametric radiation dominates over the input signal . neglecting the input , @xmath416 , we have @xmath641 in accord with eq . . then @xmath642 and @xmath643 , and the amplified vacuum noise is , @xmath644 the conditions of eq . , @xmath645 are fulfilled everywhere except of the close vicinity of the threshold . near the threshold the external noise dominates , while its role diminishes with growing pump strength . in the limit of very strong pumping , @xmath646 , eq . reduces to @xmath647 . this result can be understood from a purely hamiltonian argument . the semiclassical limit requires the quantum uncertainty of a state localized in a quantum well , @xmath648 , to be much smaller than the total phase - space volume of the well . the latter can be estimated from the separatrix area , @xmath649 , cf . second inset of fig . [ fig : homogeneous](a ) . since in the semiclassical limit tunneling between the wells is exponentially suppressed , it is consistent to treat noise as local fluctuations in each well separately . below the threshold , @xmath306 , eq . imposes constraints on the input field @xmath650 . to be consistent with the linear description of fluctuations , we consider the quasilinear limit of the classical response , @xmath651 . with this assumption , the maximum magnitude of the field in the cavity , eq . with @xmath315 , reads @xmath652 while the amplified vacuum noise is @xmath653 the number of amplified vacuum photons inside the cavity is small at weak pumping but grows and passes the one - photon level at @xmath654 , and becomes dominant while approaching the parametric threshold . using these estimates we extract from eq . the lower bound on the input signal , @xmath655 for very small pump strength , @xmath452 , the constraint ( [ eq : bsq_min1 ] ) reduces to @xmath656 , which is qualitatively similar to a high quality duffing cavity , in which the resonant field fed by the input @xmath656 achieves a large ( classical ) value , @xmath638 . the quasilinear approximation in this case , @xmath657 is valid as soon as @xmath658 , and it imposes an upper bound on the input , @xmath659 . close to the threshold , the amplified signal grows with @xmath88 more rapidly than the noise , and remains dominant at practically all input signals . this regime persists until the nonlinear effect breaks the quasilinear approximation at @xmath660 , and the signal amplitude saturates . the corresponding constraint on the input reads , @xmath661 in terms of @xmath88 , the upper bound for this regime is given by the condition @xmath662 for experimentally relevant cavity parameters , @xmath663 , our estimates for the relative noise strength are therefore valid up to @xmath664 . menzel , r. di candia , f. deppe , p. eder , l. zhong , m. ihmig , m. haeberlein , a. baust , e. hoffmann , d. ballester , k. inomata , t. yamamoto , y. nakamura , e. solano , a. marx , and r. gross , phys . lett . * 109 * , 250502 ( 2012 ) . devoret , in _ quantum entanglement and information processing _ , edited by d. esteve , j.m . raimond , and j. dalibard , proceedings of the les houches summer school of theoretical physics , lxiii , 1995 ( elsevier , amsterdam , 2004 ) .
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we develop a theory of parametric resonance in tunable superconducting cavities .
the nonlinearity introduced by the squid attached to the cavity , and damping due to connection of the cavity to a transmission line are taken into consideration .
we study in detail the nonlinear classical dynamics of the cavity field below and above the parametric threshold for the degenerate parametric resonance , featuring regimes of multistability and parametric radiation .
we investigate the phase - sensitive amplification of external signals on resonance , as well as amplification of detuned signals , and relate the amplifier performance to that of linear parametric amplifiers .
we also discuss applications of the device for dispersive qubit readout . beyond the classical response of the cavity ,
we investigate small quantum fluctuations around the amplified classical signals .
we evaluate the noise power spectrum both for the internal field in the cavity and the output field .
other quantum statistical properties of the noise are addressed such as squeezing spectra , second order coherence , and two - mode entanglement .
| 29,410 | 249 |
the equilibrium behavior and the equipartition of energy between various degrees of freedom in nonlinear , nonintegrable discrete systems has attracted considerable interest since the seminal study of fermi , pasta and ulam @xcite . in hamiltonain systems with conserved number of excitations ( waves ) the maximum entropy principle suggests that in the final state of thermal equilibrium the statistics of the system is given by grand canonical gibbs distribution with the effective `` temperature '' and `` chemical potential '' @xcite . however unlike the conventional statistical mechanics the effective temperature of this grand - canonical distribution depends on the initial position in the phase space and for certain regions can become negative making the distribution non - normalizable . such regime is commonly attributed to the emergence of stable , localized , nonlinear structures corresponding to solitons in continuous systems @xcite and discrete breathers @xcite in discrete systems . from the point of view of the wave turbulence theory @xcite the resulting equilibrium distribution provides stationary rayleigh - jeans spectra @xcite . also thermalization of light in nonlinear multimode waveguides and cavities has recently attracted attention in the context of classical optical wave condensation @xcite . here we will study the phenomenon of thermalization in the context of light propagation in a system of coupled nonlinear optical waveguides but the results can have wider applicability beyond the scope of the nonlinear optics . when the individual waveguide modes are strongly localized the nonlinear propagation of light is most commonly modelled by the discrete nonlinear schrodinger equation ( dnlse ) @xcite . in fact most studies of thermalization in nonlinear _ discrete _ systems have concentrated on dnlse in one @xcite or two @xcite dimensions . thanks the plethora of the results in the field of `` dnlse thermalization '' the structure of the final equilibrium state and the thermodynamical conditions for the occurrence of discrete breathers are now well understood . among numerous discoveries in this area we would like to point reader s attention to the universal correlations in 1d systems of optical waveguides predicted in @xcite in the limit when the nonlinearity dominates over the linear coupling . in this limit the effective dimensionless temperature turns out to be a universal constant independent on system parameters ( provided that the initial state is characterised by uniform intensities ) and the same universality is also manifested in the shape of the field correlation function . in this paper we would like to focus on a much less studied model , namely , the thermalization of two coupled fields in the presence of the four - wave mixing ( fwm ) @xcite . in the context of nonlinear optics the situation corresponds to the propagation of polarized light in the birefringent material @xcite or mode interaction from different floquet - bloch bands @xcite . here we will concentrate on the first case , however the results presented here are quite general and can be applied to other nolinearly coupled systems . in order to give reference to the real - world units we use algaas as an common example of a material with cubic symmetry and fused silica as the corresponding example of isotropic crystal . the wave dynamics of the two orthogonally polarized fields is given by the following pair of coupled equations @xcite : [ vector - dnlse ] @xmath1 in the above equations , @xmath2 and @xmath3 are slowly varying filed envelopes of the te and tm polarized waves , @xmath4 is the polarization mode dispersion constant , @xmath5 is the vacuum wave vector , @xmath6 is the linear birefringence ( @xmath7 for algaas ) , @xmath8 is the coupling constant ( @xmath9 mm@xmath10 ) , @xmath11 is the nonlinear coefficient , @xmath12 is the kerr coefficient ( @xmath13 @xmath14/w for algaas ) , @xmath15 is the linear refractive index ( @xmath16 for algaas ) . the dimensionless constants @xmath17 and @xmath18 represent the relative strength of self- and cross - phase modulation ( spm and xpm ) . if one puts @xmath19 the system ( [ vector - dnlse ] ) breaks into two independent scalar dnlse equations . we can restrict ourselves to the case of positive coupling @xmath20 since the case of negative coupling can be recovered via a standard staggered transformation @xmath21 , @xmath22 . the change of sign in the nonlinearity can be also be compensated via more complicated transformation which involves staggering , complex conjugation and swapping : @xmath23 , @xmath24 . both transformations only affect the field correlation functions and phase distributions ( and not e.g. the intensity distributions ) in a controlled way and here without loss of generality we will also restrict ourselves to the case of positive nonlinearity . in this paper we assume periodic boundary conditions although in the thermodynamic limit @xmath25 this choice is not essential . note in passing that continuous analogues of system ( [ vector - dnlse ] ) were studied in @xcite with regards to pulse propagation in optical fibers . in any chosen nonlinear medium the dimensionless xpm and fwm constants , @xmath17 and @xmath18 , are not independent and 3 possible cases of interest can be envisaged @xcite : 1 . anisotropic cubic medium ( e.g. algaas ) : @xmath26 . generic isotropic medium : @xmath27 . 3 . isotropic cubic medium ( e.g. fused silica ) : @xmath28 , @xmath29 . we will refer to cases ( a ) and ( b),(c ) as isotropic and anisotropic correspondingly . the system is hamiltonian with the hamiltonian : @xmath30 which is a natural conserved quantity in the system while the additional integral of motion is provided by the total pulse power ( proportional to the sum of local intensities ) @xmath31 additionally in the absence of the four - wave mixing ( @xmath32 ) the individual powers in each polarization @xmath33 and @xmath34 are conserved . in the state of thermodynamic equilibrium the stationary field distribution @xmath35 maximizes the entropy @xmath36 . however the nonlinear evolution always takes place on the shell @xmath37 and @xmath38 which introduces two constraints for the optimization problem . the solution then represents a grand canonical gibbs distribution ( see e.g. @xcite ) : @xmath39\ ] ] where the lagrange multipliers @xmath40 and @xmath41 play the roles of the `` inverse temperature '' and `` chemical potential '' respectively while the normalizing factor @xmath42 has the familiar meaning of the partition function . the assumption of field thermalization implies that instead of averaging the dynamics of the system ( [ vector - dnlse ] ) over a long interval of @xmath43 one can compute the same averages via the equilibrium gibbs measure ( [ gibbs ] ) . however from the point of view of the nonlinear optical waveguides it is impractical to use averaging over large distances since it requires optical waveguides that are way too long . instead the averaging can be understood as averaging over _ disordered initial conditions _ that can be be experimentally controlled @xcite . this brings about the notion of the thermalization distance @xcite @xmath44 after which the information about the initial state of the system is forgotten and the averaging over the initial conditions is equivalent to gibbs averaging . in our treatment we will assume ( unless otherwise specified ) that the initial amplitudes for both te and tm components are constant @xmath45 , @xmath46 while the phases are uncorrelated uniformly distributed random variables . such assumption is not necessary but it simplifies the calculation of the ensemble averages of the initial hamiltonian and power . the knowledge of the partition function @xmath47 $ ] allows one to calculate an average energy per waveguide @xmath48 and the average intensity per waveguide @xmath49 . on the other hand such averages must correspond to their initial values @xmath50 , @xmath51 ( averaged over the disordered initial conditions ) since for each realization of the disorder these are conserved integrals of motion . indeed , from ( [ gibbs ] ) it follows that @xmath52 for each set of phase - averaged initial conditions ( i.e. given pair @xmath53 ) the above two transcendental equations yield the effective inverse temperature @xmath40 and the chemical potential , @xmath41 . as in the scalar case @xcite the necessary condition for the gibbs distribution ( [ gibbs ] ) to be normalizable is the positiveness of the temperature , @xmath54 . thus the curve @xmath55 in the @xmath53 diagram represents a natural boundary between the conventional thermalization region ( @xmath54 ) and the area where @xmath56 and the energy is localized in a form of discrete breathers that have been observed experimentally @xcite . it is convenient to introduce complex amplitudes and phases @xmath57 , @xmath58 with @xmath59 being the phase difference between the two orthogonal components . the case @xmath60 ( @xmath61-integer ) corresponds to the linear polarization in @xmath62-the waveguide , @xmath63 , @xmath64 describe circular polarization etc . this notation corresponds to the _ jones description _ of polarization ( see e.g. @xcite ) . to calculate the partition function @xmath42 one needs to integrate the gibbs exponential in ( [ gibbs ] ) with the hamiltonian ( [ hamiltonian - vect ] ) . in the new variables the integral takes the form : @xmath65 % \end{split}\ ] ] where we have collected all the nonlinear coupling terms as well as chemical potential in the nonlinear interaction part @xmath66 with @xmath67 \\ & -\mu\,(a+b ) \end{split } \label{hamiltonian - gamma}\ ] ] while the linear waveguide coupling is given by the hamiltonian @xmath68 . the gibbs distribution ( [ gibbs ] ) is normalizable when the partition function is finite . a close inspection of eq.([z - vector ] ) reveals that in order to achieve this not only the temperature must be positive @xmath54 , but additionally the inequality @xmath69 this corresponds to possible thermalization for @xmath70 and for the anisotropic case @xmath71 this implies @xmath72 . the borderline case @xmath73 ( which includes the xpm without fwm @xmath74 , @xmath32 ) generally requires special treatment and we will not consider it here . the non - normalizable property of the gibbs distribution indicates the emergence of the localized structures , i.e the genuine equilibrium state now consist of high - amplitude discrete breather ( or several breathers if the system has not yet quite reached the equilibrium ) interacting weakly with a small quasilinear background thermalized at infinite temperature , @xmath55 @xcite . so here the statistical mechanics provides us with a clue as to the regions in the parameter space where the localized structures can be observed in principle @xcite . in this paper we will only study the thermalization regime where the inequality ( [ thermalization - ineq ] ) is fulfilled and the temperature is positive so that gibbs distribution ( [ gibbs ] ) is always normalizable . we leave the analysis of the localized structures for future studies although we will comment on these in the following section . since it is impossible to evaluate the partition function @xmath42 in the closed form we will study 3 different limiting regimes which for the scalar case were already analysed in refs . these regimes are : i ) low temperature limit @xmath75 ii ) high temperature limit @xmath76 ( which also serves as a borderline for emergence of localized structures ) and iii ) the anticontinuum ( high intensity ) regime when the effective nonlinearity parameter @xmath77 defined below in section [ sec : nonl ] is large , @xmath78 . it turns out that doubling the amount of degrees of freedom as compared to the scalar case has a significant impact on the statistical properties of system , ( [ vector - dnlse ] ) . we start our analysis with the first two regimes : @xmath75 and @xmath79 . the limit of @xmath75 corresponds to the configuration that minimizes the hamiltonian ( [ hamiltonian - vect ] ) subject to the given conserved pulse intensity per waveguide @xmath80 . if we restrict our search to a solution with constant amplitudes , phase shifts and locked state of polarization ( i.e. fixed @xmath81 ) the minimal configuration is achieved by the following field distribution : @xmath82 , @xmath83 where the amplitudes @xmath84 and @xmath85 as well as chemical potential ( i.e. the corresponding lagrange multiplier ) , @xmath41 , are given by : @xmath86 this solution exists only for not too low intensities , i.e. for @xmath87 and provides a low bound for the energy per waveguide : @xmath88 in the limit of zero birefringence , xpm and fwm , @xmath89 we get the energy as a sum of energies of two identical scalar dnlses with the average intensity per waveguide @xmath90 - which corresponds to the result of ref.@xcite . in fig . [ fig : h - p ] ( which is an analogue of fig.1 of ref.@xcite ) we plot a phase diagram in the @xmath91 space for a specific case of 16 waveguides with @xmath74 , @xmath92 . both @xmath93 and @xmath80 have been rescaled to the dimensionless multiples of @xmath94 and @xmath95 respectively . one can see that eq.([hmin ] ) provides an excellent low boundary approximation even below the critical intensity for which the solution of ( [ min - amplitudes ] ) exists ( which is @xmath96 for the chosen parameters ) . ( color online ) the different regions in @xmath91 parameter space . the ratio @xmath97 was assumed.,width=325 ] let us now turn to the opposite case of high temperatures @xmath79 assuming that the product @xmath98 remains finite . we will use the method similar to that of ref.@xcite applied earlier to the scalar case . in this limit one can neglect the linear coupling energy @xmath99 in the exponent of ( [ z - vector ] ) together with the birefringence contribution @xmath100 . the partition function is then given by @xmath101 with @xmath102 \\ \times & \exp\left[-\frac{\beta\gamma}{2}\,\left(a^2+b^2 + 2\lambda_1 \ , a \ , b \right ) -\beta\,|\mu| \,(a+b)\right ] . \end{split}\ ] ] next we assume that @xmath103 and write approximately @xmath104,\end{aligned}\ ] ] where we have used the fact that the product @xmath105 is fixed while @xmath106 tends to zero . the answer is @xmath107 finally from eqs.([h - p ] ) we obtain in the leading approximation : @xmath108 the parabola @xmath109 provides the upper boundary @xmath55 for the thermalization region in the plane of parameters @xmath91 - see fig.[fig : h - p ] . interestingly enough it does not depend on the 4-wave mixing constant @xmath18 . in the following chapters we will adopt normalized units where the propagation distance is measured in units of coupling length @xmath110 . we will also assume that the intensities of both te and tm components are @xmath111 , @xmath112 . we can also normalize the intensities of both components by the half of the initial intensity @xmath113 so that @xmath114 . in the new dimensionless units one must simply substitute @xmath115 for @xmath116 , @xmath117 for @xmath8 in the original coupled equations ( [ vector - dnlse ] ) and instead of the nonlinear coefficient @xmath118 one now has a dimensionless nonlinearity parameter @xmath119 . since in most common nonlinear materials and waveguide geometries the ratio @xmath115 is in the order of unity the parameter @xmath77 indeed gauges the relative strength of nonlinearity @xcite . in this section we will be interested in the highly nonlinear regime , @xmath78 . the motivation for this is twofold . firstly , this regime permits almost full analytical treatment which is always helpful when studying the general properties of any nonlinear system and secondly , in ref.@xcite it was shown that in the scalar case this limit corresponds to the reciprocal temperature @xmath40 that does not depend on the value of parameter @xmath77 and is a universal constant ( in dimensionless units ) . the universality of the temperature in turn gives rise to a universal shape of the field correlation function . therefore it is interesting to see how this result changes in the vector case . one should expect that the dynamics and statistics in the vector case are much richer due to the doubled number of interacting degrees of freedom . here we will show that this is indeed the case . our main objects of interest will be the statistics of the intensity and polarization state of each waveguide as well as the distribution of the phase differences ( i.e. phase gradients in the continuous limit ) between the same components ( e.g. te ) of the adjacent waveguides together with the field correlation functions . it turns out that in the strongly nonlinear regime the statistics of the phase gradient is decoupled from those of the intensity and polarization . the latter are most conveniently described by using a popular alternative to the jones description of polarization , namely by introducing the four stokes parameters @xmath120 @xcite . they are related to the jones parameters via : @xmath121 the vector @xmath122 is called a stokes vector on a poincar sphere of radius @xmath123 . the components of the stokes vector for each waveguide are related to the polarisation state of the waveguide while its magnitude provides the total intensity carried by both field components . for example , linear polarization corresponds to the equatorial plane @xmath124 while the circular clockwise and anticlockwise polarizations correspond to the north and south poles @xmath125 . in what follows we will use both jones and stokes descriptions whichever is is more convenient . we will start our analysis with the anticontinuum limit when one can completely neglect both linear coupling and birefringence in eqs.([vector - dnlse ] ) . as will be seen later this corresponds to the initial stages of evolution of field distribution towards the final state of equilibrium . the absence of linear waveguide coupling makes field dynamics in each waveguide independent from the rest . therefore instead of considering the system of coupled field equations ( [ vector - dnlse ] ) one can analyze the dynamics in a single waveguide . in particular the first three stokes parameters obey the following equations ( the waveguide - index has been omitted for convenience ) : @xmath127 this system is completely integrable and its various special cases and generalizations ( like e.g. the inclusion of the birefringence terms @xmath128 ) have been extensively studied in literature both with connection to the nonlinear polarization dynamics @xcite in general and the dynamics of the polarization - locked vector solitons @xcite in particular . the complete integrability of system ( [ stokes - dyn ] ) is due to the existence of two integrals of motion of which the first is just the intensity , i.e. the zeroth stokes parameter , @xmath129 , and the second , @xmath130 , is related to the hamiltonian of the original system @xcite : @xmath131 in other words the dynamics of the system takes place on the intersection of the poincar sphere of radius @xmath129 and a hyperbolic ( or elliptic ) cylinder in the anisotropic case or a plane in the isotropic case given by the equation @xmath132 . one can obtain an autonomous equation for @xmath133 which has the form of a duffing oscillator equation in the anisotropic case @xcite and harmonic oscillator equation in the isotropic case @xcite ( see also appendix [ sec : appendix ] ) . the other two stokes parameters are recovered from eqs.([stokes - dyn ] ) while the phase of the te component can be determined by simple integration of the original field equations . since the system in the anticontinuum limit is completely integrable it does not thermalize , i.e. formula ( [ gibbs ] ) is inapplicable . instead one must use the exact solution for the stokes vector @xmath134 and average it directly over the initial conditions . as mentioned earlier we will assume that all waveguides initially have the same set of intensities , @xmath135 , @xmath136 , @xmath137 and random , independent phases . from the definition of the stokes parameters ( [ stokes ] ) it follows that that initial value @xmath138 is fixed while the points @xmath139 are uniformly distributed on a circle of radius @xmath140 . in fig.[fig : anticont ] we show the evolution of the marginal probability density functions ( pdfs ) @xmath141 and @xmath142 for the anisotropic and anisotropic cases obtained by averaging the solution of system ( [ stokes - dyn ] ) over the initial phase distribution . the parameters @xmath143 were chosen to correspond to the algaas compounds in the anisotropic case and fused silica for the isotropic case . one can see that the structure of the histograms is very sensitive to both the values of @xmath144-coefficients and the symmetry of the initial condition . for example when the initial intensity is equally distributed between the components , i.e. @xmath145 , @xmath146 the marginal distribution @xmath147 shown in fig.[fig : anticont](a ) is 4-modal . it is largely confined to the initial circle of radius @xmath148 and the four maxima are at the points @xmath149 , @xmath150 . at the same time the marginal pdf for the first stokes parameter @xmath133 is sharply centered at zero ( fig.[fig : anticont](c ) ) i.e. for most initial realizations the symmetry between the components is preserved @xmath151 . we have also checked this property for different runs with different distances , @xmath43 ( not shown ) . this means that in the anticontinum limit for the anisotropic cubic crystal ( like e.g. algaas ) each given waveguide evolves into either linear or circular state of polarization - despite that no such preference existed in the initial conditions ( the distribution was uniform on the circle ) . in the isotropic case one can observe that although the distribution of @xmath142 is still sharply peaked around zero ( fig.[fig : anticont](c ) ) the distribution of the two remaining stokes parameters fig.[fig : anticont](b ) is only bi - modal ( and not 4-modal as in the anisotropic case ) with the two maxima at @xmath149 corresponding to circular polarization only . the positions of the maxima in both cases can be easily explained by considering the two integrals of motion ( [ stokes - integrals ] ) . as we have seen in both cases for most realizations one can neglect the value of @xmath133 and the pdf @xmath147 therefore remains confined to the circle of radius @xmath148 for all values of @xmath43 . next if one looks at the distribution of the second integral of motion in ( [ stokes - integrals ] ) , namely @xmath130 , then for the initial values @xmath139 uniformly distributed on a circle of radius 2 one can see that the distribution of @xmath130 is bimodal with the two maxima at @xmath152 and @xmath153 ) in the anisotropic case and @xmath154 in the isotropic case . the intersection points of the pair of curves @xmath155 with the circle in the @xmath156 plane are exactly the four observed maxima in the anisotropic case and two in the isotropic one . in the case when initially the symmetry between the field components is broken ( bottom row in fig.[fig : anticont ] ) the situation is much more complex owing to the multimodal features of the distribution of the parameter @xmath133 - see fig.[fig : anticont](f ) . each maximum of the @xmath142 gives rise to at least 2 maxima in the distribution @xmath141 producing complex crown - like shapes shown in fig.[fig : anticont](d),(e ) . each of the maxima now corresponds to a certain elliptically polarized state which varies from waveguide to waveguide . note also that the positions of the maxima remain fixed with the propagation distance @xmath43 while their widths experience weak periodic oscillations ( not shown ) . the full theoretical analysis of these multimodal distributions requires further study which is beyond the scope of this paper . we have seen above that in the anticontinuum limit when the birefringence and the linear waveguide coupling can be neglected completely the system is integrable and instead of reaching thermal equilibrium it experiences periodic oscillations that can be averaged directly over the initial conditions to produce multimodal distribution for the established state of polarization . the question then arises as to what will happen if both the birefringence and the coupling are taken into account . one can expect that at the initial stages of evolution when the distance is less than the thermal length , @xmath44 , the evolution of the system is close to anticontinuum limit and the state of polarization follows the statistics derived above for the anticontinuum limit . for @xmath157 however , the nonintegrability of the system becomes essential . the linear coupling between the waveguides leads to energy and power exchange between the waveguides so that eventually most of the initial peaks in the marginal pdf @xmath141 are destroyed and final thermal distribution sets in . this final distribution is characterized by the gibbs statistics ( [ gibbs ] ) with the partition function given by ( [ z - vector ] ) . we wish to calculate the partition function and the probability distributions in this final state still assuming strong nonlinearity @xmath78 but now taking into account the linear coupling and birefringence as well . let us first assume that the initial conditions are such that the system thermalizes into a state with finite temperature , i.e. @xmath158 as @xmath159 . we will shortly see that this is only possible for a very special choice of the initial conditions . then the dominating part of the exponent in ( [ z - vector ] ) is the nonlinear interaction term @xmath160 . other terms ( including the phase - dependent coupling energy ) are relatively slow functions of the amplitudes and do not therefore contribute to the field intensity pdf and the partition function . however it is important to retain the chemical potential term since as we shall see below @xmath161 always . when calculating the partition function ( [ z - vector ] ) one can resort to the saddle point approximation in @xmath162 which implies that at large values of @xmath77 the main contribution to the integral comes from absolute minimum of @xmath163 . this minimum is achieved for the values @xmath164 in other words the absolute minimum of the interaction hamiltonian subject to given total power is achieved by a circularly polarized state - the most symmetric of all . this minimum is also degenerate - for a system of @xmath165 waveguides there are @xmath166 possible choices of polarization orientation . this degeneracy is drastically reduced however if one takes into account the coupling term @xmath167 . this term favors the configuration where the phase difference @xmath168 is uniform across the waveguides which leaves only two possibilities : either all fields are clockwise polarized or they are all anticlockwise polarized . these states also corresponds to the two maxima of the distribution function @xmath169 ( or @xmath170 in stokes parameters ) one for each direction of rotation . as previously we assume here that @xmath171 which ensures the convergence of the integrals for positive temperature . the minimal value of the interaction energy ( at fixed power ) is given by @xmath172 if we assume uniform initial amplitudes @xmath135 and @xmath136 ( @xmath114 ) and neglect the terms that are not proportional to @xmath77 in the limit @xmath78 the average energy per waveguide is given by : @xmath173 . \label{def - f}\ ] ] the energy is always bound from below by @xmath174 . in the spirit of saddle point approximation we can now plug @xmath175 , @xmath176 , @xmath177 from eq.([min - thermal ] ) into linear coupling pre - factor @xmath178 when calculating the partition function . the integration over the phases @xmath179 of the te component is reduced then to calculation of the partition function of the 1d calssical xy model ( see @xcite or @xcite for details ) . in the thermodynamic limit ( @xmath180 ) the contribution of this linear coupling term to the logarithm of the partition function is @xmath181 . the remaining gaussian integration over the fluctuations around the minimum of @xmath182 eq.([min - thermal ] ) is trivial . inserting the resulting expression for the partition function @xmath42 into the system of equations ( [ h - p ] ) after some simple algebra we obtain the following transcendental equation for the inverse temperature @xmath40 : @xmath183 with @xmath50 given by eq.([def - f ] ) . for the consistency of our approximation we must demand that the inverse temperature @xmath40 remains finite as @xmath77 goes to infinity ( as is the case in the scalar dnlse @xcite ) . but this is achieved only when @xmath184 so that the l.h.s . vanishes which means that the initial input must be circularly polarised : @xmath145 , @xmath185 . in other words , in order to obtain a universal ( i.e. @xmath77-independent ) temperature constant , similar to the scalar case , the initial state must necessarily be the one that minimizes the nonlinear part of the hamiltonian , @xmath182 subject to the intensity constraint @xmath114 . the system then becomes effectively scalar and thermalization occurs only in the distribution of the te phase differences @xmath186 while the field in each waveguide remains locked in its original clock- or anti - clockwise circular state of polarization . as in ref.@xcite the gibbs distribution can be approximately factorized so that the intensity of each waveguide @xmath187 has a narrow distribution close to gaussian centered around the conserved mean value of @xmath188 . ( color online ) the statistics of the final thermal state in the limit of large nonlinearity when the initial field in _ each _ waveguide is circular clockwise polarized . ( a),(b ) the marginal distribution of stokes parameters @xmath141 for anisotropic and isotropic case respectively . ( c ) , ( d ) the statistics of the intensity and phase difference ( of the te component ) respectively for different values of the nonlinearity parameter . ( e ) cross and self field correlation functions for the te and tm components . ( f ) the evolution of the intensity pdf with distance . the thermalization length @xmath44 is determined as a point where the distribution stabilizes.,width=325 ] the results of the numerical simulations of universal correlations in the vector case of @xmath189 waveguides are presented in fig . [ fig : universal ] . as prescribed , we have started from a symmetric initial configurations @xmath145 and uniform , uncorrelated phase distribution for the te components . to ensure constant circular polarization locking the phases of the tm component were obtained by those of the te by adding @xmath190 ( corresponding to the clockwise field rotation ) . in all the figures ( unless otherwise specified ) the values of the parameters are : @xmath191 , @xmath192 and the propagation length is @xmath193 ( all in the normalized units ) . in fig.[fig : universal ] ( a),(b ) we show numerical simulations of the marginal pdf @xmath141 for the anisotropic ( @xmath74,@xmath194 ) and isotropic ( @xmath28 , @xmath29 ) cases respectively . one can see that both distributions have narrow maxima at the point @xmath195 corresponding to clockwise circular polarization ( as in the initial state ) . we have found very little qualitative difference between the anisotropic and isotropic cases so figs.[fig : universal](c)-(f ) feature only the former . in [ fig : universal](c ) we can see that the intensity pdf becomes more and more narrowly centred around the average value of 2 and the distribution is close to gaussian with the variance proportional to @xmath196 . as for the pdf for the te phase difference , @xmath197 , the theoretical prediction @xmath198 is clearly corroborated by the numerics shown in [ fig : universal](d ) : one can observe a perfect data collapse for different values of @xmath77 . similar to universal correlations in the scalar case , one can obtain universal filed correlations in the vector case . in [ fig : universal](e ) we plot the field correlation functions functions defined as @xmath199 \rangle $ ] where @xmath200 , @xmath201 denote different components of the field , te or tm , and the average is taken both over the waveguide position , @xmath62 , and the initial conditions . since in the thermal equilibrium the te and tm components are always @xmath190 out of phase the cross - field correlation @xmath202 is always zero while the self - field correlation functions have identical universal shape : @xmath203 this is illustrated in fig.[fig : universal]e , where all three components are plotted together with the theoretical fit . finally , fig.[fig : universal](f ) illustrates how quickly the final thermal distribution is achieved . the initial delta - peak in intensity distribution quickly broadens and settles to a stationary , almost gaussian distribution of the type shown in [ fig : universal](c ) already at the point @xmath204 ( in units of the coupling length ) that naturally serves as the thermalization length @xmath44 . one can see that thermalization here occurs over just a fraction of the coupling length , i.e. on the scale of @xmath205 mm for a typical algaas waveguide system . of course one can come up with a more rigorous quantitative definitions of @xmath44 , by looking e.g. at the saturation of the temperature defined from the simulated data via equipartition theorem @xcite but for the purposes of this paper we can restrict ourselves to the crude estimate above . all other distributions also stabilize at the same length . as for the value of the universal temperature constant , @xmath40 , one might think that it can be obtained from eq.([beta - saddle ] ) by putting the l.h.s . to zero . unfortunately this is not so since the resulting value of @xmath40 differs from the one observed in the numerics ( @xmath206 ) by the factor of almost 2 . the reason for such a discrepancy is that the vanishing l.h.s . of eq.([beta - saddle ] ) generally represents the main terms of approximation ( @xmath207 ) . as these terms have now cancelled each other the universal temperature @xmath208 is given by the balance of the next terms in the expansion that are of the order of unity . but it turns out that the neglected terms in the saddle point approximation of the partition function being of the order @xmath209 have nevertheless the contribution to its logarithmic derivative that are in the order of 1 , which affects the value of the reciprocal temperature @xmath40 . therefore in principle one has to consider the next terms in the saddle point approximation of ( [ z - vector ] ) . however we have opted for a different approach , namely , we have evaluated the logarithmic derivatives of the integral of @xmath210 numerically and the obtained solution of system ( [ h - p ] ) provided us a value of @xmath40 which fitted well the dependencies shown [ fig : universal](d)-(e ) ( and also the intensity pdfs in [ fig : universal](c ) ) . from a few separate runs ( not shown ) it appears also that not only the found universal value @xmath206 is independent of @xmath77 ( when @xmath78 ) but it is apparently independent of the material parameters @xmath143 as well . this fact however requires further thorough verification . in the above section we have shown that if one ensures that the initial field components are locked in the same circular state of polarization for each waveguide , the reciprocal temperature is a universal numerical constant @xmath40 , independent on the nonlinearity parameter @xmath77 which implied universal shape of the intensity and phase pdfs as well as the field correlation functions . let us now turn to the more general case when the initial field does not have any preferred state of polarization , i.e. although both te and tm components have constant intensities @xmath135 , @xmath136 ( @xmath114 ) their phases are always independent and uniformly distributed . it turns out that although the final thermal state is still given by the constrained minimum of the nonlinear energy corresponding to the circular polarization given by ( [ min - thermal ] ) , the value of the temperature @xmath40 is no longer a universal constant but depends on the nonlinearity . also there is a drastic increase in the thermalization length , sometimes by up to 3 orders of magnitude . however we shall see below that in this general case the `` universality '' is not lost altogether . rather it is the product @xmath211 and not the temperature itself that is universal so instead of e.g. universal filed correlations as in the scalar case @xcite one has _ universal intensity distribution _ @xmath212 that is not affected by the value of the nonlinearity parameter @xmath77 . the width of this distribution ( i.e. the variance of the intensity fluctuations ) is of the order of @xmath213 . it turns out , however , that this new constant @xmath211 is less `` universal '' than the temperature in the above - considered case of initial circular polarization inasmuch as it exhibits strong dependence on parameters @xmath143 . we start by noticing that from eq.([beta - saddle ] ) it follows that if the initial fields are not circularly polarised , i.e. @xmath214 the reciprocal temperature @xmath40 must necessarily be of the order @xmath215 to balance the large terms in the l.h.s . of course ( [ beta - saddle ] ) was obtained in the saddle point approximation which strictly speaking no longer applies when the product @xmath211 is in the order of unity . but as we shall presently see the relation @xmath216 also follows from the exact scaling dependence of the partition function valid beyond the saddle point approximation . we will introduce the new notations for the re - scaled temperature @xmath217 , and chemical potential @xmath218 that are now both in the order of 1 . one can notice that in this regime since @xmath40 scales as @xmath219 one can neglect the coupling energy , @xmath99 ( as well as the birefringence term @xmath220 ) in ( [ z - vector ] ) which simplifies the calculations somewhat . in fact this assumption is similar to the one used in section [ sec : boundaries ] when we determined the infinite temperature boundary @xmath55 with the only difference being that now one can not expand @xmath221 in powers of the argument ( since the latter is in the order of unity ) . regardless of the applicability of the saddle point approximation the numerical simulations demonstrate that a single waveguide probability density function factorises into the product of the pdf of the phase differences in the te polarization , @xmath222 and the pdf for the remaining three jones parameters , @xmath223 , with : @xmath224 note that since @xmath225 while @xmath226 the phase pdf @xmath222 is i ) non - universal ( i.e depends on the nonlinearity ) and ii ) approaches the uniform distribution as @xmath159 ( unlike the scalar case where it remains fixed , c.f . fig.[fig : universal](d ) ) . the intensity distribution is no longer narrow and has a finite width proportional to @xmath227 . it is convenient at this stage to change variables from jones to stokes parameters @xmath228 according to the definition ( [ stokes ] ) . the jacobian of this transformation is equal to @xmath229 and the marginal pdf for the stokes parameters has the form : @xmath230 , \end{split}\ ] ] and @xmath231 . ( color online ) the statistics of the final thermal state in the limit of large nonlinearity and no initial phase correlation . ( a)-(d ) the marginal distribution of stokes parameters @xmath141 for anisotropic ( a),(b ) and isotropic ( d , e ) cases respectively . ( a ) and ( c ) show numerical simulations while ( b ) and ( d ) demonstrate theoretically calculated profiles . ( e ) , ( f ) the statistics of the intensity for anisotropic ( e ) and isotropic ( f ) cases for different values of nonlinearity parameter @xmath77 . the phase difference distribution @xmath197 for anisotropic ( g ) and isotropic ( h ) cases for different values of the nonlinearity parameter.,width=325 ] the marginal probability for the intensity as well as the overall normalization , the temperature and the chemical potential can be established by introducing the spherical coordinates on the poincar sphere of radius @xmath129 and integrating out the angular variables . the result reads : @xmath232\times \\ & \,e^{\tilde \beta s_0 ^ 2 ( 1-\lambda_1)y/4}. \end{split}\ ] ] here @xmath233 is the normalization constant related to the overall partition function of eq.([z - vector ] ) via @xmath234 ( as both linear coupling and the birefringence can be neglected ) . the full analytical form of the intensity pdf @xmath212 can not be obtained in the general case but one can see that it is i ) asymmetric and ii ) has a gaussian asymptote @xmath235 $ ] as @xmath236 . again the condition @xmath171 ensures the convergence . the normalized inverse temperature @xmath237 and the chemical potential @xmath238 can then be determined in a standard way from eqs.([h - p ] ) . by closely inspecting the normalization integral @xmath233 one can infer that it has a self - similar form ( and so does the partition function ) : @xmath239 where the explicit form of the function @xmath240 is given below . plugging this ansatz into ( [ h - p ] ) and excluding the terms containing the derivative of @xmath241 we obtain the relation between the ( normalized ) chemical potential and the temperature : @xmath242 the temperature is then to be obtained from a single transcendental equation : @xmath243 where the derivative is taken with respect to the argument , and @xmath244 is determined via eq.([def - f ] ) . as for the function @xmath245 , it is generally only available in quadratures as @xmath246e^{(1-\lambda_1)yz/4}.\ ] ] the solution of the transcendental equation ( [ eq - beta ] ) , @xmath247 is universal inasmuch as it does not depend on the value of nonlinearity , @xmath77 . for given choice of @xmath143 it depends only on the average initial energy ( [ def - f ] ) and as long as the latter is not very close to the minimal value @xmath248 ( which corresponds to the universal regime considered in the previous section ) the solution always exists an is in the order of 1 . in our numerical simulations shown in fig.[fig : general ] we have chosen a symmetric initial condition with completely randomised phases : @xmath145 , @xmath249 . from ( [ def - f ] ) this corresponds to the value @xmath250 . for the anisotropic case ( algaas ) the numerical solution of ( [ eq - beta ] ) yields @xmath251 while for the isotropic one ( fused silica ) one gets @xmath252 . we have found that unlike in the case of universal correlations observed in section [ sec : universal ] ( or the scalar case ) , the thermalization length now depends on the temperature ( which is directly proportional to the nonlinearity ) . generally , the higher is the temperature , the longer it takes for the system to reach equilibrium . for the highest level on nonlinearity achieved in our simulations ( @xmath253 ) the thermalization length was of the order @xmath254 units of coupling length . this is of course an extreme limit and lower values of nonlinearity ( i.e. temperature ) produce lower values of the thermalization length ( typically @xmath255 ) . in [ fig : general](a)-(d ) we can see marginal distribution of the 2 stokes parameters @xmath147 for @xmath192 . theoretical profiles ( b),(d ) were obtained by integrating out the @xmath133 variable in the ( normalized ) distribution ( [ pdf - stokes ] ) while the parameters @xmath237 and @xmath238 were determined via ( [ mu - vector ] ) and ( [ eq - beta ] ) . one can observe a good agreement between theory and numerics @xcite . the distribution is bimodal but the peaks ( corresponding to circular clockwise and anticlockwise polarisation ) are wide enough ( the width is in the order of @xmath227 ) . [ fig : general](e),(f ) compare theoretical prediction given by ( [ pdf - s0 ] ) with the numerics . one can see that for different values of the nonlinearity @xmath77 the product @xmath256 ( and hence the intensity pdf @xmath212 ) indeed remains the same and instead of universal correlation functions observed in the scalar case or the case of circular initial polarization ( fig.[fig : universal](c ) ) one now has _ universal intensity distribution _ as given by ( [ pdf - s0 ] ) . the opposite case occurs with regards to the angle distribution @xmath197 , [ fig : general](g ) , ( h ) . whereas in the the scalar case and in the case of circular polarization ( fig.[fig : universal](d ) ) this pdf remains invariant , now , according to the theoretical prediction ( [ saddle - pdfs ] ) the distribution becomes closer and closer to uniform as @xmath77 increases since @xmath248 scales as @xmath77 while @xmath237 and @xmath238 remain constant . in some respect the results of ( [ fig : general])(e)-(h ) mirror those results for the circular polarization [ fig : universal](c),(d ) and the scalar case ( fig.2 of ref.@xcite ) but with the angular and intensity distribution trading places . the universal nature of the intensity pdf @xmath212 can also be explained from the energy conservation ( a similar argument for the scalar case was put forward in @xcite ) . in the limit of large @xmath77 most of the energy concentrates in the nonlinear part @xmath160 which is a sum of on - site component given by ( [ hamiltonian - gamma ] ) . we can introduce the standard deviation for intensities as @xmath257 ( where angular brackets denote both averaging over the initial conditions and over waveguides ) and rewrite the nonlinear energy in terms of @xmath258 . then the following average integral of motion exists : @xmath259=const,\ ] ] where the term @xmath260 $ ] collectively denotes the contribution from the linear coupling and birefringence . the value of the constant is determined by the initial conditions and since initially all waveguides have the amplitudes , @xmath135 , @xmath136 , @xmath114 , and random uncorrelated phases the initial variance vanishes @xmath261 and one has the following relation : @xmath262.\ ] ] this means that beyond the thermalization distance @xmath44 the first term in the r.h.s . becomes a stationary @xmath77-independent constant of the order of unity and so does the variance of the intensity distribution as clearly seen in [ fig : general](e ) . the situation changes however if one starts from the locked circular polarization state for each waveguide as in section [ sec : universal ] . then the first term in the r.h.s . of the above changes to @xmath263 . but the distribution in this case centres narrowly on the extremal values @xmath264 , @xmath265 so that the average vanishes and in this regime the variance scales inversely proportional to @xmath77 as is indeed seen in [ fig : universal](c ) . finally it is instructive to see how the distribution of the polarization state @xmath141 evolves with distance @xmath43 , passing from the initial isotropic state through the integrable regime described in section [ sec : anticont ] and eventually relaxing towards the equilibrium gibbs distribution . the successive snapshots of such evolution are given in fig.[fig : evolution ] for the anisotropic case and @xmath192 . one can see all successive stages from the uniform distribution on a circle of radius 2 onto the 4-modal distribution that occurs in a transient anticontinuum regime when the energy exchange between the waveguides is still negligible ( cf . fig.[fig : anticont](a ) ) and finally towards the thermal equilibrium state which is achieved when the weak linear inter - waveguide coupling provides uniform mixing between all the degrees of freedom . in this paper we have considered the structure of a thermalized field dynamics in discrete birefringent waveguide systems . we have defined for the first time an exact boundary in the space of the integrals of motion separating the thermal phase with positive temperature from that corresponding to localized excitations ( discrete breathers ) . we have shown that in the limit of high nonlinearity depending on the choice of the initial conditions the marginal pdf for the stokes parameters @xmath141 relaxes either to a universal , broad , bimodal distribution ( with the maxima corresponding to clock- or anti - clockwise circular polarization ) or ( if initially all waveguides are locked in the same circular polarization ) a narrow - peaked one , corresponding to the thermal fluctuation around the initial state . in both cases either the effective `` temperature constant '' @xmath40 is universal ( i.e. does not depend on the nonlinearity parameter @xmath77 ) or generally the product @xmath211 remains fixed as one increases the value of the nonlinearity . also in the limit of strong nonlinearity before reaching the final thermal state the system passes through the state where its dynamic is integrable , corresponding to the anticontinuum limit of the nonlinear polarization dynamics . in this regime the probability distribution of finding a system in a certain polarization state has a complex multimodal structure . as the thermalization sets in the different modes of the distribution gradually disappear leaving only the two maxima corresponding to the circularly polarized states achieving the global minima of the nonlinear coupling energy subject to fixed total power . as a possible continuation of this research one may suggest a more detailed study of the material dependence of the final thermal state ( here only two specific examples were considered ) as well as a full thermodynamical analysis of the breather region of the phase space as done by rumpf @xcite for the scalar case . the author would like to thank yaron silberberg and omer sidis for illuminating discussions . this work was supported by the marie curie fellowship project `` indigo '' . in this section we recall the exact results for the dynamics of the system of stokes parameters ( [ stokes - dyn ] ) ( see @xcite ) . by differentiating the first eq . in ( [ stokes - dyn ] ) and using the other two we arrive at the following system : @xmath266 with @xmath267 \\ \beta&=&\frac{1}{2}\,\gamma^2\,(2-\lambda_1)(2 - 3\lambda_1 ) , \quad \omega=2\gamma(1-\lambda_1)\,s_3(0)\end{aligned}\ ] ] subject to initial conditions @xmath268 , and @xmath269 . the coefficient @xmath40 should not be confused with the reciprocal temperature as defined in other sections . thus in the isotropic case we get an equation for a harmonic oscillator while in the anisotropic case the equation is that of the nonlinear duffing oscillator . the solution in the isotropic case is simple : @xmath270 in the anisotropic case the form of the solution depends on the sign of the coefficient @xmath40 as well as the duffing oscillator energy : @xmath271 here we will only consider the case when @xmath272 so that we have simultaneously @xmath56 , @xmath273 and @xmath274 . for algaas we have @xmath275 which corresponds exactly to this regime . moreover in this case the potential energy has 2 symmetric maxima with the values @xmath276 and one can prove that for any choice of the initial conditions @xmath277 the difference @xmath278 the effective `` particle '' oscillates in the valley between the two maxima . the solution is expressed via jacobi elliptic function : @xmath279 , \\ s_1^*&=&\sqrt{\frac{\alpha}{|\beta|}}\ , \frac{\sqrt{2}\kappa}{\sqrt{1+\kappa^2 } } , \quad \kappa=\sqrt{\frac{e_*}{e}}-\sqrt{\frac{e_*}{e}-1 } \leq 1 . \nonumber\end{aligned}\ ] ] here @xmath280 is the maximal reachable amplitude of the oscillations , @xmath281 is the elliptic sine function and @xmath282 is the incomplete elliptic integral of the first kind : @xmath283 now using either the two integrals of motion of the main system - i.e. eqs([stokes - integrals ] ) or integrating the system ( [ stokes - dyn ] ) directly one can restore the remaining stokes parameters , @xmath284 and @xmath285 . in the anti - continuum limit the averaging over the initial conditions amounts to averaging either ( [ s1-isotropic ] ) or ( [ duffing - solution ] ) for @xmath133 and using the integrals of motion to obtain the marginal pdf @xmath141 . becasue of the jacobian @xmath229 there is a narrow integrable singularity at zero both in ( [ pdf - stokes ] ) and in the marginal pdf @xmath141 , but the width of this singularity is much smaller than the computational grid and it does not feature prominently in our simulations .
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statistical mechanics of two coupled vector fields is studied in the tight - binding model that describes propagation of polarized light in discrete waveguides in the presence of the four - wave mixing .
the energy and power conservation laws enable the formulation of the equilibrium properties of the polarization state in terms of the gibbs measure with positive temperature .
the transition line @xmath0 is established beyond which the discrete vector solitons are created . also in the limit of the large nonlinearity an analytical expression for the distribution of stokes parameters is obtained which is found to be dependent only on the statistical properties of the initial polarization state and not on the strength of nonlinearity .
the evolution of the system to the final equilibrium state is shown to pass through the intermediate stage when the energy exchange between the waveveguides is still negligible .
the distribution of the stokes parameters in this regime has a complex multimodal structure strongly dependent on the nonlinear coupling coefficients and the initial conditions .
| 14,816 | 227 |
one of the most tantalizing observational discoveries of the past decade has been that the expansion of the universe is speeding up rather than slowing down . an accelerating universe is strongly suggested by observations of type ia high redshift supernovae provided these behave as standard candles . the case for an accelerating universe is further strengthened by the discovery of cosmic microwave background ( cmb ) anisotropies on degree scales ( which indicate @xmath8 ) combined with a low value for the density in clustered matter @xmath9 deduced from galaxy redshift surveys . all three sets of observations strongly suggest that the universe is permeated by a relatively smooth distribution of ` dark energy ' ( de ) which dominates the density of the universe ( @xmath10 ) and whose energy momentum tensor violates the strong energy condition ( @xmath11 ) so that @xmath12 . although a cosmological constant ( @xmath13 ) provides a plausible answer to the conundrum posed by dark energy , it is well known that the unevolving cosmological constant faces serious ` fine tuning ' problems since the ratio between @xmath14 and the radiation density , @xmath15 , is already a miniscule @xmath16 at the electroweak scale ( @xmath17 gev ) and even smaller , @xmath18 , at the planck scale ( @xmath19 gev ) . this issue is further exacerbated by the ` cosmological constant problem ' which arises because the @xmath20-term generated by quantum effects is enormously large @xmath21 , where @xmath22 gev is the planck mass @xcite . although the cosmological constant problem remains unresolved , the issue of fine tuning which plagues @xmath20 has led theorists to explore alternative avenues for de model building in which either de or its equation of state are functions of time . ( following @xcite we shall refer to the former as quiessence and to the latter as kinessence . ) inspired by inflation , the first dark energy models were constructed around a minimally coupled scalar field ( quintessence ) whose equation of state was a function of time and whose density dropped from a large initial value to the small values which are observed today @xcite . ( ` tracker ' quintessence models had the advantage of allowing the current accelerating epoch to be reached from a large family of initial conditions @xcite . ) half a decade after sne - based observations pointed to the possibility that we may be living in an accelerating universe , the theoretical landscape concerning dark energy has evolved considerably ( see the reviews * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in addition to the cosmological constant and quintessence , the current paradigm for de includes the following interesting possibilities : * * dark energy with @xmath23 * @xcite * * the chaplygin gas * whose equation of state drops from @xmath24 at high redshifts to @xmath25 today @xcite * * braneworld models * in which the source for cosmic acceleration rests in the _ gravity sector _ rather than in the matter sector of the theory @xcite * * dark energy models with negative potentials * @xcite * * interacting models of dark matter and dark energy * @xcite * * modified gravity and scalar - tensor theories * @xcite * * dark energy driven by quantum effects * @xcite * * dark energy with a late - time transition in the equation of state * @xcite * * unified models of dark energy and inflation * @xcite etc . faced with the current plethora of dark energy scenarios the concerned cosmologist is faced with two options : \(i ) she can test _ every single _ model against observations , \(ii ) she can take a more flexible approach and determine the properties of dark energy in a _ model independent manner_. in this paper we proceed along route ( ii ) and demonstrate that model independent reconstruction brings us face to face with exciting new properties of dark energy . applying the techniques developed in @xcite to a new data set consisting of @xmath26 supernovae from @xcite and an additional 22 supernovae from @xcite we show that the de equation of state which best fits the data evolves from @xmath3 at @xmath27 to @xmath28 today . _ an evolving equation of state of de is favoured by the data over a cosmological constant for a large region in parameter space . _ supernova observations during the previous decade have been pioneered by two teams : the high - z supernova search team ( hzt ) @xcite and the supernova cosmology project ( scp ) @xcite . the enormous efforts made by these two teams have changed the way cosmologists view their universe . a recent analysis @xcite of 172 type ia supernovae by hzt gives the following bounds on the cosmic equation of state ( at @xmath29 cl ) -1.48 < w < -0.72 , [ eq : state0 ] when the 2dfgrs prior @xmath30 is assumed @xcite . a similar bound w < -0.78 , [ eq : state1 ] is obtained for a new sample of high - z supernovae by scp @xcite . , the dark energy equation of state becomes virtually unbounded from below and has a @xmath31 confidence limit of being @xmath32 ! @xcite ] these results clearly rule out several de contenders including a tangled network of cosmic strings ( @xmath33 ) and domain walls ( @xmath34 ) . however a note of caution must be added before we apply ( [ eq : state0 ] ) or ( [ eq : state1 ] ) to the wider class of de models discussed in the introduction . impressive as the bounds in ( [ eq : state0 ] ) & ( [ eq : state1 ] ) are , they strictly apply only to dark energy having a _ constant equation of state _ since this prior was assumed both in the analysis of the supernova data set as well as in the 2dfgrs study @xcite . aside from the cosmological constant ( @xmath13 ) , the topological defect models alluded to earlier and the sine - hyperbolic scalar field potential @xcite no viable de models exist with the property @xmath35 . indeed , most models of dark energy ( quintessence , chaplygin gas , braneworlds , etc . ) can show significant evolution in @xmath0 over sufficiently large look back times . in this paper we shall reconstruct the properties of dark energy _ without assuming any priors _ on the cosmic equation of state . ( the dangers of imposing priors on @xmath0 have been highlighted in @xcite and several of our subsequent results will lend support to the conclusions reached in this paper . ) cosmological reconstruction is based on the observation that , in a spatially flat universe , the luminosity distance and the hubble parameter are related through the equation @xcite : [ eq : h ] h(z)=^-1 . thus knowing @xmath36 we can unambiguously determine the hubble parameter as a function of the cosmological redshift . next , the einstein equations h^2 & & _ m + _ de , + q & = & - = _ i ( _ i + 3p_i ) , [ eq : einstein ] are used to determine the energy density and pressure of dark energy : [ eq : energy ] _ de & = & _ critical - _ m = ( 1 - _ m(x ) ) , + p_de & = & ( q - ) , where @xmath37 is the critical density of a frw universe . the equation of state of de @xmath38 follows immediately @xcite [ eq : state ] w_eff(x ) = 2 q(x ) - 1 3 ( 1 - _ m(x ) ) , where @xmath39 , @xmath40 . in quintessence models and in @xmath20cdm , the equation ( [ eq : state ] ) determines the true ` physical ' equation of state of dark energy . however the subscript ` eff ' in @xmath41 stresses the fact that this quantity should be interpreted as an ` effective ' equation of state in de models in which gravity is non - einsteinian or in models in which dark energy and dark matter interact . examples of the former include braneworld models and scalar - tensor theories . it is well known that in a large class of braneworld models the hubble parameter does not adhere to the einsteinian prescription ( [ eq : einstein ] ) since it includes explicit interaction terms between dark matter and dark energy @xcite . in this case the equation of state determined using ( [ eq : state ] ) can still be used to characterize de , but physical interpretations of @xmath41 need to be treated with caution . and its derivatives . a detailed discussion of these issues can be found in @xcite . ] one route towards the meaningful reconstruction of @xmath0 lies in inventing a sufficiently versatile fitting function for either @xmath42 or @xmath43 . the parameters of this fitting function are determined by matching to supernova observations and @xmath0 is determined from ( [ eq : h ] ) and ( [ eq : state ] ) . itself @xcite . see @xcite for a summary of different approaches to cosmological reconstruction . non - parametric approaches are discussed in @xcite ; see also @xcite . ] our reconstruction exercise will be based upon the following flexible and model independent ansatz for the hubble parameter @xcite h(x ) = h_0x^3 + a_0 + a_1x + a_2 x^2^ , [ eq : taylor ] where @xmath44 . this ansatz for @xmath43 is exact for the cosmological constant @xmath13 ( @xmath45 ) and for de models with @xmath46 ( @xmath47 ) and @xmath48 ( @xmath49 ) . it has also been found to give excellent results for de models in which the equation of state varies with time including quintessence , chaplygin gas , etc . @xcite . the ansatz ( [ eq : taylor ] ) is equivalent to the following expansion for de [ eq : de1 ] _ de = ( a_0 + a_1x + a_2 x^2 + a_3x^3 ) , where @xmath50 is the present day critical density . the condition @xmath51 allows @xmath52 to _ mimic _ the properties of dark matter at large redshifts ( @xmath53 follows from large scale structure constraints ) . from ( [ eq : taylor ] ) and ( [ eq : de1 ] ) we find @xmath54 , the value of @xmath55 in ( [ eq : taylor ] ) can be slightly larger than @xmath56 in this case . substituting ( [ eq : taylor ] ) into the expression for the luminosity distance we get = _ 1 ^ 1+z . [ eq : lumdis ] the parameters @xmath57 are determined by fitting ( [ eq : lumdis ] ) to supernova observations using a maximum likelihood technique . this ansatz has only three free parameters @xmath58 since @xmath59 for a flat universe . a note of caution : since the ansatz ( [ eq : de1 ] ) is a truncated taylor expansion in @xmath60 its range of validity is @xmath61 , consequently the ansatz - derived @xmath43 and @xmath42 should not be used at higher redshifts . note that the weak energy condition for dark energy @xmath62 has the following form for the ansatz ( [ eq : taylor ] ) : [ eq : wec ] a_0 + a_1 x + a_2 x^2 0 , a_1 + 2a_2 x 0 , provided we assume that the @xmath63 term in ( [ eq : taylor ] ) is totally due to non - relativistic dark matter and does not include any contribution from dark energy . the demand that the wec ( [ eq : wec ] ) be satisfied for all @xmath64 ( i.e. in the past as well as in the future ) requires @xmath65 to be non - negative . however , the demand that the wec ( [ eq : wec ] ) be satisfied in the past ( @xmath66 ) but not necessarily in the future , leads to the somewhat weaker constraint [ eq : wec1 ] a_1 + 2a_2 0 , a_2 0 . ( models in which @xmath67 for @xmath68 and which violate the wec in the future , have been discussed in @xcite . ) = 2.4 in the presence of the term @xmath63 in ( [ eq : taylor ] ) has two important consequences : ( i ) it ensures that the the universe transits to a matter dominated regime at early times ( @xmath69 ) , ( ii ) it allows us to incorporate information ( available from other data sets ) regarding the current value of the matter density in the universe . this information can be used to perform a maximum likelihood analysis with the introduction of suitable priors on @xmath70 . in further analysis we will assume that the @xmath63 term in ( [ eq : taylor ] ) does not include any contribution from dark energy . we have also studied simple extensions of the ansatz ( [ eq : taylor ] ) by adding new terms @xmath71 and @xmath72 . the @xmath71 term allows @xmath0 to become substantially less than @xmath73 , thereby providing greater leeway to phantom models . the @xmath72 term allows de to evolve towards equations of state which are more stiff than dust ( @xmath74 ) ; its role is therefore complementary to that of @xmath71 . despite the inclusion of these new terms , our best fit to the supernova data presented below does not change significantly(choosing @xmath75 and @xmath76 ) , which points to the robustness of the ansatz ( [ eq : taylor ] ) for the given data set . we should add that our reason for choosing an ansatz to fit @xmath43 rather than some other cosmological quantity was motivated by the fact that the hubble parameter is directly related to a fundamental physical quantity the ricci tensor , and is therefore likely to remain meaningful even when other quantities ( such as the equation of state ) become ` effective ' . ( this happens for instance , in the case of the braneworld models of dark energy discussed in @xcite . ) the rationale for choosing a three parameter ansatz for @xmath43 is the following . the observed luminosity distance determined using type ia supernovae is rather noisy , therefore in order to determine the hubble parameter from @xmath77 and following that the equation of state , one must take two derivatives of a noisy quantity . this difficulty can be tackled in two possible ways : ( i ) either one smoothes the data over some interval @xmath78 ( binning is one possibility ) , or ( ii ) we may choose to smooth ` implicitly ' by parameterizing @xmath43 through an appropriate fitting function . the number of free parameters @xmath79 in the fit to @xmath43 will be related to the smoothing interval @xmath80 through @xmath81 . increasing @xmath82 implies decreasing @xmath80 which results in a rapid growth of errors through @xmath83 , and @xmath84 @xcite , therefore in order not to loose too much accuracy in our reconstruction we considered 3 parameter fits for @xmath43 in our paper ( these correspond to 2 parameter fits for @xmath0 ) . we now test the usefulness of the ansatz ( [ eq : taylor ] ) in reconstructing different dark energy models . the ansatz returns exact values for @xmath20cdm , and @xmath85 , @xmath86 quiessence models . in figure [ fig : err ] we show the accuracy of the ansatz ( [ eq : taylor ] ) when applied to several other dark energy models such as tracker quintessence , the chaplygin gas and super - gravity ( sugra ) models . we plot the deviation of @xmath87 ( which is the measured quantity for sne ) obtained with the ansatz ( [ eq : taylor ] ) from the actual model values . clearly the ansatz performs very well over a significant redshift range for @xmath88 ( also see appendix [ sec : fits ] ) . in fact , in the redshift range where sne data is available , the ansatz recovers these models of dark energy with less than @xmath89 errors . however it would be appropriate to add a note of caution at this point . although figure [ fig : err ] clearly demonstrates the usefulness of the ansatz for some de models , its performance vis - a - vis other models of de is by no means guaranteed . by its very construction the ansatz ( [ eq : taylor ] ) is expected to have limitations when describing models with a fast phase transition @xcite as well as rapidly oscillating quintessence models @xcite . ( the ansatz ( [ eq : taylor ] ) can give reasonable results even for these models provided the resulting de behaviour is suitably smoothed . ) for this reason , although the bulk of our analysis will be carried out using ( [ eq : taylor ] ) , we shall supplement it when necessary with other fitting functions , which will provide us with an independent means with which to test the robustness of our reconstruction exercise . * methodology :* for our primary reconstruction , we use a subset of 172 type ia supernovae , obtained by imposing constraints @xmath90 and @xmath91 on the 230 sne sample , as in the primary fit of @xcite . for the ansatz ( [ eq : lumdis ] ) , we require to fit four parameters : ( @xmath92 ) . we may use prior information on @xmath93 ( @xmath94 , @xcite ) and @xmath70 ( @xmath95 , @xcite ) . is not model independent since it relies on the @xmath20cdm model to project from redshift space to real space . results coming from the use of this prior should therefore not be taken too literally in the present context . see @xcite for an interesting discussion of related issues . ] the measured quantity for this data is @xmath96 , therefore the likelihood function is given by & = & n ( - ) , + ^2 & = & _ i=1 ^ 172 ( ) ^2 , where @xmath82 is a normalisation constant . therefore , the probability distribution function in the four - space @xmath97 is ( h_0 , , a_1 , a_2 ) ( - ) pr(h ) pr(h_0 ) . where @xmath98 refers to the priors applied on the parameters of the system . our goal is to reconstruct cosmological parameters such as the equation of state @xmath99 , therefore we marginalise over @xmath93 and obtain the probability distribution function in the @xmath100 space : ( , a_1 , a_2 ) = ( h_0 , , a_1 , a_2 ) dh_0 . in order to do this , we have to define the bounds of a four - dimensional volume in @xmath101 . the bounds of @xmath93 are taken at @xmath102 of the hst prior . for @xmath70 , the natural choice is @xmath103 . it is not immediately obvious what the bounds should be for @xmath104 . we choose a sufficiently large rectangular grid for @xmath105 ( roughly corresponding to @xmath106 ) which includes most known models of dark energy . this bound is merely a matter of convenience and does not affect our results in any way . after marginalisation , we have a grid in @xmath100 space on which @xmath107 is specified at each point . we may now proceed in two ways . firstly , we may choose to fix @xmath70 at a suitable constant value ( @xmath88 ) thereby obtaining a grid in the @xmath108 plane with @xmath109 ( the probability if @xmath70 is known to be an exact value ) defined at each point . for a particular redshift , we may then calculate @xmath110 at each point of the grid . this would yield results that would hold true if @xmath70 were known exactly . instead of using the exact value of @xmath70 , we may use the prior information about it available to us ( @xmath111 ) , and calculate @xmath110 at each point of a three - dimensional grid , the probability @xmath112 at each point being known . therefore , at any given redshift @xmath2 , @xmath113 can be tagged with a numerical value @xmath114 . starting from the best - fit @xmath0 ( the value at the peak of the probability distribution ) , we may move down on either side till @xmath115 of the total area is enclosed under the curve , thus obtaining asymmetric @xmath116 bounds on @xmath0 . the @xmath117 bounds can be similarly obtained . = 2.4 in = 2.4 in * results :* we first show preliminary results for which the matter density is fixed at a constant value of @xmath88 . a detailed look at the @xmath118 surface in the ( @xmath104 ) plane ( figure [ fig : chi ] ) reveals the existence of two minima in @xmath118 , a shallower one close to @xmath20cdm ( @xmath119 ) , and a deeper minimum at @xmath120 . we would like to draw the readers attention to the fact that imposing the prior @xmath121 amounts to disallowing a significant region of parameter space ( the unshaded region in figure [ fig : chi ] ) . consequently an analysis which assumes @xmath122 loses all information about the region @xmath123 around the deeper minimum ! since we have no reason ( observational or theoretical ) to favour either minimum over the other , we shall always choose the deeper minimum as our best - fit in all the subsequent calculations . = 2.4 in in the figure [ fig : dev ] , we plot the deviation of the squared hubble parameter @xmath124 from @xmath20cdm over redshift for the best - fit . we note that the quantity @xmath124 has a simple linear relationship with the parameters of the fit ( eq [ eq : taylor ] ) , therefore the errors in this quantity increase with redshift . another quantity of interest is the energy density of dark energy . for this ansatz , @xmath125 ( where @xmath126 is the present day critical density ) . the figure [ fig : dens ] shows the logarithmic variation of @xmath127 with redshift . in this figure too the errors increase with redshift . an interesting point to note is that initially , dark energy density decreases with redshift , showing the phantom - like nature ( @xmath128 ) of dark energy at lower redshifts of @xmath129 , while at higher redshifts , the dark energy density begins to track the matter density . before moving on to the second derivative of the luminosity distance ( the equation of state ) we may obtain more information from the dark energy density by considering a weighted average of the equation of state : 1+|w= ( 1+w(z ) ) , [ eq : w_avg1 ] where @xmath130 denotes the total change of the variable between integration limits . this quantity can be elegantly expressed in terms of the difference in energy densities over a range of redshift as 1+|w= . [ eq : w_avg2 ] thus the variation in the dark energy density depicted in figure [ fig : dens ] is very simply related to the weighted average equation of state ! cccc @xmath80&@xmath131&@xmath132&@xmath123 + @xmath133&@xmath134&@xmath135&@xmath136 + + @xmath137&@xmath138&@xmath139&@xmath140 + + @xmath141&@xmath142&@xmath143&@xmath144 + @xmath145 \epsfxsize=2.4 in \epsffile{h_exp_w_mi_tot.epsi } & \epsfxsize=2.4 in \epsffile{h_exp_w_tot.epsi } & \epsfxsize=2.4 in \epsffile{h_exp_w_pl_tot.epsi } \\ \mbox{\bf ( a ) } & \mbox{\bf ( b ) } & \mbox{\bf ( c ) } \end{array}$ ] . @xmath118 per degree of freedom for best - fit and @xmath20cdm models . @xmath146 is the present value of the equation of state of dark energy in best - fit models . @xmath147 refers to the best fit after imposition of the wec prior @xmath121 . [ cols="^,^,^,^,^,^,^ " , ] = 2.4 in * using priors on age of the universe :* important consistency checks on our best - fit universe may be provided by observations of the age of the universe . unfortunately , estimates of the age of the universe from different methods can produce widely varying results one reason for which is that estimates of the hubble parameter itself can vary significantly . for instance , the hst key project yields @xmath148 , while studies of the sunyaev - zeldovich effect in galaxy clusters give a significantly lower value @xmath149 @xcite . estimates of the ages of the oldest globular clusters suggest @xmath150 gyrs , at the @xmath29 confidence level @xcite and this age estimate is consistent with several other measurements including observations of eclipsing spectroscopic binaries @xcite , results from radioactive dating of a metal - poor star @xcite and wmap data @xcite ( see also @xcite ) . the results from the wmap experiment suggest @xmath151 gyrs with a hubble parameter @xmath152 , for @xmath20cdm cosmology ( which satisfies the wec ) . adding sdss and sne ia data to wmap , @xcite find an age of @xmath153 gyrs for a slightly closed @xmath20cdm universe with @xmath154 . although these results can not be carried over to evolving dark energy models including those implied by our best - fit reconstruction ( which violate the wec ) they provide an indication of the range within which the age of the universe might vary . keeping in mind these various results , we use two different priors on the hubble parameter : @xmath155 ( @xmath132 bound from hst ; @xcite ) , and @xmath156 ( approximate bound from wmap , sdss , sne ia ; @xcite ) . for each case , we choose three different gaussian priors on the present age of the universe : @xmath157 respectively , and perform the reconstruction for a @xmath88 universe . the results are shown in the figure [ fig : age_pr ] . we find that , for a hubble parameter of @xmath148 , and with an additional prior on the age of the universe @xmath158 gyrs , the best - fit remains nearly the same , showing a rapid evolution of the equation of state from @xmath159 at @xmath4 to @xmath160 at @xmath161 , and the errors become narrower . as the age is increased , the best - fit equation of state evolves more slowly , and the @xmath162 also increases ( see table [ tab : chi_pr ] ) . for the prior @xmath163 , we find that the lowest @xmath164 is obtained for the age prior of @xmath165 gyrs , which once again matches our best - fit . it should be noted that the errors are smaller in all cases , even though the @xmath118 may be larger . we must remember that the addition of a new prior which is consistent with the underlying dataset would lead to a natural reduction in errors . however , the addition of a prior inconsistent with the dataset would lead to a shift of the likelihood maximum as well as a reduction in errors , and the results would then fail to reflect the actual information present in the dataset . that this is happening here for the higher values of age can be seen from the fact that although the errors are reduced , the @xmath162 is actually larger . therefore priors from other observations should be added prudently to ensure that they do not lead to incorrect representation of the data . since there is as yet no clear model independent consensus on the age of the universe , the results we obtain in this section should be interpreted with a degree of caution . figure [ fig : dec ] shows the evolution of the deceleration parameter with redshift . we find that the behaviour of the deceleration parameter for the best - fit universe is quite different from that in @xmath20cdm cosmology . thus , the current value of @xmath166 is significantly lower than @xmath167 for @xmath20cdm ( assuming @xmath168 ) . furthermore the rise of @xmath169 with redshift is much steeper in the case of the best - fit model , with the result that the universe begins to accelerate at a comparatively lower redshift @xmath170 ( compared with @xmath171 for @xmath20cdm ) and the matter dominated regime ( @xmath172 ) is reached by @xmath4 . @xmath173 \epsfxsize=2.4 in \epsffile{h_exp_dens_hzt.epsi } & \epsfxsize=2.4 in \epsffile{h_exp_dens_scp.epsi } \\ \mbox{\bf ( a ) } & \mbox{\bf ( b ) } \end{array}$ ] @xmath173 \epsfxsize=2.4 in \epsffile{h_exp_w_hzt.epsi } & \epsfxsize=2.4 in \epsffile{h_exp_w_scp.epsi } \\ \mbox{\bf ( a ) } & \mbox{\bf ( b ) } \end{array}$ ] @xmath173 \epsfxsize=2.4 in \epsffile{h_exp_w_hzt_cons.epsi } & \epsfxsize=2.4 in \epsffile{h_exp_w_scp_cons.epsi } \\ \mbox{\bf ( a ) } & \mbox{\bf ( b ) } \end{array}$ ] based on the above analysis , it is tempting to conclude that the dominant component of the universe today is dark energy with a steeply evolving equation of state which marginally violates the weak energy condition . ( of course , the less radical possibility of weakly time dependent dark energy satisfying the weak energy condition remains an alternative , too . ) however , before any such dramatic claims are made , we need to check if our results are in any fashion a consequence of inherent bias in the statistical analysis itself , or in the sampling of the data . we therefore perform the following simple exercises to satisfy ourselves of the robustness of our results . * using different subsets of supernova data :* in an attempt to understand how the nature of the reconstructed equation of state is dependent on the distribution of data , we perform the reconstruction exercise on different samples of data . we have confined ourselves to the case where @xmath88 for these exercises . firstly , we may exclude the scp data points from the 172 sne primary fit , leading to a subsample of 130 sne . we call this the hzt sample . figures [ fig : data_dens](a ) and [ fig : data_w](a ) show the result of performing the analysis on this subsample without any constraints . the @xmath118 per degree of freedom for the best - fit is @xmath174 , which is lower than @xmath175 for this sample . in this case we find that , though the error bars are slightly larger , overall the dark energy density behaves in the same way as before ( compare figure [ fig : data_dens](a ) with figure [ fig : dens ] ) , showing phantom like ( @xmath128 ) behaviour at lower redshifts and tracking matter at higher redshifts . the equation of state of dark energy also evolves much in the same way as when the entire sample is used ( compare figure [ fig : data_w](a ) with figure [ fig : total](b ) ) , starting at @xmath176 and evolving rapidly to @xmath3 . we may also use a sample complementary to this sample , where all the scp data points published till date are considered , along with the low redshift calan - tololo sample . this leads to a sample of 58 sne @xcite , which we call the scp sample . using this sample , we obtain the figures [ fig : data_dens](b ) and [ fig : data_w](b ) . the best - fit has a chi - squared per degree of freedom : @xmath177 , lower than @xmath178 for this sample . we find that here too , the dark energy density initially decreases and then starts tracking matter . the equation of state shows signs of rising steeply at low redshifts , but since the highest redshift in this sample is @xmath179 , the behaviour of @xmath180 beyond this redshift can not be predicted , therefore the apparent flattening out of the curve beyond a redshift of one can not be seen in this case . for both these subsets of data , we may repeat the exercise using the prior @xmath181 . the results obtained for the equation of state , as seen in figures [ fig : data_c](a ) , ( b ) , are once again commensurate with the results obtained earlier for the full sample ( figure [ fig : constr](b ) ) . we may therefore conclude from this exercise that subsampling the data does not significantly affect our results , and the steep evolution of the equation of state of dark energy is not a construct of the uneven sampling of the supernovae , but rather , reflects the actual nature of dark energy . * testing our ansatz against fiducial dark energy models :* @xmath145 \epsfxsize=2.4 in \epsffile{h_exp_w_sim_w.epsi } & \epsfxsize=2.4 in \epsffile{h_exp_w_sim_cg.epsi } & \epsfxsize=2.4 in \epsffile{h_exp_w_sim_ev.epsi } \\ \mbox{\bf ( a ) } & \mbox{\bf ( b ) } & \mbox{\bf ( c ) } \end{array}$ ] @xmath173 \epsfxsize=2.4 in \epsffile{lin_w.epsi } & \epsfxsize=2.4 in \epsffile{lin_w_cons.epsi } \\ \mbox{\bf ( a ) } & \mbox{\bf ( b ) } \end{array}$ ] @xmath173 \epsfxsize=2.4 in \epsffile{bass_w.epsi } & \epsfxsize=2.4 in \epsffile{bass_w_cons.epsi } \\ \mbox{\bf ( a ) } & \mbox{\bf ( b ) } \end{array}$ ] the crucial question of course is whether the reconstructed equation of state of dark energy depends upon the ansatz which is used in the exercise , , whether the behaviour of the equation of state merely reflects a bias in the ansatz itself . in this section we show how the ansatz performs in recovering dark energy models whose equation of state is known , from simulated data . this ansatz was demonstrated to work extremely well when simulations of snap data were used @xcite . however , simulation of snap - like data is an optimistic exercise , since data of this quality is unlikely to be available in the near future . we now demonstrate the accuracy with which the ansatz can recover the fiducial background cosmological model if data is simulated using present - day observational standards . in figures [ fig : sim_err ] ( a ) , ( b ) , ( c ) , we show how well the ansatz recovers the equation of state for three fiducial models ( assuming @xmath88 ) : \(a ) a quiessence dark energy model with a constant equation of state : @xmath182 , \(b ) a generalised chaplygin gas model with @xmath183 : with @xmath184 and the present - day equation of state @xmath185 , which would give rise to an effective equation of state w(z)=- , and \(c ) a model with a linearly evolving equation of state : @xmath186 , with @xmath187 . ( for de models with @xmath188 the ansatz is exact therefore we do nt show the results for these cases . ) we find that in all three cases , the fiducial model lies within the @xmath189 confidence limits around the best - fit @xmath0 . based on this result , we claim that within the @xmath132 error bars , the reconstructed equation of state represents the true properties of dark energy when we use real data . * using other ansatz :* it is also important to check whether the results of our reconstruction can be replicated using other ansatz such as fits to the luminosity distance or the equation of state . many different fits have been suggested in the literature ( see for example @xcite , @xcite , @xcite , @xcite ) . here we choose the fit suggested in @xcite in which the equation of state of dark energy is expanded as w(z)=w_0 + . [ eq : lin ] the luminosity distance can therefore be expressed as = _ 1 ^ 1+z , + where @xmath190 $ ] . we find that for this fit , the errors in the equation of state get larger with redshift , however this fit too demonstrates that the equation of state of dark energy increases rapidly with redshift ( figure [ fig : lin](a ) ) when no priors are assumed on the equation of state ( eos ) . the @xmath118 per degree of freedom at the best - fit is @xmath191 . when the prior @xmath121 is invoked , the best - fit eos remains very close to the @xmath20cdm model ( figure [ fig : lin](b ) ) . therefore , from this ansatz , we may make the statement that at low redshifts , the equation of state of dark energy shows the same signs of rising steeply with redshift if no priors are assumed on the equation of state , thus supporting our earlier results . the large errors in the equation of state at redshifts of @xmath192 however make it difficult to make any definitive statements about the behaviour of dark energy at high redshifts . a limitation of the fit ( [ eq : lin ] ) is that it is unable to describe very rapid variations in the equation of state . an ansatz which accommodates this possibility has been suggested in @xcite w(z)=w_i+ , [ eq : bass ] where @xmath193 is the initial equation of state at high redshifts , @xmath194 is a transition redshift at which the equation of state falls to @xmath195 and @xmath130 describes the rate of change of @xmath0 . the resulting luminosity distance has the form : = _ 1 ^ 1+z , + where @xmath196 $ ] . the results for the analysis using this fit to the equation of state are shown in fig . [ fig : bass ] . we find that when the reconstruction is done without any priors on the equation of state ( figure [ fig : bass](a ) ) , the best fit is _ remarkably close _ to the result for ansatz ( [ eq : taylor ] ) ( figure [ fig : total](b ) ) . the @xmath118 per degree of freedom at the minimum is @xmath197 for this fit . the errors in this case are somewhat larger , especially at high redshift . if we constrain @xmath1 , then as before , the evolution of the equation of state is much slower ( figure [ fig : bass](b ) ) . so the reconstruction using this ansatz appears to confirm our earlier results . the above exercises lead us to conclude that our results are neither dependent on the nature of the statistical analysis nor on the manner in which the sne data is sampled . it therefore appears that dark energy with a steeply evolving equation of state provides a compelling alternative to a cosmological constant if data are analysed in a prior - free manner and the weak energy condition is not imposed by hand . as this paper was nearing completion , a new dataset consisting of 23 type ia sne was released by the hzt team @xcite . it is clearly important to check whether or not these new data points corroborate the findings reported in the previous sections . accordingly , we use a subset of 200 type ia sne with @xmath198 from the 230 sne sample of @xcite , and 22 sne with @xmath199 from the new sample to obtain a best - fit for our ansatz with @xmath88 . we then plot the magnitude deviation of our best - fit universe from an empty universe with @xmath200 in order to illustrate how well our model fits the data ( figure [ fig : hubble ] ) . for clarity , we plot the median values of the data points . we obtain medians in redshift bins by requiring that each bin has a width of at least 0.25 in log@xmath2 and contain at least 20 sne . for comparison , we also plot an @xmath20cdm @xmath201 model , as well as ocdm and scdm models . from figure [ fig : hubble ] we see that our dark energy reconstruction is a much better fit to sne beyond @xmath202 than @xmath20cdm . at low redshifts ( @xmath203 ) the agreement between data and the two models is rather marginal . we now add 22 of the new supernovae ( rejecting one with @xmath204 ) to our existing dataset of 172 supernovae and perform de reconstruction on this new dataset of 194 sne , assuming @xmath88 and no other priors . the resultant figure [ fig : w_new ] is similar to the figure [ fig : total](b ) , with slightly smaller errors and has a best - fit @xmath205 . the above exercises point to the robustness of results reported in previous sections , and indicate that evolving dark energy agrees well with the full data set containing 194 type ia sne . the energy conditions : * strong energy condition : @xmath11 ( sec ) , * weak energy condition : @xmath206 , @xmath7 ( wec ) play a vitally important role in our understanding of the accelerating universe , both in the context of inflation and dark energy . we therefore consider it worthwhile to review certain key developments which deepened our understanding of these issues . in an expanding frw universe the sec implies that the universe decelerates while the wec forbids the pressure from becoming too negative . additionally , in the 1960 s and early 1970 s it was noted that energy conditions play a crucial role in the formulation of the singularity theorems in cosmology . indeed , one of the necessary conditions for the existence of an initial / final singularity in big bang cosmology is that matter satisfies both the sec and wec @xcite . by the late 1970 s it became clear that not all forms of matter satisfy the energy conditions . perhaps the best example of a form of matter which satisfied the weak energy condition but violated the strong one is the cosmological constant , introduced into cosmology by einstein in 1917 . in addition , the vacuum expectation value of the energy momentum tensor , @xmath207 , which describes quantum effects ( particle production and vacuum polarization ) in an expanding universe could , in certain cases , violate both wec and sec @xcite . ( for certain space - times , such as de sitter space , the vacuum energy momentum tensor generates a cosmological constant since @xmath208 . ) thus by the late 1970 s it was well known that neither of the energy conditions could be held as being sacrosanct . the 1980 s , as we all know , led to great advances in the development of the inflationary paradigm . the inflaton field mimics the behaviour of a cosmological constant over sufficiently small intervals of time and therefore violates the sec . early dark energy models were based on inflaton - type scalars which coupled minimally to gravity ( quintessence ) . quintessence violates the sec but respects the wec . precisely because of the latter property , not _ any _ experimentally obtained @xmath42 is compatible with quintessence , as emphasized in @xcite . ( the same observation holds for @xmath43 , since the latter can be derived from @xmath42 using equation ( [ eq : h ] ) . ) clearly if observations do indicate that the wec is violated by de then more general ( wec - violating ) models for de should be seriously considered . one example of wec - violating de is provided by scalar - tensor gravity . scalar - tensor models contain at least two functions of the scalar field ( dilaton ) describing dark energy . as shown in @xcite , these two functions , namely , the scalar field potential and its coupling to the ricci scalar @xmath209 , are sufficiently general to explain _ any _ @xmath43 obtained from observations . the wec can also be effectively violated in de models constructed in braneworld cosmology . it has recently been shown that such models , with @xmath210 , are in excellent agreement with supernova data @xcite . since the field equations in these models are derived from a higher dimensional lagrangian the unusually rapid acceleration of the four dimensional universe arises because of the full five dimensional theory and not because of matter which continues to satisfy the energy conditions and whose density remained finite and well behaved at all times @xcite . this behaviour is in contrast to phantom , which assumes a conventional ` perfect fluid ' form for the energy - momentum tensor and therefore contains pathological features such as an energy density which diverges in the future and a sound speed which is faster than that of light @xcite . the fact that the observed luminosity distance ( derived from supernova observations ) is better fit by dark energy violating the wec than either quintessence or a cosmological constant was first noticed by @xcite . caldwell called this ` phantom energy ' and showed that larger values of @xmath70 ( @xmath211 ) implied increasingly more negative values for the equation of state ( @xmath5 ) of phantom . caldwell s results have since been confirmed by larger and better quality sne data sets for instance @xcite find that , in the absence of priors being placed on @xmath70 , the de equation of state has a @xmath31 confidence limit of being @xmath32 ! both @xcite and @xcite however work under the assumption that the equation of state of dark energy is unevolving , so that @xmath212 constant . in this paper we have shown that , suspending the wec prior and allowing the dark energy equation of state to evolve brings out dramatically new properties of dark energy . thus the dark energy model which best fits the sne observations has an equation of state which rapidly evolves from @xmath213 at present ( @xmath214 ) to @xmath215 at @xmath4 . dark energy therefore appears to have properties which interpolate between those of dark matter ( dust ) at early times and those of a ` phantom ' ( @xmath5 ) at late times . this paper reports the model independent reconstruction of the cosmic equation of state of dark energy in which no priors are imposed on @xmath0 . in the literature the imposition of various priors frequently precedes the analysis of observational data sets . such a procedure is well founded and entirely justified when priors are dictated by complementary information such as orthogonal observations coming from different data sets . however , on occasion the use of priors is justified on ` theoretical grounds ' and in this case one must be careful so as not to prejudge nature . ( compelling theoretical reasons might well reflect our own particular conditioning or set of prejudices ! ) in the case of the analysis of type ia supernova data , the priors most frequently used have been @xmath212 constant and @xmath216 . both confine de to within a narrow class of models . moreover , as shown in @xcite , the imposition of such priors on the cosmic equation of state can , on occasion , lead to gross misrepresentations of reality . in this paper we do not impose any priors on @xmath0 and reconstruct the equation of state of dark energy in a model independent manner . in this case our best fit @xmath0 evolves from @xmath5 at @xmath161 , to @xmath3 at @xmath217 ( the upper limit is set by observations ) . this result is robust to changes in the value of @xmath70 and remains in place within the broad interval @xmath218 . our reconstruction clearly favours a model of de whose equation of state metamorphoses from @xmath24 in the past to @xmath219 today . an excellent example of a model which has this property is the chaplygin gas @xcite . however , in this model dark energy does not violate the weak energy condition ( if it was not already violated initially ) . our results also lend support to the dark energy models discussed in @xcite in which the de equation of state shows a late - time phase transition . an interesting example of an evolving de model in which @xmath220 at present whereas @xmath221 at earlier times is provided by the braneworld models ( called brane1 ) examined in @xcite which have been shown to agree very well with current supernova observations @xcite . it is also conceivable that the observed rapid growth in the eos might characterise ` unified ' models of dark matter ( dm ) and dark energy ( de ) . we end this paper with a small speculation on this last possibility . since the nature of both dm and de is currently unknown , it may be that a mechanism exists which converts dm ( with @xmath24 ) into de ( with @xmath25 ) in regions with sufficiently high density contrast @xmath222 . ( this would happen if , for instance , the rate of conversion of dm into de depended upon @xmath223 , etc . ) since the conversion of dm to de is confined to high peaks of the density field this process will not occur uniformly in the entire universe but will be restricted to regions occupying a small filling fraction ( @xmath224 ) ( @xmath225 for regions with @xmath222 ; see for instance @xcite and references therein ) . this process could commence as early as @xmath226 when the first peaks in a cdm model collapse . since de does not cluster and since @xmath227 grows rapidly as the universe expands , de from high density regions ( @xmath225 ) will spread at the speed of light , percolating through the entire universe ( @xmath228 ) by @xmath4 . since the creation of de is tagged to the formation of structure , this model may not encounter the ` coincidence problem ' which plagues other scenarios of de including quintessence . ( however this model might have problems in producing a sufficiently homogeneous and isotropic distribution of dark energy on the largest scales . ) the concrete mathematical framework for a phenomenological model of this kind will be worked out in a companion paper . in summary , evolving de models have been shown to satisfy sne observations just as well ( if not better ) than the cosmological constant . our best fit equation of state , in the absence of any priors , evolves from @xmath213 at @xmath214 to @xmath215 at @xmath4 . indeed , figure [ fig : hubble ] shows that our best fit eos is better able to account for the relative brightness of supernovae at @xmath229 than @xmath20cdm . however , the evolution in @xmath0 is much weaker if the prior @xmath230 ( @xmath231 ) is imposed . due to the paucity of sne data beyond @xmath232 ( till date , there is only a single data point beyond @xmath233 , sn1999bf at @xmath234 ) it is not clear whether @xmath215 is a stable asymptotic value for the reconstructed de equation of state at high redshifts . new supernova data at @xmath235 from ongoing as well as planned surveys ( snap ) combined with data from other cosmology experiments ( cmb , lss , s - z survey s , lensing , etc . ) are bound to provide important insights on the nature of dark energy at high redshifts . our results clearly throw open exciting new possibilities for dark energy model building . @xmath236 _ acknowledgments : _ we would like to thank john tonry for several important clarifications and for help in preparing figure [ fig : hubble ] . we also thank sarah bridle , pier - stefano corasaniti , alessandro melchiorri , yuri shtanov and lesha toporenskii for their useful comments on an earlier version of the paper . one of us ( vs ) acknowledges useful discussions with salman habib and daniel holz . ua thanks the csir for providing support for this work . as was partially supported by the russian foundation for basic research , grant 02 - 02 - 16817 , and by the research program `` astronomy '' of the russian academy of sciences . alam , u. and sahni , v. , 2002 , astro - ph/0209443 . alam , u. , sahni , v. and starobinsky , a.a . , 2003 , jcap * 0304 * , 002 , [ astro - ph/0302302 ] . alam , u. , sahni , v. , saini , t. d. , and starobinsky , a.a . , 2003 , , * 344 * , 1057 [ astro - ph/0303009 ] . alcaniz , j.s . , d. and dev , a. , 2002 , , 067301 [ astro - ph/0206448 ] . amendola , l. , 2000 , , 043511 . barber , a.j . , , 2000 , , 444 . barris , b. j. , 2004 , , 571b [ astro - ph/0310843 ] . bartolo , n. and pietroni , m. 2000 , 023518 . bassett , b.a . , kunz , m. , silk , j. and ungarelli , c. , 2002 , mnras , * 336 * , 1217 [ astro - ph/0203383 ] . bertolami , o. and martins , p.j . , 2000 , , 064007 . birrell , n.d . & davies , p.c.w . , 1982 , _ quantum fields in curved space _ , cambridge university press , cambridge . boisseau , b. , esposito - farese , g. , polarski , d. and starobinsky , a.a . , 2000 , , 2236 caldwell , r.r . , dave , r. and steinhardt , p.j . , 1998 , , 1582 . caldwell , r.r . , 2002 , , 23 [ astro - ph/9908168 ] . caldwell , r.r . , kamionkowski , m. and weinberg , n.n . , 2003 , phys.rev.lett . * 91 * 071301 [ astro - ph/0302506 ] . carroll , s.m . , 2001 , living rev.rel . * 4 * 1 [ astro - ph/0004075 ] . carroll , s.m . , hoffman , m. and trodden , m. , 2003 , , 023509 [ astro - ph/0301273 ] . cayrel , r. , 2001 , nature * 409 * , 691 [ astro - ph/0104357 ] . chaboyer , b. and krauss , l.m . , 2002 , apj , * 567 * , l45 . chiba , t. and nakamura , t. , 2000 , , 121301(r ) . chiba , t. , okabe , t. and yamaguchi , m , 2000 , , 023511 . chimento , l.p . , jakubi , a.a . , pavon , d. and zimdahl , w. , 2003 , 083513 [ astro - ph/0303145 ] . copeland , e.j . , liddle , a.r . and lidsey , j.e . , 2001 , 023509 . corasaniti , p.s . and copeland , e.j . , 2003 , 063521 [ astro - ph/0205544 ] . corasaniti , p.s . , bassett , b.a . , ungarelli , c. and copeland , e.j . , 2003 , , 091303 [ astro - ph/0210209 ] . daly , r.a . and djorgovsky , s.g . , 2003 , , 9 - 20 . [ astro - ph/0305197 ] . damour , t. , kogan , i.i . and papazoglou , a. , 2002 , , 104025 [ hep - th/0206044 ] . deffayet , c. , dvali , g. and gabadadze , g. , 2002 , , 044023 [ astro - ph/0105068 ] . felder , g.n . , frolov , a. , kofman , l. and linde , a. , 2002 , phys . rev * 66 * , 023507 . frampton , p. , 2003 frampton , p. and takahashi , t. , 2003 , , 135 . freedman , w. , , 2001 , , 47 . gerke , b , & efstathiou , g. , 2002 , 33 , [ astro - ph/0201336 ] . gnedin , n , lahav , o. and rees , m.j . , astro - ph/0108034 . gratton , r. , 2003 , a & a , * 408 * , 529 , [ astro - ph/0307016 ] . grib , a.a . , mamaev , s.g . and mostepanenko , v.m . , 1980 , _ quantum effects in strong external fields _ , moscow , atomizdat ( in russian ) [ english translation : _ vacuum quantum effects in strong fields _ , friedmann laboratory publishing , st.petersburg , 1994 ] . hansen , b. , 2002 , apj * 574 * , l155 . hawking , s.w . and ellis , g.f.r . , 1973 , _ the large scale structure of space - time _ , cambridge university press . hoffman , m. , 2003 , astro - ph/0307350 . huterer , d. and starkman , g. , 2003 , , 031301 , [ astro - ph/0207517 ] . huterer , d. and turner , m.s . , 1999 , , * 60 * , 081301 . johri , v.b . , 2003 , astro - ph/0311293 . kallosh , r. , linde , a. , prokushkin , s. and shmakova , m. , 2002 , 123503 . kamenshchik , a. , moschella , u. and pasquier , v. , 2001 , 265 . knop , r.a . , et al . , 2003 , 102 [ astro - ph/0309368 ] krauss , l.m . , 2001 , in proceedings , eso - cern - esa symposium on astronomy , cosmology and fundamental physics , march 2002 . krauss , l.m . , 2001 , in international conference on the identification of dark matter , eds . n. spooner and v. kudryavtsev ( singapore , world scientific ) , 1 . krauss , l.m . and chaboyer , b. , 2001 , astro - ph/0111597 . kunz , m. , corasaniti , p. , parkinson , d. and copeland , e.j . , 2003 , astro - ph/0307346 . linder , e.v . , 2003 , 091301 , [ astro - ph/0208512 ] . maeda , k. , mizuno , s. and torii , t. , 2003 , 024033 [ gr - qc/0303039 ] . maor , i. , 2002 , 123003 , [ astro - ph/0112526 ] . marchi , g. de , , 2004 , , 971 [ astro - ph/0310646 ] . mcinnes , b. , 2002 , jhep * 0208 * , 029 [ hep - th/0112066 ] . nakamura , t. and chiba , t. , 1999 , , * 306 * , 696 . nunes , n.j . and lidsey , j.e . , 2003 , astro - ph/0310882 . padmanabhan , t. , 2003 , phys . rep . * 380 * , 235 [ hep - th/0212290 ] . parker , l. and raval , a. , 1999 , , 063512 , 123502 . peebles , p.j.e . and ratra , b. , 1998 , ap . j. lett . * 325 * , l17 . peebles , p.j.e . and ratra , b. , 2002 , rev.mod.phys . * 75 * , 559 [ astro - ph/0207347 ] . peebles , p.j.e . and vilenkin , a. , 1999 063505 . percival , w.j . , , 2001 , , 1297 . perlmutter , s.j . , , 1999 , , 565 . riess , a.g . , , 1998 , , 1009 . sahni , v. , 2002 , class . quantum grav . * 19 * , 3435 , [ astro - ph/0202076 ] . sahni , v. and habib , s. , 1998 , , 1766 , [ hep - ph/9808204 ] . sahni , v. , saini , t.d . , starobinsky , a.a . and alam , u. , 2003 , 201 [ astro - ph/0201498 ] . sahni , v. , sami , m. and souradeep , t. , 2002 , 023518 . sahni , v. and shtanov , yu.v . , 2003 , jcap * 11 * 014 , [ astro - ph/0202346 ] . sahni , v. and starobinsky , a.a . , 2000 , ijmp d * 9 * , 373 [ astro - ph/9904398 ] . sahni , v. and wang , l. , 2000 , , 103517 [ astro - ph/9910097 ] . saini , t.d . , 2003 , , 129 . saini , t.d . , raychaudhury , s. , sahni , v. and starobinsky , a.a . , 2000 , , * 85 * , 1162 . sheth , j.v . , sahni , v. , shandarin , s.f . and sathyaprakash , b.s . , 2003 , mnras * 343 * , 22 [ astro - ph/0210136 ] . singh , p. , sami , m. and dadhich , n.k . , 2003 , , 023522 [ hep - th/0305110 ] . spergel , d.n . , 2003 , suppl . * 148 * , 175 [ astro - ph/0210136 ] starobinsky , a.a . , 1998 , , 757 . tegmark , m. , 2002 , , 103507 . tegmark , m. , 2003 , astro - ph/0310723 . thomson , i.b . , 2001 , , 3089 . tonry , j.l . , et al . , 2003 , , 1 , [ astro - ph/0305008 ] . urena - lopez , l.a . , matos , t. , 2000 , , 081302 , [ astro - ph/0003364 ] . wang , y and lovelace , g , 2001 , , l115 . weinberg , s. ( 1989 ) rev . phys . * 61 * , 1 . weller , j. and albrecht , a. , 2002 , , 103512 [ astro - ph/0106079 ] . wetterich , c , 1988 , , 668 . zeldovich , ya.b . ( 1968 ) sov . uspekhi * 11 * , 381 . we have seen that the error bars on @xmath0 for the analysis using ansatz ( [ eq : taylor ] ) are non - monotonic with redshift . low redshift behaviour of the equation of state affects the luminosity distance at all higher redshifts , while high redshift behaviour effects fewer such distances . this leads to an expectation that high-@xmath2 behaviour of the equation of state should be poorly constrained as opposed to the low-@xmath2 behaviour . this seems to contradict the behaviour seen in our figures . to investigate if this could be explained by our specific method of error analysis we describe the fisher matrix error bars below and show that they are almost identical to what we obtain in our method . in an analysis which uses an ansatz with @xmath237 parameters @xmath238 , the fisher information matrix is defined to be f_ij , where @xmath239 , @xmath240 being the likelihood . for an unbiased estimator , the errors on the parameters will follow the cramr - rao inequality : @xmath241 . since the likelihood function is approximately gaussian near the maximum likelihood ( ml ) point , the covariance matrix for a maximum likelihood estimator is given by ( c^-1)_ij . the fisher information matrix is therefore simply the expectation value of the inverse of the covariance matrix at the ml - point . given the covariance matrix , the error on any cosmological quantity @xmath242 is given by : [ eq : propagate ] _ q^2 = _ i=1^n ( ) ^2 c_ii+2 _ i=1^n _ j = i+1^n ( ) ( ) c_ij . thus the nature of the errors on a quantity will depend essentially on the manner in which it is related to the parameters of the system . we now consider how errors propagate for different cosmological quantities for the polynomial fit to dark energy which we have used for most of the results in this paper : h^2/h_0 ^ 2 = x^3+a_0+a_1 x+a_2 x^2 , x=1+z , where @xmath243 . if @xmath70 is held constant then the parameters of the system are @xmath244 . = 2.4 in = 2.4 in we obtain the covariance matrix in @xmath244 from the ml analysis , and then using equation ( [ eq : propagate ] ) , calculate the errors on cosmological quantities of interest . for example , the errors on the quantity @xmath245 are given by : _ ^ 2(x ) = ( x-1)^2 [ c_11 + 2 ( x+1 ) c_12+(x+1)^2 c_22 ] . although the term @xmath246 is negative we find that @xmath247 still _ increases _ with redshift . this is shown in the figure [ fig : err_hubb ] . the errors shown are approximately similar to those obtained in figure [ fig : dev ] . the corresponding errors on the equation of state can be calculated using equations ( [ eq : state2 ] ) and ( [ eq : propagate ] ) , and has the somewhat more complicated expression : _ w^2(x ) = , where @xmath248 and @xmath105 are the mean values of the parameters . although in this case it is difficult to predict the behaviour of error bars , after substituting the numerical values we obtain the error bars that are shown in figure [ fig : err_w ] . this figure can be compared to the figure [ fig : total](b ) , having almost identical errors . this shows that the nature of our error bars is not an artifact of our specific method of error analysis . however , as shown in figure [ fig : lin ] , a two parameter expansion in @xmath0 shows monotonically deteriorating errors in @xmath0 with the redshift , while the expansion in @xmath249 shows errors that improve with redshift ( figure [ fig : total](b ) ) . this indicates that the nature of error bars might be affected by which quantity is being approximated . in the limit of infinite terms in the expansion of various quantities all the methods should produce identical result . the practical need for truncating these expansions make these approximations slightly different from each other . more specifically , we require setting of priors f(z ) & = & _ i=0^ a_n z^n + a_n & = & 0 ; ( n > n_p ) where @xmath250 could be @xmath43 , @xmath0 or any other physical quantity and @xmath251 is the chosen number of parameters . the non - linear priors in the above equation make different finite expansions inequivalent . since we do not know for certain if the underlying model for the accelerating expansion involves an energy component with negative pressure in a frw setting we are forced to choose one of the alternatives for approximations . we hope that with increasingly high quality data the effect of such truncations will eventually disappear . @xmath173 \epsfxsize=2.4 in \epsffile{h_exp_error_lin.epsi } & \epsfxsize=2.4 in \epsffile{h_exp_error_coras.epsi } \\ \mbox{\bf ( a ) } & \mbox{\bf ( b ) } \end{array}$ ] we have seen in the figure [ fig : err ] that the ansatz ( [ eq : taylor ] ) works well for several physically motivated models of quintessence , chaplygin gas and sugra . in this section we take this exercise further and see how well it can reconstruct some of the other fits to dark energy known in literature . in figures [ fig : sim ] ( a ) and ( b ) , we show results for simulations using @xmath88 and two different fits to the equation of state of dark energy : \(b ) the non - perturbative @xmath0 suggested in @xcite and @xcite , which has the parameters @xmath256 ( the dark energy equation of state today ) , @xmath257 ( the dark energy equation of state at the matter dominated epoch ) , @xmath258 ( the redshift where equation of state changes from @xmath257 to @xmath256 ) , and @xmath130 ( the width of transition ) . for the simulation we again use three sets of values in order of increasing growth rate of @xmath0 : ( a ) @xmath259 , ( b ) @xmath260 , and ( c ) @xmath261 . we find that in both cases , the ansatz recovers the measured quantity to within @xmath89 accuracy in the redshift range important for sne observations . thus we find that even for fits for which the ansatz does not return exact values , it can recover cosmological quantities to a high degree of accuracy .
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we reconstruct the equation of state @xmath0 of dark energy ( de ) using a recently released data set containing 172 type ia supernovae without assuming the prior @xmath1 ( in contrast to previous studies ) .
we find that dark energy evolves rapidly and metamorphoses from dust - like behaviour at high @xmath2 ( @xmath3 at @xmath4 ) to a strongly negative equation of state at present ( @xmath5 at @xmath6 ) .
dark energy metamorphosis appears to be a robust phenomenon which manifests for a large variety of sne data samples provided one does not invoke the weak energy prior @xmath7 .
invoking this prior considerably weakens the rate of growth of @xmath0 .
these results demonstrate that dark energy with an evolving equation of state provides a compelling alternative to a cosmological constant if data are analysed in a prior - free manner and the weak energy condition is not imposed by hand .
cosmology : theory cosmological parameters statistics
| 20,184 | 251 |
it is well known that quantum key distribution is one of the most interesting subjects in quantum information science , which was pioneered by c.bennett and g.brassard in 1984[1 ] . in the original paper of bennett , single photon communication was employed as implementation of quantum key distribution . however , despite that it is not essential in great idea of bennett , many researchers employed single photon communication scheme to realize bb84 , b92[2 ] . because of the difficulties of single photon communication in practical sense , it was discussed whether one can realize a secure key distribution guaranteed by quantum nature based on light wave communication or not . in 1998 , h.p.yuen and a.kim[3 ] proposed another scheme for key distribution based on communication theory(signal detection theory ) . this scheme corresponds to an implementation of secret key sharing which was information theoretically predicted by maurer[4 ] , et al . however , yuen s idea was found independently from maurer s discussion . in the first paper of yuen - kim[3 ] , they showed that if noises of eve(eavesdropper ) and bob(receiver ) are statistically independent , secure key distribution can be realized even if they are classical noises , in which they employed a modification of b92 protocol[2 ] . following yk s first paper , a simple experimental demonstration of yk protocol based on classical noise was reported[5 ] , and recently yk scheme with 1 gbps and 10 km long fiber system based on quantum shot noise was demonstrated[6 ] . however , these schemes are not unconditional secure . that is , ability of signal detection of eve can be superior to that of bob . as a result , an interesting question arises `` is it possible to create a system with current technology that could provide a communication in which always bob s error probability is superior to that of eve ? '' in proceedings paper of qcm and c 2002 , yuen and his coworker reported that yk protocol can be unconditional secure , even if one uses conventional optical communication system[7 ] . this is interesting result for engineer , and will open a new trend of quantum cryptography . in this report , we simulate practical feature of yuen - kim protocol for quantum key distribution with unconditional secure , and propose a scheme to implement them using our former experimental setup[6 ] . a fundamental concept of yuen - kim protocol follows the next remark . + * remark * : _ if there are statistically independent noises between eve and bob , there exist a secure key distribution based on communication . _ + they emphasized that the essential point of security of the key distribution is detectability of signals . this is quite different with the principle of bb-84 , et al which are followed by no cloning theorem . that is , bb-84 and others employ a principle of disturbance of quantum states to give a guarantee of security , but yk protocol employs a principle of communication theory . it was clarified that this scheme can be realized as a modification of b-92 . however , this scheme allows us use of classical noise , and it can not provide unconditional secure . then , yuen and his coworker showed that yk scheme is to be unconditional secure in which a fundamental theorem in quantum detection theory was used for his proof of security as follows . + * theorem * : ( helstrom - holevo - yuen ) + _ signals with non commuting density operators can not be distinguished without error . _ + so if we assign non commuting density operators for bit signals 1 and 0 , then one can not distinguish without error . when the error is 1/2 based on quantum noise , there is no way to distinguish them . so we would like to make such a situation on process between alice and eve . to do so , a new version of yk scheme was given as follows : _ _ * the sender(alice ) uses an explicit key(a short key:@xmath0 , expanded into a long key:@xmath1 by use of a stream cipher ) to modulate the parameters of a multimode coherent state . * state @xmath2 is prepared . bit encoding can be represented as follows : @xmath3 where @xmath4 . * alice uses the running key @xmath1 to specify a basis from a set of m uniformly distributed two - mode coherent state . * the message @xmath5 is encoded as @xmath6 . this mapping of the stream of bits is the key to be shared by alice and bob . because of his knowledge @xmath1 , bob can demodulate from @xmath6 to @xmath5 . here , let us introduce the original discussion on the security . the ciphering angle @xmath7 could have @xmath8 in general as discrete or continuous variable determined by distribution of keys . a ciphered two mode state may be @xmath9 the corresponding density operator for all possible choices of @xmath8 is @xmath10 , where @xmath11 or @xmath12 . the problem is to find the minimum error probability that eve can achieve in bit determination . to find the optimum detection process for discrimination between @xmath13 and @xmath14 is the problem of quantum detection theory . the solution is given by[8 ] @xmath15 as an example of encoding to create @xmath16 which is the error probability of eve , yuen et al suggested certain modulation scheme . in that case , closest values of a given @xmath8 can be associated with distinct bits from the bit at position @xmath8 , and two closest neighboring states represent distinct bits which means a set of base state . in this scheme , they assumed that one chooses a set of basis state(keying state for 1 and 0 ) for bits without overlap . the error probability for density operators @xmath13 and @xmath14 becomes 1/2 , when number of a set of basis state increases . asymptotic property of the error probability depends on the amplitude of coherent state[7][9 ] . original scheme of yk protocol in the above can be realized by practical devices . to apply them to fiber communication system , we would like to realize them by intensity modulation / direct detection scheme . if one does not want to get perfect yk scheme , one can more simplify the implementation of yk protocol . from a fundamental principle in quantum detection theory , we can construct non - commuting density operators from sets based on non - orthogonal states when one does not allow overlap of the selection of a set of basis state for 1 and 0 . on the other hand , when we allow overlap for selection of a set of basis state , one can use orthogonal state to construct the same density operators for 1 and 0 . that is , @xmath17 . however , in this case , unknown factor for eve is only an initial short key , and a stream of bits that eve observed is perfectly the same as those of alice and bob , though eve can not estimate the bits at that time . this gives still insecure situation . so , here , we employ a combination of non - orthogonality and overlap selection in order to reduce the number of basis sets . let us assume that the maximum amplitude is fixed as @xmath18 . we divide it into 2 m . so we have m sets of basis state@xmath19 . total set of basis state is given as shown in fig.1 . each set of basis state is used for @xmath20 , and @xmath21 , depending on initial keys . @xmath22 so the density operators for 1 and 0 for eve are @xmath23 for the sets of @xmath24 , @xmath25,@xmath26 , let us assign 0 and 1 by the same way as eqs(4),(5 ) . in this case , eve can not get key information , but she can try to know the information of quantum states used for bit transmission . so this is the problem for discrimination of 2 m pure states . the error probability is given by @xmath27 although we have many results for calculation of optimum detection problems[10][11][12 ] , to solve this problem is still difficult at present time , because the set of states does not have complete symmetric structure . so we here give the lower bound and tight upper bound . the lower bound is given by the minimum error probability:@xmath28 for signal set @xmath29 which are neighboring states . it is given as follows : @xmath30 } \right)\ ] ] the upper bound is given by applying square root measurement for 2 m pure states . the numerical properties are shown in fig.2-(a ) . thus if m increases , then her error for information on quantum states increases . in this case , pure guessing corresponds to @xmath31 . the error probability of bob , however , is independent of the number of set of basis state , and it is given as follows : @xmath32 we emphasize that eve can not get key information in this stage , because the information for 1 and 0 are modulated by the way of eqs(4),(5 ) . furthermore , this scheme can send 2 m bits by m sets of basis state . let us apply the original scheme such that m bits are sent by m sets of basis state . in this case , eve will try to get key information , so the density operators for eve become mixed states @xmath13 , @xmath14 consisting of set of states which send 1 and 0 , respectively . the numerical properties are shown in fig.2-(b ) . both schemes have almost same security , but the latter can only send m bits by m sets of basis state . in other word , the number of sets is reduced to 1/2 in the former scheme . in implementing yk protocol by conventional fiber communication system , we use here our proposed system . figure 3 shows the experimental setup . the laser diode serves as 1.3@xmath33 m light source . a pattern generator provides a signal pulse string to send keys . a modulator which selects basis state follows a driver of laser diode . the selector gives selection of amplitude and assignment of 1 and 0 , and is controlled by initial keys . the laser driver is driven by output signals of modulator . the optical divider corresponds to eve . the case 1 is a type of opaque " , and the case 2 is a type of translucent " . the channel consists of 10 km fiber and att . we can change the distance equivalently from 10 km to 200 km by att . the speed of pulse generator to drive laser diode is 311mbps , 622mbps , and 1.2gbps . the detector of bob is ingaas pin photo loaded by 50@xmath34 register , and it is connected to an error probability counter which can apply to 12 gbps . the dark current is 7@xmath35 and the minimum received power of our system is about -30 dbm . in this system , the problem for degree of security is only power advantage of eve which will be set in near transmitter(alice ) . when the eavesdropping is opaque , the error probability of bob increases drastically , and the error probability counter shows almost 1/2 , which means that the error of eve is also 1/2 . in this case , problem of communication distance is not so important . we can detect the existence of eve in any distance of channel . when the eavesdropping is translucent , eve has to take only few power(@xmath36 from the main stream of bits sequence in order to avoid the power level disturbance . in this case , the error of bob does not increase . as a result , alice and bob can not detect the existence of eve . the secure communication distance depends on the error probabilities of bob and eve . let @xmath37 be transparency of channel from alice to bob . the detectability for bob in this experiment setup depends on the signal distance(amplitude difference between two states as basis state ) : @xmath38 for @xmath39 , and that of eve depends on the signal distance : @xmath40 . here we assume that @xmath41 , and the total loss is 20db which corresponds to 100 km . since our receiver requires about -30 dbm , the transmitter is -10 dbm . when m increases , sufficiently the error of eve increases . we examined a simulation of yk protocol based on intensity modulation / direct detection fiber communication system , and showed a design of implementation of secure system based on our experimental setup which was used to demonstrate the first version of implementation of yk protocol . we will soon report complete demonstration in experiment by the above system .
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in this report , we simulate practical feature of yuen - kim protocol for quantum key distribution with unconditional secure . in order to demonstrate them experimentally by intensity modulation / direct detection(imdd ) optical fiber communication system , we use simplified encoding scheme to guarantee security for key information(1 or 0 ) . that is , pairwise m - ary intensity modulation scheme is employed .
furthermore , we give an experimental implementation of yk protocol based on imdd .
| 3,214 | 120 |
in this paper all groups are assumed to be finite . the problem of detecting structural properties of a finite group by looking at element orders has been considered by various authors . amiri , jafarian amiri and isaacs in @xcite proved that the sum of element orders of a finite group @xmath0 of order @xmath2 is maximal in the cyclic group of order @xmath2 . the problem of minimizing sums of the form @xmath3 , where @xmath4 is a positive integer and @xmath5 denotes the order of @xmath6 , was considered in @xcite , however there is a mistake in the proof pointed out by isaacs in @xcite . the main point of the argument in @xcite is a pointwise argument , and the strong evidence that it is true suggests to state it as a conjecture . [ mainconj ] let @xmath0 be a finite group of order @xmath2 and let @xmath7 denote the cyclic group of order @xmath2 . there exists a bijection @xmath8 such that @xmath5 divides @xmath9 for all @xmath10 . this is proved in @xcite by frieder ladisch in the case in which @xmath0 is solvable . note that the existence of a bijection as in the conjecture is equivalent to the existence of a family @xmath11 of subsets of @xmath0 with the following properties ( here @xmath12 denotes euler s totient function ) : * the sets @xmath13 are pairwise disjoint and @xmath14 . * @xmath15 for all @xmath16 , for all @xmath17 . * @xmath18 for all @xmath17 . indeed , given a bijection @xmath19 as in the conjecture , define @xmath13 to be the preimage via @xmath19 of the set of elements of @xmath7 of order @xmath20 , and given a partition as above , define @xmath19 piecewise sending @xmath13 to the set of elements of @xmath7 of order @xmath20 . the existence of such a partition is claimed in @xcite with a wrong proof , although this is not the main result of that paper . the main result of @xcite , dealing with the sum @xmath21 , is a consequence of our main result ( theorem [ mainth](1 ) for @xmath22 ) . although in this paper we do not prove conjecture [ mainconj ] , such conjecture is worth mentioning because it is very much related to our results . let @xmath12 denote euler s totient function , i.e. @xmath23 denotes the number of integers in @xmath24 coprime to @xmath2 . in this paper we consider the sum @xmath25 for @xmath26 real numbers and compare it with the case of the cyclic group of size @xmath1 . in the case @xmath27 , @xmath28 this sum equals the sum of element orders , in the case @xmath29 it equals the sum of the cyclic subgroup sizes . moreover if @xmath22 we get an extension of the case considered in @xcite and the case @xmath30 , @xmath31 gives the number of cyclic subgroups . this last case was what motivated us in the beginning , and as a particular case of our main theorem we obtain the following . let @xmath32 denote the number of positive divisors of the integer @xmath2 . [ motiv ] let @xmath0 be a finite group . then @xmath0 has at least @xmath33 cyclic subgroups and @xmath0 has exactly @xmath33 cyclic subgroups if and only if @xmath0 is cyclic . this theorem follows from corollary [ cormotiv ] . using the same techniques we also prove , in section [ sprod ] , the following : [ thprod ] let @xmath0 be a finite group of order @xmath2 and let @xmath34 . then @xmath35 with equality if and only if @xmath0 is cyclic . we also obtain a very interesting characterization of nilpotency ( theorem [ mainth](2 ) ) : let @xmath36 be a real number and let @xmath0 be a finite group of order @xmath2 . then @xmath37 and equality holds if and only if @xmath0 is nilpotent . let us be more specific about what we actually do in the paper . we prove the following result . [ mainth ] let @xmath26 be two real numbers , let @xmath0 be a finite group of order divisible by @xmath2 and let @xmath38set @xmath39 and @xmath40 . 1 . if @xmath41 and @xmath42 then @xmath43 with equality if and only if @xmath0 contains a unique cyclic subgroup of order @xmath4 , for every divisor @xmath4 of @xmath2 . 2 . if @xmath44 then @xmath43 with equality if and only if @xmath0 contains a unique subgroup of order @xmath2 and such subgroup is nilpotent . 3 . if @xmath45 and @xmath46 then @xmath47 with equality if and only if @xmath0 is cyclic . 4 . if @xmath0 is nilpotent and non - cyclic then the sign of @xmath48 equals the sign of @xmath49 . we prove this in section [ main ] . for the case @xmath50 we use lemma [ nov ] ( a combinatorial tool , which is a key result in this paper ) and for the case @xmath51 we adapt the arguments of @xcite . in section [ examples ] , for any positive integer @xmath52 , we construct infinitely many finite groups @xmath0 with exactly @xmath53 cyclic subgroups . in this section we prove theorem [ mainth ] . as usual @xmath54 denotes the set of natural numbers ( in particular @xmath55 ) . denote by @xmath56 ( the mbius function ) the map taking @xmath2 to @xmath57 if @xmath2 is divisible by a square different from @xmath58 , to @xmath58 if @xmath2 is a product of an even number of distinct primes and to @xmath59 if @xmath2 is a product of an odd number of distinct primes . the following result is well - known . let @xmath60 be two functions such that @xmath61 for all @xmath62 . then @xmath63 for all @xmath62 . an important example is the following . it is well - known that@xmath64 for all @xmath62 . this is because in the cyclic group of order @xmath2 for any divisor @xmath20 of @xmath2 there are exactly @xmath65 elements of order @xmath20 . applying the mbius inversion formula we obtain @xmath66 . the following is our key combinatorial tool . [ nov ] let @xmath67 be two functions such that @xmath68 for @xmath69 and @xmath26 two real numbers such that @xmath50 set @xmath70 then we have : 1 . write the prime factorizations of @xmath71 and @xmath4 as @xmath72 and @xmath73 where @xmath74 and @xmath75 . then @xmath76 in particular @xmath77 always and @xmath78 if and only if one of the following holds : * @xmath79 and @xmath80 . * @xmath81 and @xmath82 , i.e. @xmath71 and @xmath4 are not coprime . * @xmath83 and @xmath84 for some @xmath85 , i.e. @xmath71 is even and @xmath4 is odd . 2 . @xmath86 . 3 . suppose @xmath87 are positive functions @xmath88 with @xmath89 and @xmath90 for all divisors @xmath91 of @xmath2 . then @xmath92moreover equality holds if and only if @xmath93 for all divisors @xmath91 of @xmath2 such that @xmath94 . 1 . observe that since @xmath95 is zero whenever @xmath96 is not square - free , @xmath97 , hence we may assume @xmath98 . in the following computation the index @xmath96 will be written as @xmath99 where @xmath100 . @xmath101 since @xmath102 and @xmath42 , @xmath103 is a non - negative number , and it is zero if and only if either @xmath104 for some @xmath105 , i.e. @xmath82 and @xmath106 , or @xmath107 for some @xmath85 , i.e. @xmath30 and either @xmath28 or @xmath108 for some @xmath85 , in other words @xmath71 is even and @xmath4 is odd . we have @xmath109 3 . since by point ( 1 ) @xmath110 for all @xmath111 , applying point ( 2 ) to @xmath112 and to @xmath113 we obtain @xmath114the statement about equality follows easily . this concludes the proof . for @xmath4 a divisor of @xmath1 set @xmath115that is , the number of cyclic subgroups of @xmath0 of order @xmath4 . observe that @xmath0 has exactly @xmath116 elements of order @xmath4 . it follows that @xmath0 has exactly @xmath117 elements of order a divisor of @xmath4 , that is , elements @xmath6 with the property that @xmath118 . the following is a fundamental fact we will use in an essential way . [ frob ] let @xmath4 be a divisor of @xmath1 and let @xmath119 be the set of elements @xmath10 such that @xmath120 . then @xmath4 divides @xmath121 . thus we can write @xmath122 where @xmath123 is an integer depending on @xmath4 and @xmath0 . let @xmath26 be two real numbers , let @xmath0 be a finite group of order divisible by @xmath2 and let @xmath38set @xmath39 and @xmath40 . note that @xmath124 . we proceed with the proof of theorem [ mainth ] . we prove that if @xmath41 and @xmath42 then @xmath43 with equality if and only if @xmath0 contains a unique cyclic subgroup of order @xmath4 , for every divisor @xmath4 of @xmath2 . observe that @xmath126 apply lemma [ nov ] to @xmath127 , @xmath128 , @xmath129 , @xmath130 . since @xmath41 , if @xmath131 then @xmath43 and equality holds if and only if @xmath132 for all divisors @xmath91 of @xmath2 , i.e. @xmath0 has a unique cyclic subgroup of order @xmath91 for all divisors @xmath91 of @xmath2 . now suppose that @xmath30 . then following the above argument we find that @xmath110 with equality if and only if @xmath108 for some @xmath133 . in other words , equality holds if and only if whenever @xmath91 is a divisor of @xmath2 such that either @xmath91 is even or @xmath134 is odd , @xmath135 . we prove that @xmath135 for all divisors @xmath91 of @xmath2 , from which it follows that if @xmath4 is a divisor of @xmath2 then @xmath136 , i.e. @xmath0 has a unique cyclic subgroup of order @xmath4 . so let @xmath91 be a divisor of @xmath2 . if @xmath91 is even then @xmath135 , and the same is true if @xmath134 is odd , so now suppose that @xmath91 is odd and @xmath134 is even . in particular @xmath137 divides @xmath2 and is even , hence @xmath138 . we prove that @xmath135 . if by contradiction @xmath139 then since @xmath140 is a positive integer ( by frobenius theorem ) , @xmath141 hence @xmath142 , that is , there are at least @xmath137 elements @xmath10 such that @xmath143 . but these elements also verify @xmath144 , and @xmath145 , i.e. there are exactly @xmath137 elements @xmath6 in @xmath0 verifying @xmath144 . this means that every @xmath10 such that @xmath144 verifies @xmath146 . now since @xmath134 is even @xmath0 has an element @xmath6 of order @xmath147 , and since @xmath91 is odd @xmath148 ; on the other hand @xmath149 , a contradiction . conversely , if @xmath150 for all divisor @xmath4 of @xmath2 then if @xmath4 divides @xmath2 , @xmath151 . we prove that if @xmath152 then @xmath43 with equality if and only if @xmath0 contains a unique subgroup of order @xmath2 and such subgroup is nilpotent . as above , using lemma [ nov ] , we find @xmath43 , and equality holds if and only if @xmath132 whenever @xmath91 is a divisor of @xmath2 such that @xmath91 and @xmath134 are coprime . applying this to the case when @xmath91 is a prime power we find that writing @xmath153 , @xmath0 has a unique subgroup of order @xmath154 ( which must then be normal in @xmath0 ) for @xmath155 , thus the product of such subgroups is the unique subgroup of @xmath0 of order @xmath2 , and it is nilpotent . conversely , if @xmath0 has a unique subgroup @xmath156 of order @xmath2 and @xmath156 is nilpotent then every sylow subgroup of @xmath156 is the unique subgroup of @xmath0 of its size . indeed if @xmath157 is a sylow subgroup of @xmath156 and @xmath158 with @xmath159 and @xmath160 is some subgroup of @xmath0 such that @xmath161 then @xmath162 is a subgroup of @xmath0 of order @xmath2 ( because @xmath163 being the sylow subgroups of @xmath156 normal in @xmath0 ) , hence @xmath164 and this implies @xmath165 . hence if @xmath91 is a divisor of @xmath2 such that @xmath166 then @xmath135 . we prove that if @xmath45 and @xmath46 then @xmath47 with equality if and only if @xmath0 is cyclic . the following arguments are the natural generalization of the arguments in @xcite . for @xmath167 a subset of @xmath0 define @xmath168 . the following is lemma c in @xcite . [ largep ] let @xmath169 be the largest prime divisor of the integer @xmath170 . then @xmath171 . [ nphi ] let @xmath172 be two real numbers . let @xmath173 be a positive divisor of @xmath2 . then @xmath174 , and equality holds if and only if one of the following occurs . * @xmath175 and @xmath176 . * @xmath177 and each prime divisor of @xmath2 divides @xmath4 . since the function @xmath178 is multiplicative we may assume that @xmath179 and @xmath180 with @xmath169 a prime and @xmath181 . the inequality @xmath182 becomes @xmath183 , i.e. @xmath184 which follows from @xmath185 , @xmath172 . if equality holds and @xmath175 we find @xmath186 . [ cosets ] let @xmath157 be a cyclic normal sylow @xmath169-subgroup of @xmath0 . let @xmath10 and assume that the coset @xmath187 has order @xmath4 as an element of @xmath188 . suppose @xmath175 . then @xmath189 with equality if and only if @xmath6 centralizes @xmath157 . the case @xmath190 is clear , so now assume @xmath191 . since @xmath4 divides @xmath5 we can write @xmath192 for some integer @xmath193 . then @xmath194 and as @xmath195 , we see that @xmath193 is a power of @xmath169 . but @xmath4 divides @xmath196 , which is not divisible by @xmath169 , so @xmath193 and @xmath4 are coprime , and there exists an integer @xmath2 such that @xmath197 . now @xmath198 and we write @xmath199 so that @xmath200 because @xmath2 is coprime to @xmath4 . also @xmath201 so since @xmath157 is abelian , @xmath202 centralizes @xmath157 if and only if @xmath6 centralizes @xmath157 . we can thus replace @xmath6 by @xmath202 and assume that @xmath203 . every element of @xmath187 has the form @xmath204 for some element @xmath205 , and we argue that @xmath206 with equality if and only if @xmath6 centralizes @xmath207 . since @xmath157 is cyclic , @xmath208 is characteristic in @xmath157 , and thus @xmath209 and @xmath210 is a subgroup . now @xmath204 is an element of @xmath210 , so @xmath211 divides @xmath212 , and if equality holds then @xmath210 is cyclic hence @xmath6 centralizes @xmath207 , and conversely if @xmath6 centralizes @xmath207 then since @xmath5 and @xmath213 are coprime , @xmath214 . since @xmath175 it follows from lemma [ nphi ] that @xmath215 with equality if and only if @xmath6 centralizes @xmath207 . now @xmath216 moreover equality holds if and only if @xmath6 centralizes @xmath207 for every @xmath205 , i.e. @xmath6 centralizes @xmath157 . let @xmath157 be a cyclic normal sylow @xmath169-subgroup of @xmath0 . then @xmath217 with equality if and only if @xmath157 is central in @xmath0 . write @xmath218 to denote the order of a coset @xmath187 viewed as an element of @xmath188 . applying lemma [ cosets ] to each coset of @xmath157 in @xmath0 we have @xmath219 by lemma [ cosets ] equality holds if and only if every element @xmath10 centralizes @xmath157 , i.e. @xmath157 is central in @xmath0 . we show that if @xmath220 then @xmath0 is cyclic , and we do it by induction on @xmath1 . this is a straightforward generalization of the argument used in @xcite ( proof of the main theorem ) . assume @xmath220 , i.e. @xmath221 , which we can write as @xmath222 where @xmath223 . then averaging on @xmath0 and using the fact that @xmath7 has @xmath23 elements of order @xmath2 and lemma [ largep ] , where @xmath169 is the largest prime divisor of @xmath2 , we find @xmath224 the strict inequality comes from the fact that we did not count the contribution of the identity element of @xmath7 . this implies that there exists at least one element @xmath10 `` not below the average '' , i.e. such that @xmath225 let @xmath226 . since @xmath46 we have @xmath227 by lemma [ nphi ] , and since @xmath228 we have @xmath229 . we deduce that @xmath230hence @xmath169 does not divide @xmath231 , in other words @xmath232 contains a sylow @xmath169-subgroup @xmath157 of @xmath0 . moreover @xmath233 hence @xmath234 and it follows from the sylow theorem that @xmath235 , i.e. @xmath236 . corollary [ psigp ] then implies that @xmath217 with equality if and only if @xmath157 is central in @xmath0 . let @xmath160 be the sylow @xmath169-subgroup of @xmath237 , so that @xmath238 and @xmath239 . we have @xmath240hence @xmath241 . by the induction hypothesis we deduce that @xmath188 is cyclic . then @xmath242 and hence @xmath243 . thus equality holds in the above chain of inequalities and thus @xmath157 is central in @xmath0 . since @xmath157 is central and @xmath188 is cyclic , it follows that @xmath0 is abelian , and as @xmath157 is a sylow subgroup of @xmath0 , we know that we can write @xmath244 where @xmath245 is cyclic . thus @xmath0 is a direct product of cyclic groups of coprime orders , and thus @xmath0 is cyclic , as required . we prove that if @xmath0 is nilpotent and non - cyclic then the sign of @xmath48 equals the sign of @xmath49 . by lemma [ nov ] applied to @xmath127 , @xmath128 , we have @xmath246 , hence @xmath247 suppose first that @xmath248 with @xmath169 a prime . we then have @xmath249 where we used that @xmath250 . now , for @xmath251 we have @xmath252 . therefore @xmath253 if and only if @xmath175 , @xmath254 if and only if @xmath41 , and @xmath255 if and only if @xmath177 . since @xmath0 is non - cyclic , @xmath256 for some @xmath251 hence @xmath257 if @xmath175 , @xmath258 if @xmath259 and @xmath260 if @xmath41 . now assume that @xmath0 is any nilpotent non - cyclic group , and write @xmath261 as direct product of its sylow subgroups . note that @xmath262 is multiplicative in the sense that if @xmath112 are groups of coprime orders then @xmath263 . write @xmath264 and @xmath265 . if @xmath266 then , since @xmath0 is non - cyclic , by our above discussion of @xmath169-groups @xmath267 for some @xmath268 hence taking the product we find @xmath269 . similarly if @xmath270 then @xmath271 and if @xmath272 then @xmath273 consider now the case @xmath274 . in this case the result looks as follows . let @xmath32 denote the number of positive divisors of the integer @xmath2 . [ cormotiv ] let @xmath2 be a divisor of @xmath1 . then @xmath0 has at least @xmath32 cyclic subgroups of order a divisor of @xmath2 . moreover the following are equivalent . 1 . for all divisors @xmath4 of @xmath2 , @xmath0 has exactly @xmath4 elements @xmath6 such that @xmath120 . @xmath0 has exactly @xmath32 cyclic subgroups of order a divisor of @xmath2 . 3 . the subgroup generated by the cyclic subgroups of @xmath0 of order a divisor of @xmath2 is cyclic and has order @xmath2 . we prove that the number of cyclic subgroups of a finite group @xmath0 whose order divides @xmath2 equals @xmath275 . let @xmath276 be the distinct cyclic subgroups of @xmath0 of order a divisor of @xmath2 . then @xmath277 contains @xmath278 elements of order @xmath279 . for @xmath280 write @xmath281 if @xmath282 generate the same cyclic subgroup of @xmath0 . then @xmath283 is an equivalence relation in @xmath284 hence @xmath285now the result follows from theorem [ mainth ] choosing @xmath274 . de medts and tarnauceanu in @xcite prove that if @xmath0 is nilpotent then @xmath286 , and they conjecture that the converse holds , namely if @xmath286 then @xmath0 is nilpotent . also , they conjecture in @xcite that @xmath287 for all finite groups @xmath0 . does the equation @xmath288 detect solvability of @xmath0 for some @xmath289 ? ( this is a version of a question of thompson , cf . @xcite ) . what structural properties of @xmath0 can be detected by the equation @xmath288 ( for fixed @xmath26 ) ? is the set of pairs @xmath289 for which cyclic groups are detected by the equation @xmath288 dense in @xmath290 ? in the picture above the point @xmath291 does not detect anything , @xmath292 ( which corresponds to the number of cyclic subgroups of @xmath0 ) and @xmath293 ( which corresponds to the sum of the orders of the elements of @xmath0 ) detect cyclicity and whether @xmath294 detects nilpotency is the question of de medts and tarnauceanu . the checked zone detects cyclicity and the thick line detects nilpotency . let @xmath0 be a finite group of order @xmath2 . for a divisor @xmath20 of @xmath2 let @xmath295 be the number of elements of @xmath0 of order @xmath20 . then @xmath296 hence @xmath297 . let @xmath298 be the size of @xmath299 for @xmath300 . then @xmath301 . we compute @xmath302 where @xmath303 . clearly @xmath304 and if @xmath305 then @xmath306 . we now compute @xmath307 for @xmath305 . if @xmath71 is a power of a prime @xmath169 then @xmath308 . now assume this is not the case . let @xmath157 be the set of prime divisors of @xmath71 , so that @xmath309 . we prove that @xmath310 . @xmath311 we used that @xmath312 . in conclusion , writing @xmath313 and @xmath314 we have that @xmath315 equals @xmath316 since @xmath317 we obtain that @xmath318 since @xmath319 for all @xmath300 ( by frobenius theorem ) , we obtain @xmath320suppose now that equality holds . then @xmath321 for @xmath322 and @xmath323 . in particular if @xmath324 then @xmath325 hence there are @xmath326 elements of @xmath0 of order not dividing @xmath327 , in particular the sylow @xmath169-subgroups of @xmath0 are cyclic . in particular if @xmath0 is a @xmath169-group we are done , so now assume that @xmath2 is divisible by at least two distinct primes . let @xmath157 be a sylow @xmath169-subgroup of @xmath0 . we prove that @xmath157 is central in @xmath0 . to do this it is enough to show that for all prime @xmath193 dividing @xmath2 there is a sylow @xmath193-subgroup @xmath160 of @xmath0 which centralizes @xmath157 . if @xmath328 choose @xmath165 . now suppose @xmath329 . write @xmath330 , @xmath331 for the positive integers such that @xmath332 , @xmath333 are the largest powers of @xmath334 dividing @xmath2 , respectively . it is enough to prove that @xmath0 contains elements of order @xmath335 . there are @xmath327 elements of order dividing @xmath327 and @xmath336 elements of order dividing @xmath336 , hence there are at least @xmath337 elements of order not dividing @xmath327 and not dividing @xmath336 , in other words there are at least @xmath337 elements of order divisible by @xmath335 , a suitable power of which has order @xmath335 . we only need to make sure that @xmath338 . we have @xmath339 . since the sylow subgroups of @xmath0 are central and cyclic , @xmath0 is cyclic . the proof is completed . let @xmath52 be a fixed positive integer . in this section we construct infinitely many finite groups @xmath0 with exactly @xmath340 cyclic subgroups . this is done by applying proposition [ subcont ] with @xmath341 , @xmath342 . let @xmath343 be positive integers with @xmath344 ( since @xmath345 , @xmath346 is even ) , @xmath347 odd and @xmath348 . let @xmath349 where the action of @xmath350 on @xmath351 is given by inversion : @xmath352 . [ subcont ] the number of cyclic subgroups of @xmath0 is @xmath353 . for the record , @xmath354 where @xmath355 . we will first find the cyclic subgroups generated by elements of the form @xmath356 where @xmath357 divides @xmath347 and then we will look at the elements of the form @xmath358 . let @xmath359 and let @xmath357 be a divisor of @xmath347 . we compute @xmath360 . since @xmath347 is odd , @xmath357 is odd , hence @xmath361 hence @xmath362 . this implies that @xmath363 equals @xmath364 if @xmath2 is even and @xmath365 if @xmath2 is odd , implying that @xmath366 , which is divisible by @xmath367 . moreover clearly @xmath368 is determined by @xmath369 ( all the elements of @xmath369 outside @xmath232 are of the form @xmath365 ) and @xmath370 is determined by @xmath369 because @xmath371 . this means that different pairs @xmath372 give rise to distinct cyclic subgroups @xmath369 . hence there are @xmath373 such subgroups . hence we have found @xmath373 cyclic subgroups of the form @xmath356 where @xmath357 is a divisor of @xmath347 ( so @xmath357 is odd ) . since @xmath374 commutes with @xmath375 we have @xmath376 hence @xmath377 . moreover the elements of @xmath0 of the form @xmath358 ( i.e. @xmath368 times an even power of @xmath6 ) verify @xmath378 hence their order divides @xmath379 , on the other hand there are exactly @xmath379 such elements hence they all belong to @xmath380 . this proves that @xmath380 is the only cyclic subgroup of @xmath0 of order @xmath379 . we have found @xmath381 cyclic subgroups of @xmath0 . we have @xmath382 and @xmath383 , hence @xmath384 therefore all we are left to show is that the cyclic subgroups we listed are all the cyclic subgroups of @xmath0 . we need to show that if @xmath385 is a cyclic subgroup of @xmath0 of order divisible by @xmath367 then @xmath385 is generated by an element of the form @xmath356 with @xmath357 a divisor of @xmath347 . say @xmath385 is generated by @xmath386 . then @xmath387 is odd , indeed if @xmath387 is even then @xmath368 and @xmath388 commute hence @xmath389 , contradicting the fact that @xmath390 is divisible by @xmath367 . now , similarly as above , @xmath386 has order @xmath391 where @xmath392 is an odd divisor of @xmath346 , that is , a divisor of @xmath347 . since @xmath393 there is some integer @xmath394 such that @xmath395 and @xmath394 is coprime to @xmath396 . this implies that @xmath397 and @xmath398 equals @xmath356 if @xmath394 is odd , it equals @xmath370 if @xmath394 is even . we are really grateful to andrea lucchini and federico menegazzo for helpful discussions , comments and suggestions . we are also very grateful to the referee for his / her very careful reading of a previous version of the paper . 10 g. frobenius , uber einen fundamentalsatz der gruppentheorie , ii . , sitzungsberichte der preussischen akademie weissenstein ( 1907 ) , 428437 . i. m. isaacs and g. r. robinson , on a theorem of frobenius : solutions of @xmath399 in finite groups , amer . monthly 99 , no . 4 ( 1992 ) , 352354 . h. salmasian and nsa problems group , minimizing the sum of negative powers of orders of group elements : 10775 , amer . monthly 109 , no . 3 ( 2002 ) , t. de medts , m. tarnauceanu , an inequality detecting nilpotency of finite groups ( 2012 ) . http://arxiv.org/abs/1207.1020 t. de medts , m. tarnauceanu , finite groups determined by an inequality of the orders of their subgroups , bull . simon stevin 15 ( 2008 ) , 699704 . amiri , habib ; jafarian amiri , s. m. ; isaacs , i. m. sums of element orders in finite groups . algebra 37 ( 2009 ) , no . 9 , marius tarnauceanu , http://mathoverflow.net/questions/82547/a-question-on-the-product-of-element-orders-of-a-finite-group ( 2011 ) . t. de medts , http://mathoverflow.net/questions/104183/order-increasing-bijection-from-arbitrary-groups-to-cyclic-groups ( 2012 ) .
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we prove several results detecting cyclicity or nilpotency of a finite group @xmath0 in terms of inequalities involving the orders of the elements of @xmath0 and the orders of the elements of the cyclic group of order @xmath1 .
we prove that , among the groups of the same order , the number of cyclic subgroups is minimal for the cyclic group and the product of the orders of the elements is maximal for the cyclic group .
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let @xmath1 be two partitions of an integer @xmath2 , @xmath3 and let @xmath4 and @xmath5 be two _ coprime _ polynomials of degree @xmath2 having the following factorization patterns : @xmath6 in these expressions we consider the multiplicities @xmath7 and @xmath8 , @xmath9 , @xmath10 as being given , while the roots @xmath11 and @xmath12 are not fixed , though they must all be distinct . in this paper we study polynomials satisfying ( [ eq : p - and - q ] ) and such that _ the degree of their difference @xmath13 attains its minimum_. numerous papers , mainly in number theory , were devoted to the study of such polynomials . [ assump ] throughout the paper , we always assume that * the greatest common divisor of the numbers @xmath14 is 1 ; * @xmath15 . the case of partitions @xmath16 , @xmath17 not satisfying the above conditions can easily be reduced to this case ( see @xcite ) . in 1995 , zannier @xcite proved that under the above conditions the following statements hold : 1 . this bound is always attained , whatever are @xmath16 and @xmath17 . [ def : dz ] a pair of polynomials @xmath19 such that @xmath4 and @xmath5 are of the form and @xmath20 is called _ davenport zannier pair _ , or _ dz - pair _ for short . the pair of partitions @xmath21 is called the _ passport _ of the dz - pair . obviously , if @xmath19 is a dz - pair with a passport @xmath21 , and if we take @xmath22 , @xmath23 where @xmath24 , then @xmath25 is also a dz - pair with the same passport . we call such dz - pairs equivalent . [ def : over ] we say that a dz - pair @xmath19 is _ defined over _ @xmath0 if @xmath26 $ ] . we say that an equivalence class of dz - pairs is defined over @xmath0 if there exists a representative of this class which is defined over @xmath0 . by abuse of language , in what follows , we will use the shorter term `` dz - pair '' to denote also an equivalence class of dz - pairs . in our previous paper @xcite , using the _ theory of dessins denfants _ ( see , for example , ch . 2 of @xcite ) , we established a correspondence between dz - pairs and _ weighted bicolored plane trees_. these are bicolored plane trees whose edges are endowed with positive integral weights . the degree of a vertex is defined as the sum of the weights of the edges incident to this vertex . obviously , the sum of the degrees of black vertices and the sum of the degrees of white vertices are both equal to the total weight of the tree . let @xmath27 and @xmath28 be two partitions of the total weight @xmath2 which represent the degrees of black and white vertices respectively . the pair @xmath21 is called the _ passport _ of the tree in question . [ biject ] there is a bijection between dz - pairs with a passport @xmath21 on one hand , and weighted bicolored plane trees with the same passport on the other hand . [ uni ] a weighted bicolored plane tree such that there is no other tree with the same passport is called _ unitree_. general facts of the theory of dessins denfants imply that dz - pairs corresponding to unitrees are defined over @xmath0 . basing on our experience , we claim that this class represents a vast majority of dz - pairs defined over @xmath0 . the other examples may roughly be subdivided into two categories . the members of the first one are constructed as compositions of dz - pairs corresponding to unitrees . the second category is , in a way , a collection of exceptions . still , the latter category is no less interesting since it involves some subtle combinatorial and group - theoretic invariants of the galois action on dz - pairs and on weighted trees . * the main result of @xcite is the classification of all unitrees . the main result of the present paper is a complete list of the corresponding polynomials . * the final part of @xcite is devoted to the study of galois invariants of weighed trees . in the final part of the present paper we compute the corresponding polynomials . the class of unitrees comprises ten infinite series , denoted from @xmath29 to @xmath30 , and ten sporadic trees , denoted from @xmath31 to @xmath32 . the pictures of these trees are given below in the text . dz - pairs corresponding to the series from @xmath29 to @xmath30 are presented in sects . [ sec : a ] to [ sec : j ] ; those corresponding to the sporadic trees from @xmath31 to @xmath32 , in sect . [ sec : sporadic ] . the galois action is treated in sects . [ sec : galois ] to [ sec : other ] . for individual dz - pairs , a computation may turn out to be difficult , sometimes even extremely difficult , but the verification of the result is completely trivial . as to the infinite series , the difficulties grow as a snowball . the `` computational '' part now consists in finding an analytic expression of the polynomials in question , depending on one or several parameters , while the `` verification '' part consists in a _ proof _ , which may be rather elaborate . see a more detailed discussion below . in 1965 , birch , chowla , hall , and schinzel @xcite asked a question which soon became famous:[bchs ] _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ let @xmath29 and @xmath33 be two coprime polynomials with complex coefficients ; what is the possible minimum degree of the difference @xmath34 ? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in order for the question to be meaningful we should take @xmath35 and @xmath36 of the same degree and with the same leading coefficient . denote @xmath37 , @xmath38 , so that @xmath39 . let us start with an example . [ ex : pair - t ] in this example , @xmath40 , so that both polynomials @xmath4 and @xmath5 are of degree @xmath41 . as to their difference @xmath42 , all its coefficients of degrees from 24 down to 6 vanish , so that @xmath43 becomes a polynomial of degree 5 . @xmath44 the following two conjectures were proposed in @xcite:[init - problem ] 1 . for @xmath45 , @xmath46 , one always has @xmath47 . this bound is sharp : that is , it is attained for infinitely many values of @xmath48 . the first conjecture was proved the same year by davenport @xcite . the second one turned out to be much more difficult and remained open for 16 years : in 1981 stothers @xcite showed that the bound is in fact attained not only for infinitely many values of @xmath48 but for all of them . a far - reaching generalization of the above result was proved in 1995 by zannier @xcite . let @xmath49 and @xmath50 be two partitions of an integer @xmath2 satisfying the conditions of assumption [ assump ] , and let @xmath4 and @xmath5 be two polynomials of degree @xmath2 having the factorization pattern . then 1 . @xmath51 . this bound is always attained , whatever are @xmath16 and @xmath17 . for the case of cubes and squares considered above we have @xmath52 , @xmath53 so that @xmath54 and @xmath55 , whence @xmath56 a result equivalent to that of zannier was , in fact , proved , in a very implicit way , by boccara in 1982 @xcite ( see also @xcite , page 775 ) . the result of @xcite was purely combinatorial , and relations between combinatorics and polynomials were at the time largely overlooked . recall that a pair of polynomials @xmath19 satisfying and such that the degree of @xmath57 is _ equal _ to the minimum value @xmath58 are called davenport zannier pairs or dz - pairs ( definition [ def : dz ] ) . the theory of dessins denfants implies that dz - pairs are always defined over the field @xmath59 of algebraic numbers . however , the most interesting case is , without doubt , the one of pairs defined over @xmath0 . in 2010 , beukers and stewart @xcite undertook a study of dz - pairs of the special type @xmath60 , @xmath61 , defined over @xmath0 . in our paper we study dz - pairs of a general form defined over @xmath0 . as we have already said , the framework of our paper is the theory of dessins denfants ( see , for example , ch . 2 of @xcite ) . the main notion of this theory is that of belyi function . for a rational function @xmath62 , where @xmath63 is the riemann complex sphere , let us call @xmath64 a _ critical value _ of @xmath65 if the equation @xmath66 has multiple roots . the definition of a belyi function restricted to the planar case is as follows : [ def : belyi ] a rational function @xmath67 is a _ belyi function _ if @xmath65 has at most three critical values , namely , 0 , 1 and @xmath68 . [ th : riemann ] if @xmath62 is a belyi function then : * the preimage @xmath69)$ ] is a plane map , that is , a connected graph , which is embedded into the sphere in such a way that its edges do not intersect . * the map @xmath70 has a natural bipartite structure : its vertices may be colored in black and white in such a way that each edge would connect vertices of opposite colors . namely , black vertices of @xmath70 are the points @xmath71 , and white vertices of @xmath70 are the points @xmath72 , the vertex degrees being equal to the multiplicities of the corresponding preimages . * inside each face , there is a unique pole of @xmath65 whose multiplicity is equal to the degree of the face . here the _ degree of a face _ is defined as _ a half _ of the number of surrounding edges . we call this pole the _ center _ of the face in question . in the opposite direction , if @xmath73 is a bicolored plane map then : * there exists a belyi function @xmath65 such that @xmath73 can be realized as a preimage @xmath69)$ ] . * this function @xmath65 is unique , up to an affine change of the variable @xmath74 . * there is a uniquely defined number field @xmath31 corresponding to @xmath73 which is called the _ field of moduli _ of @xmath73 . the function @xmath65 can be realized over a number field @xmath75 . statements 4 and 5 represent a particular case of riemann s existence theorem . statement 6 follows from the rigidity of the ramified covering @xmath67 and from some general facts of the galois theory . the above theorem , being applied to the dz - pairs , gives the following statement ( see more details in @xcite ) . [ prop : dz - belyi ] a pair of complex polynomials @xmath19 is a dz - pair with a passport @xmath21 if and only if the rational function @xmath76 , where @xmath42 , is a belyi function for a bicolored plane map @xmath73 with the following characteristics : * the map @xmath73 has @xmath77 edges , @xmath78 black vertices with the degree distribution @xmath16 , and @xmath79 white vertices with the degree distribution @xmath17 . the euler formula then implies that the number of faces is @xmath80 . * all faces of @xmath73 except the outer one are of degree @xmath81 . * the number of the faces of @xmath73 of degree @xmath81 is equal to @xmath82 . in other words , the degree distribution of the faces is equal to @xmath83 where @xmath84 . furthermore , if @xmath85 is the moduli field of @xmath73 , then it is possible to find a corresponding dz - pair such that @xmath86 $ ] . in other words , in this case the realization field @xmath87 @xmath88see the last statement of theorem [ th : riemann]@xmath89 coincides with the field of moduli @xmath31 . in particular , an equivalence class of the pair @xmath19 is defined over @xmath0 if and only if the field of moduli of the map @xmath73 is @xmath90 . the characteristic which distinguishes the maps corresponding to dz - pairs from other maps is property 2 of the above theorem . we will call the faces other than the outer one _ inner faces_. the maps whose all inner faces are of degree 1 can be easily represented in the form of weighted trees : just merge every sheaf of parallel edges into one edge and indicate the number of edges merged together as the _ weight _ of the corresponding edge of the weighted tree : see fig . [ fig : map->tree ] . weighted trees are easier to work with than maps . [ def : weighted ] a _ weighted bicolored plane tree _ , or a _ weighted tree _ , or just a _ tree _ for short , is a bicolored plane tree whose edges are endowed with positive integral _ weights_. the sum of the weights of the edges of a tree is called the _ total weight _ or the _ degree _ of the tree . the _ degree _ of a vertex is the sum of the weights of the edges incident to this vertex . obviously , the sum of the degrees of black vertices , as well as the sum of the degrees of white vertices , is equal to the total weight @xmath2 of the tree . let the tree have @xmath78 black vertices , of degrees @xmath91 , and @xmath79 white vertices , of degrees @xmath92 , respectively . then the pair of partitions @xmath21 of the total weight @xmath2 of the tree is called its _ passport_. forgetting the weights and considering only the underlying plane tree , we speak of a _ topological tree_. weighted trees , all of whose edges are of weight 1 , will be called _ ordinary trees_. belyi functions for ordinary trees are polynomials ( with the only pole at infinity ) ; they are usually called _ shabat polynomials_. we call a _ leaf _ a vertex which has only one edge incident to it , whatever is the weight of this edge . by abuse of language , we will also call this edge itself a leaf . the adjective _ plane _ in the above definition means that the cyclic order of branches around each vertex of the tree is fixed , and changing this order will in general produce a different plane tree ( though the tree considered as a mere graph , without `` planar '' structure , remains the same ) . _ all trees considered in this paper will be endowed with the planar structure _ ; therefore , the adjective `` plane '' will often be omitted . the filed of moduli of a unitree is @xmath0 , see , e.g. , @xcite . therefore , the second part of theorem [ prop : dz - belyi ] implies the following statement . [ uni->q ] if a weighted bicolored plane tree is a unitree , then the corresponding equivalence class of dz - pairs is defined over @xmath0 . [ ex : pair - t.bis ] let us consider the tree shown in fig . [ fig : t ] . it has eight black vertices of degree 3 and twelve white vertices of degree 2 , so that its total weight ( or degree ) is @xmath93 . accordingly , @xmath94 , and @xmath16 and @xmath17 are the following two partitions of @xmath93 : @xmath95 in the corresponding dz - pair , the polynomial @xmath4 must have eight roots of multiplicity 3 , the polynomial @xmath5 must have twelve roots of multiplicity 2 . in other words , @xmath96 with @xmath97 , and @xmath98 with @xmath99 . the difference @xmath42 must be of degree @xmath100 . the general results formulated up to now , being applied to this particular tree , imply the following statements : * the mere existence of such a tree implies the existence of polynomials with needed properties . * the fact that there exist polynomials @xmath4 and @xmath5 _ with rational coefficients _ is a consequence of the fact that there exists a unique tree with the passport @xmath101 . all this can be affirmed without any computations , just by looking at the picture . as to the polynomials themselves , they are given in example [ ex : pair - t ] . it turns out that technically it is often much more convenient to work not with the polynomials appearing in dz - pairs but with their _ reciprocals_. [ def : reciprocal ] for a polynomial @xmath4 of degree @xmath2 , its _ reciprocal _ is @xmath102 . in many examples , the reciprocals of polynomials forming a dz - pair take the form of initial segments of power series of some special functions . after having observed this phenomenon we learned that it was ( re)discovered many times , notably in @xcite , @xcite , @xcite . assume that polynomials @xmath4 and @xmath5 form a dz - pair , so that @xmath103 and denote by @xmath104 the number of edges of the corresponding topological tree . this tree has @xmath105 vertices , therefore it has @xmath106 edges . considering @xmath4 and @xmath5 as power series we may write condition as @xmath107 for the reciprocal polynomials condition is transformed into the following one : @xmath108 where @xmath109 is a polynomial , or , equivalently , to the condition @xmath110 for instance , in the example [ ex : pair - t ] the polynomials reciprocal to and and to their difference look as follows : @xmath111 the computation of belyi functions has recently become a vast domain of research . a remarkable overview of this activity may be found in @xcite , a paper of 57 pages , with a bibliography of 176 titles . beside a direct approach , involving the solution of a system of polynomial equations , the authors of @xcite also discuss complex analytic methods , modular forms methods , and @xmath78-adic methods . in order to get an idea of the level of difficulty of such a computation let us return once again to example [ ex : pair - t ] . a naive approach would be to write down polynomials @xmath112 and @xmath113 with indeterminate coefficients @xmath114 and @xmath115 , and then equate to zero the coefficients of degrees from 6 to 24 of the difference @xmath34 . in this way we get a system of @xmath116 algebraic equations for @xmath117 unknowns . then we may set , for example , @xmath118 , @xmath119 , and @xmath120 . the system thus obtained ( 19 equations with 19 unknowns ) will be of degree @xmath121 ! obviously , this is not a clever way to proceed . by the way , the solution we are looking for is unique ; all the other solutions of this enormous system are `` parasitic '' ones . for example , the system does not give us any guarantee that the polynomials @xmath29 and @xmath33 obtained as its solution will be coprime . this condition should be added to the system , but this addition will make our situation even worse . notice , however , that , once the result is obtained , its verification is trivial . taking into account the above considerations , we would like to underline one aspect of our work : though we do compute belyi functions for certain individual dessins , the most interesting part of the paper is the computation of belyi functions for _ infinite series _ of dessins which depend on one or several parameters . for infinite series the situation is significantly more complicated than for individual dessins . usually , the first thing to do is to compute quite a few particular cases , sometimes dozens of them ( or to use other heuristics whenever possible ) . then , we need to guess a general pattern of corresponding belyi functions . and , finally , instead of a trivial verification step which was applicable to individual dessins , we should provide a _ proof _ , which may turn out to be rather laborious . in the present paper we obviously do not expose the first step of the above procedure . what we do is presenting the final results , that is , the general form of belyi functions in question , and then we give the proofs whenever they are necessary . @xmath122 as it was already said , the unitrees comprise ten infinite series , from @xmath29 to @xmath30 , and ten sporadic trees , from @xmath31 to @xmath32 . in the subsequent sections we do not strictly follow the `` alphabetic '' order of trees since we prefer to underline the structural properties of belyi functions in question . certain belyi functions are expressed in terms of jacobi polynomials ; there are others which lead to interesting differential relations ; we will also encounter compositions , pad approximants , an application to the hall conjecture , etc . our first series , called `` series @xmath29 '' in @xcite , is composed of stars - trees , see fig . [ fig : a ] . all edges except maybe one are of the same weight . this is a three - parametric series . denote the number of leaves of weight @xmath123 by @xmath48 ; then the total weight of the tree is @xmath124 . clearly , we may put the only black vertex at @xmath125 , put the white vertex of degree @xmath126 at @xmath127 , and assume that both @xmath4 and @xmath5 are monic . then @xmath128 and @xmath129 where @xmath29 is a monic polynomial of degree @xmath48 whose roots are the white vertices of degree @xmath123 . now , condition takes the form @xmath130 the only thing we need to know is the polynomial @xmath29 . the polynomial @xmath131 reciprocal to @xmath29 is the initial segment of the binomial series for @xmath132 up to the degree @xmath48 : @xmath133 [ [ proof . ] ] proof . + + + + + + let us pass to reciprocals in : we need to obtain @xmath131 such that @xmath134 let us verify that the polynomial @xmath131 defined in ( [ a - a ] ) satisfies the latter equality . we have : @xmath135 where @xmath136 therefore , @xmath137 and @xmath138^s \operatorname*{=}_{x\to 0 } 1 + o(x^{k+1})\ ] ] which concludes the proof . @xmath139 some particular cases of formula were previously found by n.adrianov ( unpublished ) . the two - parametric series of trees shown in fig . [ fig : d ] was called `` series @xmath140 '' in @xcite . this is the only infinite series of unitrees for which we were able to find the corresponding dz - pairs by computer . let us introduce the following three _ quadratic _ polynomials : [ cols= " < , < , < " , ] we have @xmath141 and @xmath142 , where @xmath143 [ [ proof.-1 ] ] proof . + + + + + + by , we must prove that @xmath144 clearly , we may assume that the sum of the roots of @xmath29 equals zero . write @xmath145 and calculate , with the help of maple , the first five coefficients of the taylor series in the left - hand side of ( [ d - maple ] ) . equate now the expressions thus obtained to zero and solve the corresponding system in the unknowns @xmath146 . maple returns two solutions : @xmath147 and @xmath148 @xmath149 rejecting the first solution , for which the roots of @xmath29 , @xmath33 and @xmath150 coincide , and making an additional normalization by setting the @xmath151 , we obtain formulas , and . @xmath139 in 1971 , m. hall , jr . @xcite suggested the following two conjectures . 1 . there exists a constant @xmath152 such that for all positive integers @xmath153 , @xmath154 , the inequality @xmath155 holds . the exponent @xmath156 in the above inequality can not be improved . namely , for every @xmath157 there exists a constant @xmath158 such that there are infinitely many pairs of integers @xmath159 satisfying the inequality @xmath160 this first conjecture is neither proved nor disproved . however , a general belief is that in order to be true it should be modified as follows : for each @xmath157 there exists a constant @xmath161 such that for all positive integers @xmath153 , @xmath154 , the inequality @xmath162 holds . in this form the conjecture is a corollary of the famous @xmath163-conjecture ( see , e.g. , @xcite , @xcite for further details ) . as to the second conjecture , in 1982 danilov @xcite proved its stronger version . his result is interesting for us since in his proof he used , in a slightly different normalization , the above polynomials @xmath164 , see , , , with the parameters @xmath165 . [ prop : hall ] there exists a constant @xmath150 such that there are infinitely many pairs of integers @xmath159 satisfying the inequality @xmath166 proof . + + + + + + specializing ( [ a - d ] ) , ( [ b - d ] ) and ( [ c - d ] ) for @xmath165 and computing the difference @xmath57 we get @xmath167 substituting @xmath168 and dividing both parts by 8 we get @xmath169 let us now consider the factor @xmath170 and try to make it a perfect square ; then will give us a relatively `` small '' difference between a cube and a square . to do that we have to solve the diophantine equation @xmath171 where @xmath172 . the last equation is a pell - like equation , that is an equation of the form @xmath173 where @xmath174 is a square - free integer and @xmath175 . for @xmath176 this equation is a usual pell equation , and it is well known that any pell equation has infinitely many integer solutions . pell - like equations not necessarily have integer solutions . however , if at least one such solution @xmath177 exists , then we can obtain infinitely many solutions @xmath178 using the following recursion : @xmath179 where @xmath180 is the minimum solution of the equation @xmath181 . in our case , @xmath182 . equation does have an integer solution @xmath183 . returning to , it is easy to verify that for all @xmath184 one has @xmath185 which proves the theorem : there are infinitely many pairs of integers @xmath159 satisfying , with the constant @xmath186 . @xmath139 the same polynomials @xmath187 with the parameters @xmath165 were used by dujella @xcite for constructing an infinite series of pairs of polynomials @xmath188 with the following properties : ( a ) @xmath189 , @xmath190 ; ( b ) @xmath4 and @xmath5 are _ not _ coprime ; ( c ) @xmath191 , so that the minimum degree @xmath192 is not attained , though the discrepancy remains bounded ; ( d ) in return , @xmath4 and @xmath5 are defined over @xmath0 . using other dz - pairs , danilov @xcite and beukers and stewart @xcite obtained results similar to proposition [ prop : hall ] for the differences between integer powers @xmath193 and @xmath194 . davenport zannier pairs for the series of trees considered in this section are expressed in terms of jacobi polynomials . the trees in question are constructed as follows . first , we take chain - trees with alternating edge weights @xmath195 , see fig . [ fig : b ] . we must distinguish chains of odd and even length since in one case both ends are of the same color while in the other case they are of different colors . then , we have a right to attach to the end - points an arbitrary number of leaves of the weight @xmath196 . in this way we obtain `` odd '' series @xmath197 and `` even '' series @xmath198 , see figs . [ fig : e1-e3 ] and [ fig : e2-e4 ] . we call these series `` double brushes '' . note that any of the parameters @xmath199 , and also both of them , may be equal to zero . thus , @xmath200 and @xmath201 are particular cases of @xmath202 , and @xmath203 and @xmath204 are particular cases of @xmath205 . there are two exceptions from the above construction . the first is when the chain part consists of a single edge , so that there is no alternance of weights . we thus obtain the series @xmath150 , see fig . [ fig : c ] . in contrast to the general case , now the weight of leaves may be smaller than the weight of the edge between the leaves . the second exception is when the chain part consists of two edges . in this case it is possible to attach exactly one leaf of weight @xmath196 to one of the ends and exactly two leaves of weight @xmath123 ( or @xmath126 , to ensure the weight alternance ) to the other end . in this way , we get the series of forks @xmath140 already studied in sect . [ sec : d ] . let us recall some general facts concerning jacobi polynomials ; for more advances and detailed treatment see , for example , @xcite or @xcite . the classical jacobi polynomials @xmath206 , @xmath207 , are defined for the parameters @xmath208 , @xmath209 , as orthogonal polynomials with respect to the measure on the segment @xmath210 $ ] , given by the density @xmath211 . the restriction @xmath209 is necessary in order to ensure the integrability . the polynomial @xmath206 can also be defined as a unique polynomial solution of the differential equation @xmath212y'+n(n+a+b+1)y { \,=\,}0,\ ] ] satisfying the condition @xmath213 or by the explicit formula @xmath214 notice that equation can be written in the form @xmath215y'+(n+1)(n+a+b)y { \,=\,}0,\ ] ] where @xmath216 , implying that the function @xmath217 satisfies . it follows from that @xmath206 are also polynomials in parameters @xmath218 and @xmath219 . therefore , their definition can be extended to arbitrary ( even complex ) values of these parameters . these _ generalized _ jacobi polynomials still satisfy , although they are no longer orthogonal with respect to a measure on the segment @xmath210 $ ] . similarly , since the function may be represented as a power series in @xmath74 whose coefficients are polynomials in @xmath153 , this function satisfies equation for arbitrary @xmath218 and @xmath219 . the following key observation will be used in subsequent proofs . if , in the differential operator , we replace @xmath2 with @xmath220 , @xmath218 with @xmath221 , and @xmath219 with @xmath222 , we get exactly the differential operator . therefore , @xmath223 along with satisfies . the last statement , however , should be taken with caution : the subscript @xmath220 must be a non - negative integer since it is the degree of a polynomial . notice that if @xmath218 and @xmath219 do not satisfy the inequalities @xmath209 , then the degree in @xmath74 of the polynomial @xmath206 defined by may drop down below @xmath2 . indeed , implies that the leading coefficient of @xmath206 is equal to @xmath224 hence , in order to obtain a polynomial of degree @xmath2 we must require that the sum @xmath225 does not take values @xmath226 , @xmath227 , , @xmath228 . in particular , this is always true if @xmath218 and @xmath219 are real and @xmath229 or , equivalently , @xmath230 . along with the density @xmath211 , which is defined on @xmath210 $ ] , we will use the multivalued complex function @xmath231 ( note the change of the sign of the term in the first parenthesis ) . clearly , this function has three ramification points @xmath232 . further , observe that if @xmath233 , then any germ of @xmath231 defined near a non - singular point @xmath234 extends to a function @xmath235 which is single - valued in any domain @xmath236 obtained from @xmath237 by removing a simple curve connecting @xmath238 and @xmath81 . indeed , in such @xmath236 the function @xmath235 may have a ramification only at infinity . on the other hand , since the analytic continuation of @xmath235 along a loop around infinity is @xmath239 , we see that @xmath68 is not a ramification point since @xmath233 . in particular , @xmath235 can be expanded into a laurent series at infinity , @xmath240 finally , if @xmath218 and @xmath219 are rational numbers , say @xmath241 then any @xmath235 as above satisfies the condition @xmath242 implying that @xmath235 is defined up to a multiplication by an @xmath104th root of unity , and that for a certain choice of this root the equality @xmath243 holds . by abuse of notation , below we will always use the expression @xmath231 to denote the function @xmath235 which satisfies the equality @xmath243 . [ lemma ] assume that @xmath218 and @xmath219 are rational numbers which satisfy the condition @xmath233 . then for any @xmath244 the equality @xmath245 holds . [ [ proof.-3 ] ] proof . + + + + + + as it was mentioned above , the function satisfies the differential equation , where the function @xmath246 is assumed to be defined on @xmath210 $ ] . however , since this function is analytic near the origin , we can consider its analytic continuation @xmath247 , and the function @xmath248 will satisfy in the domain @xmath236 as above . furthermore , if holds , then @xmath249 implying that the function @xmath250 also satisfies in @xmath236 . since the polynomial @xmath206 satisfies the differential equation , we conclude that the functions @xmath251 both satisfy the differential equation @xmath252 where @xmath253\frac{d}{d z}+(n+1)(n+a+b).\ ] ] this implies that the function @xmath254 also satisfies this equation . on the other hand , it is easy to see that if @xmath255 is a function whose laurent expansion at infinity is @xmath256 then @xmath257 where @xmath258 therefore , if @xmath259 satisfies and @xmath260 while @xmath261 , we should have either @xmath262 or @xmath263 . finally , implies that the leading terms of both @xmath264 and @xmath265 are equal to @xmath266 therefore , the degree of the leading term of their difference @xmath254 is less than @xmath220 , hence the only possible case is @xmath263 , implying . let @xmath267 be a weighted tree from the series @xmath205 or of its two particular cases @xmath204 or @xmath203 , see figs . [ fig : e2-e4 ] and [ fig : b ] . denote by @xmath268 the number of white vertices of @xmath267 which are not leaves . then the total weight of @xmath267 is equal to @xmath269 and the total number of edges is equal to @xmath270 . clearly , @xmath271 for some polynomials @xmath29 and @xmath33 with @xmath272 , @xmath273 . furthermore , by , we must have : @xmath274 where @xmath275 [ pro ] the polynomials @xmath4 and @xmath5 may be represented as follows : @xmath276 where @xmath277 is the jacobi polynomial with parameters @xmath278 and @xmath279 [ [ proof.-4 ] ] proof . + + + + + + since the polynomials @xmath29 and @xmath33 in , are defined in a unique way up to a multiplication by a scalar factor , it is enough to show that @xmath280 where @xmath218 and @xmath219 are given by , and @xmath104 , by . represent the left side of as a product of two factors using the formula @xmath281 where @xmath282 it is easy to see that both @xmath283 and @xmath284 are @xmath285 near infinity . let us consider the difference @xmath286 clearly , @xmath287 furthermore , since @xmath288 the inequality @xmath289 holds . therefore , by lemma [ lemma ] , we have : @xmath290 on the other hand , @xmath291 thus , @xmath292 as required . @xmath139 belyi functions for the series @xmath204 and @xmath205 with the parameters @xmath165 were first calculated in the thesis of nicolas magot in 1997 @xcite . a different proof , proposed by don zagier , was given in ch . 2 of @xcite . we used zagier s proof as a model for the above construction . let now @xmath267 be a weighted tree from the series @xmath202 or of its two particular cases @xmath201 and @xmath200 , see figs . [ fig : e1-e3 ] and [ fig : b ] . as above , denote by @xmath268 the number of white vertices of @xmath267 which are not leaves , so that the total weight of @xmath267 is @xmath293 , and the total number of edges is @xmath294 . now we must find polynomials @xmath4 and @xmath5 such that @xmath295 for some polynomials @xmath29 and @xmath33 with @xmath296 and @xmath297 , and @xmath274 where @xmath298 the polynomials @xmath4 and @xmath5 may be represented as follows : @xmath299 where @xmath300 is the jacobi polynomial with the parameters @xmath301 and @xmath302 [ [ proof.-5 ] ] proof . + + + + + + we must show that @xmath303 where @xmath304 and @xmath104 is defined by . equality is equivalent to the equality @xmath305 where @xmath306 on the other hand , since @xmath307 and @xmath308 it follows from lemma [ lemma ] that @xmath309 implying in the same way as in proposition [ pro ] that holds . @xmath139 the series @xmath150 is a particular case of the series @xmath310 of odd length corresponding to the case of @xmath268 equal to zero . in order to adjust the notation ( which is slightly different for the series @xmath310 and @xmath150 ) we must set @xmath311 and change @xmath123 to @xmath126 and @xmath126 to @xmath312 in formulas . thus , @xmath313 where @xmath314 is the jacobi polynomial of degree @xmath315 with parameters @xmath316 while @xmath317 finally , it is clear that the series @xmath200 and @xmath203 ( chains of odd and even length ) are particular cases of the series @xmath202 and @xmath205 , so that the davenport zannier pairs for @xmath200 and @xmath203 are obtained from those for @xmath202 and @xmath205 by setting @xmath318 the above results can be interpreted in terms of pad approximants for the function @xmath211 . recall that if @xmath319 is a formal power series , then its _ pad approximant _ of order @xmath320 $ ] at zero is a rational function @xmath321 , where @xmath322 is a polynomial of degree @xmath323 and @xmath324 is a polynomial of degree @xmath325 , such that @xmath326 defined in this way , pad approximants do not necessarily exist . however , if an approximant of a given order exists , it is unique . linearizing the problem by requiring that @xmath327 we arrive to the notion of a _ pad form _ @xmath328 of order @xmath320 $ ] . being defined by linear equations , pad forms always exist ( in general , does not imply since @xmath324 may vanish at zero ) , and the pad form of a given order is defined in a unique way up to a multiplication by a constant . keeping the notation of sect . [ sec : e - even ] we may now reformulate the condition for @xmath4 and @xmath5 to be a davenport zannier pair for the series @xmath310 of even length as follows ( a similar result is also true for the series @xmath310 of odd length ) . [ prop : pade - even ] let polynomials @xmath29 and @xmath33 be like in formulas and . then the pair of their reciprocals @xmath329 is the pad form of order @xmath330 $ ] for the function @xmath211 with parameters @xmath331 [ [ proof.-6 ] ] proof . + + + + + + since the pairs @xmath19 and @xmath332 are both defined up to a multiplication by a constant , it is enough to show that @xmath333 where @xmath334 by definition of pad forms we have : @xmath335 implying that @xmath336 ( here we use formula again though now the factors involved are series by non - negative powers of @xmath74 ) . finally , substituting @xmath337 in place of @xmath74 in and multiplying both sides by @xmath338 we obtain . [ prop : pade - odd ] let polynomials @xmath29 and @xmath33 be like in formulas and . then the pair of their reciprocals @xmath329 is the pad form of order @xmath339 $ ] for the function @xmath211 with parameters @xmath340 the proof is similar to the previous one , so we omit it . [ rem : pade ] from the computational point of view , a great advantage of pad approximants is due to the fact that the equations describing them are linear . this observation remains true even in the case like ours when the polynomials in question are known explicitly . one has to use some astute tricks in order to make maple work with jacobi polynomials whose parameters do not satisfy the condition @xmath209 . at the same time , the computation of pad approximants is instantaneous . a vast literature is devoted to the study of pad approximants for some particular functions . this is the case , for example , for the exponential function . to our surprise , we did not find any research concerning pad approximants for the function @xmath211 . by the way , our lemma [ lemma ] can also be reformulated as a result about pad forms for this function . below we find dz - pairs for the series @xmath341 and @xmath342 , see figs . [ fig : f ] and [ fig : g ] , using their relations to differential equations . for the series @xmath341 , which consists of ordinary trees , the corresponding formulas are particular cases of the formulas for _ shabat polynomials _ for trees of diameter four , first calculated by adrianov @xcite . since any tree from the series @xmath341 is ordinary , the degree of @xmath42 is zero , that is @xmath343 for some @xmath344 therefore , in order to describe the corresponding dz - pair it is enough to find @xmath4 and @xmath152 . this is equivalent to the finding of the shabat polynomial corresponding to the tree . similarly , for trees from the series @xmath342 the degree of @xmath43 is one , and it is technically easier to provide explicit formulas for @xmath4 and @xmath43 rather than for @xmath4 and @xmath5 . we start with the series @xmath342 . the polynomial @xmath4 for the series @xmath342 takes the form @xmath345 where @xmath29 is a polynomial of degree @xmath346 whose roots are the black vertices ( all of them are of degree @xmath104 ) . notice that the number of these vertices does not coincide with the degree of the central vertex since we have one `` double '' edge . we choose the normalization of @xmath4 , @xmath5 and @xmath42 in the following way : * @xmath347 where @xmath29 is monic , @xmath348 ; * the central vertex is placed at @xmath125 , so that @xmath349 where @xmath33 is monic , @xmath350 ; the roots of @xmath33 are the white vertices distinct from zero ; * @xmath351 ; this means that the pole inside the only face of degree 1 is placed at @xmath127 . thus , we get @xmath352 the polynomial @xmath29 satisfies the differential equation @xmath353 consequently , coefficients @xmath354 of @xmath355 may be found by the following backward recurrence : @xmath356 finally , @xmath357 . [ [ proof.-7 ] ] proof . + + + + + + taking the derivative of the both sides of equality we obtain the equality @xmath358 implying the equality @xmath359 substituting in the last equality the value of @xmath360 from , we obtain @xmath361-ca { \,=\,}x^{k-1}a\left(kb+xb^{\prime}\right)\ ] ] and @xmath362.\ ] ] we now observe that the degree of the left - hand side of the latter equality is @xmath346 , while its right - hand side is _ proportional _ to @xmath363 . therefore , the expression in the square brackets on the right is some constant @xmath31 , and both parts are equal to @xmath364 . the constant @xmath31 can be easily found as the leading coefficient of the left - hand side : it is equal to @xmath365 . finally , we get the equality @xmath366 which implies . substituting @xmath355 in we obtain . finally , substituting @xmath125 in we obtain @xmath367 . @xmath139 let us take @xmath368 , so that @xmath369 . then the corresponding polynomial looks as follows : @xmath370 where @xmath371 [ rem : hyp - geom ] polynomial @xmath29 also satisfies the hypergeometric differential equation @xmath372\frac{dy}{dx } - ab\cdot y { \,=\,}0.\end{aligned}\ ] ] indeed , applying the differential operator @xmath373 to both parts of equality we obtain @xmath374^{\prime}+(1-k)\left[ma^{\prime}\cdot(x-1)-a\right]=0,\ ] ] implying @xmath375a^{\prime } - \frac{(1-k)}{m}a { \,=\,}0.\ ] ] therefore , @xmath29 is a solution of the differential equation @xmath376\frac{dy}{dx } + \frac{(1-k)}{m}\,y { \,=\,}0\ ] ] which is a particular case of with @xmath377 for this series we may assume that @xmath378 here @xmath29 is monic and @xmath379 ; namely , @xmath29 is a polynomial whose roots are the black vertices of degree @xmath104 . now , @xmath33 is a polynomial whose roots are the white vertices distinct from zero , @xmath380 . the polynomials @xmath4 and @xmath5 must satisfy the condition @xmath381 where @xmath382 is a non - zero constant . the polynomial @xmath29 satisfies the differential equation @xmath383\,x^{k-1}.\end{aligned}\ ] ] consequently , coefficients @xmath354 of @xmath355 may be found by the following backward recurrence : @xmath384 finally , the value of @xmath152 in is equal to @xmath385 . [ [ proof.-8 ] ] proof . + + + + + + as above , let us take the derivative of both sides of equation . then we get @xmath386 { \,=\,}x^{k-1}\left(kb+xb^{\prime}\right).\ ] ] we observe that the polynomial @xmath363 is coprime with the factor @xmath387 in the left - hand side , and therefore it must be proportional to the factor @xmath388 which is itself a polynomial of degree @xmath346 . therefore , both of them are equal to @xmath364 where the constant @xmath31 can be found as the leading coefficient of @xmath388 ; namely , it is equal to @xmath389 . thus , holds . now , substituting @xmath355 in we obtain the recurrence , and substituting @xmath125 in we obtain the value of @xmath152 . @xmath139 here , like in the case of the series @xmath342 , the polynomial @xmath29 also satisfies the hypergeometric differential equation , and therefore it may be represented through a hypergeometric function . let us take @xmath368 , so that @xmath369 . then the corresponding polynomial looks as follows : @xmath390 where @xmath391 the above method may be applied to dz - pairs which do not necessary correspond to trees of diameter four or to unitrees . however , in general , it leads to _ differential relations _ between @xmath4 and @xmath5 . let us clarify what we mean by considering the problem of the difference between cubes and squares of polynomials , which was at the origin of the whole activity concerning dz - pairs , see @xcite , @xcite . let @xmath392 @xmath393 and @xmath43 be polynomials such that @xmath394 and @xmath395 taking the derivative of both parts of we obtain @xmath396 multiplying now the last equality by @xmath29 and substituting @xmath35 from we obtain the equality @xmath397 implying in its turn the equality @xmath398 since the degree of the right - hand side is @xmath399 while @xmath46 , the above equality implies that @xmath400 for some non - zero constant @xmath401 . the last expression is a differential equation of the first order with respect to @xmath29 as well as with respect to @xmath33 . unfortunately , both @xmath29 and @xmath33 are unknown . thus , it does not give us any immediate information about @xmath29 and @xmath33 . still , algebraic equations for coefficients of @xmath29 and @xmath33 obtained from are ( mostly ) of degree 2 while the equations obtained from are ( mostly ) of degree 3 . differentiating and writing the expression thus obtained as a differential equation with respect to @xmath29 we get : @xmath402 this differential equation is a particular case of the differential equation @xmath403 where @xmath404 is a polynomial of degree at most @xmath405 polynomial solutions of the last equation are called stieltjes polynomials . the polynomials @xmath404 for which has a polynomial solution are called van vleck polynomials . thus , @xmath33 is a van vleck polynomial , and @xmath29 is the corresponding stieltjes polynomial . writing now in the form @xmath406 we obtain that @xmath29 is a van vleck polynomial and @xmath33 is the corresponding stieltjes polynomial . the above observations show that the relations between dz - pairs and differential equations may be deeper than it seems at first glance and deserve further investigation . in this section we consider series @xmath407 ( fig . [ fig : h ] ) and @xmath408 ( fig . [ fig : i ] ) . in both cases the corresponding dz - pairs are obtained with the help of the operation of composition . notice that the trees in question are ordinary ( the weights of all edges are equal to 1 ) . as it was mentioned in definition [ def : weighted ] , belyi functions for ordinary trees are called shabat polynomials . the trees of the series @xmath407 are compositions of trees from the series @xmath150 with the parameters @xmath165 and chains of length 2 . the expressions of the shabat polynomials for the trees from the series @xmath150 in terms of jacobi polynomials are given in sect . [ sec : c ] . using the fact that @xmath165 we can also compute them directly . indeed , the trees in question have exactly two vertices of degree greater than 1 . putting them into the points @xmath125 and @xmath127 and taking into account that the degree of the corresponding shabat polynomial @xmath409 is @xmath410 , we conclude that the derivative of @xmath109 is proportional to @xmath411 . therefore , the polynomial @xmath409 itself can be written as @xmath412 then we automatically have @xmath413 , while in order to get @xmath414 we must take @xmath415 where @xmath416 is the euler beta function . then , taking the shabat polynomial for the chain with two edges and with two black vertices put to 0 and 1 , which is equal to @xmath417 we obtain the following the polynomial @xmath4 for the tree @xmath407 is equal to @xmath418 where @xmath236 is as in and @xmath109 is as in and . the proof is obvious . below are given shabat polynomials @xmath419 for the trees of the series @xmath408 . these trees are compositions of trees from the series @xmath150 with @xmath165 and @xmath420 , and the stars with three edges . thus , @xmath421 , where @xmath109 is a shabat polynomial corresponding to a tree from the series @xmath150 , and @xmath236 is a shabat polynomial corresponding to the star with three edges . however , in order to achieve the rationality of the coefficients of @xmath4 we still must find an appropriate normalization of @xmath109 . for this purpose , contrary to all traditions , let us put the vertices of degree @xmath48 of the tree from the series @xmath150 into the points @xmath422 . then the derivative of the corresponding shabat polynomials @xmath409 must be equal to @xmath423 therefore , @xmath424+b\end{aligned}\ ] ] for some @xmath425 . substituting into @xmath409 the critical points @xmath422 , we obtain the critical values @xmath426 , where @xmath427 setting @xmath428 and choosing @xmath218 in such a way that @xmath429 we obtain a polynomial @xmath430 $ ] with two critical values @xmath431 taking now @xmath432 ( we must take @xmath433 instead of @xmath434 in order to get the colors of the vertices which would correspond to fig . [ fig : i ] ) , we obtain the following the polynomial @xmath435 for the tree @xmath408 is equal to @xmath436 where @xmath236 is as in and @xmath109 is as in with @xmath218 and @xmath219 defined by conditions , , . once again , the proof is obvious . this is the last infinite series of unitrees . the degree of this tree , or its total weight , is @xmath437 . let us normalize the polynomial @xmath4 so that @xmath438 this means that the black vertex of degree 4 is put at @xmath439 , while two black vertices of degree @xmath440 are put at the points @xmath441 for certain @xmath442 , @xmath443 . all the white vertices are of degree 2 ; therefore , the polynomial @xmath5 has the form @xmath444 for some polynomial @xmath29 , @xmath445 . further , condition gives us @xmath446 here @xmath440 is the `` overweight '' of the tree ( that is , its total weight minus the number of edges of the topological tree ) . for the reciprocal polynomials this gives ( see ) @xmath447 here @xmath448 is the number of edges of the topological tree . the reciprocal polynomials @xmath449 and @xmath450 may be represented as follows : @xmath451 where @xmath131 is the initial segment of the series @xmath452 up to the degree @xmath453 : @xmath454 [ [ proof.-9 ] ] proof . + + + + + + let @xmath455 where @xmath456 computing @xmath457 we get @xmath458.\end{aligned}\ ] ] thus , for any value of the parameter @xmath218 we have @xmath459 and therefore , in order to obtain , we only have to show that for @xmath460 the constant term of @xmath461 is equal to zero , or , equivalently , the coefficient in front of @xmath462 in the series @xmath463 vanishes . let us write the second factor of the latter expression explicitly : @xmath464 notice that this series involves only even powers . multiplying it by @xmath465 we see that the coefficient in front of @xmath462 in @xmath452 is the sum of the coefficients in front of @xmath462 and @xmath466 in . therefore , we must ensure that @xmath467 collecting similar terms we get @xmath468 which gives @xmath460 . @xmath139 let us take @xmath469 . then we have : @xmath470 further , @xmath471 notice that the term with @xmath472 is missing . finally , @xmath473 as it was explained previously , in sect . [ sec : computation ] , the verification of the results given below is trivial . therefore , we present nothing else but the polynomials themselves . @xmath474 @xmath475 @xmath477 @xmath479 this tree is symmetric , with the symmetry of order 3 . therefore , @xmath4 , @xmath5 , @xmath43 are polynomials in @xmath480 . @xmath482 this triple was found in beukers and stewart @xcite ( only the polynomial @xmath4 is given in their paper , but it uniquely determines two other polynomials ) . @xmath483 once again , the answer is taken from @xcite , with a slight renormalization . @xmath484 this tree is the only sporadic tree from the adrianov s list of ordinary unitrees . correspondingly , @xmath4 is a shabat polynomial : the polynomial @xmath43 is a constant . note that the positions of certain black vertices are rational : @xmath485 @xmath486 the tree @xmath43 is the `` square '' of the tree @xmath87 : it is symmetric , with the symmetry of order 2 , and one of its `` halves '' is equal to @xmath87 . therefore , we may take the polynomials for the tree @xmath87 and insert @xmath487 instead of @xmath74 . @xmath488 @xmath489 notice that the second factor in @xmath4 , the one which is `` cubed '' , does not contain the term with @xmath74 : this is not a misprint . the picture of this tree is given in example [ ex : pair - t.bis ] , and the corresponding polynomials are given in example [ ex : pair - t ] . recall that the passport of a ( bicolored weighted plane ) tree is a pair of partitions @xmath1 , where @xmath2 is the degree ( or the total weight ) of the tree , @xmath16 represents the set of degrees of its black vertices , and @xmath17 represents the set of degrees of its white vertices . [ def : comb - orbit ] the set of weighted trees with the same passport is called _ combinatorial orbit_. unitrees represent , in fact , combinatorial orbits consisting of a unique tree . usually , a dz - pair corresponding to a tree is defined over a number field whose degree is equal to the size of the combinatorial orbit to which this tree belongs . this is why unitrees are always defined over @xmath0 . there exist , however , other galois invariants which may split a combinatorial orbit into several distinct galois orbits . in this way we may obtain certain trees which are not unitrees but which are still defined over @xmath0 . in @xcite we gave several such examples . here we present the corresponding dz - pairs . a bicolored map may be characterized by a pair of permutations acting on the set of its edges : one permutation represents the cyclic order ( in the positive direction ) of the edges around black vertices , the other one , the cyclic order around white vertices . for example , the map shown in fig . [ fig : pgl27 ] , is represented by the pair of permutations @xmath491 it turns out that the permutation group @xmath492 is equal to @xmath490 . this group , which is called _ monodromy group _ , is a galois invariant . since this tree is the only one in its combinatorial orbit whose monodromy group is @xmath490 , it is defined over @xmath0 . @xmath493 the combinatorial orbit to which this tree belongs , that is , the set of trees with the passport @xmath494 , contains six trees . the five remaining trees constitute a single galois orbit ; the corresponding dz - pairs ( or , we may say , the trees themselves ) are defined over the splitting field of the polynomial @xmath495 the combinatorial orbit corresponding to the passport @xmath496 , consists of five trees . one of them , shown in fig . [ fig : pgl27-bis ] , has the monodromy group @xmath490 . therefore , it is defined over @xmath0 . its dz - pair is given below . @xmath497 one of the trees in this combinatorial orbit is symmetric ( see fig . [ fig : sym ] ) and is therefore also defined over @xmath0 . the corresponding polynomials are @xmath498 the three remaining trees constitute a single galois orbit and are defined over the splitting field of the polynomial @xmath499 the _ duality _ for the bicolored maps is defined as follows : * a map and its dual share their white vertices ; * black vertices of each map correspond to the faces of the dual map ; * edges of the dual map connect the centers of the faces of the initial map to the white vertices which lie on the border of these faces . see details and examples in @xcite . a map is _ self - dual _ if it is isomorphic to its dual . self - duality is a galois invariant . the maps corresponding to weighted trees may well be self - dual . let @xmath500 be two positive integers , and @xmath501 . we consider the trees with the black partition @xmath502 and the white partition @xmath503 . the partition representing the face degrees is @xmath504 . we notice that @xmath505 ; therefore , the corresponding combinatorial orbit _ may _ contain self - dual trees . it is easy to verify that this combinatorial orbit consists of @xmath506 trees , and that only one of them is self - dual , namely , the tree shown in fig . [ fig : self - dual ] . therefore , this tree is defined over @xmath0 . put the white vertices at the points @xmath439 and @xmath127 so that @xmath507 ( notice that both powers are odd ) . observe now that this polynomial is `` antipalindromic '' : if we write it as @xmath508 then @xmath509 , @xmath510 , this fact trivially follows from the equality @xmath511 . because of this , the coefficient in front of the `` middle '' degree @xmath512 is zero . therefore , if we take the higher degrees from @xmath513 to @xmath105 , what will remain is a polynomial of degree @xmath514 . in other words , @xmath515 where @xmath516 . setting now @xmath517 we see that @xmath4 , @xmath5 is a dz - pair with required properties . notice that the polynomial @xmath518 is reciprocal to @xmath519 . geometrically , this means that if @xmath520 are the positions of the black vertices of degree 1 ( here @xmath521 ) , then the centers of the faces of degree 1 are @xmath522 . together with the fact that the position of the black vertex of degree @xmath105 is @xmath125 while the center of the face of degree @xmath105 is @xmath68 , this shows that the map in question is indeed self - dual . let us take , for example , @xmath523 , @xmath524 . then @xmath525 where @xmath526 and @xmath527 . the combinatorial orbit corresponding to the passport @xmath528 , @xmath529 is shown in fig . [ fig : zannier-5 ] . it consists of four trees ( recall that the invisible white vertices are middle points of the edges ) , and is divided into three galois orbits . the tree @xmath218 is the only one which is symmetric with the symmetry of order 3 . therefore , it is defined over @xmath0 . the corresponding polynomials were computed by b.birch in 1965 @xcite . they look as follows ( notice that they are polynomials in @xmath480 ) : @xmath530 the trees @xmath219 and @xmath152 are symmetric with the symmetry of order 2 with respect to an ( invisible ) white vertex . they are also mirror symmetric to each other ; therefore , the complex conjugation sends one of the trees to the other . thus , we may conclude that this couple of trees constitutes a separate galois orbit , and this orbit is defined over an imaginary quadratic field . the corresponding polynomials were computed in 2005 by shioda @xcite and , indeed , they are defined over the field @xmath531 . we do not present these polynomials here . the tree @xmath532 does not have any particular combinatorial properties . ( it is known that the mirror symmetry of a dessin is not a galois invariant . ) but _ it remains alone _ , that is , it constitutes a galois orbit containing a single element . therefore , it is defined over @xmath0 . the corresponding polynomials were computed in 2000 by n.elkies @xcite . they look as follows : @xmath533 by the way , a naive approach mentioned in sect . [ sec : computation ] , namely , taking polynomials @xmath29 and @xmath33 of degrees 10 and 15 respectively with indeterminate coefficients and equating to zero the coefficients of degrees from 7 to 30 of @xmath534 , would , this time , lead us to a system of polynomial equations of degree @xmath535 . it took 40 years ( from 1965 to 2005 ) to compute all the four dz - pairs of this example , but the fact that there are exactly four non - equivalent solutions and that two of them are defined over @xmath0 while the other two are defined over an imaginary quadratic field , can be immediately seen from the picture without any computation . all the polynomials in this section which correspond to the _ asymmetric _ trees are taken from the above - cited article @xcite . the normalization sometimes is changed . the goal of this section is to show the _ combinatorial reasons _ of appearance of these sporadic examples . the passport shows that we are treating here the problem of the minimum degree of the difference @xmath537 where @xmath538 , @xmath539 . the combinatorial orbit consists of two trees , see fig . [ fig : split-7 - 3 ] . one of them is symmetric , the other one is not ; therefore , both are defined over @xmath0 . the triple corresponding to the asymmetric tree is as follows : @xmath540 the triple corresponding to the symmetric tree may be computed as follows : 1 . compute the polynomials corresponding to a branch of this three - branch tree , that is , to a tree of the series @xmath29 ( see sect . [ sec : a ] ) with the parameters @xmath541 , @xmath542 , @xmath543 . 2 . make the change of variables @xmath544 in order to put the white vertex of degree 1 to the point @xmath125 ; thus , the polynomial @xmath435 , instead of being @xmath545 , becomes @xmath546 ; it is convenient to change its sign and to get @xmath547 . insert @xmath480 instead of @xmath74 . by pure convenience we add to the above operations one more : instead of taking @xmath548 we take @xmath549 . this permits us to avoid fractional coefficients . the resulting triple is @xmath550 the passport corresponds to the problem of the minimum degree of the difference @xmath552 where @xmath538 , @xmath553 . the combinatorial orbit consists of two trees , see fig . [ fig : split-8 - 3 ] . one of them is symmetric , the other one is not ; therefore , both are defined over @xmath0 . the triple corresponding to the asymmetric tree looks as follows : @xmath554 the triple corresponding to the symmetric tree may be computed as follows : 1 . compute the polynomials corresponding to the series @xmath205 ( see sect . [ sec : e - even ] ) with @xmath555 , @xmath556 , @xmath557 , @xmath558 . 2 . make the change of variables @xmath559 in order to move the ( left ) black vertex of degree 4 from @xmath238 to 0 . insert @xmath487 instead of @xmath74 . we omit the resulting polynomials . this time we deal with the problem @xmath561 , @xmath538 , @xmath562 . the combinatorial orbit corresponding to this passport contains three trees , see fig . [ fig : split-10 - 3 ] . these trees have three different symmetry types , hence all of them are defined over @xmath0 . the polynomials for the asymmetric tree look as follows : @xmath563 the polynomials for the tree with the symmetry of order 2 is computed in the same way as in sect . [ sec : split2 ] . the parameters of the tree of the type @xmath205 are @xmath564 , @xmath542 , @xmath557 , @xmath565 ; then we must replace @xmath74 with @xmath566 , and insert @xmath487 instead of @xmath74 . the polynomials for the tree with the symmetry of order 3 is computed in the same way as in sect . [ sec : split1 ] . the parameters of the tree of the type @xmath29 are @xmath541 , @xmath542 , @xmath469 ; then we must replace @xmath74 with @xmath567 , and insert @xmath480 instead of @xmath74 . we finish this section with an example which shows that the combinatorial methods , while being very powerful , are , however , not all - powerful . there are several trees with the passport @xmath568 , and one of them , shown in fig . [ fig : why ] , is defined over @xmath0 without any apparent reason . all known combinatorial and group - theoretic galois invariants fail to explain this phenomenon . all we can say is that the corresponding system has rational solutions `` by chance '' . the polynomials @xmath4 and @xmath5 for the tree of fig . [ fig : why ] are as follows : @xmath569 the polynomial @xmath43 here is of degree 32 , and it is too cumbersome , so we do not write it explicitly . we have already seen two examples ( see sect . [ sec : split1 ] and [ sec : split2 ] ) of combinatorial orbits of size 2 which , instead of being defined over a quadratic field , split in two orbits defined over @xmath0 because the trees in question have different orders of symmetry . here we present an infinite series of such examples . the trees in question have the passport @xmath570 for @xmath571 , see fig . [ fig : sym - and - not ] . belyi function for the symmetric tree looks as follows : @xmath572 belyi function for the asymmetric tree looks as follows : @xmath573 in both cases , the white vertex of degree 4 lies at @xmath125 . the expressions for belyi functions give us the polynomials @xmath4 and @xmath43 . in order to prove the correctness of the above expressions we need to verify two things : for both @xmath574 and @xmath575 , we have ( a ) @xmath576 ; ( b ) first three derivatives of @xmath577 at @xmath125 vanish . let us return to the problem of the minimum degree of the difference @xmath534 , the question from which this whole line of research started ( see @xcite . we have seen that when @xmath45 , @xmath46 , we have @xmath578 . for @xmath579 , the computation becomes exceedingly difficult , and there is practically no hope to find solutions defined over @xmath0 . however , if we are not so demanding and accept a solution with the degree of @xmath534 slightly greater than @xmath192 , then sometimes we can find a needed solution . let us take a polynomial @xmath29 with one double root , so that @xmath35 would have one root of multiplicity 6 and all the other roots of multiplicity 3 . the corresponding tree would have one vertex less and therefore one face more . the tree in fig . [ fig : deg9-instead-8 ] corresponds to @xmath580 . it is the `` cube '' of the tree @xmath109 , see sect . [ sec : tree - s ] . therefore , all we have to do is to insert @xmath480 instead of @xmath74 in the formulas of section [ sec : tree - s ] . @xmath581 when all the roots of @xmath29 and @xmath33 are distinct , the polynomial @xmath43 has @xmath192 distinct roots . let us accept @xmath43 with a multiple root ( thus , its degree will be greater that @xmath192 ) . the tree in fig . [ fig : deg - relaxed ] gives such and example . it corresponds to @xmath368 , and @xmath582 . the polynomials for this tree look as follows : @xmath583 fedor pakovich was partially supported by isf grants no . 639/09 and 779/13 . he is also grateful to the max plank institute for mathematics for the hospitality and support . alexander zvonkin was partially supported by the research grant graal anr-14-ce25 - 0014 . 99 handbook of mathematical functions : with formulas , graphs , and mathematical tables . dover , 1972 . on generalized chebyshev polynomials corresponding to planar trees of diameter 4 , * 13 * ( 2007 ) , no . 6 , 1933 ( in russian ) ; translation in _ j. math ( n. y. ) vol . * 158 * ( 2009 ) , no . 1 , 1121 arithmetic theory of graphs on surfaces , ph.d . thesis , moscow state university , 1997 , 116 pp . ( in russian ) . neighboring powers , _ journal of number theory _ , 2010 , vol . * 130 * , 660679 . on the difference @xmath584 , _ det kongelige norske videnskabers selskabs forhandlinger _ ( trondheim ) , 1965 , vol . * 38 * , 6569 . cycles comme produit de deux permutations de classes donnes , _ discrete math . _ , 1982 , vol . * 58 * , 129142 . the diophantine equation @xmath585 and hall s conjecture . _ matematicheskie zametki _ , 1982 , vol . * 32 * , no . 3 , 273275 . diophantine equations @xmath586 . _ matematicheskie zametki _ , 1989 , vol . * 46 * , no . 6 , 3845 . on @xmath587 , _ det kongelige norske videnskabers selskabs forhandlinger _ ( trondheim ) , 1965 , vol . * 38 * , 8687 . on hall s conjecture . _ acta arith . _ , 2011 , vol . * 147 * , no . 4 , 397 - 402 . realizability of branched coverings of surfaces , _ trans . soc . _ , 1984 , vol . * 282 * , no . 2 , 773790 . rational points near curves and small non - zero @xmath588 via lattice reduction . in : wieb bosma , ed . , algorithmic number theory , lecture notes in computer science , vol . * 1838 * , springer - verlag , 2000 , 3363 . the diophantine equation @xmath585 . in : computers in number theory , academic press , london new york , 1971 , 173198 . graphs on surfaces and their applications . springer - verlag , 2004 . old and new conjectured diophantine inequalities . _ , new ser . , 1990 , vol . * 23 * , no . 1 , 3775 . cartes planaires et fonctions de belyi : aspects algorithmiques et exprimenataux . thesis , universit bordeaux i , 1997 , 144 pp . minimum degree of the difference of two polynomials over @xmath0 , and weighted plane trees . _ selecta mathematica , new ser . _ , 2014 , vol . * 20 * , no . 4 , 10031065 . see also arxiv:1306.4141v1 . elliptic surfaces and davenport stothers triples , _ comment . univ . st . pauli _ , 2005 , vol . * 54 * , no . 1 , 4968 . on computing belyi maps , arxiv:1311.2529v3 ( may 2014 ) polynomial identities and hauptmoduln , _ quart . j. math . oxford _ , ser . 2 , 1981 , vol . * 32 * , no . 127 , 349370 . orthogonal polynomials . colloquium publications , vol . * 23 * , 1939 ( reedited in 1992 ) . on davenport s bound for the degree of @xmath589 and riemann s existence theorem , _ acta arithmetica _ , 1995 , vol . * 71 * , no . 2 , 107137 . labri , umr 5800 , universit de bordeaux , 33400 talence , france ; e - mail : zvonkin@labri.fr , and the chebyshev mathematical laboratory at the saint - petersburg state university , 29b , 14th line , vasilyevsky island , saint - petersburg 199178 , russia .
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in this paper we study pairs of polynomials with a given factorization pattern and such that the degree of their difference attains its minimum .
we call such pairs of polynomials _ davenport zannier pairs _ , or dz - pairs for short .
the paper is devoted to the study of dz - pairs _ with rational coefficients_. in our earlier paper @xcite , in the framework of the _ theory of dessins denfants _ , we established a correspondence between dz - pairs and _ weighted bicolored plane trees_. these are bicolored plane trees whose edges are endowed with positive integral weights . when such a tree is uniquely determined by the set of black and white degrees of its vertices , it is called _ unitree _ , and the corresponding dz - pair is defined over @xmath0 . in @xcite , we classified all unitrees . in this paper , we compute all the corresponding polynomials . in this way ,
the present paper is a sequel of @xcite . in the final part of the paper we present some additional material concerning the galois theory of dz - pairs and weighted trees .
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the large variety of novel and interesting phenomena of thin - film magnetism results very much from the fact that the magnetic anisotropy , which determines the easy axis of magnetization , can be one or two orders of magnitude larger than in the corresponding bulk systems@xcite . the reorientation transition ( rt ) of the direction of magnetization in thin ferromagnetic films describes the change of the easy axis by variation of the film thickness or temperature and has been widely studied both experimentally @xcite and theoretically @xcite . an instructive phenomenological picture for the understanding of the rt is obtained by expanding the free energy @xmath0 of the system in powers of @xmath1 , where @xmath2 is the angle between the direction of magnetization and the surface normal . neglecting azimuthal anisotropy and exploiting time inversion symmetry yields : @xmath3 the anisotropy coefficients of second ( @xmath4 ) and fourth ( @xmath5 ) order depend on the thickness @xmath6 of the film as well as on the temperature @xmath7 . away from the transition point usually @xmath8 holds , and , therefore , the direction of magnetization is determined by the sign of @xmath4 ( @xmath9 : out - of - plane magnetization ; @xmath10 : in - plane magnetization ) . on this basis the concept of anisotropy flow @xcite immediately tells us that the rt is caused by a sign change of @xmath4 while the sign of @xmath5 mainly determines whether the transition is continuous ( @xmath11 ) or step - like ( @xmath12 ) . in the case of a continuous transition @xmath5 also gives the width of the transition region . from the microscopic point of view we know that the magnetic anisotropy is exclusively caused by two effects , the dipole interaction between the magnetic moments in the sample and the spin - orbit coupling : @xmath13 . while the dipole interaction always favors in - plane magnetization ( @xmath14 ) due to minimization of stray fields , the spin - orbit interaction can lead to both , in - plane and out - of - plane magnetization depending sensitively on the electronic structure of the underlying sample . the spin - orbit anisotropy is caused by the broken symmetry@xcite at the film surface and the substrate - film interface as well as by possible strain@xcite in the volume of the film . it is worth to stress that a strong positive spin - orbit induced anisotropy alone opens up the possibility of an out - of - plane magnetized thin film . the rt must be seen as a competition between spin - orbit and dipole anisotropy . in many thin - film systems both thickness- and temperature - driven rts are observed . although it is clear by inspection of the corresponding phase diagrams @xcite that both types of transitions are closely related to each other , different theoretical concepts are needed to explain their physical origin . the thickness - driven rt is rather well understood in terms of a phenomenological separation of the spin - orbit induced anisotropy constant @xmath15 into a surface term @xmath16 and a volume contribution @xmath17 by the ansatz @xmath18 . experimentally , this separation seems to provide a rather consistent picture@xcite despite the fact that in some samples additional structural transitions are present@xcite which clearly restrict its validity . on the theoretical side , basically two different schemes for the calculation of @xmath19 magnetic anisotropy constants have been developed , semi - empirical tight - binding theories@xcite and spin - polarized ab initio total - energy calculations @xcite . in both approaches the spin - orbit coupling is introduced either self - consistently or as a final perturbation . however , these investigations still remain to be a delicate problem because of the very small energy differences involved . neglecting the large variety of different samples , substrates , growth conditions , etc . it is useful for the understanding of the rt to concentrate on two somewhat idealized prototype systems both showing a thickness- as well as a temperature - driven rt . the `` fe - type '' systems @xcite are characterized by a large positive surface anisotropy constant @xmath20 together with a negative volume anisotropy @xmath21 due to dipole interaction . this leads to out - of - plane magnetization for very thin films . for increasing film thickness the magnetization switches to an in - plane direction because the volume contribution becomes dominating@xcite . as a function of increasing temperature a rt from out - of - plane to in - plane magnetization is found for certain thicknesses @xcite . in the `` ni - type '' systems @xcite , the situation is different . here the volume contribution @xmath21 is positive due to fct lattice distortion @xcite , thereby favoring out - of - plane magnetization , while the surface term @xmath20 is negative . for very thin films the surface contribution dominates leading to in - plane magnetization . at a critical thickness , however , the positive volume anisotropy forces the system to be magnetized in out - of - plane direction @xcite , until at a second critical thickness the magnetization switches to an in - plane position again caused by structural relaxation effects . here a so - called anomalous temperature - driven rt from in - plane to out - of - plane magnetization was found recently by farle et al.@xcite . in this article we will focus on the temperature - driven rt which can not be understood by means of the separation into surface and volume contribution alone . here the coefficients @xmath22 and @xmath23 need to be determined for each temperature separately . experimentally , this has been done in great detail for the second - order anisotropy of ni / cu(100)@xcite . the results clearly confirm the existence and position of the rt , but , on the other hand , do not lead to any microscopic understanding of its origin . to obtain more information on the temperature - driven rt theoretical investigations on simplified model systems have proven to be fruitful . despite the fact that in the underlying transition - metal samples the spontaneous magnetization is caused by the itinerant , strongly correlated 3d - electrons , up to now heisenberg - type models have been considered exclusively @xcite . the magnetic anisotropy has been taken into account by incorporating the dipole interaction and an uniaxial single - ion anisotropy to model the spin - orbit - induced anisotropy . using appropriate @xmath19 second - order anisotropy constants as input parameters , both types of rts have been observed within the framework of a self - consistent mean field approximation @xcite as well as by first - order perturbation theory for the free energy @xcite . a continuous rt has been found for @xmath24 layers@xcite taking place over a rather small temperature range . step - like transitions occur as an exception for special parameter constellations only . the rt is attributed to the strong reduction of the surface - layer magnetization relative to the inner layers for increasing temperature leading to a diminishing influence of the surface anisotropy . since the itinerant nature of the magnetic moments is ignored completely in these calculations , it is interesting to compare these results with calculations done within itinerant - electron systems . the present work employs a similar concept but in the framework of the single - band hubbard model@xcite which we believe is a more reasonable starting point for the description of temperature - dependent electronic structure of thin transition - metal films . the paper is organized in the following way : in the next section we define our model hamilton operator . in sec . [ sec_f ] we will focus on the derivation of the free energy and the second - order anisotropy constants by use of a perturbational approach . the isotropic part of the hamilton operator is treated in sec . [ sec_sda ] . here we present a generalization of a self - consistent spectral - density approach to the film geometry . in sec . [ sec_res ] we will show and analyze the results of the numerical evaluations and discuss the possibility of a temperature - driven rt within our model system . we will end with a short conclusion in sec . [ sec_con ] . the description of the film geometry requires some care . each lattice vector of the film is decomposed into two parts @xmath25 @xmath26 denotes a lattice vector of the underlying two - dimensional bravais lattice with @xmath27 sites . to each lattice point a @xmath6-atom basis @xmath28 ( @xmath29 ) is associated referring to the @xmath6 layers of the film . the same labeling , of course , also applies for all other quantities related to the film geometry . within each layer we assume translational invariance . then a fourier transformation with respect to the two - dimensional bravais lattice can be applied . the considered model hamiltonian consists of three parts : @xmath30 @xmath31 denotes the single - band hubbard model @xmath32 where @xmath33 ( @xmath34 ) stands for the annihilation ( creation ) operator of an electron with spin @xmath35 at the lattice site @xmath36 , @xmath37 is the number operator and @xmath38 is the hopping - matrix element between the lattice sites @xmath36 and @xmath39 . the hopping - matrix element between nearest neighbor sites is set to @xmath40 . @xmath41 denotes the on - site coulomb matrix element , and @xmath42 is the chemical potential . the second term @xmath43 describes the dipole interaction between the magnetic moments on different lattice sites : @xmath44.\ ] ] here @xmath45 is the distance between @xmath36 and @xmath39 , and the unit vector @xmath46 is given by @xmath47 . @xmath48 denotes the strength of the dipole interaction , @xmath49 the lattice constant and @xmath50 the number of atoms in the corresponding bulk cubic unit cell . @xmath51 are the spin operators constructed by the pauli spin matrices @xmath52 . the expectation value of the spin operator yields the ( dimensionless ) magnetization vector @xmath53 the magnetization only depends on the layer index @xmath54 because of the assumed translational invariance within the layers . in our approach the uniaxial anisotropy caused by the spin - orbit coupling is taken into account phenomenologically by an effective layer - dependent anisotropy field coupled to the spin operator : @xmath55 the effective field @xmath56 is chosen to be parallel to the film normal @xmath57 . to ensure the right symmetry we set : @xmath58 this corresponds to a mean - field treatment of the spin - orbit - induced anisotropy@xcite . the strengths of the anisotropy fields @xmath59 enter as additional parameters and have to be fixed later . the direction of the magnetization is determined by the minimal free energy @xmath0 . the anisotropic contributions @xmath60 and @xmath43 to the hamilton operator ( [ ham_op ] ) can be considered as a small perturbation to the isotropic hubbard model ( @xmath61 ) . then we can apply a thermodynamic perturbation expansion@xcite of the free energy @xmath0 up to linear order with respect to @xmath62 : @xmath63 here @xmath64 denotes the expectation value taken within the unperturbated hubbard model @xmath31 . on the same footing we use a mean - field decoupling for the two - particle expectation values contained in @xmath65 . because @xmath0 is calculated to linear order in the anisotropy contributions , only the lowest , i.e. second - order anisotropy constants are considered , and a possible canted phase is neglected . the ratio @xmath66 and thus the width of the transition region have been found to be very small for reasonable strengths of the anisotropy contributions @xcite . within our approach it is , therefore , sufficient to consider the free - energy difference @xmath67 between in - plane and out - of - plane magnetization : @xmath68 @xmath10 and @xmath9 indicate in - plane and out - of - plane magnetization respectively . hence , the reorientation temperature @xmath69 is given by the condition @xmath70 . evaluation of @xmath67 yields : @xmath71 the constants @xmath72 contain the effective dipole interaction between the layers and can be calculated separately : @xmath73.\ ] ] @xmath74 is the angle between @xmath46 and the direction of the magnetization . the @xmath72 only depend on the film geometry . for thick films @xmath75 reduces to its continuum value @xmath76 where @xmath77 is the magnetization per atom . to calculate @xmath67 the temperature- and layer - dependent magnetizations @xmath78 of the hubbard film are needed . in this section we will focus on the evaluation of the hubbard film @xmath31 . ferromagnetism in the hubbard model is surely a strong - coupling phenomenon . the existence of ferromagnetic solutions was recently proven in the limit of infinite dimensions by quantum monte - carlo calculations@xcite . ferromagnetism is favored by a strongly asymmetric bloch density of states ( bdos ) and by a singularity at the upper band edge as it is found , e. g. , for the fcc lattice . the hubbard model constitutes a highly non - trivial many - body problem even for a periodic infinitely extended lattice . even more complications are introduced when the reduction of translational symmetry has to be taken into account additionally . one of the easiest possible approximations to treat a hubbard film is a hartree - fock decoupling , which has been applied previously @xcite . hartree - fock theory , however , is necessarily restricted to the weak - coupling regime and is known to overestimate the possibility of ferromagnetic order drastically . neglecting electron - correlation effects altogether leads to qualitatively wrong results especially for intermediate and strong coulomb interaction @xmath41 @xcite . furthermore , we did not find a realistic temperature - driven rt within this approach . thus we require an approximation scheme which is clearly beyond the hartree - fock solution and takes into account electron correlations more reasonably . on the other hand , it must be simple enough to allow for an extended study of magnetic phase transitions in thin films . for this purpose we apply the spectral - density approach ( sda)@xcite which is motivated by the rigorous analysis of harris and lange@xcite in the limit of strong coulomb interaction ( @xmath79 ) . the sda can be interpreted as an extension of their @xmath80-perturbation theory@xcite in a natural way to intermediate coupling strengths and finite temperatures and has been discussed in detail for various three - dimensional @xcite as well as infinite - dimensional@xcite lattices . at least qualitatively , it leads to rather convincing results concerning the magnetic properties of the hubbard model . a similar approach has been applied to a multiband hubbard model with surprisingly accurate results for the magnetic key - quantities of the prototype band ferromagnets fe , co , ni@xcite . recently , a generalization of the sda has been proposed to deal with the modifications due to reduced translational symmetry@xcite . in the following we give only a brief derivation of the sda solution and refer the reader to previous papers for a detailed discussion@xcite . the basic quantity to be calculated is the retarded single - electron green function @xmath81 . from @xmath82 we obtain all relevant information on the system . its diagonal elements , for example , determine the spin- and layer - dependent quasiparticle density of states ( qdos ) @xmath83 . the equation of motion for the single - electron green function reads : @xmath84 g_{lj\sigma}^{\gamma\beta}(e ) = \hbar\delta_{ij}^{\alpha\beta}.\ ] ] here we have introduced the electronic self - energy @xmath85 which incorporates all effects of electron correlations . we adopt the local approximation for the self - energy which has been tested recently for the case of reduced translational symmetry@xcite . if we assume translational invariance within each layer of the film we have @xmath86 . after fourier transformation with respect to the two - dimensional bravais lattice the equation of motion ( [ eq_motion ] ) is formally solved by matrix inversion . the decisive step is to find a reasonable approximation for the self - energy @xmath87 . guided by the exactly solvable atomic limit of vanishing hopping ( @xmath88 ) and by the findings of harris and lange in the strong - coupling limit ( @xmath89 ) , a one - pole ansatz for the self - energy @xmath87 can be motivated @xcite . the free parameter of this ansatz are fixed by exploiting the equality between two alternative but exact representations for the moments of the layer - dependent quasiparticle density of states : @xmath90_{+}\bigg\rangle_{t = t ' } .\ ] ] here , @xmath91_{+}$ ] denotes the anticommutator . it can be shown by comparing various approximation schemes @xcite that an inclusion of the first four moments of the qdos ( @xmath92 ) is vital for a proper description of ferromagnetism in the hubbard model . further , the inclusion of the first four moments represents a necessary condition to be consistent with the @xmath80-perturbation theory@xcite . the hartree - fock approximation recovers the first two moments ( @xmath93 ) only , while the so - called hubbard - i solution@xcite reproduces the third moment ( @xmath94 ) as well , but is well known to be hardly able to describe ferromagnetism . taking into account the first four moments to fix the free parameters of our ansatz we end up with the sda solution which is characterized by the following self - energy : @xmath95 the self - energy depends on the spin - dependent occupation numbers @xmath96 as well as on the so - called bandshift @xmath97 which consists of higher correlation functions : @xmath98 a possible spin dependence of @xmath97 opens up the way to ferromagnetic solutions@xcite . ferromagnetic order is indicated by a spin - asymmetry in the occupation numbers @xmath99 , and the layer - dependent magnetization is given by @xmath100 . the band occupations @xmath101 are given by @xmath102 where @xmath103 is the fermi function . the mean band occupation @xmath104 is defined as @xmath105 . although @xmath106 consists of higher correlation functions it can by expressed exactly @xcite via @xmath107 and @xmath108 : @xmath109 \rho_{\alpha\sigma}(e ) . \label{bas}\end{aligned}\ ] ] equations ( [ eq_motion ] ) , ( [ sig_sda ] ) , ( [ nas ] ) and ( [ bas ] ) build a closed set of equations which can be solved self - consistently . surely a major short - coming of the sda is the fact that quasiparticle damping is neglected completely . recently a modified alloy analogy ( maa ) has been proposed @xcite which is also based on the exact results of the @xmath80-perturbation theory but is capable of describing quasiparticle damping effects as well . for bulk systems it has been found that the magnetic region in the phase diagram is significantly reduced by inclusion of damping effects . on the other hand , the qualitative behavior of the magnetic solutions is very similar to the sda . an application of the maa to thin film systems is in preparation@xcite . the numerical evaluations have been done for an fcc(100 ) film geometry . in this configuration each lattice site has four nearest neighbors within the same layer and four nearest neighbors in each of the respective adjacent layers . we consider uniform hopping @xmath110 between nearest neighbor sites @xmath36 , @xmath39 only . energy and temperature units are chosen such that @xmath111 . the on - site hopping integral is set to @xmath112 . further , we keep the on - site coulomb interaction fixed at @xmath113 which is three times the band width of the three - dimensional fcc - lattice and clearly refers to the strong - coupling regime . let us consider the isotropic hubbard - film first . there are three model parameters left to vary , the temperature @xmath7 , the thickness @xmath6 and the band occupation @xmath114 ) . except for the last part of the discussion we will keep the band occupation fixed at the value @xmath115 and focus exclusively on the temperature and thickness dependence of the magnetic properties . in fig . [ fig_1 ] the layer - dependent density of states of the non - interacting system @xmath116 ( @xmath117 `` bloch density of states '' ( bdos ) ) is plotted for a two , three and five layer film with an fcc(100 ) geometry . the bdos is strongly asymmetric and shows a considerable layer - dependence for @xmath118 . considering the moments @xmath119 of the bdos yields that the variance @xmath120 as well as the skewness @xmath121 are reduced at the surface layer compared to the inner layers due to the reduced coordination number at the surface ( @xmath122 , @xmath123 , @xmath124 , @xmath125 for @xmath126 ) . in fig . [ fig_2 ] the layer magnetizations @xmath78 as well as the mean magnetization @xmath127 are shown as a function of temperature @xmath7 . while symmetry requires the double layer film to be uniformly magnetized , the magnetization shows a strong layer dependence for @xmath118 . the magnetization curves of the inner layers ( and for the double layer ) show the usual brillouin - type behavior . the trend of the surface layer magnetization ( @xmath24 ) , however , is rather different . note that @xmath128 depends almost linearly on temperature in the range @xmath129 and for thicknesses @xmath130 . compared to the inner layers , the surface magnetization decreases significantly faster as a function of temperature , tending to a reduced curie temperature . however , due to the coupling between surface and inner layers which is induced by the electron - hopping , this effect is delayed and a unique curie - temperature for the whole film is found . the curie temperature @xmath131 increases as a function of the film thickness @xmath6 and saturates already for film - thicknesses around @xmath132 to the corresponding bulk value . a similar behavior was found for a bcc(110 ) film geometry@xcite . the critical exponent of the magnetization ( fig . [ fig_3 ] ) is found to be equal to the mean - field value @xmath133 for all thicknesses and all other parameters considered . this clearly reveals the mean - field type of our approximation which is due to the local approximation for the electronic self - energy . the layer - dependent quasiparticle density of states ( qdos ) is shown in fig . [ fig_4 ] for three temperatures @xmath134 . two kinds of splittings are observed in the spectrum . due to the strong coulomb repulsion @xmath41 the spectrum splits into two quasiparticle subbands ( `` hubbard splitting '' ) which are separated by an energy of the order @xmath41 . in the lower subband the electron mainly hops over empty sites , whereas in the upper subband it hops over sites which are already occupied by another electron with opposite spin . the latter process requires an interaction energy of the order of @xmath41 . the weights of the subbands scale with the probability of the realization of these two situations while the total weight of the qdos of each layer is normalized to 1 . therefore , the weights of the lower and upper subbands are roughly given by @xmath135 and @xmath136 respectively . this scaling becomes exact in the strong coupling limit ( @xmath89 ) . since the total band occupation ( @xmath115 ) is above half - filling ( @xmath137 ) , the chemical potential @xmath42 lies in the upper subband while the lower subband is completely filled . for temperatures below the curie temperature @xmath131 an additional splitting ( `` exchange - splitting '' ) in majority ( @xmath138 ) and minority ( @xmath139 ) spin direction occurs , leading to non - zero magnetization @xmath140 . for @xmath19 the majority qdos lies completely below the chemical potential , the system is fully polarized ( @xmath141 ) . thus the low - energy subband of the minority spin direction disappears and the minority qdos is exactly given by the bdos of the non - interacting system ( fig . [ fig_1 ] ) due to vanishing correlation effects in the @xmath139 channel . the reduced surface magnetization at @xmath19 ( see fig . [ fig_2 ] ) is , therefore , directly caused by the layer - dependent bdos . we like to stress that the spin - splitting does not depend on the size of the coulomb interaction @xmath41 as long as @xmath41 is chosen from the strong coupling limit . contrary to hartree - fock theory the spin splitting saturates as a function of @xmath41 for values of about @xmath142 times the bandwidth of the non - interacting system . the same holds for the curie temperature @xmath131 @xcite . the temperature behavior of the qdos is governed by two correlation effects ( fig . [ fig_4 ] ) . as the temperature is increased , the spin - splitting between @xmath138 and @xmath139 spectrum decreases . this effect is accompanied by a redistribution of spectral weight between the lower and upper subbands along with a change of the widths of the subbands . for @xmath143 one clearly sees that in the minority spectrum weight has been transferred from the upper to lower subband which has reappeared due to non - saturated magnetization at finite temperatures . in the @xmath138 spectrum the opposite behavior is found . for all @xmath144 the spin splitting is significantly larger in the inner layer compared with the surface . at @xmath145 the exchange splitting has disappeared whereas the correlation - induced hubbard splitting is still present . let us consider the question why the surface magnetization shows a tendency towards a reduced curie temperature as seen in fig . [ fig_2 ] . this effect can be understood by the above mentioned moment analysis of the bdos ( fig . [ fig_1 ] ) : from bulk systems it is known that an asymmetrically shaped bdos is favorable for the stability of ferromagnetism in the hubbard model . in particular , the curie temperatures increase with increasing skewness of the bdos @xcite . the same trend shows up in the present film - system where the skewness of the bdos is higher for the inner layers compared to the surface - layer . note that this argument is somewhat more delicate than in the case of heisenberg films where the reduced surface magnetization is directly caused by the reduced number of interacting sites . in fig . [ fig_5 ] the difference between surface magnetization @xmath128 , central - layer magnetization @xmath146 , and mean magnetization @xmath147 is analyzed in more detail . for all film thicknesses where this distinction is meaningful ( @xmath118 ) , the surface magnetization is reduced with respect to the mean magnetization . this holds not only for very thin films ( @xmath132 , see also fig . [ fig_2 ] ) where some oscillations are present that are caused by the finite film thickness , but also extends to the limit @xmath148 where the two surfaces are well separated and do not interact . the surface and central - layer magnetizations already stabilize for thicknesses around @xmath149 . further , fig . [ fig_5 ] clearly shows that the reduction of @xmath128 drastically increases for higher temperatures . for @xmath143 the surface magnetization is reduced to about half the size of the magnetization in the center of the film . the charge transfer due to differing layer occupations @xmath150 is found to be smaller than @xmath151 at @xmath19 and is almost negligible for finite temperatures . we now like to focus on the magnetic anisotropy energy within the model system ( [ ham_op ] ) . the second - order anisotropy constant @xmath152 is calculated via eq . ( [ f_anis_end ] ) which needs as an input the temperature - dependent layer magnetizations of the hubbard film . the dipole constants @xmath72 for an fcc(100 ) film geometry are found to be : @xmath153 , @xmath154 , @xmath155 , and are set to zero for @xmath156 . to simulate both , surface and volume contribution of the spin - orbit induced anisotropy , we choose the effective anisotropy field ( [ anis_field ] ) in the surface - layer to be different from its value in the volume of the film : @xmath157 in the perturbational approach only the ratio @xmath158 is important . thus we are left with only two parameters ( fixed at @xmath19 ) to model the rt . in principle these constants could be taken from experiment or from theoretical ground - state calculations . in our opinion , however , this would mean to somewhat over - judge the underlying rather idealized model system . we are mainly interested in the question whether a realistic temperature - driven rt is possible at all in the hubbard film and by what mechanism it is induced . therefore , we choose the parameters @xmath159 and @xmath160 conveniently , guided however by the experimental findings in the fe - type and ni - type scenarios described in the introduction . [ fig_6 ] and fig . [ fig_7 ] show that for appropriate parameters @xmath161 both types of temperature - driven rt can be found within a three - layer film . the same is found for any film thickness @xmath24 . in the fe - type situation ( fig . [ fig_6 ] ) we consider a strong positive surface anisotropy field @xmath162 together with @xmath163 . at low temperatures the system is magnetized in out - of - plane direction . as the temperature increases , @xmath164 decreases faster than @xmath165 because of the strong reduction of the surface anisotropy and the magnetization switches to an in - plane position . in principle this kind of rt is possible for all film thicknesses @xmath24 . for thicker films @xmath159 has to be rescaled proportional to @xmath6 to compensate the increasing importance of the dipole anisotropy . the ni - type rt from in - plane to out - of - plane magnetization for a three layer system is obtained by a positive volume anisotropy field @xmath166 and a negative surface anisotropy @xmath167 . at low temperatures the dipole anisotropy as well as the negative surface anisotropy field lead to an in - plane magnetization . for higher temperatures , however , where the surface anisotropy becomes less important because of the reduced surface magnetization , the positive volume anisotropy field forces the magnetization to switch to an out - of - plane direction . the ratio @xmath168 determines for what thickness @xmath6 this type of temperature - driven rt is possible and scales like @xmath169 for thicker films . note that for both types of rt the values of @xmath159 and @xmath160 are chosen in such a way that the system is close to a thickness - driven rt . in both cases the rt is mediated by the strong decrease of the surface - layer magnetization compared to the inner layers as a function of temperature . finally , we consider the dependence on the band occupation @xmath104 . in fig . [ fig_8 ] the ratio @xmath170 of a three - layer film is plotted as a function of the reduced temperature @xmath171 for different band occupations @xmath172 . above @xmath173 and below @xmath174 the fcc(100 ) hubbard film does not have ferromagnetic solutions whereas between @xmath174 and @xmath175 there is a tendency towards first order phase transition as a function of temperature@xcite and no realistic rt is possible . the ratio @xmath170 ( fig . [ fig_8 ] ) decreases as a function of temperature for all band occupations . for @xmath115 which has been considered in fig . [ fig_6 ] and fig . [ fig_7 ] , it changes from @xmath176 at @xmath19 to @xmath177 close to @xmath131 . however , for higher band occupations this strong temperature - dependent change in @xmath170 diminishes . from the discussion above it is clear that this is unfavorable for the rt . we can thus conclude from fig . [ fig_8 ] that the possibility of a temperature - driven rt sensitively depends on the band occupation @xmath104 . in addition we plotted in fig . [ fig_8 ] the ratio @xmath170 calculated within hartree - fock theory . the result is completely different . at low temperatures we find @xmath178 whereas @xmath179 close to @xmath131 . note that the ratio @xmath170 is almost constant for the wide temperature range @xmath180 . therefore , a realistic temperature - driven rt is excluded within the hartree - fock approximation . this holds for all parameters that have been considered here . we have applied a generalization of the spectral - density approach ( sda ) to thin hubbard films . the sda which reproduces the exact results of the @xmath80-perturbation theory in the strong coupling limit , leads to rather convincing results concerning the magnetic properties . the magnetic behavior of the itinerant - electron film can be microscopically understood by means of the temperature - dependent electronic structure . for an fcc(100 ) film geometry the layer - dependent magnetizations have been discussed as a function of temperature as well as film thickness . the magnetization in the surface layer is found to be reduced with respect to the inner layers for all thicknesses and temperatures . by analyzing the layer - dependent qdos this reduction can be explained by the fact that in the free bdos both variance and skewness are diminished in the surface - layer compared to the inner layers . the inclusion of the dipole interaction and an effective layer - dependent anisotropy field allows to study the temperature - driven rt . the second - order anisotropy constants have been calculated within a perturbational approach . for appropriate strengths of the surface and volume anisotropy fields both types of rt , from out - of - plane to in - plane ( fe - type ) and from in - plane to out - of - plane ( ni - type ) magnetization are found . for the ni - type scenario the inclusion of a positive volume anisotropy is necessary . the rt in our itinerant model system is mediated by a strong reduction of the surface magnetization with respect to the inner layers as a function of temperature . here a close similarity to the model calculations within heisenberg - type systems is apparent , despite the fact that these models completely ignore the itinerant nature of the magnetic moments in the underlying transition - metal samples . contrary to heisenberg films , the band occupation @xmath104 enters as an additional parameter within an itinerant - electron model . we find that the possibility of a rt sensitively depends on the band occupation . the fact that no realistic rt is possible within hartree - fock theory clearly points out the importance of a reasonable treatment of electron correlation effects .
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the temperature - driven reorientation transition which , up to now , has been studied by use of heisenberg - type models only , is investigated within an itinerant - electron model .
we consider the hubbard model for a thin fcc(100 ) film together with the dipole interaction and a layer - dependent anisotropy field .
the isotropic part of the model is treated by use of a generalization of the spectral - density approach to the film geometry .
the magnetic properties of the film are investigated as a function of temperature and film thickness and are analyzed in detail with help of the spin- and layer - dependent quasiparticle density of states . by calculating the temperature dependence of the second - order anisotropy constants
we find that both types of reorientation transitions , from out - of - plane to in - plane ( `` fe - type '' ) and from in - plane to out - of - plane ( `` ni - type '' ) magnetization are possible within our model . in the latter case
the inclusion of a positive volume anisotropy is vital .
the reorientation transition is mediated by a strong reduction of the surface magnetization with respect to the inner layers as a function of temperature and is found to depend significantly on the total band occupation .
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large scale structures like galaxies and clusters of galaxies are believed to have formed by gravitational amplification of small perturbations @xcite . observations suggest that the initial density perturbations were present at all scales that have been probed by observations . an essential part of the study of formation of galaxies and other large scale structures is thus the evolution of density perturbations for such initial conditions . once the amplitude of perturbations at any scale becomes large , i.e. , @xmath0 , the perturbation becomes non - linear and the coupling with perturbations at other scales can not be ignored . indeed , understanding the interplay of density perturbations at different scales is essential for developing a full understanding of gravitational collapse in an expanding universe . the basic equations for this have been known for a long time @xcite but apart from some special cases , few solutions are known . a statistical approach to this problem based on pair conservation equation has yielded interesting results @xcite , and these results have motivated detailed studies to obtain fitting functions to express the non - linear correlation function or power spectrum in terms of the linearly evolved correlation function @xcite . it is well known from simulation studies that at the level of second moment , i.e. , power spectrum , correlation function , etc . , large scales have an important effect on small scales but small scales do not have a significant effect on large scales @xcite . most of these studies used power spectrum as the measure of clustering . results of these simulation studies form the basis for the use of n - body simulations , e.g. , from the above results we can safely assume that small scales not resolved in simulations do not effect power spectrum at large scales and can be ignored . substructure can play an important role in the relaxation process . it can induce mixing in phase space @xcite , or change halo profiles by introducing transverse motions @xcite , and , gravitational interactions between small clumps can bring in an effective collisionality even for a collisionless fluid @xcite . thus it is important to understand the role played by substructure in gravitational collapse and relaxation in the context of an expanding background . in particular , we would like to know if this leaves an imprint on the non - linear evolution of correlation function . effect of substructure on collapse and relaxation of larger scales is another manifestation of mode coupling . in this paper , we report results from a study of mode coupling in gravitational collapse . in particular , we study how presence of density perturbations at small scales influences collapse and relaxation of perturbations at larger scales . these effects have been studied in past @xcite but the motivation was slightly different @xcite . we believe it is important to study the issue in greater detail and make the relevance of these effects more quantitative using n - body simulations with a larger number of particles . we also study the reverse process , i.e. , how does collapse of perturbations at large scales effect density perturbations at much smaller scales . it is well known that the local geometry of collapse at the time of initial shell crossing is planar in nature @xcite , hence we model density perturbations as a single plane wave in this work . simple nature of the large scale fluctuation allows us to study interaction of well separated scales without resorting to statistical estimators like power spectrum . we are studying the same problem in a more general setting and those results will be reported in a later publication . key features of collapse of a plane wave can be understood using quasi - linear approximations , at least at a qualitative level . initial collapsing phase is well modelled by the zeldovich approximation @xcite , wherein particles fall in towards the centre of the potential well . zeldovich approximation breaks down after orbit crossing as it does not predict any change in the direction of motion for particles , thus in this approximation particles continue to move in the same direction and the size of the collapsed region grows monotonically . in a realistic situation we expect particles to fall back towards the potential well and oscillate about it with a decreasing amplitude , and the collapsed region remains fairly compact . several approximations have been suggested to improve upon the zeldovich approximation @xcite . the adhesion approximation @xcite invokes an effective viscosity : this prevents orbit crossing and conserves momentum to ensure that pancakes remain thin and matter ends up in the correct region . this changes the character of motions in dense regions ( no orbit crossing or mixing in the phase space ) but predicts locations of these regions correctly . if one assumes that the gravitational potential evolves at the linear rate @xcite , then it can be shown that the collapsed region remains confined . the effective drag due to expanding background slows down particles and they do not have enough energy to climb out of the potential well . thus the process of confining particles to a compact collapsed region results from a combination of expansion of the universe and gravitational interaction of in falling particles . none of the approximations captures all the relevant effects . therefore we must turn to n - body simulations @xcite in order to study collapse and relaxation of perturbations in a complete manner . we will consider only gravitational effects here and ignore all other processes . we assume that the system can be described in the newtonian limit . the growth of perturbations is then described by the coupled system of euler s equation and poisson equation in comoving coordinates along with mass conservation , e.g. , see @xcite . @xmath1 it is assumed that the density field is generated by a distribution of particles , each of mass @xmath2 , position @xmath3 . @xmath4 is the present value of hubble constant , @xmath5 is the present density parameter for non - relativistic matter and @xmath6 is the scale factor . in this paper we will consider the einstein - de sitter universe as the background , i.e. , @xmath7 . these can be reduced to a single non - linear differential equation for density contrast @xcite . @xmath8 where @xmath9 \delta_{\bf k ' } \delta_{{\bf k } -{\bf k ' } } \nonumber\ ] ] and @xmath10 \;\;\ ; ; \;\;\;\ ; m= \sum\limits_i m_i \nonumber\ ] ] the terms @xmath11 and @xmath12 are the non - linear coupling terms between different modes . @xmath12 couples density contrasts in an indirect manner through velocities of particles ( @xmath13 ) . the equation of motion still needs to be solved for a complete solution of this equation , or we can use some ansatz for velocities to make this an independent equation . it can be shown that individual _ virialised _ objects , i.e. , objects that satisfy the condition @xmath14 where @xmath15 is the kinetic energy and @xmath16 is the potential energy , do not make any contribution towards growth of perturbations through mode coupling @xcite at much larger scales , i.e. , the @xmath17 term is zero . the contribution of mode coupling due to interaction of such objects is not known . approximate approaches to structure formation can be developed by ignoring interaction of well separated scales . the evolution of density perturbations can be modelled as a combination of non - linear collapse at small scales , and the collapsed objects can be displaced using quasi - linear approximations @xcite . these approaches yield an acceptable description of properties of collapsed objects and their distribution for a first estimate . pinocchio @xcite provides sufficient information about halo properties and merger trees for use with semi - analytic models of galaxy formation . the efficacy of these models puts an upper bound on the effects of mode coupling that we are studying here . in this paper we simplify the system by starting with perturbations that have a non - zero amplitude only for two sets of scales . we simulate the collapse of a plane wave by starting with non - zero amplitude of perturbations for the fundamental mode of the simulation box along the @xmath18 axis , the wavenumber of the fundamental mode is denoted by @xmath19 . this serves as the large scale perturbation in our study . the amplitude for this mode is chosen so that shell crossing takes place when the scale factor @xmath20 . power spectrum for small scale fluctuations is chosen to be non - zero in a range of wave numbers @xmath21 with a constant amplitude across this window , i.e. , @xmath22 for @xmath23 . a gaussian random realisation of this power spectrum is used for small scale fluctuations . here @xmath24 is the power per logarithmic interval in @xmath25 contributed by small scales ( large @xmath25 ) and @xmath11 is the amplitude of the fundamental mode that gives rise to the plane wave . the ratio of @xmath26 at @xmath27 and for the plane wave is denoted by @xmath28 , thus when @xmath29 collapse of perturbations at these scales happens at the same time whereas for @xmath30 perturbations at small scales collapse before the plane wave collapses . we chose the ratio @xmath31 so that there is distinct separation in the scales involved . , we go from a single stream region to a three stream region and so on up to seven stream region near the centre . ] collapse of a plane wave of this kind leads to formation of a multi - stream region , we will also use the term pancake to describe this region . figure [ phase_space ] shows the phase space plot for the plane wave at late stages of collapse for the simulation pm_00l ( see table 1 for details of the simulation ) . velocity of particles is plotted as a function of position , only the @xmath18 component is plotted as there is no displacement or velocity along other directions in this simulation . regions where particles with different velocities can be found are the multi - stream regions . as we approach the centre of pancake located at @xmath32 , we go from a single stream region to a three stream region and so on up to seven stream region near the centre . in initial stages , the mass in the pancake increases rapidly as more particles fall in . figure [ den_plwave ] shows this in terms of over - density which increases sharply from @xmath20 to @xmath33 . a significant fraction of the total mass falls into the pancake and the infall velocities for the remaining matter are very small . in this regime the mass of the pancake is almost constant , this can be seen from the panels of figure [ den_adh_sub ] where the mass enclosed in the pancake region is almost constant from @xmath33 to @xmath34 . .this table lists parameters of n - body simulations we have used . all the simulations used @xmath35 particles . the first column lists name of the simulation , second column lists the code that was used for running the simulation , third column gives the relative amplitude of small scale power and the plane wave , the fourth column tells us whether the large scale plane wave was present in the simulation or not , and the last column lists the distribution of particles before these are displaced using a realisation of the power spectrum . _ grid _ distribution means that particles started from grid points . _ perturbed grid _ refers to a distribution where particles are randomly displaced from the grid points , this displacement has a maximum amplitude of @xmath36 grid points . such an initial condition is needed to prevent particles from reaching the same position in plane wave collapse as such a situation is pathological for tree codes . the treepm simulations were run with a force softening length equal to the grid length . [ cols="<,<,<,<,<",options="header " , ] in absence of any substructure the collisionless collapse retains planar symmetry and we have layers of multi - stream regions with the number of streams increasing towards the centre of the pancake . presence of small scale fluctuations can induce transverse motions and these motions are amplified in the pancake . weakly bound substructure can be torn apart due to interaction with rapidly in falling matter . on the other hand , higher average density in the multi stream region can lead to rapid growth of perturbations . it is known that pancakes are unstable to fragmentation due to growth of perturbations @xcite . the velocity field is anisotropic due to infall along one direction , hence the rate at which perturbations grows will also exhibit anisotropies . velocity dispersion along the direction of plane wave collapse is larger than the transverse direction , hence the growth of fluctuations in the transverse plane is expected to be more rapid . if the in falling material contains collapsed substructure , then gravitational interactions between these can induce large transverse velocities . this takes away kinetic energy from the direction of infall , which in turn can lead to more fragmented and thinner multi - stream region . in the following sections we describe the numerical experiments have undertaken in detail , and test the physical ideas and expectations outlined above we used a particle - mesh code @xcite and the treepm code @xcite . some simulations were run using the parallel treepm @xcite . treepm simulations used spline softening with softening length equal to the length of a grid cell in order to ensure collisionless evolution . we used force softening assuming a spline kernel @xcite . all the simulations were carried out with @xmath35 particles . table 1 lists parameters of the simulations we have used for this paper . we have used two types of initial distribution of particles . in the _ grid _ distribution particles are located at grid points before being displaced to set up the initial perturbations . _ perturbed grid _ refers to a distribution where particles are randomly displaced from the grid points @xcite , this displacement has a maximum amplitude of @xmath36 grid length . such an initial condition is needed to prevent particles from reaching the same position in plane wave collapse as such a situation is pathological for tree codes . these small displacements do not affect the power spectrum to be realised , pm_@xmath37l and t_@xmath37l were compared to test for any systematic effects . simulations t_@xmath38p and t_@xmath39p were similar to t_@xmath38l and t_@xmath39l except that the small scale fluctuations were restricted to the direction orthogonal to the direction of plane wave . thus the small scale fluctuations had the same form for all @xmath18 . these simulations are useful for differentiating between competing explanations for results outlined below . in addition to the n - body simulations listed in table 1 , we also carried out one dimensional simulations within the adhesion model @xcite with a finite viscosity following a method similar to the one outlined by weinberg and gunn ( 1990 ) . and @xmath33 from the pm_00l simulation , dashed lines show the density profile from the t_00l simulation with the same profile . ] figure [ den_plwave ] shows the density profile of the pancake for pm_00l and t_00l simulations at two epochs . these figures demonstrate that the density profiles in these simulations are almost identical , indeed the tiny differences can be attributed to the different initial distribution of particles . we have checked this assertion by running the pm_00l with the perturbed grid initial conditions . the treepm method has a slightly better resolution but it does not induce any new features . this is expected as the force softening length used in the treepm simulations is one grid length , same as the average inter - particle separation and it has been shown than such force softening does not induce two body collisions @xcite . we will mostly use treepm simulations for the remaining part of this study . an important indicator of the role played by substructure is the thickness of the pancake that forms by collapse of the plane wave . if substructure does not play an important role in evolution of large scale perturbations then the thickness of pancake should not change by a significant amount . on the other hand , if substructure does indeed speed up the process of dynamical relaxation then we should see some signature in terms of the thickness of pancake , velocity structure , or both . any such effect will be apparent only at late times as infall of matter into the pancake dominates at early times . dynamical effects of substructure will become important only at late times . figure [ pthick ] shows a slice from some of the simulations listed in table 1 . the plane wave collapses along the vertical axis . configuration at @xmath33 is shown here , the plane wave begins to collapse at @xmath20 . different panels in this figure refer to simulation t_00l , t_10l and t_40l . the boundary of the multi stream region is visible clearly in all the slices even though this region is fragmented in the last panel ( t_40l ) . it is clear that the pancake is thinner in simulations with more substructure . a more detailed comparison of simulations with different level of substructure is shown in figure [ den_all ] . the top panel of this figure shows the averaged over - density as a function of the @xmath18 coordinate , the plane wave collapses along this axis . over - density is averaged over all @xmath40 and @xmath41 for a given interval @xmath42 to obtain the averaged values plotted here . the peak over - density at the centre of the pancake is smaller in simulations with more substructure . the mass enclosed within a given distance of the centre of pancake ( defined here as the trough of the potential well of the plane wave ) is smaller for more substructure , even though the variation is very small at less than @xmath43 between the extreme cases ( see figure [ den_adh_sub ] ) . potential wells corresponding to substructure prevent infall into the pancake region . as the amount of substructure is increased , there is visible reduction in the size of the region around the pancake where density is greater than average . the visual impression of figure [ pthick ] is reinforced by the variation of over - density . the middle panel of figure [ den_all ] shows the _ rms _ velocities of particles in direction transverse to the plane wave collapse as a function of the @xmath18 coordinate . as in the top panel , averaging is done over all @xmath40 and @xmath41 for a given interval @xmath42 . this plot shows that the transverse motions are enhanced in the dense pancake region . the amplitude of transverse motions is larger in simulations with more substructure . size of the region where these motions are significant varies with the amount of substructure , as in case of over - density ( top panel ) . the _ rms _ transverse velocities do not go to zero outside the pancake region , instead these level off to a small residual value . , the direction of collapse for the plane wave . density profile has been averaged over the directions transverse to the collapse of plane wave . the curves are for @xmath33 , simulations used are t_00l ( solid line ) , t_05l ( dashed line ) , t_10l ( dot - dashed line ) , t_20l ( dotted line ) and t_40l ( dot - dot - dashed line ) . the middle panel shows the rms transverse velocities of particles at the same epoch for t_05l ( dashed line ) , t_10l ( dot - dashed line ) , t_20l ( dotted line ) and t_40l ( dot - dot - dashed line ) . the lower panel shows the rms transverse velocities of collapsed haloes at the same epoch for t_05l ( dashed line ) , t_10l ( dot - dashed line ) , t_20l ( dotted line ) and t_40l ( dot - dot - dashed line ) . ] transverse motions are due to motions of particles in clumps that constitute substructure , due to infall of particles in these clumps , and , transverse motions of clumps as they move towards each other . in order to delineate these effects , we have plotted the _ rms _ velocities for haloes in the last panel of figure [ den_all ] . these haloes were selected with the friends - of - friends ( fof ) algorithm using a linking length of @xmath44 grid length . transverse component of the velocity of centre of mass for haloes with more than @xmath45 particles was used for the figure . such a high cutoff for halo members is acceptable because typical haloes have several hundred members , see the following subsection on mass functions . differences between simulations with different amount of substructure are more pronounced than in the middle panel . for simulations with a small amount of substructure , motion of clumps is subdominant and hence the transverse motions are contributed mostly by internal motions and infall . in simulations with more substructure , motions of clumps contribute significantly to the _ rms _ transverse velocity . gravitational attraction of clumps , particularly in close encounters in the pancake region induce the transverse component . collisions are enhanced in the pancake region as the number density of clumps is higher . in order to convince ourselves that transverse motions induced by scattering / collision of clumps is the most likely reason for the reduced thickness of pancakes , we compare simulations t_@xmath38l and t_@xmath39l with t_@xmath38p and t_@xmath39p . in t_@xmath38p and t_@xmath39p simulations , the small scale fluctuations do not have any @xmath18 dependence . in these ( t_@xmath38p and t_@xmath39p ) simulations there are no clumps but streams of particles that are falling in and this reduces the number of scattering that take place no @xmath18 dependence means that dense streams run into each other head on with grazing collisions happening only rarely . of course , in the simulation the presence of the plane wave leads to breaking of these streams into clumps as the streams are stretched inhomogeneously in the @xmath18 direction . these clumps are aligned parallel to the @xmath18 axis . in the pancake region scattering of these streams occasionally leads to complex patterns . if presence of substructure and its growth in the pancake was the only cause for making the pancake thinner then pancake in these simulations should be thinner as well . figure [ pic_lp ] shows slices from simulations t_@xmath39l and t_@xmath39p for @xmath33 . a slice from the simulation pm_@xmath37l is also plotted here for reference . this visual comparison shows that the pancake is thinner in t_@xmath39l as compared to t_@xmath39p . indeed , the thickness of pancake in t_@xmath39p and pm_@xmath37l is very similar . this reinforces the point that scattering of clumps in the pancake region is the key reason for thinner pancakes . the substructure is helping to confine the pancake to a smaller region . it is interesting to study the collapse of a plane wave in an n - body simulation and compare it with the collapse in the adhesion model @xcite with a finite effective viscosity . we first study the collapse of a plane wave in absence of any substructure , n - body simulations pm_00l for comparison with numerical simulations using the adhesion model with finite effective viscosity . one dimensional adhesion simulations were done using the plane wave with the same amplitude as the n - body simulations . we use the standard method for computing the trajectories of particles in the adhesion model @xcite , a summary of the basic formalism is reproduced here for reference . in adhesion approximation , the equation of motion for a particle is replaced by the burgers equation @xcite . in the one dimensional situation we are considering here , we have : @xmath46 here @xmath47 is the velocity of particles and @xmath48 is the linear growth factor . this equation can be solved by introducing the velocity potential @xmath49 , where @xmath50 coincides with the gravitational potential at the initial time . solution has the following form . @xmath51 and , @xmath52 dq .\ ] ] here @xmath53 is the lagrangian position of the particle and @xmath40 is the eulerian position . in this method we integrate the differential equation for particle trajectories . at each time step velocity is calculated by above procedure at grid points and interpolated to particles positions . figure [ den_adh ] shows the mass enclosed within a distance @xmath54 from the centre of the pancake . the enclosed mass is defined as : @xmath55 here @xmath56 is the density at position @xmath18 and @xmath57 is the centre of the pancake . there is no ambiguity for comparing the results with n - body simulations in case of no substructure as density depends only on @xmath18 . while comparing other simulations with the adhesion solution , we will consider density averaged over @xmath40 and @xmath41 directions adhesion model is run only for the one dimensional problem . top panel of figure [ den_adh ] shows the enclosed mass @xmath58 at @xmath59 , middle panel is for @xmath60 and the lower panel is for @xmath61 . the solid curve shows the enclosed mass for pm_00l . in the region with a given number of streams , the n - body curve is smooth . jumps in mass enclosed occur at transition from single stream to multi stream region , and at other transitions where the number of streams changes within the multi stream region . all other curves show @xmath58 for adhesion model : dashed curve is for @xmath62 , dotted curve is for @xmath63 and the dot - dashed curve is for @xmath64 . there is no constant effective viscosity curve that follows the n - body curve closely through the multi stream regions . in regions with a given number of streams , the n - body curve stays around a curve for constant effective viscosity in the adhesion model . a remarkable fact is that the n - body curve for the three stream region at all the epochs follows the adhesion model curve for @xmath65 . similar behaviour is seen for the five stream region which follows @xmath66 though the range of scales and epochs over which this can be resolved is somewhat limited . addition of substructure clearly changes the character of the problem and the collapse is no longer one dimensional . however , the scale of the substructure is so small compared to the wavelength of the plane wave that the large scale collapse is still very close to planar . figure [ den_adh_sub ] shows the mass enclosed within a distance @xmath54 from the centre of the multi stream region for simulations pm_00l , t_10l and t_40l . density is averaged over all @xmath40 and @xmath41 for this plot in the same manner as for figure [ den_all ] . also plotted in the figure are curves for the adhesion model ( @xmath63 ) , where the calculation is done without taking substructure into account . the motivation for such a comparison is to see the effect of substructure on the favoured value of effective viscosity . substructure removes the sharp change in density at the boundaries of @xmath67-stream , @xmath68-stream and @xmath69-stream regions and the curves for t_10l and t_40l are smoother in the pancake region . the finite viscosity curve matches simulations with substructure over a wider range of scales than with pm_00l . there are no other noteworthy differences . mass function of collapsed haloes in these simulations can be used to understand the influence of plane wave collapse on substructure . collapsed structures form in these simulations primarily due to initial density fluctuations at small scales , with some modulation by the collapse of the plane wave . in this section we study the effect of the collapsing plane wave on the mass function of collapsed haloes . these haloes were selected with the friends - of - friends ( fof ) algorithm using a linking length of @xmath44 grid length . the initial power spectrum has a peak at the scale corresponding to @xmath70 of the simulation box , or @xmath71 grid lengths . thus typical haloes will have a lagrangian radius of about @xmath72 grid lengths and should contain about @xmath73 particles . thus a cutoff of @xmath45 or more particles for haloes is reasonable for this study . in absence of the plane wave , the only perturbations are at small scales . the small scale perturbations are concentrated around a given mass scale and the mass function is also peaked around this mass at early epochs . at late epochs mergers lead to formation of haloes with a larger mass and the range of masses is greater for models with a larger amplitude of fluctuations . figure [ slices ] shows these features in the distribution of particles . these features can also be seen in figure [ massf ] where mass fraction @xmath74 for @xmath75 , @xmath76 and @xmath77 is plotted in different panels . @xmath74 is the fraction of total mass in collapsed haloes with halo mass above @xmath78 . adding the plane wave at a much larger scale than the small scale fluctuations essentially pushes much of the mass into the pancake region , leaving a small fraction of matter in the under dense regions that occupy much of the volume . growth of small scale fluctuations in the under dense regions is inhibited whereas growth of fluctuations in the pancake region is enhanced , this is seen clearly in the slices from simulations shown in figure [ slices ] . higher background density in the pancake region leads to rapid growth of perturbations , mergers of haloes also lead to formation of massive clumps . these effects become more pronounced at late epochs and result in a shift of mass function towards larger masses , indeed haloes at two distinct mass scales are present . low mass clumps in under dense regions have the mass expected of haloes in regions where small scale fluctuation dominate whereas haloes of a much higher mass are present in the pancake region . figure [ massf ] shows these two mass scales very clearly . total mass in collapsed haloes does not change significantly with the addition of the plane wave . indeed for simulations t_40l and t_40 , mass function is the same at @xmath75 as small scales dominate . at late times ( @xmath33 ) , the effect of plane wave makes the mass function of t_05l , t_10l and t_20l similar . not surprisingly , presence of large scale power leads to formation of more massive haloes . however it does not seem to enhance the total mass in collapsed haloes . in this paper we studied the effect of substructure on collapse of a plane wave . the key conclusions of the present study of the role of substructure are : * the pancake formed due to collapse of the plane wave is thinner if the in falling material is formed of collapsed substructure . * we show that collisions between clumps lead to enhancement of velocities transverse to the direction of large scale collapse . * we show that in simulations with substructure where collisions are suppressed , pancakes are not thinner . * thus collision induced enhancement of motions transverse to the collapsing plane wave takes away kinetic energy from the direction of infall and leads to thinner pancakes . * presence of large scale power shifts the mass function towards larger masses . there is , however , no change in the total mass in collapsed haloes . the points outlined above essentially relate to coupling of density fluctuations at well separated scales . each of these points refers to a measurable effect of such a coupling . the nature of large scale fluctuation , a single plane wave , does not allow us to estimate the effect in terms of statistical indicators like the power spectrum . we plan to study these aspects with larger ( @xmath79 ) simulations where the large scale collapse will also be generic . large , high resolution studies are needed as @xmath35 simulations with particle mesh code have not shown any large effect in power spectrum at late times @xcite . from the centre of the multi stream region . the top panel shows the curves for @xmath33 . the thick solid curve is for the n - body simulation pm_00l . jumps in the mass enclosed occur at transition from multi stream region with @xmath80 streams to @xmath81 streams , with @xmath82 a non - zero positive integer . all other curves show @xmath58 for adhesion model : dashed curve is for @xmath62 , dotted curve is for @xmath63 and the dot - dashed curve is for @xmath64 . the middle panel shows the same set curves for @xmath60 and the lower panel is for @xmath34 . ] from the centre of the multi stream region . the top panel shows the curves for @xmath33 . the solid curve is for n - body simulation pm_00l . other simulations are also plotted here t_10l ( dashed curve ) and t_40l ( dot - dashed curve ) . dotted curve shows the mass enclosed in the one dimensional adhesion model with @xmath63 . the lower panel shows the same set of curves for @xmath34 and the middle panel is for @xmath60 . ] as a function of mass @xmath78 in n - body simulations . the top panel is for @xmath75 , the middle panel is for @xmath20 and the lower panel is for @xmath33 . curves are shown for t_05 ( solid curve ) , t_05l ( thick solid curve ) , t_10 ( dashed curve ) , t_10l ( thick dashed curve ) , t_20 ( dotted curve ) , t_20l ( thick dotted curve ) , t_40 ( dot - dashed curve ) and t_40l ( thick dot - dashed curve ) . ] another important point to consider is that we have considered two well separated scales for fluctuations and there is no infall once fluctuations at the larger scales collapse . numerical experiments that can shed light on effects of this feature are also required to improve our understanding of issues . we also compared the collapse of a plane wave in an n - body with the collapse in the adhesion model with a finite effective viscosity . we found that : * the adhesion model predicts the variation of density very well with a constant effective viscosity in regions with a given number of streams . * regions with a given number of streams coincide with the adhesion model with the same value of effective viscosity at all epochs . jsb thanks r.nityananda , t.padmanabhan and k.subramanian for useful discussions and suggestions . jsb also thanks varun sahni and uriel frisch for a useful discussion on related issues . numerical experiments for this study were carried out at cluster computing facility in the harish - chandra research institute ( http://cluster.mri.ernet.in ) . this research has made use of nasa s astrophysics data system . bagla j. s. and padmanabhan t. 1994 , mnras 266 , 227 bagla j. s. and padmanabhan t. 1997a , mnras 286 , 1023 bagla j. s. and padmanabhan t. 1997b , pramana 49 , 161 bagla j. s. 2002 , japa 23 , 185 bagla j. s. and ray s. 2003 , new astronomy 8 , 665 bagla j. s. 2004 , to appear in _ current science_. bernardeau f. , colombi s. , gaztanaga e. and scoccimarro r. 2002 , physics reports 367 , 1 bertschinger e. 1998 , ara&a 36 , 599 bond j. r. and myers s. t. 1996a , apjs 103 , 1 bond j. r. and myers s. t. 1996b , apjs 103 , 41 bond j. r. and myers s. t. 1996c , apjs 103 , 63 brainerd t. g. , scherrer r. j. and villumsen j. v. 1993 , apj 418 , 570 couchman h. m. p. and peebles p. j. e. 1998 , apj 497 , 499 engineer s. , kanekar n. and padmanabhan t. 2000 , mnras 314 , 279 evrard a. e. and crone m. m. 1992 , apjl 394 , 1 gurbatov s. n. , saichev a. i. and shandarin s. f. 1989 , mnras 236 , 385 hamilton a. j. s. , matthews alex , kumar p. and lu edward 1991 , apjl 374 , 1 hui l. and bertschinger e. 1996 , apj 471 , 1 jain b. , mo h. j. and white s. d. m. 1995 , mnras 276l , 25 klypin a. a. and melott a. l. 1992 , apj 399 , 397 little blane , weinberg david h. and park changbom , 1991 mnras 253 , 295 lynden - bell d. 1967 , mnras 136 , 101 ma chung - pei and boylan - kolchin michael 2004 , phys.rev.lett . 93 , 021301 ma chung - pei and bertschinger edmund 2003 , astro - ph/0311049 , apj in press . matarrese s. , lucchin f. , moscardini l. and saez d. 1992 , mnras 259 , 437 melott a. l. , shandarin s. f. , splinter r.j . and suto y. 1997 , apjl 479 , 79 monaco p. , theuns t. , taffoni g. , governato f. , quinn t. and stadel j. 2002 , apj 564 , 8 monaco p. , theuns t. and taffoni g. 2002 , mnras 331 , 587 nityananda r. and padmanabhan t. 1994 , mnras 271 , 976 padmanabhan t. 1996 , mnras 278l , 29 padmanabhan t. 2002 , _ theoretical astrophysics : galaxies and cosmology ( vol.3 ) _ , cambridge university press peacock j. a. and dodds s.j . 1996 , mnras 280 , 19 peacock j. a. 1998 , _ cosmological physics _ , cambridge university press peebles p. j. e. 1974 , a&a 32 , 391 peebles p. j. e. 1980 , _ large scale structure of the universe _ , princeton university press peebles p. j. e. 1985 , apj 297 , 350 peebles p. j. e. 1990 , apj 365 , 27 ray s. and bagla j. s. 2004 , astro - ph/0405220 sahni v. and coles p. 1995 , physics reports 262 , 1 shandarin s. f. , zeldovich y. b. , 1989 , rvmp , 61 , 185 smith r. e. et al . 2003 , mnras 341 , 1311 splinter r. j. , melott a. l. , shandarin s. f. and suto y. 1998 , apj 497 , 38 springel v. , yoshida n. and white s.d.m . 2001 , new astronomy 6 , 79 subramanian k. 2000 , apj 538 , 517 taffoni g. , monaco p. and theuns t. 2002 , mnras 333 , 623 valinia a. and shapiro p. r. and martel h. and vishniac e. t. 1997 , apj 479 , 46 weinberg d. h. and gunn j. e. 1990 , mnras 247 , 260 weinberg martin d. 2001 , mnras 328 , 311 zeldovich ya.b . 1970 , a&a 5 , 84
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we study the interplay of clumping at small scales with the collapse and relaxation of perturbations at much larger scales .
we present results of our analysis when the large scale perturbation is modelled as a plane wave .
we find that in absence of substructure , collapse leads to formation of a pancake with multi - stream regions .
dynamical relaxation of plane wave is faster in presence of substructure .
scattering of substructures and the resulting enhancement of transverse motions of haloes in the multi - stream region lead to a thinner pancake . in turn ,
collapse of the plane wave leads to formation of more massive collapsed haloes as compared to the collapse of substructure in absence of the plane wave .
the formation of more massive haloes happens without any increase in the total mass in collapsed haloes .
a comparison with the burgers equation approach in absence of any substructure suggests that the preferred value of effective viscosity depends primarily on the number of streams in a region .
gravitation cosmology : theory dark matter , large scale structure of the universe
| 10,588 | 256 |
due to the importance of the primes , the mathematicians have been investigating about them since long centuries ago . in 1801 , carl gauss , one of the greatest mathematician , submitted that the problem of distinguishing the primes among the non - primes has been one of the outstanding problems of arithmetic @xcite . proving the infinity of prime numbers by euclid is one of the first and most brilliant works of the human being in the numbers theory @xcite . greek people knew prime numbers and were aware of their role as building blocks of other numbers . more , the most natural question asked by human being was this what order prime numbers are following and how one could find prime numbers ? until this time , there have been more attempts for finding a formula producing the prime numbers and or a model for appearance of prime numbers among other numbers and although they could be more helpful for developing the numbers theory , however , the complicated structure of prime numbers could not be decoded . during last years , the prime numbers attained an exceptional situation in the field of coding . for example , `` rsa '' system is one of the most applicable system in this field used in industries relying on prime numbers . `` rsa '' system is used in most computerized systems and counted as main protocol for secure internet connections used by states and huge companies and universities in most computerized systems @xcite . on 2004 , manindra agrawal and his students in indian institute of technology kanpur could develop an algorithm called aks for detecting prime numbers @xcite . on 2006 , 2008 , 2009 and recently on 2013 , mathematics students in a project called detecting the mersenne prime numbers by computer network gimps succeeded to discover the greatest prime number . all such cases indicate the importance of mersenne theorem or any other approach for finding the largest prime numbers @xcite . generalizing the mersenne theorem , this paper could accelerate finding the largest prime numbers . in addition , there have been provided new equations and algorithm for attaining the largest primes . assume that @xmath0 is a natural number greater than 1 , @xmath1 related to n and natural numbers @xmath2 and @xmath3 are defined as below : @xmath4 if @xmath1 is a prime number , then @xmath0 is a prime number , too . if @xmath0 is not the prime number so we can write @xmath0 as the multiplication of two natural numbers except @xmath5 meaning : @xmath6 @xmath7 @xmath8 @xmath9 @xmath10 @xmath11 therefore , @xmath1 is not the prime number . so , @xmath0 must be a prime number . this theorem is a generalization for mersenne theorem in which @xmath2 and @xmath3 are arbitrary natural numbers . if in the theorem @xmath12 , c is chosen as a multiple to @xmath2 and @xmath13 , thus , @xmath1 will not be a prime number . suppose : @xmath14 therefore : @xmath15 @xmath16 @xmath17 @xmath18 @xmath19 @xmath20 the last equality shows that @xmath1 is not a prime number . suppose @xmath0 is a natural number greater than @xmath5 , function @xmath1 related to @xmath0 and natural number @xmath2 are defined as below : @xmath21 if @xmath1 is a prime number , then @xmath0 is a prime number , too . in this theorem @xmath22 , based on @xmath0 constant , please consider a sequence @xmath23 we prove that sequence @xmath24 is strictly ascending , i.e. @xmath25 to prove the last inequality , we write : @xmath26 @xmath27 @xmath28 @xmath29 status 1 . if @xmath0 is a multiple of @xmath30 : @xmath31 status 2 . if @xmath0 is not a multiple of @xmath30 : @xmath32 therefore , inequity is accepted . in this theorem , each number is higher than mersenne number , meaning : @xmath33 suppose @xmath2 be a natural number and @xmath34 are the primes smaller than or equal @xmath35 and @xmath36 , @xmath37 are natural numbers which limitations are intended for them indicated as follows : @xmath38 assume that @xmath39 is a function of @xmath40 which is displayed as bellow : @xmath41 if the @xmath42 and @xmath37 circumstances are followed , @xmath39 can obtain all the primes less than @xmath2 . knowing that @xmath39 is odd , because it is non prime , therefore it comprises from two odd numbers except @xmath5 , and because @xmath43 , @xmath39 has at least a prime factor @xmath44 . therefore , @xmath39 is divided at least on one of the prime factors @xmath45 . @xmath46 @xmath47 it is clear that above equalities are in discrepancy of the assumption of the theorem . 1 . if : @xmath48 2 . interval @xmath49 : + it is clear that by putting minimum @xmath37 in the definition @xmath50 minimum @xmath51 followed by minimum @xmath39 is obtained as below : @xmath52 according to recent equation , it is obvious that being as prime number in prime numbers smaller than @xmath53 , r may not be divided into prime factors smaller than @xmath54 . on the other hand , it is not necessary to see if prime numbers smaller than @xmath53 are divided into @xmath55 to detect it as a prime number . indeed , for obtaining the prime numbers , we only require @xmath56 in @xmath57 to enter the provision of prime factor @xmath58 . if @xmath59 is considered as a prime number bigger than @xmath60 , we could use @xmath61 instead of @xmath2 in this theorem because prime numbers smaller than @xmath35 include prime numbers smaller than @xmath62 . prime numbers smaller than 120 : @xmath63 \ { @xmath64 , prime numbers smaller than @xmath35 : @xmath65 } + @xmath66 @xmath67 @xmath68 @xmath69 @xmath70 @xmath71 @xmath72 @xmath73 @xmath74 @xmath75 suppose @xmath2 be the natural number and @xmath76 are the primes smaller than or equal @xmath35 and also consider that @xmath77 are the primes larger than @xmath35 . suppose that @xmath78 and @xmath42 be the members of the natural numbers and also @xmath79 be the members of the account numbers , these variables are selected arbitrarily . function @xmath39 related to values @xmath80 , natural numbers @xmath81 and arithmetic numbers @xmath82 are defined as below : @xmath83 @xmath84 if @xmath56 be as the natural number less than @xmath2 , then @xmath39 is a prime number . if @xmath39 is not prime number , it has a prime factor @xmath85 . on the other side , because @xmath86 , @xmath39 has at least one prime factor @xmath87 . so , it is arbitrarily supposed that @xmath39 is divisible in @xmath88 . @xmath89 @xmath90 because @xmath54 is not denominator of any @xmath91 . we have : @xmath92 @xmath93 we reached a contradiction to the assumption . thus , the theorem was verified . we obtain prime numbers smaller than 119 . + @xmath94 @xmath95 @xmath96 table 1 [ cols="^,^,^,^,^,^",options="header " , ] and continuing so suppose that : @xmath97 ( the order is considered in the primes ) general speaking , the theorem @xmath98 comes true to @xmath99 because @xmath100 includes the same primes . @xmath39 is obviously not divisible to @xmath101 and according to prime of the number @xmath39 , we have : @xmath102 to attain prime numbers , we divide the intervals as below : @xmath103 with regard to the relationship easier to be written . in example of the primes less than @xmath104 , the rang can be divided into three sections of @xmath105 , @xmath106 and @xmath107 . then , a distinct relation asserted for each . prime numbers smaller than 48 : @xmath108 @xmath109 @xmath110 @xmath111 and continuing so by integrating the relations , particularly using the relation 2 and notation 5.4 , we can attain an algorithm to obtain the largest prime number . ; one of them is as below : first of all , assign @xmath36 as one @xmath119 so as to obtain the minimum of @xmath0 . then in the following equation , give a counter to @xmath0 through the minimum @xmath0 so long as k be a member of the natural numbers @xmath120 . meanwhile , @xmath36 should not be even . @xmath121 if @xmath39 is not prime number , it has a prime factor @xmath85 . on the other side , because @xmath125 , @xmath39 has at least one prime factor @xmath126 therefore , with no interruption in the generality of the subject , we can assume that @xmath39 is divisible on @xmath127 : @xmath128 @xmath129 @xmath130 we reached a contradiction to the assumption . thus , the theorem was verified . 99 p. ribenboim , the little book of bigger primes , 2nd ed . springer science & business media , 2004 . w. stein , `` elementary number theory : primes , congruences , and secrets '' , book , pp . 1 - 172 , 2009 . r. crandall and c. b. pomerance , prime numbers : a computational perspective . springer science & business media , 2006 . m. agrawal , n. kayal , and n. saxena , `` primes is in p '' , ann . 781 - 793 , 2004 . `` list of known mersenne prime numbers - primenet '' . [ online ] . available : + http://www.mersenne.org/primes/ [ accessed : 08-apr-2015 ] .
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today , prime numbers attained exceptional situation in the area of numbers theory and cryptography . as we know , the trend for accessing to the largest prime numbers due to using mersenne theorem , although resulted in vast development of related numbers , however it has reduced the speed of accessing to prime numbers from one to five years .
this paper could attain to theorems that are more extended than mersenne theorem with accelerating the speed of accessing to prime numbers .
since that time , the reason for frequently using mersenne theorem was that no one could find an efficient formula for accessing to the largest prime numbers .
this paper provided some relations for prime numbers that one could define several formulas for attaining prime numbers in any interval ; therefore , according to flexibility of these relations , it could be found a new branch in the field of accessing to great prime numbers followed by providing an algorithm at the end of this paper for finding the largest prime numbers .
| 2,810 | 223 |
the precise determination of the cabibbo - kobayashi - maskawa ( ckm ) matrix element @xmath8 is a crucial step for @xmath9 physics to pursue phenomena beyond the standard model . in particular , the precision achieved in determining the apex of the unitarity triangle may be limited by @xmath10 , even with future high - statistics experiments . the current determination of @xmath10 @xcite is made through inclusive @xcite and exclusive @xcite @xmath9 decays . the heavy quark expansion offers a method to evaluate the hadronic transition amplitude in a systematic way . in particular , at the kinematic end point the exclusive @xmath11 matrix element is normalized in the infinite heavy quark mass limit , and the correction of order @xmath12 vanishes as a consequence of luke s theorem @xcite . it is thus possible to achieve an accuracy on @xmath10 of a few percent . calculations of the @xmath13 ( and higher order ) deviations from the heavy quark limit have previously been attempted with the non - relativistic quark model and with qcd sum rules . lattice qcd has the potential to calculate exclusive transition matrix elements from first principles . the shapes of the @xmath14 decay form factors have already been calculated successfully with propagating @xcite , static @xcite , and non - relativistic @xcite heavy quarks . on the other hand , a precise determination of the absolute normalization of the form factors has not been achieved . this paper fills that gap for the decay @xmath0 . previous lattice calculations were unable to obtain the normalization of the form factors for various reasons . first , the statistical precision of the three point function @xmath15 , which is calculated by monte carlo integration , has not been enough . second , perturbative matching between the lattice and the continuum currents has been a large source of uncertainty . since the local vector current defined on the lattice is not a conserved current at finite lattice spacing @xmath16 , the matching factor is not normalized even in the limit of degenerate quarks . although one - loop perturbation theory works significantly better with tadpole improvement @xcite , the two - loop contribution remains significant ( @xmath17 5 % ) . last , the systematic error associated with the large heavy quark mass must be understood . previous work with wilson quarks @xcite , for which the discretization error was as large as @xmath18 , could not address the @xmath12 dependence in a systematic way when @xmath19 . in this paper we present a lattice qcd calculation of the @xmath0 decay form factor . for the heavy quark we use an improved action @xcite for wilson fermions , reinterpreted in a way mindful of heavy - quark symmetry @xcite . discretization errors proportional to powers of @xmath20 do not exist in this approach . instead , discretization errors proportional to powers of @xmath21 remain , although they are intertwined with the @xmath12 expansion . the first extensive application of this approach to heavy - light systems was the calculation @xcite of the heavy - light decay constants , such as @xmath22 and @xmath23 . there the lattice spacing dependence was studied from direct calculations at several lattice spacings , and a very small @xmath16 dependence was observed . the third difficulty mentioned above is , thus , no longer a problem . to obtain better precision on the semi - leptonic form factors , we introduce ratios of three - point correlation functions . the bulk of statistical fluctuations from the monte carlo integration cancels between numerator and denominator . furthermore , the ratios are , by construction , identically one in both the degenerate - mass limit and the heavy - quark - symmetry limit . consequently , statistical and all systematic errors , as well as the signal , are proportional to the deviation from one . the first and second difficulties given above are , thus , also essentially cured . the ratio of correlation functions for the calculation of @xmath24 corresponds to the ratio of matrix elements , @xmath25 in which all external states are at rest . the denominator may be considered as a normalization condition of the heavy - to - heavy vector current , since the vector current @xmath26 with degenerate quark masses is conserved in the continuum limit , and its matrix element is , therefore , normalized . as a result the perturbative matching between the lattice and continuum currents gives only a small correction to @xmath27 . for the calculation of @xmath3 we define another ratio , corresponding to matrix elements @xmath28 where equality holds when the final - state @xmath29 meson has small spatial momentum . by construction , the ratio produces a value of @xmath30 that vanishes when the @xmath31 quark has the same mass as the @xmath32 quark , as required by current conservation . this method does not work as it stands for the @xmath33 decay form factors . the axial vector current mediates this decay , and it is neither conserved nor normalized . we will deal separately with this case in another paper . this paper is organized as follows . section [ sec : form_factors ] contains a general discussion of form factors for the exclusive decay @xmath0 . sections [ sec : hqet_and_1/m_q_expansion ] and [ sec : lattice_and_hq ] discuss heavy quark effective theory and the @xmath12 expansion in the continuum and with the lattice action used here . section [ sec : lattice_details ] contains details of the numerical calculations . sections [ sec : calculation_of_h_+][sec : heavy_quark_mass_dependence_of_h- ] present our results . sections [ sec : calculation_of_h_+ ] and [ sec : heavy_quark_mass_dependence_of_h+ ] discuss the form factor @xmath34 and its mass dependence . sections [ sec : calculation_of_h_- ] and [ sec : heavy_quark_mass_dependence_of_h- ] do likewise for @xmath30 . we compare the results from the fits of the mass dependence to corresponding results from qcd sum rules in sec . [ sec : comparison_with_the_qcd_sum_rules ] . the values of @xmath24 and @xmath35 at the physical quark masses are combined in sec . [ sec : result_for_f ] into a result for the form factor @xmath36 , which with experimental data determines @xmath10 . we give our conclusions in sec . [ sec : conclusions ] . the decay amplitude for @xmath0 is parametrized with two form factors @xmath1 and @xmath3 as @xmath37,\ ] ] where @xmath38 and @xmath39 are the velocities of the @xmath9 and @xmath29 mesons , respectively , and @xmath40 . the square of the momentum transferred to the leptons is then @xmath41 . we denote by the symbol @xmath42 the physical vector current , to distinguish it from currents in heavy quark effective theory ( hqet ) and in lattice qcd . the differential decay rate reads @xmath43 with @xmath44 at zero recoil ( @xmath45 , so @xmath2 ) one expects @xmath36 to be close to one , because of heavy quark symmetry . from ( [ eq : differential_decay_rate ] ) a determination of @xmath10 consists of the following three steps : measure @xmath46 in an experiment , extrapolate it to the zero - recoil limit assuming some functional form , and use the theoretical input of @xmath36 . in this paper we report on a new calculation of @xmath36 with lattice qcd , which is model independent , at least in principle . the present calculation includes the leading corrections to the heavy - quark limit : radiative corrections to the static limit of @xmath47 , the @xmath12 contribution of @xmath48 , and the @xmath13 contributions of @xmath49 . radiative corrections of order @xmath50 to @xmath35 are not yet available , but these and further corrections , of order @xmath51 , @xmath52 , etc . , could be included in future applications of our numerical technique , once the needed perturbative results become available . an obvious disadvantage in using the @xmath0 decay mode is that the branching fraction is much smaller than the @xmath33 mode . another , but not less important , shortcoming is that the phase - space suppression factor @xmath53 makes the extrapolation of the experimental data to @xmath2 more difficult than for @xmath33 , where the corresponding factor is @xmath54 . nevertheless , the experimental result of the cleo collaboration @xcite shows that the above method certainly works , even with current statistics . that means that future improvement of the statistics will allow a much better determination of @xmath10 , providing an important cross check against other methods . many important theoretical results have been obtained for the form factors with hqet . the lagrangian of hqet uses fields of infinitely heavy quarks , so that heavy quark symmetries are manifest . the effects of finite quark mass are included through the @xmath12 expansion and through radiative corrections . for example , at zero recoil the form factor @xmath34 is given by @xmath55,\ ] ] where @xmath56 represents a matching factor relating the vector current in ( [ eq : definition_of_the_form_factors ] ) to the current in hqet @xcite . the absence of the @xmath57 term in ( [ eq:1/m_q - expansion+ ] ) is a result of a symmetry under an interchange of initial and final states in ( [ eq : definition_of_the_form_factors ] ) , and it is known as a part of luke s theorem @xcite . the same symmetry also restricts the form of the @xmath58 terms . the matching factor , defined so that the identity @xmath59 holds for matrix elements , is an ultraviolet- and infrared - finite function of @xmath60 . through one - loop perturbation theory , @xmath61 the two - loop coefficient is also available @xcite . the vector current defined with lattice fermion fields has properties similar to @xmath62 . there is a normalization factor @xmath63 defined so that @xmath64 holds for matrix elements . the factor @xmath63 depends strongly on the ( lattice ) quark masses @xmath65 and @xmath66 @xcite , and its one - loop corrections are large . in the past , such uncertainties in the normalization prevented a calculation of @xmath47 with the sought - after accuracy . one can , however , capture most of the normalization nonperturbatively by writing , with explicit flavor indices , @xmath67 in our ratio ( [ eq : ratio_1 ] ) the flavor - diagonal factors cancel , so our method avoids the major normalization uncertainties . the remaining radiative correction @xmath68 depends on the ratio of quark masses and the lattice spacing . in the continuum limit , @xmath69 and @xmath70 with @xmath60 fixed , @xmath71 by construction . in the static limit , @xmath72 and @xmath73 with @xmath16 and @xmath60 fixed , @xmath74 because the lattice theory strictly obeys heavy - quark symmetries . in numerical work one is somewhere in between , but the limits imply that @xmath75 is never far from unity . two of us have computed @xmath75 at one loop in perturbation theory @xcite , verifying explicitly that the radiative correction is small . similarly , the ratio ( [ eq : ratio_2 ] ) is described by the expansion @xmath76 where @xmath77 is a coefficient from matching the currents in ( [ eq : ratio_2 ] ) to hqet . like @xmath78 , it is an ultraviolet- and infrared - finite function of @xmath60 , and @xmath79 at leading order . the ratio ( [ eq : ratio_2 ] ) again captures nonperturbatively most of the renormalization of the lattice currents , apart from a factor @xmath80 to compensate for the difference between the radiative corrections with a fixed lattice cutoff and with no ultraviolet cutoff . in the continuum limit @xmath81 , and in the static limit @xmath82 . again , explicit calculation verifies that the one loop contribution remains small between the limits . in the rest of this paper , we do not write the matching factors @xmath83 when there is no risk of confusion . in the final result , on the other hand , they are included . @xcite , it was shown that the usual action for light quarks @xcite can be analyzed in terms of the operators of hqet . therefore , it can be used as the basis of a systematic treatment of heavy quarks on the lattice , even when the quark mass in lattice units , @xmath20 , is not especially small . the key is to adjust the couplings in the lattice action so that operators are normalized as they are in hqet . when @xmath84 , as is the case for charmed quarks at the smaller lattice spacings in common use , this is essentially automatic , because the higher order terms of the heavy quark expansion come from the dirac term of the lattice action , as in continuum qcd . when @xmath85 , as is the case for bottom quarks , one can apply the formalism of hqet to the lattice theory to obtain the normalization conditions , as sketched below . in either case , the kinetic energy is normalized nonperturbatively by tuning the quark mass according to some physical condition . other operators are often normalized perturbatively as an initial approximation but ultimately may be normalized nonperturbatively . in the numerical calculations presented here , we use an action introduced by sheikholeslami and wohlert @xcite , @xmath86 where the index @xmath87 runs over heavy and light flavors . the hopping parameter @xmath88 is related to the bare quark mass , @xmath89 where @xmath90 is the value of @xmath91 needed to make a quark massless . the flavor - independent matrix @xmath92 vanishes except when @xmath93 , for some spacetime direction @xmath94 . the kinetic energy arises from this term . the gluons field strength @xmath95 is defined on a set of paths shaped liked a four - leaf clover , so @xmath96 is often called the `` clover '' action . with @xmath97 one has the wilson action . for the light quark the clover coupling @xmath98 can be chosen so that there are lattice artifacts of order @xmath99 . in our numerical work we take an approximation to the optimal value , leaving an artifact of order @xmath100 . for heavy quarks , the clover action ( [ eq : sw_action ] ) has the same heavy - quark spin and flavor symmetries as continuum qcd , even at nonzero lattice spacing . consequently , we can use the machinery of hqet to characterize the lattice theory . the same operators as in continuum qcd appear , but the coefficients can differ . through first order in @xmath12 there are three operators in the heavy quark effective hamiltonian , @xmath101 where @xmath102 is a heavy quark field , and the coefficients @xmath103 , @xmath104 , and @xmath105 depend on the bare mass and the gauge coupling . because the lattice breaks relativistic invariance , the three `` masses '' are not necessarily equal , except as @xmath106 . at tree level , the rest mass @xmath107 , and the ( inverse ) kinetic mass @xmath108 the first term can be traced to the dirac term of the lattice action , and the second to the wilson term . the one - loop corrections to @xmath109 and @xmath110 are also available @xcite . the chromomagnetic mass @xmath111 is considered below . in the heavy quark effective theory , the rest mass term @xmath112 commutes with the rest of the hamiltonian and , thus , decouples from the dynamics . as with decay constants @xcite , one can derive the expansions like ( [ eq:1/m_q - expansion+ ] ) and ( [ eq:1/m_q - expansion- ] ) within the lattice theory , and the rest mass disappears from physical amplitudes @xcite . on the other hand , adjusting the bare quark mass so that @xmath113 is the way to normalize the kinetic operator @xmath114 correctly . this normalization can be implemented nonperturbatively by demanding that the energy of a hadron have the correct momentum dependence . in our numerical work we use the @xmath9 and @xmath29 mesons for this purpose . furthermore , one can correctly normalize the chromomagnetic operator @xmath115 by adjusting the clover coupling @xmath98 , as a function of the gauge coupling , so that @xmath116 . for example , at tree level the desired adjustment is @xmath117 . in our numerical work , we choose @xmath98 in a way that sums up tadpole diagrams , which dominate perturbation theory . this amounts to normalizing the chromomagnetic operator perturbatively . in summary , we adjust the bare mass @xmath118 and clover coupling @xmath98 so that the leading effects of the heavy - quark expansion are correctly accounted for @xcite . previous work in the literature chose instead to adjust the bare mass until @xmath119 , which introduces an unnecessarily large error , expansion with higher orders . ] proportional to @xmath120 . under renormalization the heavy quark kinetic energy can mix with the rest mass term in a power divergent way . because the lattice action used here contains both , the rest mass fully absorbs the power divergence . a related problem is the ambiguity owing to renormalons @xcite , which appears in some quantities in hqet or nonrelativistic qcd ( nrqcd ) . it is irrelevant to our work , because we calculate physical quantities , namely the masses of the @xmath9 and @xmath29 mesons and decay amplitude for @xmath0 . to complete the correspondence of the lattice theory to hqet we must consider the vector current . at order @xmath12 of hqet @xmath121 where the coefficient @xmath122 depends on the current employed . the heavy - heavy current on the lattice is constructed by defining a rotated field @xcite , @xmath123\psi^f , \label{eq : rotation}\ ] ] where @xmath124 is the quark field in the hopping - parameter form of the action ( [ eq : sw_action ] ) . then the lattice vector current @xmath125 and @xmath126 . both @xmath127 and @xmath128 depend on the gauge coupling , the masses , and ( at higher orders ) on the dirac matrix in ( [ eq : vhh ] ) . they are adjusted so that the normalization and momentum dependence of matrix elements matches the continuum , respectively . in particular , at tree level the coefficient in ( [ eq : vhh ] ) is @xmath129 and the condition @xmath130 prescribes a condition on @xmath131 @xcite . from the properties of the operators under heavy - quark symmetry , it follows that the @xmath104 and @xmath105 terms in ( [ eq : h ] ) could give a contribution to @xmath24 , but not to @xmath35 @xcite . on the one hand , these contributions to @xmath24 must be symmetric under interchange of the initial and final states , but , on the other hand , they must vanish when the initial and final quark masses are the same . consequently , there can be no contributions linear in either @xmath104 or @xmath105 . our definition of @xmath24 enjoys this property , by construction , because ( [ eq : ratio_1 ] ) manifestly respects the interchange symmetry . similarly , the @xmath122 terms in ( [ eq : vhh ] ) give a contribution only to @xmath35 . it must be anti - symmetric under interchange of the initial and final states and must vanish when the initial and final quark masses are the same . our definition of @xmath35 , again by construction , ensures that only the combination @xmath132 appears . this feature is taken into account in sec . [ sec : heavy_quark_mass_dependence_of_h- ] . in ( [ eq:1/m_q - expansion+ ] ) and ( [ eq:1/m_q - expansion- ] ) we seek contributions of order @xmath13 . these come from double insertions of the @xmath12 terms in ( [ eq : h ] ) and ( [ eq : vhh ] ) , and from @xmath13 terms implied by the ellipses . remarkably , the latter cancel when @xmath24 and @xmath35 are defined by the double ratios ( [ eq : ratio_1 ] ) and ( [ eq : ratio_2 ] ) @xcite . this is easy to understand if one starts with the matrix elements . the @xmath133 and @xmath134 corrections to the action arise from the initial or final state only . to this order , one can factorize them . they drop out of the double ratios , because the numerator and denominator of ( [ eq : ratio_1 ] ) or ( [ eq : ratio_2 ] ) contain the same number of @xmath9 and @xmath29 factors . the same applies to @xmath133 and @xmath134 corrections to the current . there may be a nonfactorizable correction to the current with coefficient @xmath135 , where the function @xmath136 is unknown , except that in perturbation theory it starts at one loop and that @xmath137 . in the long run , one would like to pick up terms of order @xmath138 and higher . because the bottom quark is so heavy , these are dominated by the @xmath139 terms . with the normalization conditions outlined here @xcite , these come automatically from the dirac term , as in continuum qcd . in future work , at smaller lattice spacings , the dirac term will dominate , generating contributions to all orders in @xmath140 . our numerical data are obtained in the quenched approximation on a @xmath141 lattice with the plaquette gluon action at @xmath142 . we take a mean - field - improved @xcite value of the clover coupling , which on this lattice is @xmath143 . out of 300 configurations generated for our previous work @xcite , we use 200 configurations . we usually define the inverse lattice spacing through the charmonium 1s1p splitting , finding @xmath144 gev . for comparison , with the kaon decay constant @xmath145 gev , and the difference is thought to be part of the error of quenching . because the form factors are dimensionless , the lattice spacing affects them only indirectly , through the adjustment of the quark masses . to investigate the heavy quark mass dependence of the form factors we take @xmath146 , 0.089 , 0.100 , 0.110 , 0.119 and 0.125 , and consider several combinations for the heavy quarks in the initial and final states . the mass of the spectator light quark is usually taken to be close to that of the strange quark , for which @xmath147 . we examine the effect of chiral extrapolation using four @xmath148 values , 0.1405 , 0.1410 , 0.1415 , and 0.1419 , for various combinations of the initial and final heavy quark masses @xmath149 , 0.110 , and 0.119 . the critical hopping parameter is @xmath150 . for the computation of the matrix element '' instead of `` @xmath151 '' to indicate the @xmath152 meson , and we use `` @xmath9 '' or `` @xmath29 '' for any values of the heavy quark masses . ] @xmath153 we calculate the three point correlation function @xmath154 with @xmath155 from ( [ eq : v_0 ] ) and @xmath156 . the light quark propagator is solved with a source at time slice @xmath157 , and we place the interpolating field for @xmath9 at @xmath158 , where we use the source method . the interpolating fields @xmath9 and @xmath29 are constructed with the 1s state smeared source as in ref . the spatial momentum @xmath159 carried by the final state is taken to be ( 0,0,0 ) , ( 1,0,0 ) , ( 1,1,0 ) , ( 1,1,1 ) and ( 2,0,0 ) in units of @xmath160 , where @xmath161 is the physical size of the box ; in our case , @xmath162 . the numerical results presented below are obtained from uncorrelated fits to ratios of these three - point functions . the statistical errors are estimated with the jackknife method . for a subset of the data we have repeated the analysis with correlated fits and the bootstrap method . we find no statistically significant difference . in much of the numerical work presented in this paper , we set the coefficients @xmath131 of the rotation ( [ eq : rotation ] ) to zero . from the discussion following ( [ eq : vhh ] ) the dependence on @xmath131 enters directly through @xmath122 , and indirectly by changing @xmath83 . on the scattering matrix elements of the spatial current @xmath163 , this should make a small ( @xmath164 or so ) effect . on the temporal current @xmath165 , the effect should be tiny . both expectations are checked at representative choices of the heavy quark masses , and the uncertainty introduced into the spatial current is propagated to the final result . the form factor @xmath166 at zero recoil is obtained directly from the three - point correlation functions ( [ eq : three_point_correlator ] ) , setting all three momentum to be zero . we define a ratio dependence of the isgur - wise function , which is the infinite mass limit of @xmath1 . ] @xmath167 in which the exponential dependence on @xmath168 associated with the ground state masses cancels between the numerator and denominator . when the current and two interpolating fields are separated far enough from each other , the contribution of the ground state dominates and @xmath169 suppressing radiative corrections . here we use the definition ( [ eq : definition_of_the_form_factors ] ) and the unit normalization of @xmath27 in the equal mass case . thus , we expect @xmath170 to be constant as a function of @xmath168 , and its value represents the form factor squared . in fig . [ fig : r_btod ] we plot the ratio @xmath171 for two representative combinations of mass parameters . we observe a nice plateau extending over about five time slices , and our fit over the interval @xmath172 is shown by the solid line . to see if the plateau is stable under the change of the position of the interpolating field , we repeat the calculation changing the time @xmath173 of the @xmath9-meson interpolating field . the results with @xmath174 and 8 are shown in fig . [ fig : r_btod_tb ] together with the one with @xmath175 . we observe that the plateau is very stable and conclude that the extraction of the ground state is reliable . in the following analysis we use the result with @xmath176 , and the numerical data for each @xmath177 are given in table [ tab : h+_data ] . 0.3em .numerical data for @xmath170 , which corresponds to @xmath178 , at @xmath147 . rows ( columns ) are labeled by the value of @xmath177 in the initial ( final ) state . combinations without data have not been calculated in this work . the diagonal elements are 1 by construction . [ cols= " < , < , < , < , < , < , < " , ] since the systematic errors in @xmath66 and in @xmath65 are correlated , we consider the central and two limiting combinations only . the statistical errors on @xmath24 and @xmath35 are estimated with the jackknife method , so that the resulting precision is better than that obtained by adding in quadrature the errors on coefficients @xmath179 . in the physical amplitude @xmath36 , which is the linear combination of @xmath24 and @xmath35 given in ( [ eq : form_factor_relation ] ) , the uncertainty from adjusting the quark masses largely cancels , and the value of @xmath36 is very stable . to obtain the physical result , we must now fold in the radiative correction @xmath75 , relating the lattice current to the continuum . two of us recently have calculated this factor to one loop @xcite , and at @xmath180 and @xmath181 they find @xmath182 . the lepage - mackenzie scale @xmath183 for the coupling @xmath184 @xcite has also been calculated , and at the same quark masses the result is @xmath185 . at @xmath186 , @xmath187 and the correction to @xmath188 is @xmath189 , taking the error of omitting higher orders to be 20 % of the one - loop correction . a similar one - loop calculation for @xmath190 , which modifies @xmath35 , is not yet available . we allow , therefore , a systematic uncertainty for this effect . our results for the form factors are @xmath191 where the error estimates are as follows . the first error comes from statistics , after the chiral extrapolation ; the second from adjusting the heavy quark masses ; and the third error from unknown radiative corrections , two loops and higher for @xmath34 and one loop and higher for @xmath30 . the chiral extrapolations , which are shown in figs . [ fig : h+_chiral ] and [ fig : h-_chiral ] , double the statistical errors of table [ tab : form_factors ] , without changing the central values . our main result is the value of the form factor entering the decay rate , at zero recoil . inserting the physical values of the @xmath9 and @xmath29 meson masses and the results ( [ eq : result_h+ ] ) and ( [ eq : result_h- ] ) into ( [ eq : form_factor_relation ] ) , @xmath192 where errors are from statistics , heavy quark masses , and omitted radiative corrections . the last of these could be reduced substantially by calculating the radiative correction factor @xmath190 to one loop . two sources of uncertainty have yet to be investigated carefully . they are the dependence on the lattice spacing and the effects of the quenched approximation . from our experience with @xmath22 @xcite , we might suppose that these effects are a few percent and @xmath193 , respectively . the ratios have been constructed so that all sources of error , including these , vanish for equal heavy quark masses . it is , therefore , our expectation that these percentages apply not to @xmath194 but to @xmath195 . that means that these two sources of error should be under good control , just as we have found with the other sources of uncertainty . in this paper we have shown that precise lattice calculations of the zero - recoil form factors @xmath24 and @xmath35 are possible . the principal technical advance is to consider ratios of matrix elements , in which a large cancellation of statistical and systematic errors takes place . the numerical data are interpreted in a way mindful of heavy quark symmetry @xcite . we find , therefore , that the dependence of the form factors on the heavy quark mass is well described by @xmath12 expansions , and we obtain the coefficients in the expansions . our control over the heavy quark mass dependence allows us to determine the individual form factors @xmath24 and @xmath35 , as well as the physical combination @xmath36 . the main results ( [ eq : result_h+])([eq : result_f(1 ) ] ) account for most uncertainties , but not the dependence on the lattice spacing or the effect of the quenched approximation . since our method is designed to yield the deviation of @xmath36 from one , we do not expect these qualitatively to spoil the quoted precision . with the proof of principle provided by this work , it should be possible , in the short term , to obtain @xmath36 with control over all sources of uncertainty and an error bar that is small enough to be relevant to the determination of @xmath10 . we thank zoltan ligeti and ulrich nierste for comments on the manuscript . high - performance computing was carried out on acpmaps ; we thank past and present members of fermilab s computing division for designing , building , operating , and maintaining this supercomputer , thus making this work possible . fermilab is operated by universities research association inc . , under contract with the u.s . department of energy . sh is supported in part by the grants - in - aid of the japanese ministry of education under contract no . axk is supported in part by the doe oji program under contract de - fg02 - 91er40677 and through the alfred p. sloan foundation . 99 for recent reviews , see _ the babar physics book _ , edited by p.f . harrison and h.r . quinn ( slac - r-504 ) , sec . 8 ; bigi _ et al_. @xcite . p. ball , m. beneke and v.m . braun , phys . rev . * d52 * , 3929 ( 1995 ) . i. bigi , m. shifman and n. uraltsev , annu . nucl . part . sci . * 47 * , 591 ( 1997 ) . m. neubert , phys . lett . * b338 * , 84 ( 1994 ) . m. shifman , n.g . uraltsev and a. vainshtein , phys . rev . * d51 * , 2217 ( 1995 ) . m.e . luke , phys . lett . * b252 * , 447 ( 1990 ) . c.w . bernard , y. shen and a. soni , phys . lett . * b317 * , 164 ( 1993 ) . ukqcd collaboration , k.c . bowler _ et al . _ , phys . rev . * d52 * , 5067 ( 1995 ) . t. bhattacharya and r. gupta , nucl . b ( proc . suppl . ) * 42 * , 935 ( 1995 ) ; nucl . phys . b ( proc . suppl . ) * 47 * , 481 ( 1996 ) . j.e . mandula and m.c . ogilvie , nucl . b ( proc . suppl . ) * 34 * , 480 ( 1994 ) ; ` hep - lat/9408006 ` . t. draper and c. mcneile , nucl . b ( proc . suppl . ) * 47 * , 429 ( 1996 ) . c. bernard _ et al . _ , b ( proc . suppl . ) * 63 * , 374 ( 1998 ) . j. christensen , t. draper and c. mcneile , nucl . b ( proc . suppl . ) * 63 * , 377 ( 1998 ) . s. hashimoto and h. matsufuru , phys . rev . * d54 * , 4578 ( 1996 ) . lepage and p.b . mackenzie , phys . rev . * d48 * , 2250 ( 1993 ) . b. sheikholeslami and r. wohlert , nucl . phys . * b259 * , 572 ( 1985 ) . el - khadra , a.s . kronfeld and p.b . mackenzie , phys . rev . * d55 * , 3933 ( 1997 ) . el - khadra , a.s . kronfeld , p.b . mackenzie , s.m . ryan and j.n . simone , phys . rev . * d58 * 14506 ( 1998 ) . jlqcd collaboration , s. aoki _ et al . lett . * 80 * , 5711 ( 1998 ) . cleo collaboration , m. athanas _ et al . * 79 * , 2208 ( 1997 ) . m. neubert , nucl . b371 * , 149 ( 1992 ) . a. czarnecki , phys . lett . * 76 * , 4124 ( 1996 ) ; + a. czarnecki and k. melnikov , nucl . b505 * , 65 ( 1997 ) . a.s . kronfeld and s. hashimoto , nucl . b ( proc . suppl . ) * 73 * , 387 ( 1999 ) ; and ( work in progress ) . b.p.g . mertens , a.s . kronfeld , and a.x . el - khadra , phys . rev . * d58 * , 034505 ( 1998 ) . kronfeld , nucl . b ( proc . suppl . ) * 42 * 415 ( 1995 ) . a.s . kronfeld , fermilab - conf-99/251-t , hep - lat/9909085 . see , for example , m. beneke , phys . rep . * 317 * , 1 ( 1999 ) . l. randall and m.b . wise , phys . b303 * , 135 ( 1993 ) . boyd and b. grinstein , nucl . * b451 * , 177 ( 1995 ) . falk and m. neubert , phys . rev . * d47 * , 2965 ( 1993 ) . t. mannel , phys . rev . * d50 * , 428 ( 1994 ) . m. neubert and c.t . sachrajda , nucl . phys . * b438 * , 235 ( 1995 ) . + the borel transforms of wilson coefficients presented there imply a large contribution of order @xmath196 [ z. ligeti , ( private communication ) ] . a further implication is that the brodsky - lepage - mackenzie ( blm ) scale @xcite is _ tiny_. s.j . brodsky , g.p . lepage , and p.b . mackenzie , phys . rev . * d28 * , 228 ( 1983 ) . m. neubert , phys . rev . * d46 * , 3914 ( 1992 ) . z. ligeti , y. nir and m. neubert , phys . rev . * d49 * , 1302 ( 1994 ) . m. neubert , phys . lett . * b306 * , 357 ( 1993 ) .
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a lattice qcd calculation of the @xmath0 decay form factors is presented .
we obtain the value of the form factor @xmath1 at the zero - recoil limit @xmath2 with high precision by considering a ratio of correlation functions in which the bulk of the uncertainties cancels .
the other form factor @xmath3 is calculated , for small recoil momenta , from a similar ratio . in both cases , the heavy quark mass dependence is observed through direct calculations with several combinations of initial and final heavy quark masses .
our results are @xmath4 and @xmath5 . for both
the first error is statistical , the second stems from the uncertainty in adjusting the heavy quark masses , and the last from omitted radiative corrections . combining these results
, we obtain a precise determination of the physical combination @xmath6 , where the mentioned systematic errors are added in quadrature . the dependence on lattice spacing and the effect of quenching
are not yet included , but with our method they should be a fraction of @xmath7 .
| 10,787 | 263 |
non - relativistic quantum fluids ( fermions or bosons ) constrained by periodic structures , such as layered or tubular , are found in many real or man - made physical systems . for example , we find electrons in layered structures such as cuprate high temperature superconductors or semiconductor superlattices , or in tubular structures like organo - metalic superconductors . on the experimental side , there are a lot of experiments around bosonic gases in low dimensions , such as : bec in 2d hydrogen atoms @xcite , 2d bosonic clouds of rubidium @xcite , superfluidity in 2d @xmath1he films @xcite , while for in 1d we have the confinement of sodium @xcite , to mention a few . meanwhile , for non - interacting fermions there are only a few experiments , for example , interferometry probes which have led to observe bloch oscillations @xcite . to describe the behavior of fermion and boson gases inside this symmetries , several works have been published . for a review of a boson gas in optical lattices see @xcite , and for fermions @xcite is very complete . most of this theoretical works use parabolic @xcite , sinusoidal @xcite and biparabolic @xcite potentials , with good results only in the low particle energy limit , where the tight - binding approximation is valid . although in most of the articles mentioned above the interactions between particles and the periodic constrictions are taken simultaneously in the system description , the complexity of the many - body problem leads to only an approximate solution . so that the effects of interactions and constrictions in the properties of the system , are mixed and indistinguishable . in this work we are interested in analyzing the effect of the structure on the properties of the quantum gases regardless of the effect of the interactions between the elements of the gas , which we do as precisely as the accuracy of the machines allows us to do . this paper unfolds as follows : in sec . 2 we describe our model which consists of quantum particles gas in an infinitely large box where we introduce layers of null width separated by intervals of periodicity @xmath2 . in sec . 3 we obtain the grand potential for a boson and for a fermion gas either inside a multilayer or a multitube structure . from these grand potentials we calculate the chemical potential and specific heat , which are compared with the properties of the infinite ideal gas . in sec . 4 we discuss results , and give our conclusions . we consider a system of @xmath3 non - interacting particles , either fermions or bosons , with mass @xmath4 for bosons or @xmath5 for fermions respectively , within layers or tubes of separation @xmath6 , @xmath7 = @xmath8 or @xmath9 , and width @xmath10 , which we model as periodic arrays of delta potentials either in the @xmath11-direction and free in the other two directions for planes , and two perpendicular delta potentials in the @xmath8 and @xmath9 directions and free in the @xmath11 one for tubes . the procedure used here is described in detail in refs . @xcite and @xcite for a boson gas , where we model walls in all the constrained directions using dirac comb potentials . in every case , the schrdinger equation for the particles is separable in @xmath8 , @xmath9 and @xmath11 so that the single - particle energy as a function of the momentum @xmath12 is @xmath13 . for the directions where the particles move freely we have the customary dispertion relation @xmath14 , with @xmath15 , @xmath16 , and we are assuming periodic boundary conditions in a box of size @xmath17 . meanwhile , in the constrained directions , @xmath11 for planes and @xmath18 for tubes , the energies are implicitly obtained through the transcendental equation @xcite @xmath19 with @xmath20 , and the dimensionless parameter @xmath21 represents the layer impenetrability in terms of the strength of the delta potential @xmath22 . we redefine @xmath23 , where @xmath24 is the thermal wave length of an ideal gas inside an infinite box , with @xmath25 the fermi energy and @xmath26 is the density of the gas . the energy solution of eq . ( [ kpsol ] ) for has been extensively analized in refs . @xcite and @xcite , where the allowed and forbidden energy - band structure is shown , and the importance of taking the full band spectrum has been demonstrated . every thermodynamic property may be obtained starting from the grand potential of the system under study , whose generalized form is @xcite @xmath27\bigr\ } , \label{omega}\ ] ] where @xmath28 for bosons , 1 for fermions and 0 for the classical gas , @xmath29 is the kronecker delta function and @xmath30 . the ground state contribution @xmath31 , which is representative of the bose gas , is not present when we analyze the fermi gas . for a boson gas inside multilayers we go through the algebra described in @xcite , and taking the thermodynamic limit one arrives to @xmath32\bigr ) \nonumber \\ & & -\frac{1}{\beta ^{2}}\frac{l^{3}m}{\left ( 2\pi \right)^{2}\hbar ^{2 } } { \int_{-\infty } ^{\infty } dk_{z}}g_{2 } \bigl\{\exp [ -\beta ( \varepsilon _ { k_{z}}-\mu)]\bigr\}. \label{omegaboson}\end{aligned}\ ] ] meanwhile , for a fermion gas we get @xmath33\bigr\ } , \label{omegafermion}\ ] ] where @xmath34 and @xmath35 are the bose and fermi - dirac functions @xcite . the spin degeneracy has been taken into account for the development of eq . ( [ omegafermion ] ) . on the other hand , for a multitube structure we have @xmath36 \nonumber \\ & & -\frac{l^{3}m^{1/2}}{\left ( 2\pi \right ) ^{5/2}\hbar } \frac{1}{\beta ^{3/2 } } \int_{-\infty } ^{\infty } \int_{-\infty } ^{\infty } dk_{x}\ dk_{y}g_{3/2}(e^{-\beta ( \varepsilon _ { k_{x}}+\varepsilon _ { k_{y}}-\mu ) } ) \label{tubosboson}\end{aligned}\ ] ] for a boson gas , and @xmath37\bigr\ } \label{tubosfermion}\ ] ] for a fermion gas . for calculation matters , it is useful to split the infinite integrals into an number @xmath38 of integrals running over the energy bands , taking @xmath38 as large as necessary to acquire convergence . for a gas inside a multilayer structure , the particle number @xmath3 is directly obtained from eqs . ( [ omegaboson ] ) and ( [ omegafermion ] ) . important characteristics can be extracted , such as the critical temperature for a condensating boson gas and the influence of the system parameters on it , @xmath2 and @xmath39 , already reported in refs . but for the case of a fermion gas , we focus on the chemical potential since it is closely related to the fermi energy of the system . in this case the number equation is @xmath40\ \bigr\ } , \label{numfer}\ ] ] from which we are able to numerically extract the fermi energy of the system , which corresponds to the chemical potential for @xmath41 , over the fermi energy of the ifg @xmath42 , namely @xmath43 , as a function of the impenetrability parameter @xmath39 , whose behavior corresponds to a monotonically increasing curve as @xmath39 increases , being more evident for smaller values of @xmath44 . another important feature is the chemical potential of the system over its fermi energy , @xmath45 as a function of the temperature in units of the fermi temperature @xmath46 , fig [ muvst ] , for a given value of @xmath47 . there is a special interest in fig [ muvst ] , since for certain geometrical configurations the chemical potential shows an anomalous behavior , as will be shown later . also , in this last figure one may notice that for @xmath48 the 3d ifg behavior for the chemical potential is recovered , and that the curve crosses the @xmath8 axis in @xmath49 , as has been reported in @xcite . meanwhile , as @xmath50 , we have a fermion gas inside a two dimension structure , giving a zero chemical potential at @xmath51 , as expected . for planes with @xmath52 and different values of @xmath44 . ] [ muvst ] for multitubes with @xmath53 and different values of @xmath39 . ] we make a similar procedure for the boson an fermi gases inside a multitube structure , the first one being reported in @xcite . but for a fermion gas we start from the equation @xmath54\bigr\ } \label{fermtubos}\ ] ] and extract the chemical potential over the fermi energy of the system , @xmath45 , which is probably the feature that attracts greater attention due to its anomalous behavior shown in fig . [ muvsttubos ] , which shows up as an unexpected small hump . another interesting characteristic is that the chemical potential over the ifg fermi energy in every case is lifted as @xmath39 increases due to the presence of the layers , in the same way as the chemical potential of the boson gas started above zero . the _ specific heat _ of a boson gas has been reported in ref . @xcite where we can observe a transition from a 3d system to a 2d one , which becomes evident for certain parameter values and sufficiently high temperatures . at this point is where the advantages of summing over a great amount of allowed energy bands shows its relevance . meanwhile , the specific heat for a fermion gas inside layered arrays is obtained going through the derivatives of eqs . ( [ omegaboson ] ) and ( [ tubosfermion ] ) , leading , after some algebra , to @xmath55\bigr\ } \nonumber \\ & & + 2{\int_{-\infty } ^{\infty } dk_{z } } \ln \bigl\{1+\exp [ -\beta ( \varepsilon _ { k_{z}}-\mu ) ] \bigr\ } \{2\varepsilon _ { k_{z}}-\mu + t\frac{d\mu } { dt}\ } \nonumber \\ & & + 2\beta { \int_{-\infty } ^{\infty } dk_{z}}\frac{\varepsilon _ { k_{z } } \bigl\{\varepsilon_{k_{z}}-\mu + t\frac{d\mu } { dt}\bigr\}}{\exp \bigl\{\beta ( \varepsilon_{k_{z}}-\mu ) \bigr\}+1 } \bigr\ } \label{cvfermiones}\end{aligned}\ ] ] for multiplanes , and @xmath56 for multitubes . for a fermion gas in multilayers . ] for a fermion gas in multitubes . ] in figs . [ cvlayers ] and [ cvtubos ] we show the behavior of the specific heat of layers with a separation among layers of @xmath57 and several impenetrability intensities @xmath39 , as a function of the temperature over the system s fermi temperature . it may be observed that , as the barriers disappear ( @xmath48 ) , one recovers the ifg specific heat classical value , @xmath58 . it is also noticeable that as the value of @xmath59 diminishes , a dimensional crossover signature from 3d to 2d becomes evident , since the the first minimum ( going from right to left ) in the specific heat deepens towards a value @xmath60 and broadens as @xmath39 increases . in fig . [ cvtubos ] the dimensional crossover from 3d to 1d is very noticeable as the mentioned minimum drops down to the value @xmath61 which corresponds to one - dimension . in summary , we have calculated the thermodynamic properties of ideal boson and fermion gases inside periodical structures . in our model the multilayers and multitubes are generated with dirac - comb potentials in either one or two directions , while the particles are free in the remaining directions . just by introducing the planes , the translational symmetry of the particles is broken . this fact reflects in every thermodynamic property of the constrained system . in particular , fermions in multi - tubes progress from a 3d behavior to that 2d and finally to 1d as the wall impenetrability is increased , which is observed in the curves of the specific heat as a function of temperature . there is a critical value of the wall impenetrability for which the system begins to behave in dimensions less than two , which is signaled by the appearance of an anomalous chemical potential . bosons in multitubes show similar dimensional crossover like that expressed by fermions , in addition to the bose - einstein condensation at temperatures below the critical temperature of a ideal bose - gas with the same particle density . + * acknowledgements . * we acknowledge partial support from unam - dgapa - papiit ( mxico ) # in105011 and in111613 - 3 , and conacyt 104917 .
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we report the thermodynamic properties of bose and fermi ideal gases immersed in periodic structures such as penetrable multilayers or multitubes simulated by one ( planes ) or two perpendicular ( tubes ) external dirac comb potentials , while the particles are allowed to move freely in the remaining directions .
although the bosonic chemical potential is a constant for @xmath0 , a non decreasing with temperature anomalous behavior of the fermionic chemical potential is confirmed and monitored as the tube bundle goes from 2d to 1d when the wall impenetrability overcomes a critical value . in the specific heat curves
dimensional crossovers are very noticeable at high temperatures for both gases , where the system behavior goes from 3d to 2d and latter to 1d as the wall impenetrability is increased .
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the discontinuous galerkin method @xcite nowadays is a well - established method for solving partial differential equations , especially for time - dependent problems . it has been thoroughly investigated by cockburn and shu as well as hesthaven and warburton , who summarized many of their findings in @xcite and @xcite , respectively . concerning maxwell s equations in time - domain , the dgm has been studied in particular in @xcite . the former two apply tetrahedral meshes , which provide flexibility for the generation of meshes also for complicated structures . the latter two make use of hexahedral meshes , which allow for a computationally more efficient implementation @xcite . in @xcite the authors state that the method can easily deal with meshes with hanging nodes since no inter - element continuity is required , which renders it particularly well suited for @xmath2-adaptivity . indeed , many works are concerned with @xmath0- , @xmath1- or @xmath2-adaptivity within the dg framework . the first published work of this kind is presumably @xcite , where the authors consider linear scalar hyperbolic conservation laws in two space dimensions . for a selection of other publications see @xcite and references therein . the latter three are concerned with the adaptive solution of maxwell s equations in the time - harmonic case . in this article , we are concerned with solving the maxwell equations for electromagnetic fields with arbitrary time dependence in a three - dimensional domain @xmath3 . they read [ eq : maxwell ] @xmath4 with the spatial variable @xmath5 and the temporal variable @xmath6 subject to boundary conditions specified at the domain boundary @xmath7 and initial conditions specified at time @xmath8 . the vectors of the electric field and flux density are denoted by @xmath9 and @xmath10 and the vectors of the magnetic field and flux density by @xmath11 and @xmath12 . the electric current density is denoted by @xmath13 . however , we assume the domain to be source free and free of conductive currents ( @xmath14 ) . furthermore , we assume heterogeneous , linear , isotropic , non - dispersive and time - independent materials in the constitutive relations @xmath15 the material parameters @xmath16 and @xmath17 are the magnetic permeability and dielectric permittivity . at the domain boundary , we apply either electric ( @xmath18 ) or radiation boundary conditions ( @xmath19 ) , where @xmath20 denotes the local speed of light @xmath21 . we also introduce the electromagnetic energy @xmath22 contained in a volume @xmath23 obtained by integrating the energy density @xmath24 as @xmath25 this paper focuses on a general formulation of the dgm on non - regular hexahedral meshes as well as the projection of solutions during mesh adaptation . the issues of optimality of the projections and stability of the adaptive algorithm are addressed . special emphasis is put on discussing the computational efficiency . to the best of our knowledge , this is the first publication dealing with dynamical @xmath2-meshes for the maxwell time - domain problem employing the dg method in three - dimensional space . as they are key aspects of adaptive and specifically @xmath2-adaptive methods , we will also address the issues of local error and smoothness estimation . this includes comments on the computational efficiency of the estimates . as estimators are not at the core of this article the discussion is , however , rather short . we perform a tesselation of the domain of interest @xmath26 into @xmath27 hexahedra @xmath28 such that the tesselation @xmath29 is a polyhedral approximation of @xmath26 . the tesselation is not required to be regular , however , it is assumed to be derivable from a regular root tesselation @xmath30 by means of element bisections . the number of element bisections along each cartesian coordinate , which is required to an obtain element @xmath31 of @xmath32 is referred to as the refinement levels @xmath33 . as we allow for anisotropic bisecting the refinement levels of one element may differ . in case of isotropic refinement we simply use @xmath34 . the intersection of two neighboring elements @xmath35 is called their interface , which we denote as @xmath36 . as we consider non - regular grids , every face @xmath37 of a hexahedral element may be partitioned into several interfaces depending on the number of neighbors @xmath38 such that @xmath39 . the face orientation is described by the outward pointing unitary normal @xmath40 . the union of all faces is denoted as @xmath41 , and the internal faces @xmath42 are denoted as @xmath43 . finally , the volume , area and length measures of elements , interfaces , faces and edges are referred to as @xmath44 , @xmath45 , @xmath46 and @xmath47 , where @xmath48 denotes any of the cartesian coordinates . every element of the tesselation @xmath32 is related to a master element @xmath49 ^ 3 $ ] through the mapping @xmath50 @xmath51 where @xmath52 denotes the element center . multiplying maxwell s equations ( [ eq : maxwell ] ) by a test function @xmath53 , integrating over @xmath54 and performing integration by parts yields [ eq : weakmaxwell ] @xmath55 where the explicit dependencies of @xmath56 and @xmath6 have been omitted . equations ( [ eq : weakmaxwell ] ) constitute the generic weak formulation of the time - dependent maxwell s equations . in the following , we will replace the exact field solutions @xmath9 and @xmath11 by approximations using the discontinuous galerkin framework . the space and time continuous electromagnetic field quantities are approximated on @xmath32 as @xmath57 where @xmath58 . the element - local approximation @xmath59 reads @xmath60 with the polynomial basis functions @xmath61 and the time - dependent vector of coefficients @xmath62^\text{t } , \ ] ] representing the numerical degrees of freedom . the basis functions are defined with element - wise compact support , which is an essential property of dg methods @xmath63 we define the basis functions on the master element @xmath64 and obtain the element - specific basis through the mapping @xmath50 as @xmath65 we employ cartesian grids and tensor product basis functions of the form @xmath66 where @xmath1 is a multi - index obtained from all @xmath67 . we denote by @xmath68 the local maximum approximation orders of element @xmath28 . the finite element space ( fes ) @xmath69 spanned by the basis functions is given by the tensor product of the respective one - dimensional spaces @xmath70 the approximation may , thus , make use of different orders @xmath71 in each of the coordinate directions , where we drop the subscript if they are equal . the basis functions are legendre polynomials scaled such that @xcite @xmath72 in the following the dependence of the spatial and temporal variable is not written down explicitly . if now we were to substitute the exact electromagnetic field solution @xmath73 for its approximation the surface integral term of ( [ eq : weakmaxwell ] ) can not be evaluated straightforwardly at the internal faces @xmath43 . this is due to the ambiguity of the dg approximation at any interface as a result of ( [ eq : approx ] ) and ( [ eq : basisfcts ] ) . weak continuity at internal faces is obtained locally by introducing numerical interface fluxes as @xmath74 where @xmath75 is a unique interface value computed solely from @xmath76 and @xmath77 , where @xmath78 is a neighboring element . common choices include centered and upwind fluxes . the centered interface value is given as @xmath79 computing the upwind value is more involved . it is obtained as the exact solution of maxwell s equations for piece - wise constant initial data after an infinitesimal time span , which is referred to as the riemannian problem @xcite . for the @xmath80-component of the electric and magnetic field at an interface with normal @xmath81 they read [ eq : upw ] @xmath82 with the intrinsic impedance and admittance @xmath83 other components are obtained by cycling the component indices and signs . note that centered fluxes preserve the hamiltonian structure of maxwell s equations while this property does not carry over to the semi - discrete equations when applying upwind fluxes due to the mixing of electric and magnetic quantities in ( [ eq : upw ] ) . consequently , an energy conservation property @xcite can be obtained with the centered flux formulation only , determining the kind of time integration schemes to be used as well @xcite . our implementation includes both flux types . having resolved the ambiguity at interfaces , we insert the approximations ( [ eq : approx ] ) into the weak formulation ( [ eq : weakmaxwell ] ) and follow the galerkin procedure yielding the semi - discrete dg formulation [ eq : dgmaxwell ] @xmath84 @xmath85 , @xmath86 . the volume integrals are referred to as the mass and stiffness terms , the surface integrals represent face fluxes . note that no assumptions on the grid regularity have been made in the derivation . due to the strictly element - local support of the basis and test functions , the dgm is highly suited for the application on non - regular grids . the actual difference of the refinement levels @xmath87 and @xmath88 of neighboring elements , i.e. , the level of hanging nodes , plays a minor role as shown in the following . inspecting equations ( [ eq : dgmaxwell ] ) it is seen that the mass and stiffness terms are not affected by the grid regularity as they are strictly local to the element @xmath28 . the flux term , however , involves neighboring elements as well . decomposing the surface integral into the six contributing face integrals @xmath89 and considering centered fluxes for brevity each of these can be expressed as @xmath90.\ ] ] accounting for the kind of non - regular grids described above , i.e. grids obtained from a regular root tesselation , requires no more than summing up the contributions of all neighboring elements to the total flux . this is independent of the hanging node levels as well as the actual number of neighboring elements . inserting the approximation ( [ eq : elocal ] ) into ( [ eq : faceinterfaces ] ) yields @xmath91.\ ] ] again , the first integral term does not depend on the grid regularity . assuming @xmath92 to be aligned with the @xmath93-coordinate and to point towards positive direction it amounts to @xmath94 due to the basis function scaling ( [ eq : orthogonalbasis ] ) . the second integral term can be expressed as @xmath95 in this case the orthogonality property of the basis functions is lost due to non - identical supports of @xmath96 and @xmath97 . we gather the terms ( [ eq : faceinterfaceregular ] ) and ( [ eq : faceinterfaceirregular ] ) in the interior and exterior flux matrices @xmath98 and @xmath99 . following to the usual notation the sign indicates the evaluation from the interior and exterior side of the interface . any non - regularity of the grid is now concealed within @xmath100 , which reduces to the standard form on regular grids . for high level hanging nodes the number of integrals to compute quickly becomes large , imposing a heavy computational burden if integration is performed at run time . however , as the integrals @xmath101 in ( [ eq : faceinterfaceirregular ] ) do not include the actual approximation but basis functions only , they can be precomputed analytically ( making use of the master basis functions ) and stored in tabulated form in the code . this has to be done for all combinations of @xmath102 and @xmath103 as well as for each possible edge overlap according to the respective difference in the refinement levels @xmath104 ( cf . [ fig : nonregularinterface ] ) . the number of possible overlaps grows as @xmath105 . we tabulated the integrals up to @xmath106 and for basis functions up to order six , yielding 247 matrices @xmath107 of size @xmath108 . in the isotropic refinement case @xmath106 corresponds to one element interfacing with @xmath109 neighbors . in the case of even larger differences in the refinement levels of neighboring elements , which are unlikely to occur a numerical integration is invoked at run time . if the neighboring element has a smaller instead of higher refinement level the respective transposed matrix @xmath110 is applied . for upwind fluxes , the interior and exterior flux matrices do not change , however , they are applied to both , the electric and the magnetic field due to ( [ eq : upw ] ) . -axis for a better visualization . the left hand root element has been refined several times , the right hand element is at root level . the interface i connects an element of refinement levels @xmath111 with the root level element . the tick marks indicate possible locations for the imprint of elements of these refinement levels . the actual imprint on the root element face fills the first and sixth slab along the @xmath80- and @xmath112-axis , respectively . the interface ii fills the respective second slab along the @xmath80-axis.,scaledwidth=60.0% ] in order to further enhance computational performance , all combinations of @xmath113 and the integrals @xmath114 arising form the stiffness terms of ( [ eq : dgmaxwell ] ) are evaluated and tabulated as well . precomputing the interface integrals maintains the high computational efficiency of the dg methods also for non - regular grids . using matrix notation , the semi - discrete dg maxwell equations ( [ eq : dgmaxwell ] ) read @xmath115 where * s * and * z * denotes the stiffness and impedance matrix . the matrix operator on the right hand side of ( [ eq : maxwellsemimatrix ] ) represents a weak dg curl operator . choosing @xmath116 as either zero or one yields centered or upwind fluxes , respectively . by applying centered fluxes the hamiltonian structure of maxwell s equations in continuum is preserved , whereas upwind fluxes lead to a mixed form . symplectic explicit time integration can be applied in the former case but not in the latter one @xcite . for examples of symplectic time integration for maxwell s equations in the dg framework see , e.g. , @xcite . in @xcite upwind fluxes and runge - kutta schemes are applied for the time integration , where the latter one is concerned with maxwell s equations . the adaptation techniques presented in the following are based on projections between the finite element spaces introduced in ( [ eq : dgtensorspace ] ) . the projection operators have been introduced in @xcite , however , they are included for completeness . also , we address the issue of stability in depth and amended this section with examples . the approximation @xmath117 to a given function @xmath118 in the fes @xmath69 is obtained by performing an orthogonal projection . the projection is carried out in an element - wise manner , by means of the projection operator @xmath119 given by @xmath120 where @xmath121 denotes the inner product @xmath122 on the element @xmath28 with the associated 2-norm @xmath123 . when applied successively to all elements and all components of given initial conditions of the electric field , @xmath124 , and the magnetic field , @xmath125 , the respective dg approximations @xmath126 and @xmath127 are obtained . these approximations are optimal in the sense that the projection errors @xmath128 are orthogonal to the space of basis functions @xmath69 @xmath129,\ , \varphi_i^p = \hat{\varphi}^p \circ g_i^{-1},\ , \varphi^p \in \mathcal{v}^p\ ] ] as stated above @xmath0-refinement is achieved by means of element bisections along the coordinate directions , where we allow for anisotropic refinements . the refined elements are referred to as the left and right hand side element @xmath130 and @xmath131 with basis functions denoted as @xmath132 and @xmath133 spanning the spaces @xmath134 and @xmath135 in a full analogy to @xmath69 defined in ( [ eq : dgtensorspace ] ) . the approximation orders @xmath136 and @xmath137 in each child element do not have to be identical , neither are they required to be equal to the respective order @xmath71 of the parent element . the direct sum of the spaces @xmath138 and @xmath139 is denoted by @xmath140 @xmath141 in the following , the projection can be applied in order to project an approximation given in an element @xmath28 to the fes associated with an @xmath0-refined or @xmath0-reduced element . for @xmath0-refinement this yields @xmath142 due to the tensor product character of the basis , this can be expressed as @xmath143 for the left and right child , respectively . if refinement is carried out along one coordinate only , e.g. @xmath80 , this further simplifies to @xmath144 where @xmath145 denotes the kronecker delta . note that above we loop over @xmath146 , whereas in ( [ eq : projsolutionrefine2 ] ) the loop parameter is @xmath1 . as , moreover , @xmath147 vanishes for any @xmath148 , we can limit the above loop to the range @xmath149 $ ] , which reduces the number of addends to the minimum possible . for the merging of elements , the approximation within the parent element , @xmath28 , is considered to be given piece - wise within its child elements . the projection reads @xmath150 where the simplifications ( [ eq : projsolutionrefine2 ] ) and ( [ eq : projsolutionrefine3 ] ) apply . for the case of @xmath1-enrichments , the local fes are amended with the @xmath151 order basis functions @xmath152 where any ( non - zero ) number of the local maximum approximation orders @xmath71 may be increased . also , an enrichment by more than one higher order basis function is possible . formally , we perform the orthogonal projection ( [ eq : projectionoperator ] ) , however , due to the orthogonality property of the basis functions the coefficients @xmath153 remain unaltered under a projection from @xmath154 to @xmath155 . practically , we simply extend the local vectors of coefficients @xmath156 with the new coefficients @xmath157 , which are initialized to zero . conversely , for the case of a @xmath1-reduction , the local fes is reduced by discarding the @xmath71-order basis functions @xmath158 again , by virtue of the orthogonality , we find that the coefficients @xmath159 are deleted from the local vectors of coefficients while the coefficients @xmath160 remain unaltered . we denote by @xmath161 the projection of the global approximation @xmath162 from the current discretization to another one obtained by local @xmath0- and @xmath1-adaptations . an approximation @xmath163 with coefficients according to is optimal in the sense of ( [ eq : dgapproxerror ] ) . the approximations within refined and merged elements with coefficients obtained through the orthogonal projections and are , hence , optimal in the same sense . if @xmath164 and @xmath165 holds true for all @xmath48 , the fes @xmath166 is a subspace of @xmath140 ( cf . ( [ eq : legendrespaceunion ] ) ) and every function of @xmath166 is representable in @xmath140 but not vice versa . in this case , a given approximation is exactly represented within an element under @xmath0-refinement but not under @xmath0-reduction . see fig . [ fig : adaplegendreremarks ] for an example . [ cc](a ) [ cc](b ) [ cc]@xmath167 [ cc]@xmath168 [ cc]@xmath169 [ cc]@xmath170 [ cc]@xmath171 [ lb]@xmath80 [ cb]@xmath172 and @xmath173 , the approximations of the parent and child elements agree point - wise . the projection to a merged element shown in ( b ) can , in general , not be exact due to the discontinuity . , title="fig:",scaledwidth=95.0% ] since the projections ( [ eq : projsolutionrefine ] ) for performing @xmath0-refinement are independent of the actual approximation , we also tabulated the projection operators @xmath174 and @xmath175 ( expressed in master basis functions ) yielding the matrix operators @xmath176 and @xmath177 , where the superscript denotes that the refinement level @xmath178 is increased . accordingly , we make use of the matrix operators @xmath179 and @xmath180 for evaluating the projections of ( [ eq : projsolutionreduce ] ) in the case of element merging . the matrix operators are related as @xmath181 this allows for the computation of the approximations within adapted elements by means of efficient matrix - vector multiplications . as all projection matrices are triangular the evaluation can be carried out as an in - place operation requiring no allocation of temporary memory . the global approximation associated with an adapted grid is computed as @xmath182 . it can be considered as initial conditions applied on the new discretization obtained by performing the refinement operations . assuming stability of the time stepping scheme ( cf . @xcite ) , it is sufficient to show that the application of the projection operators at some time @xmath183 does not increase the electromagnetic energy associated with the approximate dg solution , i.e. , @xmath184 in this case it follows @xmath185 and , thus , stability of the adaptive scheme . following ( [ eq : energydensity ] ) the energy associated with element @xmath28 is given as @xmath186 as a consequence of ( [ eq : projsolutionrefine2 ] ) , it is sufficient to show that the energy ( [ eq : dgenergy ] ) is non - increasing during any adaptation involving one coordinate only . [ [ sec : h - refinement-1 ] ] @xmath0-refinement + + + + + + + + + + + + + + + + + + + + + + + + + + for the following discussion of stability it is assumed that refinement is carried out along the @xmath80-coordinate . also we assume the maximum approximation orders @xmath187 and @xmath188 to be identical . it is clarified later , that this does not pose a restriction to the general validity of the results . in the case of @xmath0-refinement , the operators @xmath189 and @xmath190 project from the space @xmath166 to the larger space @xmath140 . following the argument of paragraph [ sec : optimality ] on optimality , any function defined in the space @xmath166 is exactly represented in @xmath140 . the conservation of the discrete energy is a direct consequence as the approximation in the parent and child elements are point - wise identical . we find the following relation for the 2-norms of the respective local vectors of coefficients @xmath191 the exemplary parent element approximation plotted in fig . [ fig : adaplegendreremarks]a has a maximum order of @xmath192 with all coefficients equal to one . the coefficients of the child element approximations and the square values of their 2-norms are given in tab . [ tab : adapdgcoeffsr ] . if the vector @xmath193 is considered to be either the vector of coefficients of the electric field @xmath194 or the magnetic field @xmath195 the result agrees with ( [ eq : dgenergy1drefine ] ) . .parent and child element coefficients of the function plotted in fig . [ fig : adaplegendreremarks]a [ cols="^,^,^,^,^,^,^,^,>",options="header " , ] we chose a maximum @xmath0-refinement level of two and the local element order to vary in between zero and four . the local energy density , introduced in ( [ eq : energydensity ] ) , serves as the criterion for controlling the adaptation procedure . denoting by @xmath196 the average energy density of element @xmath31 and by @xmath197 the normalized energy density with @xmath198 , we assigned the local refinement levels according to @xmath199 and polynomial orders as @xmath200 with @xmath201 . the initial discretization consisted of @xmath202 elements . during the simulation the number of elements varies and grew strongly after scattering from the reflector took place , when it reached close to 800,000 elements corresponding to slightly more than 55 million dof . for comparison , we note that employing the finest mesh resolution globally as well as fourth order approximations uniformly would lead to approximately 7.5 billion ( @xmath203 ) dof . this corresponds to a factor of approximately 130 in terms of memory savings . we emphasize that the simulations were carried out on a single machine . the implementation takes full advantage of multi - core capabilities through openmp parallelization . the numerous run - time memory allocations and deallocations are handled through a specialized memory management library based on memory blocking , which we implemented for supporting the main code @xcite . figure [ fig : eygrids ] depicts cut - views of the @xmath112-component of the electric field and the respective @xmath2-mesh at three instances in time . note that the scaling of the electric field differs for every time instance , which is necessary to allow for a visual inspection . the enlargement shows details of the computational grid . all elements are of hexahedral kind , however , we make use of the common tensor product visualization technique ( cf . @xcite ) using embedded tetrahedra for displaying the three tensor product orders ( out of which only @xmath204 and @xmath205 are visible in the depicted @xmath206-plane ) . as we employed isotropic @xmath0- as well as @xmath1-refinement all tetrahedra associated with one element share the same color . figure [ fig : signals ] shows plots of the outgoing and reflected waveform recorded along the waveguide center . the blue dashed line was obtained with the commercial cst microwave studio software @xcite on a very fine mesh and serves as a cross comparison result . -component of the electric field ( top panel ) and the computational grid ( bottom panel ) at three instances in time . the enlargement shows details of the grid . we employ hexahedral elements for the computation but make use of embedded tetrahedra for displaying the tensor product orders in the grid view . as isotropic @xmath1-refinement was employed in this examples all tetrahedra associated with one element share a common color . note that different scalings are used for the time instances in the top panel.,scaledwidth=100.0% ] we presented a discontinuous galerkin formulation for non - regular hexahedral meshes and showed that hanging nodes of high level can easily be included into the framework . in fact , any non - regularity of the grid can be included in a single term reflecting the contribution of neighboring elements to the local interface flux . we demonstrated that the method can be implemented such that it maintains its computational efficiency also on non - regular and locally refined meshes as long as the mesh is derived from a regular root tesselation by means of element bisections . this is achieved by extensive tabulations of flux and projection matrices , which are obtained through ( analytical ) precomputations of integral terms . we also presented local refinement techniques for @xmath0- and @xmath1-refinements , which are based on projections between finite element spaces . these projections were shown to guarantee minimal projection errors in the @xmath207-sense and to lead to an overall stable time - domain scheme . local error and smoothness estimates have been addressed , both of them relate to the size of the interface jumps of the dg solution . we considered the simulation of a smooth and a non - smooth waveform in a one - dimensional domain for validating the error and smoothness estimates . as an application example in three - dimensional space the backscattering of a broadband waveform from a radar reflector was considered . in this example the total wave propagation distance corresponds to approximately sixty wavelengths involving thousand of local mesh adaptations . as the implementation of the derived error and smoothness estimates for three - dimensional problems is subject of ongoing work , we chose to drive the grid adaptation using the energy density as refinement indicator . crosschecking with a result obtained using a commercial software package showed good agreement . l. fezoui , s. lanteri , s. lohrengel , s. piperno , convergence and stability of a discontinuous galerkin time - domain method for the 3d heterogeneous maxwell equations on unstructured meshes , esaim - math model num 39 ( 6 ) ( 2005 ) 11491176 . d. wirasaet , s. tanaka , e. j. kubatko , j. j. westerink , c. dawson , a performance comparison of nodal discontinuous galerkin methods on triangles and quadrilaterals , int j numer meth fluids 64 ( 10 - 12 ) ( 2010 ) 13361362 . l. krivodonova , j. xin , j. remacle , n. chevaugeon , j. flaherty , shock detection and limiting with discontinuous galerkin methods for hyperbolic conservation laws , appl numer math 48 ( 3 - 4 ) ( 2004 ) 323338 . smove , a program for the adaptive simulation of electromagnetic fields and arbitrarily shaped charged particle bunches using moving meshes , technical documentation : _ - school - ce.de / files2/schnepp / smove/_.
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a framework for performing dynamic mesh adaptation with the discontinuous galerkin method ( dgm ) is presented .
adaptations include modifications of the local mesh step size ( @xmath0-adaptation ) and the local degree of the approximating polynomials ( @xmath1-adaptation ) as well as their combination .
the computation of the approximation within locally adapted elements is based on projections between finite element spaces ( fes ) , which are shown to preserve an upper limit of the electromagnetic energy .
the formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency .
error and smoothness estimates based on interface jumps are presented and applied to the fully @xmath2-adaptive simulation of two examples in one - dimensional space .
a full wave simulation of electromagnetic scattering form a radar reflector demonstrates the applicability to large scale problems in three - dimensional space .
discontinuous galerkin method , dynamic mesh adaptation , @xmath2-adaptation , maxwell time - domain problem , large scale simulations 65m60 , 78a25
| 8,206 | 266 |
let us consider the following question : a group of people , living in the same country , but in different places , want to move in such a way that the distance among any two of them strictly decreases . unfortunately they can not all fit into one single house . hence , the constraints are : a. anyone can either move from his place to another one s ( no matter what the owner of the `` destination house '' does ) or stay where he is , b. they can not all move to the same place . is it possible for them to move ? this question is a particular case of a more interesting problem involving discrete , possibly infinite , metric spaces ( we will come back to the original problem in section [ sec : conclusions ] at the end of the paper ) . we say that a couple @xmath0 is a metric space if @xmath1 is a set and @xmath2 is real function , called _ distance _ , defined on @xmath3 such that ( 1 ) @xmath4 if and only if @xmath5 ( for all @xmath6 ) and ( 2 ) @xmath7 for all @xmath8 . we do not require the distance to be finite ; if @xmath9 we imagine that @xmath10 and @xmath11 belong to two disjoint components of the space ( see below a more detailed discussion on this , in the case of a graph ) . from ( 2 ) , with @xmath12 , and ( 1 ) we have that @xmath13 ( for all @xmath14 ) hence @xmath2 is nonnegative . moreover , again from ( 2 ) , using @xmath15 , and ( 1 ) , we have that @xmath16 ( for all @xmath6 ) , thus , by symmetry , we have @xmath17 . for some basic properties of metric spaces see for instance ( * ? ? ? * chapter 2 ) . an example of a discrete metric space is given by a ( nonoriented ) graph ( see for instance @xcite ) . roughly speaking , a graph is a ( finite or infinite ) collection of points , called _ vertices _ , along with a set of pairs of vertices , called _ edges_. we say that it is possible to move ( in one step ) from a vertex @xmath10 to a vertex @xmath11 if and only if @xmath18 is an edge . a _ path _ is a concatenation of edges , and the minimum number of steps required to go from @xmath10 to @xmath11 , say @xmath19 , is called _ natural distance _ from @xmath10 to @xmath11 . to be precise , in order to have an actual distance we have to assume that if we can move in one step from @xmath10 to @xmath11 then we can also go back in one step , that is , if @xmath18 is an edge then @xmath20 is an edge ( in this case the graph is called _ nonoriented _ ) . if we can go , in a finite number of steps , from @xmath10 to @xmath11 and back then we say that @xmath10 and @xmath11 _ communicate_. the set of vertices which communicate with @xmath10 is called the _ connected component _ containing @xmath10 ; if there is only one connected component , we say that the graph is _ connected_. of course , if two vertices belong to two disjoint connected components then the distance between them is infinite . in figure [ fig:1 ] there is an example of an infinite graph which is connected if the dashed arrow is an edge and it has two connected components otherwise ( the double arrows mean that you can move forth and back between the points , thus the graph is nonoriented ) . the central concept in this paper is the one of _ contractive map _ ( see also @xcite ) . let @xmath0 and @xmath21 be two metric spaces ; a function @xmath22 is _ contractive _ if and only if for any @xmath6 , such that @xmath23 , we have @xmath24 . the map is called a _ contraction _ if there exists @xmath25 such that @xmath26 for all @xmath6 . roughly speaking a map is contractive if it ( stricly ) reduces the distances between points ; if the ratio of reduction has an upper bound which is ( stricly ) smaller than 1 , then we have a contraction . clearly , any contraction is a contractive map and the two classes coincide on finite metric spaces . we are ( mainly ) interested in the case @xmath27 and @xmath28 . observe that in our original problem , if we denote by @xmath29 the new position of the person who was at @xmath10 before , what we are looking for are nonconstant , contractive maps . any constant map is trivially contractive ; when the converse is true ? the main result theorem [ main ] characterizes all the discrete metric spaces , satisfying a certain property ( see below ) ) , which have nonconstant contractive maps . as a simply consequence of theorem [ main ] we will see that every contractive map on a graph with the natural distance is a constant function if and only if the graph is connected . let us consider a discrete metric space @xmath0 ( which is a metric space such that for any given point @xmath30 it is possible to find @xmath31 such that @xmath32 for all @xmath33 ) and let us assume that @xmath34 ( the existence of the minimum is implicitly assumed ) . roughly speaking , according to equation , the distance either equals @xmath35 or it is at least @xmath36 ; this implies that if @xmath37 then @xmath5 . we define , once and for all , @xmath38 and we introduce the equivalence relation @xmath39 defined by @xmath40 if and only if there exists @xmath41 such that @xmath42 , @xmath43 and @xmath44 for any @xmath45 ( if @xmath46 ) . hence , two points @xmath10 and @xmath11 are equivalent if and only if we can reach @xmath11 from @xmath10 by performing a finite number of jumps of length @xmath36 . we denote by @xmath47 $ ] , the equivalence class induced by @xmath48 , that is , @xmath47:=\{y : y \sim x\}$ ] . as usual , the _ quotient space _ @xmath49 is the set of the equivalence classes . it is easy to show that , for any @xmath6 such that @xmath23 , we have @xmath50 $ ] if and only if there exists @xmath51 $ ] satisfying @xmath52 . we denote by @xmath53 the metric on @xmath49 defined by @xmath54,[y]):= \inf_{x^\prime \in [ x],y^\prime \in [ y ] } d(x , y)$ ] . examples of discrete metric spaces satisfying equation are finite metric spaces and ( nonoriented ) graphs with their natural distance ; in the last case we have @xmath55 and the quotient space @xmath49 is the set the connected components of the graph . we are ready to state and prove the main result of this paper . [ main ] let @xmath0 be a metric space satisfying equation ; then tfae a. there exists a nonconstant contractive map on @xmath1 ; b. there exists @xmath30 such that @xmath47 \not = x$ ] ; c. for every @xmath30 , we have @xmath47 \not = x$ ] ; d. @xmath56 . moreover there is a one - to - one map from the set of contractive maps @xmath57 into the set @xmath58 of contractive maps from @xmath59 to @xmath1 . @xmath60 . if @xmath61 is contractive and @xmath62 then @xmath63 which implies , by equation , @xmath64 ; hence @xmath65}$ ] , the function @xmath61 restricted to the class @xmath47 $ ] , is a constant function for any @xmath30 . hence , if @xmath61 is nonconstant we have @xmath47 \neq x$ ] . just remember that @xmath67\}_{x\in x}$ ] is a partition of @xmath1 ; hence for all @xmath68 we have either @xmath69=[x]$ ] or @xmath69 \subseteq [ x]^c \subsetneq x$ ] . we have that @xmath71 \not = \emptyset$ ] ; let us define a function @xmath61 by @xmath72 \\ y_0 & \hbox{if } x\in y , \\ \end{cases}\ ] ] where @xmath73 . it is just a matter of easy computation to show that this is a nonconstant contractive map ( indeed for any @xmath74 $ ] , @xmath75 we have @xmath76 ) . it is an easy consequence of the fact that for all @xmath78 , @xmath79 $ ] if and only if @xmath69 \neq [ x]$ ] which , in turn , is equivalent to @xmath69 \cap [ x]=\emptyset$ ] . finally , it is easy to show that , given a contractive map @xmath61 , the map @xmath80 is well defined by @xmath81):=f(x)$ ] ( since @xmath61 is constant on every class @xmath47 $ ] ) . it is straightforward to show that @xmath82 is injective and , clearly , being @xmath61 a contractive map , @xmath83 ) , \phi(f)([y]))=d(f(x),f(y))=d(f(x^\prime),f(y^\prime ) ) \le d(x^\prime , y^\prime)$ ] for all @xmath84 and @xmath85 . this implies that @xmath83 ) , \phi(f)([y ] ) ) \le d_\sim([x],[y])$ ] , thus @xmath80 is a contractive map from @xmath59 to @xmath1 . let us observe that if the range of the distance @xmath86 is a finite set ( take for instance @xmath1 finite ) then any contractive map @xmath61 is actually a contraction and there is @xmath87 such that the @xmath88-th iteration @xmath89 is a constant map for all @xmath90 and @xmath91 . indeed , @xmath92 , hence if @xmath93 then @xmath94 is constant ( say , @xmath95 for all @xmath48 ) . if @xmath96 then @xmath97 . clearly @xmath98 , that is , @xmath99 is ( the unique ) fixed point of @xmath61 . indeed , it is possible to prove ( see ( * ? ? ? * theorem 9.3 ) ) that for any contraction @xmath61 in a ( complete ) metric space there exists @xmath100 exists and it is the unique fixed point for @xmath61 , that is , @xmath101 . a generic contractive map @xmath61 has at most one fixed point , since @xmath102 , @xmath103 and @xmath104 implies @xmath105 which is a contradiction . nevertheless , if @xmath86 is not a finite set then the set of fixed points of @xmath61 might be empty as example [ ex : empty ] shows . in the language of our original problem , @xmath10 is a fixed point if and only if the member of the group living at @xmath10 does not move . moreover if @xmath106 then the three classes of contractive maps @xmath57 , contractions @xmath107 and constant functions @xmath108 coincide . [ cor : main ] let @xmath0 be a metric space such that there are just two couples satisfying equation ( namely @xmath109 and @xmath110 ) ; if @xmath111 then there exists a non constant contractive map . [ ex : empty ] consider @xmath112 and define the distance as follows @xmath113 where @xmath114 ( see figure [ fig:2 ] for a picture of a finite portion of this space along with the distances between `` consecutive '' points ) . in this case it is easy to prove that the following is a contracting map without fixed points @xmath115 in the following we compare the relations between @xmath57 , @xmath107 and @xmath108 . clearly @xmath116 ; we provide examples to show that , even if equation holds , all cases are possible . given @xmath117 , by @xmath118 we mean the usual characteristic function of @xmath119 ( which equals @xmath120 on @xmath119 and @xmath35 elsewhere ) . the reader is encouraged to verify that , in the following examples , the spaces we define are indeed metric spaces . [ ex1 ] let @xmath121 and @xmath2 be defined by @xmath122 , @xmath123 , @xmath124 and @xmath35 otherwise ; then @xmath125 is a nonconstant contraction.another example ( in the infinite case ) is the following . let @xmath112 and define the distance by @xmath126 then @xmath127 is a nonconstant contraction . in the previous examples @xmath128 since the range of the distance is finite . [ ex2 ] let @xmath129 and let @xmath2 be defined as follows @xmath130 in this case it is easy to check that this is a distance and that @xmath131 is a contractive map which is not a contraction while @xmath132 is a ( nonconstant ) contraction with @xmath133 equal to @xmath134 . hence @xmath135 . in theorem [ main ] , to prove that every contractive map is a constant map , we require that all the points belong to the same class , that is , we can reach @xmath11 from @xmath10 with a finite number of consecutive steps of length @xmath36 ( for any choice of @xmath6 , @xmath104 ) . to prove an analogous result for contractions we do not need such a strong property : we simply require that any two different points can be joined by a sequence of consecutive steps whose lengths are arbitrarily close to @xmath36 . the following result is used in example [ ex3 ] to construct a space where every contraction is a constant map but there are nonconstant contractive maps . [ pro : contrconst ] if equation holds and for all @xmath136 , @xmath137 there exists @xmath138 such that , @xmath42 , @xmath139 and @xmath140 for all @xmath141 then any contraction from @xmath1 into itself is constant . suppose that @xmath142 for all @xmath6 where @xmath143 . define @xmath144 ; it is clear that for all @xmath45 we have that @xmath145 which implies @xmath146 and @xmath147 . thus @xmath148 . [ ex3 ] let @xmath129 and let @xmath2 be defined as follows @xmath149 it is easy to check that this is a distance and that the hypotheses of proposition [ pro : contrconst ] are satisfied ( since @xmath150 , we can always take @xmath151 and @xmath152 sufficiently large ) . hence in this case any contraction is a constant function . nevertheless @xmath131 is a ( nonconstant ) contractive map . hence @xmath153 . let us consider now a nonoriented graph along with its natural distance @xmath2 . if @xmath117 , @xmath154 is the natural distance restricted to @xmath155 and @xmath156 , then condition ( ii ) of theorem [ main ] is equivalent to the existence , given any couple of vertices @xmath157 , of a finite sequence @xmath158 of vertices in @xmath119 such that @xmath159 for all @xmath160 . in particular if @xmath161 then any contractive map on @xmath119 is constant if and only if @xmath119 is a connected subgraph . by taking @xmath162 we have that there are no nonconstant , contractive maps on @xmath1 if and only if @xmath1 is a connected graph . we come back to our original question : may the group of people move according to the rules ( i ) and ( ii ) as stated in section [ sec : intro ] ? we may reasonably suppose that the group is finite and with cardinality strictly greater than @xmath163 ( if there are just @xmath163 people then any contractive map is constant and they can not move ) . hence the metric space is finite and equation is fulfilled . moreover one can assume that to different couples of places correspond different distances ( i.e. if @xmath164 and @xmath165 then @xmath166 unless @xmath5 and @xmath167 ) . under these assumptions , according to corollary [ cor : main ] , we know that the group can move . moreover , since the contractive map in this case is a contraction , there exists one ( and only one ) person who does not move at all ( this is the fixed point of the contraction ) .
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we consider discrete metric spaces and we look for nonconstant contractions . we introduce the notion of contractive map and we characterize the spaces with nonconstant contractive maps .
we provide some examples to discussion the possible relations between contractions , contractive maps and constant functions .
finally we apply the main result to the subgraphs of a nonoriented , connected graph .
.6 cm * keywords * : discrete metric space , contractive map , contraction , fixed point , nonoriented graph * ams subject classification * : 47h09 , 54e99 .
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incidence of energetic ion - beam causes the displacement and even removal of atoms near the surface . these whole processes are conventionally referred to as ion - beam - sputtering ( ibs ) . simultaneously , the healing kinetics proceeds via mass transport to minimize the surface free energy of the modified surface . those two competing processes often produce patterns of nano - dots / holes or ripples depending on the incidence angle of the ion - beam . ibs is one of the most versatile tools to fabricate nano patterns in various sizes and shapes by controlling physical variables , and applicable to a wide range of materials from metals , insulators to organic materials . ibs has , thus , drawn attention as a representative method for triggering physical self - assembly.@xcite continuum models have mostly elucidated the pattern formation by ibs . in their seminal work , bradley and harper ( bh)@xcite proposed a model of the pattern evolution ; they took the erosion into account according to sigmund picture@xcite in the linear approximation , while the diffusion of adatoms according to mullin s model.@xcite according to sigmund , the sputter yield at a position on the surface is proportional to the energy deposited at that position by an incident ion . the energy depends on gaussian ellipsoidal function of the distance from a terminal position of an incident ion to the position at the surface.@xcite according to mullin s picture , the adatom current is proportional to the gradient of the surface free energy which is in proportion to the curvature at the position of interest .@xcite bh model and its non - linear extensions@xcite that incorporate surface - confined mass flow , redeposition and surface damping well reproduce , although qualitatively yet , many features of the pattern formation by ibs . those models , though , tacitly assume amorphous surface . refined models have also been proposed taking the crystallinity@xcite of the substrate and anisotropy@xcite of the surface and sputter geometry combined into account . polycrystals consist of the grains whose mean size can be comparable with characteristic size of the structures formed by ibs.@xcite polycrystals offer unique environment for the pattern formation by ibs contrasted with the homogeneous substrates such as the amorphous and the single crystals . a challenging question is , then , whether and in the case how the grained structure of the polycrystalline substrate affects the pattern formation and its temporal evolution . recently , toma _ et al._@xcite have studied the ripple evolution of polycrystalline au films by ibs . with continued sputtering , the initially rough surface leveled off , and the pattern evolved as if it did on single crystalline au surface in regards to the surface width and ripple wavelength . this observation leads them to the conclusion that the polycrystallinity did not influence the pattern evolution . on the other hand , kere _ et al._@xcite reported that the variation of the sputter yield depending on the orientation of the grains induced the surface instability during ibs , and well reproduced the morphological evolution of polycrystalline ni films without invoking the instability owing to the curvature dependent erosion@xcite . the two conclusions on the effects of the polycrystallinity contradict to each other . highly oriented pyrolytic graphite ( hopg ) is polycrystalline and composed of the large grains , compared with the metallic films such as au@xcite and ni@xcite . hopg would , thus , offer the opportunity to study the effects of the grained structure on the pattern formation in the limit opposite to that of the metallic films . several groups have previously worked on patterning hopg by ibs.@xcite their studies are , however , little motivated by the interests in the effects of polycrystallinity on pattern formation . in this work , we study the effects of the grained structure on the pattern formation , employing two different kinds of graphites , hopg and natural graphite ( ng ) . the mean grain size of ng is distinctly larger than that of hopg . this controlled experiment clearly tells that the grain boundaries play a critical role in determining the mean uninterrupted length of the ripples along their ridges or the coherence length @xmath0 and the surface width @xmath1 , while they little influence the ripple wavelength . this highly anisotropic effects on the ripple evolution are attributed to the intricate roles of the grain boundaries in the temporal evolution of the primordial islands to the ripples during the pattern formation . the ion - beam - sputtering of both hopg ( zya grade , spi ) and ng ( donated from union carbide ) samples were performed in a high vacuum chamber whose base pressure was 5@xmath2 10@xmath3 torr . the ion - irradiated surface is characterized _ ex situ _ by atomic force microscopes ( afm ) in both the contact ( afm , psi , autoprobe cp ) and the noncontact modes ( xei-100 , park systems ) . sputtering is performed by irradiation of ar@xmath4 ion - beam with its beam diameter @xmath5 mm , incident ion energy @xmath6 , 2 kev , at a polar angle of incidence @xmath7 , 78@xmath8 from the global surface normal . the partial pressure of ar@xmath4 , @xmath9 and the ion flux @xmath10 are 1.2 @xmath2 10@xmath11 torr and 0.3 ions nm@xmath12 s@xmath13 , respectively . the sample temperature is kept around room temperature by limiting each sputter period to 1 minute with interval for 10 minutes , unless mentioned otherwise . the raman spectra of the samples were obtained by using a micro - raman spectroscopy system.@xcite the 514.5 nm ( 2.41 ev ) line of an ar ion laser was used as the excitation source and the laser power was kept below 1mw to avoid unintentional heating . the laser beam was focused - spot size @xmath14m - onto the graphite sample by a x 50 microscope objective lens ( 0.8 n.a . ) , and the scattered light was collected and collimated by the same objective . the collected raman scattered light was dispersed by a jobin - yvon triax 550 spectrometer and detected by a liquid - nitrogen - cooled charge - coupled - device detector . the spectral resolution was about 1 cm@xmath13 . the grazing incidence x - ray diffraction ( gid ) of the samples was performed with 20 kev photons ( @xmath16 = 0.62 @xmath17 ) in the 5a beam line of pohang light source in korea . the incident angle of the x - ray was kept to be 0.1@xmath8 from the global sample plane to reduce both the beam penetration depth and bulk diffuse scattering . in fig . 1(a ) , the raman spectrum of hopg before ibs shows three bands , labeled as g , g * and 2d , typical of @xmath18 hopg.@xcite after an extended sputtering with an ion fluence @xmath19 , flux times total sputter time , @xmath19 = 5625 ions nm@xmath12 , its surface is patterned by ripples as shown in the inset of fig . 1 . two new bands d and d develop as shown in fig . 1(b ) , while both g and 2d bands notably weaken as previously reported.@xcite both d and d bands originate from defective carbon atoms.@xcite the observed spectral changes tell that sputtering produces defective carbons at the cost of the @xmath18 carbons . the sample , however , largely remains crystalline ; the spectral shape of the _ amorphous _ carbon shows broad adjoined peaks of d and g bands,@xcite while the present spectrum in fig . 1 shows a well - defined d band with still intense and sharp g band . since the raman spectrum could also sample the @xmath18 layers beneath the damaged surface layers , we exfoliated the surface layers of the hopg after transferring it to silicon oxide substrate . 1(c ) is a raman spectrum of the exfoliated surface layers that should be thinner than 10 layers , because the intensity of the 2d band is now comparable with that of the g band.@xcite in fig . 1(c ) , we still find a sharp g band with negligible d band , confirming the crystallinity of the sputtered surface . et al._@xcite also observed the development of both d and d bands upon ibs by ar@xmath4 . their intensities monotonically increase with the increase of @xmath6 ( @xmath20 10 kev ) for the same @xmath19 , indicating that the ion - beam creates defects in the deeper region and/or more effectively for the larger @xmath6 . for the present sputtering condition , @xmath6 is 2 kev , small to critically damage the surface region , and leaves the sample in largely crystalline state . 2(a ) shows omega(@xmath21)-rocking profiles of ( 0,0,6 ) peak for both @xmath22 hopg and ng . the profile from the hopg reveals a broad peak , indicating wide angular distribution of the grains . in contrast , ng shows sharp peaks that originate from large crystalline grains , indicating single crystal - like character of ng . ( see also afm images in figs . 3(a ) and ( b ) . ) c|c|c|c|p22mm|p22 mm & & & & & & & & lateral & vertical hopg & 2.4600 & 6.7119 & @xmath23 & @xmath24 & @xmath25 ng & 2.4605 & 6.7082 & @xmath26 & n / a & @xmath27 table 1 summarizes the lateral and vertical correlation lengths of both the @xmath22 hopg and ng . they are obtained by williamson - hall ( wh ) plots@xcite(plots not shown ) of both peak width and peak rocking width versus the position of ( 0,0,l ) peaks . the lateral correlation length of hopg is @xmath28 350 nm , comparable with the grain size observed in fig . 3(a ) , a typical afm image of a @xmath18 hopg . the mean grain size of hopg is still quite large compared with those of metallic films by an order of magnitude.@xcite the lateral correlation length of ng can not be determined by wh plots , because the peak width becomes smaller than the resolution limit of the detector as l of ( 0,0,l ) peaks becomes small . this result indicates that the lateral correlation length of ng should be much larger than that of hopg , as also suggested by the afm images of both hopg and ng respectively in figs . 3(a ) and ( b ) . each grain of ng should behave as single crystalline graphite . 2 ( b ) and ( c ) show reciprocal space maps ( rsms ) around ( 101 ) peak of a @xmath22 hopg with @xmath19 = 5625 ions nm@xmath12 , for q@xmath29 ( reciprocal vector ) ( b ) parallel to the ridge of the ripple and ( c ) perpendicular to the ridge . the blue dashed lines in both maps are equi - q plots , originating from the angular ( rotational ) spread of the grain orientations analogous to the powder diffraction pattern . note that a feature indicated by the white line is observed in the map perpendicular to the ripple , fig . 2(c ) , but not for the map along the ripple . this is attributed to the formation of high index facets along the side wall of the ripples with the facet normal , @xmath30 from the global surface normal . such a facet formation points to the crystalline surface layers , consistently with the conclusion from the raman spectroscopy in fig.1 . figures 3(a ) and ( b ) show ripple patterns on ( a ) hopg(0001 ) and ( b ) ng(0001 ) , formed after extended ibs ( @xmath31 ions nm@xmath12 ) under the same sputter condition . the mean uninterrupted ripple lengths or the coherence lengths @xmath0s show notable difference between hopg and ng , 0.41 @xmath32 0.06 @xmath33 m for hopg in fig . 3(a ) , much shorter than that for ng , @xmath34 @xmath33 m ( not properly determined by the limited image size ) in fig . the error limit is set from the standard deviation of the measured @xmath0s that are obtained by manually measuring the uninterrupted lengths of the ripples along their ridges from the images . in addition , the surface width @xmath1 of the patterned hopg in fig . 3(a ) is @xmath3513.46 nm , notably larger than that of ng in fig . 3(b ) , @xmath35 9.6 nm for the same sputter condition . that difference in @xmath1 is much larger than the difference between the initial surface widths of hopg ( @xmath36 nm ) and ng ( @xmath37 nm ) , and should have resulted from ibs . since the conspicuous difference between the hopg and ng lies in their mean grain sizes , the grained structure of the graphite should have affected both @xmath0 and @xmath1 . in contrast , @xmath16s are almost identical , @xmath38 nm and @xmath39 nm , respectively for hopg and ng . ( taking line profiles from each image , we obtain the mean ripple wavelength @xmath16 . ) the error limit is set from the standard deviation of the ripple wavelengths taken from the line profiles . @xmath16s are smaller than the grain size , @xmath40 350 nm for hopg . still 2 out of the 7 ripples of hopg meet the grain boundary for hopg , while the portion of the ripples neighboring the grain boundaries should be much smaller for ng due to much larger grain size than that for hopg . the present observation that @xmath16 is independent of the grain size indicates that the grain boundaries little affect coarsening of the ripple or that the grain boundaries parallel to the ridge direction do not hinder the mass transport across them . similar conclusion is also drawn for polycrystalline metal films.@xcite a common feature of the ripples on both hopg and ng is that the pronounced protuberances or the buds terminate ripples on the side facing the ion - beam . ( also see the inset of fig . 1 and the following figures . ) the other carbon allotropes such as diamond and tetrahedral amorphous carbon do not show such a bud structures upon ibs.@xcite polymer(pdms ) show a labyrinthine pattern under the same sputter condition . 4(a ) ) layered structure may be a requirement for the formation of such protuberances . muscovite mica , however , shows ripple pattern with no such protuberances under the same sputtering condition . 4(b ) ) mica is another layered material , but the interlayer bonding is stronger and constituent atoms are heaver than for graphite . mica should then be less modified than graphite by the incidence of the same ion - beam . since the pronounced protuberance needs high sputter yield producing large mass transport , we tentatively conclude that the bud termination of the ripples requires layered structure with weak interlayer bonding and/or light constituents of each layer . the growth kinetics during the initial pattern formation on hopg offers insights on how the grain boundaries influence the growth of the ripples in such a highly anisotropic fashion as observed in fig . 5(a ) shows an image at an incipient stage of pattern formation on hopg with @xmath41 ions nm@xmath12 . the surface is covered by the oval - shaped islands elongated along the ion - beam direction . some of the islands have already developed tails or the incipient ripples along the ion - beam direction . the islands look distributed randomly , indicating little preference in the island formation between the grain boundaries and the terraces , which is also the case for ng . ( figure not shown ) as sputtering proceeds with @xmath42 ions nm@xmath12 , we observe that the islands have followed two different paths in their temporal evolution . 5(b ) ) one kind of islands have grown to pronounced protuberances in the shape of buds as observed in fig . note that the buds always form in the side of the ripple facing the ion - beam . their tails have also grown to form segmented ripples that are terminated by the adjacent buds along their ridges . each bud , thus , works as the birth place of the segmented ripple and simultaneously as the terminus of the adjacent ripple growing along the ion - beam direction . ripple growth via elongation of the tails has also been observed for the polycrystalline ni films.@xcite the other kind of islands become affiliated with the neighboring ripples , and barely idetifiable as nodes along their ridges.(fig . 5(b ) ) only the islands in the form of the buds remain distinguished , and the density of the islands looks significantly diminished as compared with that in fig . note that the growth of the ripple occurs via the linkage - which can occur multiply - of the ripples along the ion - beam direction , for which the nodes work as linker . due to the linkage of the neighboring ripples that do not meet along a line , the elongated ripples slightly meander throughout their length as shown in fig . 5(b ) and also in fig . 3(b ) . with the continued sputtering with @xmath43 ions nm@xmath12 , each ripple in fig . 5(c ) has grown longer , wider and shows the less modulation in both the height and width than the ripples in fig . the buds have also become further enlarged . contrastingly , the nodes are not identified any more along the ridge , indicating the the mass redistribution around each node is very efficient . with the further sputtering with @xmath44 ions nm@xmath12 , ripples are so coarse that adjacent ripples are almost in touch . as a result , the side - by - side coalescence of adjacent ripples is frequently observed as seen in the circled area in fig . such coalescence of the ripples improves the order of the ripple pattern , because ibs drives the coalescence of the adjacent ripples , leading to the alignment of the merged ripples along the ion - beam direction . et al._@xcite also observe in their kinetic monte carlo simulation that the coalescence improves the order of the ripples by eliminating defects in the pattern . note that the distribution of the buds is not so random in fig . 5(d ) as that of the incipient islands in fig . the buds often locally group or separated by region depleted of the buds , conspicuously in fig . 5(c ) and also in fig . such grouping of the islands are more and more frequently observed with the larger @xmath19 . the distance between the two neighboring groups in figs . 5(c ) and ( d ) is around 400 nm that is similar to the coherence length @xmath0 of the ripple pattern and also the mean grain size . this suggests the close correlation between the grouping of the islands and the grained structure . from the temporal evolution of the patterns in fig . 5 , the growth kinetics of the two kinds of primordial islands can be summarized as follows ; the @xmath45 kind of islands or 1 ) the buds form only on the side of each ripple facing the ion - beam . 2 ) the bud grows up as sputter proceeds . 3 ) the buds tend to locally group . 4 ) the mean distance between the neighboring buds along the ion - beam direction is similar to the mean grain size . the @xmath46 kind of islands or 1 ) the nodes reside in the middle of the ripples or each of the nodes stay inbetween two neighboring buds defining a ripple . 2 ) the nodes swiftly die out as sputter proceeds , indicating efficient diffusion processes of adspecies around them . upon those observation , we propose the following picture for the ripple growth on hopg(0001 ) ; the buds form at the grain boundaries , while the nodes on the terraces of the grains . in the early stage of sputtering , the grain boundaries are not influential on the formation of the primordial islands because the roughness of the @xmath18 surface is of the atomic scale as shown in figs . 3(a ) and ( b ) . with the continued sputtering , however , the initially embedded side walls of the grains become exposed . the side wall - formed of the edges of the graphenes - is more easily modified by the incident ion - beam than the surface of the graphite surface . in addition , the ion - beam shed the enhanced ion flux on the side walls , especially those facing the incident ion - beam due to the reduced angle from the local surface normal , and there . with all the effects in synergy , the growth of the islands becomes escalated near the grain boundaries facing the incident ion - beam , and the buds selectively decorate them . on the other hand , the diffusion of adspecies is highly efficient on the terrace due to strong intraplanar @xmath47 bond of the graphite surface.@xcite then , the islands on the terrace becomes linked to the adjacent ripple to form the nodes as seen in fig . 5(b ) that becomes swiftly transformed to be a part of the respective , now elongated ripple as in figs . 5(c ) and 5(d ) . in the long run , each ripple extends via the linkage of the ripples from a bud on a grain boundary facing the ion - beam to the other bud on the adjacent grain boundary . @xmath0 of the ripple pattern should , thus , be congruent to the mean grain size , as actually observed in the experiments . the present picture can also elucidate the experimental observation that the surface width @xmath1 of the patterned hopg is always larger than that of ng for the same sputter condition . in fig . 3 , for example , @xmath48 of the patterned hopg is @xmath49 nm , larger than that of ng , @xmath50 nm . the mean height of the nano buds from the adjacent ridge of the ripple is 9.9 @xmath32 0.7 nm for both hopg and ng , and is a major source of the surface width . since the density of the nano buds is higher for hopg due to the smaller grain size than that for ng , @xmath1 is , thus , larger for hopg than for ng . this observation assures the significant role of the grain boundaries in the morphological evolution of the ripple pattern . experimental observation of the similar @xmath16s for both hopg and ng in fig . 2 indicates that the grain boundaries do not affect the coarsening of the ripple . under the present experimental condition , the mass transport across the grain boundary , transversally to the ion - beam is not driven by thermal diffusion that should be seriously hampered by the grain boundary . instead , athermal processes such as sputter - induced 1 ) solid flow@xcite and/or 2 ) ballistic diffusion@xcite must play the major roles . such athermal diffusion could be substantial for graphite , because 1 ) the small mass of constituent of hopg or carbon compared with that of the projectile or argon leads to large momentum transfer from the incident ar ion . habenicht@xcite observed linear dependence of @xmath16 on incident ion energy , consistent with a prediction for the ion - induced solid flow for hopg@xcite recently , amorphous carbon sputtered by xe@xmath51 of 0.2 to 10 kev showed significant mass redistribution by the incident ion , which was also found the major source of the surface instability.@xcite those results support the argument of the significant contribution from the sputter - induced mass transport to the pattern formation on hopg(0001 ) . recently , kere _ et al._@xcite proposed a model for the pattern formation of polycrystalline ( ni ) films . the variation of the sputter yield depending on the orientations of the grains forming the film can well reproduce the observed pattern evolution on ni films formed under various sputter conditions without invoking the well - known instability due to curvature dependent erosion of surface by ibs.@xcite the @xmath18 hopg is largely formed of the grains differing only in relative azimuthal angle , but sharing ( 0001 ) basal plane with minute mosaic estimated from the omega - rocking width in table i. the sputter yields among grains should not significantly differ . with continued sputtering , however , the side walls at the grain boundaries in the direction facing the ion - beam becomes exposed , and have distinctly larger sputter yield than ( 0001 ) basal plane . this - dynamically driven - inhomogeneous surface with respect to the sputter yield is similar to the situation for the polycrystalline ni film , and actually leads to an instability or to the formation of the buds near the grain boundaries . this is an aspect in common with the model of kere _ et al._@xcite . away from the grain boundary or on terrace , however , the ripple formation should follow the same mechanism for both hopg and ng as expected for single crystalline surface . in the previous work with the polycrystalline au films@xcite , the large height fluctuation of au grains leads to shadowing instability . the present hopg(0001 ) is distinct from the metallic film , because the lateral grain size is much larger than the characteristic wavelength and the initial surface width is very small , less than 1 nm . due to the small surface height fluctuation , the shadowed area is negligible . for the ni film whose initial surface width is also less than 1 nm , kere _ et al._@xcite also observe the rapid reduction of the shaded area as sputter proceeds , and the negligible effects of shadowing . accordingly , the temporal evolution of hopg do not show any sign of the initial smoothening observed for the au film , but show monotonic increases of both @xmath1.@xcite instead , the grain boundaries play significant role in the determination of both @xmath0 and @xmath1 of the ripple pattern in contrast to the case of the au films . we investigated the effects of the grained structure on the pattern formation by ion - beam - sputtering ( ibs ) , employing two different graphites : highly oriented pyrolytic graphite ( hopg ) and natural graphite ( ng ) whose mean grain size is distinctly larger than that of hopg . each ripple runs from a pronounced protuberance or a bud at one grain boundary to that at the adjacent boundary . the buds originate from the side walls of the grains - formed of the edges of the graphenes , thus having high sputter yield- that become exposed at the grain boundaries facing the ion - beam by continued sputtering . since hopg is composed of the smaller grains than ng , so is the mean uninterrupted ripple length of hopg . due to the higher bud density on hopg , the surface width of the patterned hopg is larger than that of ng , well elucidating the experimental observations . on the other hand , the wavelengths on both hopg and ng are similar , indicating that the ripple coarsening across the grain boundary proceeds via athermal processes such as sputter - induced viscous flow or ballistic diffusion.@xcite in short , the grain boundary of the graphite significantly affects the morphological evolution of the ripple pattern on graphite , but in highly anisotropic fashion . m. stepanova and s. k. dew , j. phys . : condens . matter * 21 * , 224014 ( 2009 ) . m. heggie , b. r. eggen , c. p. ewels , p. leary , s. ali , g. jungnickel , r. jones , and p. r. briddon , electrochem . 98 * , 60 ( 1998 ) . i. v. lebedeva , a. a. knizhnik , a. m. popov , o. v. ershova , y. e. lozovik , and b. v. potapkin , j. chem . phys . * 134 * , 104505 ( 2011 ) .
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employing graphites having distinctly different mean grain sizes , we study the effects of polycrystallinity on the pattern formation by ion - beam - sputtering . the grains influence the growth of the ripples in highly anisotropic fashion ; both the mean uninterrupted ripple length along its ridge and the surface width depend on the mean size of the grains , which is attributed to the large sputter yield at the grain boundary compared with that on terrace .
in contrast , the ripple wavelength does not depend on the mean size of the grains .
coarsening of the ripples - accompanying the mass transport across the grain boundaries - should not be driven by thermal diffusion , rather by ion - induced processes .
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cadmium arsenide , known for decades as an inverted - gap semiconductor , has recently been shown to be a three - dimensional dirac semimetal . these materials , with a massless dirac dispersion throughout the bulk , are the 3d analogs of graphene , and cd@xmath0as@xmath1 is foremost among them : stable , high - mobility , and nearly stoichiometric . it displays giant magnetoresistance , hosts topologically nontrivial fermi - arc states on its surface , and is predicted to serve as a starting point from which to realize a weyl semimetal , quantum spin hall insulator , or axion insulator . ultrafast spectroscopy , which monitors changes in a sample s optical properties after excitation by a short laser pulse , has in many materials provided a time - resolved probe of basic carrier relaxation processes such as electron - electron and electron - phonon scattering and carrier diffusion . calculations for dirac and weyl semimetals predict that photoexcited electrons will , anomalously , cool linearly with time once their elergy drops below that of the lowest optical phonon . nothing , however , is known of cadmium arsenide s ultrafast properties . here we use the transient - grating method , which measures both the magnitude and phase of the complex change of reflectance . our measurements reveal two processes , distinct in lifetime and in phase , by which the sample s reflectance recovers after photoexcitation . analysis of the signal s phase allows us to identify changes in both the real and the imaginary parts of the index of refraction , @xmath2 . the fastest response , with a lifetime of 500 fs , is a reduction in the absorptive part , @xmath3 , which we attribute to photoexcited electrons filling of states near the excitation energy . the longer - lived response is an increase in @xmath4 and arises from the filling of states at much lower energy . these observations reveal a two - stage cooling process , which we suggest may proceed first through optical phonons , then through acoustic . we measured two samples of cd@xmath0as@xmath1 . sample 1 had well - defined crystal facets and measured a few millimeters in each dimension . it was grown by evaporation of material previously synthesized in argon flow and was annealed at room - temperature for several decades . such annealing is known to increase electron mobility and to decrease electron concentration . indeed , hall measurements on a sample of the same vintage give electron density @xmath5 @xmath6 ( roughly independent of temperature ) , metallic resistivity , and mobility @xmath7 @xmath8 at 12 k. x - ray powder diffraction gives lattice parameters in agreement with previous reports . sample 2 was grown in an argon - purged chamber by cvd in the form of a platelet ; the surface was microscopically flat and uniform . the ratio of the main cd and as peaks seen in energy - dispersive x - ray spectroscopy corresponds to cd@xmath0as@xmath1 , indicating proper stoichiometry . though its transport was not unambiguously metallic , in our experiment samples 1 and 2 behaved identically . this is consistent with the interpretation given below , that our ultrafast signal arises from the dynamics of high - energy electrons . we use the transient - grating method to measure the change , @xmath9 , in reflectance after photoexcitation . a pair of pump pulses interfere at the sample , exciting electrons and holes in a sinusoidal pattern . the sinusoidal variation in @xmath10 caused by this excitation is the `` grating . '' time - delayed probe pulses reflect and diffract off of the grating . the experimental geometry is shown in fig . we use a diffractive - optic beamsplitter to generate the pair of pump pulses . as these pulses converge on the sample , they make angles @xmath11 with the surface normal , creating a grating of wavevector @xmath12 . ( here @xmath13 is the light s wavelength . ) two probe pulses are incident on the sample at the same angles , @xmath11 . the difference in their wavevectors equals @xmath14 , so when each probe diffracts off of the grating , it is scattered to be collinear with the other probe . orders . a concave mirror ( ccm ) focuses the two probes onto the sample ( s ) , at an angle @xmath15 from the normal . diffracted beams ( dashed ) scatter through @xmath16 , so that each diffracted probe is collinear with the opposite reflected probe . pump beams ( not shown ) follow the same paths . however , pump beam paths are tipped slightly out of the page , and probe beams slightly into the page . thus the pumps are not collinear with the probes , nor are the reflected beams collinear with the incident ones . , width=2,height=0 ] this geometry allows for simple heterodyne detection of the diffracted probe : rather than provide a separate `` local oscillator '' beam , the reflected beam from one probe acts as a local oscillator for the diffracted beam from the other probe . if an incident probe has electric field @xmath17 , then the reflected and diffracted probe fields are , respectively , @xmath18 here @xmath19 is the complex reflectance , @xmath20 is the order of diffraction , and @xmath21 is a geometric phase due to the grating s spatial location . @xmath21 can not be measured , but it can be changed controllably . heterodyne detection of @xmath22 improves signal , and we suppress noise by modulation of @xmath21 and lock - in detection . the transient - grating signal is proportional to @xmath23 each measurement is repeated with the grating shifted by a quarter wavelength , giving the real and imaginary parts of @xmath24 . in the absence of measurable diffusion , as seen here , @xmath25 . the laser pulses have wavelength near 810 nm , duration 120 fs , repetition rate 80 mhz , and are focused to a spot of diameter 114 @xmath26 m . the pump pulses have fluence @xmath27 at the sample of @xmath28 @xmath26j/@xmath29 ; the probe pulses are a factor of 10 weaker . at 810 nm cd@xmath0as@xmath1 has index of refraction @xmath30 , giving @xmath31 . the absorption length is of order 45 nm and the reflectivity is 35% , so at our highest fluence each pair of pump pulses photoexcites electrons and holes at a mean density of @xmath32 @xmath6 . measurements were taken at temperatures @xmath33 k and 80 k , and one at 115 k. @xmath33 k , @xmath34 @xmath26m@xmath35 , @xmath36 @xmath26j/@xmath29 . * ( a ) , ( b ) * are @xmath37 , @xmath38 diffracted orders , respectively.,width=3,height=1 ] examples of the data obtained appear in fig . [ transients ] . all of our data fit well to the form : @xmath39 the data s three most salient features are each evident . first , the signal returns to equilibrium through two distinct decay processes , the first with @xmath40 fs and the second with @xmath41 ps . second , the two decay processes differ distinctly in complex phase . finally , as shown in fig . [ taus ] , the decays are insensitive to both @xmath14 and @xmath27 . of these observations , the complex phase will play the key role in our identification , below , of the causes of the two decay processes . and @xmath42 are roughly constant _ vs_. pump fluence . * ( c ) * and * ( d ) * : @xmath43 and @xmath44 are roughly constant _ vs_. @xmath45 . @xmath42 is consistent with diffusion coefficients from @xmath46 ( horizontal line ) to @xmath47 @xmath29/s ( sloped line).,width=3,height=1 ] in fact , the transient reflectance is even less sensitive to experimental conditions than fig . [ taus ] indicates . we varied the conditions sample , @xmath48 , @xmath27 , @xmath20 , and @xmath14to measure 32 distinct @xmath9 curves ; we saw little variation in any of the fitting parameters of eq . [ twoexp ] . the relative size of the two decay processes is constant , @xmath49 . the constant term increases from @xmath50 at 80 k to @xmath51 at 295 k , but always remains small . we attribute the @xmath52 term to lattice heating , for which we present qualitative evidence in the supplemental material . transient - grating experiments are often used to measure the diffusivity @xmath53 of photoexcited species . in the presence of diffusion , the diffracted signal @xmath24 decays faster than @xmath9 because carriers diffuse from the grating s peaks to its troughs . this effect is stronger at higher @xmath14 , because the peak - to - trough distance is shorter . however , fig . [ taus ] ( d ) shows that @xmath42 is independent of @xmath14 , consistent with @xmath46 . we caution against assigning too much weight to this negative result . the sloped line in fig . [ taus ] ( d ) shows that our data exclude only @xmath54 @xmath29/s a distinctly high upper bound . so the carriers likely do diffuse , but relax so quickly that they do not diffuse through an appreciable fraction of the grating s wavelength . the situation for @xmath55 is similar : fig . [ taus ] ( c ) . our typical measurement , of @xmath37 , is not sensitive to the multiplication of eq . [ twoexp ] by an overall phase . however , by additionally measuring @xmath56 , it is possible to determine the absolute phase of @xmath57 . we have done several such measurements on each sample ; one appears in fig . [ transients ] ( b ) . we can then calculate @xmath58 and similarly for the signal s @xmath59 and @xmath52 components . though the half - angle in eq . [ halfangle ] can take two values differing by @xmath60 , this ambiguity is easily resolved . the photoinduced change in reflectivity is @xmath61 ; we measure @xmath62 and choose the angles @xmath63 to reproduce its sign , shown in figs . [ phasefig ] ( a ) and [ phasefig ] ( b ) . , measured . * ( b ) * : @xmath62 , calculated from our mean fit parameters . the sign of each component is chosen to match the shape of the measured curve . * ( c ) * : transient change , @xmath64 , in index of refraction calculated from our mean fit parameters . imaginary part , dashed , accounts for most of the fast decay . real part , solid , accounts for most of the slow decay and the constant term.,width=2,height=1 ] we now use these angles to determine the photoinduced change in @xmath10 . the reflectance changes after photoexcitation by @xmath65\delta n(t)$ ] . for cadmium arsenide , the bracketed factor has argument @xmath66 , so @xmath67 . we obtain , finally , @xmath68 , @xmath69 , and @xmath70 . this result is surprisingly simple . the signal s faster component results from a negative @xmath71a reduction in absorption and the slower from a positive @xmath72a decrease in the light s phase velocity . the calculated @xmath73 appears in fig . [ phasefig ] ( c ) . for cd@xmath0as@xmath1 , both the real and imaginary parts of @xmath64 appear in @xmath74 , and they may be distinguished by the time - scales of their decays . the key questions in interpreting these two decay processes are what has been excited , and by what means it relaxes . our excitation energy @xmath75 ev is well beyond the region of cadmium arsenide s dirac - like dispersion , and , though optical transitions near 1.5 ev are believed to occur at the @xmath76 point , transitions are allowed between electrons and holes of several different bands . cadmium arsenide s large unit cell hosts over 200 phonon branches ; infrared and raman measurements detect a few dozen , with energies from 3.2 mev to 49 mev . ( the deficit of detected branches is attributed to a weak polarizability . ) considering the abundance of excited states and relaxation pathways available , we can not hope to identify precise processes of excitation or relaxation . nonetheless , the optical signal s phase constrains our interpretation significantly . photoexcitation changes a sample s reflectance by changing its frequency - dependent absorption coefficient . leaving aside the possibility of changes to the band structure , it does so either by occupying excited states or by changing the free carriers absorption . our experiment s probe photons have the same energy @xmath77 as those of the pump . therefore excited electrons fill phase - space effectively , reducing absorption at @xmath77 , and causing the negative @xmath71 observed in our fast decay process . this picture remains valid even as electrons scatter away from their initial excited energy @xmath78 . carrier - carrier scattering gradually creates a thermal distribution of electrons at elevated temperature . if this process is fast compared to the carriers energy loss , their mean energy remains nearly @xmath78 , and they occupy states both below and above @xmath78 . such a distribution results in @xmath79 , just as does the conceptually simpler case of phase - space filling exactly at @xmath78 . our signal s slower component has @xmath80 , which , according to the kramers - kronig relation , must result either from increased absorption at @xmath81 or from decreased absorption at @xmath82 . we can eliminate the former as the cause of our signal . if absorption increases at all , it should do so at low frequency due to enhanced free - carrier ( intraband ) conductivity ; this would cause a negative @xmath72 that we do not observe . on the other hand , there is a straightforward mechanism for decreased absorption at @xmath82 : as electrons and holes lose their excess energy , they fill phase space at progressively lower energies . kramers - kronig analysis using a simplified density of states suggests that , by the time @xmath83 becomes mostly real , the carriers mean energy should be @xmath84 or less ; our data show that cooling of this magnitude occurs within 500 fs . we attribute this cooling to phonons rather than to carrier - carrier scattering , since there are too few cool , background electrons compared to the hot , photoexcited ones ( an order of magnitude fewer for sample 1 and at our highest fluence ) . the subsequent dynamics of @xmath72 indicates that once carriers reach low energy , their relaxation slows to give @xmath85 ps . possibly cooling slows when the carriers excess energy falls below that of the lowest optical phonon , as occurs in graphene and as recently preicted for weyl and 3d dirac semimetals . however , for cd@xmath0as@xmath1 this energy is just 15 mev . other possible relaxation processes include electron - impurity scattering or electron - electron scattering with plasmon emission . however , we suggest that after the initial 500-fs cooling the carriers and optical phonons have equilibrated ; further cooling requires the slower emission of acoustic phonons . this picture fits the measured time - scale : electron - lattice cooling in bismuth , a semimetal , occurs in 5 ps . we may gain insight into the two decay processes we observe in cadmium arsenide by considering another dirac semimetal , graphene . photoexcitation of graphene initially produces electrons and holes with separate chemical potentials . within the pulse duration , these carriers partially equilibrate with optical phonons ; they then quickly occupy the dirac cone and enhance the intraband conductivity , and recombine in less than a picosecond . the chemical potential reverts to its original level , but because carriers are still hot they continue to occupy high - energy states , filling phase - space and reducing optical absorption . these hot carriers finally relax _ via _ optical , then acoustic , phonons . our measurements indicate that some of the same processes occur in cadmium arsenide , but possibly not all . we do not know whether carriers relax into the dirac cone , but the weakness of cadmium arsenide s photoluminescence suggests that many do . we also can not conclude that , as in graphene , photoexcitation produces electrons and holes with separate chemical potentials ; time - resolved photoemission and thz could more directly detect changes in carrier population and conductivity . in conclusion , we have shown that after photoexcitation cadmium arsenide relaxes in two distinct stages , irrespective of sample , fluence , and temperature . first , carriers fill phase - space at the pump energy , but relax within 500 fs to lower energy . these low - energy carriers relax further with a time - scale of 3.1 ps ; the lattice finally reaches high temperature . this result may guide further ultrafast measurements on cd@xmath0as@xmath1 and other dirac and weyl semimetals . this work was supported by the national science foundation grant no . dmr-1105553 . here we describe further our two samples of cd@xmath0as@xmath1 . fig . [ xray ] shows the x - ray powder diffraction pattern from samples of the same vintage as sample 1 . the data were fit using rietveld refinement , giving lattice parameters @xmath86 and @xmath87 with space group symmetry @xmath88 , consistent with other recent experiments . no peak corresponding to an impurity phase was detected . [ samplediagnostics ] ( a ) shows the resistivity of a sample of the same vintage . the resistivity is metallic , and at low temperature is nearly as small as that measured in samples exhibiting confirmed dirac - semimetal behavior . for a sample of type 1 . for a sample of type 2 , we show * ( b ) * the zero - bias resistance @xmath89 and * ( c ) * the differential resistance @xmath90 _ vs_. voltage @xmath91 , averaged over the temperatures measured.,width=3,height=1 ] for a sample of the same vintage as sample 2 , the current - voltage characteristics were measured by placing it across a gap formed by two indium electrodes . the dc photoconductivity at 300 k was negative , consistent with the optical heating of free carriers . the differential resistance , fig . [ samplediagnostics ] ( b ) and ( c ) , is not unambiguously metallic : it depends non - monotonically on temperature and is bias - dependent . nonetheless , in our experiment samples 1 and 2 behaved identically . though transient - grating spectroscopy has a long history , the method has advanced considerably in recent decades . here we provide additional details of the method and analysis used in this work . using the notation of eq . [ notation ] , the reflected and diffracted fields are , respectively , @xmath92 it is instructive to compare heterodyne detection to the traditional , homodyne - detected transient - grating signal , in which one simply measures the diffracted beam , @xmath93 . the homodyne signal is second - order in @xmath24 . unfortunately , the photoinduced changes in a sample s optical response@xmath24 and @xmath9are typically quite small . one advantage of heterodyne detection is a large increase in signal , for it has several terms of first or zeroth order : latexmath:[\[\begin{aligned } indeed , the second - order terms are negligible , giving latexmath:[\[\begin{aligned } the term of interest is the one proportional to @xmath24 ; it is also the only one depending on @xmath21 . to isolate this term , we use the coverslip discussed below to modulate @xmath21 at 95 hz , and filter our signal through a lock - in amplifier . this procedure acts as a derivative @xmath95 , giving a signal proportional to @xmath96 equivalent to eq . [ eq2 ] . key to the heterodyne detection of the transient grating is the ability to control and modulate @xmath21 , by controlling the grating s spatial position . we do this by passing one of the incident pump beams obliquely through a thin , glass coverslip . fine adjustments of the coverslip s angle change the beam s path - length , adding or subtracting phase relative to the other pump beam . the coverslip is mounted on both a torsional oscillator and a stepping rotation stage . the former allows us to modulate the coverslip s angle rapidly and sinusoidally , for lock - in detection ; the latter allows us to change the angle in calibrated increments . below , when we discuss measurement `` at '' a particular coverslip position , we mean the coverslip s _ central _ position , about which it oscillates at 95 hz . to maintain the spatial and temporal overlap of the beams that converge on the sample , we introduce three similar coverslips into the paths of both probe beams and of the other pump . their orientations are fixed , but are similar to that of the modulated coverslip . to obtain data such as that shown in fig . [ transients ] ( a ) , we set the coverslip to a position corresponding to an arbitrary , unknown @xmath21 , and measure using the @xmath37 diffracted probe . we then shift to @xmath97 and measure again . we obtain @xmath98 and @xmath99 and define these , respectively , as the real and imaginary parts of our signal : re@xmath100 and im@xmath100 . fitting to the form : @xmath101 defines the set of angles @xmath102 , @xmath103 , @xmath104 . the superscript indicates that @xmath37 . hereafter we consider just one of the signal s @xmath105 , @xmath59 , and @xmath52 components ; the same equations apply to each . comparison of eqs . [ real ] , [ imaginary ] , and [ twoexp ] shows that @xmath106 because re@xmath107 and im@xmath108 . up to this point , @xmath109 is arbitrary , because @xmath21 is arbitrary . we next describe how measurement of the @xmath56 diffracted order allows us to eliminate @xmath21 and to determine @xmath63 . our measurements of the @xmath56 diffracted order are done at the same coverslip positions as for the @xmath110 order . these correspond to grating phases of @xmath21 and @xmath111 . our transient - grating signals are , respectively , @xmath112 and @xmath113 from which @xmath114 we can then calculate @xmath115 equivalent to eq . [ halfangle ] . above , we state that @xmath117 because the diffusion is negligible . here we clarify the reasoning , which may otherwise appear circular . @xmath9 must equal the @xmath116 limit of @xmath24 . we observe that , within experimental error , @xmath24 is the same at all @xmath14 , including at some rather low @xmath14 . therefore our measured @xmath24 does in fact equal @xmath9 . in other words , we never assume that the diffusion is negligible ; we observe it . nm and @xmath47 @xmath29/s . if @xmath53 is lower , the density will be even more nearly constant.),width=2,height=1 ] because carriers are photoexcited within an absorption length @xmath118 of just 45 nm , they will diffuse away from the sample s surface . as their density at the surface drops , @xmath64 will decrease . surprisingly , however , this effect has little influence on the time constants @xmath55 and @xmath42 measured in this experiment . the carriers initial exponential distribution quickly evolves to be nearly gaussian , and diffusion broadens a gaussian s width only as @xmath119 . ( see , for instance , sheu _ et al . _ , particularly eq . 3 and fig . 5a . ) fig . [ inward ] makes this argument more quantitative . we used the diffusion equation with no relaxation term_i.e . _ for conserved particle number to model carriers diffusion away from the sample s surface , and we plot the density at the surface as a function of time . the largest drop in density occurs at very early times , before any of the data shown in fig . [ transients ] . the drop at later times is not nearly enough to account for the experimentally observed decays . for this reason we conclude that our relaxation rates are little influenced by inward diffusion . above , we measure the photoinduced change of reflectance , @xmath9 , and argue that it is related to the cooling of optically heated electrons . here we describe qualitative evidence for the electrons high temperature . when the laser was incident on the sample , we saw that the illuminated spot glowed with broadband visible light ; it looked like incandescence , of a reddish hue . some locations on the sample surface glowed more than others ; however , we excluded surface contamination as a cause of the light emission by visual examination of the sample and by cleaning with acetone and methanol . et al_. measured a similar effect in graphene , caused by thermal emission by electrons heated to several thousand kelvin . these electrons were partially equilibrated with the optical phonons . after equilibration with all phonon modes , the lattice temperature was estimated to be around 700 k. the more complex band structure of cd@xmath0as@xmath1 precludes the quantitative analysis of lui _ et al_. , but we expect that emission from our sample is caused by similarly heated electrons . above , we attribute our signal s small , constant component @xmath52 to lattice heating . it is unremarkable that energy deposited by the laser should eventually find its way to the lattice . however , given the modest optical power used in our experiment tens of milliwats for the pump beams this heating maifests itself in a surprising way that may serve to caution future experimenters . we observed that the direction of the probe beam s specular reflection from the sample s surface could vary by about @xmath120 , depending on whether the more powerful pump beam was incident on the sample or blocked . this change was reproducible over dozens of cycles , and occurred with a time constant of several seconds . reflection remained specular , but the orientation of the sample s surface evidently shifted . after many cycles , sample 1 s surface showed small cracks . we explain this observation as follows . the thermal conductivity of cadmium arsenide is low , of order 1 w / k - m , leading to large temperature gradients . the material suffers several structural phase transitions at elevated temperature , the lowest at 503 k ; combined with temperature gradients , these could create strains that move the sample s surface slightly . note that a transient - grating signal can not be measured when the reflected beam is shifting . we were able to obtain data because samples neither shifted nor glowed when exposed to atmosphere , perhaps due to convective cooling . for low - temperature measurement under vacuum , sample 2 glowed and shifted only rarely , evidently depending on which part of the sample was illuminated . the sample was never measured while incandescing ; nonetheless , it is likely that even when the sample was cooled to 80 k , the measured spot was much hotter . both a second platelet - like sample and sample 1 glowed and shifted more consistently and could not be measured under vacuum . these observations suggest the value of thin - film samples or of laser systems with lower average power . 10 [ 2]#2 [ 1]#1 [ 1]#1 , , , , and , `` , '' ( 12 ) , ( ) . , , , , , , , , , , , , , `` , '' ( ) . , , , , , , , , , , , , , , , , , , `` , '' ( 7 ) , ( ) . , , , , , and , `` , '' ( 2 ) , ( ) . , , , , , and , ( ) . , , , , , and , , ( ) . , , , , , , , , , , , , , , , , , , , , , , , `` , '' , , and , , , ( ) . , , , and , `` , '' ( 20 ) , ( ) r. lundgren and g. a. fiete , `` electronic cooling in weyl and dirac semimetals . '' arxiv:1502.07700 ( 2015 ) . , `` , '' ( 2 - 3 ) , ( ) . and , `` , '' , ( ) . see appendices for the samples transport properties , additional description of the transient grating , and a discussion of optical heating of the samples . , , and , `` ultrafast heterodyne - detected transient- grating spectroscopy using diffractive optics '' ( 6 ) ( ) . , , and , `` , '' ( 16 ) ( ) . and , `` , '' ( 18 ) , ( ) . , , and , `` , '' ( 3 ) , ( ) . errors and error - bars are standard deviation of the mean . the lack of measurable diffusion is perhaps to be expected . photoexcited electrons and holes must move together to preserve local charge neutrality , resulting in ambipolar diffusion . in an @xmath10-type sample , the ambipolar diffusivity nearly equals the holes diffusivity , which is likely much less than the electrons. and , `` , '' , ( ) . , , and , `` , '' , 2 , ( ) . , , , and , `` , '' ( 2 ) , ( ) . , , , and , `` , '' ( 4 ) , ( ) . note that electronic energy of @xmath78 corresponds to optical transitions at @xmath77 . and , `` , '' ( 20 ) , ( ) . , , , , , and , `` , '' ( 11 ) , ( ) . , , , , and , `` free - carrier relaxation and lattice heating in photoexcited bismuth '' , ( ) . , , , , , , , and , `` , '' ( 1 ) , ( ) . , , , and , `` , '' ( 12 ) , ( ) . , , , , , , , , , and , `` , '' ( 12 ) , ( ) . , , , , , , , and , `` , '' ( 15 ) , ( ) . , , and , `` , '' , ( ) . , , , , , and , `` the crystal and electronic structures of cd@xmath0as@xmath1 , the three - dimensional analogue of graphene '' , ( ) . , , , , , , and , `` quantum transport evidence for a three - dimensional dirac semimetal phase in cd@xmath0as@xmath1 '' , ( ) . , , and , ( , ) . and , `` transient grating optical heterodyne detected impulsive stimulated raman - scattering in simple liquids '' ( 9 ) , ( ) . , , and , `` optical heterodyne - detection of impulsive stimulated raman - scattering in liquids '' ( 20 ) ( ) . , , and , `` how to make femtosecond pulses overlap '' ( 17 ) ( ) . , , , , , and , `` observation of spin coulomb drag in a two - dimensional electron gas '' ( 7063 ) , ( ) . the characteristic time - scale of the diffusive process is @xmath121 . in this experiment @xmath122 ps . indeed , this is the _ only _ time - scale in the problem , so that the @xmath123-axis in fig . [ inward ] may we re - scaled by the factor @xmath124 to adapt it to other experiments . and , `` , '' ( 11 ) , ( ) . and , `` , '' ( 1 ) , ( ) .
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we report ultrafast transient - grating measurements of crystals of the three - dimensional dirac semimetal cadmium arsenide , cd@xmath0as@xmath1 , at both room temperature and 80 k. after photoexcitation with 1.5-ev photons , charge - carriers relax by two processes , one of duration 500 fs and the other of duration 3.1 ps . by measuring the complex phase of the change in reflectance
, we determine that the faster signal corresponds to a decrease in absorption , and the slower signal to a decrease in the light s phase velocity , at the probe energy .
we attribute these signals to electrons filling of phase space , first near the photon energy and later at lower energy .
we attribute their decay to cooling by rapid emission of optical phonons , then slower emission of acoustic phonons .
we also present evidence that both the electrons and the lattice are strongly heated .
+ + the following article appeared in _ applied physics letters _ and may be found at + http://scitation.aip.org/content/aip/journal/apl/106/23/10.1063/1.4922528 + ( this version of the article differs slightly from the published one . ) + + _ copyright 2015 american institute of physics .
this article may be downloaded for personal use only .
any other use requires prior permission of the author and the american institute of physics . _
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critical properties of structurally disordered magnets remain a problem of great interest in condensed matter physics , as far as real magnetic crystals are usually non - ideal . commonly , in the theoretical studies , as well as in the mc simulations , one considers point - like uncorrelated quenched non - magnetic impurities @xcite . however , in real magnets one encounters non - idealities of structure , which can not be modeled by simple point - like uncorrelated defects . indeed , magnetic crystals often contain defects of a more complex structure : linear dislocations , planar grain boundaries , three - dimensional cavities or regions of different phases , embedded in the matrix of the original crystal , as well as various complexes ( clusters ) of point - like non - magnetic impurities @xcite . therefore , a challenge is to offer a consistent description of the critical phenomena influenced by the presence of such complicated defects . different models of structural disorder have arisen as an attempt to describe such defects . in this paper we concentrate on the so - called long - range - correlated disorder when the point - like defects are correlated and the resulting critical behaviour depends on the type of this correlation . several models have been proposed for description of such a dependence @xcite , a subject of extensive analytical and numerical @xcite treatment . a common outcome of the above studies is that although the concentration of non - magnetic impurities is taken to be far from the percolation threshold , in the region of weak dilution , the impurities make a crucial influence on an onset of ordered ferromagnetic phase . given that the pure ( undiluted ) magnet possesses a second - order phase transition at certain critical temperature @xmath0 , an influence of the weak dilution may range from the decrease of @xmath0 to the changes in the universality class and even to the smearing off this transition @xcite . moreover , the critical exponents governing power low scaling in the vicinity of @xmath0 may depend on the parameters of impurity - impurity correlation . to give an example , the harris criterion , which holds for the energy - coupled uncorrelated disorder @xcite is modified when the disorder is long - range correlated @xcite . in particular , when the impurity - impurity pair correlation function @xmath1 decays at large distances @xmath2 according to a power law : @xmath3 the asymptotic critical exponents governing magnetic phase transition ( and hence the universality class of the transition ) do change if @xcite @xmath4 where @xmath5 is the correlation length critical exponent of the undiluted magnet . the above condition ( [ 2 ] ) holds for @xmath6 , @xmath7 being the space ( lattice ) dimension . for @xmath8 the usual harris criterion @xcite is recovered and condition ( [ 2 ] ) is substituted by @xmath9 . the fact , that the power of the correlation decay might be a relevant parameter at @xmath6 can be easily understood observing an asymptotics of the fourier transform @xmath10 of @xmath1 at small wave vector numbers @xmath11 . from ( [ 1 ] ) one arrives at @xmath12 , which for @xmath6 leads to a singular behaviour at @xmath13 . as far as the small @xmath11 region defines the criticality , the systems with @xmath6 are good candidates to manifest changes in the critical behaviour with respect to their undiluted counterparts . on contrary , impurity - impurity correlations at @xmath8 do not produce additional singularities with respect to the uncorrelated point - like impurities , therefore they are referred to as the short - range correlated . in turn , the disorder characterized by eq . ( [ 1 ] ) with @xmath6 is called the long - range correlated . there are different ways to model systems with the long - range - correlated disorder governed by eq . ( [ 1 ] ) . the most direct interpretation relies on the observation that the integer @xmath14 in eq . ( [ 1 ] ) corresponds to the large @xmath2 behaviour of the pair correlation function for the impurities in the form of points ( @xmath15 ) , lines ( @xmath16 ) , and planes ( @xmath17 ) @xcite . since the last two objects extend in space , the impurities with @xmath6 sometimes are called the extended ones . note that the isotropic form of the pair correlation function ( [ 1 ] ) demands random orientation of such spatially extended objects . non - integer @xmath14 sometimes are treated in terms of a fractal dimension of impurities , see e.g. @xcite . besides energy - coupled disorder , the power - low correlation decay ( [ 1 ] ) is relevant for the thermodynamic phase transition in random field systems @xcite , percolation @xcite , scaling of polymer macromolecules at presence of porous medium @xcite , zero - temperature quantum phase transitions @xcite . our paper was stimulated by the observations of obvious discrepancies in the state - of - the - art analysis of criticality in three - dimensional ising magnets with the long - range - correlated disorder governed by eq . ( [ 1 ] ) . indeed , since for the pure @xmath18 ising model @xmath19 @xcite , the long - range correlated disorder should change its universality class according to eq . ( [ 2 ] ) . whereas both theoretical and numerical studies agree on the validity of extended harris criterion ( [ 2 ] ) and bring about the new universality class @xcite , the numerical values of the critical exponents being evaluated differ essentially . we list the values of the exponents found so far by different approaches in table [ tab1 ] and refer the reader to the section [ ii ] for a more detailed discussion of this issue . here , we would like to point out that presently the results of each of existing analytical approaches ( refs . @xcite and @xcite ) is confirmed by only one numerical simulation ( refs . @xcite and @xcite , respectively ) . to resolve such a bias , we perform mc simulations of a @xmath18 ising model with extended impurities and evaluate critical exponents governing ferromagnetic phase transition . as it will become evident from the further account , our estimates for the exponents differ from the results of two numerical simulations performed so far @xcite and are in favour of a non - trivial dependency of the critical exponents on the peculiarities of long - range correlations decay . lllll reference & @xmath20 & @xmath21 & @xmath22 & @xmath23 + + weinrib , halperin , @xcite & 1 & 2 & 1/2 & 0 + prudnikov _ et al . _ , @xcite & 0.7151 & _ 1.4449 _ & _ 0.3502 _ & -0.0205 + ballesteros , parisi , @xcite@xmath24 & 1.012(10 ) & _ 1.980(16 ) _ & _ 0.528(7 ) _ & 0.043(4 ) + @xcite@xmath25 & 1.005(14 ) & _ 1.967(23)_&_0.524(9 ) _ & 0.043(4 ) + prudnikov _ et al . _ , @xcite@xmath24 & 0.719(22 ) & _ 1.407(24 ) _ & 0.375(45 ) & _ 0.043(93 ) _ + @xcite@xmath25 & 0.710(10 ) & 1.441(15)&0.362(20 ) & _ -0.030(7 ) _ + the outline of the paper is the following . in the next section we make a brief overview of the results of previous analysis of the 3d ising model with long - range - correlated impurities paying special attention to the former mc simulations . details of our mc simulations are explained in sections [ iii ] and [ iv ] . there , we formulate the model and define the observables we are interested in . we analyze statistics of typical and rare events taking the magnetic susceptibility as an example in section [ iv ] . the numerical values of the exponents are evaluated there by the finite - size scaling technique . section [ v ] concludes our study . currently , there exist two different analytical results for the values of critical exponents of the 3d ising model with long - range - correlated impurities . the first one is due to weinrib and halperin , who formulated the model of a @xmath26-vector magnet with quenched impurities correlated via the power law ( [ 1 ] ) @xcite . they used the renormalzation group technique carrying out a double expansion in @xmath27 , @xmath28 , considering @xmath29 and @xmath30 to be of the same order of magnitude , and estimating values of the critical exponents to the first order in this expansion . further , they conjectured the obtained first order result for the correlation length critical exponent @xmath31 to be an exact one and to hold for any value of spin component number @xmath26 provided that @xmath32 . by complementing eq . ( [ 3 ] ) with the first order value of the pair correlation function critical exponent , @xmath33 @xcite , the other critical exponents can be obtained via familiar scaling relations . for @xmath34 the exponents are listed in table [ tab1 ] . the second theoretical estimate was obtained by prudnikov _ @xcite in the field theoretical renormalization group technique by performing renormalization for non - zero mass at fixed space dimension @xmath18 . their two - loop calculations refined by the resummation of the series obtained bring about a non - trivial dependence of the critical exponents both on @xmath26 and on @xmath14 . we list their values of the exponents for @xmath35 and @xmath34 in the second row of table [ tab1 ] . in particular , one observes that the correlation length exponent @xmath20 obtained in ref . @xcite differs from the value predicted by ( [ 3 ] ) by the order of 25 % . as for the reason of such discrepancy the authors of ref . @xcite point to a higher order of the perturbation theory they considered together with the methods of series summation as well as to taking into account the graphs which are discarded when the @xmath36-expansion is being used . let us note however , the qualitative agreement between the above analytical results : both renormalization group treatments , refs . @xcite and @xcite , predict that the new ( long - range - correlated ) fixed point is stable and reachable for the condition considered and hence the 3d ising model with long - range - correlated impurities belongs to a new universality class . these are only the numerical values of the exponents which call for additional verification . in such cases one often appeals to the numerical simulations . indeed , two simulations were performed , and , strangely enough , the results again split into two groups : whereas the simulation of ballesteros and parisi @xcite is strongly in favour of the theoretical results of weinrib and halperin , the simulation of prudnikov _ et al . _ @xcite almost exactly reproduces former theoretical results of this group @xcite as one can see from table 1 . there , we give the results for the exponents obtained in simulations complementing them by those that follow if one uses familiar scaling relations ( the last are shown in italic ) . since our own study will rely on the simulation technique as well , we describe an analysis performed in refs . @xcite in more details . in ref . @xcite the simulations were done for two different types of the long - range - correlated disorder referred by the authors as the gaussian and the non - gaussian one . in the first case , the point - like defects are scattered directly on the sites of a 3d simple cubic lattice according to the desired decay of the pair correlation function ( [ 1 ] ) , @xmath34 . in the second case , the defects form the lines directed along the randomly chosen axes and , as was mentioned before , their impurity - impurity pair correlation function should also decay at large @xmath2 as @xmath37 ) . the mc algorithm used in ref . @xcite is a combination of a single - cluster wolff method and a swendsen - wang algorithm . after a fixed number of a single - cluster updates one swendsen - wang sweep is performed . the procedure was called a mc step ( mcs ) . the number of single cluster flips was chosen such that the autocorrelation time @xmath38 was typically 1 mcs . for thermalization of the system @xmath39 mcss were performed and then various observables were measured . the simulation was done on the lattice of sizes @xmath40 and @xmath41 . then @xmath39 mcss have been used for measurements on @xmath42=20000 different samples for @xmath43 and on @xmath42=10000 samples for @xmath44 . the procedure seems to be quite safe . values of the critical exponents @xmath20 and @xmath23 obtained in ref . @xcite in the case of gaussian disorder ( point - like long - range - correlated defects ) are given in table [ tab1 ] for the impurities concentrations @xmath45 , @xmath46 . it was stated in the paper , that for the non - gaussian disorder ( lines of defects ) the estimates of the exponents are comparable with analytic calculations of weinrib and halperin @xcite . in ref . @xcite the mc simulation of the critical behavior of the three - dimensional ising model with long - range - correlated disorder at criticality was performed by means of the short - time dynamics and the single - cluster wolff method . randomly distributed defects had a form of lines and resembled the `` non - gaussian disorder '' of ref . however , in contrast to ref . @xcite , a condition of lines mutual avoidance was implemented . according to ref . @xcite , a situation when crossing of the lines of defects is allowed is not described by the weinrib - halperin model . in the short - time critical dynamics method , the concentration of spins was chosen @xmath45 and the cubic lattices of sizes @xmath47 were considered . as a mc method was used a metropolis algorithm . resulting numbers for the exponents are given in the 5th row of table [ tab1 ] . additional simulations in the equilibrium state were performed in ref . @xcite to verify the reliability of results obtained by means of the short - time critical dynamics . the single - cluster wolff algorithm was used for simulation and a finite size scaling for evaluation of the critical exponents . one mcs was defined as 5 cluster flips , @xmath48 mcss were discarded for equilibration and @xmath49 for measurement . disorder averaging was typically performed over @xmath48 samples . again , the procedure seems to be safe . the values of critical exponents obtained in the simulations are quoted in the 6th row of table [ tab1 ] . with the above information at hand we started our numerical analysis as explained in the forthcoming sections . we consider a @xmath18 ising model with non - magnetic sites arranged in a form of randomly oriented lines ( see fig . [ fig_1 ] ) . the hamiltonian reads : @xmath50 where @xmath51 means the nearest neighbour summation over the sites of a s.c . lattice of linear size @xmath52 , @xmath53 is the interaction constant , ising spins @xmath54 , @xmath55 is the occupation number for the @xmath56-th site and periodic boundary conditions are employed . non - magnetic sites ( @xmath57 ) are quenched in a fixed configuration and form lines , as shown in fig . [ fig_1 ] . concentration @xmath58 of the magnetic sites is taken to be far above the percolation threshold . [ fig_1 ] shows one possible configuration for non - magnetic impurity lines . during the simulations , one generates a large number of such disorder realizations this is done in our study using the following procedure . the lines of impurities are generated one by one , each along randomly chosen axis and the final disorder realization is accepted with the probability @xmath59 using rejection method . here @xmath60 is a required value for the concentration of magnetic sites and @xmath61 is the dispersion of the resulted gaussian distribution in @xmath58 ( see fig . [ hist ] ) . the average value for the impurity concentration is equal to @xmath62 . in the simulations presented here we paid special attention to the width of this distribution and consider both broad and narrow cases , see sections [ ivb ] and [ ivc ] ) . the observables saved during each step of the mc simulations for a given disorder realization will be the instantaneous values for the internal energy @xmath63 and magnetisation @xmath64 per spin , defined as @xmath65 @xmath66 where number of magnetic sites is @xmath67 , total number of sites is @xmath68 . using the histogram reweighting technique @xcite we compute the following expectation values at the critical temperature region for a given disorder realization : @xmath69 as functions of an inverse temperature @xmath70 . in ( [ 99 ] ) the angular brackets @xmath71 stand for the thermodynamical averaging in one sample of disorder . for a given disorder configuration , one can evaluate the temperature behaviour of the magnetic susceptibility @xmath72 and magnetic cumulants @xmath73 , @xmath74 : @xmath75 in order to refer to the physical quantities , the observables are to be averaged over different disorder configurations , denoted hereafter by an overline : @xmath76 . two ways of averaging can be found in the literature , which we will consider for the case of finding the maximum value for the susceptibility . at each disorder realization an individual curve for the susceptibility @xmath72 as a function of the temperature @xmath77 is obtained ( using histogram reweighting technique ) . in the first method of averaging , depicted in fig . [ fig_11],a the maximal values of each individual curves are averaged over all disorder realizations . this type of averaging , referred hereafter as _ averaging a _ , will be denoted as @xmath78 . \a ) _ averaging a _ b ) _ averaging b _ an alternative method is shown in fig . [ fig_11],b : at first one performs a configurational averaging of the susceptibility , i.e. the single averaged curve is evaluated for @xmath79 . then the maximal value for this curve , @xmath80 , is found . hereafter , this method will be referred as _ averaging b_. both _ averaging a _ and _ averaging b _ will be exploited in our analysis . to evaluate the critical exponents , a number of characteristics will be used with known finite size scaling : * _ averaging a _ : @xmath81 here , @xmath78 , @xmath82 , and @xmath83 are the averaged over disorder configurations maximal values of the magnetic susceptibility and magnetic cumulants temperature derivatives . the value @xmath84 is calculated for the temperature that corresponds to the magnetic susceptibility maximum . * _ averaging b _ : + @xmath85 + @xmath86 + where @xmath87 note , that quantities ( [ 111a ] ) are ill - defined if _ averaging a _ is considered . moreover , as one can see from the definitions ( [ 22a ] ) , ( [ 22b ] ) there are different ways to define magnetic cumulants for the disordered system ( see refs . @xcite for more discussion ) . we performed our numerical simulations using the swendsen - wang cluster algorithm . the main reason is , that with the diluted system one often performs simulations at the temperature which is far from the native phase transition point of each particular disorder realization . as the result , the single clusters can be of rather small size and the wolff one algorithm , used by us before @xcite is found to be less effective . the following set of lattice sizes @xmath88 ( @xmath89 in some cases ) is used and performing the finite - size - scaling analysis is performed . the magnetic site concentration is chosen to be @xmath45 which is far from the percolation threshold and this also allows comparison with the previous simulations @xcite . another reason for such a choice is that for the 3d ising model with uncorrelated impurities the correction to scaling terms were found to be minimal at @xmath45 @xcite , ( obviously , this does not mean that @xmath45 minimizes correction - to - scaling terms at any level of impurities correlation ) . the number of samples ranges from @xmath90 to @xmath48 for all lattice sizes . the simulations are performed at the finite - size critical temperature @xmath91 which is found from the maximum of magnetic susceptibility @xmath72 at each lattice size @xmath52 and the reweighting technique for the neighbouring temperatures is used . note , that in refs . @xcite all simulation were performed at the critical temperature of an infinite system . similarly to our other previous papers @xcite we start simulation from estimating the critical temperature of a finite - size system as a function of the lattice size @xmath52 , @xmath91 . for the smallest lattice size @xmath92 , the preliminary simulation is performed at @xmath93 first ( @xmath94 is the critical temperature for the pure system ) . the @xmath95 is located then from the maximum of the susceptibility . at the same time , we estimate various autocorrelation times to control the error due to time correlations . for the next larger lattice size , e.g. @xmath96 , preliminary simulation is performed at @xmath95 and then @xmath97 is located again from the maximum of the susceptibility at @xmath96 . the process is repeated for all lattice sizes . in this way we estimate @xmath91 and the energy autocorrelation time @xmath98 . the results are given in table [ tab2 ] . one can see that the finite size system critical temperature obtained via _ averaging a _ differs from those obtained via and _ averaging b_. the difference is not too large , the next simulations were performed at the critical temperature obtained by _ averaging a_. in evaluation of the critical indices we followed the standard finite - size - scaling scheme as described e. g. in ref . the estimates for the correlation length , magnetic susceptibility , and spontaneous magnetization critical exponents @xmath20 , @xmath21 and @xmath22 can be obtained from the fss behaviour of the following quantities : * _ averaging a _ : @xmath99 in eqs . ( [ 111b ] ) , ( [ 111c ] ) , the correction - to - scaling terms have been omitted . taking them into account the fss behaviour of the quantity @xmath100 attains the form : @xmath101 where @xmath102 is the leading exponent , @xmath103 is the correction - to - scaling exponent , and @xmath104 is a correction - to - scaling amplitude . hereafter , we perform the finite - size - scaling analysis in several ways . first , assuming the correction - to - scaling terms to be small we will fit the data for the observables to the leading power laws ( [ 111b ] ) , ( [ 111c ] ) only . besides that , we will also perform the fits with correction - to - scaling terms ( [ 111d ] ) as explained below . in this section , applying the above described formalism , we give preliminary estimates of the critical exponents . as we will see , the estimates for the same exponent obtained on the base of fss of different quantities differ from each other . moreover , similar differences arise when the results obtained via _ averaging a _ and _ averaging b _ are compared . we will discuss possible reasons for such behaviour and will analyze them further in the forthcoming section . the starting point of the analysis performed in this section is connected with the fact that at the same concentration of impurities , different samples vary configurationally ( geometrically ) as well as in their thermodynamical characteristics . in systems with uncorrelated dilution these variations are also present ( see e.g. ref . @xcite ) although , as we will see below , these are less pronounced than in the case of the long - range - correlated disorder . moreover , although the mean concentration of magnetic sites is well - defined for an ensemble samples ( and chosen to be @xmath45 ) , the concentration of magnetic sites in each sample vary from the mean one . such deviations are caused by possible intersections of impurity lines in different samples , besides not for all lattice sizes one can reach given concentration of impurities by an integer number of lines . therefore , to start an analysis one has to make a choice of a set of sample specifying allowed values of the concentration dispersion . we start an analysis by dealing with the set of samples that are characterized by dispersion of concentrations @xmath105 . the sample - to - sample variations increase with the increase of the lattice size . as an example , we show in fig . [ fig2 ] the values of the susceptibility for a given disorder realization calculated via _ averaging a _ ( @xmath78 , fig . 3a-3c ) and _ averaging b _ ( @xmath106 , fig . 3d-3f ) for three typical lattice sizes @xmath107 . each point in these plots represents the data obtained for a given sample ( i.e. given disorder realization ) . solid lines are made up of the average values , where at each @xmath108 the averaging is done on the interval @xmath109 . one can see that the averages saturate at @xmath108 larger than two hundred samples . from this in particular one can conclude that considered here number of samples @xmath110 is quite sufficient for further analysis . one observes that for some samples @xmath78 and @xmath106 differ significantly from the average values . when this is the case , we will call such an event a rare one . the values that are close to the maximal one will be referred to as typical events . we would like to stress that for _ averaging b _ for the lattice size @xmath111 most of rare events produce the values for the susceptibility below the average value , while for @xmath112 the situation is reversed , see figure [ fig2 ] . at @xmath113 the distribution of rare events is aproximately symmetric . the situation is more homogenous for the _ averaging a_. the averaging over too small number of disorder realizations leads to typical ( i.e. most probable ) values instead of averaged ones . indeed , as can be seen in fig . [ fig3 ] for @xmath112 , the probability distributions of @xmath78 and @xmath106 possess long tails of rare events with larger values of susceptibility . the samples which correspond to the rare events give a large contribution to the average and shift it far from the most probable values . the value of @xmath106 calculated with the complete probability distribution in shown in fig . [ fig3 ] by a dashed line ( blue on - line ) . another characteristic value shown in fig . [ fig3 ] by lines ( solid ) are the median values @xmath114 , defined as the values of @xmath72 where the integrated probability distributions are equal to 0.5 . a distance between the average and median values may serve to quantify the asymmetricity of the probability distribution . figures [ fig5a ] and [ fig6a ] show contributions of typical and rare events to the temperature behaviour of the observables under discussion . as an example of typical events , we plot in fig . [ fig5a ] the magnetic susceptibility @xmath72 , binder s cummulant @xmath74 , and magnetisation @xmath115 selecting the samples where susceptibilities have the same values as @xmath116 at @xmath117 for the largest lattice size @xmath112 . few examples of rare events corresponding to large values of @xmath72 are shown in fig . [ fig6a ] . the bold lines in the figures show averages over all disorder realizations ( all samples ) . the thin lines ( color on - line ) correspond to different samples , as shown in the legends to the plots [ fig5a]a , [ fig6a]a . it follows from the above analysis that the rare events are obviously present in the random variables distributions and give a contribution to the average values of physical quantities . law to choose the samples during the mc experiment ( as it was done at the initial phase of our analysis , see ref . @xcite ) modifies the distribution . such modifying leads to different results for the exponents . we thank the referee for the comments on this point . ] before passing to evaluation of the critical exponents let us recall that as it was noted at the beginning of section [ v ] we perform the simulations for the samples with varying concentration of magnetic sites , keeping the mean concentration equal to @xmath45 and limiting dispersion of concentrations by @xmath105 ( [ probp ] ) . following refs . @xcite it is natural to call such situation the grand canonical disorder. afterwards , in section [ ivc ] we will consider the case of canonical disorder , when we will choose the samples much more close in the concentration of magnetic sites ( taking @xmath118 ) . in the finite - size - scaling technique @xcite , the critical exponents are calculated by fitting the data set for the observables evaluated at various lattice sizes to the power laws ( [ 111b ] ) , ( [ 111d ] ) as was already mentioned in section [ iii ] . this procedure will be used for the evaluation of the critical exponents @xmath20 , @xmath22 , and @xmath21 based on the observables measured during the mc simulations . corresponding finite - size - scaling plots are shown in figs . [ fig5 ] , [ fig7 ] , the numbers are given in table [ tab3 ] . in fig . [ fig5 ] , we give the log - log plots of the maximum values of the configurationally averaged derivatives of binder cumulants @xmath119 and @xmath120 . the scaling of these quantities is governed by the correlation length critical exponent @xmath20 , see eqs . ( [ 111b ] ) . [ fig7 ] shows the size dependence of the configurationally averaged magnetic susceptibility @xmath78 ( fig . [ fig7]a ) and magnetisation @xmath84 ( fig . [ fig7]b ) . these values are expected to manifest a power - law scaling with exponents @xmath121 and @xmath122 , see eq . ( [ 111b ] ) .
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we analyze a controversial topic about the universality class of the three - dimensional ising model with long - range - correlated disorder . whereas both theoretical and numerical studies agree on the validity of extended harris criterion ( a. weinrib , b.i .
halperin , phys .
rev .
b 27 ( 1983 ) 413 ) and indicate the existence of a new universality class , the numerical values of the critical exponents found so far differ essentially . to resolve this discrepancy
we perform extensive monte carlo simulations of a 3d ising model with non - magnetic impurities being arranged in a form of lines along randomly chosen axes of a lattice .
the swendsen - wang algorithm is used alongside with a histogram reweighting technique and the finite - size scaling analysis to evaluate the values of critical exponents governing the magnetic phase transition .
our estimates for these exponents differ from both previous numerical simulations and are in favour of a non - trivial dependency of the critical exponents on the peculiarities of long - range correlations decay . , , , random ising model , long - range - correlated disorder , monte carlo , critical exponents 05.10.ln , 64.60.fr , 75.10.hk
| 8,203 | 336 |
the x - ray source is a 7.7 s pulsar in a highly compact binary system of orbital period 2485 s. it is unusual in that it is one of the few low mass x - ray binary systems to contain an x - ray pulsar . while the x - ray emission is strongly modulated by the pulsar , there is no evidence for doppler shifts induced by the orbital motion of the source , despite extensive searches . the implies that the projected orbital radius of the neutron star is small , i.e. , @xmath2sin @xmath3 @xmath4 13 m - lt - s ( levine et al . optical pulsations were first detected by ilovaisky et al . ( 1978 ) and interpreted as x - ray re - processing near to , or along , the line of sight to the x - ray source . middleditch et al . ( 1981 ) found a single low frequency side - lobe which they interpret as arising from the optical re - processing of the primary x - rays on the companion star . assuming the pulsar spins in the same sense as the orbital motion , these photons will be shifted to a lower frequency by the rotation frequency of the binary orbit . from the observed frequency shift of 0.4 mhz an orbital period of 2485 s and a projected semimajor axis of 0.4 lt - s is inferred . the current picture of is of a highly compact system comprising a neutron star of mass @xmath11 m@xmath5 , with a 0.08@xmath6 main sequence or 0.02@xmath6 white dwarf companion ( verbunt et al . 1990 ) . for the first decade after its discovery was rapidly spinning - up at a rate of @xmath7@xmath8 yr@xmath9 . however , long term monitoring by the burst and transient source experiment ( batse ) on - board the compton gamma - ray observatory beginning in 1991 april found that @xmath10 , and hence the accretion torque , had changed sign , resulting in a spin - down at nearly the same rate ( wilson et al . it is estimated that the reversal occurred in mid-1990 . observations of during the earlier spin - up phase found that the phase averaged spectrum could be modeled by a blackbody of temperature , kt , @xmath10.6 kev together with a power - law of photon index , @xmath11 , of @xmath11 ( e.g. , pravdo et al . 1979 ; kii et al . 1986 ) . in the 210 kev energy range the pulse profile consisted of a narrow pulse with a `` notch '' , while at higher and lower energies this evolved into a roughly sinusoidal shape ( levine et al . 1988 ; mavromatakis 1994 ) . this strong energy dependence may result from anisotropic radiative transfer in a strongly magnetized plasma ( kii et al . 1986 ) . in addition to periodic pulsations , also exhibits quasi periodic behavior . both the x - ray and optical intensities show correlated flaring on timescales of @xmath11000 s ( joss et al . 1978 ) . the origin of this behavior is unknown . a 40 mhz quasi periodic oscillation ( qpo ) has been detected in x - rays ( shinoda et al . 1990 ) and more recently in the optical band ( chakrabarty et al . 1997 ) . finally , the recent observation of an emission line complex near 1.0 kev by angelini et al . ( 1995 ) is particularly interesting . this emission is interpreted as arising primarily from ne k rather than from fe l , based on the measured line energies and intensities . this result suggests that the companion star has evolved past its hydrogen burning stage . ne is a by - product of he burning and therefore its overabundance suggests that the star is burning , or has burnt , he . the low - energy concentrator spectrometer ( lecs ) is one of 5 scientific instruments on - board the _ bepposax _ satellite ( boella et al . 1997 ) . the lecs is described in parmar et al . ( 1997 ) and consists of a nest of conical approximation to wolter i x - ray mirrors which produce a focused beam of x - rays on an imaging gas scintillation proportional counter . the detector employs a driftless design which , in conjunction with an extremely thin entrance window , ensures an extended low energy response . the nominal energy range of the instrument is 0.110 kev and the full width at half maximum ( fwhm ) energy resolution is 19% at 1 kev . the field of view ( fov ) is 37@xmath12 diameter and the spatial resolution is 5.1@xmath12 fwhm at 1 kev . _ bepposax _ was launched into a 600 km equatorial orbit on 1996 april 30 . was observed twice by the lecs during the science verification phase on 1996 august 6 from 12:00 ut to 21:50 ut and on august 9 02:25 ut to august 11 00:00 ut . the total source exposure is 40 ks and the mean lecs count rate is @xmath13 s@xmath9 . because of the large spatial extent of the point spread function , background data were acquired separately by viewing blank regions of sky . background subtraction is not critical for such a bright source . source events were obtained from both observations using the standard extraction radius of 8@xmath12 , centered on the source centroid . the extracted data were rebinned to have bin widths of 0.25 @xmath14 the fwhm energy resolution . this is to help ensure an unbiased fit across the energy range , while preserving sensitivity to line features . since the lecs response is dependent on both position within the fov and extraction radius , the appropriate matrix was created using saxledas version 1.4.0 ( lammers 1997 ) . the spectrum was first fit with an absorbed blackbody plus power - law model yielding a @xmath15 of 115 for 84 degrees of freedom ( dof ) . inspection of the residuals ( see fig . [ 4uspec ] ) shows an excess centered around 1 kev , similar to that seen by angelini et al . ( 1995 ) with the asca solid - state imaging spectrometer ( sis ) . the addition of a gaussian line feature to the model results in a @xmath15 of 92 for 81 dof . an f - test shows that the improvement is significant at the @xmath16 99.99% level . the best - fit line energy and flux are 1.05 @xmath17 kev and ( 4.58 @xmath18 ) @xmath19 photons @xmath20 s@xmath9 , respectively . ( all uncertainties are quoted at 68% confidence . ) the equivalent width ( ew ) is 47.6 @xmath0 13.5 ev . the spectrum is shown in fig . [ 4uspec ] with the line intensity set to zero . the best - fit values of kt and @xmath11 are 0.33 @xmath0 0.02 kev and 0.61 @xmath0 0.02 , respectively . the derived absorption column is ( 6.9 @xmath0 2.0)@xmath21 atoms @xmath20 , consistent with the interstellar value in the direction of ( daltabuit & meyer 1972 ; dickey & lockman 1990 ) . the best - fit line energy agrees well with that expected from a blend of 80% ne ly-@xmath11 ( 1.021 kev ) and 20% ne he-@xmath22 ( 1.084 kev ) emission , consistent with the asca result of angelini et al . the values of kt and @xmath11 are also similar to those measured using asca , but both a factor of 2 lower than earlier heao-1 ( pravdo et al . 1979 ) and @xmath23 ( kii et al . 1986 ) values . the blackbody radius , r@xmath24 , of 1.0 d@xmath25 km , where d@xmath26 is the distance in kpc , is also consistent with asca . the 0.510 kev luminosity is 2.0 @xmath14 10@xmath27 ergs s@xmath9 d@xmath28 which is @xmath140% lower than measured by asca three years earlier , and a factor of 6 lower than that derived using the _ einstein _ solid state spectrometer in 1979 march ( angelini et al . 1994 ) . in the case of asca , angelini et al . ( 1995 ) attribute the spectral and luminosity differences to the torque reversal , since the heao-1 , @xmath23 , and _ einstein _ measurements were carried out prior to mid-1990 . the asca and _ bepposax _ measurements were performed during the present spin - down phase . ll parameter & value + + n@xmath29 & 1.10 @xmath0 0.20 + @xmath11 & 0.64 @xmath0 0.02 + kt & 0.29 @xmath0 0.01 + r@xmath24 & 1.4 @xmath0 0.8 + @xmath15/@xmath30 & 115/84 + + n@xmath29 & 0.69 @xmath0 0.20 + @xmath11 & 0.61 @xmath0 0.02 + kt & 0.33 @xmath0 0.02 + r@xmath24 & 1.0 @xmath0 0.5 + e@xmath31 & 1.05 @xmath32 + @xmath33 & 0.04 @xmath0 0.04 + ew & 48 @xmath0 14 + flux@xmath31 & 4.6 @xmath18 + @xmath15/@xmath30 & 92/81 + + n@xmath29 & 0.81 @xmath0 0.26 + @xmath11 & 0.62 @xmath0 0.02 + kt & 0.33 @xmath0 0.02 + r@xmath24 & 1.0 @xmath0 0.6 + o he-@xmath11 0.568 kev flux & 5.3 @xmath34 + ew & 34 @xmath0 34 + ne ly-@xmath11 1.021 kev flux & 3.2 @xmath0 1.5 + ew & 28 @xmath0 12 + ne he-@xmath22 1.084 kev flux & 1.5 @xmath0 1.3 + ew & 13 @xmath0 10 + fe k-@xmath11 6.400 kev flux & 0.7 @xmath0 0.5 + ew & 38.6 @xmath0 22.9 + @xmath15/@xmath30 & 88/80 + in order to investigate whether the single narrow feature seen by the lecs is consistent with the emission line complex observed by the asca sis , the feature at 1.05 kev was replaced by the blend of lines used in the asca fit ( see table 1 of angelini et al . each line was added separately with its energy and width fixed at the measured asca values . the only @xmath35 for inclusion in the final fit was that @xmath15 must reduce . only three lines satisfied this criteria ; the ne ly-@xmath11 line at 1.021 kev , the ne he-@xmath22 line at 1.084 kev , and the o he-@xmath11 line at 0.568 kev . surprisingly , the fit did not require an ne he-@xmath11 line and we derive an upper flux limit at the 90% confidence level of 2.2 @xmath14 10@xmath36 photons @xmath20 s@xmath9 . phase - resolved asca spectra reveal the presence of a weak iron k feature over a narrow range of pulse phases ( angelini et al . including such a line in the current model gives a @xmath15 of 88 for 80 dof . best - fit spectral parameters are listed in table 1 . the derived o he-@xmath11 and ne he-@xmath22 line intensities are consistent with the asca values reported in angelini et al . ( 1995 ) . however , the ne ly-@xmath11 line intensity is a factor of 3 lower and the 90% confidence upper flux limit to any ne he-@xmath11 emission is lower by a factor of @xmath1 6 . [ 4u10per ] we next investigated an alternate spectral model for in which the power - law plus blackbody is supplemented by emission from an optically - thin collisionally ionized plasma ( specifically the vmekal model in xspec v.9.01 ) . in principle , this would allow us to estimate the elemental abundances necessary to produce the excess emission around 1 kev , in a similar manner to angelini et al . the abundances of ne and fe were allowed to vary while the abundances of the other elements were fixed at the photospheric values of anders & grevesse ( 1989 ) . both high ne and fe over - abundances gave acceptable fits to the data with @xmath15 s comparable to the `` ne complex '' fit . however , the lecs spectrum is of insufficient quality to determine meaningful limits to these abundances . the barycentric pulse period during the lecs observations of @xmath37 s is in good agreement with the predicted value of 7.667943 @xmath0 0.000006 s derived from batse data ( chakrabarty 1997 ) . the pulse profiles in the energy ranges 0.51 , 1.03.0 , and 3.010 kev , are shown in fig . [ 4u10per ] . the overall shape and energy dependence are similar to those seen by asca . the pulse profile in the lowest energy band in fig . [ 4u10per ] may be consistent with that seen during spin - up ( e.g. , pravdo et al . 1979 ) , but with a reduced amplitude . [ 4ups ] figure [ 4ups ] shows a white noise subtracted , power density spectrum of the lecs data from which quasi periodic oscillations ( qpos ) are apparent . the center frequency is 0.049 @xmath0 0.002 hz and the fwhm is 0.015 @xmath38 hz . the qpo strength is 18 @xmath39% _ rms _ of the mean count rate . within uncertainties , the qpo amplitude is the same in the 0.52.0 kev and 2.010.0 kev energy ranges . the qpo centroid frequency , width , and amplitude are consistent with the asca measurements ( angelini et al . the amplitude and width are also consistent with the @xmath40 values ( shinoda et al . 1990 ) , but the centroid frequency is not . this change may be related to the torque reversal . the lecs data confirm the recent asca detection of excess emission near 1 kev from . in addition , the best - fit spectral parameters confirm that the spectral shape remains changed following the mid-1990 torque reversal . both kt and @xmath11 decreased by a factor of @xmath12 , while the x - ray luminosity decreased by a factor of 6 . as before the reversal , the new parameters appear stable with time . chakrabarty et al . ( 1997 ) report that the intensity of the source is steadily decreasing with time . the lecs results support this , since the 0.510 kev source intensity is 0.6 that measured by asca . extrapolating from previous measurements ( see fig . 8 of chakrabarty et al . 1997 ) , the expected decrease is a factor of @xmath10.7 . the shape of the pulse profile is also different from that measured before the reversal . prior to the reversal , the profile was strongly energy and phase dependent ( e.g. , see levine et al . 1988 ) , while the lecs pulse profile has the same shape , but a variable amplitude , over the 1.010.0 kev energy range ( see fig . we thank chris butler , luigi piro , and the staff of the _ bepposax _ science data center in rome . lorella angelini is thanked for discussions and deepto chakrabarty for providing the batse pulse period . t. oosterbroek acknowledges an esa research fellowship . the _ bepposax _ satellite is a joint italian and dutch programme .
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we report on observations of the x - ray pulsar by the lecs instrument on - board _
bepposax_. we confirm the recent asca discovery of excess emission near 1 kev ( angelini et al .
1995 ) .
the pulse period of 7.66794 @xmath0 0.00004 s indicates that the source continues to spin - down .
the phase averaged spectrum is well fit by an absorbed power - law of photon index 0.61 @xmath0 0.02 and a blackbody of temperature 0.33 @xmath0 0.02 kev , together with an emission feature at 1.05 @xmath0 0.02 kev .
this spectral shape is similar to that observed by asca during the spin - down phase , but significantly different from measurements during spin - up .
this suggests that the change in spectrum observed by asca may be a stable feature during spin - down intervals .
the source intensity is a factor @xmath12 lower than observed by asca three years earlier , confirming that continues to become fainter with time .
| 4,454 | 261 |
chiral perturbation theory is a low energy effective field theory of the strong interaction . the work @xcite presents analytic expressions for the two - loop contribution to the pion mass and decay constant in su(3 ) chiral perturbation theory with suitable expansions in powers of @xmath3 . in an upcoming work @xcite , we will present analogous expressions for the pion decay constant . work is also underway to find similar simple analytic representations for the kaon and eta mass and decay constants to two loops . due to the goldstone nature of the particles involved , scalar , tensor and derivatives of sunset diagrams appear in these calculations , with various mass configurations and with up to three distinct masses . much work has been done on sunset diagrams ( an incomplete list is given in references @xcite-@xcite ) , and a variety of analytic results exist in the literature for the one - and two - mass scale configurations @xcite . papers directly relevant to this work are the following . in @xcite , analytic results have been given for the master integrals at the pseudothreshold @xmath4 and threshold @xmath5 , the former of which may be used to obtain the single , and many of the double , mass scale analytic expressions . gasser and saino @xcite use integral representations to give results in closed form for several basic two - loop integrals appearing in chpt , including the sunset , with one mass - scale . for unequal masses , fully analytic results are given by @xcite gives in terms of newly defined elliptic generalizations of the clausen and glaisher functions , but the application of methods or approximation schemes that give the three mass scale sunsets as expansions in powers of the mass ratio allow for a more transparent interpretation of the results being considered . in @xcite , just such an expansion is given for the most general sunset integral in terms of lauricella functions . however , none of the series presented in @xcite converge for the physical values of the meson masses . the interest in analytic or semi - analytic expressions arises from the desire to make as direct a contact as possible with results in lattice field theories . recent advances in lattice qcd now allow for quark masses in these theories to be varied independently , allowing for realistic quark masses . the availability of analytic results for pseudo - scalar masses and decay constants , for example , would allow for easy and computationally efficient comparison with lattice results . aside from the derivation of analytic expressions for the pseudo - scalar meson masses and decay constants to two - loops , the application of sunset diagrams to chiral perturbation theory is also of general interest . in this context , sunset diagrams have been studied quite early ( @xcite ) , where not only the single mass scale sunset ( which appears in su(2 ) chiral perturbation theory ) is considered , but also the cases with more than one mass scale which are common in the su(3 ) theory . in su(3 ) chiral perturbation theory , the sunset is the simplest diagram that appears at two loops , and a careful study of it paves the way for the study of the other diagrams that appear at this order ( i.e. vertices , boxes and acnodes ) . the work @xcite gives a terse but comprehensive summary of results . another possible use of the sunsets is to expand them out using methods such as expansion in regions @xcite , and then use this to reduce the su(3 ) low energy constants to the su(2 ) ones . the process of relating the su(3 ) to su(2 ) low - energy - constants has been done using an alternative method in @xcite but it has not yet been done for the full set of low - energy - constants at next - to - next - to - leading order . it must be noted in the context of @xcite that the sunset technology is also important when considering vertices , as many of the latter get related to the sunsets when using , for example , the method of expansion by regions . in this paper , we use the mellin - barnes method to derive results for all the single and double mass scale integrals . it has been shown in @xcite that the mellin - barnes method is an efficient one for obtaining expansions in ratios of two mass scales should they appear in feynman diagrams in general . this work therefore serves as an independent verification of the existing results in the literature . the mellin - barnes method is also an appropriate tool for chiral perturbation theory applications as it ab initio allows us to express the integrals as expansions in mass ratios . a further reason for mellin - barnes as our tool of choice is the availability of powerful public computer packages in this approach . the availability of such codes has made such a study of sunsets ( and two - loop diagrams in general ) in chiral perturbation theory much more accessible . the mathematica based package ` tarcer ` @xcite applies the results of tarasov s work @xcite to recursively reduce all sunset diagrams to the master integrals . several packages @xcite have automatized many aspects of the application of mellin - barnes methods to feynman integrals . the sunsets appearing in chiral perturbation theory have been implemented numerically in the package ` chiron ` @xcite using the methods of @xcite . one of the goals of the present work is to improve on this implementation . in addition , there are two other packages ` bokasum ` @xcite and ` tsil ` @xcite that can be used to numerically calculate sunset integrals . we present along with this paper several mathematica notebooks ( lodged as ancillary files along with the arxiv submission ) which contain the details of our calculations , as well as a demonstration of how to apply the above packages to the calculation of sunset integrals . the notebooks are thoroughly annotated , and can be used in a stand - alone capacity , or in conjunction with this note . these may also serve as a pedagogical introductions to the analytic evaluation of sunsets diagram . the primary goal of this paper is to show the use of the packages of @xcite ; the results as presented here have been checked in a number of other ways as well . the relations from @xcite have been implemented independently using ` form ` @xcite . the expansions around @xmath6 were also derived using the methods of @xcite and numerical results have been compared with the results from analytical expressions of @xcite . this paper is organized as follows . in section 2 , we give the five different sunset configurations that will be explicitly considered in this work , and show from where they arise . in section 3 we give an overview of the sunset integrals , their divergences , and their renormalization in chiral perturbation theory . in section 4 , we briefly discuss the mellin - barnes method of evaluating feynman integrals . in section 5 , we demonstrate the use of the package ` tarcer ` @xcite to reduce the tensor and derivatives of the sunsets to master integrals . in section 6 , we explain the use of the packages @xcite to derive the results for the one - mass scale master integral . we also explain how the ` tarcer ` package @xcite alone can be used to derive this result . in section 7 , we describe briefly the two different categories of two - mass scale sunset diagrams and their evaluation , and present a complete set of results in appendix a. in section 8 , we explain how three mass scale sunsets can be handled either by means of an expansion in the external momentum , or by a more sophisticated application of the mellin - barnes method to . in section 9 , we present a one - dimensional integral representation of an important configuration that arises in the su(2 ) chiral perturbation theory , and in section 10 with a discussion of some numerical issues of the new results presented herein . we conclude in section 11 with a discussion of the import and limitations of this work , and possible future work in this field . in appendix b , we give a brief description of all the public codes used in this work , and in appendix c , we present a dictionary that allows for an easy translation between the definition used in this work for the sunset and other integrals , and those used in the various programs and papers . in appendix d , we list the ancillary files provided with this paper . expressions for the pseudoscalar meson masses and decay constants in two loop chiral perturbation theory are given in appendix a of @xcite . as a concrete example , the pion mass is given by : @xmath7 where @xmath8 is the bare mass , @xmath9 is the one - loop contribution , @xmath10 is the two - loop model - dependent counterterm contribution , and @xmath11 is the chiral loop contribution . it is in this last term that the sunset integrals appear : @xmath12 the @xmath13 in the above expression refer to the scalar sunset integral @xmath14 as defined in eq.([sunsetdef ] ) of section 3 , where the first three arguments pertain to the masses entering the propagators , and the last is the square of the energy entering the loop . the @xmath15 and @xmath16 are the scalar integrals that make up the passarino - veltman decomposition of vector and tensor sunsets , and are defined precisely in eq.([h1 ] ) and eq.([h21 ] ) respectively . in the case of the meson decay constants , in addition to the variety of sunset integrals appearing above , also appear derivatives of the sunsets ( i.e.@xmath17 , @xmath18 and @xmath19 ) . the work of finding an analytic expression for the pion mass ( as well as the other pseudoscalar meson masses and decay constants ) reduces to analytically evaluating these sunset integrals . in the subsequent sections of this paper , we explain how to analytically evaluate each of the different types of integrals appearing in expressions such as eq.([pionmass ] ) above . in particular , we show in detail how to evaluate the following integrals as representative of the different types of integrals and the different types of mass configurations that may appear in expressions for the pseudoscalar masses and decay constants : ' '' '' * integral * & * characteristic * + ' '' '' @xmath20 & one mass scale + ' '' '' @xmath21 & two mass scales + ' '' '' @xmath22 & three mass scales with smallest parameter as external momentum + ' '' '' @xmath23 & three mass scales with an internal mass as smallest parameter + ' '' '' @xmath24 & tensor sunset derivative + the evaluation of all these integrals requires writing them in terms of master integrals , and then analytically evaluating the master integrals . this is explained in greater detail in the next section . the analytic evaluation of the master integrals can be done using a variety of methods , and many of these have previously been used to derive the plethora of results that exist in the literature . in this paper , we use the mellin - barnes approach , which appears to be the most efficient method by which to evaluate the three mass scale integrals , such as @xmath23 that appears in the expressions for eta mass and decay constant . the integrals given in the table above are all amenable to a mellin - barnes treatment . however , for @xmath22 , we instead take an expansion in the external momentum @xmath25 , as it provides a result that is as accurate as a mellin - barnes expansion ( to the same order ) but that is much easier to calculate . a similar expansion can not be done for @xmath26 in either the external momentum @xmath27 due to poor convergence , or in @xmath3 as it gives rise to an infrared divergence . the sunset integral , shown in fig.([sunset ] ) , is defined as : @xmath28^{\alpha } [ r^2-m_2 ^ 2]^{\beta } [ ( q+r - p)^2-m_3 ^ 2]^{\gamma } } \label{sunsetdef}\end{aligned}\ ] ] vector and tensor sunset integrals have four - momenta , such as @xmath29 or @xmath30 , sitting in the numerator . two tensor integrals that appear in the calculation of meson masses and decay constants in chiral perturbation theory are : @xmath31^{\alpha } [ r^2-m_2 ^ 2]^{\beta } [ ( q+r - p)^2-m_3 ^ 2]^{\gamma } } \nonumber \\ \nonumber \\ h_{\mu \nu}^d \{m_1,m_2,m_3 ; p^2\ } = \frac{1}{i^2 } \int \frac{d^dq}{(2\pi)^d } \frac{d^dr}{(2\pi)^d } \frac{q_{\mu } q_{\nu}}{[q^2-m_1 ^ 2]^{\alpha } [ r^2-m_2 ^ 2]^{\beta } [ ( q+r - p)^2-m_3 ^ 2]^{\gamma } } \nonumber\end{aligned}\ ] ] these may be decomposed into linear combinations of scalar integrals via the passarino - veltman decomposition as : @xmath32 to obtain the scalar integral @xmath33 , we take the scalar product of @xmath34 with @xmath35 : @xmath36^{\alpha } [ r^2-m_2 ^ 2]^{\beta } [ ( q+r - p)^2-m_3 ^ 2]^{\gamma } } \equiv \frac{1}{p^2 } \langle \langle q.p \rangle \rangle \label{h1}\end{aligned}\ ] ] where we have defined @xmath37 as the scalar sunset diagram with unit powers of the propagators , and with @xmath38 in the numerator . similarly , @xmath16 may be expressed as : @xmath39 in @xcite tarasov has shown by using the method of integration by parts that all sunset diagrams , including those of higher than @xmath40 dimensions , may be rewritten as linear combinations of a set of four master integrals and bilinears of one - loop tadpole integrals . these basic integrals are @xmath41 and the one - loop tadpole integral : @xmath42 application of tarasov s relations becomes crucial when evaluating another class of integrals that show up in chiral perturbation theory calculations , namely the derivatives of scalar and tensor sunsets ( e.g. @xmath43 ) . these may be evaluated by means of the following well - known formula relating derivatives and integrals in different dimensions @xcite : @xmath44 the mathematica package ` tarcer ` @xcite automatizes the reduction of any sunset integral to the master integrals . many results exist in the literature regarding these master integrals . one result that we use frequently in the subsequent sections is that of the two - mass scale master integral with zero external momentum @xmath45 . this is given in @xcite as : @xmath46 - \frac{x}{2 } \text { ln}^2 \left [ x \right ] + \left ( 2 + x \right ) \left [ \frac{\pi^2}{12 } + \frac{3}{2 } \right ] \bigg\ } \nonumber \\ & - ( \mu^2)^{-2\epsilon } \bigg\ { m^2 \log \left ( \frac{m^2}{\mu^2 } \right ) \left [ 1 - \log \left ( \frac{m^2}{\mu^2 } \right ) \right ] + 2 m^2 \log \left ( \frac{m^2}{\mu^2 } \right ) \left [ 1 - \log \left ( \frac{m^2}{\mu^2 } \right ) \right ] \bigg\ } \nonumber \\ & + \frac{m^2}{2 } \bigg\ { \bigg [ 2 + x \bigg ] \frac{1}{\epsilon^2 } + \bigg [ x \left ( 1 - 2 \log \left ( \frac{m^2}{\mu^2 } \right ) \right ) + 2 \left ( 1 - 2 \log \left ( \frac{m^2}{\mu^2 } \right ) \right ) \bigg ] \frac{1}{\epsilon } \bigg\ } + \mathcal{o}(\epsilon ) \label{k111}\end{aligned}\ ] ] where @xmath47 , \qquad \sigma = \sqrt{1-\frac{4}{x } } \end{aligned}\ ] ] eq.([k111 ] ) above is given in the modified version of the @xmath48 scheme normally used in chiral perturbation theory ( @xmath49 ) . the change from the minimal subtraction ( ms ) scheme to @xmath49 involves making the replacement : @xmath50 analytical expressions for the divergent parts of the sunset master integrals have been derived in @xcite , amongst other places . the following are the divergent parts of the master integrals in the @xmath49 scheme : @xmath51 \frac{1}{\epsilon^2 } \nonumber \\ & + \left [ m_1 ^ 2+m_3 ^ 2+m_3 ^ 2 - \frac{s}{2 } - 2m_1 ^ 2 \log \left(\frac{m_1 ^ 2}{\mu ^2}\right ) - 2m_2 ^ 2 \log \left(\frac{m_2 ^ 2}{\mu ^2}\right)- 2m_3 ^ 2 \log \left(\frac{m_3 ^ 2}{\mu ^2}\right ) \right ] \frac{1}{\epsilon } \bigg\ } \nonumber \\ h_{\{2,1,1\}}^{div } & \{m_1,m_2,m_3;s\ } = \frac{1}{512 \pi^4 } \bigg\ { \frac{1}{\epsilon^2 } - \left [ 1 + 2 \log \left(\frac{m_1 ^ 2}{\mu ^2}\right ) \right ] \frac{1}{\epsilon } \bigg\ } \nonumber \\ h_{\{1,2,1\}}^{div } & \{m_1,m_2,m_3;s\ } = \frac{1}{512 \pi^4 } \bigg\ { \frac{1}{\epsilon^2 } - \left [ 1 + 2 \log \left(\frac{m_2 ^ 2}{\mu ^2}\right ) \right ] \frac{1}{\epsilon } \bigg\ } \nonumber \\ h_{\{1,1,2\}}^{div } & \{m_1,m_2,m_3;s\ } = \frac{1}{512 \pi^4 } \bigg\ { \frac{1}{\epsilon^2 } - \left [ 1 + 2 \log \left(\frac{m_3 ^ 2}{\mu ^2}\right ) \right ] \frac{1}{\epsilon } \bigg\ } \label{divparts}\end{aligned}\ ] ] in the remainder of this paper , unless explicitly stated , @xmath52 will be used to denote the finite part of the sunset integral in the @xmath49 scheme . eq.([chiralmsbar ] ) may be reverse engineered and used in combination with eq.([divparts ] ) to find the full results in any other subtraction scheme . we give a brief overview of the basic mellin - barnes approach to feynman integrals here . for a more comprehensive overview see @xcite . the mellin transform is defined as follows : @xmath53(s ) = \int\limits_0^\infty f(t)t^{s-1}dt , \hspace{0.2 in } s \in \mathcal{c}\end{aligned}\ ] ] its inverse is given by : @xmath54(x ) = \frac{1}{2\pi i } \int\limits_{c - i\infty}^{c+i\infty } x^{-s}g(s)ds\end{aligned}\ ] ] the following formula derived from the inverse mellin transform is used in high energy physics to write massive propagators as combinations of massless propagators : @xmath55 the expression obtained after application of this formula and evaluation of the momentum integral is known as the mellin - barnes representation of a feynman integral . in some cases , it may be possible to simplify the mellin - barnes representation of an integral by the application of the following two barnes lemmas @xcite : @xmath56 where @xmath57 the evaluation of the mellin - barnes integrals may then be performed either numerically , or analytically by the addition of residues . in case of multiple mellin - barnes parameters , results from the theory of several complex variables may have to be used for analytic evaluation @xcite . in this section , we demonstrate how to handle both the tensor sunset integrals , as well as the derivatives of the sunsets , by reducing them to master integrals . in particular , we show how to evaluate the integral @xmath59 , by making extensive use of the package ` tarcer ` @xcite . the computer implementation of what follows is given in the ancillary file ` reductiontomi.nb ` . the first step is to decompose @xmath60 into master integrals . from eq.([h21 ] ) , we have : @xmath61 differentiating with respect to @xmath62 gives : @xmath63 the next step involves evaluating the scalar sunset integrals with @xmath64 and @xmath65 in the numerator . the following command allows us to express the first of these integrals in terms of the master integrals . + ` tarcerrecurse[tfi[d , s , { 0 , 0 , 2 , 0 , 0 } , { { 1 , mpi } , { 0 , 0},{0 , 0},{1 , mk},{1 , mk } } ] ] ` + the output , @xmath66 , is a function of the dimensional parameter @xmath40 , the external momentum @xmath62 , the masses @xmath67 and @xmath68 , the integrals @xmath69 , @xmath70 , @xmath71 , @xmath72 and @xmath73 . this expression is then differentiated with respect to @xmath62 , the resulting expression , @xmath74 , also being a function of the same parameters and integrals as @xmath66 , but in addition also being a function of the differentiated master integrals @xmath75 , @xmath76 , @xmath77 . each of these differentiated master integrals can be expressed as a sunset integral in a higher ( @xmath78 ) dimension by use of eq.([dh ] ) , and each of these higher dimensional sunsets can in turn be expressed in terms of the @xmath40 dimensional master integrals by further use of ` tarcer ` . for example , the integral @xmath79 is equal to @xmath80 . by use of the command : + ` tarcerrecurse[tfi[d+2 , s , { { 3 , mpi } , { 0 , 0},{0 , 0},{2 , mk},{2 , mk } } ] ] ` + we get an expression for @xmath81 in terms of @xmath40 dimensional master integrals . we repeat this process for each of the differentiated master integrals that appear , and substitute them ( and @xmath25 ) into the expression for @xmath74 . we can similarly obtain an expression for @xmath82 and @xmath83 , and substituting all these expressions into eq.([h21prime ] ) with @xmath25 gives us our desired expression for @xmath59 . the expressions we obtain for @xmath84 and @xmath19 , given in the notebook ` reductiontomi.nb ` , have been positively checked against expressions obtained from a direct differentiation of eq.(2.13 ) and eq.(2.14 ) respectively of @xcite . all one mass scale case sunset integrals can be reduced to a single master integral , namely @xmath86 where @xmath87 is the mass in question . below , we show how to evaluate the one mass scale sunset integral @xmath88 , and therefore give a pedagogical demonstration of the use of the mellin - barnes approach to evaluating feynman integrals . we also demonstrate the use of the public packages @xcite and @xcite . the accompanying mathematica notebook ` onemassmb.nb ` has a detailed computer implementation of what follows . we begin by applying eq.([mbformula ] ) to the definition of the sunset integral eq.([sunsetdef ] ) , and then evaluating the integrals over the loop momenta , to get the following mellin - barnes representation : @xmath89 to make contact with results in the literature , we extract a factor of @xmath90 . the above is also obtained automatically by use of the public code @xcite . the next step is to resolve ( i.e separate ) the singularities in @xmath91 and the finite part by shifting the contour across the points @xmath92 and @xmath93 . this can be done in an automatic manner by use of the package @xcite . the result is an expression consisting of two terms : @xmath94 the first term contains the divergences , and the second piece is a finite one - fold contour integral which is to be evaluated by adding up residues . since the singularities in @xmath91 have been extracted , we can set @xmath91 to 0 in the second term . expressing the divergent piece as a laurent series around @xmath95 , we get : @xmath96 the convergent piece is calculated by summing up the residues at the points @xmath97 . the residues at non - zero integers @xmath98 for @xmath99 are given by : @xmath100 summing this up from @xmath101 to @xmath102 gives : @xmath103 the residue at @xmath104 = 0 is : @xmath105 combining the convergent and divergent pieces , we get the full result , expressed as a laurent series in @xmath91 : @xmath106 by pulling out a factor of @xmath107 and setting @xmath87 to 1 , this can be expressed more succinctly as : @xmath108 this reproduces the result derived in eq.(13 ) of @xcite . the result given in eq.([gs1mass ] ) above is prior to the application of any subtraction procedure . the @xmath49 scheme may be applied by multiplying both terms of eq.([intmbdef ] ) by the additional factor : @xmath109^{2\epsilon}\end{aligned}\ ] ] the inclusion of these two factors gives the following result for the @xmath49 subtracted single mass scale sunset integral : @xmath110 the ` tarcer ` package @xcite has the added functionality of performing a laurent series expansion in the small parameter @xmath111 for the master integrals . the command for such an expansion is : ` tarcerexpand [ expression , d \rightarrow 4- 2\epsilon ] ` for one mass - scale sunsets , using this feature , ` tarcer ` can be used directly to derive expressions for the integrals @xmath14 , @xmath33 , @xmath112 , @xmath113 , @xmath18 , @xmath114 , i.e. for all the sunset results that appear in @xcite . this has been demonstrated in the notebook ` onemasstarcer.nb ` , in which is derived a very comprehensive set of relations with detailed annotations , and completely verifies all the sunset relations in @xcite . note that the ` tarcerexpand ` command has been found to work for all the cases of interest , since this is a pure single mass scale example . we find that for other more complicated mass configurations , including the case when we have a single mass scale with @xmath6 , this command is unable to reproduce the laurent expansion of the integral . however , that ` tarcer ` can reproduce all the results for the sunsets in @xcite so efficiently indicates the power and utility of this package . there are eight possible independent mass configurations of the sunset master integrals with two masses . three of these fall into the pseudothreshold configurations , in which @xmath116 . in the two - loop calculation of the pseudoscalar meson masses and decay constants , these are the only two - mass configurations that arise . results for the pseudothresholds , calculated directly using an integral representation of the sunsets , are given in @xcite . we rederived the three pseudothreshold results @xmath117 , @xmath118 and @xmath119 using mellin - barnes representations , and expressions for these are given below : @xmath120\end{aligned}\ ] ] @xmath121\end{aligned}\ ] ] @xmath122\end{aligned}\ ] ] where @xmath123 . these results are valid for all real values of @xmath124 . the other two mass pseudothreshold expressions may be obtained from the above by a simple re - ordering of the masses and indices . in the notebook ` twomasspt.nb ` , we demonstrate the above calculations by means of the example @xmath125 . the evaluation of non - pseudothreshold two mass sunset configurations results in three complications that do not arise in the pseudothreshold case . firstly , their mellin - barnes representation is a linear combination of complex - plane integrals of which at least one is two - fold , and which therefore requires a more sophisticated approach in its evaluation . these two - fold mellin - barnes integrals result in nested infinite sums , many of which can not be expressed as common analytic functions . therefore , completely analytic expressions for these integrals can not be obtained easily , and we are forced instead to take as many terms of these sums as yields the degree of accuracy we desire . secondly , the specific form of these infinite series depends on the numerical values of the two masses @xmath87 and @xmath126 , or more specifically their ratio @xmath127 . thirdly , there exists a range of values of @xmath128 for which it is not possible to use the mellin - barnes method ( given the current state of the art ) to evaluate these integrals . for these values of @xmath128 , recourse must be had to other techniques , such as expansion in the external momentum @xmath62 . the non - pseudothreshold mass configurations do not appear in the calculation of the pseudoscalar meson masses and decay constants to two - loops in chiral perturbation theory , but they may appear elsewhere . thus for completeness we provide results for these as well in appendix a. the notebook ` twomassresults.nb ` contains all the pseudothreshold and non - pseudothreshold two mass scale sunset integrals . three mass scale sunset integrals result in two - fold mellin barnes representations , which can be evaluated using the method of @xcite . however , for purposes of evaluating the pion mass and decay constant , we take an expansion in the external momentum @xmath62 : @xmath130 for the pion mass and decay constant the external momentum is always @xmath131 , which is much smaller than the @xmath132 and @xmath133 that can appear in the propagators . therefore , the above series converges fairly fast , and only a few of higher order terms are required . for integrals with @xmath134 or @xmath27 , the mellin - barnes approach may be more suitable . the derivatives of the integrals above can be evaluated using a combination of eq.(6 ) and ` tarcer ` @xcite . it turns out that derivatives to all orders of the sunset integral with @xmath6 can be expressed in terms of the single master integral @xmath135 given in eq.([k111 ] ) . [ [ two - fold - mellin - barnes - representations - hd_111-left - m_k2-m_k2-m_pi2-m_eta2-right ] ] two - fold mellin - barnes representations : @xmath136 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ for the three mass scale sunset integrals in which the external momentum is not the smallest parameter , such as those that appear in the kaon and eta masses and decay constants , the expansion in @xmath62 does not converge well . an expansion in one of the propagator masses must also be precluded as they lead to infrared divergences . the simplest method by which to obtain analytic expressions for these integrals to the order desired is by evaluating their two - fold mellin - barnes representation , a detailed explanation of which is given in @xcite . in this section , we list the main intermediate results in the evaluation of @xmath137 to exemplify the method in brief . the first step is to find the mellin - 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we demonstrate the use of several code implementations of the mellin - barnes method available in the public domain to derive analytic expressions for the sunset diagrams that arise in the two - loop contribution to the pion mass and decay constant in three - flavoured chiral perturbation theory .
we also provide results for all possible two - mass configurations of the sunset integral , and derive a new one - dimensional integral representation for the one mass sunset integral with arbitrary external momentum .
thoroughly annotated mathematica notebooks are provided as ancillary files , which may serve as pedagogical supplements to the methods described in this paper .
lu tp 16 - 43 + august 2016 * an analytic approach to sunset diagrams in chiral perturbation theory : theory and practice * + @xmath0 centre for high energy physics , indian institute of science , + bangalore-560012 , karnataka , india + @xmath1department of astronomy and theoretical physics , lund university , + slvegatan 14a , se 223 - 62 lund , sweden + @xmath2department of physics and astronomy , university of delaware , + newark , de 19716 , usa +
| 11,714 | 318 |
the development of coherent electromagnetic sources of higher and higher intensity with increasing photon energy up to the hard x - ray range motivates the theoretical study of the change of the processes of strongly bound systems , such as nuclear processes , by these intense fields ledingham . in this paper , the change of the nuclear photoeffect due to the presence of an intense coherent electromagnetic field is studied . this process is analogous to the laser - assisted x - ray photo effect ( x - ray absorption ) , a process which was already discussed @xcite in the late 80 s taking into account gauge invariance @xcite , @xcite . the laser - assisted nuclear photoeffect ( lanp ) and the laser - assisted x - ray photo effect ( x - ray absorption ) are laser - assisted bound - free transitions . the difference between them lies in the charged particle ( proton or electron , respectively ) which takes part in these processes . although the lanp was recently investigated far from the threshold and neglecting the effect of the coulomb field of the remainder nucleus @xcite , in the case of the laser - assisted x - ray absorption processes it was found that the most interesting changes due to the presence of the laser field appear near the threshold @xcite , @xcite . thus , applying the results of @xcite , the lanp is reexamined in a gauge invariant manner and near the threshold , where the hindering effect of the coulomb field of the remainder nucleus is very large so that it must be taken into account . the effect of the coulomb field of the remainder nucleus on the transition rate is approximately taken into account . the laser - modified differential cross section is compared to the laser - free differential cross section , and it is shown that their ratio does not depend on nuclear parameters in the two types of initial nuclear states investigated and on the state of polarization of the @xmath0 radiation , but it has only a laser parameter dependence . the process investigated can be symbolically written as @xmath1where @xmath2 denotes the target nucleus of mass number @xmath3 and of charge number @xmath4 . the target nucleus absorbs a @xmath0 photon symbolized by @xmath5 , and @xmath6 laser photons take part in the process which is symbolized by @xmath7 . @xmath8 and @xmath9 correspond to @xmath10 laser photon emission and absorption , respectively . as a result , a free proton @xmath11 is emitted and the remainder nucleus is @xmath12 . the calculation is made in the radiation @xmath13 gauge , and in the long wavelength approximation ( lwa ) of the electromagnetic fields , the recoil of the remainder nucleus and the initial momentum carried by the laser and @xmath0 fields are neglected . in the case of a circularly polarized monochromatic wave for the vector potential of a laser field , @xmath14 $ ] is used . @xmath15 is the angular frequency of the laser . the amplitude of the corresponding electric field @xmath16 . the frame of reference is spanned by the unit vectors @xmath17 , @xmath18 and @xmath19 . the vector potential describing the gamma radiation is @xmath20 , with @xmath21 the energy and @xmath22 the unit vector of the state of polarization of the @xmath0 photon , and @xmath23 the volume of normalization . it is shown in @xcite that the electromagnetic transition amplitudes of a particle ( proton ) of rest mass @xmath24 and of charge @xmath25 in the presence of a laser field are determined by the matrix elements of the operator @xmath26 with the eigenstates of the instantaneous energy operator @xmath27 in both ( @xmath28 and @xmath29 ) gauges . ( @xmath25 is the elementary charge and the superscript @xmath30 refers to the gauge . ) accordingly , the gauge - independent s - matrix element can be written as@xmath31where @xmath32 and @xmath33 are the initial and final states of the proton in the same gauge and @xmath34 is the reduced planck constant . our calculation is carried out in the radiation @xmath35 gauge because of the choice of the final state of the proton ( see below ) . the initial state of the proton has the form @xmath36where @xmath37 is a stationary nuclear state of separation energy @xmath38 of the proton . the @xmath39 factor , where@xmath40 , appears because of gauge transformation since @xmath41 is the eigenfunction of the instantaneous energy operator , @xmath42 in the @xmath28 gauge . @xmath43 is the nuclear potential and @xmath44 is the coulomb potential felt by the proton initially , and the superscript @xmath45 refers to the @xmath28 gauge . the modification of the initial state due to the laser field is neglected since the direct effect of the intense laser field on the nucleus has been found to be negligible @xcite at the laser parameters discussed . it is also supposed that the initial nucleus does not have an excited state which is resonant or nearly resonant with the applied @xmath0 radiation . if similarly to @xcite the modification of the final state due to the strong interaction is neglected , then in the final state and in the @xmath29 gauge the instantaneous energy operator @xmath46 reads @xmath47where the superscript @xmath48 refers to the radiation @xmath35 gauge and @xmath49 is the coulomb potential of the remainder nucleus . an approximated solution of @xmath50 , i.e. an approximated time dependent state of a particle in the laser plus coulomb fields , is the coulomb - volkov solution of a proton of wave number vector @xmath51 @xcite , @xcite : @xmath52 here @xmath53 is the coulomb function , i.e. the wave function of a free proton in a repulsive coulomb field of charge number @xmath54 , @xmath23 denotes the volume of normalization , @xmath55 is the relative coordinate of the two particles . @xmath56 ) , \label{hyperg}\]]where @xmath57is the sommerfeld parameter , with @xmath58 the fine structure constant , and it is supposed that @xmath24 is much less than the rest mass of the remainder nucleus . @xmath59 is the confluent hypergeometric function and @xmath60 is the gamma function @xcite . the function @xmath61 \label{ft}\]]where @xmath62here the polar angles of the wave number vector @xmath63 of the outgoing proton are @xmath64 and @xmath65 , i.e. they are the polar angles of the direction in which the proton is ejected . in the low - energy range ( @xmath66 , where @xmath48 is the radius of a nucleon ) and for @xmath67 the long wavelength approximation yields @xmath68 -1 } } , \label{cb2}\]]which is the square root of the so - called coulomb factor . ( the coulomb factor @xmath69 describes well e.g. the coulomb correction to the spectrum shape of beta decay @xcite . ) for the final state of a proton of wave number vector @xmath63 , the lwa of the nonrelativistic coulomb - volkov solution @xmath70 is used . @xmath71with @xmath72 , that is the energy of the outgoing proton in the intense field , where @xmath73 is the ponderomotive energy . substituting @xmath74 , @xmath75 into @xmath76 and using @xmath77 one can obtain the following form of the gauge - independent s - matrix element as @xmath78f^{\ast } ( t)\frac{\partial } { \partial t}g\left [ \overrightarrow{q}\left ( t\right ) \right ] dt , \label{sfi}\]]where @xmath79is the fourier transform of the initial stationary nuclear state @xmath37 of the proton and @xmath80(equation @xmath81 can be obtained directly with the aid of eq.(27 ) of @xcite . ) using the @xmath82 identity@xmath83with @xmath77 , i.e. @xmath84 . the @xmath85 term of the last factor of @xmath86 can be neglected if the pure intense field induced proton stripping process is negligible since this term describes the process without the gamma photon . furthermore , the ratio of the amplitudes of @xmath87 and @xmath88 equals @xmath89 . therefore the @xmath90 approximation is justified to use , where @xmath91 . the relative strength of the @xmath92 and @xmath93 terms is characterized by the parameter @xmath94 . in the laser free case @xmath95 , where @xmath96 is the difference of the photon energy and the proton separation energy . numerical estimation shows that @xmath97 near @xmath98 @xmath99 used here and in the case of laser photon energy and intensity values discussed . therefore the @xmath100 term is the leading one in the last factor of @xmath86 . as to the radiation field dependence of @xmath101 , the effect of @xmath102 is negligible in @xmath103 and thus @xmath104 . it was shown above that the amplitude of oscillation of @xmath105 due to the intense field can be neglected . therefore @xmath106 can be used in @xmath101 , and @xmath107 results @xmath108 using the jacobi - anger formula in the fourier series expansion of @xmath109 @xcite the s - matrix element can be written as@xmath110i}{v}\chi _ { c}(q)\times \label{sfi2 } \\ & & \times \left ( \frac{\partial } { \partial q}g\right ) _ { q = q}\frac{e\sqrt{2\pi \hbar \omega _ { \gamma } } } { \hbar } \xi j_{n}(\alpha ) e^{-in\eta _ { 0 } } , \notag\end{aligned}\]]where @xmath111 , @xmath112 is a bessel function of the first kind , and @xmath113 the terms , that are small if the @xmath114 condition is fulfilled ( @xmath115 and @xmath116 see below ) , were neglected in the calculation . in this paper two cases of initial nuclear energy @xmath117 of the initial state having a different type of space - dependent part @xmath118 are considered in order to show the general nature of the effect of the laser on the process . the one case is the @xmath119 one - proton halo isotope of separation energy @xmath120 @xmath121 firestone and of initial state @xmath122 , with @xmath123 and @xmath24 the rest mass of the proton @xmath124 . although the proton rest mass is more than 12 % of the total rest mass of @xmath119 and the approximation , the fact that @xmath24 is much less than the rest mass of the remainder nucleus , is not very good , but following @xcite we investigate @xmath119 . in the other case the initial state is the @xmath125 shell model state @xcite of the form @xmath126 , with @xmath127 ^{1/2}/\gamma ( \frac{3}{2})$ ] where @xmath128 is the quantum number of the nuclear shell model and @xmath129 denotes the gamma function . @xmath130 is the confluent hypergeometric function , @xmath131 , @xmath132 , @xmath24 is the nucleon rest mass and @xmath133 is the shell model angular frequency ( @xmath134 @xmath121 , @xmath135 is the nucleon number @xcite , and @xmath136 @xmath137 ) . the differential cross section of lanp has the form@xmath138where @xmath139 is the differential solid angle around the direction of the outgoing proton . @xmath140 is the smallest integer fulfilling the @xmath141 condition and @xmath142 is the same in both cases of the initial state . ( the cases @xmath8 and @xmath9 correspond to @xmath10 laser photon emission and absorption , respectively . ) the partial differential cross section @xmath143with@xmath144 _ { q = q_{n}}^{2}.\]]here @xmath145 } /\hbar $ ] , @xmath146 , @xmath147 is the reduced compton wavelength of the proton and @xmath148.@xmath149@xmath150 is a bessel function of the first kind , with @xmath151 . in the variable @xmath152 the coulomb factor @xmath153 reads as@xmath154 -1\right ] } , \label{fjk}\]]where @xmath155 with @xmath156 the charge number of the remainder nucleus . the coulomb factor causes a strong hindering of the effect in both the laser assisted and laser free cases . near the threshold @xmath157 the @xmath158 _ { q_{n}}^{2}\varpropto \varepsilon _ { n}^{2}$ ] in the case of the two types of initial state discussed and @xmath159here @xmath160 is constant , it depends on the form of the initial state and@xmath161 in the laser free case @xmath162 using @xmath163 , @xmath164 and @xmath165 at @xmath166 , the differential cross section near above the threshold@xmath167with @xmath168 ^{-1}$ ] . the ratio @xmath48 of the laser - assisted and the laser - free differential cross sections , @xmath169where @xmath170 [ @xmath171 divided by @xmath172 , equals the ratio of the rates of the corresponding processes in an elementary solid angle in a given direction of the outgoing proton . the rate of change in one channel , @xmath173with @xmath174 in the variable @xmath152 . so @xmath175 and @xmath48 describe the change caused by the intense coherent field independently of the state of @xmath0 polarization and the initial states applied . in our numerical calculation , the laser photon energy @xmath176 @xmath177 . first the case in which the outgoing proton moves in the plane of polarization of the laser beam ( @xmath178 @xmath179 ) is investigated in the case of @xmath180 . figure 1 shows the laser photon number dependence of @xmath175 at @xmath98 @xmath99 with laser intensity @xmath181 @xmath182 . the intensity dependence of @xmath48 has been investigated with @xmath98 @xmath99 . @xmath48 increases linearly from @xmath183 at @xmath184 @xmath182 up to @xmath185 at @xmath181 @xmath182 . figure 2 depicts the @xmath142 dependence of @xmath48 with @xmath181 @xmath182 . figure 3 shows the @xmath64 dependence of @xmath175 in the case @xmath181 @xmath182 at ( a ) @xmath186 ( b ) @xmath187 ( c ) @xmath188 and ( d ) @xmath189 . finally fig . 4 depicts the @xmath64 dependence of @xmath175 at @xmath188 with different laser intensities : ( a ) @xmath190 @xmath182 ( b ) @xmath191 @xmath182 ( c ) @xmath181 @xmath182 . the numerical calculation in all the cases discussed above has been repeated at @xmath192 , and negligible change has been found . to compare our results and the results of @xcite we investigate our formulas in the @xmath193 limit . in this case @xmath194 , @xmath195 , @xmath196and @xmath197thus in this limit the @xmath6 dependence disappears from the argument of the bessel function and @xmath198 the cross sections obtained in @xcite are symmetric in @xmath6 around @xmath199 ( see the figures of @xcite ) and the total cross section is asserted to be unaffected by the laser radiation . this corresponds to @xmath200 . in contrast , our result is significantly asymmetric in @xmath6 ( see fig . 1 . ) , which is a consequence of the @xmath152 dependence of @xmath175 [ see @xmath201 . the change ( increase ) of the kinetic energy of the proton is manifested in the increase of @xmath152 from @xmath202 up to @xmath203 . the increase of @xmath152 with increasing @xmath6 causes an asymmetry in the @xmath6 dependence of fig . the sum of the changes ( increments ) in the different channels results in a moderate increment of @xmath48 ( @xmath204 ) as can be seen in fig . 2 . summarizing , one can say that near the threshold , @xmath48 which measures the change of the rate of the lanp , has minor @xmath142 and intensity dependence , and negligible @xmath54 dependence . furthermore , @xmath175 , which is the rate of change in one channel ( at a definite laser photon number ) , has a significant laser photon number and @xmath64 dependences . @xmath48 and @xmath175 are the same in the cases of the two different initial states considered . since the numerical results obtained seem to be independent of the initial nuclear states chosen , it can be expected that @xmath48 and @xmath175 have a minor dependence from the form of the initial state in general . regarding the experimental situation at such a high intensity , it is hard to distinguish photo - protons from background protons that arise as a result of interaction with the hot electron plasma created by the intense laser field . we also have to mention that in an experiment the gamma ray pulse from an accelerator must be synchronized with an intense ( e.g. attosecond ) pulse of a laser system . moreover , in the case of @xmath119 , which has a short half - life of about @xmath205 @xmath206 , the @xmath119 nuclei must be created in situ in the laser beam by a nuclear reaction . fortunately , most of the heavier nuclei have protons of @xmath125 shell model state ( the other case investigated ) in their stable ground state . the wispy target determined by the focal spot of the focused intense laser beam , the low repetition rate of the laser system , and the angular resolution of the proton detector together make it very challenging to carry out a successful near - threshold , laser - modified proton emission experiment that could produce significant counting statistics .
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the change of the probability of proton emission in the nuclear photoeffect due to an intense coherent ( laser ) field is discussed near the threshold , where the hindering effect of the coulomb field of the remainder nucleus is essential .
the ratio of the laser - assisted and laser - free differential cross section is deduced and found to be independent of the polarization state of the @xmath0 field and the two types of initial nuclear state considered .
the numerical values of this ratio are given at some characteristic parameters of the intense field .
| 5,058 | 121 |
the study of tachyons and their condensation processes has been a longstanding challenge in string theory . during recent years , the interest in branes has added significantly to the relevance of this issue since open string tachyons occur in various configurations of stable branes , the most important being the brane - antibrane system . fortunately , unstable brane configurations turned out to be much more tractable than bulk backgrounds with instabilities such as e.g. the 26-dimensional bosonic string . in fact , various different approaches have been employed over the last five years , including string field theory ( see e.g. @xcite and @xcite for a review ) , effective field theory models ( see @xcite for some early work ) , and world - sheet conformal field theory ( see e.g. @xcite and references therein ) . in @xcite , sen initiated the study of _ exact time - dependent solutions_. one goal of his work was to find a world - sheet description of so - called s - branes @xcite , i.e. of branes that are localized in time . in this context he proposed to add a @xmath1-shaped boundary interaction to the usual world - sheet model of d - branes in a flat space - time , @xmath2 \ = \ \frac{1}{4\pi } \int_\sigma d^2 z \ , \eta_{\mu\nu } \partial x^\mu \bar \partial x^\nu + \int_{\partial \sigma } d u \ , \lambda \cosh x^0(u ) \ \ .\ ] ] here , @xmath3 denotes the time - coordinate and we have chosen units in which @xmath4 . with such an interaction term the open string tachyon becomes large at early and late times and hence , according to an earlier proposal of sen @xcite , is believed to dissolve the brane for @xmath5 . unfortunately , the world - sheet model ( [ senact ] ) appears to be ill - defined without an additional prescription to select the proper vacuum . to resolve this issue , sen argued that the appropriate choice for super - string computations would be determined by wick rotation from the corresponding euclidean theory . this suggestion relates the study of open string tachyon condensation to the so - called boundary sine - gordon model @xmath6 \ = \ \frac{1}{4\pi } \int_\sigma d^2 z\ , \partial x \bar \partial x + \int_{\partial \sigma } du \ , \lambda \cos x(u)\ ] ] in which @xmath7 is a field with space - like rather than time - like signature . all spatial coordinates @xmath8 have been suppressed here since their contribution to the model is not affected by the interaction . the boundary sine - gordon theory has been studied rather intensively , see e.g. @xcite . let us briefly review some of the most important results . to begin with , we point out that the boundary interaction is exactly marginal so that the theory is conformal for all values of @xmath9 . properties of the model , however , depend crucially on the strength @xmath9 of the boundary potential . in fact , it is well known that variations of @xmath9 allow to interpolate smoothly between neumann and dirichlet boundary conditions . the former appear for all integer values of @xmath9 ( in particular for @xmath10 , of course ) while the latter are reached when @xmath11 . at these points , the theory describes a one - dimensional infinite lattice of d0 branes . from the geometric pictures we can infer that the spectra of boundary excitations must also depend drastically on the parameter @xmath9 . in fact , at the points @xmath12 with neumann boundary conditions , the open string spectrum is continuous . if we now start tuning @xmath9 away from these special values , the spectrum develops band gaps which become wider and wider until we reach the dirichlet points @xmath13 at which the spectrum is discrete . the first computation of the spectrum for generic values of @xmath9 can be found in @xcite ( see also @xcite for a much more elegant derivation ) . despite of these significant insights into the structure of the boundary sine - gordon model , there are several important quantities that remain unknown . this applies in particular to the boundary 2- and 3-point functions and the bulk - boundary operator product expansions . in the string theoretic context , these missing data determine the time - dependence of open string couplings on a decaying brane and the back - reaction to the bulk geometry . our desire to understand such important quantities is one of the main motivations to study the following , closely related world - sheet theory @xcite , @xmath14 \ = \ \frac{1}{4\pi } \int_\sigma d^2 z \ , \eta_{\mu\nu } \partial x^\mu \bar \partial x^\nu + \int_{\partial \sigma } d u \ , \mu_b \exp x^0(u ) \ \ .\ ] ] this model has been named time - like boundary liouville theory . since the tachyon vanishes in the far past , the model seems to describe a half - brane , i.e. a brane that dissolves as time progresses . after wick rotation @xmath15 , the theory of the time coordinate becomes . this might be relevant for comparisons with previous results in models with exponential interactions . ] @xmath16 \ = \ \left ( \frac{1}{4\pi } \int_\sigma d^2 z\ , \partial x \bar \partial x + \mu \exp 2b x(z,{\bar{z } } ) + \int_{\partial \sigma } du \ , \mu_b \exp b x(u)\right)^{\mu=0}_{b = i } \ \ .\ ] ] we have written the interaction term for general parameters @xmath17 and also added a similar interaction in the bulk , mainly to emphasize the relation with boundary liouville theory . what makes this relation so valuable for us is the fact that boundary liouville theory has been solved over the last years @xcite . needless to stress that the solution includes explicit formulas for the bulk - boundary structure constants @xcite and the boundary 3-point couplings @xcite . there is one crucial difference between liouville theory and the model we are interested in : whereas the usual liouville model is defined for real @xmath17 , our application requires to set @xmath18 . a continuation of results in liouville theory from @xmath19 to @xmath0 ( i.e. @xmath18 ) might appear to be a rather daring project , even more so as a naive inspection of the world - sheet action would suggest the @xmath0 model could not possibly be unitary . nevertheless , we shall show below that the theory is entirely well - defined and unitary . for the pure bulk theory , a similar result was established in @xcite . it was shown there that the bulk 3-point couplings of liouville theory possess a @xmath18 limit which is well - defined for real momenta of the participating closed strings . the @xmath18 theory , however , is no longer analytic in the momenta . furthermore , the limit turned out to agree with the @xmath0 limit of unitary minimal models which was constructed by runkel and watts in @xcite . our analysis here extends these findings to the boundary theory . we shall see that the latter is considerably richer than the bulk model . some qualitative properties of the @xmath0 boundary model can be understood using no more than a few general observations . to begin with , let us reconsider the simpler bulk model . recall from ordinary liouville theory that it has a trivial dependence on the coupling constant @xmath20 . since any changes in the coupling can be absorbed in a shift of the zero mode , one can not vary the strength of the interaction . this feature of liouville theory persists when the parameter @xmath17 moves away from the real axis into the complex plane . as we reach the point @xmath18 , our model seems to change quite drastically : at this point , the ` liouville wall ' disappears and the potential becomes periodic . standard intuition therefore suggests that the spectrum of closed string modes develops gaps at @xmath18 . but since the strength of the interaction can not be tuned in the bulk theory , the band gaps must be point - like . though our argument here was based on properties of the classical action which we can not fully trust , the point - like band - gaps are indeed a characteristic property of the @xmath0 bulk theory @xcite . these band - gaps also explain why the couplings of liouville theory cease to be analytic when we reach @xmath0 . for the boundary theory , we can go through a similar argument , but the consequences are more pronounced . in the presence of a boundary , liouville theory contains a second coupling constant @xmath21 which controls the strength of an exponential interaction on the boundary of the world - sheet and is a real parameter of the model . in fact , the freedom of shifting the zero mode can only be used for one of the couplings @xmath20 or @xmath21 . once more , the boundary potential becomes periodic at @xmath18 and hence the open string spectrum should develop gaps , as in the case of the bulk model . but this time , the width of these gaps can be tuned by changes of the parameter @xmath21 . hence , we expect the boundary liouville theory to possess gaps of finite width at @xmath0 . our exact analysis will confirm this outcome of the discussion . after these more qualitative remarks on the model we are about to construct , we shall summarize our main results in more detail . in section 3 we shall provide the boundary states for a family of boundary conditions of the @xmath0 liouville theory that is parametrized by one parameter @xmath22 . formulas for these boundary states are trivially obtained from the corresponding expressions in @xmath23 liouville theory and , as we shall also explain , through particular limits of boundary theories in minimal models . we shall then use both approaches to establish the existence of finite band gaps in the boundary spectrum of the @xmath0 boundary theories . more precisely , we show that boundary fields @xmath24 in the boundary theory with parameter @xmath25 carry a momentum whose allowed values are taken from the following set @xmath26 when @xmath25 is a half - integer , the set @xmath27 fills the entire real line . as we vary @xmath25 away from such half - integer values , the allowed momenta are restricted to intervals of decreasing width until we reach integer values of @xmath25 where the spectrum of boundary fields becomes discrete . in this sense the parameter @xmath25 interpolates between neumann - type and dirichlet - type boundary conditions . let us note that the boundary conditions with @xmath28 are related to those constructed by runkel and watts in @xcite . all other boundary theories , however , are new . it is worthwhile pointing out how closely the @xmath0 limit of liouville theory mimics the behavior of the boundary sine - gordon model . in both theories we can interpolate smoothly between continuous and discrete boundary spectra . even the geometric interpretation of the dirichlet points is very similar : in the context of the @xmath0 liouville model they describe a semi - infinite array of point - like branes from which individual branes can be removed through shifts of the boundary parameter @xmath25 . let us point out that finite arrays of such point - like branes are described by the @xmath0 limit of zz branes . that parametrize the zz branes correspond to the transverse position and the length of the finite array . ] hence , the picture we suggest for the dirichlet points of the @xmath0 fzzt branes provides a nice geometric interpretation for the well - known relation of fzzt and zz branes in liouville theory @xcite . there are two important quantities that we shall construct for all these new boundary theories . these are the boundary 2-point function and the bulk - boundary 2-point function . for the former we obtain @xmath29 \\[3 mm ] \mbox { where } & & \r^{s_1s_2}_{c=1}(p ) \ = \ \r^{is_1\ is_2}_{c=25 } ( 1-p)^{-1}\ \ . \end{aligned}\ ] ] detailed formulas for @xmath30 ( see eq . ( [ 2ptfctres1 ] ) ) and for the so - called reflection amplitude @xmath31 are spelled out in section 4 . here , we have expressed the result in terms of the reflection amplitude for the theory at @xmath32 . it is quite remarkable that the reflection amplitudes of the two models are related in such a simple way . let us point out , though , that the boundary reflection amplitude for the @xmath0 theory is only defined within a subset @xmath33 of momenta @xmath34 . it is more difficult to write down the bulk - boundary coupling @xmath35 of the @xmath0 model , i.e. the coefficient in the 2-point function of a bulk field @xmath36 with one of the allowed boundary fields , @xmath37 we shall prove below that this coupling @xmath35 is given by @xmath38 & & \hspace*{-3.5cm}\nonumber \times \,\bigg ( \frac{e^{-2\pi ip_{\alpha } ( s+1)}}{1-e^{-4\pi ip_{\alpha } } } \frac{g(y_0)^2 e^{-\frac{ih ( y_{0})}{2\pi}}}{y_0^{2 } \ , h''(y_0)}\ , \exp \int_0^\infty \frac{dt}{t } \ \frac{\left(e^{-p_\b t } - e^{2p_\a t}\right ) \sinh^2\frac{p_{\b}t}{2}}{\sinh^2\frac{t}{2}}-\ ( p_{\alpha}\rightarrow -p_{\alpha } ) \bigg ) \end{aligned}\ ] ] where @xmath39 and the functions @xmath40 and @xmath41 are defined through @xmath42 h ( y)&= & { \text{li}_{2}}(a_{0})-{\text{li}_{2}}(a_{0}y)+{\text{li}_{2}}(b_{0})-{\text{li}_{2}}(b_{0}y)\nonumber \\ & & -{\text{li}_{2}}(c_{0})+{\text{li}_{2}}(c_{0}y)-{\text{li}_{2}}(1)+{\text{li}_{2}}(y)+ \log y \log z_{0 } \ \ .\end{aligned}\ ] ] here , @xmath43 denotes the dilogarithm and we have abbreviated @xmath44 and the parameter @xmath45 is one of the two solutions of the following quadratic equation @xmath46 for more details and explicit formulas , see section 5 . we also show that the coefficients in the bulk - boundary operator product expansion vanish whenever the open string momentum @xmath47 lies within the band gaps , i.e. when @xmath48 . this provides another non - trivial consistency check for the couplings we propose . let us add that correlators with boundary insertions in a similar model where also discussed recently in @xcite . the techniques used there , however , only allowed to determine such correlations functions for a discrete set of boundary momenta @xmath49 . in this section we shall mainly review some results on the @xmath0 limit of bulk liouville theory @xcite ( see also @xcite ) . our derivation of the 3-point couplings , however , is different from the one given in @xcite . the new construction is simpler and uses some of the same techniques that we shall also employ to analyze the boundary model later on . to begin with , let us recall a few standard facts about the usual @xmath50 liouville theory . as in any bulk conformal field theory , the exact solution of the liouville model is entirely determined by the structure constants of the 3-point functions for the ( normalizable ) primary fields @xmath51 where @xmath52 . these fields are primaries with conformal weight @xmath53 under the action of the two virasoro algebras whose central charge is @xmath54 . for real values of @xmath17 , the couplings of three such fields are given by the following expression @xmath55 where @xmath56 and @xmath57 are linear combinations of @xmath58 , @xmath59 @xmath60 is a quotient of ordinary @xmath61-functions and the function @xmath62 is defined in terms of barnes double gamma function @xmath63 ( see appendix a.2 ) by @xmath64 the solution ( [ bulktpf ] ) was first proposed several years ago by h. dorn and h.j . otto @xcite and by a. and al . zamolodchikov @xcite . crossing symmetry of the conjectured 3-point function was then checked analytically in two steps by ponsot and teschner @xcite and by teschner @xcite . it is well known that barnes double gamma function is analytic for @xmath65 . in the limit @xmath18 , however , the function becomes singular . nevertheless , the combinations of double gamma functions that appear in the various couplings of liouville theory turn out to be well defined , though they are no longer analytic . most of our analysis of the @xmath18 limit is based on the following relation between double gamma functions with parameter @xmath17 and @xmath66 , @xmath67 which holds for @xmath68 . here we have introduced @xmath69 and the so - called q - pochhammer symbols @xmath70 ( see appendix a.1 ) . relation ( [ dgdgrel ] ) follows from a rotation of the contour in the integral representation ( [ barnesg ] ) of the double gamma function . the factor involving the q - pochhammer symbols assembles contributions from the poles in the integrand of the double gamma function ( see appendix a.2 for details ) . using formula ( [ pochasymp ] ) for the behavior of our q - pochhammer symbols at @xmath18 we conclude @xmath71 - 1/2 ) } \right)^{-(p+1-n)/2 } \ , \times \\[2 mm ] & & \hspace*{2 cm } \times \ , e^{\frac{1}{\e } \left ( li_2(e^{-2\pi i p } ) - li_2(1)\right ) } \gamma_2^{-1}(1-p|1 ) \ , \left(1 + o(\e^{0})\right ) \label{gasym } \nonumber\end{aligned}\ ] ] with @xmath72 and @xmath73 $ ] denoting the integer part of @xmath34 . note that the double gamma function diverges as we send @xmath74 ( or @xmath75 ) with a divergent factor involving the dilogarithm @xmath76 . in the expression for closed string couplings , barnes double gamma functions appears only through @xmath62 , and using a standard dilogarithm identity ( see appendix a.3 ) one may show that @xmath77 - 1/2 ) } \ , \upsilon ( 1-p|1)^{-1 } \ \ .\ ] ] here , @xmath78 is a continuous periodic function which is quadratic on each interval @xmath79 $ ] for @xmath80 , @xmath81 - 1/2)^{2 } \ \ .\ ] ] observe that the divergent factor in @xmath62 is much simpler than for the double gamma functions it is built from . in the function @xmath62 , the divergence comes from a product of q - pochhammer symbols . the latter is closely related to jacobi s function @xmath82 and its divergence may be controlled more directly using modular properties of @xmath83-functions ( see @xcite ) . using once more the formula ( [ dgdgrel ] ) and some simple facts about barnes double gamma function is is not difficult to show that @xmath84 with the help of the two asymptotic expressions and , we find @xmath85 - 1/2 ) } \prod _ { j=1}^{3 } \frac{e^{\pi i p_{j}}e^{-\pi i ( 2p_{j}-1 ) ( p_{j}-[2p_{j}]-1/2)}}{e^{-\pi i 2\tilde{p}_{j } ( \tilde{p}_{j}-[2\tilde{p}_{j}]-1/2)}}\\ & \ \times \ \frac{\upsilon ( 1 - 2\tilde{p}|1)}{\upsilon ( 1|1 ) } \prod _ { j=1}^{3}\frac{\upsilon(1 - 2\tilde{p}_{j}|1 ) } { \upsilon ( 1 - 2p_{j}|1)}\ ( 1+o ( \epsilon^{0 } ) ) \label{bulktpfasymp}\end{aligned}\ ] ] where @xmath86 it can be shown that @xmath87 , so that the exponential is never diverging , but it can suppress the whole three - point coupling . we write @xmath88+f_{j}$ ] and find that @xmath89 is zero if either @xmath90+[2p_{2}]+[2p_{3}]\ \ \text{even , and } \ |f_{1}-f_{2}|\leq f_{3}\leq \min \{f_{1}+f_{2},2-f_{1}-f_{2 } \}\ \ , \ ] ] or @xmath91+[2p_{2}]+[2p_{3}]\ \ \text{odd , and } \ |f_{1}-f_{2}|\leq 1-f_{3}\leq \min \ { f_{1}+f_{2},2-f_{1}-f_{2}\ } \ \ , \ ] ] otherwise @xmath89 is strictly greater than zero and the three - point coupling vanishes for @xmath92 . in more formal terms , the function @xmath93 assumes the value @xmath94 when one of the above conditions is fulfilled and it vanishes otherwise . with the new function @xmath95 being introduced , we can use eq . to recast the 3-point couplings in the form @xmath96 } \ , p ( 2p_{1},2p_{2},2p_{3 } ) \times \nonumber\\[2 mm ] & & \hspace*{3 cm } \times \ \frac{\upsilon ( 1 - 2\tilde{p}|1)}{\upsilon ( 1|1 ) } \prod _ { j=1}^{3}\frac{\upsilon ( 1 - 2\tilde{p}_{j}|1)}{\upsilon ( 1 - 2p_{j}|1 ) } \ \ . \label{bulk3ptfunc}\end{aligned}\ ] ] in our conventions , @xmath97 , so up to a factor @xmath98 we reproduce precisely the result of @xcite ( see eqs.(4.8 ) and ( 5.3 ) of that article ) . to obtain a finite limit , the bulk fields should be renormalized by a factor @xmath99 and correlators on the sphere receive an additional factor @xmath100 due to a rescaling of the vacuum ( see section 4.2 in @xcite for a related discussion ) . this prescription produces a finite 3-point function and it does not affect the 2-point function . at first , the form of the couplings , and in particular the non - analytic factor @xmath95 , may seem a bit surprising . it is therefore reassuring that exactly the same couplings emerged several years ago through a @xmath0 limit of unitary minimal models @xcite . runkel and watts investigated the consistency of these couplings and demonstrated that crossing symmetry holds when all the half - integer momenta @xmath34 are removed from the spectrum . as we have argued in the introduction , the appearance of such point - like band - gaps is rather natural from the point of view of liouville theory . now we are ready to analyze the boundary model . we shall begin with the so - called fzzt boundary conditions of liouville theory . they describe extended branes with an exponential potential for open strings ( see eq . ( [ blact ] ) ) . taking the @xmath0 limit of their 1-point function is straightforward and results in an expression with analytic dependence on the closed string momentum . these @xmath0 boundary states are parametrized through one real parameter @xmath25 and for non - integer values of @xmath25 they can be reproduced by taking an appropriate limit of boundary conditions in unitary minimal models . the latter construction will provide a first derivation of the spectrum of boundary vertex operators , including the precise position and width of the band - gaps . when @xmath28 , the bands become point - like . at these special values of the parameter @xmath25 , our two constructions through liouville theory and minimal models do not produce the same boundary states , though they still give closely related results . from minimal models we obtain the discrete set of boundary conditions that is already contained in the work of runkel and watts . these agree with the @xmath0 limit of zz branes in liouville theory and they possess a simple geometric interpretation as finite arrays of point - like branes . the relation between zz and fzzt branes in liouville theory then implies that we can interpret fzzt branes with @xmath28 as half - infinite arrays of point - like branes in the @xmath0 model . it is well known that for real @xmath17 , liouville theory admits a 1-parameter family of boundary conditions which correspond to branes stretching out along the real line . the boundary state of this theory was found in @xcite , @xmath101 & & \mbox{where } \ \ \ \mu_b^2 \sin \pi b^2 \ = \ \mu \cosh^2 \pi s b \nonumber \end{aligned}\ ] ] and @xmath102 , as usual . note that the coupling on the right hand side is analytic in @xmath17 and hence there is no problem to extend the fzzt brane boundary state into the @xmath0 theory , @xmath103 this boundary state is related to its lorentzian analogue @xcite by analytic continuation . it is , however , not entirely obvious that eq . ( [ 1ptfceq1 ] ) really provides the one - point function of a consistent boundary conformal field theory . we shall argue below that this is the case , at least as long as the parameter @xmath25 remains real . let us now see how such a boundary state of the @xmath0 model can arise from a limit of boundary minimal models . to begin with , we need to set up a few notations . we shall label minimal models by some integer @xmath104 so that @xmath105 . for representations of the corresponding virasoro algebra we use the kac labels @xmath106 with @xmath107 and @xmath108 along with the usual identification @xmath109 . recall that the boundary conditions of the theory are in one to one correspondence with the representations , i.e. that they are also labeled by kac labels @xmath110 . the associated one - point functions are given by @xmath111 where @xmath112 and @xmath113 is the conformal dimension of the primary field with kac labels @xmath106 , @xmath114 here , we have introduced the new quantity @xmath115 that can be regarded as a discrete analogue of the continuous momentum @xmath34 when @xmath116 . in taking the limit @xmath117 , the bulk fields @xmath118 of minimal models approach the fields @xmath119 in the continuum theory . the momentum variable @xmath34 is related to the kac - labels by @xmath120 in other words , the integer part of the quantity @xmath121 is determined by the distance of the kac label @xmath106 from the diagonal @xmath122 , whereas the fractional part @xmath123 $ ] corresponds to the ( rescaled ) position along the diagonal . for the a - series of minimal models , the bulk spectrum is obtained by filling every point of the kac table above the diagonal . when we send @xmath124 to infinity , approximately the same number of fields appears at any given distance from the diagonal of the kac table and these are distributed almost homogeneously along the diagonal @xmath125 . hence , the spectrum of @xmath34 in the @xmath0 limit is homogeneous . only integer values of @xmath121 are not part of this spectrum since the kac labels satisfy @xmath126 . this leads to the point - like band - gaps in the spectrum of bulk fields that we also saw emerging from liouville theory . to obtain a one - parameter family of branes with 1-point function given by eq ( [ 1ptfceq1 ] ) we propose the following ansatz , @xmath127 + [ s ] + 1 \ , , \ , [ f_s m]+ 1 \ , ) \ \ .\ ] ] here , @xmath22 is the parameter of the resulting limiting boundary condition . @xmath128 $ ] and @xmath129 denote the integer and fractional part of @xmath25 , respectively . the prescription ( [ prescript ] ) instructs us to pick a boundary condition that is represented by a point at distance @xmath128 $ ] from the diagonal of the kac table and to scale its projection to the diagonal with @xmath124 . our first aim now is to show that this recipe reproduces the 1-point function ( [ 1ptfceq1 ] ) in the @xmath0 liouville theory . the main idea is to rewrite the product of @xmath130-functions in the numerator of eq . ( [ 1ptfmm ] ) as follows @xmath131+[s]+1 ) ( r / t - r ' ) \ , \sin \pi ( [ f_s m ] + 1)(r - r't ) \nonumber \\[2 mm ] & & \hspace*{3 cm } \stackrel{m\rightarrow \infty}{\sim } \ \cos 2 \pi p s - \cos 4 \pi p ( [ f_s m]+1+[s]/2 ) + \dots \nonumber \end{aligned}\ ] ] with @xmath132 being the momentum in the @xmath0 theory . provided @xmath133 , the second term in the previous formula oscillates rapidly when we send @xmath124 to infinity and hence we can drop it in the limit ( recall that all correlation functions should be understood as distributions in momentum space ) . together with the other factors in the 1-point function ( [ 1ptfmm ] ) of minimal models , a short analysis similar to the one carried out in @xcite gives @xmath134 the result agrees with the 1-point function ( [ 1ptfceq1 ] ) of the @xmath0 liouville model up to some prefactors which may be absorbed by an appropriate renormalization of the bulk fields . if , on the other hand , the label @xmath25 is an integer , then our limit of boundary conditions in minimal models becomes @xmath135 the last identification with the @xmath0 limit of boundary states in liouville theory holds up to some factors that are due to the slightly different normalizations of bulk fields in minimal models and liouville theory ( see @xcite ) . our result ( [ mml ] ) is very reminiscent of a similar relation between localized and extended branes in liouville theory @xcite and we shall comment on the precise connection and its geometric interpretation in the @xmath0 model below . now that we have identified the construction of the boundary state from minimal models , it is instructive to study the associated spectrum of boundary fields . according to the usual rules @xcite , the boundary spectrum of the brane @xmath110 contains fields with kac labels @xmath136 and @xmath137 . using the identification @xmath138 we can assume that @xmath139 . hence , the kac labels of boundary fields fill every second lattice point within two triangles in the kac table ( see figure 1 ) . in the first triangle with corner at @xmath140 , the difference @xmath141 of kac labels is even . for kac labels in the second triangle , on the other hand , @xmath141 is an odd integer since the reflection @xmath142 shifts the difference of kac labels by one unit . recall that the quantity @xmath141 measures the distance of a point in the kac table from the diagonal and thereby determines the integer part of @xmath121 in the @xmath0 theory . we conclude that the two triangles contribute boundary fields with @xmath143 $ ] being even or odd integer , respectively . furthermore , points from the first triangle are uniformly distributed in the direction along the diagonal up to a maximal value @xmath144 . a similar observation holds true for the second triangle . after we have sent @xmath124 to infinity , this implies that the spectrum of @xmath34 is given by @xmath145 hence , the momentum @xmath34 has bands of width @xmath146 centered around integer momentum @xmath34 . the gaps between these bands widen while we vary @xmath25 between a half - integer and an integer value . once we reach a point @xmath147 , the boundary spectrum becomes discrete . branes in the @xmath0 limit of minimal models with a discrete open string spectrum have also been constructed by runkel and watts ( see also @xcite ) . actually , the branes that we obtained through the limit of minimal models when @xmath147 are identical to the limit of @xmath148 branes in minimal models and hence to the discrete boundary conditions that were found by runkel and watts . we wish to point out that these branes may also be obtained from the so - called zz branes of liouville theory . in general , the zz branes are labeled by a pair @xmath149 of positive integers and they possess the following 1-point functions , @xmath150 if we send the parameter @xmath17 to @xmath18 , these 1-point functions assume the following form @xmath151 we can rewrite the coupling on the right hand side with the help of the trigonometric identity @xmath152 to read off the following geometric interpretation of the zz brane with label @xmath149 in the @xmath0 theory : it corresponds to an array of @xmath153 point - like branes that are distributed equidistantly such that the rightmost brane is located at @xmath154 . this interpretation agrees with the geometry of branes that emerges from the coset construction of minimal models ( see @xcite ) . with this in mind , we shall also be able to interpret our fzzt branes at the dirichlet points @xmath28 . let us recall that the @xmath0 limit of fzzt branes with integer label @xmath25 differs from the limit of minimal model branes . the precise relation between their two boundary states is given in eq . ( [ mml ] ) . on its right hand side we consider the difference of one - point couplings for two fzzt branes whose labels differ by two units . this is claimed to be the same as the one - point coupling to the discrete brane with label @xmath148 . therefore , the brane configuration that is associated to the fzzt brane with integer label @xmath25 contains an additional point - like object with transverse position parameter @xmath25 along the real line . the latter gets removed as we shift from @xmath25 to @xmath155 . our observation suggests to think of the @xmath0 fzzt branes with integer @xmath25 as being built up from discrete objects , or more specifically , as some half - infinite array of point - like objects . they occupy points with even or odd integer position @xmath156 in target space , depending on whether @xmath25 is odd or even . changes of @xmath25 by one unit should therefore correspond to a simple translation of the entire array in the target space . at least qualitatively , such an interpretation in terms of point - like branes appears consistent with our previous study of brane spectra on fzzt branes . in fact , we noticed before that their bands shrink to points when @xmath25 reaches integer values . let us also stress that our geometric picture of the branes with integer @xmath25 is very similar to the interpretation of the dirichlet points in the boundary sine - gordon model ( see introduction ) . more support for the proposed geometric pictures comes from considering spectra involving the zz branes in the @xmath0 theory . the annulus amplitude for two zz branes with labels @xmath149 and @xmath157 was found in @xcite to be of the form @xmath158 \mbox{where } \ \ \chi_{m , n}(q ) & = & \eta^{-1}(q ) \ , \left ( q^{-(m / b+nb)^2/4}-q^{-(m / b - nb)^2/4}\right ) \\[2 mm ] & \stackrel{b = i}{= } & \eta^{-1}(q ) \ , \left ( q^{(m - n)^2/4}-q^{(m+n)^2/4}\right)\ \ . \end{aligned}\ ] ] note that in the @xmath0 theory , excitations of the zz branes have non - negative conformal dimensions . since @xmath159 are characters of irreducible virasoro representations at @xmath0 , the expression for the annulus amplitude coincides with what is obtained from the limit of cardy type branes in unitary minimal models . a particularly simple case arises from setting @xmath160 and @xmath161 , @xmath162 according to our previous discussion , this amplitude encodes the spectrum of open strings between a single point - like brane and an array of length @xmath163 . for the annulus amplitude between a zz brane @xmath149 and an fzzt brane at parameter @xmath25 one has @xmath164 \mbox{where } \ \ \chi_p(q ) & = & \eta^{-1}(q ) \ , q^{p^2}\ \ . \nonumber \end{aligned}\ ] ] here , @xmath165 denotes a summation in steps of two . let us point out that once more , the open string excitations between zz and fzzt branes have real , non - negative conformal dimensions when @xmath18 . this is in contrast to the situation for @xmath23 where the spectrum contains fields with complex dimensions if @xmath166 or @xmath167 . the spectrum between the @xmath140 branes and an fzzt brane of parameter @xmath25 contains a single primary field with conformal dimension @xmath168 , in perfect agreement with the construction from unitary minimal models . in fact , the construction of the @xmath0 brane we have proposed along with cardy s rule implies that there appears a single primary with kac label given by eq . ( [ prescript ] ) in the corresponding spectrum of the @xmath169 minimal model . when we send @xmath124 to infinity , this field approaches a @xmath0 primary field with momentum @xmath170 + [ s ] - [ f_s m ] + \frac{1}{2(m+1 ) } \ , ( 2[f_s m ] + [ s]+2 ) + \mathcal{o}(1/m^2 ) \ \stackrel{m \rightarrow \infty}{\longrightarrow } \ [ s ] + f_s \ = \ s \ \ .\ ] ] hence , there is perfect agreement between the annulus amplitudes in the @xmath0 limit of unitary minimal models and of liouville theory . let us finally return to the analysis of our fzzt branes at the dirichlet points . since we already understand the effect of shifts in @xmath25 , we can restrict the following discussion to the case @xmath171 . we would like to probe this fzzt brane with the zz brane @xmath140 . the amplitude ( [ zzfzzt ] ) then becomes @xmath172 in the second equality , we inserted formula ( [ ann1l ] ) . this result nicely confirms our interpretation of the fzzt brane with label @xmath171 as an infinite array of point - like branes . in this section we shall discuss how the boundary 2-point function of fzzt branes in liouville theory can be continued to @xmath0 . this will allow us to derive the exact boundary spectrum , in particular the position and width of the expected band gaps , from liouville theory . our results will confirm nicely the outcome of our previous discussion in the context of unitary minimal models . let us recall that the boundary 2-point functions for the fzzt branes in @xmath50 liouville theory are given by @xmath173 \ \ .\ ] ] the coefficient in front of the first term has been set to @xmath174 by an appropriate normalization of the boundary fields . once this freedom of normalization has been fixed , the coefficient of the second term is entirely determined by the physics . it takes the form @xmath175 where @xmath176 and @xmath177 . the special function @xmath178 is constructed as a ratio of two barnes double gamma functions ( see appendix a.2 for details ) . @xmath31 is known as the _ reflection amplitude _ since it describes the phase shift of wave functions that occurs when open strings spanning between the fzzt branes with parameters @xmath179 and @xmath180 are reflected by the liouville potential . our aim now is to continue these expressions to the @xmath0 model ( see also @xcite for a first attempt in this direction ) . using once more the general formula ( [ gasym ] ) for the behavior of barnes double gamma function at @xmath181 , it is not hard to see that the boundary reflection amplitude @xmath31 of liouville theory vanishes outside a discrete set of open string momenta . this behavior is unacceptable since it does not allow for a sensible physical interpretation . actually , it does not originate from the physics of the model but rather from an inappropriate choice of the limiting procedure . to see this we recall that the boundary spectrum of the theory is expected to develop gaps of finite width . for such gaps to emerge from liouville theory , it is necessary that the corresponding states in the liouville model become non - normalizable . but our normalization of boundary fields was chosen such that all states in the boundary theory have unit norm . hence , the normalization we have chosen above is clearly not appropriate for the limit we are about to take . there exists a distinguished normalization of the boundary fields in which the reflection amplitude of the boundary theory becomes trivial . since the usual ` liouville wall ' ceases to exist at the point @xmath18 , the trivialization of the boundary reflection amplitude seems quite well adapted to our physical setup . explicitly , the new normalization of the boundary fields is given by @xcite , @xmath182 with @xmath183 it is straightforward to rewrite the 2-point function in terms of the fields @xmath184 . the result is @xmath185 & & \hspace*{-4 cm } \times\ , \frac{\gamma(1 + 2ibp_1 ) \gamma(2ib^{-1 } p_1 ) } { \gamma(1-ibs_1)\ga(-ib^{-1}s_1)\ga(1-ibs_2)\ga(-ib^{-1}s_2 ) } \ \left[\ , \delta(q_{b}- \b_1 - \b_2 ) + \delta(\b_1-\b_2)\ , \right ] \ \ . \nonumber \end{aligned}\ ] ] here we use @xmath186 and the term @xmath187 in the denominator denotes the bulk 3-point couplings of liouville theory evaluated at the points @xmath188 and @xmath189 . in writing this expression we have omitted a factor that is constant in the momentum @xmath190 and invariant under the exchange of @xmath179 and @xmath180 since this can easily be absorbed in a redefinition of the boundary fields . when rewritten in the new normalization , the only nontrivial term in the expression ( [ new2ptf ] ) is the bulk 3-point function . hence , we can use the results of @xcite ( see also section 2 ) to continue the boundary 2-point function to @xmath0 . note , however , that the bulk 3-point coupling needs to be inverted . this is possible whenever it is non - zero . since the coupling @xmath191 vanishes whenever the factor @xmath192 does , we conclude that boundary fields @xmath193 exist for @xmath194 where @xmath195 fields which are labeled by momenta outside this range @xmath33 do not correspond to normalizable states of the model . on a single brane with label @xmath196 we obtain in particular @xmath197 here , @xmath198 $ ] denotes the fractional part of the boundary parameter as before . this is exactly the same spectrum we found in our discussion of boundary conditions in the @xmath0 limit of unitary minimal models ! our result for the 2-point functions of boundary fields in the spectrum of the model is given by @xmath199 \nonumber \\[2 mm ] \mbox { where } \ \ e^{q(\a_1,\a_2,\a_3 ) } & : = & \frac{{\upsilon}(1 + 2i\ta|1)}{{\upsilon}(1|1 ) } \ \prod_{j=1}^3 \ , \frac{{\upsilon}(1 + 2i \ta_j)}{{\upsilon}(1 + 2i\a_j)}\ \ . \nonumber \end{aligned}\ ] ] in a final step , we change the normalization of the fields @xmath200 in the @xmath0 theory again so that they possess unit norm , @xmath201 with @xmath202 in an abuse of notation , we have denoted the normalized boundary fields of the @xmath0 model by the same letter @xmath203 as in @xmath50 liouville theory . nevertheless it important to keep in mind that they are not justs limits of the corresponding fields in ordinary liouville theory . after this final change of normalization , the boundary 2-point functions take the form so that the limit of the properly renormalized 2-point function is finite . ] @xmath204 \\[3 mm ] & & \hspace*{-.5 cm } \mbox { where } \ \ \r^{s_1s_2}_{c=1}(p ) \ = \ \r^{is_1\ is_2}_{c=25 } ( 1-p)^{-1 } \ \ \mbox{and } \ \ { \cal{n}}^{s_1s_2}(p ) \ = \ -2\epsilon p_1 ( -1)^{[2p_1]+[s_1]+[s_2]}\nonumber \end{aligned}\ ] ] for all @xmath205 . note that we have been able to express the `` boundary reflection amplitude '' of the @xmath0 theory through the corresponding quantity of the @xmath206 model . we shall comment on this interesting outcome of our computation in the section 6 . our final aim is to compute the bulk - boundary couplings of the @xmath0 boundary liouville models . for liouville theory with real @xmath17 , expressions for these couplings were provided by hosomichi in @xcite . we shall depart from these formulas to derive the corresponding couplings in the @xmath0 model . the analysis turns out to be rather intricate . nevertheless , the final answer is in some sense simpler than for @xmath50 . let us begin by reviewing hosomichi s formula for the bulk - boundary correlation function latexmath:[\[\langle \phi_{\alpha } ( z,\bar{z})\tilde \psi^{ss}_{\beta } ( u)\rangle \ = \ |z-\bar{z}|^{h_{\beta}-2h_{\alpha } } labeled by a parameter @xmath25 . in our normalization ( [ normalisation ] ) for the boundary fields @xmath208 the couplings @xmath209 read @xmath210 \times \ \frac{\gamma_{2 } ( q_{b}-\beta|b)\gamma_{2 } ( 2q_{b}-2\alpha -\beta|b)\gamma_{2 } ( 2\alpha -\beta|b)\gamma_{2 } ( q_{b}+is|b)\gamma_{2 } ( q_{b}-is|b)}{\gamma_{2 } ( q_{b}-\beta -is|b)\gamma_{2 } ( q_{b}-\beta + is|b)\gamma_{2 } ( \beta|b)\gamma_{2 } ( 2\alpha|b)\gamma_{2 } ( q_{b}-2\alpha|b ) } \label{bbcorrf}\end{gathered}\ ] ] where @xmath211 is given by the integral @xmath212 while sending @xmath17 to @xmath213 , an infinite number of poles and zeroes of the integrand approach the integration path . it turns out to be more convenient to evaluate the integral by cauchy s theorem . for @xmath214 we can close the integration contour and rewrite the sum over residues as a combination of basic hypergeometric series ( see appendix a.1 ) , @xmath215 & \quad \times \ { } _ { 2}\phi_{1 } \big ( e^{-2\pi ib^{-1 } ( \beta + 2\alpha -q_{b})},e^{-2\pi ib^{-1}\beta},e^{-4\pi i b^{-1}\alpha};e^{-2\pi ib^{-2}};e^{2\pi ib^{-1 } ( \beta -q_{b}+is)}\big ) \nonumber \\[2 mm ] & \quad \times \ e^{-\pi s ( \beta + 2\alpha -q_{b})}s ( \beta|b)s ( \beta + 2\alpha -q_{b}|b)s ( q_{b}-2\alpha|b ) + \ ( \alpha\rightarrow q_{b}-\alpha)\ \ .\label{integral}\end{aligned}\ ] ] our previous results allow us to estimate the behavior of the double gamma functions and its close relative @xmath178 in the vicinity of @xmath18 . but the function @xmath216 has not appeared in our discussion before . hence , in order to continue our analysis , we need to study the basic hypergeometric series in the limit when the parameter @xmath217 approaches @xmath218 . we shall begin with a separate treatment of the two factors @xmath219 before we combine the results to address the full coupling @xmath209 . in dealing with one of the basic hypergeometric series , the basic idea is to approximate the series ( we denote the summation parameter by @xmath220 ) through an integral and to evaluate the latter using a saddle - point analysis . we will not give a rigorous derivation but shall content ourselves with a rough sketch of the argument . this will help to understand the main features of our final formulas . let us introduce the variable @xmath221 . using the asymptotic behavior of the q - pochhammer symbols ( see appendix b ) we obtain @xmath222 where @xmath223 & & -{\text{li}_{2}}(c)+{\text{li}_{2}}(cy)-{\text{li}_{2}}(1)+{\text{li}_{2}}(y)+\log y \ , \log z \ \ , \end{aligned}\ ] ] and @xmath224 the asymptotics of the integral in eq . ( [ qhypasym ] ) can be determined by the method of steepest descent . the leading contribution comes from saddle - points @xmath225 , i.e. from points satisfying the condition @xmath226 . an elementary computation shows that such saddle - points are obtained as a solution of the quadratic equation @xmath227 once we have understood which saddle - points @xmath225 contribute to the integral , we end up with an expansion of the form @xmath228 the most difficult part , however , is to find the right saddle - points . to begin with , the saddle - point equation ( [ saddlepeq ] ) has two different solutions , at least for generic choice of the momenta . but this is not the full story since the function @xmath41 is a complex function with branch - cuts . hence , the description of the saddle point @xmath229 is not complete before we have specified on which branch the relevant saddle - point is located . we were not able to solve the problem in full generality , but for the combination of parameters in the problem at hand , we could find the saddle - points through a comparison with numerical studies ( mathematica ) . let us state the result for the first q - hypergeometric function appearing in eq . ( [ integral ] ) . here , the parameter @xmath217 is @xmath230 and thus the prefactor @xmath231 of @xmath41 in the exponent becomes @xmath232 , where @xmath233 as introduced in section 2 . note that the arguments @xmath234 and @xmath235 depend on @xmath17 and thus on @xmath99 and hence they need to be expanded around @xmath236 . we shall use the symbols @xmath237 and @xmath238 to denote the leading terms , i.e. @xmath239 where we have set @xmath240 and @xmath241 . since the function @xmath41 in the exponential comes with an extra factor @xmath242 , the sub - leading term in the @xmath99-expansion of @xmath41 contributes an extra factor which we shall combine with @xmath243 into a new function @xmath40 . before we spell out our results on the saddle points , we also have to specify the branch we use for @xmath244 . we found it most convenient to choose @xmath245-s+[s])$ ] . in the end we obtain , @xmath246 with @xmath247 the two solutions of the saddle - point equation are denoted by @xmath248 , and they are given explicitly by @xmath249 & \ \ = \ \left\{\begin{array}{l } e^{\pi i ( 2p_{\alpha } + p_{\beta } ) } \frac{\cos ( 2\pi p_{\alpha})\sin ( \pi s)}{\sin ( \pi ( p_{\beta}+s))}\bigg ( 1\mp i\tan ( 2\pi p_{\alpha})\sqrt{1-\frac{\sin ( \pi p_{\beta})^{2}}{\sin ( \pi s)^{2}\sin ( 2\pi p_{\alpha})^{2 } } } \bigg)\\[1 mm ] \quad \quad \text{for}\ |\sin ( \pi p_{\beta})|<|\sin ( \pi s)\sin ( 2\pi p_{\alpha})|\\[3 mm ] e^{\pi i ( 2p_{\alpha } + p_{\beta } ) } \frac{\cos ( 2\pi p_{\alpha})\sin ( \pi s)}{\sin ( \pi ( p_{\beta}+s))}\bigg(1\mp \sqrt{\frac{\sin^{2 } ( \pi p_{\beta})-\sin^{2 } ( 2\pi p_{\alpha})\sin^{2 } ( \pi s)}{\sin^{2}(\pi s)\cos^{2 } ( 2\pi p_{\alpha } ) } } \bigg)\\[1 mm ] \quad \quad \text{for}\ |\sin ( \pi s)|\geq |\sin ( \pi p_{\beta})|\geq |\sin ( \pi s)\sin ( 2\pi p_{\alpha})| \\[3 mm ] e^{\pi i ( 2p_{\alpha } + p_{\beta } ) } \frac{\cos ( 2\pi p_{\alpha})\sin ( \pi s)}{\sin ( \pi ( p_{\beta}+s))}\bigg(1\mp \frac{\sin ( \pi ( s+2p_{\alpha}))}{\sin ( \pi s)\cos ( 2\pi p_{\alpha } ) } \sqrt{\frac{\sin^{2 } ( \pi p_{\beta})-\sin^{2 } ( 2\pi p_{\alpha})\sin^{2 } ( \pi s)}{\sin ^{2 } ( \pi ( s+2p_{a } ) ) } } \bigg)\\[1 mm ] \quad \quad \text{for}\ |\sin ( \pi p_{\beta})|\geq |\sin ( \pi s)| \end{array } \right.\end{aligned}\ ] ] the last of these three different cases concerns the band gaps , i.e. momenta @xmath250 . the first two cases , on the other hand , apply to the bands so that all bands appear to be split into an inner ( first case ) and an outer ( second case ) region , depending on the value of the bulk momentum @xmath251 . analyzing the limit in the outer part of the bands turns out to be the most difficult task because the two saddle - points @xmath252 are of the same order , i.e. @xmath253 , and there are at least some subsets in momentum space where they both contribute . within the inner region and the gap , the saddle - point @xmath254 dominates so that only the summand @xmath255 remains in our asymptotic expansion . the exact description of our findings for the intermediate region is rather cumbersome . fortunately , the distinction between inner and outer parts of the band disappears once we combine our two basic hypergeometric series into the bulk - boundary coupling . since our main goal is to provide a formula for the @xmath0 limit of @xmath209 , we shall not attempt to present our findings for @xmath256 in the outer region . instead , let us recall from above that we also need to specify the branch on which the saddle - points sit . the prescription is as follows : if @xmath257 , then we choose the branch by moving it from the fundamental branch first to @xmath258 , infinitesimally above / below the real axis depending on whether the fractional part @xmath259 $ ] is greater or smaller than @xmath260 . then we move it on the shortest arc to @xmath225 . having completed the analysis for the first hypergeometric function we now turn to the second for which an analogous investigation gives the following results , @xmath261 and @xmath262 note that in this case @xmath263 is directly given by @xmath264 . the relevant saddle - points are @xmath265 our discussion of the three different cases and the choice of branches carries over from the previous discussion without any significant changes . before we return to the study of @xmath209 it is necessary to make one more important observation which relates the two saddle - points @xmath266 and @xmath267 . it is not difficult to see that @xmath268 solves the saddle - point equation ( [ saddlepeq ] ) of @xmath266 . indeed we find for @xmath269 that @xmath270 . this fact along with some simple formulas in appendix a.3 allows to re - express the final formulas entirely in terms of @xmath271 . now we have all ingredients at our disposal to analyze the asymptotic behavior of the bulk - boundary correlator . we must combine the divergences coming from the q - hypergeometric functions with the divergences of the prefactors in eq . ( [ bbcorrf ] ) . since we know already that boundary fields with @xmath272 do not exist , we can restrict our evaluation of @xmath209 to the inner and outer band regions ( see discussion above ) . in the inner region , only one saddle - point contributes and the divergent terms may be determined easily from the formulas we have provided . it then turns out that they cancel each other thanks to a rather non - trivial dilogarithm identity which is attributed to ray @xcite . for convenience we state ray s identity in appendix a.3 ( eq.([5term ] ) ) . hence , we are left with a finite result . after a tedious but elementary computation we obtain @xmath273 where @xmath274 . in the derivation we used the fact that @xmath275 and @xmath276 are the two solutions of the same quadratic saddle - point equation along with a few elementary identities which we list in appendix a.3 . within the outer regions of the bands , the q - hypergeometric functions can have contributions from both saddle - points . it is possible to show that out of the four possible terms in our product of two q - hypergeometric functions , only two appear at any point in momentum space . moreover , after the divergences of the prefactors have been taken into account , only one of the two terms is free of divergences . the other keeps a rapidly oscillating factor . the cancellation is again due to ray s identity and the finite contributions come from the combination @xmath277 . in the end , the result for the outer region of the band is therefore given by the same formula as for the inner region . it now remains to change the normalization of our boundary fields . we pass from the fields @xmath208 to @xmath203 using the prescription ( [ newnormalisation ] ) , as before . our result then reads @xmath278 & \quad \times \ \big(\frac{e^{-2\pi i p_{\alpha } ( s+1)}}{1-e^{-4\pi ip_{\alpha } } } \frac{g ( y_{+})^{2}e^{-\frac{ih ( y_{+})}{2\pi}}}{y_{+}^{2}h '' ( y_{+})}\frac{\gamma_2 ( 1|1)\gamma_2 ( 1 + 2p_{\beta}|1)\gamma_2 ( 1 - 2p_{\alpha}|1)^{2}}{\gamma_2 ( 1 - 2p_{\alpha}-p_{\beta}|1)\gamma_2 ( 1 - 2p_{\alpha}+p_{\beta}|1)\gamma_2 ( 1+p_{\beta } |1)^{2 } } \\ & \quad \quad \quad \ -\ ( p_{\alpha}\to -p_{\alpha})\big)\ \ .\end{aligned}\ ] ] this is the formula we anticipated in the introduction . for a direct comparison one has to employ the integral representation ( [ barnesg ] ) of barnes double gamma function . before we conclude this section , we would like to comment briefly on the band - gaps . actually , we can use the analysis of the bulk - boundary couplings to confirm our previous statement that boundary fields @xmath279 with @xmath280 decouple from the theory . to this end , we shall focus on the coefficients @xmath281 of the bulk boundary operator product expansion rather than the couplings @xmath35 . the former are related to latter by multiplication with the boundary 2-point function . our claim is that the coefficients @xmath282 vanish within the band - gaps . as we have seen before , the 2-point function contributes a factor that diverges when @xmath272 . with the help of our asymptotic expansion formulas in this section it is possible to show that the couplings @xmath35 always diverge slower hence , boundary fields whose momenta lie in the gaps can not be excited when a bulk field approaches the boundary . in this work we have constructed boundary conditions of the euclidean @xmath0 liouville model . in particular we argued that there exists a family of boundary conditions that is parametrized by one real parameter @xmath22 . we provided explicit expressions for their boundary states ( see eq . ( [ 1ptfceq1 ] ) ) , the boundary 2-point function ( [ 2ptfctres1],[reflb ] ) , and the bulk - boundary coupling ( [ bcoupl ] ) . the only quantity that we are missing for a complete solution of the model is the boundary 3-point coupling . in the case of liouville theory , the corresponding formulas have been found by ponsot and teschner in @xcite . we believe that the analysis of their @xmath0 limit can proceed along the lines of the studies we have presented in section 5 , but obviously this program remains to be carried out in detail . a crucial ingredient in our study was the formula ( [ gasym ] ) for the asymptotics of barnes double gamma function . in section 2 , the latter enabled us to provide a new derivation of the bulk couplings in the @xmath0 liouville model . our approach here is simpler and more general than the one that was developed in @xcite . with this technical progress it is now also possible to calculate the bulk couplings of non - rational conformal field theories with @xmath116 . in contrast to the @xmath0 case , the models with @xmath116 are certainly non - unitary . we will comment on the results and their relation with minimal models elsewhere . possible applications of such developments include the 2-dimensional cigar background with @xmath283 @xcite and similar limits of the associated boundary theories @xcite . all this research , however , was mainly motivated by the desire to construct an exact conformal field theory model for the homogeneous condensation of open string tachyons , such as the tachyon on an unstable d0 brane in type iib theory . as we explained in the introduction , the underlying world - sheet theory is a lorentzian version of the model we have considered in this work . but since the couplings of our theory are not analytic in the momenta ( with the boundary state being the only exception ) , this lorentzian theory can not be obtained by a simple wick rotation from the solution we described here . instead , it was suggested in @xcite to perform the wick rotation before sending @xmath17 to its limiting value @xmath18 . such a prescription makes sense because the couplings of liouville theory are analytic in the momenta as long as @xmath17 is not purely imaginary . it is certainly far from obvious that the wick rotated couplings again possess a well - defined @xmath18 limit . preliminary studies of this issue show that our analysis of the boundary 2-point function extends to the lorentzian case . in the time - like model , the spectrum of boundary fields is continuous ( there are no gaps ) and their 2-point function is still given by eq . ( [ 2ptfctres1 ] ) with a reflection amplitude @xmath284 this result does not agree with the proposal in @xcite . note , however , that in the lorentzian case the correlation functions can depend on the details of how @xmath17 approaches @xmath18 . such a behavior might be related to a choice of vacuum . let us also anticipate that consistency of our expression for the boundary 2-point coupling with the half - brane boundary states @xcite can be checked by means of the modular bootstrap . a related observation , though with a somewhat problematic domain of open string momenta , has also been made in @xcite . concerning the bulk - boundary coupling , our investigations are not complete yet . but before having worked out the expression in the lorentzian model , it is worthwhile comparing our formula ( [ bcoupl ] ) for the bulk - boundary coupling with a corresponding expression in the time - like theory that was suggested recently in @xcite ( eq . [ 4.14 ] of that paper ) . in fact , the exponential in the second line of our formula is identical to a corresponding factor in the work of balasubramanian et . it will be interesting to determine the other factors through our approach . we shall return to this issue in a forthcoming publication . * acknowledgements : * we would like to thank v. balasubramanian , p. etingof , j. frhlich , m.r . gaberdiel , k. graham , g. moore , b. ponsot , i. runkel , j. teschner and g. watts for very helpful discussions and some crucial remarks . part of this research has been carried out during a stay of vs with the string theory group of rutgers university and of both authors at the erwin schroedinger institute for mathematical physics in vienna . we are grateful for their warm hospitality during these stays . in this first appendix we collect a few results on the special functions which appear in the analysis of the @xmath0 boundary liouville model . we shall start with some q - deformed special functions and then introduce barnes double gamma functions and certain closely related special functions . dilogarithms and some of their properties are finally reviewed in the third subsection . one of the most basic objects in the theory of q - deformed special functions is the finite q - pochhammer symbol ( see e.g. @xcite ) @xmath285 its limit for @xmath286 exists for @xmath287 and is denoted by @xmath288 the tilde is used to avoid confusion with our convention for @xmath289 in the rest of the text . with the help of the the finite q - pochhammer symbol we can now introduce the basic hypergeometric series @xmath290 . this q - deformation of the hypergeometric function @xmath291 is defined as @xmath292 the interested reader can find many basic properties of these functions in the literature ( see e.g. @xcite ) . all the properties we need in the main text are stated and derived there . barnes double @xmath61-function @xmath293 is defined for @xmath294 and complex @xmath17 with @xmath295 ( see @xcite ) , and can be represented by an integral ( for @xmath296 ) , @xmath297 whenever @xmath298 . here we have also introduced the symbol @xmath299 throughout the main text , we often use the following special combinations of double gamma functions , @xmath300 s ( x|b ) & = & \gamma_{2 } ( x|b ) \ , \gamma_{2 } ( q_{b}-x|b)^{-1}\ \ .\end{aligned}\ ] ] for the latter , we would also like to spell out an integral representation that easily follows from the formula ( [ barnesg ] ) above , @xmath301 this representation is valid for @xmath302 . let us also note that @xmath178 is unitary in the sense that @xmath303 the function @xmath178 is also closely related to ruijsenaars hyperbolic gamma function @xmath304 ( see @xcite ) , @xmath305 many further properties of double gamma functions , in particular on the position of their poles and shift properties , can be found in the literature ( see e.g. @xcite ) . here we would like to prove one relation that involves the q - pochhammer symbols we introduced in the previous subsection . to this end , we depart from the integral representation ( [ barnesg ] ) . let us denote the integrand in formula ( [ barnesg ] ) by @xmath306 , s.t . the function @xmath243 has the property @xmath308 we evaluate the integral by closing the contour in the first quadrant , @xmath309 & = & -\log \gamma_{2 } ( -ib+ix|-ib ) - \int_{0}^{\infty } \frac{dt}{2 t } ( \tfrac{q_{b}}{2}-x)^{2 } ( e^{-it}-e^{-t})-2\pi i \sum_{\text{poles}\ t_{n}}\!\!\text{res}_{t = t_{n } } f ( t , x , b ) \nonumber\end{aligned}\ ] ] where we have used the formula ( [ frot ] ) in passing to the second line . the integral in the second term is @xmath310 the poles appearing in the sum over residues are at @xmath311 with @xmath312 hence we find @xmath313 replacing @xmath314 through @xmath315 and changing the order of summation , we obtain @xmath316 with @xmath317 . we can now re - express the infinite product with the help of q - pochhammer symbols ( see appendix a.1 ) . inserting this result and eq . ( [ hilfsintegral ] ) into eq . ( [ tempresult ] ) we finally arrive at @xmath318 this formula is the starting point for our evaluation of the asymptotic behavior of @xmath319 near @xmath320 . we shall return to this issue in appendix b. there is one more special function that plays an important role in our analysis : euler s dilogarithm ( l. euler 1768 ) . it is defined by @xmath321 for @xmath322 . at @xmath323 , we find @xmath324 . the dilogarithm has the integral representation @xmath325 it can be analytically continued with a branch cut along the real axis from 1 to @xmath326 . at @xmath323 , the dilogarithm is still continuous , but not differentiable . properties of dilogarithms are used frequently to derive the formulas that appear in the main text . here we list the most relevant properties @xmath327 { \text{li}_{2}}(z)+{\text{li}_{2}}(1-z ) & = & { \text{li}_{2}}(1)-\log ( z)\ , \log ( 1-z)\\[2 mm ] { \text{li}_{2}}(-z)+{\text{li}_{2}}(-1/z ) & = & 2{\text{li}_{2}}(-1)-\frac{1}{2}\log ^{2 } ( z)\\[2 mm ] \label{liclosetoone } { \text{li}_{2}}(e^{-\epsilon } ) & = & { \text{li}_{2}}(1)- ( 1-\log \epsilon)\epsilon + o ( \epsilon ) \ \ . \end{aligned}\ ] ] many more properties of the dilogarithm can be found in the literature ( see e.g. @xcite ) . of particular relevance for our analysis of the bulk - boundary 2-point function is the following equality @xmath328 & & \hspace*{-3 cm } = \ { \text{li}_{2}}(z)-{\text{li}_{2}}(z\tfrac{\eta_{1}\eta_{2}}{\zeta_{1 } \zeta_{2}})+\log z \log ( x_{1}x_{2}\zeta_{1}\zeta_{2 } ) \ \ \end{aligned}\ ] ] here , @xmath329 are the two solutions of the quadratic equation @xmath330 equation ( [ 5term ] ) is derived from the usual 5-term relation of the dilogarithm ( see e.g. @xcite ) with the help of the following list of relations that hold for any two solutions @xmath331 of the equation ( [ quad ] ) , @xmath332 \left(1-z\ , \frac{\eta_1\eta_2}{\zeta_1\zeta_2}\right ) ( 1-\zeta_i x_1 ) ( 1-\zeta_i x_2 ) & = & \frac{(\zeta_i - \eta_1)(\zeta_i-\eta_2)}{\zeta_i } \label{eq2 } \\[2 mm ] x_1 x_2 & = & \frac{1-z}{\zeta_1\zeta_2 - z \eta_1 \eta_2 } \label{eq3 } \ \ . \end{aligned}\ ] ] we leave the details to the reader . let us note that the properties ( [ eq1 ] ) to ( [ eq3 ] ) are also used frequently to simplify the final formula for the bulk - boundary 2-point function . in order to evaluate the behavior of the double gamma function near @xmath18 one can start from our equation ( [ dgqpoch ] ) . since the double gamma function that appears on the right hand side of this equation is analytic at @xmath18 , the main issue is to understand the asymptotic behavior of the q - pochhammer symbol . this is what we are concerned with here . to begin with , we shall parametrize @xmath217 by @xmath333 . in general , the limit can depend on the way we send @xmath334 to zero . we start by taking the logarithm of the definition , @xmath335)\frac{d}{dx}\big(\log ( 1-a\,\tilde{q}^{z/2+x } ) \big)\big|_{x = t}\ , dt \ \ .\end{aligned}\ ] ] here , we used the euler - maclaurin sum formula to re - express the sum as an integral , @xmath336 is the first bernoulli polynomial , @xmath337 note that the branch of the logarithm @xmath338 is chosen such that it approaches @xmath339 when we send @xmath156 along the real axis toward @xmath326 . the first integral is given by the dilogarithm ( see appendix [ dilog ] for the definition and properties ) , so we obtain @xmath340 where @xmath341 denotes the integral containing the bernoulli polynomial , @xmath342 ) \frac{(-a)\log ( \tilde{q})\ , \tilde{q}^{z/2+t}}{1-a\ , \tilde{q}^{z/2+t}}\ , dt \ \ .\ ] ] as long as @xmath343 , @xmath344 is suppressed by @xmath345 . on the other hand , if @xmath346 , then the integral gives a non - trivial contribution . let us explain that in more detail . we rewrite the integral as @xmath347 ) \frac{1}{a^{-1}\ , e^{\epsilon z/2}\ , e^{t}-1}dt \ \ .\ ] ] when @xmath99 goes to zero , the strongly oscillating sign of the bernoulli polynomial suppresses the integral , so that generically @xmath344 vanishes in that limit . the leading contribution comes from the pole of the integrand , and if it approaches the path of integration when sending @xmath99 to zero , we can get a finite contribution . for @xmath343 , the pole is always outside of the integration region , but for @xmath348 , the pole might come close to the real axis . we can take this effect into account by extracting the pole part from the integrand , @xmath349 the regular part will not contribute to the integral in the limit @xmath92 , so we can rewrite @xmath344 as @xmath350)\frac{1}{t-\frac{\log a}{\epsilon}+\frac{z}{2 } } + o ( \epsilon^{0})\ \ .\ ] ] the remaining integral is given by the difference of @xmath351 and the first terms of its stirling series , @xmath352 as long as @xmath353 and @xmath354 , the limit of @xmath355 vanishes because the @xmath61-function approaches its sterling approximation . if @xmath346 , we find @xmath356 \ \ .\ ] ] this formula together with describe the asymptotics of q - pochhammer symbols in the cases we shall need for our applications . in the following we want to apply the gained insight in a number of special cases . first , let us write down the full asymptotics for @xmath346 , @xmath357 where we used from appendix [ dilog ] to expand the dilogarithm close to 1 . we observe that this expression has zeroes for @xmath358 which is consistent with the general definition . for integer @xmath235 these asymptotics can alternatively be derived using modular properties of dedekind s @xmath359-function . we are often interested in q - pochhammer - symbols of the form @xmath360 with some function @xmath361 . in most cases , it is enough to expand @xmath243 to first order around @xmath362 , and we find @xmath363 this relation breaks down if @xmath364 and @xmath365 for some @xmath366 . let us look at an example . set @xmath367 and @xmath368 for @xmath369 . then we obtain @xmath370 by carefully choosing the correct branch of the logarithm we can rewrite the result as @xmath371 - 1/2 ) } \big)^{(1-z - p)/2 } \ \big ( 1+o ( \epsilon^{0 } ) \big ) \ \ , \ ] ] where @xmath73 $ ] denotes the largest integer smaller or equal @xmath34 . let us finally look at the behavior of @xmath372 when @xmath373 is close to 1 . obviously @xmath374 , and the derivative is easily obtained as @xmath375 similar formulas for the asymptotics of q - pochhammer symbols have been derived before ( see in particular @xcite ) . a. sen and b. zwiebach , _ tachyon condensation in string field theory _ , jhep * 0003 * ( 2000 ) 002 [ hep - th/9912249 ] . w. taylor and b. zwiebach , _ d - branes , tachyons , and string field theory _ , [ hep - th/0311017 ] . m. r. garousi , _ tachyon couplings on non - bps d - branes and dirac - born - infeld action _ b * 584 * ( 2000 ) 284 [ hep - th/0003122 ] . a. sen , _ tachyon matter _ , jhep * 0207 * ( 2002 ) 065 [ hep - th/0203265 ] . j. a. harvey , d. kutasov and e. j. martinec , _ on the relevance of tachyons _ , [ hep - th/0003101 ] . v. schomerus , _ lectures on branes in curved backgrounds _ , class . * 19 * ( 2002 ) 5781 [ hep - th/0209241 ] . a. sen , _ rolling tachyon _ , jhep * 04 * ( 2002 ) 048 [ hep - th/0203211 ] . m. gutperle and a. strominger , _ spacelike branes _ , jhep * 0204 * ( 2002 ) 018 [ hep - th/0202210 ] . a. sen , _ universality of the tachyon potential _ , jhep * 9912 * ( 1999 ) 027 [ hep - th/ 9911116 ] . j. callan , curtis g. , i. r. klebanov , a. w. w. ludwig and j. m. maldacena , _ exact solution of a boundary conformal field theory _ , nucl . ( 1994 ) 417448 [ hep - th/9402113 ] . j. polchinski and l. thorlacius , _ free fermion representation of a boundary conformal field theory _ * d50 * ( 1994 ) 622626 [ hep - th/9404008 ] . p. fendley , h. saleur and n. p. warner , _ exact solution of a massless scalar field with a relevant boundary interaction _ , nucl . * b430 * ( 1994 ) 577596 [ hep - th/9406125 ] . a. recknagel and v. schomerus , _ boundary deformation theory and moduli spaces of d - branes _ , nucl . phys . * b545 * ( 1999 ) 233282 [ hep - th/9811237 ] . m. r. gaberdiel and a. recknagel , _ conformal boundary states for free bosons and fermions _ , jhep * 11 * ( 2001 ) 016 [ hep - th/0108238 ] . m. gutperle , a. strominger , _ timelike boundary liouville theory _ , [ hep - th/0301038 ] . i. affleck , w. hofstetter , d. r. nelson and u. schollwock , _ non - hermitian luttinger liquids and flux line pinning in planar superconductors _ , [ cond - mat/0408478 ] . h. dorn and h. j. otto , _ two and three point functions in liouville theory _ , nucl . * b429 * ( 1994 ) 375388 [ hep - th/9403141 ] . a. b. zamolodchikov and a. b. zamolodchikov , _ structure constants and conformal bootstrap in liouville field theory _ , nucl . * b477 * ( 1996 ) 577605 [ hep - th/9506136 ] . b. ponsot and j. teschner , _ liouville bootstrap via harmonic analysis on a noncompact quantum group _ , [ hep - th/9911110 ] . j. teschner , _ liouville theory revisited _ , class . * 18 * ( 2001 ) r153r222 [ hep - th/0104158 ] . j. teschner , _ a lecture on the liouville vertex operators _ , [ hep - th/0303150 ] . v. fateev , a. b. zamolodchikov and a. b. zamolodchikov , _ boundary liouville field theory . i : boundary state and boundary two - point function _ , [ hep - th/0001012 ] . j. teschner , _ remarks on liouville theory with boundary _ , [ hep - th/0009138 ] . k. hosomichi , _ bulk - boundary propagator in liouville theory on a disc _ , jhep * 11 * ( 2001 ) 044 [ hep - th/0108093 ] . b. ponsot and j. teschner , _ boundary liouville field theory : boundary three point function _ , nucl . phys . * b622 * ( 2002 ) 309327 [ hep - th/0110244 ] . a. b. zamolodchikov and a. b. zamolodchikov , _ liouville field theory on a pseudosphere _ , [ hep - th/0101152 ] . v. schomerus , _ rolling tachyons from liouville theory _ , jhep * 0311 * ( 2003 ) 043 [ hep - th/0306026 ] . i. runkel and g. m. t. watts , _ a non - rational cft with c = 1 as a limit of minimal models _ , jhep * 09 * ( 2001 ) 006 [ hep - th/0107118 ] . e. j. martinec , _ the annular report on non - critical string theory _ , [ hep - th/0305148 ] . j. teschner , _ on boundary perturbations in liouville theory and brane dynamics in noncritical string theories _ , jhep * 0404 * ( 2004 ) 023 [ hep - th/0308140 ] . k. r. kristjansson and l. thorlacius , _ c = 1 boundary conformal field theory revisited _ , class . * 21 * , s1359 ( 2004 ) [ hep - th/0401003 ] . a. strominger and t. takayanagi , _ correlators in timelike bulk liouville theory _ , [ hep - th/0303221 ] . f. larsen , a. naqvi and s. terashima , _ rolling tachyons and decaying branes _ , jhep * 02 * ( 2003 ) 039 [ hep - th/0212248 ] . j. l. cardy , _ boundary conditions , fusion rules and the verlinde formula _ , nucl . b * 324 * ( 1989 ) 581 . k. graham , i. runkel and g. m. t. watts , _ minimal model boundary flows and c = 1 cft _ , nucl . b * 608 * ( 2001 ) 527 [ hep - th/0101187 ] . s. fredenhagen and v. schomerus , _ d - branes in coset models _ , jhep * 0202 * ( 2002 ) 005 [ hep - th/0111189 ] . g. ray , _ multivariable polylogarithm identities _ , in _ structural properties of polylogarithm _ , l. lewin ( ed ) , math . surveys and monographs 37 ( 1991 ) 123 . y. hikida and t. takayanagi , _ on solvable time - dependent model and rolling closed string tachyon _ , [ hep - th/0408124 ] . s. ribault and v. schomerus , _ branes in the 2-d black hole _ , jhep * 0402 * , 019 ( 2004 ) [ hep - th/0310024 ] . s. fredenhagen and v. schomerus , _ on minisuperspace models of s - branes _ , jhep * 0312 * , 003 ( 2003 ) [ hep - th/0308205 ] . j. l. karczmarek , h. liu , j. maldacena and a. strominger , _ uv finite brane decay _ , jhep * 0311 * , 042 ( 2003 ) [ hep - th/0306132 ] . v. balasubramanian , e. keski - vakkuri , p. kraus and a. naqvi , _ string scattering from decaying branes _ , [ hep - th/0404039 ] . g. e. andrews , r. askey and r. roy , _ special functions_. cambridge university press , cambridge , 1999 . m. jimbo and t. miwa , _ qkz equation with @xmath376 and correlation functions of the xxz model in the gapless regime _ , j. phys . * a29 * ( 1996 ) 29232958 [ hep - th/9601135 ] . a. n. kirillov , _ dilogarithm identities _ , prog . * 118 * ( 1995 ) 61 [ hep - th/9408113 ] . r. j. mcintosh , _ some asymptotic formulae for q - shifted factorials _ , the ramanujan journal * 3 * ( 1999 ) 205 .
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the @xmath0 liouville theory has received some attention recently as the euclidean version of an exact rolling tachyon background . in an earlier paper
it was shown that the bulk theory can be identified with the interacting @xmath0 limit of unitary minimal models . here
we extend the analysis of the @xmath0-limit to the boundary problem .
most importantly , we show that the fzzt branes of liouville theory give rise to a new 1-parameter family of boundary theories at @xmath0 .
these models share many features with the boundary sine - gordon theory , in particular they possess an open string spectrum with band - gaps of finite width .
we propose explicit formulas for the boundary 2-point function and for the bulk - boundary operator product expansion in the @xmath0 boundary liouville model . as a by - product of our analysis
we also provide a nice geometric interpretation for zz branes and their relation with fzzt branes in the @xmath0 theory .
| 25,889 | 247 |
wolf - rayet ( wr ) stars represent the final phase in the evolution of very massive stars prior to core - collapse , in which the h - rich envelope has been stripped away via either stellar winds or close binary evolution , revealing products of h - burning ( wn sequence ) or he - burning ( wc sequence ) at their surfaces , i.e. he , n or c , o ( crowther 2007 ) . wr stellar winds are significantly denser than o stars , as illustrated in fig . [ wrross ] , so their visual spectra are dominated by broad emission lines , notably heii @xmath04686 ( wn stars ) and ciii @xmath04647 - 51 , ciii @xmath05696 , civ @xmath05801 - 12 ( wc stars ) . the spectroscopic signature of wr stars may be seen individually in local group galaxies ( e.g. massey & johnson 1998 ) , within knots in local star forming galaxies ( e.g. hadfield & crowther 2006 ) and in the average rest frame uv spectrum of lyman break galaxies ( shapley et al . 2003 ) . in the case of a single massive star , the strength of stellar winds during the main sequence and blue supergiant phase scales with the metallicity ( vink et al . consequently , one expects a higher threshold for the formation of wr stars at lower metallicity , and indeed the smc shows a decreased number of wr to o stars than in the solar neighbourhood . alternatively , the h - rich envelope may be removed during the roche lobe overflow phase of close binary evolution , a process which is not expected to depend upon metallicity . wr stars represent the prime candidates for type ib / c core - collapse supernovae and long , soft gamma ray bursts ( grbs ) . this is due to their immediate progenitors being associated with young massive stellar populations , compact in nature and deficient in either hydrogen ( type ib ) or both hydrogen and helium ( type ic ) . for the case of grbs , a number of which have been associated with type ic hypernovae ( galama et al . 1998 ; hjorth et al . 2003 ) , a rapidly rotating core is a requirement for the collapsar scenario in which the newly formed black hole accretes via an accretion disk ( macfadyen & woosley 1999 ) . indeed , wr populations have been observed within local grb host galaxies ( hammer et al . 2006 ) . at solar metallicity , single star models predict that the core is spun down either during the red supergiant ( via a magnetic dynamo ) or wolf - rayet ( via mass - loss ) phases . the tendency of grbs to originate from metal - poor environments ( e.g. stanek et al . 2006 ) suggests that stellar winds from single stars play a role in their origin since roche lobe overflow in a close binary evolution would not be expected to show a strong metallicity dependence . in this article , evidence in favour of a metallicity dependence for wr stars is presented , of application to the observed wr subtype distribution in local group galaxies , plus properties of wr stars at low metallicity including their role as grb progenitors . historically , the wind properties of wr stars have been assumed to be metallicity independent ( langer 1989 ) , yet there is a well known observational trend to earlier , higher ionization , wn and wc subtypes at low metallicity as illustrated in fig . [ wrpop ] , whose origin is yet to be established . mass - loss rates for wn stars in the milky way and lmc show a very large scatter . the presence of hydrogen in some wn stars further complicates the picture since wr winds are denser if h is absent ( nugis & lamers 2000 ) . this is illustrated in fig . [ wn_mdot ] , which reveals that the wind strengths of ( h - rich ) wn winds in the smc are lower than corresponding h - rich stars in the lmc and milky way ( crowther 2006 ) . [ wc_mdot ] shows that the situation is rather clearer for wc stars , for which lmc stars reveal @xmath10.2 dex lower mass - loss rates than milky way counterparts ( crowther et al . 2002 ) the observed trend to earlier subtypes in the lmc ( fig . [ wrpop ] ) was believed to originate from a difference in carbon abundances relative to galactic wc stars ( smith & maeder 1991 ) , yet quantitative analysis reveals similar carbon abundances ( koesterke & hamann 1995 ; crowther et al . 2002 ) . theoretically , nugis & lamers ( 2002 ) argued that the iron opacity peak was the origin of the wind driving in wr stars , which grfener & hamann ( 2005 ) supported via an hydrodynamic model for an early - type wc star in which lines of fe ix - xvii deep in the atmosphere provided the necessary radiative driving . vink & de koter ( 2005 ) applied a monte carlo approach to investigate the metallicity dependence for cool wn and wc stars revealing @xmath2 where @xmath3=0.86 for wn stars and @xmath3=0.66 for wc stars for 0.1 @xmath4 . the weaker wc dependence originates from a decreasing fe content and constant c and o content at low metallicity . empirical results for the solar neighbourhood , lmc and smc presented in figs . [ wn_mdot][wc_mdot ] are broadly consistent with theoretical predictions , although detailed studies of individual wr stars within galaxies broader range in metallicity would provide stronger constraints . theoretical wind models also predict smaller wind velocities at lower metallicity , as is observed for wo stars , which are presented in fig . [ wo ] ( crowther & hadfield 2006 ) . the impact of a metallicity dependence for wr winds upon spectral types is as follows . at high metallicity , recombination from high to low ions ( early to late subtypes ) is very effective in very dense winds , whilst the opposite is true for low metallicity , low density winds . the situation is illustrated in the upper panel of fig . [ wc_wo ] , where we present synthetic wc spectra obtained from identical models except that their wind densities differ by a factor of 10 , and the weak wind model is assumed to be extremely fe - poor ( adapted from crowther & hadfield 2006 ) . the high wind density case has a wc4 spectral type whilst the low wind density case has an earlier wo subtype . crowther et al . ( 2002 ) noted that a further increase in wind density by a factor of 2 predicts a wc7 subtype . stellar temperatures further complicates this picture , such that the spectral type of a wr star results from a subtle combination of ionization and wind density , in contrast with normal stars . the effect of reduced wr wind densities at low metallicity on wr populations is as follows . wr optical recombination lines will ( i ) decrease in equivalent width , since their strength scales with the square of the density , and ( ii ) decrease in line flux , since the lower wind strength will reduce the line blanketing , resulting in an increased extreme uv continuum strength at the expense of the uv and optical . the equivalent widths of optical emission lines in smc wn stars are well known to be lower than milky way and lmc counterparts ( conti et al . 1989 ) . to date , the standard approach for the determination of unresolved wr populations in external galaxies has been to assume metallicity independent wr line fluxes obtained for milky way and lmc stars ( schaerer & vacca 1998 ) regardless of whether the host galaxy is metal - rich ( mrk 309 , schaerer et al . 2000 ) or metal - poor ( i zw18 , izotov et al . 1997 ) . ideally , one would wish to use wr template stars appropriate to the metallicity of the galaxy under consideration . unfortunately , this is only feasible for the lmc , smc and solar neighbourhood , since it is challenging to isolate individual wr stars from ground based observations in more distant galaxies , which span a larger spread in metallicity . line luminosities for optical emission lines in lmc and smc wr stars are compared in table [ flux ] , illustrating significantly lower ( factor of 56 ) luminosities for the lower metallicity of the smc . .mean heii @xmath04686 line luminosities for magellanic cloud wn stars including known binaries ( crowther & hadfield 2006 ) [ cols="<,^,>,^ , > " , ] reduced wr line fluxes are also predicted for wr atmospheric models at low metallicity if one follows the metallicity dependence from vink & de koter ( 2005 ) , such that wr populations inferred from schaerer & vacca ( 1998 ) at low metallicity may underestimate actual populations by an order of magnitude . this is potentially problematic for single star evolutionary models at very low metallicities ( @xmath5 ) since the wr populations inferred for i zw18 and sbs0335 - 052e using milky way line fluxes compare well with evolutionary models ( e.g. izotov et al . 1997 ; papaderos et al . if wr populations are in fact a factor of @xmath110 larger , similar to that of the smc , close binary evolution would represent the most likely origin for such large wr populations . cc + & + for magellanic cloud metallicity starburst galaxies , one may employ appropriate template spectra ( e.g. crowther & hadfield 2006 ) to reproduce wr features , as shown in the upper panel of fig . [ ngc3125 ] for the starburst galaxy ngc 3125 . indeed , consistent fits to the blue and yellow wr bumps may be achieved for the a1 cluster within ngc 3125 using lmc template wr stars , as shown in the lower panels of fig . [ ngc3125 ] ( hadfield & crowther 2006 ) . schmutz et al . ( 1992 ) demonstrated that the ionizing fluxes from wr stars soften as wind density increases . consequently , a metallicity dependence for wr wind strengths implies that wr ionizing flux distributions soften at increased metallicity , as demonstrated by smith et al . indeed , relatively soft ionizing fluxes are observed in the super - solar metallicity wr starburst galaxy ngc 3049 ( gonzalez delgado et al . 2002 ) . at low metallicities , one anticipates a combination of weak uv and optical spectral lines from wr stars ( i.e. weak stellar heii @xmath04686 ) but very strong h and he lyman continua ( i.e. strong nebular heii @xmath04686 ) , as is indicated in the lower panel of fig . [ wc_wo ] . indeed , low metallicity star forming galaxies display strong nebular heii @xmath04686 , although shocks from supernovae remnants may also contribute to nebular emission . the typically environment of nearby ( @xmath6 ) long duration grbs is unusually metal - poor , as emphasized by stanek et al . ( 2006 ) with respect to star forming galaxies from the sloan digital sky survey . reduced wr mass - loss rates at low metallicity will lead to reduced densities in the immediate environment of grbs with respect to typical wr stars , as is observed ( chevalier et al . in addition , massive single stars undergoing homogeneous evolution in which wr mass - loss rates are low may maintain their rapidly spinning cores through to core - collapse ( yoon & langer 2005 ; langer & norman 2006 ) . observational and theoretical evidence supports reduced wind densities and velocities for low metallicity wr stars , which addresses the relative wr subtype distribution in the milky way and magellanic clouds , plus the reduced wr line strengths in the smc with regard to the galaxy and lmc . the primary impact at low metallicity is as follows ; ( a ) an increased wr population due to lower line fluxes from individual stars , of particular relevance to i zw18 and sbs0335 - 052e ; ( b ) harder ionizing fluxes from wr stars , potentially responsible for the strong nebular heii @xmath04686 seen in low metallicity hii galaxies ; ( c ) responsible for the reduced density of grb environments with respect to solar metallicity wr counterparts . finally , a metallicity dependence for wr winds may help to reconcile the relative number of wn to wc stars observed in surveys ( e.g. massey & johnson 1998 ) with evolutionary predictions . evolutionary models for which rotational mixing is included yet metallicity dependent wr winds are not ( meynet & maeder 2003 ) fail to predict the high n(wc)/n(wn ) ratio observed at high metallicities ( hadfield et al . 2005 ) , whilst models which account for the vink & de koter ( 2005 ) wr wind dependence compare much more favourably with observations ( eldridge & vink 2006 ) , in spite of the neglect of rotational mixing . chevalier , r.a . li , z .- y . & fransson , c. 2004 the diversity of gamma - ray burst afterglows and the surroundings of massive stars _ apj _ * 606 * , 369 conti , p.s . garmany , c.d . & massey p. , 1989 spectroscopic studies of wolf - rayet stars . v - optical spectrophotometry of the emission lines in small magellanic cloud stars _ apj _ * 341 * , 113 crowther , p.a . 2006 observed metallicity dependence of winds from wr stars . in _ stellar evolution at low metallicity : mass - loss , explosions , cosmology _ ( eds : h. lamers , n. langer & t. nugis ) , asp conf series in press crowther , p.a . 2007 physical properties of wolf - rayet stars _ _ in preparation crowther , p.a . & hadfield , l.j . 2006 reduced wolf - rayet line luminosities at low metallicity _ a&a _ * 449 * , 711 crowther , p.a . dessart , l. hillier , d.j . abbott j.b . & fullerton a.w . 2002 stellar and wind properties of lmc wc4 stars . a metallicity dependence for wolf - rayet mass - loss rates _ a&a _ * 392 * , 653 eldridge , j.j . & 2006 implications of the metallicity dependence of wolf - rayet winds _ a&a _ * 452 * , 295 galama t.j . 1998 an unusual supernova in the error box of the gamma - ray burst of 25 april 1998 _ nat _ * 395 * , 670 gonzalez delgado , r.m . leitherer , c. stasinska , g. & heckman , t.m . 2002 the massive stellar content in the starburst ngc 3049 : a test for hot - star models _ apj _ * 580 * , 824 grfener , g. & hamann , w .- r . 2005 hydrodynamic model atmospheres for wr stars . self - consistent modeling of a wc star wind _ a&a _ * 432 * , 633 hadfield , l.j . & crowther , p.a . 2006 how extreme are the wolf - rayet clusters in ngc3125 ? _ mnras _ * 368 * , 1822 hammer , f. , flores , h. , schaerer , d. , dessauges - zavadsky , m. , le floch , e. & puech , m. 2006 detection of wolf - rayet stars in host galaxies of gamma - ray bursts ( grbs ) : are grbs produced by runaway massive stars ejected from high stellar density regions ? _ a&a _ , in press ( astro - ph/0604461 ) hjorth , j. , sollerman , j. , moller , p. et al . 2003 a very energetic supernova associated with the -ray burst of 29 march 2003 _ nat _ * 423 * , 847 izotov , y.i . foltz , c.b . green , r.f . guseva , n.g . & thuan , t.x . 1997 i zw 18 : a new wolf - rayet galaxy _ apj _ * 487 * , l37 koesterke , l. & hamann , w .- r . 1995 spectral analyses of 25 galactic wolf - rayet stars of the carbon sequence _ a&a _ * 299 * , 503 langer , n. , 1989 mass - dependent mass loss rates of wolf - rayet stars _ a&a _ * 220 * , 135 langer , n. & norman , c. 2006 on the collapsar model of long gamma - ray bursts : constraints from cosmic metallicity evolution _ apj _ * 638 * , l63 macfadyen , a.i . & woosley s.e . 1999 collapsars : gamma - ray bursts and explosions in `` failed supernovae '' _ apj _ * 524 * , 262 massey , p. & johnson , o. , 1998 evolved massive stars in the local group . ii . a new survey for wolf - rayet stars in m33 and its implications for massive star evolution : evidence of the `` conti scenario '' in action _ apj _ * 505 * , 793 meynet , g. & maeder a. 2005 stellar evolution with rotation . xi . wolf - rayet star populations at different metallicities _ a&a _ * 429 * 581 nugis , t. & lamers , h.j.g.l.m . , 2000 mass - loss rates of wolf - rayet stars as a function of stellar parameters _ a&a _ * 360 * , 227 nugis , t. & lamers , h.j.g.l.m . 2002 the mass - loss rates of wolf - rayet stars explained by optically thick radiation driven wind models _ a&a _ * 389 * , 162 papaderos , p. , izotov , y.i . , guseva , n.g . , thuan t.x . & fricke k.j . 2006 oxygen abundance variations in the system of the two blue compact dwarf galaxies sbs 0335 - 052e and sbs 0335 - 052w _ a&a _ in press ( astro - ph/0604270 ) schaerer , d. & vacca , w.d . 1998 new models for wolf - rayet and o star populations in young starbursts _ apj _ * 497 * , 618 schaerer , d. , guseva , n.g . , izotov , y.i . & thuan , t.x . 2000 massive star populations and the imf in metal - rich starbursts _ _ * 362 * , 53 schmutz , w. leitherer , c. & gruenwald , r. 1992 theoretical continuum energy distributions for wolf - rayet stars _ pasp _ * 104 * , 1164 shapley , a.e . , steidel , c.s . , pettini , m. & adelberger , k.l . 2003 rest - frame ultraviolet spectra of z 3 lyman break galaxies _ apj _ * 588 * , 65 smith , l.f . & maeder , a. 1991 comparison of predicted and observed properties of wc stars - explanation of the subtype gradient in galaxies _ a&a _ * 241 * , 77 smith , l.j . norris , r.p.f . & crowther , p.a . 2002 realistic ionizing fluxes for young stellar populations from 0.05 to 2@xmath7 _ mnras _ * 337 * , 1309 stanek , k.z . , gnedin , o.y . beacom , j.f . et al . 2006 protecting life in the milky way : metals keep the grbs away _ apj _ in press ( astro - ph/0604113 ) vink , j.s . & de koter a. 2005 on the metallicity dependence of wolf - rayet winds _ a&a _ * 442 * , 587 vink , j.s . , de koter , a. & lamers , h.j.g.l.m . 2001 mass - loss predictions for o and b stars as a function of metallicity _ a&a _ , * 369 * , 574 yoon , s - c & langer , n. 2005 evolution of rapidly rotating metal - poor massive stars towards gamma - ray bursts _ a&a _ * 443 * , 643
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observational and theoretical evidence in support of metallicity dependent winds for wolf - rayet stars is considered .
well known differences in wolf - rayet subtype distributions in the milky way , lmc and smc may be attributed to the sensitivity of subtypes to wind density .
implications for wolf - rayet stars at low metallicity include a hardening of ionizing flux distributions , an increased wr population due to reduced optical line fluxes , plus support for the role of single wr stars as gamma ray burst progenitors .
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quantum walks @xcite have been proposed as potentially useful components of quantum algorithms @xcite . in recent years these systems have been studied in detail and some progress has been made in developing new quantum algorithms using either continuous @xcite or discrete @xcite versions of quantum walks . the key to the potential success of quantum walks seems to rely on the ability of the quantum walker to efficiently spread over a graph ( a network of sites ) in a way that is much faster than any algorithm based on classical coin tosses . quantum interference plays an important role in quantum walks being the crucial ingredient enabling a faster than classical spread . for this reason , some effort was made in recent years in trying to understand the implications of the process of decoherence for quantum walks @xcite . decoherence , an essential ingredient to understand the quantum classical transition @xcite , could turn the quantum walk into an algorithm as inefficient as its classical counterpart . the models studied in this context can be divided in two classes depending on how the coupling with an external environment is introduced . in fact , a quantum walk consists of a quantum particle that can occupy a discrete set of points on a lattice . in the discrete version , the walker carries a quantum coin , which in the simplest case can be taken as a spin-@xmath0 degree of freedom . the algorithm proceeds so that the walker moves in one of two possible directions depending on the state of the spin ( for more complex regular arrays , a higher spin is required ) . so , in this context it is natural to consider some decoherence models where the spin is coupled to the environment and others where the position of the walker is directly coupled to external degrees of freedom . the specific system in which the algorithm is implemented in practice will dictate which of these two scenarios is more relevant . several experimental proposals to implement discrete quantum walks in systems such as ion traps @xcite , cavity qed @xcite , and optical lattices @xcite have been analyzed ( see also ref . @xcite for a recent nmr implementation of a continuous quantum walk ) . the main effect of decoherence on quantum walks is rather intuitive : as the interaction with the environment washes out quantum interference effects , it restores some aspects of the classical behavior . for example , it has been shown that the spread of the decohered walker becomes diffusion dominated proceeding slower than in the pure quantum case . this result was obtained both for models with decoherence in the coin and in the position of the walker @xcite . however , it is known that classical correspondence in these systems has some surprising features . for example , for models with some decoherence in the quantum coin the asymptotic dispersion of the walker grows diffusively but with a rate that does not coincide with the classical one @xcite . also , a small amount of decoherence seems to be useful to achieve a quantum walk with a significant speedup @xcite . in this work we will revisit the quantum walk on a cycle ( and on a line ) considering models where the quantum coin interacts with an environment . the aim of our work is twofold . first we will use phase - space distributions ( i.e. , discrete wigner functions ) to represent the quantum state of the walker . the use of such distributions in the context of quantum computation has been proposed in ref . @xcite , where some general features about the behavior of quantum algorithms in phase space were noticed . a phase - space representation is natural in the case of quantum walks , where both position and momentum play a natural role . our second goal is to study the true nature of the transition from quantum to classical in this kind of model . we will show that models where the environment is coupled to the coin are not able to induce a complete transition to classicality . this is a consequence of the fact that the preferred observable selected by the environment is the momentum of the walker . this observable , which is the generator of discrete translations in position , plays the role of the `` pointer observable '' of the system @xcite . therefore , as we will see , the interaction with the environment being very efficient in suppressing interference between pointer states preserves the quantum interference between superpositions of eigenstates of the conjugate observable to momentum ( i.e. , position ) . again , the use of phase - space representation of quantum states will be helpful in developing an intuitive picture of the effect of decoherence in this context . the paper is organized as follows : in sec . ii we review some basic aspects of the quantum walk on the cycle . we also introduce there the phase - space representation of quantum states for the quantum walk and discuss some of the main properties of the discrete wigner functions for this system . in sec . iii we introduce a simple decoherence model and show the main consequences on the quantum walk algorithm . in sec . iv we present a summary and our conclusions . the quantum walks on an infinite line or in a cycle with @xmath1 sites are simple enough systems to be exactly solvable . for the infinite line the exact solution was presented in ref . the case of the cycle was first solved in ref . however , the exact expressions are involved enough to require numerical evaluation to study their main features . here we will review the main properties of this system presenting them in a way which prepares the ground to use phase - space representation for quantum states ( we will focus on the case of a cycle , the results for the line can be recovered from ours with @xmath2 ) . for a quantum walk in a cycle of @xmath1 sites , the hilbert space is @xmath3 , where @xmath4 is the space of states of the walker ( an @xmath1-dimensional hilbert space ) and @xmath5 is the two dimensional hilbert space of the quantum coin ( a spin @xmath0 ) . the algorithm is defined by a unitary evolution operator which is the iteration of the following map : @xmath6 here @xmath7 is the hadamard operator acting on the hilbert space of the quantum coin [ @xmath8 , @xmath9 being the usual @xmath10 pauli matrices ] . the operator @xmath11 is the cyclic translation operator for the walker , which in the position basis is defined as @xmath12 ( mod @xmath1 ) . it is worth stressing that the operator @xmath13 is nothing but a spin controlled translation acting as @xmath14 . so , the map @xmath15 consists of a spin controlled translation preceded by a coin toss , which is implemented by the hadamard operation ( the use of the hadamard operator in this context is not essential and can be replaced by almost any unitary operator on the coin @xcite ) . the notion of phase - space is natural in the context of this kind of quantum walk . in fact , the position eigenstates @xmath16 form a basis of the walkers hilbert space @xmath4 . the conjugate basis is the so called momentum basis @xmath17 . position and momentum bases are related by the discrete fourier transform , i.e. , @xmath18 the cyclic translation operator @xmath11 that plays a central role in the quantum walk is diagonal in momentum basis since @xmath19 . this simply indicates that momentum is nothing but the generator of finite translations . as a consequence of this , the total unitary operator defining the quantum walk algorithm is also diagonal in such basis . this fact , which was noticed before by several authors , enables a simple exact solution of the quantum walk dynamics . indeed , we can write the initial state of the system using the momentum basis of the walker as ( below @xmath20 denotes the total density matrix of the system formed by the walker and the coin ) @xmath21 where @xmath22 is the initial state of the quantum coin ( we assume that the initial state of the combined system is a product , but this assumption can be relaxed ) . after @xmath23 iterations of the quantum map the reduced density matrix of the walker ( denoted as @xmath24 ) is @xmath25,\end{aligned}\ ] ] where the operator @xmath26 is defined as @xmath27 all the temporal dependence is contained in the function @xmath28 which can be exactly computed in a straightforward way : one should first expand the initial state @xmath22 in a ( nonorthogonal ) basis of operators of the form @xmath29 ( @xmath30 ) , where @xmath31 are the eigenstates of the operator @xmath26 ( i.e. , @xmath32 ) . the explicit expressions for the eigenstates @xmath31 and the eigenvalues @xmath33 will not be given here since they can be found in the literature ( see , for example , @xcite ) . after doing this the evolution of the quantum state is fully determined by the equation @xmath34 below we will describe the properties of this solution using a phase - space representation for the quantum state of the system . wigner functions @xcite are a powerful tool to represent the state and the evolution of a quantum system . for systems with a finite - dimensional hilbert space , the discrete version of wigner functions was introduced using different methods ( see ref . @xcite ) . we will follow the approach and notation used in ref . @xcite , where these phase - space distributions were applied to study properties of quantum algorithms . for completeness , we will give here the necessary definitions and outline some of the most remarkable properties of the discrete wigner functions . for a system with an @xmath1-dimensional hilbert space the discrete wigner function can be defined as the components of the density matrix in a basis of operators defined as @xmath35 these are the so called phase - space point operators . they are defined in terms of the cyclic shift @xmath11 ( which in the position basis acts as @xmath36 ) , the reflection operator @xmath37 ( which in the position basis acts as @xmath38 ) , and the momentum shift @xmath39 ( which generates cyclic displacements in the momentum basis , i.e. , @xmath40 ) . phase - space operators are unitary , hermitian and form a complete orthogonal basis of the space of operators ( they are orthogonal in the hilbert schmidt inner product since they satisfy that @xmath41=n\delta_{q , q'}\delta_{p , p'}$ ] ) . expanding the quantum state in the @xmath42 basis as @xmath43 the coefficients @xmath44 are the discrete wigner functions of the quantum state , which are obtained as @xmath45 . \label{wigner}\ ] ] this function has three remarkable properties that almost give it the status of a probability distribution . the first two properties are evident : wigner functions are real numbers ( a consequence of the hermiticity of the phase - space operators ) and they provide a complete description of the quantum state ( a consequence of the completeness of the basis of such operators ) . the third property is less obvious : marginal probability distributions can be obtained by adding values of the wigner function along lines in phase - space . for this to be possible , it turns out that the phase - space has to be defined as a grid of @xmath46 points where @xmath44 is given at each point precisely by eq . ( [ wigner ] ) . thus , adding the values of the wigner function over all points satisfying the condition @xmath47 one gets the probability to detect an eigenstate of the operator @xmath48 with eigenvalue @xmath49 ( the sum is equal to zero if such eigenvalue does not exist ) . in particular , adding the wigner function along vertical lines @xmath50 one obtains the probability to detect eigenstates of the operator @xmath51 , with eigenvalues given by @xmath49 . these numbers are equal to zero if @xmath52 is odd and they are equal to the probability for measuring the position eigenstate @xmath53 when @xmath52 is even . complementary , adding values of the wigner function along horizontal lines enables us to compute the probability to detect a momentum eigenstate . a final remark about properties of the discrete wigner function is in order . figure [ wigner - basicas ] shows the wigner function of a position eigenstate @xmath54 and of a superposition of two position eigenstates , such as @xmath55 . as we see , in the first case the function is positive on a vertical line located at @xmath56 and is oscillatory on a vertical line located at @xmath57 . the interpretation of these oscillations is clear . it is well known that wigner functions display oscillatory regions whenever there is interference between two pieces of a wave packet . in this case , the cyclic boundary conditions we are imposing ( that originate from the fact that @xmath11 and @xmath39 are cyclic shift operators ) generate a mirror image for every phase - space point . thus , the oscillating strip can be interpreted as the interference between the positive strip and its mirror image . for the case of a quantum state which is a superposition of two position eigenstates , we observe two positive vertical lines with the usual interference fringes in between them . all these vertical lines have their corresponding oscillatory counterparts originated from the boundary conditions which are located at a distance @xmath1 . in what follows we will show wigner functions for typical states of a quantum walker . for a localized state ( up ) and for a delocalized superposition state ( down ) . the dimension of the hilbert space is @xmath58 . the horizontal ( vertical ) axis corresponds to position ( momentum ) . color code is such that red ( blue ) regions correspond to positive ( negative ) values while white corresponds to zero ( @xmath59 , @xmath60 , @xmath61).,title="fig:",scaledwidth=28.0% ] + for a localized state ( up ) and for a delocalized superposition state ( down ) . the dimension of the hilbert space is @xmath58 . the horizontal ( vertical ) axis corresponds to position ( momentum ) . color code is such that red ( blue ) regions correspond to positive ( negative ) values while white corresponds to zero ( @xmath59 , @xmath60 , @xmath61).,title="fig:",scaledwidth=28.0% ] ' '' '' for an initial state where the walker starts at a given position and the spin is initially unbiased [ @xmath62 , the behavior of the quantum walker starts to deviate from its classical counterpart at early times ( in this paper we will only consider unbiased initial states for the quantum coin ) . the phase - space representation of the state is shown in fig . [ wigner - local - nodeco ] and makes evident that a peculiar pattern of quantum interference fringes develops between the different pieces of the wave packet . the consequence of these interference effect is evident also when one looks at the probability distribution for different positions . this distribution is shown in fig . [ probab - nodeco ] and has been previously studied in the literature ( see @xcite ) . like its classical counterpart , at a given time @xmath23 the state initially located at @xmath63 has support only on states @xmath64 satisfying that @xmath65 adds up to an even number . however , in general the quantum distribution differs from the classical one , exhibiting peaks located at @xmath66 and a plateau of height @xmath67 around @xmath63 . after some time the wigner function of the quantum walker develops a shape that resembles a thread , as it is clear in the pictures . for this reason we will call this a thread state . + iterations for an initially localized walker with unbiased spin . @xmath68 , @xmath69 . we only plot the function for sites such that @xmath65 adds to an even number ( solid ) , and also include the classical distribution ( dotted).,scaledwidth=30.0% ] ' '' '' + a ) @xmath70 + b ) @xmath71 + c ) @xmath72 ' '' '' it is also interesting to analyze the evolution of the quantum walk for delocalized initial states . in particular , we will consider an initial state that is a coherent superposition of two position eigenstates ( whose wigner function was already displayed in fig . [ wigner - basicas ] ) . we find a wigner function that develops into a sum of two threads with a region in between where interference fringes are evident . this is displayed in fig . [ wigner - delocal - nodeco ] . some properties of the quantum walk for this kind of delocalized initial states were analyzed in ref . @xcite where it was noticed that the asymptotic probability distribution can be rather different from the one obtained from a localized initial state . below , we will show that the process of decoherence affects localized and delocalized initial states in a rather different way . + a ) @xmath70 + b ) @xmath73 + c ) @xmath74 ' '' '' we will consider a quantum walk where the quantum coin couples to an environment . to describe such coupling we will use a model which was introduced and studied in detail in ref . @xcite . in that paper it was shown that one can mimic the coupling to an external environment by a sequence of random rotations applied to the quantum coin , which have the effect of scrambling the spin polarization . more precisely , these kicks will be generated by the evolution operator @xmath75 where the angles @xmath76 take random values and @xmath77 is a fixed vector specifying the rotation axis . the virtue of this model is not only its simplicity but also the fact that can be experimentally implemented in a controllable manner using , for example , nmr techniques . after the application of one step of the quantum walk algorithm and one kick the evolution of the total system is @xmath78 to obtain a closed expression for the reduced density matrix of the walker for an ensemble of realizations of the random variables @xmath76 we follow the method proposed in ref . @xcite ( see ref . @xcite for a similar approach ) : assuming that these angles are randomly distributed in an interval ( @xmath79,@xmath80 ) , this density matrix is @xmath81\ \label{rodet}\end{aligned}\ ] ] expanding the initial state in the momentum basis as before enables us to simplify this expression . in fact , after doing this one can integrate over the random variables to find @xmath82 . \label{fkkp}\end{aligned}\ ] ] here @xmath83 is a superoperator acting on the spin state ( depending on direction @xmath77 of the kicks ) as @xmath84 where @xmath85 is a parameter related to the strength of the noise ( notice that @xmath86 corresponds to unitary evolution , i.e. , to @xmath87 ) . one can find a simple matrix representation for the superoperator @xmath83 for different choices of the rotation axis by writing @xmath22 in the basis formed by the identity and the pauli operators . in the appendix we show the explicit form of this matrix representation , which is helpful in finding exact and numerical solutions to the problem . in what follows we will show results for the case @xmath88 ( the other cases are qualitatively similar ) . to find the state of the walker at arbitrary times we simply need to find eigenstates and eigenvalues of the superoperator @xmath83 . this can always be done numerically and also analytically in the interesting case of @xmath89 , which can be denoted as `` total decoherence '' . in such case , the exact solution turns out to be @xmath90 several features of the decoherence effect are evident in the above formula . the environment produces a tendency towards diagonalization of the density matrix of the walker in the momentum basis ( matrix elements with @xmath91 are maximally suppressed ) . the decay of nondiagonal elements is exponential in time , as already discussed in ref . it is also clear that momentum eigenstates are not affected by the interaction since they are eigenstates of the full evolution [ in fact , from eqs . ( [ fkkp ] ) and ( [ superop ] ) follows @xmath92 . in this sense they are perfect pointer states for this model . in what follows we will present results concerning the evolution of several initial quantum states . + a ) @xmath93 + b ) @xmath94 + c ) @xmath95 ' '' '' + a ) @xmath96 + b ) @xmath97 + c ) @xmath98 ' '' '' the effect of decoherence on the evolution of states which are initially localized in position has been analyzed elsewhere @xcite . as shown in fig . [ wigner - local - deco ] , the wigner function of the evolved quantum state gradually loses its oscillatory nature . thus , instead of a thread state the interaction with the environment gradually produces a mixed state with a binomial distribution in the position direction ( which has an approximately gaussian shape for large @xmath23 ) but remains constant along the momentum direction . it is worth noticing that for any value of @xmath99 the resulting state has support only on position eigenstates satisfying that the sum @xmath100 is equal to an even number , as it was already pointed out for both the classical distribution ( @xmath89 ) and the purely quantum one ( @xmath86 ) in the preceding section . it is interesting to notice that the process of decoherence has a rather simple interpretation when represented in phase - space : decoherence in phase - space is roughly equivalent to diffusion in the position direction . this is not unexpected : in fact , in ordinary quantum brownian motion models a coupling to the environment through position ( momentum ) gives rise to a momentum ( position ) diffusion term in the evolution equation for the wigner function . the situation here is quite similar , since the walker effectively couples with the environment through its momentum . this is indeed the case because the environment interacts with the quantum coin which is itself coupled with the walker through the displacement operator which is diagonal in the momentum basis . therefore , the decoherence effect on the wigner function is expected to correspond to diffusion along the position direction . ' '' '' as noticed before , if one considers initial states where the walker is localized in a well defined position , one can see that the probability distribution for the different positions of the walker gradually tends to the classical one by increasing the coupling strength @xmath101 from @xmath87 ( no kicks ) to @xmath102 . this is shown in fig . [ probab - deco ] . + from the above analysis we could be tempted to conclude that the interaction between the quantum coin and the environment induces the classicalization of the walker . however , this is not the case . the process of decoherence induced in this way is not complete . this is most clearly seen by analyzing how it is that the interaction with the environment affects initial states of the quantum walker which are not initially localized . in fig . [ wigner - delocal - deco ] we show the wigner function of an initially delocalized state ( shown in fig . [ wigner - basicas ] ) under full decoherence . we can clearly see that decoherence does not erase all quantum interference effects . in fact , as mentioned above , the interaction with the environment induces diffusion along the position direction . therefore , interference fringes which are aligned along the position direction are immune to decoherence . thus , the final state one obtains from a superposition of two position eigenstates is not the mixture of two binomial states but a coherent superposition of them . this peculiar behavior is easily understood by noticing that this is a simple consequence of the fact that momentum eigenstates are pointer states : decoherence is effective in destroying superpositions of pointer states but highly inefficient in destroying superpositions of eigenstates of the conjugate observable ( position ) . by analyzing the entropy of the reduced density matrix of the walker one can get a more quantitative measure of the degree of decoherence achieved as a consequence of the interaction with the environment . for convenience we will not examine the von neumann entropy @xmath103 but concentrate on the linear entropy defined as @xmath104)$ ] , which is easier to calculate . this entropy provides a lower bound to @xmath103 @xcite . it is possible to show that almost no entropy is produced by the decay of the coherence present in the initially delocalized superposition state . in fact , this can be seen by comparing the entropy produced from the initially delocalized superposition and the one originated from an initial state in which the walker is prepared in an equally weighted mixture of two positions . these entropies can be seen in fig . [ entropy ] . the initial entropy of the mixture is 1 bit [ @xmath105 . it is quite clear from the curves shown in such figure that the entropy arising from the initial mixture remains to be 1 bit higher than the one originated from the initial coherent superposition . thus , the quantum coherence present in the initial state is robust under the interaction with the environment and does not decay at all . [ entropy ] shows another interesting feature : one would naively expect a monotonic dependence of the entropy with the coupling to the environment ( which is parametrized by @xmath101 ) . however this is not the case since the curves in fig . [ entropy ] intersect . this peculiar effect is made more evident in fig . [ entropy - rate ] where we study the entropy at a fixed time as a function of the coupling strength . in this figure a clear indication of a nontrivial behavior is seen : for early times the entropy grows slowly with @xmath101 and exhibits a flat plateau for large values of @xmath101 . however , as time progresses a peak develops : the largest value of entropy at a given time is not achieved by the largest coupling . to the contrary , the largest entropy is attained by an intermediate coupling @xmath106 , whose value decreases with time . the fact that for a given time the maximal entropy is not achieved by the maximal coupling to the environment is counterintuitive . as entropy is a measure of the spread of a distribution , this strange behavior can be rephrased as a manifestation of the counterintuitive fact that the decohered state ( which is approximately diagonal in position basis ) has a probability distribution that is more spread for @xmath107 than for @xmath108 . a possible explanation for this peculiar behavior is the following : for high values of the coupling to the environment the state rapidly becomes classical and the spread in position grows diffusively , as in the classical random walk . when the coupling to the environment is not strong , our result seems to indicate that the state of the walker remains `` quantum '' for a longer time during which it spreads at a rate faster than classical . when this quantum state finally decoheres it may end up having a larger entropy than the one attained for high coupling simply because it is spread over a wider range of positions . we speculate that there could be a relation between this peculiar feature and the properties that make some degree of decoherence useful for quantum walks as discussed by kendon and tregenna in @xcite . the value of the @xmath106 introduced above depends on both @xmath1 and @xmath23 and could be related to the position of the minima reported in refs . @xcite . for example , in a cycle regime ( @xmath109 ) @xmath106 diminishes with increasing @xmath1 as it is also the case for the position of the minima of the so called _ quantum mixing time _ @xcite . ( solid ) , @xmath110 ( dotted ) , @xmath111 ( dash dotted ) , @xmath112 ( dashed ) . the top ( down ) plot corresponds to an initial state which is an equally weighted mixture ( superposition ) of two position eigenstates . @xmath113.,title="fig:",scaledwidth=32.0% ] + ( solid ) , @xmath110 ( dotted ) , @xmath111 ( dash dotted ) , @xmath112 ( dashed ) . the top ( down ) plot corresponds to an initial state which is an equally weighted mixture ( superposition ) of two position eigenstates . @xmath113.,title="fig:",scaledwidth=32.0% ] ' '' '' for various values of time : @xmath114 ( solid ) , @xmath115 ( dashed ) , @xmath116 ( dotted ) , @xmath117 ( @xmath118 ) , @xmath119 ( @xmath120 ) , @xmath121 ( dash dotted ) . the initial state of the walker is well localized and the initial state of the quantum coin is unbiased . @xmath122 and all the curves are below the saturation regime [ @xmath123 . it is evident that , after some time , the maximum value of the entropy is not achieved by the maximum value of the coupling strength.,scaledwidth=32.0% ] ' '' '' the use of phase - space representation enables us to develop some intuition about the nature of the decoherence process in the kind of quantum walk analyzed in this paper . by coupling the quantum coin to an environment we obtain a decoherence model which is roughly equivalent to position diffusion . as we mentioned above , this is a natural result whose origin can be traced back to the way in which the system effectively couples to the environment ( via the momentum operator ) . the relation between decoherence and position diffusion can also be established by analyzing in more detail the structure of the superoperator @xmath83 [ given in eq . ( [ superop ] ) ] . let us consider the form of the superoperator after @xmath23 iterations . if we use eq . ( [ superop ] ) we can easily see that , as each iteration doubles the number of terms , we will have an expression with @xmath124 terms each one of which has pauli operators applied at different times . to obtain the function @xmath125 one should compute the trace over the quantum coin . in each of the @xmath124 terms we can move the pauli operator @xmath126 towards the outside of the expression and cancel them due to the cyclic property of the trace . for the case @xmath88 it is easy to show that the only remaining effect of the pauli operators ( that in this case anti commute with the hadamard operator ) is to reverse the direction of the rotation in @xmath26 defined in eq . ( [ mofk ] ) . the final expression can be shown to be @xmath127 where @xmath128 and @xmath129 ( we use the convention @xmath130 ) . therefore , the superoperator is the sum of @xmath124 terms each one of which contains a contribution that is identical to that of a quantum walk where the direction of the walker is chosen at random after the first step . each of the @xmath124 terms is labeled by a @xmath23bit string @xmath131 and corresponds to a quantum walk where the direction of the @xmath132th step ( @xmath133 ) is reversed if and only if @xmath134 is odd . in the limit of total decoherence each of these terms has equal weight . therefore the final state is simply the average over an ensemble where each member corresponds to each of the @xmath135 possible choices of two directions ( forward or backward ) for the @xmath136 steps ( notice that the direction of the first step is not affected by the decoherence model we chose ) . for this type of decoherence it is clear that the quantum walk becomes a random walk . the relation between decoherence and position diffusion is quite evident in this way . it is worth pointing out that similar models of decoherence were considered in ref . @xcite in a different context . other decoherence models have been analyzed for quantum walks @xcite , where the effective coupling to the environment is through the position observable . in such case , we expect decoherence to correspond to diffusion along the momentum direction . combining the two types of decoherence ( i.e. , considering coupling to the environment via the quantum coin and the position of the quantum walker ) the initial state corresponding to a superposition of two positions would finally decay into a mixture of two binomial states ( see ref . @xcite for similar results obtained when studying decoherence models with a natural phase - space representation in a finite quantum system evolving under various quantum maps ) . the above conclusions are generic for any model in which decoherence is due to the coupling of the quantum coin to an environment . an interesting class of models , based on the use of quantum multi baker maps @xcite , has been studied . in such models one replaces the quantum coin with a quantum system with a higher - dimensional hilbert space . the total space of states is then @xmath137 . here @xmath138 is the dimensionality of the system which plays the role of the quantum coin and is considered to be an even number ( @xmath139 , so that we can always consider @xmath140 ) . the dynamics for a quantum multi baker map is defined in terms of the unitary operator [ that replaces eq . ( [ onestep ] ) ] : @xmath141 where @xmath142 is the unitary operator defining the so called `` quantum baker map '' ( see refs . @xcite ) and @xmath143 is a pauli operator acting on the hilbert space @xmath5 ( the most significant qubit of the internal space @xmath140 ) . the properties of the operator @xmath142 have been widely studied in the literature @xcite : the map faithfully represents a classically chaotic system ( in the large @xmath138 limit ) . from the point of view of the quantum walker the situation is quite similar to the one we studied in this paper . one can describe a quantum multi baker map as an ordinary quantum walk where the quantum coin ( whose hilbert space is @xmath5 ) interacts with an environment ( whose hilbert space is @xmath144 ) . the interaction is modeled by the quantum baker map acting on the total internal hilbert space @xmath145 , which also replaces the usual hadamard step in eq . ( [ onestep ] ) . as the quantum baker map is chaotic , the state of the quantum coin will be roughly randomized after each iteration . thus , the effect should be similar to the one we described here ( where the quantum coin is subject to a noisy evolution ) . however , after a large number of iterations ( of the order of @xmath138 ) all the possible orthogonal directions available in the internal space of the quantum coin would have been explored . one should therefore expect that this model will stop being effective in producing decoherence after such time . recent studies of quantum multi baker systems agree with these expectations ( see ref . @xcite where a transition from diffusive to ballistic behavior after a time of the order of @xmath138 has been analyzed ) . in any case , based on the results of our work we believe that in quantum multi baker systems the relative stability of initially delocalized superpositions will also be observable . we thank marcos saraceno and augusto roncaglia for useful discussions and assistance during several stages of this work . was partially supported by funds from fundacin antorchas , anpcyt and ubacyt . a matrix representation for the superoperator @xmath83 can be obtained for @xmath146 . we will write this superoperator in the basis formed by the identity and the pauli matrices ( we use the standard ordering of the basis as @xmath147 ) . in such basis the matrix of @xmath83 is @xmath148 where [ @xmath149 . notice they all converge to the same matrix when @xmath86 ( no decoherence ) . + to compute eq . ( [ fkkplanteodeco ] ) we need to diagonalize @xmath83 . although the matrix representation of the superoperator is rather sparse , the eigenvalues and eigenvectors are quite cumbersome for an arbitrary value of @xmath99 ( including @xmath86 , no decoherence ) , so it was more convenient to use numerical techniques . however , for the special case of complete decoherence ( @xmath89 ) it is possible to obtain a simple formula and the final result for @xmath125 . the result for @xmath154 is given in eq . ( [ fkkyz ] ) . n. shenvi , j. kempe , k. birgitta whaley , phys . a * 67 * , 052307 ( 2003 ) . v. kendon and b. tregenna , phys . a * 67 * , 042315 ( 2003 ) . brun , h.a . carteret , a. ambainis , phys . a * 67 * , 032304 ( 2003 ) . v. kendon and b. tregenna , e - print quant - ph/0301182 . for a review see j.p . paz and w.h . zurek , in _ coherent matter waves , les houches session lxxii _ , edited by r. kaiser , c. westbrook and f. david ( springer verlag , berlin , 2001 ) , pp . 533614 , e - print quant - ph/0010011 . w. dr , r. raussendorf , v.m . kendon , h.j . briegel , phys . a * 66 * , 052319 ( 2002 ) . j. du , h. li , x. xu , m. shi , j. wu , x. zhou , r. han , phys . rev . a * 67 * , 042316 ( 2003 ) . c. miquel , j.p . paz , m. saraceno , phys . a * 65 * , 062309 ( 2002 ) . zurek , phys . d , * 24 * , 15161525 ( 1981 ) . a. nayak and a. vishwanath , e - print quant - ph/0010117 ; see also a. ambainis , e. bach , a. nayak , a. vishwanath , j. watrous , proc . 33rd annual acm symposium on theory of computing ( stoc 2001 ) , 3749 . m. scully , m. hillery and e. wigner , phys . rep . * 106 * , 121 ( 1984 ) . g. teklemariam , e.m . fortunato , c.c . lpez , j. emerson , j.p . paz , t.f . havel , d.g . cory , phys . a * 67 * , 062316 ( 2003 ) . the linear entropy @xmath155)$ ] shares two important properties of the von neumann entropy @xmath156 $ ] : ( @xmath157 ) it is non - negative and vanishes only if @xmath20 is pure ; ( @xmath158 ) its maximum value is @xmath159 and it is reached iff @xmath160 . in addition to this , @xmath161 provides a lower bound to @xmath103 since @xmath162 . the use of this entropy can be found , for example , in @xcite and also in i. garca - mata , m. saraceno , m.e . spina , phys . lett . * 91 * , 064101 ( 2003 ) ; d. monteoliva and j.p . paz , phys . * 85 * , 3373 ( 2000 ) .
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we analyze the quantum walk on a cycle using discrete wigner functions as a way to represent the states and the evolution of the walker .
the method provides some insight on the nature of the interference effects that make quantum and classical walks different .
we also study the behavior of the system when the quantum coin carried by the walker interacts with an environment .
we show that for this system quantum coherence is robust for initially delocalized states of the walker .
the use of phase - space representation enables us to develop an intuitive description of the nature of the decoherence process in this system .
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radiation pressure on spectral lines ( line force ) driving a wind from an accretion disk is the most promising hydrodynamical ( hd ) scenario for agn outflows . within this framework , a wind is launched from the disk by the local disk radiation at radii where the disk radiation is mostly in the ultraviolet ( uv ; e.g. , shlosman , vitello & shaviv 1985 ; murray et al . 1995 , mcgv hereafter ) . such a wind is continuous and has mass loss rate and velocity which are are capable of explaining the blueshifted absorption lines observed from many agn , if the ionization state is suitable . such winds have the desirable feature that they do not rely on unobservable forces or fields for their motive power . however , detailed tests of this idea via modelling is challenging . the wind dynamics are coupled to the ionization and opacity properties of the gas , and the location and nature of the radiation sources is not well understood . previous models by us ( proga stone and kallman 2000 , hereafter psk ) relied on uv radiation from the black hole in order to make the flow steady and to impart a strong radial component . in this paper we report new calculations which relax this requirement , and which demonstrate quantitative consistency between disk winds and observations . uv driven disk winds in agn are motivated by analogy with winds from hot stars , which have been explored in great detail ( eg . lamers and cassinelli 1999 and references therein ) . but they differ in important ways , including the role of rotation near a keplerian disk , the non - uniform disk temperature distribution , and the influence of the strong x - ray flux from the inner disk and black hole . rotation acts to make the vertical component of gravity increase with height near the disk plane , and can also affect the wind trajectory when height is comparable to the radius . close to the disk plane the wind is driven by photons emitted locally , but at greater heights the radiation spectrum and the driving momentum depend on position ; the increase in disk temperature at small radius can result in a net outward component to the momentum . the wind density and mass loss rate can be estimated from the observed agn uv luminosity . when compared with the x - ray flux from the agn , the density is low enough that the gas is predicted to be highly ionized . if so , the opacity needed for efficient driving and for line formation will be very small and the wind will fail . on the other hand , estimates for the column density in the radial direction close to the disk surface are high enough that a portion of the wind can be shielded from this ionization . viable models for disk winds must account self - consistently for the wind driving , ionization , and self shielding . mcgv pointed this out and postulated the existence of ` hitchhiking ' gas which is not driven by uv but which provides the shielding . their calculation was a one - dimensional time independent quasi - radial flow . proga , stone & kallman ( 2000 , psk hereafter ) were able to to relax some of the mcgv assumptions , and explored consequences of a radiation driven disk wind model by performing 2.5 dimensional time - dependent hd simulations . the most challenging component of fluid dynamics calculations is treatment of the radiative transfer . in the case of disk wind calculations this limitation forces an inexact treatment of the radiative transfer , by assuming parameterized values for the x - ray and uv continuum opacities . psk found that a disk accreting onto a @xmath0 black hole at the rate of 1.1 @xmath1 ( @xmath2 ) can launch a wind at @xmath3 cm from the central engine . the x - rays from the central object are significantly attenuated by the disk atmosphere so they can not prevent the uv radiation from pushing matter away from the disk . however , the x - rays can overionize the gas in the flow high above the disk and decrease the wind velocity . for a reasonable x - ray opacity , e.g. , @xmath4 , the disk wind can be accelerated by the central uv radiation to velocities of up to 15000 @xmath5 at a distance of @xmath6 cm from the central engine . the covering factor of the disk wind is @xmath7 . the wind is unsteady and consists of an opaque , slow vertical flow near the disk that is bounded on the polar side by a high - velocity stream . a typical column density radially through the fast stream is a few @xmath8 so that the stream is optically thin to the uv radiation but optically thick to the x - rays . a key issue in modeling radiation pressure driven winds from disks is the relative importance of the driving by photons emanating from near the black hole , i.e. at radii smaller than the computational grid , compared with photons emitted by the disk at radii within the computational grid . models for cv disk winds , which resemble agn disk winds except for an absence of strong x - rays , show that strong steady flows require a central source of uv . motivated by this , psk as well as mcgv considered the situation where the central radiation in the uv accelerates the gas in the radial direction . a consequence of this assumption is that an efficiently driven wind must attenuate the x - rays , in order to avoid over - ionization , but must transmit the uv , in order to be driven . this places an intrinsic limit on the radial column density , and hence on the mass loss rate , and can also influence the geometry of the flow . our goal in the present paper is to explore what happens when the assumption of a strong central flux is relaxed in the psk models , and to examine other consequences of the line - driven disk wind model . in particular , we check how robust is the disk wind solution and whether the solution predicts synthetic line profiles capable of reproducing line profiles observed in agn . in 2 , we summarize the key elements of our calculations . we present the results from disk wind simulations and synthetic line profile calculations in 2.1 and 2.2 , respectively . the paper ends in 4 , with our conclusions and discussion . in this paper we extend work by psk by relaxing some of their assumptions and simplifications . our 2.5-dimensional hd numerical method is in most respects as described by psk . here we only describe the key elements of the method and list the changes we made . we refer a reader to psk for details . as in psk , we apply line - driven stellar wind models ( castor , abbott & klein 1975 , hereafter cak ) to winds driven from accretion disks ( see also proga , stone & drew 1998 ; 1999 , psd98 and psd99 hereafter ) . we specify the radiation field of the disk by assuming that the temperature follows the radial profile of the optically thick accretion disk ( shakura & sunyaev 1973 ) . we account for some of the effects of photoionization . in particular , we calculate the gas temperature assuming that the gas is optically thin to its own cooling radiation . we also take into account some of the effects of photoionization on the line force . in particular , we compute the parameters of the line force using a value of the photoionization parameter , @xmath9 and the analytical formulae due to stevens & kallman ( 1991 ) . this procedure is computationally efficient and gives approximate estimates for the number and opacity distribution of spectral lines for a given @xmath9 without detail information about the ionization state ( see stevens & kallman 1991 ) . additionally , we take into account the attenuation of the x - ray radiation by computing the x - ray optical depth in the radial direction . we modify psk s method as follows : ( i ) we decrease the inner and outer radius of the computational domain ( see 3 for details ) ; ( ii ) to compute the radiation force due to lines , we use the intensity of the radiation integrated over the uv band only ( i.e. , between @xmath10 and @xmath11 , see also proga 2003 ) . ( in psk we took the uv flux to be a constant fraction of the flux from the black hole ) ; ( iii ) we exclude entirely the central object radiation force ; ( iv ) we compute the x - ray optical depth in the radial direction by considering only electron scattering ( in psk we took the x - ray opacity to be @xmath12 @xmath13 ) ; ( v ) we do not explicitly treat continuum opacity for photons emitted from the disk , although the self - shielding of the lines is taken into account by our sobolev treatment of the line force ( in psk we allowed the disk radiation to be attenuated in the radial direction with an opacity @xmath14 g@xmath15 @xmath13 ) . the changes in our calculations extend the range of validity of calculations of psk . the decrease of the inner radius of the computation domain allows us to capture as much as possible of the uv emitting disk within our grid . the second change makes our calculations as self - consistent as possible without making them computationally prohibitive , i.e. , there is no wind launching from very large radii where the disk is cold and radiates few uv photons , and from very small radii where the disk effective temperature or gas temperature , or both , are too high to permit enough spectral lines . the third change was motivated by the results from psk : we want to test the importance of attenuation of the central uv flux as a limit to the acceleration of gas to high velocities , and also the extent to which the central flux is needed to make a strong and steady flow . by neglecting the radiation force from the central object we are exploring relatively unfavorable conditions for wind acceleration . the only force that can accelerate the wind is the radiation force due to the disk . the fourth change was also motivated by our wish to explore relatively unfavorable conditions for attenuation of x - rays by disk winds . using this treatment , our method gives a lower limit for the optical depth to ionizing photons . the fifth change to the calculations means that we allow all disk photons emitted toward a given point in the wind to reach this point . this may correspond to an overestimate of the disk radiation force . however , exact transfer of uv continuum is computationally prohibitive and is treated in the sobolev approximation even in one dimensional hot star wind models . we briefly discuss some of the consequences of our changes in .4 . we present here results for the same model parameters as in psk with only two exceptions ( see below ) . we assume the mass of the non - rotating black hole , @xmath16 . to determine the radiation field from the disk , we assume the mass accretion rate @xmath17 m@xmath18 yr@xmath15 . these system parameters yield the disk luminosity , @xmath19 of 50% of the eddington luminosity and the disk inner radius , @xmath20 cm , where @xmath21 is the schwarzschild radius of a black hole . the radiation field from the central engine is specified by its luminosity @xmath22 with @xmath23 set to 1 , its fraction in the uv band ( @xmath24 ) and its fraction in the x - rays ( @xmath25 ) . in psk , @xmath26 and these are the only changes to the model parameters we made . the spectral energy distribution of the ionizing radiation is not well known , our choice of values for @xmath27 and @xmath28 is guided by the results from zheng et al . ( 1997 ) and laor et al . ( 1997 ) [ e.g. , see figure 6 in the latter , for a comparison of the spectral energy distributions found for various samples of qsos . ] as mentioned in 2 , we include the central radiation only as a source of ionizing photons and exclude its contribution to the radiation force . there are two definitions of the photoionization parameter used in literature : @xmath9 and @xmath29 ( e.g. , krolik 1999 ) . the former is based on the ionizing flux while the latter on the number density of the ionizing photons . for the adopted spectral energy distribution , the conversion between the two is as follows : @xmath30 . our computational domain is defined to occupy the radial range @xmath31 , and the angular range @xmath32 . the @xmath33 domain is discretized into zones . our numerical resolution consists of 100 and 140 zones in the @xmath34 and @xmath35 directions , respectively . we use fixed zone size ratios , @xmath36 and @xmath37 . figure 1 shows the instantaneous density , temperature and photoionization parameter distributions and the poloidal velocity field of the model . the wind speed at the outer boundary is 2000 to 12000 km @xmath38 . this corresponds to a dynamical time of @xmath390.2 yrs for the material at @xmath40 . figure 1 shows results at the end of the simulation after @xmath41 years . although the flow is still weakly time - dependent after this time has elapsed , the gross properties of the flow ( e.g. , the mass loss rate and the radial velocity at the outer boundary ) , settle down to steady time - averages . as in the flow found by psk , the wind has 3 components : ( i ) a hot , low density flow in the polar region ( ii ) a dense , warm and fast equatorial outflow from the disk , ( iii ) a transitional zone in which the disk outflow is hot and struggles to escape the system . the main difference with the results of psk is that here the transitional zone is much more prominent , and it occupies a large fraction of the computational domain . in the polar region , the density is very small and close to the lower limit that we set on the grid , i.e. , @xmath42 g @xmath43 . the line force is negligible because the matter is highly ionized as indicated by a very large photoionization parameter ( @xmath44 ) . the gas temperature is close to the compton temperature of the x - ray radiation . the matter in the polar region is pulled by the gravity from the outer boundary , which is an artifact of the boundary conditions ( e.g. , we do not model a jet which is likely to propagate through the polar region ) . however , this region is sometimes filled with gas which is launched from the disk with a large vertical velocity in an episodic manner . the outflowing wind itself has distinct two components : hot and warm . the two components are launched from the disk by the line force . the gas density at the disk atmosphere and wind base is @xmath45 , so the photoionization parameter is low ( log(@xmath9)@xmath46 - 5 ) despite the strong central radiation . however as the flow from the inner part of the disk is accelerated by the line force its density decreases . for the disk wind launched at small radii ( @xmath47 ) , this decrease of the density causes an increase in the gas temperature and the photoionization parameter . as this process proceeds the gas becomes fully ionized and loses all driving lines . the wind speed when this happens is not generally great enough to allow the gas to escape , and it tends to fall back toward the disk . this ` failed wind ' has an effect on the remainder of the flow , both due to its shielding effect on the central x - rays and due to its pressure as it falls toward the disk . the disk wind launched at large radii does not become overionized downstream despite a density decrease . the photoionization parameter in this wind is kept low because of the large column density toward the source of the ionizing photons . consequently , the outer disk wind is not only launched but also accelerated by the disk line force . we find that the outer wind becomes radial relatively close to the disk . this may seem surprising because we set to zero the radiation force in the radial direction due to the central object . there are three effects responsible for the flow to be equatorial : ( i ) overionization of the vertical part of the wind by the central object radiation ; ( ii ) the ram pressure of the failed wind launched at smaller radii that falls back after it is overionized ; and ( iii ) the line force due to disk photons emitted interior to a given point in the wind . the two first effects dominate at smaller radii and close to the interface between the hot and warm flow whereas the third effect dominates above the disk at large radii where the fore - shortening of the disk radiation weakens . the flows we find are time dependent and have density and velocity fluctuations of @xmath48 even after the flow has evolved for many dynamical times . in particular , there are occasionally regions along the equator with the gas densities lower than those for hydrostatic equilibrium . from those low density regions , the radiation force can launch a wind with an acceleration length scale shorter than for a cak - like wind [ the line force is a strong function of the gas density ( cak ) ] . in fact , we observe that the line acceleration can be so efficient that some gas can reach velocities comparable to the escape velocity before it is eventually overionized . this flow behavior manifests itself as erratic high velocity ejections of gas from the inner disk . figure 2 presents the run of the density , radial velocity , mass flux density , accumulated mass loss rate , photoionization parameter and column density as a function of the polar angle , @xmath35 , at the outer boundary , @xmath49 cm from figure 1 . the accumulated mass loss rate , @xmath50 and the column density , @xmath51 are computed as in psk ( see eq . 13 in psd 98 and eq . 24 in psk for the definitions of @xmath50 and @xmath51 , respectively ) . the gas density is very low , i.e. , @xmath52 , for @xmath35 between @xmath53 and @xmath54 . then the gas density increases with @xmath35 between @xmath54 and @xmath55 to the level of a few @xmath56 . for @xmath57 , the density decreases to the level of @xmath52 at @xmath58 . then the density increases again with @xmath35 and reaches the maximum of a few @xmath56 at @xmath59 . this is followed by a decrease of the density with the minimum of a few @xmath60 at @xmath61 . for , @xmath62 , the density sharply increases , as might be expected of a density profile determined by hydrostatic equilibrium . the radial velocity is @xmath63 km @xmath38 for @xmath64 and has a broad flat maximum at the level of @xmath65 km @xmath38 for @xmath35 between @xmath54 and @xmath66 . for @xmath35 between @xmath66 and @xmath67 , the radial velocity is negative and then gradually increases to @xmath68 km @xmath38 at @xmath69 . this is followed by a drop of @xmath70 to a very small value near the equator . we note that @xmath70 stays nearly constant at the level of @xmath71 km @xmath38 for @xmath72 . the accumulated mass loss rate is negligible for @xmath73 because of the very low gas density and velocity . for @xmath74 , @xmath75 increases to @xmath76 at @xmath77 . for @xmath35 between @xmath55 and @xmath78 , the accumulated mass loss rate stays nearly constant . for @xmath79 , the accumulated mass loss rate increases to @xmath80 at @xmath81 . the column density in the wind increases gradually with @xmath35 . in particular , it increases from @xmath82 at @xmath83 to @xmath84 at @xmath85 . for @xmath86 , @xmath51 continues to increase and indicates that the wind is optically thick to electron scattering at @xmath87 for the central object radiation . the column densities greater than @xmath88 are effectively infinite , and represent complete obscuration of the central object . winds with such high @xmath51 can still be radiation driven owing to the difference between the column the disk radiation sees and the column the central source radiation sees . we expect that the disk wind obscuration can change the ratio between the number of all qsos and bal qsos , i.e. , qsos , viewed by an observer almost edge - on , would not be detected or identified as qso . the photoionization parameter is very high , @xmath44 , for @xmath40 because of the very low density in the polar region . however , over next @xmath89 , @xmath9 drops by many order of magnitude owing to the increase of the column density . the @xmath35 profiles of the flow properties show that the overoinized outflow can contribute to the total mass loss rate at the level of a few per cent . our new simulations of agn disk wind confirm the main result from psk : the disk atmosphere can shield itself so that the local disk radiation can launch gas off the disk photosphere . here we find that this results holds even when the condition for shielding are unfavorable : the ionizing photons are only scattered on electrons but not absorbed by the shielding gas . our new simulations also provide some new insights to the acceleration . in particular , we find the local disk force suffices to accelerate the disk wind to high velocities in the radial direction provided the wind does not change the geometry of the disk radiation by continuum scattering and absorption processes . the main difference between our disk wind solution and the solution presented in psk is the properties and behavior of the hot component of the disk outflow which struggles to escape the system . in psk , this component occupied a relatively narrow @xmath35 range ( @xmath90 ) above the disk wind , whereas here @xmath91 . the key reason for this difference is the strength of the radial line force relative to the latitudinal line force acting on the inner disk wind . in psk , the radial force is strong and accelerates the inner wind on a relatively long length scale ( of order of the wind launch radius ) . a small fraction of the inner flow is overionized and fails to develop into a strong outflow . this overionized gas falls on the freshly launched disk wind and introduces weak perturbations to the disk wind so that the wind is not significantly slowed down nor disrupted . here , on the other hand , the radial force is relatively weak and the flow from the innermost disk tends to be vertical . the flow geometry is important for how long the flow can stay shielded : for the vertical geometry , the flow height can be larger than the height of the shielding gas whereas for the radial geometry , the flow height can stay smaller than the height of the shielding gas even at very large distances from the launch point . therefore vertical flow is more susceptible to overionization than radial flow ; a much larger fraction of the inner flow is overionized in the vertical flow than in the radial flow . in our simulations the overionized flow is relatively dense and as it falls on the disk it disrupts the freshly launched the disk wind . we note that the mass loss rate in our simulations is smaller than in psk s simulation by a factor of @xmath39 2 . this is an unexpected result because here we allow launching a wind from radii smaller than those in psk and a simple scaling law for line - driven disk wind models suggests that mass loss rate should increase with decreasing radius ( e.g. , psd98 ; proga & kallman 2002 ) . additionally , one can argue that because of this scaling , overionization should become less of a problem with decreasing radius ( proga & kallman 2002 ) . what is then responsible for this low mass loss rate ? our analysis of the flow properties and time evolution shows that at the early phase of the evolution , the mass flux density is consistent with the expectations based on the above arguments . however , as the inner disk wind becomes overionized and starts falling back toward the equator the mass loss rate decreases because the average gas density at the wind base increases which reduces the driving line force . additionally , the infalling gas is dense and can significantly slow the outflow which it encounters on its way toward the disk . there are then a few factors which reduce the mass loss rate but the primary factor is the flow overionization . our time - dependent disk wind solutions differ from those found by psd98 , in which it was shown that cv disk winds are unsteady in time when the wind is driven solely by disk radiation . in the latter , the flow could also fall back on the disk , but it was not strong enough to cause a significant decrease in the mass loss rate . furthermore , in the psd98 simulations there was no radiation capable of fully ionizing the wind so that regions of relatively low density , above the falling gas , could be accelerated to high velocities and contribute to the total mass loss rate . simulations presented in psk and here can serve as a proof - of - concept for the radiation driven disk wind model of outflows in agn . some consequences of the radiation driven disk wind model have been explored in the context of cv and low mass x - ray binaries ( lmxb ; e.g. , see psd98 and proga & kallman 2002 , respectively ) . generally , the insights from those models allow us to explain systems with strong evidence for uv absorbing disk wind such as cvs and systems without uv absorbing disk wind such as lmxb . in the case of cvs , the model also predicts uv line profiles which are capable of reproducing observations ( proga 2003 ) . in this section , we present synthetic line profiles predict by our model and briefly discuss their relevance to agn observations . we present line profiles for the c iv @xmath92 transition . we compute the profiles by integrating flux from the entire disk at a given orientation ( i.e. , as in the dynamical models , the continuum source has a finite size ) . the line profile calculation performed here are exactly as in proga et al . ( 2002 ) with the exception that we set the c iv abundance to one in the regions with the gas temperature from 8000 k to 420000 k and to zero elsewhere . for the adopted model parameters , the upper limit for the gas density corresponds to the disk effective temperature at @xmath93 . we refer a reader to proga et al . ( 2002 ) for details on line - profile calculations . the c iv abundance could be in principle , computed self - consistently for given wind structure and radiation field . however , we save this for a future paper and focus on gaining some insights from the multidimensional models of disk winds and predicted profiles . to illustrate the effect of viewing angle on the line profiles , figure 3 shows line profiles for five inclination angles : @xmath94 , and @xmath95 . we limit our line profile calculation to the absorption contribution only . modeling of the emission component depends on the influence of thermal line emission , and we will discuss this in a future paper . figure 3 clearly shows that the line profiles are very sensitive to the inclination angle . in particular , for @xmath96 the line absorption is broad and nearly symmetric whereas for @xmath97 just @xmath54 higher a strong blueshifted absorption dominates the line profile ( compare figs 3c and 3d ) . we note that for @xmath98 the profiles is more symmetric than for @xmath96 . in the latter case , the blueshifted absorption is somewhat stronger than the redshifted absorption . overall our line profiles are representative of generic line profiles for a bipolar wind from a rotating disk ( e.g. proga et al . 2002 and references therein ) . for example , for a relatively high inclination angles ( @xmath99 ) the profiles are affected by the absorption in the rotating base of the wind , so blueshifted as well as redshifted absorption is present ( see proga et al . 2002 , and references therein ) . the presence of the expanding wind manifests itself in the profiles by a blueshifted absorption for inclination angles where the continuum source is viewed through the expanding wind . in our wind model , the disk wind with a low @xmath9 ( i.e. , high c iv abundance ) has a half - opening angle of @xmath100 . therefore only for @xmath101 , our line profiles show blueshifted absorption stronger than redshifted absorption . in general , the line profiles predicted by our models are similar to those predicted by the model for cv winds ( proga 2003 ) . however , there are two important differences : ( i ) for low @xmath97 , cv wind models predict a blueshifted absorption whereas here little absorption is present at those angles and ( ii ) for all @xmath97 , cv wind models predict that the maximum of the blueshifted absorption is line - centered whereas here the maximum can be blueshifted by as much as @xmath102 km @xmath38 . the reason for these differences is the fact that in cv wind models , the base of the wind covers the whole continuum source whereas here the continuum source extends inside the computational domain and therefore the continuum is only partially covered by the flow . our line profiles are consistent with many of the properties of profiles observed in broad absorption line qsos ( e.g. , korista et al . 1993 ; hall et al 2002 ) . in particular , the synthetic line profiles can be as strong and broad ( up to 30000 km @xmath38 ) as those observed . additionally , for certain inclination angles the maximum of the blueshifted absorption is far from the line center - this is consistent with so - called detached troughs observed in some qsos . some of the profiles shown in fig . 3 display prominent redshifted absorption . in observed lines , this component is likely to be affected by the emission contribution to the line ( e.g. , proga 2003 ) . the line emission can fill in the redshifted absorption and also some of the highly blueshifted absorption produced by the wind near the equator where the rotational velocity exceeds the expansion velocity . for example , proga ( 2003 ) showed that for high inclinations ( @xmath103 ) , the profile has double - peaked emission ( figs . 1e and 1j in proga 2003 ) . in this paper we have presented numerical simulations of the radiation - driven disk wind scenario for agn outflows . these models provide insights into the wind s capability to shield itself from ionizing radiation . we performed axisymmetric time - dependent hydrodynamic calculations similar to those of proga , stone and kallman ( 2000 ) . we studied the robustness of the radiation launching and acceleration of the wind for relatively unfavorable conditions . in particular , we have taken into account the central engine radiation as a source of ionizing photons but neglected its contribution to the radiation force . additionally , we have accounted for the attenuation of the x - ray radiation by computing the x - ray optical depth in the radial direction assuming that only electron scattering contributes to the opacity . our new simulations confirm the main result from psk : the disk atmosphere can shield itself from external x - rays so that the local disk radiation can launch gas off the disk photosphere . we have also found that the local disk force suffices to accelerate the disk wind to high velocities in the radial direction . this is true provided the wind does not change significantly the geometry of the disk radiation by continuum scattering and absorption processes , and we discuss the plausibility of this requirement . overall our models are consistent with observations of broad absorption line ( bal ) qsos . additionally , synthetic profiles of a typical resonance ultraviolet ( uv ) line , predicted by our models , promise to reproduce observed bals and it is likely that the model can account for other agn outflows observed in uv as well as in x - rays . strong ionizing radiation has long been recognized as a serious challenge to theories of agn outflows absorbing the uv radiation . radiation - driven disk wind models are particularly affected by this problem because the overionization of the wind can prevent the wind from being driven . but in other models , including a magnetic one , overionization also needs to be addressed owing to the requirement for sufficient un - ionized gas to account for the observed lines . here we find that an indirect effect of overionization can reduce but not prevent the development of a wind , for our choice of model parameters : the so - called failed inner disk wind . if the density of the failed wind is relatively high then the ram pressure of such a wind can seriously slow down the disk wind which otherwise could be driven by radiation . we emphasize that this damaging effect of the failed wind can be important to explain agn winds within the radiation driven disk wind scenario rather than being used as an argument against the scenario . in particular , in this scenario the accretion disk , with parameters as adopted here ( see 3 ) , should produce a powerful wind from radii as small as @xmath104 because the radiation at those radii is mostly in the uv . however , such a wind would be too fast to be consistent with agn observations . one of plausible solutions of this problem is to allow for the energy dissipation at small radii to occur not only inside but also outside the disk ( i.e. , in the disk corona ) . consequently , the uv radiation from the innermost disk would be reduced while ionizing radiation would be increased . the two factors would reduce launching of the wind . a complimentary solution is that the failed wind can disrupt development of a wind up to a radius of a few @xmath105 . we have performed many simulations to study the sensitivity of disk wind solutions to the black hole mass and disk luminosity . these results will appear in proga et al.(2004 ) ( in preparation ) . here we only mention that the wind solution is very sensitive to the luminosity compared to the eddington limit . in particular , we find that for the model parameters as in .3 but with the luminosity reduced by a factor of 5 ( from 50% to 10% of the eddington limit ) there is no disk wind . the primary reason for this luminosity sensitivity is the fact that the mass flux density of the wind decreases strongly with decreasing disk luminosity and the wind is more subject to overionization ( e.g. proga & kallman 2002 ) . our simulations show also that for a fixed eddington fraction ( e.g. , 50% ) it is easier to produce a wind for @xmath106 than for @xmath107 . this result is a consequence of the decrease of the uv contribution to the disk total radiation with decreasing mass of the bh in the shakura sunyaev disk model for a fixed eddington fraction . we have also performed many simulations to study the sensitivity of our main results to the model parameters . the most important parameter of the model is the x - ray opacity . this quantity could be computed from first principles for given spectral energy distribution , strength and geometry of the radiation , and the structure and chemical composition of the flow . however , such an approach to @xmath108 s calculations is not computationally feasible and therefore we have explored asymptotic approximations , e.g. , assuming that @xmath109 ( corresponding to the thomson cross section ) . as we discussed above , such a simplistic approach yields rather conservative results for the wind strength . in our sensitivity studies , we have computed wind models using a more realistic @xmath108 . in particular , we computed @xmath110 based on the analytic fit to the detailed photoionization calculations using the photoionization code xstar . we find that @xmath110 can be expressed a function of the photoionization parameter , @xmath9 in the following way : @xmath111 , in units of @xmath112 . we found that as expected a wind with this higher @xmath110 is stronger ( e.g. , in terms of @xmath113 and the opening angle ) the model solution described here . the radiative transfer of continuum photons inside the disk wind is as critical as the wind photoionization structure in modeling disk winds . to accelerate the wind to velocities comparable to the escape velocity we should not rely on the radiation which originates interior to the wind zone because the wind will shield itself from this radiation in the same way as it shields itself from the external ionizing radiation . here we explore a configuration where the ionizing radiation comes from a point source located at the center while the driving radiation comes from the extended source , ( i.e. , the uv emitting disk ) . calculations of the radiative transfer of the former is relatively simple as it can be approximated by 1d solutions for the optical depth . however , the radiative transfer of the latter is intrinsically 3d and computationally expensive . we argue that the uv continuum photons emitted from the disk into the overlying disk wind can propagate through the disk without large changes of the net radiation flux , hence our assumption that the wind is optically thin to the disk photons . on the other hand , x - rays which are emitted outside the disk wind suffer many scatterings and absorptions once they reach the wind zone . consequently , the x - rays do not penetrate deep inside the wind zone . the x - ray transfer in the case we consider is analogous to the problem of disk irradiation by an external source whereas the uv transfer is analogous to the problem of radiation transfer in an extended atmosphere . our simulations show how the line force can produce a fast , highly ionized wind : if the density near the wind base is relatively low and the base is shielded over long enough length scale above the disk then the local disk radiation can accelerate gas to relatively high velocities ( even exceeding the escape velocity ) before the gas becomes overionizated . at large radii , such a wind would be transparent to the uv radiation but may be able to absorb x - rays . we plan to present models for the x - ray properties of disk winds in a future paper . we thank n. arav , m.c . , begelman , j.r . gabel , p.b . hall , g.t . richards , and j.m . stone for useful discussions . we also thank an anonymous referee for comments that helped us clarify our presentation . we acknowledge support provided by nasa through grant hst - ar-09947 from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas5 - 26555 . dp also acknowledges support from nasa ltsa grants nag5 - 11736 and nag5 - 12867 . we acknowledge support from the w. m. keck foundation , which purchased the jila 74-processor keck cluster .
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we explore consequences of a radiation driven disk wind model for mass outflows from active galactic nuclei ( agn ) .
we performed axisymmetric time - dependent hydrodynamic calculations using the same computational technique as proga , stone and kallman ( 2000 ) .
we test the robustness of radiation launching and acceleration of the wind for relatively unfavorable conditions .
in particular , we take into account the central engine radiation as a source of ionizing photons but neglect its contribution to the radiation force .
additionally , we account for the attenuation of the x - ray radiation by computing the x - ray optical depth in the radial direction assuming that only electron scattering contributes to the opacity .
our new simulations confirm the main result from our previous work : the disk atmosphere can shield itself from external x - rays so that the local disk radiation can launch gas off the disk photosphere .
we also find that the local disk force suffices to accelerate the disk wind to high velocities in the radial direction .
this is true provided the wind does not change significantly the geometry of the disk radiation by continuum scattering and absorption processes ; we discuss plausibility of this requirement .
synthetic profiles of a typical resonance ultraviolet line predicted by our models are consistent with observations of broad absorption line ( bal ) qsos .
| 10,117 | 324 |
we consider @xmath0 diffeomorphisms on riemannian surfaces . our goal is to study the hyperbolic properties of a class of maps exhibiting a homoclinic tangency associated to a fixed saddle point @xmath1 , as in figure [ map ] . homoclinic tangencies inside the limit set , width=226 ] we assume without loss of generality that we are working on @xmath2 and in the standard euclidean norm . we recall that a compact invariant set @xmath3 is * uniformly hyperbolic * if there exist constants @xmath4 and a continuous , @xmath5-invariant , decomposition @xmath6 of the tangent bundle over @xmath7 such that for all @xmath8 and all @xmath9 we have @xmath10 by standard hyperbolic theory , every point @xmath11 in @xmath7 has stable and unstable manifolds @xmath12 tangent to the subspaces @xmath13 and @xmath14 respectively , and thus in particular _ transversal _ to each other . the presence of the tangency therefore implies that the dynamics on @xmath15 can not be uniformly hyperbolic . we emphasize at this point that , in the case we are considering , the homoclinic tangency is accumulated by transverse homoclinic orbits which in turn are accumulated by periodic points . thus it constitutes an intrinsic obstruction to uniform hyperbolicity which can not be resolved by simply ignoring the orbit of tangency . most of the classical theory of homoclinic bifurcations for diffeomorphisms ( see @xcite and references therein ) considers the unfolding of homoclinic tangencies _ external _ to the set @xmath7 to which they are associated , thus causing no real issues with the hyperbolicity at the bifurcation parameter . the main goal of such a theory has often been to study the hyperbolicity and the occurrence of tangencies in a neighbourhood of the orbit of tangency _ after _ the bifurcation . the presence of an internal tangency gives rise to a more subtle situation and it has only recently been shown that this can actually occur as a _ first bifurcation _ part of the motivation of the present paper is to study the global dynamics and hyperbolicity _ at this bifurcation parameter_. another part of the motivation for this result is to give an example of a compact invariant set which is as `` uniformly '' hyperbolic as possible in the ergodic theory sense , but still not uniformly hyperbolic . to formulate this result precisely , let @xmath16 denote the set of all @xmath17-invariant probability measures @xmath18 on @xmath7 . by the classical multiplicative ergodic theorem of oseledet s there is a well defined set @xmath19 of _ lyapunov exponents _ associated to the measure @xmath18 , we give the precise definitions below . we say that the measure @xmath20 is _ hyperbolic _ if all the lyapunov exponents are non - zero . the existence of an invariant measure with non - zero lyapunov exponents indicates a minimum degree of hyperbolicity in the system . a stronger requirement is that all invariant measures @xmath18 are hyperbolic and of course an even stronger requirement is that they are all `` uniformly '' hyperbolic in the sense that all lyapunov exponents are uniformly bounded away from 0 . this condition is clearly satisfied for uniformly hyperbolic systems but , as we show in this paper , it is strictly weaker . the class of examples we are interested in were first introduced in @xcite and constitute perhaps the simplest situation in which an internal tangency can occur as a first bifurcation . in section [ maps ] we give the precise definition of this class . for this class we shall then prove the following all lyapunov exponents of all measures in @xmath21 are uniformly bounded away from zero . as an immediate corollary we have the following statement which is in itself non - trivial and already remarkable . @xmath22 is uniformly hyperbolic on periodic points . we recall that uniform hyperbolicity on periodic points means that there exists constants @xmath23 such that for each periodic point @xmath24 of period @xmath25 , the derivative @xmath26 has two distinct real eigenvalues @xmath27 with @xmath28 notice that the bounds for the eigenvalues are exponential in @xmath29 . as far as we know this is the first known example of this kind , although it is possible , and indeed even likely , that such a property should hold for more complex examples such as benedicks - carleson and mora - viana parameters in hnon - like families @xcite and horseshoes at the boundary of the uniform hyperbolicity domain of the hnon family . the weaker result on the uniform hyperbolicity of periodic points has been proved recently for both cases in @xcite and @xcite respectively . . other known examples of non - uniformly hyperbolic diffeomorphisms include cases in which the lack of uniformity comes from the presence of `` neutral '' fixed or periodic points . in these cases , the dirac-@xmath30 measures on such periodic orbits are invariant and have a zero lyapunov exponent . it is interesting to view our result in the light of some recent work which appears to go in the opposite direction : if a compact invariant set @xmath31 admits an invariant splitting @xmath32 such that , for a total measure set of points @xmath33 , the lyapunov exponents are positive in @xmath34 and negative in @xmath35 , then @xmath31 is uniformly hyperbolic , @xcite . here , the lyapunov exponents are not even required to be uniformly bounded away from zero . thus the existence of at least one orbit as which the splitting degenerates is a necessary condition for a situation such as the one we are considering , in which the lypaunov exponents are all non - zero but @xmath7 is strictly not uniformly hyperbolic . the concept of uniform hyperbolicity of the periodic points and of measures plays an important role in the general theory of one - dimensional dynamics . in some situations , such as for certainclasses of smooth non - uniformly expanding unimodal maps these notions have been been shown to be equivalent to each other and to various other properties usually associated to uniform hyperbolicity such as exponential decay of correlations @xcite . higher dimensional cases are generally much harder due to the complexity of the geometrical and dynamical structure of the examples and progress is only starting to be made . after defining the class of systems of interest in section [ maps ] , in section [ splitting ] we construct a field of cones which are invariant under the derivative map . in section [ periodicpoints ] we show that some * non - uniform hyperbolicity * is satisfied off the orbit of tangency : the expanding and contracting directions of points which are very close to the orbit of tangency are almost aligned and therefore the contraction and/or expansion may take arbitrarily long time to become effective . we formalize this by saying that there exists a constant @xmath36 , _ depending on the point _ @xmath11 ( compare with ) , such that for all @xmath11 not in the orbit of tangency we have @xmath37 with @xmath38 arbitrarily small near the points of the orbit of tangency , i.e. @xmath39 one way to show that the non - uniformity is not too extreme is to show that there is a _ uniformly hyperbolic core _ in the following sense : there exists a region @xmath40 containing one point of the orbit of tangency , and there exists constants @xmath23 such that , for each point @xmath41 that returns to @xmath40 at the @xmath42-th iterate , we have @xmath43 in fact , we show that this uniform hyperbolicity occurs well before the return of the orbit of @xmath44 to @xmath40 . those exponential estimates are valid from the moment that the orbit of @xmath44 leaves a certain neighborhood of the fixed point @xmath1 , and they keep working as long as the orbit wanders around @xmath45 . finally , in section [ lyapunov ] we show that this implies the statement of our main theorem . we start with a geometrical definition of the class of maps under consideration . let @xmath46 be a @xmath47 diffeomorphism and let @xmath48\times [ 0,1 ] $ ] denote the unit square in @xmath49 . we suppose that there exist 5 `` horizontal '' regions in @xmath50 which are mapped as in figure [ the map ] : @xmath51 for @xmath52 . the map @xmath53,width=340 ] we suppose that regions @xmath54 are mapped affinely to their images , with derivative @xmath55 in @xmath56 and @xmath57 respectively , for two constants @xmath58 satisfying @xmath59 and @xmath60 . more explicitly we suppose that @xmath61 in in @xmath62 ( @xmath63 ) , @xmath64 in @xmath57 ( @xmath65 ) and that @xmath66 is a vertical strip parallel to @xmath67 . in particular there is a hyperbolic fixed point at the origin @xmath68 whose stable and unstable manifolds contain the lower and left hand side of the square @xmath50 respectively . we emphasize that the explicit nature of the map in these regions is for simplicity only , and to allow us to concentrate on the strategy for dealing with the tangency . this could be weakened significantly , for example by assuming only some uniformly hyperbolic structure in regions @xmath69 . region @xmath70 is mapped outside the square @xmath50 and thus we do not need to make any particular assumption on its form . region 4 is mapped to a `` fold '' @xmath71 which contains a point of the orbit of tangency . here we make some `` non - degeneracy assumptions '' essentially stating that vertical lines in @xmath72 are mapped to non - degenerate parabolas . more precisely , we suppose that there is a region @xmath73 bounded by two disjoint curves so that points in @xmath74 , are mapped inside region @xmath75 with second coordinate greater than @xmath76 . then , for each @xmath77 in @xmath78 $ ] we have that @xmath79 is contained in the graph of the map @xmath80 for some fixed @xmath81 and some @xmath82 . thus all vertical lines in @xmath83 are mapped to parabolas with constant curvature @xmath84 . again , this could be weakened significantly although we do assume for less trivial reasons that the curvature @xmath84 is sufficiently large in relation to the constants @xmath85 and @xmath86 . notice that the point @xmath87 is a point of tangency of the stable and unstable manifolds of the fixed point at the origin . to control the global properties of the family of parabolas we also assume that @xmath88 and that , for every @xmath89 , @xmath90 a completely explicit example of a map @xmath17 satisfying all these conditions is given in @xcite . we now define the limit set @xmath91 it follows from the construction that @xmath3 is non empty and contains ( at least ) one orbit of tangency between stable and unstable manifolds . our assumptions make it easy to control the dynamics in regions @xmath69 and therefore our main objective is to control the dynamics of points returning to the fold @xmath92 and to a neighbourhood of the tangency . thus we concentrate on the _ first return map _ from @xmath93 to itself and show that this map satisfies some strong hyperbolicity estimates from which we can deduce our main estimate on the hyperbolicity of periodic points . consider the foliation @xmath94 , of @xmath95 , whose leaves are images of vertical lines by the map @xmath96 ( in @xmath97 , the leaves are parabolas ) . to each point @xmath98 we associate the tangent direction , @xmath99 , to the leaf of @xmath94 which passes through @xmath98 , at this point ( @xmath99 is parallel to @xmath100 . let @xmath101 be the angle between @xmath99 and the horizontal . there is only one point for which @xmath102 and this is the point of tangency @xmath103 . at all other points we have with @xmath104 . then , we divide @xmath7 into 3 regions , according to the value of @xmath105 : @xmath106 notice that @xmath107 for each point @xmath108 we define a _ cone _ @xmath109 of vectors in the tangent space @xmath110 as follows : let @xmath111 , and @xmath112 for any integer @xmath113 . all cones will be centered on the vertical and , for convenience , we define a _ standard cone _ @xmath114 for @xmath115 in @xmath116 we simply let @xmath117 . for @xmath118 , we define cones which extend to each side of the vertical by three times the ( cotangent of the ) angle of the line @xmath119 defined above . more precisely , recall that @xmath120 is tangent to the graph of @xmath121 , see , and therefore @xmath122 . then we let @xmath123 before defining the cones in the remaining points in @xmath124 we show that the conefield defined above in @xmath125 is invariant under the first return map to @xmath125 . as a first step towards proving this invariance we estimate the first return time . [ return ] consider @xmath126 , and @xmath127 the smallest positive integer such that @xmath128 . then @xmath129 note that , since @xmath130 , we have @xmath131 , provided that @xmath132 . in fact , the images @xmath133 of the point @xmath115 will stay in region @xmath134 while @xmath113 is such that @xmath135 . since @xmath136 , there must be at least one more linear iterate of @xmath133 before its orbit visits the pre - image of @xmath137 . so , we have @xmath138 . analogously , using the inverse map , we have @xmath139 . we shall show below that these estimates mean that cones are sufficiently contracted before returning , guaranteeing that they are mapped back strictly into existing cones even if they started out very wide . in particular we shall use the following simple [ returncor ] consider @xmath126 , and @xmath127 the smaller positive integer such that @xmath128 . then for this @xmath127 we have @xmath140 and @xmath141 the first inequality follows from which gives @xmath142 the second follows also by by a similar straightforward calculation . we are now ready to prove the invariance of the conefield for the first return map . [ inclusion ] there exists @xmath143 such that , if @xmath144 in the definition of @xmath96 , @xmath145 and @xmath146 is the first positive iterate of @xmath115 in @xmath147 , then @xmath148 since @xmath149 is centered in the vertical line , we have , after applying @xmath150 to the vectors of @xmath151 , that @xmath152 is a cone centered in the vertical line at the point @xmath153 whose width is @xmath154 times the width of @xmath151 . after applying @xmath155 to this cone , since @xmath136 , we obtain a cone centered in @xmath156 with width smaller than @xmath157 times the original width . there are many cases to be considered depending on the location of @xmath115 and @xmath146 in @xmath147 . here we present the case where @xmath115 and @xmath146 are both in @xmath158 . the other cases are made following the same steps . let @xmath115 and @xmath146 be in @xmath158 . then @xmath159 and @xmath160 since @xmath161 , it satisfies @xmath162 this implies that a cone centered in @xmath163 with width @xmath164 will be properly contained in @xmath165 . then , in order to obtain @xmath166 we need @xmath167 or , equivalently , @xmath168 we now distinguish two cases . in the case where @xmath169 we have @xmath170 and therefore it is enough to show that @xmath171 . by this follows as long as we have @xmath172 for all @xmath173 ( remember that we are working with points of @xmath78\times[0,1]$ ] ) . since @xmath174 is fixed and negative , the condition holds if @xmath175 is big enough . in the case where @xmath176 we argue in the same way and , using , reduce the problem to showing that @xmath177 for any @xmath178 . again this follows as long as @xmath84 is sufficiently large . the other cases follow analogously . we can now extend the conefield to the set @xmath179 , in a natural way by considering the images of of all cones in @xmath125 and taking slightly wider cones at each point . in this way we obtain a conefield defined at every point outside the orbit of tangency such that @xmath180 for every point @xmath181 . notice that these cones can be arbitrarily wide close to the orbit of tangency . for points @xmath181 which do not enter @xmath125 in either forward or backward time , we simply let @xmath109 be the standard cone @xmath182 , see . for points @xmath181 which intersect @xmath183 only in forward time , we define @xmath184 where @xmath185 is the first time for which @xmath186 and @xmath187 is that standard cone . the stable cone field is defined assigning to each @xmath188 , the closure of the complement of @xmath189 , and satisfies the inclusion condition for the inverse map . we fix a neighbourhood @xmath40 of the tangency point @xmath190 of radius @xmath191 . let @xmath192 . we start with a simple estimate of the number of iterations @xmath127 it takes before @xmath146 falls outside the set @xmath78\times[0,1/3]$ ] ( @xmath127 is the escape time of @xmath115 ) . [ pontos ] we have @xmath193 . note that @xmath40 is contained in the domain where @xmath194 is linear , and , as long as the second coordinate of the image of a point of @xmath31 is less than @xmath195 , it is contained in @xmath134 , the second coordinate of @xmath115 being multiplied by @xmath196 at each iteration . for @xmath197 as above , let @xmath198 be a vector contained in @xmath151 and @xmath199 . [ vetores ] we have @xmath200 as we saw in the last section , since @xmath201 is linear in @xmath40 , we have @xmath202 [ expansao]with the notation above , if @xmath175 is sufficiently big , then @xmath203 since @xmath204 , we have that @xmath205 using lemmas [ vetores ] and [ pontos ] , we obtain @xmath206 by lemma [ pontos ] we have that @xmath207 and therefore @xmath208 since @xmath209 , we can write @xmath210 the result follows if @xmath175 is sufficiently large ( greater than @xmath211 in this case ) . let @xmath212 be the connected component of @xmath213 containing @xmath214 , and @xmath215 . in this section we estimate the growth of the unstable vectors of points in @xmath216 outside @xmath217 , under the action of @xmath155 . we also compute bounds for the lyapunov exponents of @xmath96 , and prove the main result . first we claim that , if @xmath219 , we have @xmath222 . indeed , by construction , if @xmath223 and at distance at least @xmath191 from the tangency point @xmath190 , then we have that @xmath224 , and the width of @xmath151 is less than @xmath225 . notice that vectors in @xmath226 grow by at least a constant factor @xmath218 at each iteration where @xmath218 is the rate of growth of the vector @xmath227 by the linear map @xmath228 ) . if @xmath115 is a point outside the set @xmath217 , then we have already @xmath229 , and the estimate applies to the unstable vectors of @xmath115 . let @xmath230 . until now we proved that , if the orbit of @xmath115 visits the set @xmath40 , then when it leaves the set @xmath217 , it has accumulated exponential growth to the unstable vectors by a factor @xmath231 as we have seen in lemma [ expansao ] . as shown above , iterates outside @xmath217 contribute as well with the same factor . similar calculations hold for the stable vectors , and we define @xmath232 through the analogous estimates for @xmath233 . we remark that in particular these estimates improve significantly on the growth estimates of @xcite which already imply the existence of a non - uniformly hyperbolic splitting as defined in the introduction . [ sequencia ] if @xmath234 does not belong to the orbit of homoclinic tangency , @xmath235 , and @xmath236 , then there exist sequences @xmath237 and @xmath238 , and a positive number @xmath239 such that @xmath240 and @xmath241 . let @xmath234 be a point in @xmath31 outside the orbit of homoclinic tangency . for @xmath244 , let @xmath245 and @xmath246 . let @xmath247 . if @xmath248 is non - empty and infinite , let @xmath249 and consider @xmath250 as the smaller natural number bigger than @xmath251 such that @xmath252 . in this case , we have @xmath253 , for any @xmath254 . as a consequence of the estimates in lemmas ( [ expansao ] ) and ( [ first ] ) , we have that if @xmath248 is empty , then , there are three cases to consider . if the orbit of @xmath234 is not in @xmath217 , then we have exponential growth beginning at the first iterate . if @xmath263 and leaves it eventually , we wait until it does so to have exponential growth , and argue as before . if @xmath264 and never leaves it , then @xmath265 for all @xmath244 , meaning that @xmath234 is in the local stable manifold of @xmath266 . in this case , any non - horizontal vector based in @xmath234 will eventually grow exponentially by a factor @xmath267 . let @xmath269 be an invariant measure , and @xmath275 the subset of @xmath276 for which the lyapunov exponents exist . then , by the oseledet s theorem @xmath277 . by lemma ( [ sequencia ] ) all lyapunov exponents are outside the interval @xmath278 , for all @xmath279 .
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we study the hyperbolicity of a class of horseshoes exhibiting an _ internal _ tangency , i.e. a point of homoclinic tangency accumulated by periodic points .
in particular these systems are strictly _ not _ uniformly hyperbolic .
however we show that all the lyapunov exponents of all invariant measures are uniformly bounded away from 0 .
this is the first known example of this kind .
| 6,257 | 114 |
the spectroscopic signature of the presence of @xmath1li in the atmospheres of metal - poor halo stars is a subtle extra depression in the red wing of the @xmath2li doublet , which can only be detected in spectra of the highest quality . based on high - resolution , high signal - to - noise vlt / uves spectra of 24 bright metal - poor stars , ( * ? ? ? * asplund ( 2006 ) ) report the detection of @xmath1li in nine of these objects . the average @xmath1li/@xmath2li isotopic ratio in the nine stars in which @xmath1li has been detected is about 4% and is very similar in each of these stars , defining a @xmath1li plateau at approximately @xmath5li@xmath6 ( on the scale @xmath7h@xmath8 ) . a convincing theoretical explanation of this new @xmath1li plateau turned out to be problematic : the high abundances of @xmath1li at the lowest metallicities can not be explained by current models of galactic cosmic - ray production , even if the depletion of @xmath1li during the pre - main - sequence phase is ignored ( see reviews by e.g. ( * ? ? ? * christlieb 2008 ) , ( * ? ? ? * cayrel 2008 ) , prantzos 2010 [ this volume ] and references therein ) . a possible solution of the so - called ` second lithium problem ' was proposed by ( * ? ? ? * cayrel ( 2007 ) ) , who point out that the intrinsic line asymmetry caused by convection in the photospheres of metal - poor turn - off stars is almost indistinguishable from the asymmetry produced by a weak @xmath1li blend on a presumed symmetric @xmath2li profile . as a consequence , the derived @xmath1li abundance should be significantly reduced when the intrinsic line asymmetry in properly taken into account . using 3d nlte line formation calculations based on 3d hydrodynamical model atmospheres computed with the co@xmath0bold code ( ( * ? ? ? * freytag 2002 ) , ( * ? ? ? * wedemeyer 2004 ) , see also http://www.astro.uu.se/@xmath9bf/co5bold_main.html ) , we quantify the theoretical effect of the convection - induced line asymmetry on the resulting @xmath1li abundance as a function of effective temperature , gravity , and metallicity , for a parameter range that covers the stars of the ( * ? ? ? * asplund ( 2006 ) ) sample . a careful reanalysis of individual objects is under way , in which we consider two alternative approaches for fixing the residual line broadening , @xmath10 , the combined effect of macroturbulence ( 1d only ) and instrumental broadening , for given microturbulence ( 1d only ) and rotational velocity : ( i ) treating @xmath10 as a free parameter when fitting the li feature , ( ii ) deriving @xmath10 from additional unblended spectral lines with similar properties as lii@xmath4 . we show that method ( ii ) is potentially dangerous , because the inferred broadening parameter shows considerable line - to - line variations , and the resulting @xmath1li abundance depends rather sensitively on the adopted value of @xmath10 . + the hydrodynamical atmospheres used in the present study are part of the cifist 3d model atmosphere grid ( ( * ? ? ? * ludwig 2009 ) ) . they have been obtained from realistic numerical simulations with the co@xmath0bold code which solves the time - dependent equations of compressible hydrodynamics in a constant gravity field together with the equations of non - local , frequency - dependent radiative transfer in a cartesian box representative of a volume located at the stellar surface . the computational domain is periodic in @xmath11 and @xmath12 direction , has open top and bottom boundaries , and is resolved by typically 140@xmath13140@xmath13150 grid cells . the vertical optical depth of the box varies from @xmath14 ( top ) to @xmath15 ( bottom ) , and the radiative transfer is solved in 6 or 12 opacity bins . further information about the models used in the present study is compiled in table[tab1 ] . each of the models is represented by a number of snapshots , indicated in column ( 6 ) , chosen from the full time sequence of the corresponding simulation . these representative snapshots are processed by the non - lte code nlte3d that solves the statistical equilibrium equations for a 17 level lithium atom with 34 line transitions , fully taking into account the 3d thermal structure of the respective model atmosphere . the photo - ionizing radiation field is computed at @xmath16 frequency points between @xmath17 and 32407 , using the opacity distribution functions of @xcite to allow for metallicity - dependent line - blanketing , including the hi h@xmath18 and hi hi quasi - molecular absorption near @xmath19 and @xmath20 , respectively . collisional ionization by neutral hydrogen via the charge transfer reaction h(@xmath21 ) + li(@xmath22 ) @xmath23 li@xmath18(@xmath24 ) + h@xmath25 is treated according to @xcite . more details are given in @xcite . finally , 3d nlte synthetic line profiles of the lii @xmath26 doublet are computed with the line formation code linfor3d ( http://www.aip.de/@xmath9mst/linfor3d_main.html ) , using the departure coefficients @xmath27=@xmath28 provided by nlte3d for each level @xmath29 of the lithium model atom as a function of geometrical position within the 3d model atmospheres . as demonstrated in fig.[fig1 ] , 3d nlte effects are very important for the metal - poor dwarfs considered here : they strongly reduce the height range of line formation such that the 3d nlte equivalent width is smaller by roughly a factor 2 compared to 3d lte . ironically , the line strength predicted by standard 1d mixing - length models in lte are close to the results obtained from elaborate 3d nlte calculations . we note that the half - width of the 3d nlte line profile , fwhm(nlte)=8.5 km / s , is larger by about 10% : fwhm(lte)=@xmath30 and @xmath31 km / s , respectively , before and after reducing the li abundance such that 3d lte and 3d nlte equivalent widths agree . this is because 3d lte profile senses the higher photosphere where both thermal and hydrodynamical velocities are lower . however , the nlte line profile is significantly less asymmetric than the lte profile , even if the latter is broadened to the same half - width ( fig.[fig1 ] , bottom panel ) . + .list of models used in the present study . columns ( 2)-(6 ) give effective temperature , surface gravity , metallicity , number of opacity bins used in the radiation hydrodynamics simulation , and number of snapshots selected for spectrum synthesis . the equivalent width of the synthetic 3d non - lte @xmath2li doublet at @xmath26 , assuming a(li)=2.2 and no @xmath1li , is given in column ( 7 ) . columns ( 8) and ( 9 ) show @xmath32(li ) and @xmath33(li ) , the @xmath1li/@xmath2li isotopic ratio inferred from fitting this 3d non - lte line profile with two different kinds of 1d profiles , in each case assuming a rotational broadening of @xmath34 = 0.0 / 2.0 km / s , respectively ( see text for details ) . [ cols="^,^,^,^,^,^,^,^,^ " , ] the @xmath1li/@xmath2li isotopic ratio derived from fitting of the lii doublet with 3d nlte synthetic line profiles is shown to be about 1% to 2% lower than what is obtained with 1d lte profiles . based on this result , we conclude that only @xmath35 out of the @xmath36 stars of the @xcite sample would remain significant @xmath1li detections when subjected to a 3d non - lte analysis , suggesting that the presence of @xmath1li in the atmospheres of galactic halo stars is rather the exception than the rule , and hence does not necessarily constitute a _ cosmological _ @xmath1li problem . if we adopt the approach by @xcite , relying on additional spectral lines to fix the residual line broadening , the difference between 3d nlte and 1d lte results increases even more , as far as we can judge from our case study hd74000 . until the 3d nlte effects are fully understood for all involved lines , we consider this method as potentially dangerous .
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based on 3d hydrodynamical model atmospheres computed with the co@xmath0bold code and 3d non - lte ( nlte ) line formation calculations , we study the effect of the convection - induced line asymmetry on the derived @xmath1li abundance for a range in effective temperature , gravity , and metallicity covering the stars of the ( * ? ? ?
* asplund ( 2006 ) ) sample .
when the asymmetry effect is taken into account for this sample of stars , the resulting @xmath1li/@xmath2li ratios are reduced by about 1.5% on average with respect to the isotopic ratios determined by ( * ? ? ?
* asplund ( 2006 ) ) .
this purely theoretical correction diminishes the number of significant @xmath1li detections from 9 to 4 ( 2@xmath3 criterion ) , or from 5 to 2 ( 3@xmath3 criterion ) . in view of this result
the existence of a @xmath1li plateau appears questionable .
a careful reanalysis of individual objects by fitting the observed lithium @xmath4 doublet both with 3d nlte and 1d lte synthetic line profiles confirms that the inferred @xmath1li abundance is systematically lower when using 3d nlte instead of 1d lte line fitting .
nevertheless , halo stars with unquestionable @xmath1li detection do exist even if analyzed in 3d - nlte , the most prominent example being hd84937 .
| 2,465 | 409 |
carbon chains play a central role in the chemistry and spectroscopy of interstellar space . the detection of cyanopolyacetylenes in dense interstellar clouds @xcite led to the suggestion by @xcite that carbon chain species be considered as candidates for the diffuse interstellar bands ( dibs ) , which are found in the 4000 - 8500 spectral region of stars reddened by interstellar dust @xcite . since then , many other molecules with a carbon chain backbone have been identified at radio frequencies in dense clouds @xcite . meanwhile , advances in laboratory measurements have provided an understanding of the types and sizes of carbon chains which have strong electronic transitions in the dib range @xcite . it is thus somewhat surprising that as yet among the bare carbon species only diatomic c@xmath11 has been identified in interstellar clouds where dib are detected . the 4052 electronic band system of c@xmath0 was first detected in comets @xcite and then in circumstellar shells by infrared spectroscopy @xcite . most recently c@xmath0 was identified in a dense cloud using sub - mm measurements of its low frequency bending mode and n@xmath12(c@xmath0)@xmath13 @xmath7 was estimated @xcite . @xcite established an upper limit of 5@xmath1410@xmath15 @xmath7 for the column density of c@xmath0 in the direction of @xmath2 oph , some two orders of magnitude lower than that set by @xcite . @xcite made a tentative detection of c@xmath0 towards an eighth magnitude star in the same part of the sky , hd 147889 , at a column density of 4@xmath1410@xmath16 @xmath7 . unfortunately , the star turned out to be a ( previously unknown ) double - lined spectroscopic binary which limited their sensitivity . this letter presents the detection of c@xmath0 towards three stars and infers the column densities in the diffuse clouds . although c@xmath17 , c@xmath18 and c@xmath19 were not detected , upper limits for their column densities are estimated . diatomic species , such as ch , cn , c@xmath11 , and ch@xmath20 , have been detected towards two of the stars chosen and their column densities are considered standards with which to compare models for the physical and chemical processes in diffuse regions @xcite . the four bare carbon chains , c@xmath0 , c@xmath17 , c@xmath18 and c@xmath19 , were selected for the present search because gas phase electronic transitions for these species have been identified in the laboratory in the 4000 - 5500 region and their oscillator strengths are known ( see table [ results ] ) . observations of the reddened stars @xmath2 oph ( hd 149757 ) , @xmath2 per ( hd 24398 ) and 20 aql ( hd 179406 ) were made with the gecko echellette spectrograph on 2000 july 16 and 19 , fiber fed from the cassegrain focus of the canada - france - hawaii 3.6-m telescope ( cfht ) @xcite . all three stars have a visual extinction , a@xmath21 , near 1 and were chosen because they are bright with sharp interstellar k i lines indicating either single clouds or little doppler distortion ( in the case of @xmath2 oph , @xcite resolved the c@xmath11 at 8756 into two close velocity components separated by 1.1 km s@xmath22 ) . the detector was a rear illuminated eev1 ccd ( 13.5 @xmath23m@xmath24 pixels ) and the spectral regions were centered at 4047 in the 14th order , and at 5060 and 5400 in the 11th and 10th orders , respectively . the ultraviolet gecko prism was used to isolate the 14th order , the blue grism for the 11th order , while the stock cfht filter # 1515 was used for the 10th order observations . individual spectra had exposure times ranging from 5 to 20 minutes and were obtained with continuous fiber agitation to overcome modal noise . the resulting combined spectra for the individual stars at each wavelength had unusually high signal - to - noise ratios ( s / n@xmath5800 - 4000 ) for ccd observations . the th / ar comparison arc spectra , taken before and after each spectrograph wavelength reconfiguration , had a typical fwhm of 2.8 pixels , which corresponds to resolutions of @xmath25 = 121000 , 113000 and 101000 at 4047 , 5060 and 5400 , respectively . processing of the spectra was conventional . groups of biases were taken several times throughout each night and at each grating setting a series of flat - field spectra of a quartz - iodide lamp were recorded . the biases and appropriate flats were averaged and used to remove the zero - level offset and pixel - to - pixel sensitivity variations of the ccd . one - dimensional spectra were extracted using standard iraf routines . spectra of vega and other unreddened stars were used to search for contaminating telluric water vapor lines and stellar photospheric features . heliocentric corrections were applied to each spectrum . the observations are summarised in table [ observations ] which lists exposure times and s / n per pixel for each spectral region . the final column gives the radial velocities measured from the interstellar k i 4044.1 and 4047.2 lines . these velocities have been applied to each spectrum to put the interstellar features on a laboratory scale . the @xmath3 origin band of c@xmath0 is quite clearly detected towards all three stars . figure [ figure1 ] compares the observed spectra with a simulated c@xmath0 absorption spectrum based on the spectrograph resolution and assuming a boltzmann distribution in the ground state rotational levels with a temperature of 80 k. the continuum noise level in the observations is @xmath50.1 % . low order polynomials have been applied to the stellar data to give a level continuum ( base line ) and , in the case of @xmath2 per , a weak , broad stellar feature at 4053.2 has been removed . residual broad features in the final spectra are only a few tenths of a percent deep , much less than in the original , and they in no way mask the sharp c@xmath0 lines . in the simulation the rotational line intensities were calculated using the hnl - london factors , while the line positions were taken from the laboratory measurements ( this avoids the problem of a perturbation affecting the low @xmath4 ground state levels which is not accounted for by the fitted spectroscopic constants ) . the individual rotational p , q and r lines are clearly resolved in the spectra of all three stars . table [ lines ] lists the observed positions and equivalent widths of each rotational line assigned in the spectra of the three stars . the table also gives the corresponding positions measured for these transitions in the laboratory ( gausset et al . the positions of some 30 lines in the spectrum of @xmath2 oph agree with the laboratory data to within 0.1 cm@xmath22 providing an unambiguous identification of c@xmath0 in these diffuse clouds . figures [ figurec2p ] , [ figurec2 m ] , and [ figurec3 m ] show the results of equivalent searches for @xmath27 origin band of c@xmath17 at 5078.1 ( @xmath2 oph only ) , the @xmath28 origin band of c@xmath18 at 5415.9 and the @xmath29 origin band of c@xmath19 at 4040.4 , together with simulated spectra for these transitions at 80 k based upon the published spectroscopic constants ( @xcite , @xcite , @xcite ) . in the cases of c@xmath30 and c@xmath31 the linewidth was assumed to be determined by the spectrograph resolution . for c@xmath32 the excited state has been identified as a short - lived feshbach resonance and the measured natural linewidth of @xmath51 cm@xmath22 is employed in the simulation . weak telluric lines were been removed from the c@xmath17 and c@xmath18 observations using standard procedures . unlike the 4050 region for c@xmath0 , each of these spectral regions is contaminated by weak stellar features . nonetheless , for the c@xmath17 and c@xmath18 ions , there are no sharp features corresponding to the rotational lines in the simulations . on the other hand , for c@xmath32 there are features in the spectrum of @xmath2 per and ( much less convincing ) in the magnified plot for @xmath2 oph which appear to coincide in position and shape with the simulated band heads . it is unlikely that these are due to c@xmath32 because the stellar lines in @xmath2 per have exactly the same shape as the coincident features . for @xmath2 oph the whole c@xmath32 spectrum sits within a weak stellar feature ( the lines are broadened by rapid rotation @xmath5400 km s@xmath22 ) . the photospheric lines in @xmath2 oph show nonradial pulsation ` ripples ' which will be washed out to some extent by the long exposure time employed . the spectrum of 20 aql , which normally has the strongest interstellar lines of the three , is free of stellar features but has no features coincident with the c@xmath32 simulation . it is concluded that , while interstellar c@xmath32 might be absent for 20 aql and present for the other two stars , it is more likely that in the latter cases the features are instead stellar . table [ results ] gives the measured equivalent widths for the most intense c@xmath0 line ( q(8 ) ) for each star together with an 1@xmath33 error estimate . for c@xmath17 , c@xmath18 and c@xmath19 , 3@xmath33 detection limits are given . the 1@xmath33 level errors and detection limits are derived from : where the 1@xmath33 limiting equivalent width , @xmath35 , and the fwhm of the feature , @xmath36 , are both measured in , the spectrograph dispersion , @xmath37 , in pixel@xmath22 , and s / n is the signal to noise per pixel . from the simulations , @xmath36 = 0.045 , 1.0 , 0.13 and 0.045 for c@xmath0 , c@xmath19 , c@xmath17 and c@xmath18 , respectively . in the case of c@xmath0 , equivalent widths , @xmath35 , were determined for each rotational line ( varying between 0.3 - 2.7@xmath38 ) and , in combination with the transition oscillator strength , f@xmath39 , and hnl - london factors , the column densities , @xmath40(c@xmath0 ) , of each rotational level ( @xmath4 ) in the ground electronic state were calculated @xcite . in cases where several rotational lines originating from the same level were assigned ( e.g. p(8 ) , q(8 ) , r(8 ) ) the mean of the determined column densities was taken . figure [ figure5 ] shows a boltzmann plot of ln(@xmath41 ) vs. the rotational energy @xcite where the slope is inversely proportional to the rotational temperature . among the lowest rotational levels ( @xmath4214 ) the populations are reasonably approximated by a distribution at 50 - 70 k , whereas the higher rotational levels correspond to a temperature of 200 - 300 k. the simulation in figure 1 uses 80 k as this represents an average temperature for the entire rotational population and allows both the high and low @xmath4 lines to be identified . the high temperature component of the distribution is apparent in the astronomical spectra where the r band head and the higher q lines are more intense than in the simulation ( figure 1 ) . @xcite also found a bimodal population distribution for c@xmath11 in diffuse clouds , with similar characteristic temperatures for the low and high @xmath4 values . the lower temperature is interpreted as the kinetic energy of the cloud and for both c@xmath11 and c@xmath0 the values obtained are comparable to those used in models of diffuse clouds @xcite . the higher temperature component is attributed to repopulation of the levels in the ground electronic state by radiative pumping from excited states . in the case of c@xmath0 it is expected that both the @xmath43 state and the higher lying @xmath44 state will contribute to the radiative pumping . the sensitivity of these measurements is such that @xmath40(c@xmath0 ) in the range 0.2 - 2@xmath45 @xmath7 is determined for rotational levels up to @xmath4=30 . the @xmath40 values were summed to give the estimated lower limits in the range 1 - 2@xmath6 @xmath7 for the total column density , @xmath46(c@xmath0 ) , in table [ results ] . a previous search for c@xmath0 in the direction of @xmath2 oph did not identify the molecule @xcite . it is unclear why this was the case as , in the light of the present observations , the signal - to - noise quoted for these measurements was adequate and the upper limit given was some thirty times lower than the column density reported here . the present measurements for the column densities of c@xmath0 are of the same order of magnitude as the tentative estimate for a translucent cloud @xcite and can be compared to those of other polyatomic molecules observed in diffuse interstellar clouds . column densities ( also towards @xmath2 oph ) in the 10@xmath47 @xmath7 range have been inferred for hco@xmath20 , c@xmath11h and c@xmath0h@xmath11 from observations in the mm region by @xcite , while h@xmath48 has been identified in diffuse regions and n@xmath12(h@xmath48 ) estimated @xmath49 @xmath7 by @xcite . column densities of c@xmath11 towards @xmath2 oph and @xmath2 per have been determined in the 2 - 3@xmath50 @xmath7 range by @xcite . a current model of the diffuse clouds by @xcite predicts n@xmath51(c@xmath11)/n@xmath12(c@xmath0 ) @xmath5 20 ( on a 10@xmath52 year time scale ) , implying n@xmath12(c@xmath0)@xmath53 @xmath7 , in agreement with the values deduced from the table [ results ] . the main production route to c@xmath0 is presumed to be the dissociative recombination process : c@xmath0h@xmath20 + @xmath54 @xmath55 c@xmath0 + h , where c@xmath0h@xmath20 is produced from smaller species by c@xmath20 ion insertion . under conditions where ultra - violet radiation penetrates , photodissociation of c@xmath0 takes place at a threshold of 1653 . as the strong @xmath44@xmath56@xmath57 electronic transition of c@xmath0 is predicted to occur around 1700 @xcite , the dissociation process , c@xmath0 @xmath55 c@xmath11 + c , may be an important destruction pathway in diffuse clouds . although only upper limits for the column densities of c@xmath19 , c@xmath17 and c@xmath18 could be presently established , these species are of interest as small ionic carbon fragments play a crucial role in the ion molecule schemes for diffuse cloud chemistry @xcite . the c@xmath17 ion is the only bare carbon cation for which the gas phase electronic spectrum is known @xcite . in diffuse clouds it is supposed to be the product of the fundamental step : c@xmath20 + ch @xmath55 c@xmath17 + h. its main destruction mechanism is hydrogenation : c@xmath17 + h@xmath11 @xmath55 c@xmath11h@xmath20 + h , which dominates over recombination with electrons in diffuse regions . the diffuse cloud model @xcite predicts a c@xmath17 abundance a factor of 10@xmath58 lower than c@xmath11 , implying a column density @xmath59 @xmath7 , in accord with the upper limit in table [ results ] . published models for diffuse regions do not include the smallest pure carbon anions , c@xmath18 and c@xmath19 in their reaction libraries . unlike c@xmath17 , c@xmath18 does not react with h@xmath11 so its main destruction mechanism is expected to be photodetachment . the similar rotational line widths and oscillator strengths of the c@xmath18 and c@xmath17 transitions lead to similar upper limits for their total column densities . the width of the unresolved bands for c@xmath19 and the presence of weak stellar features in this spectral region means that a higher column density of this ion could have escaped detection . the detection of c@xmath0 provides a powerful incentive for the laboratory study of the electronic transitions of longer carbon chains in the gas phase with the aim of comparison with dib data . the question as to what types and sizes of carbon chains will have strong transitions in the 4000 - 9000 range has already been answered : for example , @xmath44@xmath56@xmath57 transitions of the odd - number bare chains , c@xmath60 , @xmath61=8 - 30 @xcite . the existence of linear carbon chains up to c@xmath62 has been confirmed by the observation of their electronic spectra in neon matrices @xcite . as the oscillator strength scales almost linearly with the length of the molecule , one can expect f@xmath6310 - 20 for these carbon chains . with such an oscillator strength , a species with a column density @xmath64 @xmath7 would be enough to give rise to a strong dib , with an equivalent width of 1 . in view of the column density @xmath510@xmath16 @xmath7 for c@xmath0 determined in this work for three diffuse clouds , this appears to be a reasonable expectation . the support of the swiss national science foundation ( project no . 20 - 055285.98 ) , the canadian natural sciences and engineering research council and the national research council of canada is gratefully acknowledged . the authors thank the staff of the cfht for their care in setting up the fiber feed and agitator , thereby making such high signal - to - noise spectra possible . clccccccccl @xmath2 per & b1 ib & 2.85 & 0.28 & 4800 & 1200 & & & 2700 & 1900 & + 13.91 @xmath650.24 + @xmath2 oph & o9.5 v & 2.56 & 0.30 & 5400 & 2400 & 5400 & 4000 & 3000 & 2200 & @xmath5614.53 @xmath650.18 + 20 aql & b3 v & 5.36 & 0.27 & 10800 & 800 & & & 8400 & 900 & @xmath5612.53 @xmath650.08 + cc|lc|lc|lc 4049.784 & r(22 ) & & & 4049.770@xmath66&1.016 & 4049.795@xmath66&1.708 + 49.770 & r(24 ) & & & 49.770@xmath66&1.016 & 49.795@xmath66&1.708 + 49.810 & r(20 ) & 4049.782@xmath66 & 1.658 & 49.807@xmath66&1.162 & & + 49.784 & r(26 ) & 49.782@xmath66 & 1.658 & 49.807@xmath66&1.162 & & + 49.861 & r(18 ) & 49.865 & 0.309 & 49.877 & 0.511 & 49.865 & 0.821 + 49.963 & r(16 ) & 49.959 & 0.726 & 49.961 & 0.773 & & + 50.081 & r(14 ) & 50.079 & 0.792 & 50.091 & 0.759 & & + 50.206 & r(12 ) & 50.198 & 0.920 & 50.198 & 0.680 & 50.211 & 1.165 + 50.337 & r(10 ) & 50.329 & 1.034 & 50.342 & 0.679 & 50.339 & 1.450 + 50.495 & r(8 ) & 50.483 & 1.525 & 50.497 & 0.383 & 50.489 & 2.204 + 50.670 & r(6 ) & 50.669 & 1.562 & 50.662 & 0.943 & 50.667 & 2.330 + 50.865 & r(4 ) & 50.863 & 1.018 & 50.864 & 1.068 & 50.853 & 2.225 + 51.069 & r(2 ) & 51.073 & 0.896 & & & 51.074 & 1.657 + 51.309 & r(0 ) & 51.267@xmath67 & 0.371 & & & 51.386@xmath67&0.730 + 51.461 & q(2 ) & 51.457 & 1.045 & 51.457 & 0.923 & 51.455 & 1.180 + 51.521 & q(4 ) & 51.515 & 2.187 & 51.518 & 0.752 & 51.506 & 1.876 + 51.590 & q(6 ) & 51.586 & 2.719 & 51.588 & 2.183 & 51.583 & 1.965 + 51.682 & q(8 ) & 51.679 & 2.336 & 51.680 & 2.016 & 51.669 & 2.229 + 51.793 & q(10 ) & 51.788 & 2.138 & 51.795 & 2.294 & 51.787 & 2.508 + 51.929 & q(12 ) & 51.930 & 1.060 & 51.922 & 2.020 & 51.929 & 2.186 + 52.062 & p(4 ) & 52.074@xmath66&1.266 & 52.085@xmath66&1.043 & 52.065@xmath66 & 3.130 + 52.089 & q(14 ) & 52.074@xmath66&1.266 & 52.085@xmath66&1.043 & 52.065@xmath66 & 3.130 + 52.271 & q(16 ) & 52.262 & 1.217 & & & 52.271 & 1.821 + 52.424 & p(6 ) & & & & & 52.456 & 2.499 + 52.473 & q(18 ) & 52.466 & 1.233 & & & & + 52.698 & q(20 ) & 52.701 & 0.923 & & & & + 52.792 & p(8 ) & 52.784 & 0.605 & & & 52.772 & 1.592 + 52.900 & q(22 ) & 52.929 & 0.985 & & & 52.939 & 1.122 + 53.180 & p(10 ) & 53.197@xmath66&1.883 & & & + 53.207 & q(24 ) & 53.197@xmath66&1.883 & & & + 53.590 & p(12 ) & 53.593 & 0.678 & & & 53.588 & 1.237 + 53.795 & q(28 ) & 53.786 & 0.469 & & & 53.781 & 1.098 + 54.112 & q(30 ) & 54.113 & 0.523 & & & & + 54.459 & p(16 ) & 54.445 & 0.870 & & & & + 54.908 & p(18 ) & 54.904 & 1.122 & & & & + c@xmath0&@xmath3&4051.5&0.016&@xmath2 oph&2.3@xmath650.1&20&1.6 + & & & & 20 aql&2.2@xmath650.3&19&2.0 + & & & & @xmath2 per & 2.0@xmath650.2&17&1.0 + c@xmath19&@xmath29&4040.4&0.04&@xmath2 oph&6.0&&@xmath680.3 + & & & & 20 aql&20&&@xmath681.2 + & & & & @xmath2 per&12&&@xmath680.7 + c@xmath17&@xmath27&5066.9&0.025&@xmath2 oph&0.35&@xmath681.1&@xmath680.04 + c@xmath18&@xmath28&5408.6&0.044&@xmath2 oph & 0.30&@xmath680.5&@xmath680.02 + & & & & 20 aql&0.55&@xmath680.9&@xmath680.03 + & & & & @xmath2 per&0.35&@xmath680.6&@xmath680.02 +
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the smallest polyatomic carbon chain , c@xmath0 , has been identified in interstellar clouds ( a@xmath11 mag ) towards @xmath2 ophiuchi , 20 aquilae , and @xmath2 persei by detection of the origin band in its @xmath3 electronic transition , near 4052 .
individual rotational lines were resolved up to @xmath4=30 enabling the rotational level column densities and temperature distributions to be determined .
the inferred limits for the total column densities ( @xmath51 to 2@xmath6 @xmath7 ) offer a strong incentive to laboratory and astrophysical searches for the longer carbon chains .
concurrent searches for c@xmath8 , c@xmath9 and c@xmath10 were negative but provide sensitive estimates for their maximum column densities .
| 6,932 | 220 |
the existence of feeble magnetic fields of several microgauss in our galaxies @xcite , as well as of gigagauss in intense laser - plasma interaction experiments @xcite and of billions of gauss in compact astrophysical objects @xcite ( e.g. super dense white dwarfs , neutron stars / magnetars , degenerate stars , supernovae ) is well known . the generation mechanisms for seed magnetic fields in cosmic / astrophysical environments are still debated , while the spontaneous generation of magnetic fields in laser - produced plasmas is attributed to the biermann battery @xcite ( also referred to as the baroclinic vector containing non - parallel electron density and electron temperature gradients ) and to the return electron current from the solid target . computer simulations of laser - fusion plasmas have shown evidence of localized anisotropic electron heating by resonant absorption , which in turn can drive a weibel - like instability resulting in megagauss magnetic fields @xcite . there have also been observations of the weibel instability in high intensity laser - solid interaction experiments @xcite . furthermore , a purely growing weibel instability @xcite , arising from the electron temperature anisotropy ( a bi - maxwellian electron distribution function ) is also capable of generating magnetic fields and associated shocks @xcite . however , plasmas in the next generation intense laser - solid density plasma experiments @xcite would be very dense . here the equilibrium electron distribution function may assume the form of a deformed fermi - dirac distribution due to the electron heating by intense laser beams . it then turn out that in such dense fermi plasmas , quantum mechanical effects ( e.g. the electron tunneling and wave - packet spreading ) would play a significant role @xcite . the importance of quantum mechanical effects at nanometer scales has been recognized in the context of quantum diodes @xcite and ultra - small semiconductor devices @xcite . also , recently there have been several developments on fermionic quantum plasmas , involving the addition of a dynamical spin force @xcite , turbulence or coherent structures in degenerate fermi systems @xcite , as well as the coupling between nonlinear langmuir waves and electron holes in quantum plasmas @xcite . the quantum weibel or filamentational instability for non - degenerate systems has been treated in @xcite . in this work , we present an investigation of linear and nonlinear aspects of a novel instability that is driven by equilibrium fermi - dirac electron temperature anisotropic distribution function in a nonrelativistic dense fermi plasma . specifically , we show that the free energy stored in electron temperature anisotropy is coupled to purely growing electromagnetic modes . first , we take the wigner - maxwell system @xcite with an anisotropic fermi - dirac distribution for the analysis of the linearly growing electromagnetic perturbations as a function of the physical parameters . second , we use a fully kinetic simulation to assess the saturation level of the magnetic fields as a function of the growth rate . the treatment is restricted to transverse waves , since the latter are associated with the largest weibel instability growth rates . the nonlinear saturation of the weibel instability for classical , non - degenerate plasmas has been considered elsewhere @xcite . it is well known @xcite that a dense fermi plasma with isotropic equilibrium distributions does not admit any purely growing linear modes . this can be verified , for instance , from the expression for the imaginary part of the transverse dielectric function , as derived by lindhard @xcite , for a fully degenerate non - relativistic fermi plasma . it can be proven ( see eq . ( 30 ) of @xcite ) that the only exception would be for extremely small wavelengths , so that @xmath0 , where @xmath1 is the wave number and @xmath2 the characteristic fermi wave number of the system . however , in this situation the wave would be super - luminal . on the other hand , in a classical vlasov - maxwell plasma containing anisotropic electron distribution function , we have a purely growing weibel instability @xcite , via which dc magnetic fields are created . the electron temperature anisotropy arises due to the heating of the plasma by laser beams @xcite , where there is a signature of the weibel instability as well . in the next generation intense laser - solid density plasma experiments , it is likely that the electrons would be degenerate and that electron temperature anisotropy may develop due to an anisotropic electron heating by intense laser beams via resonant absorption , similar to the classical laser plasma case @xcite . in a dense laser created plasma , quantum effects must play an important role in the context of the weibel instability . in order to keep the closest analogy with the distribution function in phase space for the classical plasma , we shall use the wigner - maxwell formalism for a dense quantum plasma @xcite . here the distribution of the electrons is described by the wigner pseudo - distribution function @xcite , which is related to the fermi - dirac distribution widely used in the random phase approximation @xcite . proceeding with the time evolution equation for the wigner function ( or quantum vlasov equation @xcite ) , we shall derive a modified dispersion relation accounting for a wave - particle duality and an anisotropic wigner distribution function that is appropriate for the fermi plasma . the results are consistent with those of the random phase approximation , in that they reproduce the well - known transverse density linear response function for a fully degenerate fermi plasma @xcite . consider linear transverse waves in a dense quantum plasma composed of the electrons and immobile ions , with @xmath3 , where @xmath4 is the wave vector and @xmath5 is the wave electric field . following the standard procedure , one then obtains the general dispersion relation @xcite for the transverse waves of the wigner - maxwell system @xmath6 where @xmath7 is the frequency , @xmath8 is the speed of light in vacuum , @xmath9 is the planck constant divided by @xmath10 , @xmath11 the rest electron mass , @xmath12 the unperturbed plasma number density , @xmath13 the electron plasma frequency , @xmath14 is the velocity vector , and @xmath15 is the equilibrium wigner function associated to fermi systems . for spin @xmath16 particles , the equilibrium pseudo distribution function is in the form of a fermi - dirac function . here we allow for velocity anisotropy and express @xmath17 + 1 } \,,\ ] ] where @xmath18 is the chemical potential , @xmath19 the boltzmann constant , and the normalization constant is @xmath20 here @xmath21 is a polylogarithm function @xcite . also , @xmath22 $ ] , where @xmath23 and @xmath24 are related to velocity dispersion in the direction perpendicular and parallel to @xmath25 axis , respectively . in the special case when @xmath26 , the usual fermi - dirac equilibrium is recovered . the chemical potential is obtained by solving the normalization condition ( [ e3 ] ) , yielding , in particular , @xmath27 in the limit of zero temperature , where @xmath28 is the fermi energy . also , the fermi - dirac distribution @xmath29 , where @xmath4 is the appropriated wave vector in momentum space , is related to the equilibrium wigner function ( [ e2 ] ) by @xmath30 , with the factor @xmath31 coming from spin @xcite . however , these previous works refer to the cases where there is no temperature anisotropy . notice that it has been suggested @xcite that in laser plasmas the weibel instability is responsible for further increase of @xmath24 with time . inserting ( [ e2 ] ) into ( [ e1 ] ) and integrating over the perpendicular velocity components , we obtain @xmath32 where @xmath33\bigr\ } \nonumber \\ & - & { \rm li}_{2}\bigl\{-\exp\bigl[-\bigl(\nu - \frac{h}{2}\bigl)^2 + \beta\mu\bigl]\bigr\}\biggr ) \ , . \nonumber \end{aligned}\ ] ] in ( [ e5 ] ) , @xmath34 is the dilogarithm function @xcite , @xmath35 is a characteristic parameter representing the quantum diffraction effect , @xmath36 , and @xmath37 , with @xmath38 . in the simultaneous limit of a small quantum diffraction effect ( @xmath39 ) and a dilute system ( @xmath40 ) , it can be shown that @xmath41 , where @xmath42 is the standard plasma dispersion function @xcite . it is important to notice that either ( [ e1 ] ) or ( [ e4 ] ) reproduces the transverse dielectric function calculated from the random phase approximation for a fully degenerate quantum plasma @xcite , in the case of an isotropic system . the simple way to verify this equivalence is to put @xmath43 in ( [ e1 ] ) and then take the limit of zero temperature , so that @xmath44 for @xmath45 , and @xmath46 otherwise , where @xmath47 is the fermi velocity . however , to the best of our knowledge , there is no corresponding calculation for an anisotropic fermi equilibrium , as necessary in laser - solid interaction experiments with an anisotropic electron heating due to resonant absorption . also notice that in this letter we are mainly interested in the real part of the transverse response function , since we are looking for purely growing instabilities ( @xmath48 ) , so that the contribution from the poles at ( [ e4 ] ) is not relevant . ( @xmath49 ) and @xmath50 , relevant for the next generation inertially compressed material in intense laser - solid density plasma interaction experiments . the temperature anisotropies are @xmath51 ( dashed line ) , @xmath52 ( solid line ) and @xmath53 ( dotted line ) , yielding , respectively , @xmath54 , @xmath55 and @xmath56 . , width=321 ] ( @xmath49 ) . here the temperature anisotropy is @xmath52 . we used @xmath57 ( dashed line ) , @xmath50 ( solid line ) and @xmath58 ( dotted line ) , yielding @xmath59 , @xmath60 and @xmath61 , respectively.,width=321 ] we next solve our new dispersion relation ( [ e4 ] ) for a set of parameters that are representative of the next generation laser - solid density plasma interaction experiments . the normalization condition ( [ e3 ] ) can also be written as @xmath62=(4/3\sqrt{\pi})(\beta{\cal e}_f)^{3/2}$ ] , which is formally the same relation holding for isotropic fermi - dirac equilibria @xcite . for a given value on the product @xmath63 and the density , this relation yields the value @xmath64 , from which the temperatures @xmath65 and @xmath66 can be calculated , if we know @xmath67 . consider only purely growing modes . from the definition ( [ e5 ] ) , one can show that @xmath68 when @xmath69 for a finite wavenumber @xmath1 . from ( [ e4 ] ) we then obtain the maximum wavenumber for instability as @xmath70 . when @xmath71 , the range of unstable wavenumbers shrinks to zero . in figs . 1 and 2 , we have used the electron number density @xmath72 , which can be obtained in laser - driven compression schemes . the growth rate for different values on @xmath67 is displayed in fig . we see that the maximum unstable wavenumber is @xmath70 , as predicted , and that the maximum growth rate occurs at @xmath73 . figure 1 also reveals that the maximum growth rate of the instability is almost linearly proportional to @xmath74 . in fig . 2 , we have varied the product @xmath63 , which is a measure of the degeneracy of the quantum plasma . we see that for @xmath63 larger than @xmath75 , the instability reaches a limiting value , which is independent of the temperature , while thermal effects start to play an important role for @xmath63 of the order unity . ( top panel ) and @xmath76 ( bottom panel ) as a function of space and time , for @xmath50 and @xmath77 . the magnetic field has been normalized by @xmath78 . we see a nonlinear saturation of the magnetic field components at an amplitude of @xmath79.,width=321 ] , over the simulation box ( top panel ) , and the logarithm of the magnetic field maximum ( bottom panel ) as a function of time , for @xmath52 and @xmath50 . the magnetic field has been normalized by @xmath78 . from the logarithmic slope of the magnetic field in the linear regime we find @xmath80 . , width=321 ] from several numerical solutions of the linear dispersion relation , we have been able to deduce an approximate scaling law for the instability as @xmath81 , where the constant is approximately @xmath82 . using that @xmath83 , we have @xmath84 for the maximum growth rate of the weibel instability in a degenerate fermi plasma . this scaling law , where the growth rate depends on the fermi energy and the temperature anisotropy , should be compared to that of a classical plasma @xcite , where the growth rate depends on the thermal energy and the temperature anisotropy . for a maxwellian plasma , it has been found @xcite that the weibel instability saturates nonlinearly once the magnetic bounce frequency @xmath85 has increased to a value comparable to the linear growth rate . in order to assess the nonlinear behavior of the weibel instability for a degenerate plasma , we have carried out a kinetic simulation of the wigner - maxwell system . we have assumed that the quantum diffraction effect is small , so that the simulation of the wigner equation can be approximated by simulations of the vlasov equation by means of an electromagnetic vlasov code @xcite . as an initial condition for the simulation , we used the distribution function ( 2 ) . in order to give a seed for any instability , the plasma density was perturbed with low - frequency fluctuations ( random numbers ) . the results are displayed in figs . 3 and 4 , for the parameters @xmath50 and @xmath52 , corresponding to the solid lines in figs . 1 and 2 . figure 3 shows the magnetic field components as a function of space and time . we see that the magnetic field initially grows , and saturates to steady state magnetic field fluctuations with an amplitude of @xmath86 . the maximum amplitude of the magnetic field over the simulation box as a function of time is shown in fig . 4 , where we see that the magnetic field saturates at @xmath87 , while the linear growth rate of the most unstable mode is @xmath88 . similar to the classical maxwellian plasma case @xcite , we can thus estimate the magnetic field ( in tesla ) as @xmath89 for a degenerate fermi plasma . for our parameters parameters relevant for intense laser - solid interaction experiments , we will thus have magnetic fields of the order @xmath90 ( one gigagauss ) . in conclusion , we have demonstrated the existence of the weibel instability for a wigner - maxwell dense quantum plasma , taking into account an anisotropic fermi - dirac equilibrium distribution function and the quantum diffraction effect . numerically solving the dispersion relation for transverse waves , we found the dependence of the growth rate on the fermi energy and the temperature anisotropy . the nonlinear saturation level of the magnetic field was found by means of kinetic simulations , which show a linear dependence between the growth rate and the saturated magnetic field . the present results may account for intense magnetic fields in dense quantum plasmas , such as those in the next generation of intense laser - solid density plasma interaction experiments . 99 l. w. wildrow , rev . phys . * 74 * , 775 ( 2002 ) ; 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we present an investigation for the generation of intense magnetic fields in dense plasmas with an anisotropic electron fermi - dirac distribution . for this purpose , we use a new linear dispersion relation for transverse waves in the wigner - maxwell dense quantum plasma system .
numerical analysis of the dispersion relation reveals the scaling of the growth rate as a function of the fermi energy and the temperature anisotropy .
the nonlinear saturation level of the magnetic fields is found through fully kinetic simulations , which indicates that the final amplitudes of the magnetic fields are proportional to the linear growth rate of the instability .
the present results are important for understanding the origin of intense magnetic fields in dense fermionic plasmas , such as those in the next generation intense laser - solid density plasma experiments .
| 5,909 | 186 |
in industrial inspection , there is an ever - growing demand for highly accurate , non - destructive measurements of three - dimensional object geometries . a variety of optical sensors have been developed to meet these demands @xcite . these sensors satisfy the requirements at least partially . numerous applications , however , still wait for a capable metrology . the limitations of those sensors emerge from physics and technology the physical limits are determined by the wave equation and by coherent noise , while the technological limits are mainly due to the space - time - bandwidth product of electronic cameras . closer consideration reveals that the technological limits are basically of information - theoretical nature . the majority of the available optical 3d sensors need large amounts of raw data in order to obtain the shape . a lot of redundant information is acquired and the expensive channel capacity of the sensors is used inefficiently @xcite . a major source of redundancy is the shape of the object itself : if the object surface @xmath0 is almost planar , there is similar height information at each pixel . in terms of information theory the surface points of such objects are `` correlated '' ; their power spectral density @xmath1 decreases rapidly . in order to remove redundancy , one can apply spatial differentiation to whiten the power spectral density ( see fig . [ spectra ] ) . fortunately , there are optical systems that perform such spatial differentiation . indeed , sensors that acquire just the local slope instead of absolute height values are much more efficient in terms of exploiting the available channel capacity . further , reconstructing the object height from slope data reduces the high - frequency noise since integration acts as a low - pass filter . there are several sensor principles that acquire the local slope : for _ rough surfaces _ , it is mainly the principle of shape from shading @xcite . for _ specular surfaces _ , there are differentiating sensor principles like the differential interference contrast microscopy or deflectometry @xcite . deflectometric scanning methods allow an extremely precise characterization of optical surfaces by measuring slope variations as small as @xmath2arcsec @xcite . full - field deflectometric sensors acquire the two - dimensional local gradient of a ( specular ) surface . using `` phase - measuring deflectometry '' ( pmd ) @xcite , for example , one can measure the local gradient of an object at one million sample points within a few seconds . the repeatability of the sensor described in @xcite is below @xmath3arcsec with an absolute error less than @xmath4arcsec , on a measurement field of @xmath5 mm @xmath6 @xmath5 mm and a sampling distance of @xmath7 mm . in several cases it is sufficient to know the local gradient or the local curvature ; however , most applications demand the height information as well . as an example we consider eyeglass lenses . in order to calculate the local surface power of an eyeglass lens by numerical differentiation , we only need the surface slope and the lateral sampling distance . but for quality assurance in an industrial setup , it is necessary to adjust the production machines according to the measured shape deviation . this requires height information of the surface . another application is the measurement of precision optics . for the optimization of these systems sensors are used to measure the local gradient of wavefronts @xcite . to obtain the shape of these wavefronts , a numerical shape reconstruction method is needed . in the previous section we stated that measuring the gradient instead of the object height is more efficient from an information - theoretical point of view , since redundant information is largely reduced . using numerical integration techniques , the shape of the object can be reconstructed locally with high accuracy . for example , a full - field deflectometric sensor allows the detection of local height variations as small as _ a few nanometers_. however , if we want to reconstruct the global shape of the object , low - frequency information is essential . acquiring _ solely the slope _ of the object reduces the low - frequency information substantially ( see fig . [ spectra ] ) . in other words , we have a lot of _ local _ information while lacking _ global _ information , because we reduced the latter by optical differentiation . as a consequence , small measuring errors in the low - frequency range will have a strong effect on the overall reconstructed surface shape . this makes the reconstruction of the global shape a difficult task . furthermore , one - dimensional integration techniques can not be easily extended to the two - dimensional case . in this case , one has to choose a path of integration . unfortunately , noisy data leads to different integration results depending on the path@xcite . therefore , requiring the integration to be path independent becomes an important condition ( `` integrability condition '' ) for developing an optimal reconstruction algorithm ( see sections [ problem ] and [ optimal ] ) . we consider an _ object surface _ to be a twice continuously differentiable function @xmath8 on some compact , simply connected region @xmath9 . the integrability condition implies that the gradient field @xmath10 is _ curl free _ , i.e. every path integral between two points yields the same value . this is equivalent to the requirement that there exists a potential @xmath11 to the gradient field @xmath12 which is unique up to a constant . most object surfaces measurable by deflectometric sensors fulfill these requirements , or at least they can be decomposed into simple surface patches showing these properties . measuring the gradient @xmath12 at each sample point @xmath13 yields a discrete vector field @xmath14 . these measured gradient values usually are contaminated by noise the vector field is not necessarily curl free . hence , there might not exist a potential @xmath11 such that @xmath15 for all @xmath16 . in that case , we seek a least - squares approximation , i.e. a surface representation @xmath11 such that the following error functional is minimized @xcite : @xmath17 ^ 2 + \left[z_y({\bf x}_i)-q({\bf x}_i)\right]^2.\ ] ] in the case of one - dimensional data , integration is a rather straightforward procedure which has been investigated quite extensively @xcite . in case of two - dimensional data , there exist mainly two different approaches to solve the stated problem @xcite . on the one hand , there are _ local methods _ : they integrate along predetermined paths . the advantage of these methods is that they are simple and fast , and that they reconstruct small local height variations quite well . however , they propagate both the measurement error and the discretization error along the path . therefore , they introduce a global shape deviation . this effect is even worse if the given gradient field is not guaranteed to be curl free : in this case , the error also depends on the chosen path . on the other hand , there are _ global methods _ : they try to minimize @xmath18 by solving its corresponding euler - lagrange equation@xcite @xmath19 where @xmath20 and @xmath21 denote the numerical @xmath22- and @xmath23-derivative of the measured data @xmath24 and @xmath25 , respectively . the advantage of global methods is that there is no propagation of the error ; instead , the error gets evenly distributed over all sample points . unfortunately , the implementation has certain difficulties . methods based on finite differences are usually inefficient in their convergence when applied to strongly curved surfaces @xcite . therefore , they are mainly used for nearly planar objects . another approach to solve eq . ( [ poisson ] ) is based on fourier transformation @xcite . integration in the fourier domain has the advantage of being optimal with respect to information - theoretical considerations @xcite . however , fourier methods assume a periodic extension on the boundary which can not be easily enforced with irregularly shaped boundaries in a two - dimensional domain . in general , it is crucial to note that the reconstruction method depends on the slope - measuring sensor and the properties of the acquired data . for example , slope data acquired by shape from shading is rather noisy , exhibits curl , and is usually located on a full grid of millions of points . here , a fast subspace approximation method like the one proposed by frankot and chellappa @xcite is appropriate . on the other hand , wavefront reconstruction deals with much smaller data sets , and the surface is known to be rather smooth and flat . in this case , a direct finite - difference solver can be applied @xcite . deflectometric sensors deliver a third type of data : it consists of very large data sets with rather small noise and curl , but the data may not be complete , depending on the local reflectance of the measured surface . furthermore , the measuring field may have an unknown , irregularly shaped boundary . these properties render most of the aforementioned methods unusable for deflectometric data . in the following sections , we will describe a surface reconstruction method which is especially able to deal with slope data acquired by sensors such as phase - measuring deflectometry . the desired surface reconstruction method should have the properties of both _ local and global _ integration methods : it needs to _ preserve local details _ without propagating the error along a certain path . it also needs to minimize the error functional of eq . ( [ approx_functional ] ) , hence yielding a _ globally optimal solution _ in a least - squares sense . further , the method should be able to deal with _ irregularly shaped boundaries _ , _ missing data points _ , and it has to be able to reconstruct surfaces of a large variety of objects with steep slopes and _ high curvature values_. it should also be able to _ handle large data sets _ which may consist of some million sample points . we now show how to meet these challenges using an analytic interpolation approach . a low noise level allows _ interpolation _ of the slope values instead of approximation . interpolation is a special case which has the great advantage that we can ensure that small height variations are preserved . in this paper we will only focus on the interpolation approach as analytic reconstruction method . for other measurement principles like shape from shading , an approximation approach might be more appropriate . the basic idea of the integration method is as follows : we seek an analytic interpolation function such that its gradient interpolates the measured gradient data . once this interpolation is determined , it uniquely defines the surface reconstruction up to an integration constant . to obtain the analytic interpolant , we choose a generalized hermite interpolation approach employing _ radial basis functions _ ( rbfs ) @xcite . this method has the advantage that it can be applied to _ scattered data_. it allows us to integrate data sets with holes , irregular sampling grids , or irregularly shaped boundaries . furthermore , this method allows for an optimal surface recovery in the sense of eq . ( [ approx_functional ] ) ( see section [ optimal ] below ) . in more detail : assuming that the object surface fulfills the requirements described in section [ problem ] , the data is given as pairs @xmath26 , where @xmath27 and @xmath28 are the measured slopes of the object at @xmath29 in @xmath22- and @xmath23-direction , respectively , for @xmath30 . we define the interpolant to be @xmath31 where @xmath32 and @xmath33 , for @xmath34 , are coefficients and @xmath35 is a radial basis function . hereby , @xmath36 and @xmath37 denote the analytic derivative of @xmath38 with respect to @xmath22 and @xmath23 , respectively . this interpolant is specifically tailored for gradient data @xcite . to obtain the coefficients in eq . ( [ interpolant ] ) we match the analytic derivatives of the interpolant with the measured derivatives : @xmath39 this leads to solving the following system of linear equations @xcite : @xmath40 \\[-0.7em ] $ \phi_{x y}(\mathbf{x}_i-\mathbf{x}_j)$ & $ \phi_{y y}(\mathbf{x}_i-\mathbf{x}_j)$\\ \end{tabular } } \right)}_{a \ , \in \ , m^{2n\times2n } } \underbrace{\left ( \mbox{\begin{tabular}{c } $ \alpha_i$\\ \\[-0.7em ] \\[-0.7em ] $ \beta_i$ \end{tabular } } \right)}_{\alpha \ , \in \ , m^{2n\times1 } } = \underbrace{\left ( \mbox{\begin{tabular}{c } $ p(\mathbf{x}_j)$\\ \\[-0.7em ] \\[-0.7em ] $ q(\mathbf{x}_j)$ \end{tabular } } \right)}_{d \ , \in \ , m^{2n\times1}}.\ ] ] + using the resulting coefficients @xmath41 we then can apply the interpolant in eq . ( [ interpolant ] ) to reconstruct the object surface . for higher noise levels an _ approximation _ approach is recommended . in this case , we simply reduce the number of basis functions so that they do not match the number of data points any more . the system @xmath42 in eq . ( [ linear_system ] ) then becomes overdetermined and can be solved in a least - squares sense . the interpolation approach employing radial basis functions has the advantage that it yields a _ unique _ solution to the surface recovery problem : within this setup , the interpolation matrix in eq . ( [ linear_system ] ) is always symmetric and positive definite . further , the solution satisfies a _ minimization principle _ in the sense that the resulting analytic surface function has minimal energy @xcite . we choose @xmath38 to be a wendland s function @xcite , @xmath43 , with @xmath44 this has two reasons : first , wendland s functions allow to choose their continuity according to the smoothness of the given data . the above wendland s function leads to an interpolant which is three - times continuously differentiable , hence guaranteeing the integrability condition . second , the compact support of the function allows to adjust the support size in such a way that the solution of eq . ( [ linear_system ] ) is stable in the presence of noise . it turns out that the support size has to be chosen rather large in order to guarantee a good surface reconstruction @xcite . the amount of data commonly acquired with a pmd sensor in a single measurement is rather large : it consists of about one million sample points . this amount of data , which results from a measurement with high lateral resolution , would require the inversion of a matrix with @xmath45 entries ( eq . ( [ linear_system ] ) ) . since we choose a large support size for our basis functions to obtain good numerical stability the corresponding matrix is not sparse . it is obvious that this large amount of data can not be handled directly by inexpensive computing equipment in reasonable time . to cope with such large data sets we developed a method which first splits the data field into a set of overlapping rectangular patches . we interpolate the data on each patch separately . if the given data were height information only , this approach would yield the complete surface reconstruction . for slope data , we interpolate the data and obtain a surface reconstruction _ up to a constant of integration _ ( see fig . [ mirror_fit](a ) ) on each patch . in order to determine the missing information we apply the following fitting scheme : let us denote two adjacent patches as @xmath46 and @xmath47 and the resulting interpolants as @xmath48 and @xmath49 , respectively . since the constant of integration is still unknown the two interpolants might be on different height levels . generally , we seek a correcting function @xmath50 by minimizing @xmath51 this fitting scheme is then propagated from the center toward the borders of the data field to obtain the reconstructed surface on the entire field ( see fig . [ mirror_fit](b ) ) . in the simplest case , the functions @xmath52 are chosen to be constant on each patch , representing the missing constant of integration . if the systematic error of the measured data is small , the constant fit method is appropriate since it basically yields no error propagation . for very noisy data sets it might be better to use a planar fit , i.e. @xmath53 , to avoid discontinuities at the patch boundaries . this modification , however , introduces a propagation of the error along the patches . the correction angle required on each patch to minimize eq . ( [ lsqfit ] ) depends on the noise of the data . numerical experiments have shown that in most cases the correction angle is at least ten times smaller than the noise level of the measured data . using this information we can estimate the global height error which , by error propagation , may sum up toward the borders of the measuring field @xcite : @xmath54 where @xmath55 is the standard deviation of the correction angles , @xmath56 is the patch size , and @xmath57 is the number of patches . suppose we want to integrate over a field of @xmath5 mm ( which corresponds to a typical eyeglass diameter ) , assuming a realistic noise level of @xmath58arcsec and a patch size ( not including its overlaps ) of @xmath59 mm . with our setup , this results in @xmath60 patches , with a maximal tipping of @xmath61arcsec per patch . according to eq . ( [ globaltilt ] ) , the resulting global error caused by propagation of the correction angles is _ only @xmath62nm_. we choose the size of the patches as big as possible , provided that a single patch can still be handled efficiently . for the patch size in the example , @xmath63 points ( including @xmath64 overlap ) correspond to a @xmath65 interpolation matrix that can be inverted quickly using standard numerical methods like cholesky decomposition . a final remark concerning the runtime complexity of the method described above : the complexity can be further reduced in case the sampling grid is regular . since the patches all have the same size and the matrix entries in eq . ( [ linear_system ] ) only depend on the distances between sample points , the matrix can be inverted once for all patches and then applied to varying data on different patches , as long as the particular data subset is complete . note that eq . ( [ interpolant ] ) can be written as @xmath66 , where @xmath67 is the evaluation matrix . then , by applying eq . ( [ linear_system ] ) we obtain @xmath68 , where the matrix @xmath69 needs to be calculated only once for all complete patches . if samples are missing , however , the interpolation yields different coefficients @xmath70 and hence forces to recompute @xmath69 for this particular patch . using these techniques , the reconstruction of @xmath71 surface values from their gradients takes about @xmath72 minutes on a current personal computer . first , we investigated the stability of our method with respect to noise . we simulated realistic gradient data of a sphere ( with @xmath5 mm radius , @xmath5 mm @xmath6 @xmath5 mm field with sampling distance @xmath73 mm , see figure [ noise_sphere](a ) ) and added uniformly distributed noise of different levels , ranging from @xmath74 to @xmath75arcsec . we reconstructed the surface of the sphere using the interpolation method described in section [ method ] . hereby , we aligned the patches by only adding a constant to each patch . the reconstruction was performed for @xmath76 statistically independent slope data sets for each noise level . depicted in figure [ noise_sphere](b ) is a cross - section of the absolute error of the surface reconstruction from the ideal sphere , for a realistic noise level of @xmath58arcsec . the absolute error is less than @xmath77 nm _ on the entire measurement field_. the local height error corresponding to this noise level is about @xmath78 nm . this demonstrates that the dynamic range of the global absolute error with respect to the height variance ( @xmath79 mm ) of the considered sphere is about @xmath80 . the graph in figure [ noise_sphere](c ) depicts the mean value and the standard deviation ( black error bars ) of the absolute error of the reconstruction , for @xmath76 different data sets and for different noise levels . it demonstrates that for increasing noise level the absolute error grows only linearly ( linear fit depicted in gray ) , and even for a noise level being the _ fifty - fold of the typical sensor noise _ the global absolute error remains in the _ sub - micrometer regime_. this result implies that the reconstruction error is smaller than most technical applications require . another common task in quality assurance is the detection and quantification of surface defects like scratches or grooves . we therefore tested our method for its ability to reconstruct such local defects that may be in a range of only a few nanometers . for this purpose , we considered data from a pmd sensor for small , specular objects . the sensor has a resolvable distance of @xmath81 m laterally and a local angular uncertainty of about @xmath76arcsec @xcite . in order to quantify the deviation of the perfect shape , we again simulated a sphere ( this time with @xmath76 mm radius and @xmath82 mm @xmath6 @xmath82 mm data field size ) . we added parallel , straight grooves of varying depths from @xmath83 to @xmath4 nm and of @xmath84 m width and reconstructed the surface from the modified gradients . the perfect sphere was then subtracted from the reconstructed surface . the resulting reconstructed grooves are depicted in figure [ groove_sphere](a ) . the grooves ranging from @xmath4 down to @xmath72 nm depth are clearly distinguishable from the plane . figure [ groove_sphere](b ) shows that all reconstructed depths agree fairly well with the actual depths . note that each groove is determined by only @xmath72 inner sample points . the simulation results demonstrate that our method is almost free of error propagation while preserving small , local details of only some nanometers height . so far , we used only simulated data to test the reconstruction . now , we want to demonstrate the application of our method to a real measurement . the measurement was performed with a phase - measuring deflectometry sensor for very small , specular objects . it can laterally resolve object points with a distance of @xmath85 nm , while having a local angular uncertainty of about @xmath86arcsec . the object under test is a part of a wafer with about @xmath87 nm height range . the size of the measurement field was @xmath4 m @xmath6 @xmath5 m . depicted in figure [ wafer ] is the reconstructed object surface from roughly three million data values . both the global shape and local details could be reconstructed with high precision . we motivated why the employment of optical slope - measuring sensors can be advantageous . we gave a brief overview of existing sensor principles . the question that arose next was how to reconstruct the surface from its slope data . we presented a method based on radial basis functions which enables us to reconstruct the object surface from noisy gradient data . the method can handle large data sets consisting of some million entries . furthermore , the data does not need to be acquired on a regular grid it can be arbitrarily scattered and it can contain data holes . we demonstrated that , while accurately reconstructing the object s global shape , which may have a _ height range of some millimeters _ , the method preserves _ local height variations on a nanometer scale_. a remaining challenge is to improve the runtime complexity of the algorithm in order to be able to employ it for inline quality assurance in a production process . the authors thank the deutsche forschungsgemeinschaft for supporting this work in the framework of sfb 603 . i. weingrtner and m. schulz , `` novel scanning technique for ultra - precise measurement of slope and topography of flats , aspheres , and complex surfaces , '' in _ optical fabrication and testing _ , ( 1999 ) , no . 3739 in proc . spie , pp . 27482 . m. c. knauer , j. kaminski , and g. husler , `` phase measuring deflectometry : a new approach to measure specular free - form surfaces , '' in _ optical metrology in production engineering _ , , w. osten and m. takeda , eds . ( 2004 ) , no . 5457 in proc . spie , pp . 366376 . j. kaminski , s. lowitzsch , m. c. knauer , and g. husler , `` full - field shape measurement of specular surfaces , '' in _ fringe 2005 , the 5th international workshop on automatic processing of fringe patterns _ , , w. osten , ed . ( springer , 2005 ) , pp . 372379 . s. lowitzsch , j. kaminski , m. c. knauer , and g. husler , `` vision and modeling of specular surfaces , '' in _ vision , modeling , and visualization 2005 _ , , g. greiner , j. hornegger , h. niemann , and s. m. , eds . ( akademische verlagsgesellschaft aka gmbh , 2005 ) , pp . 479486 . j. pfund , n. lindlein , j. schwider , r. burow , t. blumel , and k. e. elssner , `` absolute sphericity measurement : a comparative study of the use of interferometry and a shack - hartmann sensor , '' opt . * 23 * , 742744 ( 1998 ) . a. agrawal , r. chellappa , and r. raskar , `` an algebraic approach to surface reconstruction from gradient fields , '' in `` ieee international conference on computer vision ( iccv ) , '' , vol . 1 ( 2005 ) , vol . 1 , pp . 174181 . c. elster and i. weingrtner , `` high - accuracy reconstruction of a function @xmath88 when only @xmath89 or @xmath90 is known at discrete measurement points , '' in `` x - ray mirrors , crystals , and multilayers ii , '' ( 2002 ) , no . 4782 in proc . spie , pp . 1220 . s. ettl , j. kaminski , and g. husler , `` generalized hermite interpolation with radial basis functions considering only gradient data , '' in _ curve and surface fitting : avignon 2006 _ , , a. cohen , j .- l . merrien , and l. l. schumaker , eds . ( nashboro press , brentwood , 2007 ) , pp . 141149 . of a typical smooth object and ( b ) its derivative @xmath91 , together with ( c ) the power spectral density of the surface @xmath1 and ( d ) of its slope @xmath92 ( all in arbitrary units ) . the power spectral density shows that differentiation reduces redundancy , contained in the low frequencies . , width=627 ] nm down to @xmath83 nm . after the reconstruction , the sphere was subtracted to make the grooves visible . the actual , reconstructed grooves are depicted in ( a ) full - field and in ( b ) cross - section.,width=529 ]
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we present a novel method for reconstructing the shape of an object from measured gradient data .
a certain class of optical sensors does not measure the shape of an object , but its local slope .
these sensors display several advantages , including high information efficiency , sensitivity , and robustness . for many applications , however , it is necessary to acquire the shape , which must be calculated from the slopes by numerical integration .
existing integration techniques show drawbacks that render them unusable in many cases .
our method is based on approximation employing radial basis functions .
it can be applied to irregularly sampled , noisy , and incomplete data , and it reconstructs surfaces both locally and globally with high accuracy .
| 7,292 | 166 |
in lattice qcd , the finite lattice spacing and finite lattice volume effects on the gluon propagator can be investigated with the help of lattice simulations at several lattice spacings and physical volumes . here we report on such a calculation . for details on the lattice setup see @xcite . in figure [ fig : gluevol ] , we show the renormalized gluon propagator at @xmath0 gev for all lattice simulations . note that we compare our data with the large volume simulations performed by the berlin - moscow - adelaide collaboration @xcite see @xcite for details . in each plot we show data for a given value of @xmath1 , i.e. data in the same plot has the same lattice spacing . the plots show that , for a given lattice spacing , the infrared gluon propagator decreases as the lattice volume increases . for larger momenta , the lattice data is less dependent on the lattice volume ; indeed , for momenta above @xmath2900 mev the lattice data define a unique curve . we can also investigate finite volume effects by comparing the renormalized gluon propagator computed using the same physical volume but different @xmath1 values . we are able to consider 4 different sets with similar physical volumes see figure [ fig : gluespac ] . although the physical volumes considered do not match perfectly , one can see in figure [ fig : gluespac ] that for momenta above @xmath2 900 mev the lattice data define a unique curve . this means that the renormalization procedure has been able to remove all dependence on the ultraviolet cut - off @xmath3 for the mid and high momentum regions . however , a comparison between figures [ fig : gluevol ] and [ fig : gluespac ] shows that , in the infrared region , the corrections due to the finite lattice spacing seem to be larger than the corrections associated with the finite lattice volume . in particular , figure [ fig : gluespac ] shows that the simulations performed with @xmath4 , i.e. , with a coarse lattice spacing , underestimate the gluon propagator in the infrared region . in this sense , the large volume simulations performed by the berlin - moscow - adelaide collaboration provide a lower bound for the continuum infrared propagator . we also aim to study how temperature changes the gluon propagator . at finite temperature , the gluon propagator is described by two tensor structures , @xmath5 where the transverse and longitudinal projectors are defined by @xmath6 the transverse @xmath7 and longitudinal @xmath8 propagators are given by @xmath9 @xmath10 on the lattice , finite temperature is introduced by reducing the temporal extent of the lattice , i.e. we work with lattices @xmath11 , with @xmath12 . the temperature is defined by @xmath13 . in table [ tempsetup ] we show the lattice setup of our simulation . simulations in this section have been performed with the help of chroma library @xcite . for the determination of the lattice spacing we fit the string tension data in @xcite in order to have a function @xmath14 . note also that we have been careful in the choice of the parameters , in particular we have only two different spatial physical volumes : @xmath15 and @xmath16 . this allows for a better control of finite size effects . .lattice setup used for the computation of the gluon propagator at finite temperature . [ cols="^,^,^,^,^,^",options="header " , ] [ tempsetup ] figures [ fig : transtemp ] and [ fig : longtemp ] show the results obtained up to date . we see that the transverse propagator , in the infrared region , decreases with the temperature . moreover , this component shows finite volume effects ; in particular , the large volume data exhibits a turnover in the infrared , not seen at the small volume data . the longitudinal component increases for temperatures below @xmath17 . then the data exhibits a discontinuity around @xmath18 , and the propagator decreases for @xmath19 . the behaviour of the gluon propagator as a function of the temperature can also be seen in the 3d plots shown in figure [ fig:3dtemp ] . as shown above , data for different physical ( spatial ) volumes exhibits finite volume effects . this can be seen in more detail in figure [ fig : finvoltemp ] , where we show the propagators for two volumes at t=324 mev . moreover , we are also able to check for finite lattice spacing effects at t=305 mev , where we worked out two different simulations with similar physical volumes and temperatures , but different lattice spacings . for this case , it seems that finite lattice spacing effects are under control , with the exception of the zero momentum for the transverse component see figure [ fig : lattspactemp ] . our results show that a better understanding of lattice effects is needed before our ultimate goal , which is the modelling of the propagators as a function of momentum and temperature . paulo silva is supported by fct under contract sfrh / bpd/40998/2007 . work supported by projects cern / fp/123612/2011 , cern / fp/123620/2011 and ptdc / fis/100968/2008 , projects developed under initiative qren financed by ue / feder through programme compete .
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we study the landau gauge gluon propagator at zero and finite temperature using lattice simulations .
particular attention is given to the finite size effects and to the infrared behaviour .
| 1,477 | 47 |
one of the first striking applications of gromov s theory of pseudoholomorphic curves @xcite was that a closed exact lagrangian immersion @xmath4 inside a liouville manifold must have a double - point , given the assumption that it is hamiltonian displaceable . gromov s result has the following contact - geometric reformulation , which will turn out to be useful . consider the so - called _ contactisation _ @xmath5 of the liouville manifold @xmath6 , which is a contact manifold with the choice of a contact form . recall that a ( generic ) exact lagrangian immersion @xmath4 lifts to a legendrian ( embedding ) @xmath7 . one says that @xmath1 is _ horizontally displaceable _ given that @xmath8 is hamiltonian displaceable . the above result thus translates into the fact that a horizontally displaceable legendrian submanifold @xmath1 must have a _ reeb chord _ for the above standard contact form i.e. a non - trivial integral curve of @xmath9 having endpoints on @xmath1 . a similar result holds for legendrian submanifolds of boundaries of subcritical weinstein manifolds , as proven in @xcite by mohnke . in the spirit of arnold @xcite , the following conjectural refinement of the above result was later made : the number of reeb chords on a chord - generic legendrian submanifold @xmath7 whose lagrangian projection is hamiltonian displaceable is at least @xmath10 . however , as was shown by sauvaget in @xcite by the explicit counter - examples inside the standard contact vector space @xmath11 , @xmath12 , the above inequality is not true without additional assumptions on the the legendrian submanifold ; also , see the more recent examples constructed in @xcite by ekholm - eliashberg - murphy - smith . the latter result is based upon the h - principle proven in @xcite by eliashberg - murphy for lagrangian cobordisms having loose negative ends in the sense of murphy @xcite . on the positive side , the above arnold - type bound has been proven using the legendrian contact homology of the legendrian submanifold , under the additional assumption that the legendrian contact homology algebra is sufficiently well - behaved . legendrian contact homology is a legendrian isotopy invariant independently constructed by chekanov @xcite and eliashberg - givental - hofer @xcite , and later developed by ekholm - etnyre - sullivan @xcite . this invariant is defined by encoding pseudoholomorphic disc counts in the legendrian contact homology differential graded algebra ( dga for short ) which usually is called the _ chekanov - eliashberg algebra _ of the legendrian submanifold . in the case when the chekanov - eliashberg algebra of a legendrian admits an augmentation ( this should be seen as a form of non - obstructedness for its floer theory ) , the above arnold - type bound was proven by ekholm - etnyre - sullivan in @xcite and by ekholm - etnyre - sabloff in @xcite . in @xcite , the authors generalised this proof to the case when the chekanov - eliashberg algebra admits a finite - dimensional matrix representation , in which case the same lower bound also is satisfied . the above arnold - type bound is also related to the one regarding the number of hamiltonian chords between the zero - section in @xmath13 ( or , more generally , any exact closed lagrangian submanifold of a liouville manifold ) and its image under a generic hamiltonian diffeomorphism . namely , such hamiltonian chords correspond to reeb chords on a legendrian lift of the union of the lagrangian submanifold and its image under the hamiltonian diffeomorphism . in fact , as shown by laudenbach - sikorav in @xcite , the number of such chords is bounded from below by the stable morse number of the zero - section ( and hence , in particular , it is bounded from below by half of the betti numbers of the disjoint union of _ two _ copies of the zero - section ) . arnold originally asked whether this bound can be improved , and if in fact the _ morse number _ of the zero - section is a lower bound . however , this question seems to be out of reach of current technology . on the other hand , we note that the stable morse number is equal to the morse number in a number of cases ; see @xcite as well as section [ sec : gendefs ] below for more details . finally , we mention the remarkable result by ekholm - smith in @xcite , which shows that the smooth structure itself can predict the existence of more double points than the original bound given in terms of the homology . namely , a @xmath14-dimensional manifold @xmath15 for @xmath16 that admits a legendrian embedding having precisely one transverse reeb chord in the standard contact space must be _ diffeomorphic _ to the standard sphere unless @xmath17 . also see @xcite for similar results in other dimensions . in this paper , we will explore a priori lower bounds for the number of reeb chords on a legendrian submanifold @xmath7 , given that it admits an exact lagrangian filling @xmath18 inside the symplectisation . recall that the condition of admitting an exact lagrangian filling is invariant under legendrian isotopy ; see e.g. @xcite . the bound will be given in terms of the simple homotopy type of @xmath19 . first , we recall that such a legendrian submanifold automatically has a well - behaved chekanov - eliashberg algebra ; namely , an exact lagrangian filling induces an augmentation by @xcite . in the case when the projection of @xmath1 to @xmath20 is displaceable , the aforementioned result can thus be applied , giving the above arnold - type bound . however , in this case , there are even stronger bounds that can be obtained from the topology of the exact lagrangian filling @xmath19 ( and without the assumption of horizontal displaceability ) . see section [ sec : wrapped ] below for previous such results as well as an outline of the proof , which is based upon seidel s isomorphism in wrapped floer homology . this is also the starting point of the argument that we will use in order to prove our results here . in the following we assume that a legendrian submanifold @xmath21 is chord - generic and has an exact lagrangian filling @xmath22 . here @xmath23 denotes the coordinate on the first @xmath24-factor . in particular , the set of reeb chords @xmath25 of @xmath1 is finite . further , the set of reeb chords @xmath26 in degree @xmath27 will be denoted by @xmath28 , where the grading is induced by the conley - zehnder index modulo the maslov number @xmath29 of @xmath19 as defined in @xcite . observe that @xmath30 in particular implies that the first chern class of @xmath6 vanishes on @xmath31 . for a group @xmath32 being the epimorphic image of @xmath33 , consider the morse homology complex @xmath34),\partial_f)$ ] of @xmath19 with coefficients in the group ring @xmath35 $ ] twisted by the fundamental group , where @xmath36 is a unital commutative ring and @xmath37 is a morse function satisfying @xmath38 outside of a compact set . ( the generators of this complex are graded by the morse index , and the differential counts negative gradient flow lines . ) [ mainthmsmplehomeqnts ] let @xmath22 be an exact lagrangian filling of an @xmath39-dimensional closed legendrian submanifold @xmath40 with fundamental group @xmath41 and maslov number @xmath29 . * in the case when the filling is spin and when @xmath30 , the morse homology complex @xmath42),\partial_f)$ ] is simple homotopy equivalent to a @xmath43$]-equivariant complex @xmath44\langle \mathcal{q}_{n-\bullet}(\lambda ) \rangle,\partial)$ ] ; * in the general case , it follows that the complex @xmath45),\partial_f)$ ] is homotopy equivalent in the category of @xmath32-equivariant complexes to a complex @xmath46\langle \mathcal{q}_{n-\bullet}(\lambda ) \rangle,\partial)$ ] with grading in @xmath47 . here we can always take @xmath48 , while we are free to choose an arbitrary unital commutative ring in the case when @xmath19 is spin . we prove theorem [ mainthmsmplehomeqnts ] in section [ maintheoremanditsconsequences ] . now let @xmath49 denote the stable morse number of a manifold @xmath50 with possibly non - empty boundary , see definition [ defn : stablemorse ] . using theorem [ mainthmsmplehomeqnts ] and the adaptation of ( * ? ? ? * theorem 2.2 ) to the case of manifolds with boundary ( see proposition [ prop : morse ] ) , the following result is immediate : [ maininequalitystablemorsenumberofafilling ] suppose that @xmath51 is a chord - generic closed legendrian submanifold admitting an exact lagrangian filling @xmath19 which is spin and has vanishing maslov number . it follows that the bound latexmath:[\ ] ] where @xmath604 is a subgroup of @xmath605 $ ] generated by @xmath606-[\phi(h)]$ ] . the proof can , for example , be deduced from ( * ? ? ? * proposition 29 ) . then we describe a series of finite solvable groups @xmath607 with the property that @xmath608 , @xmath609)=1 $ ] . let @xmath610 be a finite field with @xmath611 elements , where @xmath612 is a prime number . we define a group @xmath613 , where @xmath614 acts on the additive group @xmath615 by @xmath616 , @xmath617 , @xmath618 . it is easy to see that @xmath607 is a solvable group . this follows from the existence of the following subnormal series @xmath619 where @xmath620 and @xmath621 . then we observe that from formula [ absemiddirprreltohsh ] it follows that @xmath622\simeq { \mathbb{f}}^{\ast}_q$ ] is a non - trivial cyclic group . therefore , @xmath609)=1 $ ] . finally we prove that @xmath608 . since @xmath623 and @xmath624 , we get that @xmath625 . we first show that @xmath626 . assume that @xmath607 has a generating set @xmath627 with @xmath628 , @xmath629 and @xmath630 . then , note that every element in the group generated by @xmath631 has a form @xmath632 , where @xmath633 , @xmath634 , and @xmath635 . this leads to the contradiction with the fact that @xmath623 . then we take a set of generators @xmath631 and @xmath636 with the property that @xmath330 is a generator of @xmath637 ( @xmath637 is a cyclic group ) . such an element definitely exists since if all the elements of @xmath631 are of the form @xmath638 , where @xmath496 is not a generator of @xmath637 , then @xmath631 is not a generating set of @xmath607 . again , @xmath637 is a cyclic group of order @xmath639 , and the order of @xmath330 that we denote by @xmath640 is coprime to @xmath641 . this implies that @xmath642 is coprime to @xmath641 , and hence none of the primes which divide @xmath643 will divide @xmath644=|{\mathbb{f}}^m_q|$ ] . let @xmath282 be the set of primes which divide @xmath640 . then @xmath645 and @xmath646 are two hall @xmath282-subgroups , and hence by theorem [ conjpihallsubgr ] they are conjugate by some element @xmath647 . this implies that @xmath631 , after conjugation , contains an element @xmath648 , where @xmath649 is a generator of @xmath637 . we also would like to mention that it is possible to find @xmath173 explicitly without relying on the theory of hall @xmath282-subgroups . then , already knowing that @xmath626 , we can apply the previous argument and see that , in fact , @xmath650 . together with the fact that @xmath625 we get that @xmath608 . using proposition [ constrrealizoffundgroup ] , we construct an exact lagrangian filling @xmath651 of a legendrian @xmath652 inside the standard contact vector space . then theorem [ intrineqabssi ] tells us that @xmath653 is satisfied for all representatives . on the other hand , the bound given by seidel s isomorphism is @xmath654 finally , note that the difference between the previous two bounds gets arbitrarily large as @xmath655 . we would like to thank franois charette and jarek kdra for very helpful conversations and interest in our work . in addition , we are grateful to the referee of an earlier version of this paper for many valuable comments and suggestions . b. chantraine , g. dimitroglou rizell , p. ghiggini , and r. golovko , _ floer homology and lagrangian concordance _ , proceedings of 21st gkova geometry - topology conference 2014 , 76113 , gkova geometry / topology conference ( ggt ) , gkova , 2015 . t. ekholm , _ rational sft , linearized legendrian contact homology , and lagrangian floer cohomology _ , perspectives in analysis , geometry , and topology . on the occasion of the 60th birthday of oleg viro , volume 296 , pages 109145 . springer , 2012 . k. fukaya , p. seidel and i. smith , _ the symplectic geometry of cotangent bundles from a categorical viewpoint _ , homological mirror symmetry , lecture notes in phys . 757 , springer , berlin , 2009 , 126 .
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assume that we are given a closed chord - generic legendrian submanifold @xmath0 of the contactisation of a liouville manifold , where @xmath1 moreover admits an exact lagrangian filling @xmath2 inside the symplectisation . under the further assumptions that this filling is spin and has vanishing maslov class , we prove that the number of reeb chords on @xmath1 is bounded from below by the stable morse number of @xmath3 . given a general exact lagrangian filling @xmath3
, we show that the number of reeb chords is bounded from below by a quantity depending on the homotopy type of @xmath3 , following ono - pajitnov s implementation in floer homology of invariants due to sharko .
this improves previously known bounds in terms of the betti numbers of either @xmath1 or @xmath3 .
| 4,194 | 243 |
the top quark is the heaviest standard model ( sm ) particle which has been discovered so far , and might be the first place in which new physics effects could appear . while the top quark has been discovered and studied in some details at the tevatron , many of its properties are being studied at the lhc with high precision . since new physics can show up in the couplings of the top quark with other sm particles , in particular gauge bosons , the precise measurement of the top couplings among the top quark and gauge bosons is important . in particular , the top quark is copiously produced at the lhc , the lhc experiments are the places to probe these couplings@xcite . at the lhc , top quarks are produced primarily via two independent mechanisms . the dominant production mechanism is the pair production processes @xmath1 , @xmath2 and second is single top production via electroweak interactions involving the @xmath3 vertex . single top quarks at the lhc are produced in three different modes : the s - channel ( the involved @xmath4 is time - like ) , the t - channel mode ( the involved @xmath4 is space - like ) , and the @xmath0 production ( the @xmath4 boson is real ) . despite single top has a smaller cross section than top pair production , it can play an important role in top quark physics at lhc because this channel has potential to allow a direct measurement of @xmath5 ckm matrix elements as well as its sensitivity to various new physics models . however sufficient integrated luminosity and improved method of analysis can help us achieve detection of single top events at the lhc . the first observable in single top study is the total cross section and measurement of any possible deviation from predicted value by the sm . therefore , it is worthwhile to investigate the effects of physics beyond sm on single top quark production . as it is well - known sm has been successfully predicted experimental measurements with a great precision . nevertheless , it is commonly accepted that it is a valid effective lagrangian which is applicable at low energies . in many beyond sm theories which have been studied to date , reduction to the sm at low energies proceeds via decoupling of heavy particle with masses of order @xmath6 . there have been many attempts to study the sensitivity of the lhc observables to various effective operators @xcite . our goal in this paper is to study the effect of anomalous couplings of the top quark with gluon via @xmath0-channel of single top at the lhc . we will assume that new physics effects in @xmath0 single top production are induced by consideration of an effective lagrangian . here , we confine our studies to interaction of mass dimension @xmath7 after spontaneous symmetry breaking . the total statistical and systematic uncertainties in the measurement of cross section for this process at the lhc is about @xmath8 with an integrated luminosity of @xmath9@xcite . the top quark is the heaviest quark , therefore effect of new physics on its coupling are expected to be larger than for any other fermions and deviation with respect to the sm predictions might be detectable . the rest of this paper is organized as follows : in the next section , including the effective lagrangian for @xmath10 coupling , we calculate the analytical expression for the single @xmath0 top cross section production at the lhc . in section 3 , we present the dependency of observables which we study at the lhc . then , we find the allowed regions in parameters space of our effective lagrangian and compare our results with the results obtained from observables in production of @xmath11 at lhc and edm of top quark . the conclusions are given in section 4 . in this section , we introduce a model independent effective lagrangian for the vertex of @xmath10 and look for any possible deviation from sm prediction in production of @xmath12-channel at the lhc . in this approach , we assume that sm modified by an addition 5-dimensional lagrangian which include interaction of top pair and gluon and consider the effect of this lagrangian in production of @xmath0 single top production . coefficients of this lagrangian parameterize the low energy effects of the underlying high scale physics . as it is well - known , the gauge invariant effective lagrangian for the interactions between the top quark and gluons which include the cedm and cmdm form factors is given by : @xmath13 t g^{a}_{\nu}\end{aligned}\ ] ] where @xmath14 are the @xmath15 color matrices and @xmath16 and @xmath17 are , respectively , the cmdm and cedm forms factors of the top quark . notice that a sizable non - zero cedm would be the signal of a new type of cp - violating interaction beyond ckm phase and can contribute to electric dipole moment ( edm ) of neutron . for this reason , experimental upper bound on edm of neutron constraint this coupling @xcite,@xcite . in the following , we use this constraint on parameters space of above lagrangian and compare these bounds with our result which arise from @xmath12 single top production at lhc . assuming @xmath18 , where @xmath6 is the scale of new physics , the form factors can be approximated by @xmath19 where @xmath20 and @xmath21 are independent of @xmath22 . the cmdm of the top quark is then given by @xmath23,while cedm is @xmath24 . = 1.6 in as it is shown in fig . [ twfig ] , @xmath10 effective lagrangian can contribute to the right diagram in @xmath0 of single top production at the lhc . we calculate the amplitude of @xmath25 process including cedm and cmdm effect . this amplitude is given by : @xmath26 \\ \nonumber\end{aligned}\ ] ] where @xmath27 and @xmath28 are respectively weak and strong coupling constants and @xmath29 , @xmath30 are the masses of top and w gauge boson and @xmath5 is ckm matrix element . the explicit forms for @xmath31 ( @xmath32 ) in terms of mandelstam variables , @xmath21 and @xmath33 are given in eqs . ( [ f1]-[f6 ] ) of the appendix . in this section , we study the total cross section of @xmath0 single top production at the lhc and study the effect of cedm and cmdm couplings on it . recently , @xmath34 collaboration reported the measured value of cross section of @xmath0 single top production at the center - of - mass energy @xmath35 with an integrated luminosity of @xmath36@xcite : @xmath37.\label{exp}\end{aligned}\ ] ] this measurement is in agreement with the sm expectation @xmath38 @xcite . the hadronic cross section for production of @xmath0 can be obtained by integrating over the parton level cross section convoluted with the parton distribution functions : @xmath39 where @xmath40 are the parton structure functions of proton . the parameters @xmath41 and @xmath42 are the parton momentum fractions and @xmath43 is the factorization scale . in this paper , we obtain the direct constraints on dipole operators including top quark in the above effective lagrangian approach . we consider the total cross section of @xmath0 single top production at the lhc and study the effect of cmdm and cedm coupling on it . for this study , we consider the relative change in cross section which is defined as : @xmath44 where @xmath45 is total cross section in the presence of cmdm and cedm couplings . the relative change in cross section of the single top production at the lhc are shown in fig . [ deltacrossk ] and [ deltacrosskt ] . in these figures , we consider that only cmdm or cedm coupling exists and display the relative change in cross section of @xmath46 versus @xmath21 and @xmath20 . in fig . [ deltacrossk]-a , we have set @xmath47 . in fig . [ deltacrossk]-b , as explained in the caption , we have set @xmath48 which might be measured in ongoing run of the lhc . in the center of mass of energy @xmath49 , leading order sm cross section of @xmath50 have been obtained @xmath51 @xcite . to calculate @xmath46 , we have used the cteq6.6 m @xcite , mstw2008 @xcite and alekhin2 @xcite as the parton structure functions ( pdf ) . the green curve ( dashed ) , the pink curve ( line ) and blue curve ( dotted ) are corresponding to cteq6.6 m , mstw2008 and alekhin2 structure functions , respectively . an interesting observation from fig . [ deltacrossk ] and [ deltacrosskt ] is that the correction to @xmath0-channel cross section due to cmdm and cedm is sensitive to the choice of parton distribution function , in particular at large values of @xmath21 and @xmath20 . as it is seen in these figures , different structure functions change the value of @xmath46 more than @xmath52 for large values of @xmath21 and @xmath20 . considering the cteq pdf , the presence of @xmath21 or @xmath20 can change total cross section more than @xmath53 . at small value in the range of @xmath54 $ ] , @xmath55/@xmath56 is almost robust against the choice of pdf . /@xmath56 versus @xmath21 . in this figure , we have set @xmath47 . the green curve ( dashed ) , the pink curve ( line ) and blue curve ( dotted ) respectively correspond to cteq6.6 m @xcite , mstw2008 @xcite and alekhin2 @xcite structure functions . b ) similar to fig . a except that @xmath48 . the horizontal small dotted ( yellow dark ) , violet and dot - dashed ( green ) lines respectively correspond to @xmath57 , @xmath58 and @xmath59 uncertainties in the measurement of @xmath50.,title="fig:"]/@xmath56 versus @xmath21 . in this figure , we have set @xmath47 . the green curve ( dashed ) , the pink curve ( line ) and blue curve ( dotted ) respectively correspond to cteq6.6 m @xcite , mstw2008 @xcite and alekhin2 @xcite structure functions . b ) similar to fig . a except that @xmath48 . the horizontal small dotted ( yellow dark ) , violet and dot - dashed ( green ) lines respectively correspond to @xmath57 , @xmath58 and @xmath59 uncertainties in the measurement of @xmath50.,title="fig : " ] ( a)(b ) /@xmath56 versus @xmath20 for the center - of - mass energy of @xmath60 ( left ) and @xmath61 ( right ) . input parameters are similar to fig . [ deltacrossk].,title="fig:"]/@xmath56 versus @xmath20 for the center - of - mass energy of @xmath60 ( left ) and @xmath61 ( right ) . input parameters are similar to fig . [ deltacrossk].,title="fig : " ] ( a)(b ) fig . [ deltacrossk ] demonstrates that the effect of the presence of cmdm can change the total cross section more than @xmath62 for @xmath63 . the horizontal small dotted ( yellow dark ) , violet and dot - dashed ( green ) lines respectively correspond to @xmath57 , @xmath58 and @xmath59 uncertainty in the measurement of @xmath50 . as it was mentioned , the total statistical and systematic uncertainties in the measurement of cross section for @xmath0 single top production at the lhc is about @xmath8 with an integrated luminosity of @xmath9 at center - of - mass energy @xmath64 . with @xmath52 uncertainty in future measurement of the total cross section , from fig . [ deltacrossk ] ( [ deltacrosskt ] ) , we can put upper bound on @xmath21 ( @xmath20 ) down to @xmath65 ( @xmath66 ) . if in forthcoming run of lhc ( in @xmath61 ) , we measure the total cross section even with @xmath67 uncertainty , we can constrain @xmath21 ( @xmath20 ) down to @xmath68 ( @xmath69 ) . as it is seen in fig . [ deltacrosskt ] , in spite of cmdm ( @xmath21 ) , the cross section is symmetric with respect to @xmath20 because cedm coupling enters in the cross section in even powers when @xmath70 . therefore , in this situation , the cross section is not a cp violating observable . another observation from fig . [ deltacrossk ] is that the effect of presence of cmdm ( @xmath21 ) , in spite of cedm ( @xmath20 ) , can be destructive . if the total uncertainties in the measurement of cross section is less than @xmath67 , we can distinguish between cedm and cmdm effects . in figs . [ scater ] , red area depicts ranges of parameters space in cmdm ( @xmath21 ) and cedm ( @xmath20 ) couplings plane for which prediction of effective lagrangian ( equation [ lag ] ) on @xmath0 single top production at lhc are consistent with experimental measurements . when performing such study , one should take into account constraints from other studies . there exist many direct and indirect constraints on dipole operator . presently , the most sensitive observable obtained from mercury and neutron edm . in @xcite , it is shown that the neutron edm constrains the top cedm to be @xmath71 . moreover , the constraints from @xmath72 and the top electric dipole moment , provide weaker bound on @xmath20 . furthermore , the cedm and cmdm couplings of the top quark directly affect on top pair production at hadron colliders @xcite . we have borrowed the results of lhc constraints which come from @xmath73 total cross section on cedm and cmdm of top from reference @xcite . these results have been shown in fig . [ scater ] . in this figure green line depicts neutron edm constraint on @xmath20 . hatched cyan shaded area depicts the allowed region which is consistent with @xmath73 total cross section at lhc . yellow area shows allowed region which is consistent with spectrum measurement of @xmath74 at lhc . it is remarkable that the allowed regions of top pair production cross section and spectrum measurement of @xmath74 at the lhc overlap with allowed region of @xmath0 single top production at lhc . as it can be seen in this figure , @xmath0 single top constraints on @xmath21 and @xmath20 are stronger than direct constraints which come from @xmath73 total cross section and spectrum measurement of @xmath74 at lhc . in this paper , based on the effective lagrangian approach , we modify the sm by the additional 5-dimensional operators which include the interaction of top quark with gluon and consider the effect of this lagrangian in production of @xmath0 single top . coefficients of this lagrangian are related to cedm and cmdm form factors . we consider the total cross section of @xmath0 single top production at the lhc and study the effect of cedm and cmdm couplings on it . we have found the allowed regions in parameters space of cedm and cmdm in such a way that the experimental measurement of @xmath46 is satisfied . we also investigate the effect of different pdfs on @xmath0 single top production at lhc as a function of @xmath21 and @xmath20 . we have shown that deviation of the @xmath0-channel single top cross section from the sm value is significant . we consider constraints on cedm and cmdm couplings which come from @xmath73 total cross section and spectrum measurement of @xmath74 at the lhc and compare them with our results . we have shown that constraints on @xmath21 and @xmath20 which arise from @xmath0 single top production at lhc are comparable with the ones coming from @xmath73 production at the lhc . it is notable that with the current @xmath0 cross section precision measurement , tight bounds are obtained and therefore with more precision measurements in future even more stringent limits than @xmath73 cross section could be achieved . we would like to thank h. kanpour for helping us in technical issues . here , we list the formulas of @xmath31 which have been applied in calculation of single top production cross section . notice that mass dimension of @xmath31 are not equal . @xmath75 @xmath76 @xmath77 @xmath78 @xmath79 @xmath80\end{aligned}\ ] ] where @xmath81 and @xmath82 are mandelstam variables , @xmath29 , @xmath30 are the masses of top and w gauge boson . the parameters @xmath20 and @xmath21 are , respectively cedm and cmdm couplings . f. -p . schilling , int . j. mod . a * 27 * ( 2012 ) 1230016 [ arxiv:1206.4484 [ hep - ex ] ] . w. buchmuller and d. wyler , nucl . b * 268 * ( 1986 ) 621 ; c. arzt , m. b. einhorn and j. wudka , nucl . b * 433 * ( 1995 ) 41 [ hep - ph/9405214 ] ; g. j. gounaris , m. kuroda and f. m. renard , phys . d * 54 * ( 1996 ) 6861 [ hep - ph/9606435 ] ; a. cordero - cid , m. a. perez , g. tavares - velasco and j. j. toscano , phys . d * 70 * ( 2004 ) 074003 [ hep - ph/0407127 ] ; b. grzadkowski , z. hioki , k. ohkuma and j. wudka , nucl . b * 689 * ( 2004 ) 108 [ hep - ph/0310159 ] ; b. grzadkowski , m. iskrzynski , m. misiak and j. rosiek , jhep * 1010 * ( 2010 ) 085 [ arxiv:1008.4884 [ hep - ph ] ] ; r. torre , arxiv:1005.4801 [ hep - ph ] . s. chatrchyan _ et al . _ [ cms collaboration ] , phys . [ arxiv:1209.3489 [ hep - ex ] ] . e. o. iltan , phys . d * 65 * ( 2002 ) 073013 [ hep - ph/0111038 ] . j. f. kamenik , m. papucci and a. weiler , phys . d * 85 * ( 2012 ) 071501 [ arxiv:1107.3143 [ hep - ph ] ] . n. kidonakis , phys . d * 82 * ( 2010 ) 054018 [ arxiv:1005.4451 [ hep - ph ] ] . d. atwood , a. kagan and t. g. rizzo , phys . d * 52 * ( 1995 ) 6264 [ hep - ph/9407408 ] ; p. haberl , o. nachtmann and a. wilch , phys . d * 53 * ( 1996 ) 4875 [ hep - ph/9505409 ] ; c. degrande , j. -m . gerard , c. grojean , f. maltoni and g. servant , jhep * 1103 * ( 2011 ) 125 [ arxiv:1010.6304 [ hep - ph ] ] z. hioki and k. ohkuma , eur . phys . j. c * 65 * ( 2010 ) 127 [ arxiv:0910.3049 [ hep - ph ] ] ; z. hioki and k. ohkuma , eur . j. c * 71 * ( 2011 ) 1535 [ arxiv:1011.2655 [ hep - ph ] ] ; s. s. biswal , s. d. rindani and p. sharma , top quark using its polarization in pair production at hadron colliders , arxiv:1211.4075 [ hep - ph ] . j. pumplin , d. r. stump , j. huston , h. l. lai , p. m. nadolsky and w. k. tung , jhep * 0207 * ( 2002 ) 012 [ hep - ph/0201195 ] . a. d. martin , w. j. stirling , r. s. thorne and g. watt , eur . j. c * 63 * ( 2009 ) 189 [ arxiv:0901.0002 [ hep - ph ] ] . s. alekhin , j. blumlein , s. klein and s. moch , arxiv:0908.3128 [ hep - ph ] .
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we study the effects of chromoelectric and chromomagnetic dipole moments ( cedm and cmdm ) on the production cross section of single top @xmath0-channel at the lhc based on the effective lagrangian approach .
we show that the impact of cedm and cmdm could be large . using the experimental measurement of the @xmath0-channel cross section ,
constraints on cedm and cmdm are extracted .
these constraints are comparable with the ones obtained from the top pair analysis .
0.65 cm 0.5 cm * seyed yaser ayazi , hoda hesari and mojtaba mohammadi najafabadi*0.5 cm school of particles and accelerators , institute for research in fundamental sciences ( ipm ) , p.o .
box 19395 - 5531 , tehran , iran
| 5,898 | 219 |
the theme of @xmath0 ( parity time ) symmetric systems was initiated in the works of bender and collaborators @xcite as an alternative to the standard quantum theory , where the hamiltonian is postulated to be hermitian . the principal conclusion of these works was that @xmath0-invariant hamiltonians , which are not necessarily hermitian , may still give rise to completely real spectra , thus being appropriate for the description of physical settings . in terms of the schrdinger - type hamiltonians , which include the usual kinetic - energy operator and the potential term , @xmath3 , the @xmath0-invariance admits complex potentials , subject to constraint that @xmath4 . recent developments in optics have resulted in an experimental realization of the originally theoretical concept of the @xmath0-symmetric hamiltonians , chiefly due to the work by christodoulides and co - workers christo1 ( see also @xcite ) . it has been demonstrated that the controllable imposition of symmetrically set and globally balanced gain and loss may render optical waveguiding arrays a fertile territory for the construction of @xmath0-symmetric complex potentials . the first two such realizations made use of couplers composed of two waveguides with and without loss @xcite ( so - called passive @xmath5couplers ) , or , in more standard " form , a pair of coupled waveguides , one carrying gain and the other one loss @xcite . in fact , more general models of linearly coupled active ( gain - carrying ) and passive ( lossy ) intrinsically nonlinear waveguides , without imposing the condition of the gain - loss balance , were considered earlier , and stable solitons were found in them @xcite , including exact solutions @xcite ( see also a brief review in ref . recently , an electronic analog of such settings has also been implemented @xcite . configurations with a hidden @xmath6 symmetry have been identified also in fine - tuned parameter regions of microwave billiards @xcite . effects of the nonlinearity in a gross - pitaevski equation on the @xmath7 properties of a bose - einstein condensate have been analyzed in @xcite . the possibility to engineer @xmath0-symmetric _ oligomers _ ( coupled complexes of a few loss - and gain - carrying elements ) @xcite , which may include nonlinearity , was an incentive to a broad array of additional studies on both the few - site systems and entire @xmath0-symmetric lattices @xcite . more recently , nonlinear @xmath0-symmetric systems , incorporating @xmath0-balanced nonlinear terms , have drawn considerable interest too @xcite-@xcite . most of the @xmath0-invariant systems considered thus far have been one - dimensional ( 1d ) in their nature , although the stability of solitons in 2d periodic @xmath0-symmetric potentials has also been recently investigated @xcite . actually , 2d arrays of optical waveguides can be readily built @xcite ( the same is true about other quasi - discrete systems , including electrical ones ) , hence , a natural question is whether @xmath0-symmetric oligomers ( and ultimately lattices built of such building blocks ) can be created in a 2d form . this work aims to make a basic step in this direction , by introducing fundamental 2d plaquettes consisting , typically , of four sites ( in one case , it will be a five - site cross ) . these configurations , illustrated by fig . [ modes ] , are inspired by earlier works on 2d hamiltonian lattices described by discrete nonlinear schrdinger equations @xcite , where diverse classes of modes , including discrete solitary vortices malomed , pelin , have been predicted and experimentally observed moti , yuri . the plaquettes proposed herein should be straightforwardly accessible with current experimental techniques in nonlinear optics , as a straightforward generalization of the coupler - based setting reported in ref . we start from the well - established hamiltonian form of such plaquettes in the conservative form , gradually turning on the gain - loss parameter ( @xmath2 ) , as the strength of the @xmath0-invariant terms , to examine stationary states supported by the plaquettes , studying their stability against small perturbations and verifying the results through direct simulations . actually , in this work we focus on those ( quite diverse , although , obviously , not most generic ) modes that can be found in an analytical form , while their stability is studied by means of numerical methods . the analytical calculations and the manifestations of interesting features , such as a potential persistence past the critical point of the linear @xmath0 symmetry , are enabled by the enhanced symmetry of the modes that we consider below . it is conceivable that additional asymmetric modes may exist too within these 2d configurations . our principal motivation for studying the above systems stems from the fact that realizations of @xmath0-symmetry e.g. within the realm of nonlinear optics will be inherently endowed with nonlinearity . hence , it is only natural to inquire about the interplay of the above type of linear systems with the presence of nonlinear effects . in addition to this physical argument , there exists an intriguing mathematical one which concerns the existence , stability and dynamical fate of the nonlinear states in the presence of @xmath0-symmetric perturbations . in particular , previous works @xcite point to the direction that neither the existence , nor the stability of @xmath0-symmetric nonlinear states mirrors that of their linear counterparts ( or respects the phase transition of the latter generically ) . the presentation of our results is structured as follows . section [ symmetry ] contains a part of the analytical results , including a detailed analysis of the @xmath8symmetry properties of the nonlinear schrdinger type model , as well as the spectral properties of the linear hamiltonian subsystems . section [ numerics ] is devoted to the existence , stability and dynamics of stationary modes in the nonlinear systems . beside analytical results , it contains a detailed presentation of the numerical findings . in section [ conclu ] we summarize conclusions and discuss directions for future studies . the dynamics of the 2d plaquettes that we are going to consider is described by a multicomponent nonlinear schrdinger equation ( nlse ) @xmath9built over a transposition - symmetric linear @xmath10 hermitian matrix hamiltonian @xmath11 and an additional nonlinear @xmath10 matrix operator , @xmath12 . to understand the symmetry properties of this nlse , we first analyze the associated linear problem @xmath13and check then whether the symmetry is preserved by the nonlinear term , @xmath14 . the analysis can be built , in a part , on techniques developed for other nonlinear dynamical systems with symmetry preservation nonlin-1,nonlin-2,nonlin-3,nonlin-4,nonlin-5,nonlin-6,nonlin-7,nonlin-8 . for the present setups , the time reversal operation @xmath15 can be defined as the combined action of a scalar - type complex conjugation @xmath16 , @xmath17 , and the sign change of time , @xmath18 , in full accordance with wigner s original work which introduced these concepts @xcite . for the linear schrdinger equation ( [ a2 ] ) and its solutions @xmath19this implies @xmath20where the overbar denotes complex conjugation . from the actual form of the gain - loss arrangements in the 2d plaquettes we can conjecture the existence of certain plaquette - dependent parity operators @xmath21 , with @xmath22 , which will render the hamiltonians @xmath8symmetric , @xmath23=0 $ ] . to find an explicit representation of these parity operators @xmath21 , we use the following ansatz , @xmath24=0 \label{a2 - 3}\]]together with the pseudo - hermiticity condition @xmath25the latter follows trivially from eq . ( [ a2 - 3 ] ) , @xmath23=0 $ ] , and the transposition symmetry , @xmath11 . these parity operators @xmath21 will be used to check whether the corresponding nonlinear terms @xmath26 satisfy the same @xmath27symmetry . in contrast to linear setups with the @xmath28symmetry being either exact ( @xmath29=0 $ ] , @xmath30 ) or spontaneously broken ( @xmath29=0 $ ] , @xmath31 ) , the nonlinear setups considered in the present paper allow for sectors of exact @xmath28symmetry ( @xmath32=0 $ ] , @xmath30 ) and of broken @xmath28symmetry ( @xmath31@xmath33@xmath34\neq 0 $ ] ) , as it is common for nonlinear @xmath6-symmetric systems @xcite-@xcite . we start from the 2d plaquette of 0 + 0- type depicted as configuration ( a ) in fig . [ modes ] . this plaquette has only two ( diagonally opposite ) nodes carrying the gain and loss , while the other two nodes bear no such effects . the corresponding dynamical equations for the amplitudes at the four sites of this oligomer are @xmath35where @xmath36 is the above - mentioned gain - loss coefficient , and @xmath37 is a real coupling constant . the nonlinearity coefficients are scaled to be @xmath38 ( we use time @xmath39 as the evolution variable , although in the mathematically equivalent propagation equations for optical waveguides @xmath39 has to be identified with the propagation distance , @xmath40 ) . denoting @xmath41 , the matrices @xmath42 and @xmath26 in ( [ a1 ] ) take the form of @xmath43to find the parity matrix @xmath21 which renders the linear hamiltonian @xmath8symmetric , @xmath23=0 $ ] , we use the pseudo - hermiticity condition ( [ a2 - 4 ] ) and notice that @xmath44obviously , the following relations should hold : @xmath45=0 \nonumber \\ \mathcal{p}h_{l,1}\mathcal{p}=-h_{l,1 } & \qquad \longrightarrow \qquad & \left\ { \mathcal{p},h_{l,1}\right\ } = 0 . \label{a6}\end{aligned}\]]the first of these conditions together with @xmath22 , @xmath46 reduces the possible form of the parity transformation to one of the three types , @xmath47where @xmath48 are the usual pauli matrices . taking into account that @xmath49 , the anti - commutativity condition in eqs . ( [ a6 ] ) singles out the only possible parity matrix : @xmath50for the linear transformation , i.e. , the matrix interchanging @xmath51 and @xmath52 , as well as @xmath53 and @xmath54 . one then immediately checks that @xmath55hence , in contrast to the linear component @xmath42 , the nonlinear terms @xmath26 corresponding to eqs . ( [ zpzm1 ] ) are not @xmath8symmetric , in the usual matrix sense . rather , the symmetry properties of the nonlinear terms have to be considered in the context of the nonlinear schrdinger equation itself . acting with @xmath56 on eq . ( [ a1 ] ) we observe that @xmath57 , \notag \\ i\partial _ { t}(\mathcal{p}\mathbf{t}\mathbf{u } ) & = & h_{l}\mathcal{p}\mathbf{t}\mathbf{u}+h_{nl}(\mathcal{p}\mathbf{t}\mathbf{u})(\mathcal{p}\mathbf{t}\mathbf{u})\label{a13}.\end{aligned}\ ] ] hence , the full nlse system ( [ zpzm1 ] ) remains invariant if we define the @xmath58 transformation of the vectorial wave function obeying this system as follows : @xmath59 this is in full analogy to the condition of exact @xmath27symmetry for the corresponding linear schrdinger equation , nonlinearity matrices @xmath60 of more general type than that in ( [ nl-1 ] ) and ( [ nl-2 ] ) may be envisioned which may produce @xmath61symmetric solutions @xmath62 with less simple time dependence . a detailed analysis of such systems will be presented elsewhere . ] . but the condition of _ spontaneously broken _ @xmath28symmetry ( @xmath29=0 $ ] , @xmath31 ) is replaced by the condition of _ completely broken _ @xmath28symmetry ( @xmath31@xmath63@xmath64\neq 0 $ ] ) . in contrast to the present 2d plaquettes , which are mainly motivated by feasible experimental realizations , one can envision more sophisticated setups with @xmath65 . this will lead to a new type of partial ( or intermediate ) @xmath28symmetry ( to be considered elsewhere ) , which for solutions @xmath66 with broken @xmath28symmetry will keep the nonlinear term @xmath26 explicitly @xmath8symmetric ( @xmath67pseudo - hermitian ) in the matrix sense , but not @xmath28symmetric ( under inclusion of the explicit time reversal @xmath18 ) in the sense of the nlse system . for configurations ( b ) and ( c ) in fig . [ modes ] , @xmath68 and @xmath26 are still given by eqs . ( [ a3 ] ) and ( [ a5 ] ) , but with @xmath69respectively . hence , relations ( [ a6 ] ) are valid for both configurations ( b ) and ( c ) as well . using ( [ a7 ] ) in @xmath70 , we find a richer variety of parity operators @xmath21 than for configuration ( a ) . configuration ( b ) allows for @xmath71whereas configuration ( c ) may be associated with @xmath72for configuration ( d ) , we have @xmath73and simple computer algebra gives again two possible parity operators : @xmath74 in strong structural analogy to configuration ( b ) . one verifies that @xmath75 holds also for configurations ( b ) , ( c ) and ( d ) , hence all 2d plaquettes considered in the present paper are not @xmath28symmetric in the usual matrix sense . next , we turn to the eigenvalue problems of the linear setups associated with plaquettes ( a ) - ( d ) , i.e. , to solutions of the equation @xmath76from the corresponding characteristic polynomials , @xmath77 , @xmath78 = 0 , \notag \\ ( \mathrm{b}):\qquad \qquad & ( e^{2}+\gamma ^{2})\left [ e^{2}-(4k^{2}-\gamma ^{2})\right ] = 0 , \notag \\ ( \mathrm{c}):\qquad \qquad & e^{4}-2(2k^{2}-\gamma ^{2})e^{2}+\gamma ^{4}=0 , \notag \\ ( \mathrm{d}):\qquad \qquad & e(e^{2}+\gamma ^{2})\left [ e^{2}-(4k^{2}-\gamma ^{2})\right ] = 0,\end{aligned}\]]we find @xmath79obviously , the matrix hamiltonians @xmath42 for plaquettes ( a ) and ( d ) are not of full rank . for plaquette ( a ) we find @xmath80 , and @xmath42 has a two - dimensional kernel space , @xmath81 . for plaquette ( d ) we find @xmath82 and @xmath83 , where @xmath84 . moreover , we see that the spectrum for plaquette ( d ) , up to the additional eigenvalue @xmath85 , coincides with that for ( b ) . the different eigenvalues of the 4-node plaquettes displayed in eqs . ( [ a15 - 2 ] ) show that these plaquettes are also physically not equivalent . equivalence classes of nonlinear 4-node plaquettes with isospectral linear hamiltonians @xmath86 but different pairwise couplings have been considered , e.g. , in @xcite . for plaquettes ( a ) , ( b ) and ( d ) an exceptional point ( ep ) occurs at @xmath87 , being associated with a branching of the eigenvalue pair @xmath88 @xmath89 in the case of plaquette ( a ) , all four eigenvalues are involved in the branching at @xmath90 , where @xmath91 . via jordan decomposition ( e.g. , with the help of the corresponding linear algebra tool of mathematica ) we find that @xmath92i.e . , a spectral degeneration of the type @xmath93 in arnold s notation @xcite , or , in other words , a third - order ep with a single decoupled fourth mode . hence , plaquettes of type ( a ) may serve as an easily implementable testground for the experimental investigation of third - order eps ( see e.g. @xcite ) . for plaquettes ( b ) and ( d ) we have second - order eps at @xmath87 , similar as for plaquette ( c ) where a pair of second - order eps occurs at @xmath94 with @xmath95 , @xmath96 . from the eigenvalues in ( [ a15 - 2 ] ) we read off the @xmath27symmetry content of the four types of plaquettes . the sector of exact @xmath28symmetry ( i.e. , the sector with all eigenvalues purely real , @xmath97 ) corresponds to @xmath98i.e . , for plaquettes ( b ) and ( d ) the @xmath28symmetry is spontaneously broken as soon as the gain - loss coupling is switched on , namely for @xmath99 . in this section , we seek stationary solutions of the type @xmath100constructed over constant vectors @xmath101 . according to ( [ a13 ] ) and ( [ a13 - 2 ] ) such solutions will be @xmath61symmetric provided it holds @xmath102 for some @xmath103 . we will test these symmetry properties for the solutions to be obtained . we note that restricting the explicit analysis to stationary solutions of the type ( [ e ] ) we by construction exclude from this analysis @xmath61violating solutions with @xmath104 which are necessarily non - stationary . a useful technical tool to facilitate the explicit derivation of stationary solutions @xmath105 are conservation equations of the type @xmath106constructed from eq . ( [ a1 ] ) and its adjoint , where @xmath107 denotes an arbitrary constant matrix . the most simplest of them can be found via eqs . ( [ a2 - 4 ] ) , ( [ a3 ] ) and eq . ( [ a5 ] ) to be @xmath108 \mathbf{u}. \label{a18 - 2}\end{aligned}\]]for stationary equations @xmath109 the time - dependent phase factors @xmath110 cancel so that the left - hand - sides of these relations vanish , yielding simple algebraic constraints on the right - hand - sides . from eq . ( [ a18 - 2 ] ) we see that for stationary solutions the @xmath6 inner product quantum mechanics ( ptqm ) the @xmath7 inner product was introduced first by znojil in @xcite in 2001 . immediately afterwards , it was interpreted by japaridze as indefinite inner product @xcite in a krein space and generalized by mostafazadeh to the @xmath111metric in the context of pseudo - hermitian hamiltonians @xcite . finally , it was used by bender , brody and jones in 2002 to construct the positive definite @xmath112 inner product @xcite . for oligomer settings ( of plaquettes or other few site configurations ) , it can be employed , e.g. , to derive a simple algebraic constraint or as a criterion of the numerical accuracy of the evolutionary dynamics ( especially since the solutions rapidly acquire very large amplitudes when unstable , as will be seen below ) . it also turned out useful in @xcite . ] will remain conserved ( @xmath113const ) regardless of the violated @xmath114pseudo - hermiticity , @xmath115 , characteristic for of our specific nonlinear plaquette couplings ( see eq . ( [ nl-2 ] ) ) . subsequently , we first derive classes of stationary solutions @xmath105 explicitly . then , we analyze the stability of small perturbations over these stationary solutions by the linearization , via ansatz @xmath116 + o(\delta ^{2}),\qquad |\delta |\ll 1,\]]where @xmath117 is the small amplitude of the perturbation . exponents @xmath118 can be defined as wick - rotated eigenvalues from the corresponding @xmath119 perturbation matrix @xmath120 ( see , e.g. , @xcite for more details ) : @xmath121 where @xmath122\mathbf{u}\]]characterizes the stationary problem , and the elements of the matrix @xmath120 are evaluated at @xmath123 . linear stability is ensured for @xmath124 , whereas @xmath125 corresponds to growing and decaying modes , i.e. , exponential instabilities . substituting ansatz ( [ e ] ) for the stationary solutions in eqs . ( [ a1 ] ) , ( [ a18 - 1 ] ) and ( [ a18 - 2 ] ) we obtain the following algebraic equations : @xmath126@xmath127and @xmath128 \mathbf{u , } \notag \label{a31 } \\ 0 & = & \left ( |a|^{2}-|c|^{2}\right ) ( \bar{a}c-\bar{c}a)+\left ( equations can be analyzed via the madelung substitution ( i.e. , via amplitude - phase decomposition ) , @xmath129without loss of generality , we may fix @xmath130 . for arbitrary phase factors in ( [ aa ] ) , eqs . ( [ a30 ] ) and ( [ a31 ] ) are satisfied by @xmath131 and @xmath132 . using this condition in eq . ( [ zpzm2 ] ) and dividing each equation ( [ zpzm2 ] ) by the phase factor on its left - hand side , one obtains the imaginary parts of the resulting equations : @xmath133 = 2kb\sin \left ( \frac{\phi _ { b}+\phi _ { d}}{2}-\phi _ { a}\right ) \cos \left ( \frac{\phi _ { b}-\phi _ { d}}{2}\right ) , \notag \\ \gamma b & = & ka\left [ \sin ( \phi _ { a}-\phi _ { b})+\sin ( \phi _ { c}-\phi _ { b})\right ] = 2ka\sin \left ( \frac{\phi _ { a}+\phi _ { c}}{2}-\phi _ { b}\right ) \cos \left ( \frac{\phi _ { a}-\phi _ { c}}{2}\right ) , \notag \\ 0 & = & kb\left [ \sin ( \phi _ { b}-\phi _ { c})+\sin ( \phi _ { d}-\phi _ { c})\right ] = 2kb\sin \left ( \frac{\phi _ { b}+\phi _ { d}}{2}-\phi _ { c}\right ) \cos \left ( \frac{\phi _ { b}-\phi _ { d}}{2}\right ) , \notag \\ -\gamma b & = & ka\left [ \sin ( \phi _ { a}-\phi _ { d})+\sin ( \phi _ { c}-\phi _ { d})\right ] = 2ka\sin \left ( \frac{\phi _ { a}+\phi _ { c}}{2}-\phi _ { d}\right ) \cos \left ( \frac{\phi _ { a}-\phi _ { c}}{2}\right ) . \notag \\ & & \label{zpzm_add}\end{aligned}\]]for @xmath130 the first of these equations implies @xmath134 , hence either @xmath135 ( case 1 ) or @xmath136 ( case 2 ) . in case 1 , we conclude from the third equation that either @xmath137 and @xmath138 ( case 1a ) , or @xmath139 and @xmath140 is arbitrary ( case 1b ) . in case 2 the third equation is satisfied automatically . in all the three cases , the second and the fourth equation are compatible . they give @xmath141returning to the phase - factor divided equations ( [ zpzm2 ] ) and considering their real parts , we find @xmath142 + a^{3 } , \notag \label{a33 } \\ eb & = & ka\left [ \cos ( \phi _ { a}-\phi _ { b})+\cos ( \phi _ { c}-\phi _ { b})\right ] + b^{3 } , \notag \\ ea & = & kb\left [ \cos ( \phi _ { b}-\phi _ { c})+\cos ( \phi _ { d}-\phi _ { c})\right ] + a^{3 } , \notag \\ eb & = & ka\left [ \cos ( \phi _ { a}-\phi _ { d})+\cos ( \phi _ { c}-\phi _ { d})\right ] + b^{3}.\end{aligned}\]]the pairwise compatibility of the first and third , as well as of the second and fourth equations requires @xmath143for case 1a , these conditions are trivially satisfied , whereas for the remaining cases they lead to further restrictions : @xmath144 in this way the phase angles are fixed for all the three cases and we can turn to the amplitudes . the corresponding equation sets reduce to @xmath145 in the latter two cases ( 1b and 2 ) the amplitudes and phases completely decouple and we have @xmath146case 1a allows for a richer behavior . equating the terms @xmath147 in the upper two equations ( [ a36 ] ) leads to the constraint @xmath148which can be resolved by @xmath149 ( case 1aa ) as well as by @xmath150 ( case 1ab ) . the analysis of these two cases can be completed with the help of the relation @xmath151 from eq . ( [ a32 ] ) . as result we obtain the following set of stationary solutions : @xmath152 from eq . ( [ a31 ] ) , it can also be seen that either @xmath131 or if @xmath153 , then @xmath154 must be true . here , we use the information available so far to check the @xmath28symmetry content of the solutions ( [ a39 - 1aa ] ) - ( [ a39 - 2 ] ) explicitly . for stationary solutions @xmath155 , @xmath156 the @xmath28symmetry condition ( [ a13 - 2 ] ) implies @xmath157taking into account that @xmath16 acts as complex conjugation , we see from the explicit structure of @xmath158 in eq . ( a8 ) that a stationary solution is @xmath28symmetric if , with @xmath130 , it has @xmath138 and @xmath159 ( up to a common phase shift ) . additionally , the amplitudes have to coincide pairwise : @xmath131 , @xmath132 . for eqs . ( [ a39 - 1aa ] ) - ( [ a39 - 2 ] ) this means that all case-1 stationary solutions with @xmath138 are @xmath28symmetric in their present form . the case-2 mode becomes explicitly @xmath8symmetric after a global @xmath160 multiplication by a phase factor : @xmath161 ^{t } , \notag \\ \mathcal{p}\mathcal{t}\mathbf{v}_{0 } & = & \mathbf{v}_{0},\end{aligned}\]]where @xmath162 has to be chosen in eq . ( [ a39 - 2 ] ) . we note that this procedure is effectively equivalent to a redefinition of the original phase constraint : @xmath163 at the very beginning of the calculations in eq . ( [ aa ] ) . the linear stability analysis was performed numerically . subsequently we present corresponding graphical results . the plaquettes ( b ) - ( d ) can be analyzed in a similar way . for brevity s sake , in fig . [ figzpzm1 ] we present only the basic numerical results , by means of the following symbols : [ figzpzm1 ] * case 1aa with @xmath164 blue circles ; * case 1aa with @xmath165 red crosses ; * case 1ab green stars ; * case 2 black squares ; * case 1b is not depicted explicitly because it corresponds to point configurations without gain - loss ( @xmath166 ) and to exceptional point configurations @xmath90 . figure [ figzpzm1 ] presents the mode branches ( their amplitudes , phases , and also their stability ) over the gain - loss parameter @xmath2 , starting from the conservative system at @xmath166 . the same symbols are used in fig . [ figzpzm2 ] , which displays typical examples of the spectral plane @xmath167 for stability eigenvalues @xmath168 of the linearization ; recall that the modes are unstable if they give rise to @xmath169 . explicitly we observe the following behavior . * case 1aa with @xmath164 blue circlesaccording to fig . [ figzpzm2 ] , the present solution is stable . notice that , although featuring a phase profile , it can not be characterized as a vortex state ( the same is true for some other configurations carrying phase structure ) . interestingly , the relevant configuration is generically stable bearing two imaginary pairs of eigenvalues . * case 1aa with @xmath165 red crosses.obviously , this kind of solutions as well as the previous one exist up to the exceptional point @xmath90 of the @xmath0-symmetry breaking in the linear system , where the two branches collide and disappear ( leave the stationary regime and become nonstationary ) . as seen in fig . figzpzm2 , the present branch has two eigenvalue pairs which are purely imaginary for small @xmath2 , but become real ( rendering the configuration unstable ) at @xmath170 and then @xmath171 , respectively . ultimately , these pairs of unstable eigenvalues collide at the origin of the spectral plane with those of the previous branch ( blue circles ) . * case 1ab green stars.this stationary solution has a number of interesting features . firstly , it is the only one among the considered branches which has two unequal amplitudes . secondly , it exists past the critical point @xmath90 of the linear system , due to the effect of the nonlinearity ( the extension of the existence region for nonlinear modes was earlier found in 1d couplers @xcite and oligomers @xcite ) . furthermore , this branch has three non - zero pairs of stability eigenvalues , two of which form a quartet for small values of the gain - loss parameter , while the third is imaginary ( i.e. , the configuration is unstable due to the real parts of the eigenvalues within the quartet ) . at @xmath172 , the eigenvalues of the complex quartet collapse into two imaginary pairs , rendering the configuration stable , in a narrow parametric interval . at @xmath173 , the former imaginary pair becomes real , destabilizing the state again , while subsequent bifurcations of imaginary pairs into real ones occur at @xmath174 and @xmath175 ( at the latter point , all three non - zero pairs are real ) . shortly thereafter , two of these pairs collide at @xmath176 and rearrange into a complex quartet , which exists along with the real pair past that point . * case 2 black squares . in contrast to all other branches , this one is _ always _ unstable . one of the two nonzero eigenvalue pairs is always real ( while the other is always imaginary ) , as seen in fig . this branch also terminates at the exceptional point @xmath90 , as relation @xmath177 can not hold at @xmath178 . this branch collides with the two previous ones via a very degenerate bifurcation ( that could be dubbed a double saddle - center " bifurcation ) , which involves 3 branches instead of two as in the case of the generic saddle - center bifurcation , and two distinct eigenvalue pairs colliding at the origin of the spectral plane . by means of direct simulations , we have also examined the dynamics of the modes belonging to different branches in fig . [ stabzpzm ] . the stable blue - circle branch demonstrates only oscillations under perturbations . this implies that , despite the presence of the gain - loss profile , none of the perturbation eigenmodes grows in this case . nevertheless , the three other branches ultimately manifest their dynamical instability , which is observed through the growth of the amplitude at the gain - carrying site [ b , in fig . [ modes](a ) ] at the expense of the lossy site ( d ) . that is , the amplitude of the solution at the site with the gain grows , while the amplitude of the solution at the dissipation site loses all of its initial power . depending on the particular solution , passive sites ( the ones without gain or loss , such as a and c ) may be effectively driven by the gain ( as in the case of the black - square - branch , where the site a is eventually amplified due to the growth of the amplitude at site b ) or by the loss ( red - cross and green - star branches , where , eventually , the amplitudes at both a and c sites lose all of their optical power ) . we now turn to the generalized ( not exactly @xmath56-symmetric ) configurationsymmetry , spontaneously broken @xmath61symmetry and completely broken @xmath61symmetry see the discussion of eqs . ( [ a13 ] ) and ( [ a13 - 2 ] ) . ] featuring the alternation of the gain and loss along the plaquette in panel ( b ) of fig . [ modes ] . indeed , the absence of @xmath0-symmetry in this case is mirrored in the existence of imaginary eigenvalues in the linear problem of eqs . ( [ a15 - 2 ] ) , as soon as @xmath99 . the corresponding nonlinear solutions ( with @xmath104 ) are not covered by the stationary solution ansatz ( [ e ] ) . stationary solutions ( with @xmath179 ) solely belong to dynamical regimes below the concrete @xmath61thresholds . apart from the two @xmath61symmetry violating solutions , there should exist at least two stationary solutions which we construct in analogy to [ cf . ( [ zpzm2 ] ) ] from @xmath180 substituting the madelung representation ( [ aa ] ) and setting @xmath181 ( for illustration purposes , we focus here only on this simplest case ) , we obtain @xmath182further , fixing @xmath183 , eqs . ( [ phi1 ] ) and ( [ phi2 ] ) yield @xmath184obviously , the solution terminates at point @xmath90 . similar to what was done above , the continuation of this branch and typical examples of its linear stability are shown in figs . [ figpmpm2 ] and [ figpmpm3 ] , respectively . from here it is seen that the blue - circle branch , which has a complex quartet of eigenvalues , is always unstable . in fact , the gain - loss alternating configuration is generally found to be more prone to the instability . the red - cross branch is also unstable via a similar complex quartet of eigenvalues . this quartet , however , breaks into two real pairs for @xmath185 , and , eventually , the additional imaginary eigenvalue pair becomes real too at @xmath186 , making the solution highly unstable with three real eigenvalue pairs . the manifestation of the instability is shown in fig . [ stabpmpm ] , typically amounting to the growth of the amplitudes at one or more gain - carrying sites . [ figpmpm3 ] we now turn to the plaquette in fig . [ modes](c ) , which involves parallel rows of gain and loss . in this case , the stationary equations are @xmath187 in this case too , we focus on symmetric states of the form of @xmath181 [ see eq . ( [ aa ] ) ] , which gives rise to two solutions displayed in fig . figppmm , represented by the following analytical solutions : @xmath188 @xmath189the analysis demonstrates that the branch with the upper sign in eq . ( + -1 ) is always unstable ( through two real pairs of eigenvalues ) , as shown by blue circles in fig . [ figppmm ] . on the other hand , the branch denoted by the red crosses , which corresponds to the lower sign in eq . ( [ + -1 ] ) is stable up to @xmath190 , and then it gets unstable through a real eigenvalue pair . the black - squares branch with the upper sign in eq . ( + -3 ) is always stable , while the green - star branch with the lower sign in eq . ( [ + -3 ] ) is always unstable . at the linear-@xmath0-symmetry breaking point @xmath191 , we observe a strong degeneracy , since all the three pairs of eigenvalues for two of the branches ( in the case of the blue circles , two real and one imaginary , and in the case of red crosses one real and two imaginary ) collapse at the origin of the spectral plane . on the other hand , the black - squares branch is always stable with three imaginary eigenvalue pairs , while the green - star branch has two imaginary and one real pair of eigenvalues . between the latter two , there is again a collision of a pair at the origin at the critical condition , @xmath191 . direct simulations , presented for @xmath192 in fig . [ stabppmm ] , demonstrate the stability of the lower - sign black - squares branch , while the instability of the waveform associated with the blue circles and the green stars leads to the growth and decay of the amplitudes at the sites carrying , respectively , the gain and loss . notice that at the parameter values considered here , the red - cross branch is also dynamically stable as shown in the top right panel of fig . [ stabppmm ] . [ figppmm ] lastly , motivated by the existence of known cross "- shaped discrete - vortex modes in 2d conservative lattices , in addition to the fundamental discrete solitons @xcite , we have also examined the five - site configuration , in which the central site does not carry any gain or loss , while the other four feature a @xmath0-balanced distribution of the gain and loss , as shown in panel ( d ) of fig . [ modes ] . seeking for stationary states with propagation constant , @xmath193 [ instead of @xmath194 in eq . ( [ e ] ) , as in this case we reserve label @xmath194 for one of the sites of the 5-site plaquette in fig . [ modes](d ) ] , we get : @xmath195similarly as before , we use the madelung decomposition @xmath196 , cf . ( [ aa ] ) , and focus on the simplest symmetric solutions with @xmath197 . without the loss of generality , we set @xmath138 , reducing the equations to @xmath198 we report here numerical results for parameters @xmath199 ( smaller @xmath193 yields similar results but with fewer solution branches ) . we have identified five different solutions in this case , see figs . [ figpmzpm1 ] and figpmzpm2 for the representation of the continuation of the different branches , and for typical examples of their stability ( the latter is shown for @xmath200 , @xmath201 and @xmath202 ) . there are two branches ( green stars and black squares ) that only exist at @xmath203 colliding and terminating at that point . one of them has three real eigenvalue pairs and one imaginary pair , while the other branch has two real and two imaginary pairs . two real pairs and one imaginary pair of green stars collide with two real pairs and one imaginary pair of black squares , respectively , while the final pairs of the two branches ( one imaginary for the green stars and one real for the black squares ) collide at the origin of the spectral plane . these collisions take place at @xmath204 , accounting for the saddle - center bifurcation at the point where those two branches terminate . on the other hand , there exist two more branches ( red crosses and magenta diamonds in fig . [ figpmzpm1 ] ) , which collide at @xmath205 . one of these branches ( the less unstable one , represented by magenta diamonds ) bears only an instability induced by an eigenvalue quartet , while the highly unstable branch depicted by the red crosses has four real pairs ( two of which collide on the real axis and become complex at @xmath206 ) . last but not least , the blue circles branch does not terminate at @xmath207 , but continues to larger values of the gain - loss parameter , @xmath208 . it is also unstable ( as the one represented by the magenta diamonds ) due to a complex quartet of eigenvalues . the dynamics of the solutions belonging to these branches is shown in fig . [ stabpmzpm ] . for the branches depicted by black squares and green stars ( recall that they disappear through the collision and the first saddle - center bifurcation at @xmath204 ) , the perturbed evolution is fairly simple : the amplitudes grow at the gain - carrying sites and decay at the lossy ones , while the central passive site ( c ) stays almost at zero amplitude . for the other branches , the amplitudes also grow at the two gain - carrying sites and decay at the lossy elements , while the passive site may be drawn to either the growth or decay . [ figpmzpm1 ] [ figpmzpm2 ] in the present work , we have proposed generalizations of the one - dimensional @xmath0-symmetric nonlinear oligomers into two - dimensional plaquettes , which may be subsequently used as fundamental building blocks for the construction of @xmath0-symmetric two - dimensional lattices . in this context , we have introduced four basic types of plaquettes , three of which in the form of four - site squares . the final one was in the form of the five - site cross , motivated by earlier works on cross - shaped ( alias rhombic or site - centered ) vortex solitons in the discrete nonlinear schrdinger equation . our analysis was restricted to modes which could be found in the analytical form , while their stability against small perturbations was analyzed by means of numerical methods . even within the framework of this restriction , many effects have been found , starting from the existence of solution branches that terminate at the critical points of the respective linear @xmath0-symmetric systems e.g. , in the settings corresponding to plaquettes ( a ) and ( c ) in fig . [ modes ] . the bifurcation responsible for the termination of the pair of branches may take a complex degenerate form [ such as the one in the case of setting ( a ) ] . other branches were found too , that continue to exist , due to the nonlinearity , past the critical points of the underlying linear systems . in addition , we have identified cases [ like the gain - loss alternating pattern ( b ) or the cross plaquette of type ( d ) ] when the @xmath0 symmetry is broken immediately after the introduction of the gain - loss pattern . the spectral stability of the different configurations was examined . most frequently , the stationary modes are unstable , although stable branches were found too [ e.g. , in settings ( a ) and ( c ) ] . we have also studied the perturbed dynamics of the modes . the evolution of unstable ones typically leads to the growth of the amplitudes at the gain - carrying sites and decay at the lossy ones . it was interesting to observe that the passive sites , without gain or loss , might be tipped towards growth or decay , depending on the particular solution ( and possibly on specific initial conditions ) . the next relevant step of the analysis may be to search for more sophisticated stationary modes ( that plausibly can not be found in an analytical form ) , produced by the _ symmetry breaking _ of the simplest modes considered in this work , cf . ref . the difference of such modes from the @xmath0-symmetric ones considered in the present work is the fact that modes with the unbroken symmetry form a continuous family of solutions , with energy @xmath194 depending on the solution s amplitude , see eq . ( [ e ] ) . this feature , which is generic to conservative nonlinear systems , is shared by @xmath0-symmetric ones , due to the automatic " balance between the separated gain and loss . on the other hand , the breaking of the symmetry gives rise to the typical behavior of systems with competing , but not explicitly balanced , gain and loss , which generate a single or several _ attractors _ , i.e. , _ isolated _ solutions with a single or several values of the energy , rather than a continuous family . a paradigmatic example of the difference between continuous families of solutions in conservative models and isolated attractors in their ( weakly ) dissipative counterparts is the transition from the continuous family of solitons in the usual nlse to a pair of isolated soliton solutions , one of which is an attractor ( and the other is an unstable solution playing the role of the separatrix between attraction basins , the stable soliton and the stable zero solution ) in the complex ginzburg - landau equation , produced by the addition of the cubic - quintic combination of small dissipation and gain terms to the nlse @xcite . as concerns the systems considered in the present work , in the context of the breaking of the @xmath0 symmetry it may also be relevant to introduce a more general nonlinearity , which includes @xmath0-balanced cubic gain and loss terms , in addition to their linear counterparts ( cf . @xcite and @xcite ) . nevertheless , it should also be noted that the issue of potential existence of isolated solutions versus branches of solutions in @xmath0-symmetric systems is already starting to be addressed in the relevant literature ( including in plaquette - type configurations ) , as in the very recent work of @xcite . moreover , the present work may pave the way to further considerations of two - dimensional @xmath0-symmetric lattice systems , and even three - dimensional ones . in this context , the natural generalization is to construct periodic two - dimensional lattices of the building blocks presented here , and to identify counterparts of the modes reported here in the infinite lattices , along with new modes which may exist in that case . on the other hand , in the three - dimensional realm , the first step that needs to be completed would consist of the examination of a @xmath0-symmetric cube composed of eight sites , and the nonlinear modes that it can support . this , in turn , may be a preamble towards constructing full three - dimensional @xmath0-symmetric lattices . these topics are under present consideration and will be reported elsewhere . ug thanks holger cartarius and eva - maria graefe for useful discussions . pgk gratefully acknowledges support from the national science foundation under grant dms-0806762 and cmmi-1000337 , as well as from the alexander von humboldt foundation and the alexander s. onassis public benefit foundation . pgk and bam also acknowledge support from the binational science foundation under grant 2010239 . 99 c. m. bender and s. boettcher , phys . rev . lett . * 80 * , 5243 ( 1998 ) ; c. m. bender , s. boettcher and p. n. meisinger , j. math . phys . * 40 * , 2201 ( 1999 ) . z. h. musslimani , k. g. makris , r. el - ganainy and d. n. christodoulides , phys . lett . * 100 * , 030402 ( 2008 ) ; k. g. makris , r. el - ganainy , d. n. christodoulides and z. h. musslimani , phys . a * 81 * , 063807 ( 2010 ) . b. a. malomed and h. g. winful , phys . rev . e * 53 * , 5365 ( 1996 ) ; h. sakaguchi and b. a. malomed , physica d * 147 * , 273 ( 2000 ) ; w. j. firth and p. v. paulau , eur . j. d * 59 * , 13 ( 2010 ) ; p. v. paulau , d. gomila , p. colet , n. a. loiko , n. n. rosanov , t. ackemann , and w. j. firth , opt . express * 18 * , 8859 ( 2010 ) ; a. marini , d. v. skryabin , and b. a. malomed , _ ibid_. * 19 * , 6616 ( 2011 ) ; p. v. paulau , d. gomila , p. colet , b. a. malomed , and w. j. firth , phys . rev . e * 84 * , 036213 ( 2011 ) . m. znojil , rendic . circ . mat . palermo , ser . ii , suppl . * 72 * ( 2004 ) , 211 - 218 , math - ph/0104012 . g. s. japaridze , j. phys . a : math . theor . * 35 * , 1709 - 1718 ( 2002 ) , quant - ph/0104077 . a. mostafazadeh , j. math . phys . * 43 * , 205 - 214 ( 2002 ) , math - ph/0107001 . c. m. bender , d. c. brody , and h. f. jones , phys . lett . * 89 * , 270401 ( 2002 ) , quant - ph/0208076 .
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we introduce four basic two - dimensional ( 2d ) plaquette configurations with onsite cubic nonlinearities , which may be used as building blocks for 2d @xmath0-symmetric lattices . for each configuration
, we develop a dynamical model and examine its @xmath1symmetry .
the corresponding nonlinear modes are analyzed starting from the hamiltonian limit , with zero value of the gain - loss coefficient , @xmath2 .
once the relevant waveforms have been identified ( chiefly , in an analytical form ) , their stability is examined by means of linearization in the vicinity of stationary points .
this reveals diverse and , occasionally , fairly complex bifurcations .
the evolution of unstable modes is explored by means of direct simulations . in particular ,
stable localized modes are found in these systems , although the majority of identified solutions is unstable .
| 14,120 | 212 |
multiple - input , multiple - output(mimo ) wireless transmission systems have been intensively studied during the last decade . the alamouti code @xcite for two transmit antennas is a novel scheme for mimo transmission , which , due to its orthogonality properties , allows a low complexity maximum - likelihood ( ml ) decoder . this scheme led to the generalization of stbcs from orthogonal designs @xcite . such codes allow the transmitted symbols to be decoupled from one another and single - symbol ml decoding is achieved over _ quasi static _ rayleigh fading channels . even though these codes achieve the maximum diversity gain for a given number of transmit and receive antennas and for any arbitrary complex constellations , unfortunately , these codes are not @xmath2 , where , by a @xmath2 code , we mean a code that transmits at a rate of @xmath3 complex symbols per channel use for an @xmath4 transmit antenna , @xmath5 receive antenna system . the golden code @xcite is a full - rate , full - diversity code and has a decoding complexity of the order of @xmath6 for arbitrary constellations of size @xmath7 the codes in @xcite and the trace - orthogonal cyclotomic code in @xcite also match the golden code . with reduction in the decoding complexity being the prime objective , two new full - rate , full - diversity codes have recently been discovered : the first code was independently discovered by hottinen , tirkkonen and wichman @xcite and by paredes , gershman and alkhansari @xcite , which we call the htw - pga code and the second , which we call the sezginer - sari code , was reported in @xcite by sezginer and sari . both these codes enable simplified decoding , achieving a complexity of the order of @xmath8 . the first code is also shown to have the non - vanishing determinant property @xcite . however , these two codes have lesser coding gain compared to the golden code . a detailed discussion of these codes has been made in @xcite , wherein a comparison of the codeword error rate ( cer ) performance reveals that the golden code has the best performance . in this paper , we propose a new full - rate , full - diversity stbc for @xmath9 mimo transmission , which has low decoding complexity . the contributions of this paper may be summarized ( see table [ table1 ] also ) as follows : * the proposed code has the same coding gain as that of the golden code ( and hence of that in @xcite and the trace - orthonormal cyclotomic code ) for any qam constellation ( by a qam constellation we mean any finite subset of the integer lattice ) and larger coding gain than those of the htw - pga code and the sezginer - sari code . * compared with the golden code and the codes in @xcite and @xcite , the proposed code has lesser decoding complexity for all complex constellations except for square qam constellations in which case the complexity is the same . compared to the htw - pga code and the sezginer - sari codes , the proposed code has the same decoding complexity for all non - rectangular qam [ fig [ fig2 ] ] constellations . * the proposed code has the non - vanishing determinant property for qam constellations and hence is diversity - multiplexing gain ( dmg ) tradeoff optimal . the remaining content of the paper is organized as follows : in section [ sec2 ] , the system model and the code design criteria are reviewed along with some basic definitions . the proposed stbc is described in section [ sec3 ] and its non - vanishing determinant property is shown in section [ sec4 ] . in section [ sec5 ] the ml decoding complexity of the proposed code is discussed and the scheme to decode it using sphere decoding is discussed in section [ sec6 ] . in section [ sec7 ] , simulation results are presented to show the performance of the proposed code as well as to compare with few other known codes . concluding remarks constitute section [ sec8 ] . _ notations : _ for a complex matrix @xmath10 the matrices @xmath11 , @xmath12 and @xmath13 $ ] denote the transpose , hermitian and determinant of @xmath10 respectively . for a complex number @xmath14 @xmath15 and @xmath16 denote the real and imaginary part of @xmath14 respectively . also , @xmath17 represents @xmath18 and the set of all integers , all real and complex numbers are denoted by @xmath19 @xmath20 and @xmath21 respectively . the frobenius norm and the trace are denoted by @xmath22 and @xmath23 $ ] respectively . the columnwise stacking operation on @xmath24 is denoted by @xmath25 the kronecker product is denoted by @xmath26 and @xmath27 denotes the @xmath28 identity matrix . given a complex vector @xmath29^t,$ ] @xmath30 is defined as @xmath31^t\ ] ] and for a complex number @xmath32 , the @xmath33 operator is defined by @xmath34.\ ] ] the @xmath33 operator can be extended to a complex @xmath35 matrix by applying it to all the entries of it . a finite set of complex matrices is a stbc . a @xmath35 linear stbc is obtained starting from an @xmath35 matrix consisting of arbitrary linear combinations of @xmath36 complex variables and their conjugates , and letting the variables take values from complex constellations . the rate of such a code is @xmath37 complex symbols per channel use . we consider rayleigh quasi - static flat fading mimo channel with full channel state information ( csi ) at the receiver but not at the transmitter . for @xmath9 mimo transmission , we have @xmath38 where @xmath39 is the codeword matrix , transmitted over 2 channel uses , @xmath40 is a complex white gaussian noise matrix with i.i.d entries , i.e. , @xmath41 and @xmath42 is the channel matrix with the entries assumed to be i.i.d circularly symmetric gaussian random variables @xmath43 . @xmath44 is the received matrix . [ def1]@xmath45 if there are @xmath36 independent information symbols in the codeword which are transmitted over @xmath46 channel uses , then , for an @xmath47 mimo system , the code rate is defined as @xmath48 symbols per channel use . if @xmath49 , where @xmath50 , then the stbc is said to have @xmath51 @xmath52 . considering ml decoding , the decoding metric that is to be minimized over all possible values of codewords @xmath53 is given by @xmath54 [ def2]@xmath55 the ml decoding complexity is given by the minimum number of symbols that need to be jointly decoded in minimizing the decoding metric . this can never be greater than @xmath36 , in which case , the decoding complexity is said to be of the order of @xmath56 . if the decoding complexity is lesser than @xmath56 , the code is said to admit simplified decoding . [ def3]@xmath57 for any stbc @xmath53 that encodes @xmath36 information symbols , the @xmath58 matrix @xmath59 is defined by the following equation @xmath60 where @xmath61^t$ ] is the information symbol vector the code design criteria @xcite are : ( i ) @xmath62 @xmath63 to achieve maximum diversity , the codeword difference matrix @xmath64 must be full rank for all possible pairs of codewords and the diversity gain is given by @xmath65 ( ii ) @xmath66 @xmath63 for a full ranked stbc , the minimum determinant @xmath67 , defined as @xmath68\ ] ] should be maximized . the coding gain is given by @xmath69 , with @xmath4 being the number of transmit antennas . for the @xmath9 mimo system , the target is to design a code that is full - rate , i.e transmits 2 complex symbols per channel use , has full - diversity , maximum coding gain and allows low ml decoding complexity . in this section , we present our stbc for @xmath1 mimo system . the design is based on the class of codes called co - ordinate interleaved orthogonal designs ( ciods ) , which was studied in @xcite in connection with the general class of single - symbol decodable codes and , specifically for 2 transmit antennas , is as follows . the ciod for @xmath0 transmit antennas @xcite is + @xmath70\ ] ] where @xmath71 are the information symbols and @xmath72 and @xmath73 are the in - phase ( real ) and quadrature - phase ( imaginary ) components of @xmath74 respectively . notice that in order to make the above stbc full rank , the signal constellation @xmath75 from which the symbols @xmath76 are chosen should be such that the real part ( imaginary part , resp . ) of any signal point in @xmath75 is not equal to the real part ( imaginary part , resp . ) of any other signal point in @xmath75 @xcite . so if qam constellations are chosen , they have to be rotated . the optimum angle of rotation has been found in @xcite to be @xmath77 degrees and this maximizes the diversity and coding gain . we denote this angle by @xmath78 the proposed @xmath79 stbc @xmath80 is given by @xmath81 where * the four symbols @xmath82 and @xmath83 , where @xmath75 is a @xmath84 degrees rotated version of a regular qam signal set , denoted by @xmath85 which is a finite subset of the integer lattice , and @xmath86 to be precise , @xmath87 * @xmath88 is a permutation matrix designed to make the stbc full rate and is given by @xmath89.$ ] * the choice of @xmath90 in the above expression should be such that the diversity and coding gain are maximized . a computer search was done for @xmath90 in the range @xmath91 $ ] . the optimum value of @xmath90 was found out to be @xmath92 . explicitly , our code matrix is @xmath93 @xmath94 \\\ ] ] the minimum determinant for our code when the symbols are chosen from qam constellations is @xmath95 , the same as that of the golden code , which will be proved in the next section . the generator matrix for our stbc , corresponding to the symbols @xmath76 , is as follows : @xmath96\ ] ] it is easy to see that this generator matrix is orthonormal . in @xcite , it was shown that a necessary and sufficient condition for an stbc to be _ information lossless _ is that its generator matrix should be unitary . hence , our stbc has the _ information losslessness _ property . in this section it is shown that the proposed code has the non - vanishing determinant ( nvd ) property @xcite , which in conjunction with full - rateness means that our code is dmg tradeoff optimal @xcite . the determinant of the codeword matrix @xmath53 can be written as @xmath97.\ ] ] using @xmath98 and @xmath99 in the equation above , we get , @xmath100\\ & = & \big((s_1+s_2)+(s_1-s_2)^*\big)\big((s_1+s_2)-(s_1-s_2)^*\big ) { } \nonumber\\ & & { } -j[\big((s_3+s_4)+(s_3-s_4)^*\big)\big((s_3+s_4)-(s_3-s_4)^*\big)].\end{aligned}\ ] ] since @xmath101 , with @xmath102 , @xmath103 , a subset of @xmath104 $ ] , defining @xmath105 , @xmath106 , @xmath107 and @xmath108 , with @xmath109 and @xmath110 $ ] , we get @xmath111\\ & = & e^{j2\theta_g}a^2-e^{-j2\theta_g}b^2 - j[e^{j2\theta_g}c^2-e^{-j2\theta_g}d^2].\end{aligned}\ ] ] since @xmath112 , we get @xmath113 for the determinant of @xmath53 to be 0 , we must have @xmath114 the above can be written as @xmath115 where @xmath116 and clearly @xmath117 $ ] . it has been shown in @xcite that holds only when @xmath118 , i.e. , only when @xmath119 . this means that the determinant of the codeword difference matrix is 0 only when the codeword difference matrix is itself 0 . so , for any distinct pair of codewords , the codeword difference matrix is always full rank for any constellation which is a subset of @xmath104 $ ] . also , the minimum value of the modulus of r.h.s of can be seen to be @xmath120 . so , @xmath121 . in particular , when the constellation chosen is the standard qam constellation , the difference between any two signal points is a multiple of 2 . hence , for such constellations , @xmath122 , where @xmath53 and @xmath123 are distinct codewords . the minimum determinant is consequently 16/5 . this means that the proposed codes has the non - vanishing determinant ( nvd ) property @xcite . in @xcite , it was shown that full - rate codes which satisfy the non - vanishing determinant property achieve the optimal dmg tradeoff . so , our proposed stbc is dmg tradeoff optimal . in this section , it is shows that sphere decoding can be used to achieve the decoding complexity of @xmath8 . it can be shown that can be written as @xmath147 where @xmath148 is given by @xmath149 with @xmath150 being the generator matrix as in and @xmath151^t\ ] ] with @xmath152 drawn from @xmath75 , which is a rotation of the regular qam constellation @xmath85 . let @xmath153^t$ ] then , @xmath154 where @xmath155 is @xmath156 $ ] with @xmath157 being a rotation matrix and is defined as follows @xmath158.\ ] ] so , can be written as @xmath159 where @xmath160 . using this equivalent model , the ml decoding metric can be written as @xmath161 on obtaining the qr decomposition of @xmath162 , we get @xmath163= @xmath164 , where @xmath165 is an orthonormal matrix and @xmath166 is an upper triangular matrix . the ml decoding metric now can be written as @xmath167 if @xmath168 $ ] , where @xmath169 are column vectors , then @xmath170 and @xmath171 have the general form obtained by @xmath172 process as shown below @xmath173\ ] ] where @xmath174 are column vectors , and @xmath175\ ] ] where @xmath176 , @xmath177 @xmath178 @xmath179 it can be shown by direct computation that @xmath171 has the following structure @xmath180\ ] ] where @xmath181 stands for a possibly non - zero entry . the structure of the matrix @xmath182 allows us to perform a 4 dimensional real sphere decoding ( sd ) @xcite to find the partial vector @xmath183^t$ ] and hence obtain the symbols @xmath124 and @xmath125 . having found these , @xmath126 and @xmath127 can be decoded independently . observe that the real and imaginary parts of symbol @xmath126 are entangled with one another because of constellation rotation but are independent of the real and imaginary parts of @xmath127 when @xmath124 and @xmath125 are conditionally given . having found the partial vector @xmath184^t$ ] , we proceed to find the rest of the symbols as follows . we do two parallel 2 dimensional real search to decode the symbols @xmath126 and @xmath127 . so , overall , the worst case decoding complexity of the proposed stbc is 2@xmath8 . this is due to the fact that 1 . a 4 dimensional real sd requires @xmath185 metric computations in the worst possible case . two parallel 2 dimensional real sd require @xmath186 metric computations in the worst case . this decoding complexity is the same as that achieved by the htw - pga code and the sezginer - sari code . though it has not been mentioned anywhere to the best of our knowledge , the ml decoding complexity of the golden code , dayal - varanasi code and the trace - orthogonal cyclotomic code is also @xmath146 for square qam constellations . this follows from the structure of the @xmath171 matrices for these codes which are counterparts of the one in . the @xmath171 matrices of these codes are similar in structure and as shown below : @xmath187\ ] ] table [ table1 ] presents the comparison of the known full - rate , full - diversity @xmath1 codes in terms of their ml decoding complexity and the coding gain . fig [ 4qam ] shows the codeword error performance plots for the golden code , the proposed stbc and the htw - pga code for the 4-qam constellation . the performance of the proposed code is the same as that of the golden code . the htw - pga code performs slightly worse due to its lower coding gain . fig [ 16qam ] , which is a plot of the cer performance for 16-qam , also highlights these aspects . table [ table1 ] gives a comparison between the well known full - rate , full - diversity codes for @xmath9 mimo . [ cols="^,^,^,^,^ " , ] in this paper , we have presented a full - rate stbc for @xmath9 mimo systems which matches the best known codes for such systems in terms of error performance , while at the same time , enjoys simplified - decoding complexity that the codes presented in @xcite and @xcite do . recently , a rate-1 stbc , based on scaled repetition and rotation of the alamouti code , was proposed @xcite . this code was shown to have a hard - decision performance which was only slightly worse than that of the golden code for a spectral efficiency of @xmath188 , but the complexity was significantly lower . this work was partly supported by the drdo - iisc program on advanced research in mathematical engineering . j. c. belfiore , g. rekaya and e. viterbo , `` the golden code : a @xmath9 full rate space - time code with non - vanishing determinants , '' _ ieee trans . inf . theory _ , 51 , no . 4 , pp . 1432 - 1436 , april 2005 . j. paredes , a.b . gershman and m. gharavi - alkhansari , a @xmath9 space - time code with non - vanishing determinants and fast maximum likelihood decoding , " in proc _ ieee international conference on acoustics , speech and signal processing(icassp 2007 ) , _ vol . 2 , pp.877 - 880 , april 2007 . s. sezginer and h. sari , `` a full rate full - diversity @xmath9 space - time code for mobile wimax systems , '' in proc . _ ieee international conference on signal processing and communications _ , dubai , july 2007 . p. elia , k. r. kumar , s. a. pawar , p. v. kumar and h. lu , explicit construction of space - time block codes : achieving the diversity - multiplexing gain tradeoff , _ ieee trans . inf . theory _ , 52 , pp . 3869 - 3884 , sept . 2006 . v.tarokh , n.seshadri and a.r calderbank,"space time codes for high date rate wireless communication : performance criterion and code construction , _ ieee trans . inf . theory _ , 744 - 765 , 1998 . h. yao and g. w. wornell , `` achieving the full mimo diversity - multiplexing frontier with rotation - based space - time codes , '' in _ proc . allerton conf . on comm . control and comput . , _ monticello , il , oct . 2003 .
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this paper presents a low - ml - decoding - complexity , full - rate , full - diversity space - time block code ( stbc ) for a @xmath0 transmit antenna , @xmath0 receive antenna multiple - input multiple - output ( mimo ) system , with coding gain equal to that of the best and well known golden code for any qam constellation .
recently , two codes have been proposed ( by paredes , gershman and alkhansari and by sezginer and sari ) , which enjoy a lower decoding complexity relative to the golden code , but have lesser coding gain .
the @xmath1 stbc presented in this paper has lesser decoding complexity for non - square qam constellations , compared with that of the golden code , while having the same decoding complexity for square qam constellations .
compared with the paredes - gershman - alkhansari and sezginer - sari codes , the proposed code has the same decoding complexity for non - rectangular qam constellations .
simulation results , which compare the codeword error rate ( cer ) performance , are presented .
| 5,753 | 303 |
the recent enormous improvement of our knowledge of the neutrino oscillation parameters suggests a detailed investigation of the current constraints on the neutrino mass matrix . most of these constraints depend on the assumed nature of neutrinos ( dirac or majorana ) . the structure of this paper is as follows . after a brief general discussion of the light - neutrino mass matrix in section [ section - neutrino_mass_matrix ] , we will investigate the implications of the currently available data on the majorana neutrino mass matrix in section [ section - majorana ] . in section [ section - dirac ] we will discuss constraints on the neutrino mass matrix in the dirac case . finally we will conclude in section [ section - conclusions ] . in this paper we assume that there are exactly three light neutrino mass eigenstates with masses smaller than @xmath0 , _ i.e. _ we assume that there are no light sterile neutrinos . by the term `` neutrino mass matrix '' we thus always mean the @xmath1 mass matrix of the three light neutrinos . if neutrinos are _ majorana particles _ , we assume that there is a ( possibly effective ) mass term @xmath2 where @xmath3 is a complex symmetric @xmath1-matrix . such a mass term directly arises from the type - ii seesaw mechanism and can be effectively generated via the seesaw mechanisms of type i and iii . if neutrinos are dirac particles , _ i.e. _ if the total lepton number is conserved , we assume the existence of three right - handed neutrino fields @xmath4 leading to the mass term @xmath5 where @xmath6 is an arbitrary complex @xmath1-matrix . before we can discuss any constraints on the neutrino mass matrix , we have to specify a basis in flavour space . in models involving flavour symmetries , the chosen matrix representations of the flavour symmetry group specify the basis . since we will at this point not assume any flavour symmetries in the lepton sector , we are free to choose a basis . for simplicity we will always choose a basis in which the charged - lepton mass matrix is given by @xmath7 in the basis specified by equation ( [ mldiag ] ) the majorana neutrino mass matrix has the form @xmath8 where @xmath9 is the lepton mixing matrix and the @xmath10 ( @xmath11 ) are the masses of the three light neutrinos . as any unitary @xmath1-matrix , @xmath9 can be parameterized by six phases and three mixing angles . we will use the parameterization @xmath12 with @xmath13 the phases @xmath14 and @xmath15 are unphysical since they may be eliminated by a suitable redefinition of the charged - lepton fields . on the contrary , @xmath16 and @xmath17 are physical in the case of majorana neutrinos and are therefore referred to as the majorana phases . @xmath18 denotes the well - known unitary matrix @xmath19 where @xmath20 and @xmath21 are the sines and cosines of the three mixing angles , respectively . the phase @xmath22 is responsible for a possible cp violation in neutrino oscillations ( also in the dirac case ) and is therefore frequently referred to as the dirac cp phase . the fact that the neutrino masses are the _ singular values _ of @xmath3 allows to derive a generic upper bound on the absolute values @xmath23 . from linear algebra it is known that the absolute value of an element of a matrix is smaller or equal its largest singular value . for the neutrino mass matrix this implies @xcite @xmath24 since this bound is valid for _ any _ matrix , it holds also for dirac neutrinos . the strongest bounds on the absolute neutrino mass scale come from cosmology , where the sum of the masses of the light neutrinos is usually constrained to be at most of the order @xmath25see _ e.g. _ the list of upper bounds in @xcite . from this we deduce the approximate upper bound @xmath26 leading to @xmath27 in @xcite also an analytical lower bound on the @xmath23 is provided . defining @xmath28 one can show that @xmath29 note that this lower bound is independent of the majorana phases @xmath16 and @xmath17 . unlike the generic upper bound discussed before , the lower bound ( [ lowerbound ] ) is valid only for majorana neutrinos . numerically evaluating this lower bound using the results of the global fits of oscillation data of @xcite only for two matrix elements leads to non - trivial lower bounds . the lower bounds in units of ev for these matrix elements are given by @xcite : [ cols="<,<,<,^,^,^ " , ] all of these five correlations may be subsumed as `` if one matrix element is small , the other one must be large . '' an example for such a correlation plot can be found in figure [ fig : m11-m33normal ] . in the case of an inverted neutrino mass spectrum , there are no correlations manifest at the @xmath30-level . it is important to note that while at the @xmath30-level the correlation plots based on the global fits of @xcite and @xcite agree , this is not true at the @xmath31-level for further details see @xcite . in analogy to the majorana case , we will study the @xmath1 dirac neutrino mass matrix @xmath6 in the basis where the charged - lepton mass matrix is diagonal see equation ( [ mldiag ] ) . in this basis @xmath6 takes the form @xmath32 where @xmath33 is a unitary @xmath1-matrix . @xmath33 can be eliminated by considering the matrix @xmath34 since all observables accessible by current experimental scrutiny are contained in @xmath35 , all matrices @xmath6 leading to the same @xmath35 are _ indistinguishable _ from the experimental point of view . therefore , the nine parameters of @xmath36 have to be treated as _ free _ parameters . consequently , in stark contrast to the majorana case , in the dirac case the neutrino mass matrix has at least nine free parameters ( even if the mixing matrix and the neutrino masses are known ) . this freedom of choosing @xmath33 has important consequences for the analysis of @xmath6 . obviously it is much harder to put constraints on the elements of @xmath6 than in the majorana case . the freedom of choosing @xmath33 even allows to set several elements of @xmath6 to zero without changing the physical predictions . this directly follows from the fact that every matrix can be decomposed into a product of a unitary matrix and an upper triangular matrix . thus there is a choice of @xmath33 such that @xmath37 similarly , by multiplication of @xmath6 by one of the six @xmath1 permutation matrices generated by @xmath38 from the left , one can arbitrarily permute the rows of @xmath6 without changing any physical predictions @xcite . however , it is most important to note that the freedom of choosing @xmath33 holds only as long as we do not impose a symmetry in the lepton sector . namely , a flavour symmetry which acts non - trivially in the neutrino sector imposes constraints on the form of the neutrino mass matrix @xmath6 , _ i.e. _ not only on the neutrino masses and @xmath9 but also on @xmath33 . consequently , a choice of @xmath33 , _ e.g. _ such that @xmath6 is upper triangular , will in general be incompatible with the flavour symmetry . nevertheless , if we want to set bounds on the elements of @xmath6 ( without introducing flavour symmetries ) , we indeed have the freedom of arbitrarily choosing the matrix @xmath33 . therefore , examining bounds and correlations of the elements of @xmath6 is much less elucidating than in the majorana case . that said , studies of @xmath6 such as the question for the allowed cases of texture zeros in @xmath6 are still of great interest . in the following we will shortly comment on the allowed cases of texture zeros in @xmath6 under the assumption that @xmath39 is diagonal . a detailed analysis has been done by hagedorn and rodejohann in @xcite , which provides a classification of all possible texture zeros in this framework . we repeated the analysis of @xcite of the allowed cases of five , four and three texture zeros are investigated , the result being that all these cases are allowed and do not show any relations among the observables . ] in @xmath6 based on the global fit results of @xcite . our numerical results are in perfect agreement with the analysis of @xcite . however , there are some previously allowed cases of texture zeros which can be excluded due to the new data , namely precisely those which lead to a vanishing or too small value ( @xmath40 ) of @xmath41 , @xmath42 , @xmath43 , @xmath44 , @xmath45 , @xmath46@xmath47 , @xmath48@xmath49 , @xmath50 ( inverted spectrum ) and @xmath51 ( inverted spectrum ) in the notation of @xcite . consequently , all cases of five texture zeros in @xmath6 are now excluded , and among the cases of four texture zeros only @xmath50 ( normal spectrum ) , @xmath51 ( normal spectrum ) as well as @xmath52@xmath53 remain valid . in the case of majorana neutrinos , the absolute values of the elements of the light - neutrino mass matrix @xmath3 can be described by nine parameters , of which seven are constrained by experiments / observations . the by now very precise knowledge of the oscillation parameters therefore allows detailed studies of the elements of @xmath3 , including their allowed ranges and their correlations . the situation is quite different in the case of dirac neutrinos , where the neutrino mass matrix @xmath6 is by far not uniquely determined , even if the neutrino masses and the mixing matrix are known . therefore putting bounds on the elements of @xmath6 is much harder than in the majorana case . nevertheless studies of @xmath6 are possible , for example the analysis of texture zeros in @xmath6 . we reinvestigated the allowed texture zeros of @xmath6 in the basis where the charged - lepton mass matrix is diagonal . our results agree with the original analysis @xcite , the only difference being that by now we know that @xmath54 , which excludes some previously viable types of texture zeros . * acknowledgments : * the author wants to thank the organizers for the invitation to the delightful and interesting conference . this work is supported by the austrian science fund ( fwf ) , project no . p 24161-n16 .
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we analyse the mass matrix of the three light neutrinos in the basis where the charged - lepton mass matrix is diagonal and discuss constraints on its elements for the majorana and the dirac case .
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the comparison between chemical abundances of deuterium , helium , and lithium predicted by bbn models with current empirical estimates is one of the most viable method to constrain the physical mechanisms and the cosmology which governed the nucleosynthesis of primordial abundances ( olive , steigman , & walker 2000 ) . as far as the primordial he content is concerned , current empirical estimates are mainly based on measurements of nebular emission lines in low - metallicity , extragalactic hii regions ( izotov , thuan , & lipovetsky 1997 ; olive , steigman , & skillman 1997 ) . recent he determinations present small observational errors ( @xmath7% ) , but large uncertainties between independent measurements : @xmath8 by olive & steigman ( 1995 ) against @xmath9 by izotov & thuan ( 1998 ) . this evidence suggests that current he abundances are still dominated by systematic errors . in fact , viegas , gruenwald , & steigman ( 2000 ) and gruenwald , steigman , & viegas ( 2001 ) in two detailed investigations on the ionization correction for unseen neutral and doubly - ionized he in hii regions , found that he estimates should be reduced by 0.006 ( @xmath10 ) , a quantity which is a factor of 2 - 3 larger than typical statistical errors quoted in the literature . moreover and even more importantly , pistinner et al . ( 1999 ) on the basis of a new grid of stellar atmosphere models for ob stars found that the inclusion of both nlte and metal - line blanketing effects causes an increase of the order of 40% in the ratio of he to h ionizing photons . this evidence together with uncertainties due to the occurrence of stellar winds , shocks , temperature fluctuations ( izotov , thuan , & lipovetsky 1997 ; pistinner et al . 1999 ; peimbert , peimbert , & luridiana 2001 ; sauer , & jedamzik 2001 , and references therein ) and of peculiar nebular dynamics certainly affects the he abundance estimates based on giant extragalactic hii regions . in addition it is worth mentioning that the hii regions used for determining the cosmological helium abundance could have been somewhat polluted by the stellar yields of the pristine type ii supernovae , and in turn the empirical he abundances in these stellar systems should be corrected for self - pollution by massive stars . a plain evidence of this occurrence has been recently provided by aloisi , tosi , & greggio ( 1999 ) , and stlin ( 2000 ) . on the basis of deep hst optical and nicmos data they have resolved the stellar content of i zw 18 and found evidence that this blue compact galaxy hosts a relatively old population of asymptotic giant branch stars ( @xmath11 0.1 - 5 gyr ) . on the other hand , the comparison between star counts of horizontal branch ( hb , central he burning phase ) and red giant ( rg , h shell burning phase ) stars in galactic globular clusters ( ggcs ) with the lifetimes predicted by evolutionary models , the so - called r parameter ( iben 1968 ) , supplies upper limits to primordial he mass fraction of the order of 0.20 ( sandquist 2000 ; zoccali et al . 2000 ) . however , such estimates should be cautiously treated ( bono et al . 1995 ; cassisi et al . 1998 ) , since they are hampered by current uncertainties on the nuclear cross - section of the @xmath12 reaction ( buchmann 1996 ) . note that spectroscopic measurements of he abundances in low - mass population ii stars are useless for constraining the primordial he content , because the he lines are either too faint ( low - temperature stars ) or affected by gravitational settling such as high temperature hb stars ( giannone & rossi 1981 ; moheler et al . . however , empirical and statistical errors affecting abundance determinations of primordial deuterium , @xmath13he , and lithium could be significantly larger than for he ( sasselov & goldwirth 1995 ; olive et al . moreover , the primordial he content plays a paramount role in constraining both stellar ages and cosmic distances , since the mass - luminosity ( m / l ) relation of low and intermediate - mass stars during h and he burning phases depends on @xmath0 ( bono et al . 2000 ) . at the same time , at fixed he to metal enrichment ratio the he abundance adopted to model evolutionary and pulsational properties of metal - rich stellar structures does depend on @xmath0 as well ( bono et al . 1997 ; zoccali et al . 2000 ) . the physical baryon density of the universe is one of the observables that can be determined with high accuracy using measurements of cmb anisotropies at intermediate and small angular scales ( see e.g. , hu et al . 2000 , and references therein ) . it goes without saying that this observable plays a key role not only to assess the plausibility of the physical assumptions adopted in bbn models ( tegmark & zaldarriaga 2000 ) but also for constraining the intrinsic accuracy of current primordial abundance estimates . according to the joint analysis of both boomerang and maxima-1 data , it has been estimated at 68% confidence level a baryon density @xmath14 ( jaffe et al . 2001 ) . on the basis of this observable esposito et al . ( 2001 ) found that the new cmb measurements are inconsistent at more that 3@xmath4 with both standard and degenerate bbn models . on the other hand , the latest analysis of the boomerang results , which has improved the removal of systematics from the data ( netterfield et al . 2001 ) , found @xmath15 ( de bernardis et al . 2001 ) , in very good agreement with the bbn value . the same conclusion has been derived from the analysis of the ground - based cmb observations performed by the dasi interferometer ( halverson et al . 2001 ) , which also found @xmath15 ( pryke et al . this notwithstanding , the new analysis of the maxima data ( lee et al . 2001 ) , which extended the high @xmath5 coverage of the power spectrum measurement , still points towards somewhat higher values of the physical baryon density : @xmath16 ( stompor et al . 2001 ) . finally , we mention the measurements of the cmb power spectrum at @xmath171000 by the cosmic background imager . from these observations , and by assuming a flat cosmological model , the likelihood for the physical baryon density is found to peak at @xmath18=0.009 ( padin et al . current physical baryon densities based on cmb measurements and bbn models would imply that @xmath0 might range from roughly 0.24 to approximately 0.26 . the main aim of this investigation is to constrain @xmath0 on the basis of recent cmb measurements and two stellar observables that depend on @xmath0 , namely the trgb luminosity and the zahb luminosity . in 2 we discuss in detail the adopted theoretical framework as well as the comparison between predicted and empirical observables . the effect of a change in @xmath0 abundance on the _ uv - upturn _ as well as on cmb anisotropies are presented in 3 and 3.1 respectively . our conclusions and final remarks are briefly mentioned in 4 . on the basis of the new cmb measurements and bbn models esposito et al . ( 2000 ) estimated at 68% confidence level a @xmath0 abundance ranging from 0.249 to 0.254 . however , recent investigations ( tegmark & zaldarriaga 2000 ; padin et al . 2001 ; stompor et al . 2001 ; tegmark , zaldarriaga , & hamilton 2001 ) suggest on the basis of cmb measurements and of the simplest flat inflation model that the baryon density should range from 0.009 to 0.045 . the upper limit taken at face value would imply a larger primordial he content . as a generous but still plausible primordial he content we adopted @xmath3 . to assess the impact that such a determination has on stellar structures we selected two observables , namely the luminosity of zahb stars and the luminosity of the tip of trgb stars . the previous observables refer to stars belonging to ggcs . the reasons why we selected these observables are the following : 1 ) the stellar population in ggcs are among the oldest stars in the galaxy , and therefore they are the best laboratory to investigate the primordial he content . the comparison between theory and observations in ggcs is more predictable when compared with field , halo stars , since it relies on stars that are coeval , located at the same distance , and chemically homogeneous , since the latter was somewhat contaminated by the debris of the first stellar generation . ] . 2 ) the zahb in ggcs marks the phase in which the stars are mainly supported by @xmath19 reaction in the stellar center and it is a well - defined observational feature . the zahb luminosity depends on the he core mass and an increase of 15% in y causes an increase in the luminosity of approximately 0.16 mag ( sweigart & gross 1976 ; raffelt 1990 ) . 3 ) according to current evolutionary prescriptions the trgb phase marks the onset of central he burning ( he - core flash ) in low - mass stars and its luminosity strongly depends on the he - core mass , and in turn on the initial he content . in fact , sweigart & gross ( 1978 ) found that an increase of 15% in y causes a decrease in the trgb luminosity of 0.1 mag . finally , we mention that the previous observables are virtually unaffected by cluster age ( @xmath20 gyr ; vandenberg , stetson , & bolte 1996 ) , since for stellar ages larger than @xmath116 gyr both the zahb and the trgb luminosities do not depend on age ( castellani , deglinnocenti , & luridiana 1993 ; lee , freedman , & madore 1993 ; cassisi & salaris 1997 ; salaris & cassisi 1998 ) . 1 shows the comparison between predicted and observed zahb luminosity at the rr lyrae effective temperature ( @xmath21 ) as a function of global metallicity abundances ( salaris et al . 1993 ; vandenberg et al . 2000 ) . ] . the approach adopted to estimate the empirical bolometric magnitudes as well as their errors have been discussed by de santis & cassisi ( 1999 ) . the solid and the dashed line show the zahb luminosities predicted by cassisi & salaris ( 1997 ) and by vandenberg et al . the comparison between theory and observations suggests that hb models constructed by the previous authors are in good agreement with empirical data . on the other hand , the zahb luminosities predicted by hb models constructed by assuming @xmath3 ( dotted line ) are @xmath22 brighter than the observed ones . note that current predictions on zahb luminosities are still controversial , since there is a mounting evidence that recent hb models based on new input physics ( equation of state , neutrino energy loss rates , conductive opacities ) are systematically brighter than observed and predicted by canonical hb models ( cassisi et al . 1999 ; bono , castellani , & marconi 2000 ) . however , such a conundrum does not affect our conclusion , because the systematic shift in the luminosity showed by he - enhanced hb models is a differential effect . 2 shows the comparison between theory and observations for the trgb bolometric magnitudes in a sample of 12 ggcs as a function of global metallicity . apparent bolometric magnitudes were estimated by frogel , persson , & cohen ( 1983 ) and by ferraro et al . ( 2000 ) . the distance moduli adopted to derive the absolute magnitudes were estimated by comparing predicted zahb luminosities at fixed chemical composition , derived by adopting the same theoretical framework , with the observed distribution of hb stars in the color - magnitude diagram of each individual cluster . theoretical predictions ( solid line ) refer to models constructed by adopting a canonical initial he content of 0.23 and global metallicities ranging from [ m / h]=-2.4 to -0.4 . the top panel shows that the predicted luminosities are systematically brighter than the observed ones . this mismatch between theory and observations is expected and caused by the fact that the empirical estimates are hampered by sample size and also by the decrease in the lifetimes of the stellar structures approaching the trgb . by taking into account the typical sample sizes of cluster rgb stars , salaris & cassisi ( 1997 ) found that the 30% of clusters should lay within 0.1 mag ( dotted line ) the predicted trgb luminosity and the 70% within 0.3 mag ( dashed line ) the predicted ones . data plotted in the top panel support , within current uncertainties , this prediction . as a matter of fact , more than 70% of the empirical trgb bolometric magnitudes lay within the expected range ( dashed and dotted lines ) and only a few clusters attain magnitudes close to predicted @xmath23 values . the bottom panel of fig . 2 shows the same empirical data plotted in the top panel . theoretical predictions on trgb bolometric magnitudes are based on evolutionary models constructed by adopting an initial he content of 0.26 . at the same time , the absolute bolometric magnitudes were estimated by adopting the distance moduli obtained by comparing observed hb stars with the zahb luminosities predicted by hb models constructed by adopting @xmath3 , i.e. the same he content adopted in he enhanced models of fig . a glance at the data plotted in this panel shows that more than 70% of the clusters in our sample attain bolometric magnitudes similar or even brighter than predicted by he enhanced models . this finding is at odds with the straightforward statistical arguments mentioned above , and indeed only two measurements lay within 0.3 mag from the predicted trgb bolometric magnitudes . stellar observables discussed in the previous section can be extended over local group ( lg ) galaxies or slightly beyond , i.e. on scales of the order of a few mpc . moreover , current he estimates are based on spectroscopic measurements of hii regions in extremely metal - poor blue compact galaxies . these systems are located outside the lg and their typical distances are , within the uncertainties , of the order of 200 - 300 mpc . this means that previous stellar observables and hii regions can hardly be adopted on gpc scale even by using the largest telescopes . however , the he abundance on these scales can be probed on the basis of the _ uv - upturn_. this phenomenon shows up as a sharp rise in the spectra of elliptical and s0 galaxies for wavelengths smaller than 2500 @xmath24 . according to both theoretical ( greggio & renzini 1990 ; castellani & tornamb 1991 ; castellani et al . 1994 ) and empirical ( burstein et al . 1988 ; brown et al . 2000 ) evidence the _ uv - upturn _ is driven by the progeny of extreme and hot hb stars , namely agb - manqu and post - early - agb ( castellani & tornamb 1991 ) . this observable presents a strong dependence on age , since he abundance affects the mass of main sequence turn off stars , the hb morphology , and in turn the uv emission of old stellar populations . as a consequence , the _ uv - upturn _ is a powerful age indicator in elliptical galaxies ( tantalo et al . 1996 ; greggio & renzini 1999 ) . however , as already mentioned in the previous section the evolutionary properties of hb stars do depend on the primordial he content . in particular , an increase in y causes , at fixed age , an increase in the zahb luminosity . current hb evolutionary models for metal - poor and metal - rich structures ( @xmath25 ) constructed ( zoccali et al . 2000 ) by adopting different he contents ( @xmath26 ) suggest that the hb luminosity at @xmath21 scales with the initial he content according to the following relation : @xmath27 . a quite similar value was also suggested by raffelt ( 1990 ) on the basis of hb models computed by sweigart & gross ( 1976 , 1978 ) . thus suggesting that evolutionary predictions on this ratio are quite robust . this means that in metal - rich populations , which is typical of e and s0 galaxies , an increase in y of a factor of two ( 0.23 vs 0.46 ) causes an increase in @xmath28 of the order of 0.40 dex . this result is a rough estimate of the impact of y on the hb luminosity , it can not be easily extrapolated to the _ uv - upturn_. in fact , the crucial parameters for this phenomenon are the changes in hb and post - hb evolutionary lifetimes as well as the uv flux and not the bolometric one . this means that a detailed quantitative estimate does require synthetic models which simultaneously account for both h and he burning phases , and in turn for the change in the spectral energy distribution ( sed ) of the entire field population ( brown et al . this notwithstanding , we are interested in estimating the dependence of the _ uv - upturn _ on he content at intermediate redshifts , and therefore we constructed four metal - poor ( z=0.0001 ) extreme hb structures at different he contents , namely 0.15 , 0.23 , 0.35 , and 0.5 ( see fig . the he - core masses of these structures were estimated by constructing h - burning evolutionary tracks that reach the rgb tip with an age of @xmath11 10 gyr . we find that the he - core masses for the previous compositions are 0.523 , 0.511 , 0.481 , and 0.433 @xmath29 respectively . to estimate the impact of he content on the uv emission of agb - manqu stars we selected a typical effective temperature for these structures , namely @xmath30 . one finds that the total mass of hb structures at the selected compositions range from @xmath31 0.53 ( @xmath32 ) to 0.45 ( @xmath33 ) . the evolution was followed from central he - burning till the beginning of the white dwarf cooling sequence ( see fig . 3 ) , and then we estimated the total spectral energy distribution ( sed ) for the four structures according to the individual evolutionary lifetimes . it turns out that the uv emission of extreme horizontal branch ( ehb ) structures is relatively sensitive to the he content , and indeed an increase from @xmath34 to @xmath35 and @xmath33 causes a decrease in the uv emission by roughly 10% and 22% respectively . at the same time , a decrease from @xmath34 to @xmath32 causes an increase by @xmath119% . this effect is due to the fact that among ehb structures an increase in y causes , at fixed age and effective temperature , a decrease in the he - core mass and in turn in the zahb luminosity . this means that he - rich ehb structures spend a substantial portion of their central he - burning lifetime at lower luminosities . therefore their total uv emission decreases when compared with he - poor structures . moreover , we also find that an increase in the he content causes in metal - poor structures a decrease in the range of stellar masses evolving at high temperatures during he - burning . in fact , we find that the largest mass that evolve as agb - manqu slightly decreases from 0.54 at @xmath32 to 0.52 at @xmath35 . this finding further strengthens the evidence that in metal - poor structure an increase in the he abundance causes a decrease in the uv emission . note that such a trend is at odds with the behavior in metal - rich structure , and indeed dorman et al . ( 1993 ) found that an increase in y causes an increase in the largest mass evolving as agb - manqu , and in turn in the uv emission . the difference seems to be due to the fact that the evolution of metal - poor ehb structures is mainly governed by central he burning , and therefore by the he - core mass at the tip of the rgb . on the contrary the h - burning shell is more efficient in metal - rich ehb structures . this means that the he - core mass in these structures undergoes a mild increase during the central he burning phase , and therefore the range of stellar masses evolving as agb - manqu increases as well . once again we note that current arguments are preliminary but plausible speculations of the impact of he - content on the _ uv - upturn_. however , a firm quantitative evaluation requires the calculation of synthetic population models that account for both agb - manqu and post - early - agb structures . this seems a promising result , since a substantial increase / decrease in the he content should cause , at fixed look - back time and similar star formation histories , a decrease / increase in the scatter of the empirical average restframe 1550-v colors , i.e. the fingerprint of the _ uv - upturn_. moreover and even more importantly , present - day instrumentation allowed the detection and measurement of the _ uv - upturn _ in intermediate redshift ( @xmath36 ) e galaxies ( brown et al . 2000b ) . by assuming a hubble constant @xmath37 km s@xmath38 mpc@xmath38 and an eistein - de sitter cosmological model one finds that the comoving distance @xmath39 and the look - back time @xmath40 of this e galaxy are @xmath41 gpc and @xmath42 gyr respectively . on the other hand , if we assume a high lambda cosmological model ( @xmath43 @xmath44 ) one finds @xmath45 gpc and @xmath46 gyr . this is the reason why we constructed ehb structures by adopting he - core masses at evolutionary ages of approximately 10 gyr . therefore this evidence and our finding seem to support the use of the _ uv - upturn _ to trace the he content up to gpc scales . as it has been emphasized countless times in the literature ( see e.g. , jungman et al . 1996 ; hu & white 1996 ; kamionkowski & kosowsky 1999 ) , observations of cmb anisotropies provide a powerful way of setting tight constraints on the value of most cosmological parameters . in particular , the angular power spectrum of cmb temperature fluctuations depends both on the primordial fluctuations which seeded structure formation in the universe and on the physical processes occurring before the recombination in the baryon - photon plasma . within the framework of inflationary adiabatic models , these processes leave a characteristic imprint in the angular power spectrum in the form of a series of harmonic peaks ( usually named `` acoustic peaks '' in the literature ) whose height and position is sensitively dependent on the parameters of the cosmological model . while the parameter which is most robustly determined from measurements of the cmb power spectrum is undoubtedly the total energy density of the universe , many other parameters are measurable with striking precision . among them , the physical density of baryonic matter in the universe , whose value affects the height ratio of odd and even peaks in the spectrum : in particular , a high baryon density enhances the height of the first peak with respect to the second , and vice - versa . this effect is quite relevant even for small variations of @xmath18 , as shown in fig . the dependence of the cmb anisotropy on the primordial he mass fraction is , on the contrary , quite weak . as shown in figure 5 , the effect on the first peak is just about @xmath47 , even for an unrealistically large range of values @xmath48 . one has to go beyond the second peak in order to obtain effects larger than @xmath49 . we stress the fact that the effect of both the baryon density and the primordial he abundance on the cmb pattern is only relevant at small angular scales ( corresponding to @xmath6 in the power spectrum ) . these scales are only weakly affected by the uncertainty deriving from the limited coverage of the observed region , and from the so called `` cosmic variance '' resulting from the fact that we can only observe one statistical realization ( our sky ) drawn from the underlying cosmological model . on the other hand , the high @xmath5 end of the spectrum is unfortunately the one which is currently most at risk of being affected by unknown systematics such as pointing inaccuracies , poorly known beam pattern , residual instrumental noise , etc . until recent times , no high resolution observation of the cmb anisotropy pattern was available . consequently , the structure of peaks in the power spectrum could not be resolved with the accuracy needed to obtain a precise measurement of the baryon density . the situation has dramatically changed after recent observations . the maxima and boomerang experiments ( hanany et al . 2000 ; de bernardis et al . 2000 ) produced the first high - resolution maps of the cmb , and measured the cmb angular power spectrum on a wide range of @xmath5 : @xmath50 , corresponding to angular scales @xmath51 . this has provided tight constraints on the main parameters of the inflationary adiabatic model ( balbi et al . 2000 ; lange et al . 2000 ; jaffe et al . in particular , the constraints on the physical baryon density from the joint analysis of the maxima and boomerang data , @xmath14 , was found to be higher than the one derived from primordial nucleosynthesis considerations @xmath52 ( see e.g. , burles , nollett & turner 2001a ; burles et al . 1999 ) although the bbn value fall within the cmb 95% confidence interval . this stirred some discussion about the existence of a conflict between cmb and bbn and possible explanation for it ( for example , burles , nollett & turner 2001b ; kurki - suonio & sihvola 2001 ; esposito et al . 2001 ; di bari & foot 2001 ; lesgourgues & peloso 2000 ) . it is remarkable , however , that these first limits from the cmb agree , within 2@xmath4 , with those from the bbn , which are derived using a different set of measurements . recent measurements of the cmb anisotropy provided the unique opportunity to evaluate several fundamental cosmological parameters and to supply for all of them a preliminary but plausible estimate of their error budget . the impact of these new measurements on cosmological models uncorked a flourishing literature . however , it is not easy to assess on a quantitative basis to what extent current differences in the physical baryon density derived from cmb observations are caused by deceptive systematic errors . as a matter of fact , cmb measurements of @xmath53 range from 0.009 ( padin et al . 2001 ) to @xmath54 ( 95% confidence level , stompor et al . 2001 ) . according to bbn models the new measurements imply that @xmath0 might range from approximately 0.24 to roughly 0.26 . by adopting the upper limit on @xmath0 we investigated the impact of the change on two stellar observables , namely the zahb luminosity and the luminosity of the tip of the rgb . the main outcome of our analysis is that an increase in the primordial he content from the canonical @xmath34 to @xmath3 does not seem to be supported by the comparison between current theoretical predictions and empirical data . we found that the _ uv - upturn _ can be adopted to estimate the primordial he content . in fact , numerical experiments suggest that an increase of @xmath0 from 0.23 to 0.50 causes a decrease in the uv emission at least of the order of 20% . this is a preliminary rough estimate based on the assumption that agb - manqu structures are the main sources of the _ uv - upturn_. an interesting feature of this observable is that current instruments can allow us to measure the _ uv - upturn _ up to distances of the order of gpcs . note that to supply quantitative estimates of @xmath0 on the basis of the comparison between synthetic and observed _ uv - upturns _ it is necessary to account for the sed typical of complex stellar populations as a function of redshift ( tantalo et al . 1996 ; yi et al . however , theoretical predictions should be cautiously treated , since uv flux when moving from low to high metal contents strongly depends on the efficiency of the mass loss as well as on the he to metal enrichment ratio ( greggio & renzini 1999 ) . the scenario has been further complicated by recent spectroscopic measurements of hot hb stars ( @xmath55 k ) in metal - poor ggcs ( m15 , m13 ) . in fact , behr et al . ( 2000 ) and behr , cohen , & mccarthy ( 2000 ) found that in these stars the iron abundance is enhanced by 1 - 2 order of magnitudes , whereas the he content is depleted by at least one order of magnitude respect to solar abundance . unfortunately , we still lack quantitative estimates of the impact that such a peculiarities have on the uv emission . theoretical and empirical arguments support the evidence that the density of baryons in the universe is homogeneous ( copi , olive , & schramm 1995 ) . the same outcome applies to large scale chemical inhomogeneities ( copi , olive , & schramm 1996 ) . however , it has been recently suggested by dolgov & pagel ( 1999 , hereinafter dp ) a new cosmological model that predicts a substantial spatial variation in the primordial chemical composition and a small baryon density variation . this investigation was triggered by a difference of one order of magnitude in the deuterium abundance of damped @xmath56 systems along the line of sight of high - redshift ( @xmath57 ) qsos ( dodorico et al . 2001 ; steigman et al . the scenario developed by dp relies on a model of leptogenesis ( dolgov 1992 ) in which takes place a large lepton asymmetry and this asymmetry undergoes strong changes on spatial scales ranging from mega to giga pcs . the key feature of this model is to predict a large and varying lepton asymmetry and a small baryon asymmetry . within this theoretical framework the he mass fraction in deuterium - rich regions should range from 35% to 60% , while the @xmath58 one should increase up to @xmath59 , while the variation of the photon temperature should be @xmath60 . obviously , the hypothesis that current changes in baryon density are due to real spatial variations is premature as any further speculative issue . future full - sky cmb observations from space missions such as nasa s map ( wright 1999 ) and esa s planck ( mandolesi et al . 1998 ) will play a crucial role to properly address the problem of the spatial variation , since they will supply a larger sensitivity up to very high @xmath5 ( @xmath61 ) and an improved control on systematics . we warmly thank claudia maraston for kindly providing us the spectral energy distribution of current hb models . we also ackowledge an anonymous referee for his / her useful suggestions that improved the readability of the paper . this work was supported by murst / cofin2000 under the project : `` stellar observables of cosmological relevance '' ( g. b. , r. b. , & s. c. ) . aloisi , a. , tosi , m. , & greggio , l. 1999 , aj , 118 , 302 balbi , a. , et al . , 2000 , apjl , 545 , l1 behr , b. b. , cohen , j. g. , mccarthy , j. k. 2000 , apj , 531 , l37 behr , b. b. , djorgovski , s. g. , cohen , j. g. , mccarthy , j. k. , cot , p. , piotto , g. , zoccali , m. 2000 , apj , 528 , 849 bono , g. , caputo , f. , cassisi , s. , incerpi , r. , & marconi , m. 1997 , apj , 483 , 811 bono , g. , caputo , f. , cassisi , s. , marconi , m. , piersanti , l. , & tornamb , a. 2000 , apj , 543 , 955 bono , g. , castellani , v. , deglinnocenti , s. , & pulone , l. 1995 , a&a , 297 , 115 bono , g. , castellani , v. , & marconi , m. 2000 , apj , 532 , l129 brown , t. m. , bowers , c. w. , kimble , r. a. , sweigart , a. v. , & ferguson , h. c. 2000 , apj , 532 , 308 brown , t. m. , ferguson , h. c. , stanford , s. a. , & deharveng , j 1998 , apj , 504 , 113 burles , s. , nollett , k.m . , & turner , m. 2001a , apjl , 552 , l1 burles , s. , nollett , k.m . , & turner , m. 2001b , phys . d , 63 , 063512 burles , s. , et al . , 1999 , phys . lett . , 82 , 4176 burstein , d. , bertola , f. , buson , l. m. , faber , s. m. , & lauer , t. r. 1988 , apj , 328 , 440 cassisi , s. , castellani , v. , deglinnocenti , s. , & weiss , a. 1998 , a&as , 129 , 267 cassisi , s. , castellani , v. , deglinnocenti , s. , salaris , m. , & weiss , a. 1999 , a&as , 134 , 103 cassisi , s. , & salaris , m. 1997 , mnras , 285 , 593 castellani , m. , castellani , v. , pulone , l. , & tornamb , a. 1994 , a&a , 282 , 711 castellani , v. , deglinnocenti , s. , & luridiana , v. 1993 , a&a , 272 , 442 castellani , m. , & tornamb , a. 1991 , apj , 381 , 393 copi , c. j. , olive , k. a. , & schramm , d. n. 1995 , apj , 451 , 51 copi , c. j. , olive , k. a. , & schramm , d. n. 1996 , astro - 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we present the results of a joint investigation aimed at constraining the primordial he content ( @xmath0 ) on the basis of both the cosmic microwave background ( cmb ) anisotropy and two stellar observables , namely the tip of the red giant branch ( trgb ) and the luminosity of the zero age horizontal branch ( zahb ) .
current baryon density estimates based on cmb measurements cover a wide range values @xmath1 , that according to big bang nucleosynthesis ( bbn ) models would imply @xmath2 .
we constructed several sets of evolutionary tracks and hb models by adopting @xmath3 and several metal contents .
the comparison between theory and observations suggests that zahb magnitudes based on he - enhanced models are 1.5@xmath4 brighter than the empirical ones .
the same outcome applies for the trgb bolometric magnitudes .
this finding somewhat supports a @xmath0 abundance close to the canonical 0.23 - 0.24 value .
more quantitative constraints on this parameter are hampered by the fact that the cmb pattern shows a sizable dependence on both @xmath0 and the baryon density only at small angular scales , i.e. at high @xmath5 in the power spectrum ( @xmath6 ) .
however , this region of the power spectrum could be still affected by deceptive systematic uncertainties . finally , we suggest to use the _ uv - upturn _ to estimate the he content on gpc scales .
in fact , we find that a strong increase in @xmath0 causes in metal - poor , hot hb structures a decrease in the uv emission .
| 12,460 | 426 |
a diffusive electron - electron interaction correction ( eec ) to the conductivity was predicted theoretically @xcite about 30 years ago . in 2d system it is proportional to @xmath5 ( @xmath6 is the momentum relaxation time ) and grows in amplitude as temperature decreases . a way to experimentally single - out eec among other numerous effects is based on its property not to affect hall component of magnetoconductivity tensor @xmath7 in perpendicular magnetic field @xcite . eec therefore gives birth to temperature - dependent and parabolic with field contribution to the diagonal magnetoresistance @xmath8 and correction to the hall coefficient @xmath9 , both being proportional to @xmath5 . the predicted features were observed in numerous experiments , mainly with n - type gaas - based 2d systems @xcite . however , the quantitative level of agreement between theory and experiment was achived only in the 2000s by minkov group @xcite from simultaneous analysis of both hall and diagonal components of resistivity tensor . the suggested method was later approbated by others @xcite . we note , that zeeman splitting effects were negligible in most of the studied systems . zeeman splitting was predicted to decrease the eec value @xcite , the physical interpretation of this effect introduced later in ref . @xcite consists in decreasing the effective number of triplet channels with field . for the diffusive regime @xmath10 , the eec is predicted to be quadratic - in - field in the low field limit , and proportional to logarithm of field in the high field limit . experimentally , however , the effect of zeeman splitting on eec in the diffusive regime was only briefly considered in refs . @xcite . for the most ubiquitous 2d system known , 2deg in si - mosfet , which fits well all theory requirements , no convincing measurements of the eec have been done so far . at the same time this system demonstrates positive magnetoresistance in parallel field , the behavior expected for eec . in the 1980-s there were attempts to reveal eec in si from temperature and magnetic field dependences @xcite of resistivity ; these attempts were based on not yet developed theoretical concepts and did not lead to a self - consistent picture of magnetotransport . interest to zeeman splitting effects was resumed in 1997 with observation of a huge rise in resistivity of 2deg in clean si - mosfets in parallel magnetic field @xcite , close to metal - to - insulator transition . the interest further increased with interpretation of this magnetoresistance as a signature of magnetic quantum phase transition @xcite . in the 2000s several attempts to treat the parallel field magnetoresistance ( mr ) in terms of renormalization - group approach were taken both theoretically @xcite and experimentally @xcite . this approach is in fact self - consistent generalization of the eec for arbitrary interaction strength and conduction . independently , another theoretical approach was developed in refs . @xcite , and successfully applied @xcite , which accounts for resistivity increase with field simply by renormalization of the density of states and single impurity scattering time . the latter effect is essentially different from logarithmic eec which emerges from multiple electron - impurity scattering . the experimental situation , however , is more complicated : studies @xcite showed a strong effect of disorder on the parallel field magnetoresistance , that was discussed in terms of the band tail effects in refs . moreover , detailed studies of the mr on different material systems @xcite did demonstrate quantitative disagreement between the fitted - to - eec theory temperature- and magnetic field dependences of the conductivity . to summarize the present state of the field , there seems to be a common agreement on the zeeman nature of parallel field mr in 2d carrier systems . however , two conceptually different underlining mechanisms of mr were put forward : ( i ) eec ( multiple - scattering effect ) and ( ii ) screening change in magnetic field ( single - scattering effect ) . which of them is responsible for the experimentally observed strong mr in parallel field ? the answer is especially crucial in the vicinity of the metal - to - insulator transition , where the mr is dramatically strong . unfortunately , both theories become inapplicable in this regime of small conductances @xmath11 . to address this issue , we have chosen to approach the problem from the large conductance regime , where both theories have solid ground , though the mr is low . in our paper we contest possible origin of the parallel field magnetoresistance of weakly interacting 2d electron gas . in order to study eec , we take detailed measurements of the magnetoresistance tensor in tilted field , and analyze the data using the procedure developed in refs . we stress that our approach does not rely on any particular microscopic theory , rather , it is _ ab - initio _ phenomenological and uses only general property of the eec in the diffusive regime to affect @xmath12 solely . for the experiments we have chosen the simplest model system , the 2d electron gas in si in diffusive regime @xmath13 , @xmath14 . to vary the strength of the zeeman splitting , and thus , the eec magnitude , we tilted magnetic field with respect to the 2d plane . this procedure allowed us to extract eec on top of other magnetoresistivity effects and to establish two principally different regions : ( i ) high - field region , where eec depends on total field and quantitatively agrees with the theoretically predicted @xmath15 asymptotics , and ( ii ) low - field region , where eec unexpectedly depends on perpendicular field component , grows with field and does not match existing theories . our observations suggest a new insight on the origin of the parallel field mr : ( i ) the high - field small and @xmath16independent eec @xmath17 , revealed in the current study , can not be responsible for large and @xmath18-dependent parallel field mr , and ( ii ) application of even low @xmath19 perpendicular field component strongly suppresses mr . the latter fact proves not purely spin nature of the parallel field mr and points at the incompleteness of the existing theory of mr . the paper is organized as follows : after a brief description of experimental details in section ii , we give theoretical background in section iii , and describe the results in section iv , first on the eec , then on the experimental proof of the zeeman origin of the magnetic field effect on eec , and further we compare the known eec with the parallel field magnetoresistance to show that the eec is not the main origin of the parallel field magnetoresistance . in section v we discuss the obtained result and suggest possible directions of further development . the ac - measurements ( 13 to 73hz ) of the resistivity were performed at temperatures 0.3 - 25k in magnetic fields up to 15 t with two ( 100 ) si - mos samples with 200 nm oxide thickness : si-40 , ( peak mobility @xmath20m@xmath21/vs at @xmath22 k ) , si-24 ( 0.22m@xmath23/vs ) . the samples were lithographically defined as rectangular hall bars of the size 0.8@xmath245 mm@xmath21 . to obtain the data for different orientations of magnetic field relative to the 2d plane , the sample platform was rotated _ in situ _ at low temperature using a step motor . because of the smallness of the studied effects and unavoidable misalignment of potential contacts there always was some asymmetry ( within less than a few percents ) of both @xmath25 and @xmath26 with respect to field reversal at a constant @xmath27 ( @xmath28 na ) current direction . to compensate this asymmetry the field was swept from positive to negative values and the data were symmetrized . the alignment of the sample parallel to magnetic field was done using the resistivity peak due to weak localization . the carrier density @xmath29 was varied by the gate voltage in the range @xmath30 @xmath31 . the field was tilted in the @xmath32-plane , always perpendicular to the current direction , which is however not crucial for si because of weak spin - orbit coupling @xcite . we have selected samples with moderate carrier mobility in order to ensure studies deep in the diffusive regime @xmath33 , where eec theory should be applicable , and at the same time to achieve @xmath34 in available magnetic field . other features which make si - mos system preferable for this study are ( i ) short - range and uncorrelated scatterers which make motion diffusive , ( ii ) large conductance @xmath35 , which ensure quantum correction theory applicable , ( iii ) very thin potential well @xmath36 nm which excludes orbital effects in parallel magnetic fields up to 20 t , and ( iv ) filling of the lowest size quantization subband solely for @xmath37@xmath31 with effective mass @xmath38 @xcite . the idea of the present study is based on a property of the e - e correction to affect only diagonal component of the conductivity tensor @xcite : @xmath39 correspondingly , the procedure of the eec extraction @xcite for arbitrary ( @xmath40 ) value includes the following steps : ( i ) reversing the measured resistivity tensor one calculates the conductivity tensor . ( ii ) from @xmath7 one finds the mobility @xmath41 value , using experimentally determined density @xmath29 @xcite . ( iii ) subtracting @xmath42 from @xmath12 one finds the correction @xmath43 . the mobility @xmath44 value is a by - product of this algorithm . if the conductivity is large , @xmath45 , the above algorithm leads to the following expressions for the resistivity components : @xmath46 \label{aamrtensor}\ ] ] @xmath47 as we checked , these formulae are valid for all our data with @xmath48 kohm . in practice , both @xmath49 and mobility @xmath41 in eqs . ( [ aamctensor],[aamrtensor],[simpleds ] ) might depend on @xmath50 and @xmath18 thus incorporating other possible magnetoresistance effects . nevertheless , as follows from eq.([simpleds ] ) , no matter how the mobility depends on field , the correction to the hall coefficient arises from eec solely . depending on the geometry of experiment the following cases are possible : ( i ) the field is directed normally to the sample plane , @xmath51 , during the sweep ( this case refers to most of the previous studies ) . ( ii ) the field is inclined by angle @xmath52 relative to the 2d plane , in order to enhance zeeman effects which depend on @xmath53 . ( iii ) the field magnitude @xmath54 remains constant while sample is rotated relative to the field direction ; in the latter case one expects the zeeman effects to remain unchanged , thus leading to the angle - independent @xmath49 . we note here that in the case of geometry ( ii ) data processing requires knowledge of the ratio @xmath55 solely , rather than the tilt angle . according to our definition of density @xcite , the @xmath55 ratio is obtained from the measured linear - in - total - field high temperature limit of the hall resistance @xmath56 . we find this limit by linear extrapolation of @xmath57 to the highest temperatures ( typically 15k ) , and by ignoring low - field effects ( see below ) . making use of geometries ( ii ) and ( iii ) is the key feature of the present study . we note here that the @xmath3-factor in si is large ( @xmath58 ) and isotropic , which insures the zeeman effects to be tilt independent . in systems with anisotropic @xmath3-factor , like holes in gaas @xcite one should take into account different components of the @xmath3-factor tensor , which complicates the problem . according to the theory of interaction corrections @xcite in its modern form @xcite , the eec value at zero magnetic field in the diffusive regime is @xcite . @xmath59 where @xmath60 is a fermi - liquid constant , @xmath61 is the number of triplet channels of interaction , @xmath62- valley degeneracy ( in the original formula @xcite @xmath63 ) . in ( 001 ) si - mosfets , electron system is 2-fold valley degenerate , and the degeneracy , if perfect , should increase @xmath64 to 15 @xcite . in fact , however , this degeneracy is never perfect , because of the two sample - dependent parameters , a finite valley splitting @xmath65 @xcite , and intervalley scattering time @xmath66 k@xmath67 for the sample si-40 @xcite . both effects decrease the number of triplet terms , which was described theoretically @xcite and studied experimentally @xcite . the problem with mosfets is that @xmath65 ( which is typically less than @xmath68 ) can hardly be measured directly in zero and low magnetic fields , because valley splitting of shubnikov - de haas oscillations does not exceed level broadening and hence can not be resolved @xcite . one could treat @xmath65 as an additional free parameter , which vary in the range from 0 to @xmath68 , strongly affecting predictions of the e - e interaction correction theory @xcite . within the present study , uncertain @xmath65 value affects only the effective number of valleys @xmath62 that can vary from @xmath69 to @xmath70 . we therefore can use @xmath62 as adjustable parameter , which quantifies effective degree of valley multiplicity . in 2001 , the eec was recalculated by zala et al @xcite and a new `` ballistic '' contribution was introduced , that gave explanation to @xmath71 dependence observed in different 2d systems in the regime @xmath72 @xcite . it was shown experimentally @xcite that the ballistic and diffusive corrections differ fundamentally : diffusive eec does not affect hall component of conductivity tensor ( see eq . [ aamctensor ] ) , whereas ballistic contribution is basically renormalization of the single impurity scattering time or mobility . in the present study we do not consider ballistic contribution , rather , we extract from the experimental data and analyse the diffusive part of the eec solely . theoretical prediction for the zeeman splitting dependence of the eec was first given in @xcite : @xmath73 where @xmath74 , and the two asymptotics for @xmath75 function are : @xmath76 @xmath77 within the same theoretical formalism , the zeeman splitting effect on ballistic and diffusive corrections to magnetoresistance in parallel field was recalculated in ref . @xcite . although the theory was successfully used to fit some data on parallel field magnetoresistance @xcite , the procedure of the comparison with theory is ill - defined and requires careful separation of the diffusive and ballistic eec contributions in the crossover regime . indeed , theoretical eec is a sum of two contributions with absolutely different structure : ( i ) a ballistic one which comes from the renormalization of single impurity scattering and ( ii ) diffusive eec for which @xmath78 . since our method catches only diffusive part , we briefly discuss below modern theoretical expressions in the diffusive regime @xmath13 solely @xmath79 this low field limit is close to @xcite for small @xmath80 . in the high field limit , the diffusive contribution from ref . @xcite is given by : @xmath81 noteworthy , functional dependence on @xmath60 is the same for eqs . ( [ znamaineq ] ) and ( [ znabpar1 ] ) . this fact has a transparent physical meaning : application of high field @xmath82 suppresses temperature dependence of only @xmath83 triplets with @xmath84 . this suppresion comes from expansion of the @xmath85 . correspondingly , if we define @xmath86 we may rewrite the theoretical expectation for the low temperature ( @xmath87 ) high field ( @xmath88 ) asymptotics : @xmath89 \label{lowtbasympt}\ ] ] here @xmath90 is a @xmath16 and @xmath91 independent term . this expression allows one to compare experimental data on low - temperature high - field asymptotics of the eec with the microscopic theory predictions . it s meaning is as follows : in the high field limit @xmath82 , the magnetic field dependence is not affected by temperature . in appendix a we show that from the practical point of view this limit is achieved already for @xmath92 . to conclude this section , studying magnetoresistance in purely parallel field is insufficient to disentangle the ballistic and eec contributions . tilting the field enables one to overcome this drawback . as shown in the next section , application of high tilted fields allows us to achieve in experiment the asymptotical @xmath93 behavior , for which there is a firm theory prediction . this section is organized as follows : in the first subsection we discuss phenomenology , low - field regime and experimental proof of the eec dependence on modulus of magnetic field rather than on its direction . in the second subsection we consider high - field asymptotics of the eec , and compare them with the fermi - liquid theory expectations . in the third subsection we discuss magnetoresistance in parallel and tilted fields . we explore mr in the range of fields @xmath94 , i.e. in the domain between weak localization , @xmath95 ) , and shubnikov - de haas oscillations ( here @xmath96 is the transport mean free path ) . figure [ perpfield]a shows magnetoresistance @xmath0 and hall resistance @xmath1 versus @xmath97 in field up to @xmath98 t . the data were collected at various temperatures ( 0.6 - 4.2k ) in a standard geometry ( magnetic field perpendicular to the sample plane ) for sample si-40 at @xmath99@xmath31 . the set of curves for si [ fig . [ perpfield](a ) ] does not look similar to numerous data for n - gaas 2d systems @xcite . indeed , there are two important features : ( i)unlike gaas , for si in high fields ( @xmath100 t in fig . 1(a ) ) the mr gets weaker and ultimately even changes sign to positive as temperature decreases ; ( ii ) the shubnikov - de haas oscillations start to appear already in fields @xmath101 , due to short - range disorder . because of the above features , the eec is not seen straightforwardly from the data . therefore , in order to extract it we use the procedure described in the previous section . at the first step we invert the resistivity tensor and obtain the conductivity one ( solid lines in figs . [ perpfield]c , [ perpfield]e ) . hall component of the conductivity tensor allows one to calculate mobility @xmath41 using eq . ( [ aamctensor ] ) ( shown in fig . [ perpfield]d ) . surprisingly , the mobility appears to be field - dependent ( unlike that in experiments with gaas @xcite ) : in strong fields ( @xmath102 t ) , the higher is the temperature the stronger is the field dependence . this effect is beyond the scope of the present study , we note only that the sign of the @xmath103-dependence is in - line with expectations from memory effects @xcite though such effects do not produce temperature dependence . field dependent mobility was suggested to arise also from ballistic correction @xmath104 @xcite . even though such correction looks qualitatively similar to our data , it is hardly relevant , because originates from multiple cyclotron returns and should be valid for @xmath105 , which is not the case . in weak fields , the hall mobility increases as @xmath50 decreases due to increase of the hall slope . having the mobility known , we calculate the drude part of @xmath12 , @xmath106 [ shown by dotted lines in fig . [ perpfield](e ) ] . correction to conductivity @xmath49 is calculated as a difference between @xmath12 and its drude expectation ( see inset to fig . [ perpfield]e for the graphical definition ) . the resultant @xmath107 is shown by solid lines in fig . [ perpfield]f . as it is clear from eq . ( [ simpleds ] ) , the correction resembles behavior of the hall coefficient @xmath9 ( shown in fig . [ perpfield]f ) : the larger the hall coefficient , the less the correction . all @xmath108 dependences ( collected at different temperatures , densities , tilt angles and for different samples ) manifest similar behavior : in low fields ( region a in fig . [ perpfield]g ) @xmath49 grows as @xmath50 increases , then reaches a maximum and decreases in the high field region b. in the low field region a , the feature in @xmath49 originates from non - linearity of the hall resistance with field ( as seen from fig . [ perpfield]f ) . similar low - field feature was observed in numerous previous studies with various 2d systems @xcite ; it is still poorly understood . in appendix b we summarize our observations on the low - field hall nonlinearity and argue that this low - field feature does not follow contemporary theories . empirically , the boundary between the regions a and b ( a point where @xmath49 is maximal ) roughly follows the equation @xmath109 . the correction decreasing with increasing field in the region b fig . [ perpfield]g is qualitatively in line with zeeman splitting effect @xcite . the latter should be direction independent , as discussed above . to check this property we set the tilt angle to 45@xmath110 , measured both @xmath0 and @xmath1 for the same temperatures and recalculated the eec . the resultant eec versus _ total _ magnetic field is shown in fig . [ perpfield]g by dashed lines . in the high - field region b , the monotonic parts of the eec data ( ignoring shubnikov - de haas oscillations ) in perpendicular and tilted field are quantitatively similar to each other , even though the perpendicular component of the field differs by a factor of @xmath111 . this observation proves the zeeman nature of the eec in the high - field region . another instructive way to check whether @xmath49 depends on the _ total _ field is to rotate the sample in constant field . during these measurements , the tilt angle @xmath52 was slowly swept at a constant rate using a step motor . we used positions of the specific sharp mirror symmetric features in the @xmath112 dependence at @xmath113 , @xmath114 , @xmath115 to calibrate angle and calculate perpendicular component of the field . figure [ spining ] shows @xmath116 and @xmath117 for the same electron density and different temperatures . the inset to fig . [ spining ] shows the corresponding eec . it is easy to see that gradual changes of @xmath49 with @xmath118 , which were seen in fig . [ perpfield]g almost disappeared in the inset to fig . 2 ; this demonstrates that eec remains unchanged in constant total field . we stress the importance of this `` rotating field '' experiment ; indeed , the coincidence of @xmath107 curves in fig . 1f for different tilt angles is not a complete proof for direction - independence of the eec , because it does not exclude corrections to conductivity @xmath119 ( see e.g. ref.@xcite ) . in the latter case one would see just a constant shift @xmath120 in tilted field sweep experiment ( fig . 1f ) with no visible change in functional form of the @xmath121 dependence . in contrast , in the rotating field experiment such corrections would reveal themselves as inclined @xmath122-lines , which is not the case in the insert to fig . the discussed above field direction independence of the eec was tested with two samples in the density range @xmath123 @xmath31 which corresponds to @xmath124 @xmath125 , and in the field range @xmath126 . the data in the high field region b ( @xmath127 ) should follow the asymptotics of eq.([lowtbasympt ] ) . the field range for observing the logarithmic - in - field behavior , however , is limited on the low - field side by low - field hall feature ( at @xmath128 t ) and , on the high field side , by the onset of shubnikov - de haas oscillations ( at @xmath129 t ) . for this reason , the field range for fitting the data with @xmath4-dependence is less than one decade . we performed detailed measurements of magnetoresistivity tensor at different tilt angles in magnetic fields up to 15 t. the high - field asymptotes were fitted with a function @xmath130 where @xmath131 and @xmath132 are two positive adjustable parameters , common for all curves . according to eq.([lowtbasympt ] ) : @xmath133 and @xmath134 . example of the data and corresponding eec is shown in fig . [ 15tesla ] . one can see that the @xmath107 dependencies for different temperatures follow almost parallel to each other being only shifted vertically ; exactly such behavior should be expected for the eec correction according to eq.([10 ] ) . the temperature prefactor @xmath131 can be easily found from the corresponding @xmath135- dependence ( see the inset to fig . [ 15tesla]b ) ; for the given dataset @xmath136 . the data in the inset also demonstrates that the obtained @xmath131 value is field - independent for @xmath137 t , which confirms that the analyzed data follow the high - field asymptotic behavior . as for the field - dependence prefactor @xmath132 , since the accessible field interval is less than a decade and we do nt know how far the low - field correction ( caused by the hall anomaly ) may extend , we can only roughly estimate @xmath138 . its lower bound is found from the low - field , @xmath139 t , data and the upper bound - from the high field ( @xmath140 t ) data . the next logical step would be to analyze the density dependences of the two prefactors , @xmath131 and @xmath132 , and to check their consistency with each other and with a microscopic theory . figure [ vg21 ] shows magnetoresistance and the corresponding eec for elevated density @xmath141 @xmath31 and for the same tilt angle . since conductivity increases here by a factor of 2.5 , the effect of eec becomes less pronounced on top of drude conductivity , nevertheless both temperature and magnetic field dependences of the eec remain approximately the same : @xmath142 ; @xmath143 . the resulting @xmath131 and @xmath132-values are summarized in table [ f0summarytable ] . as we argued above , these parameters do not demonstrate a pronounced density dependence . this fact is reasonable because in the studied range of densities ( i ) interaction parameter @xmath144 @xcite is rather small @xmath145 , and ( ii ) conductivity is large compared to the quantum unit to ensure smallness of the renormalization effects@xcite . in table [ f0summarytable ] we present @xmath60 calculated from @xmath131 and @xmath132- values using eqs . ( [ eq : f0s]),([lowtbasympt ] ) for the effective valley multiplicity @xmath70 and 2 . for comparison , we also show the @xmath60-values experimentally determined from shubnikov - de haas oscillations @xcite and from @xmath146 ballistic dependences @xcite . firstly , @xmath147 extracted from @xmath131 decreases with density . secondly , @xmath147 values extracted from @xmath131 for @xmath69 is rather close to the earlier measured values . if we adopt the effective valley multiplicity @xmath62 to be somewhat smaller than 2 , the agreement will become even better . when making such comparison with earlier data one should keep in mind that previous results were obtained in high - temperature ballistic regime , where both valleys contribute equally and @xmath69 exactly . one should also take into account that the @xmath147 values for the diffusive ( this work ) and ballistic ( previous data @xcite ) regimes should not coincide , for the reasons discussed in ref . we therefore conclude the temperature dependence of the diffusive eec ( i.e. , @xmath131-values ) to agree quantitatively with earlier data and to agree qualitatively with theory . possible reasons for large uncertainty in @xmath132-values and their poor consistency with @xmath131-values may be ( i ) too narrow range of fields accessible for identifying @xmath4 dependence ( see above ) , ( ii ) the effective number of valleys ( @xmath148 ) may be different in the field and temperature dependences . for the lowest studied density @xmath149 @xmath31 , the low - field feature broadens and obscures observation of the decreasing logarithmic @xmath121 dependence ; this prevented us from measuring the respective @xmath132 values . to summarize this section , we presented above three firm experimental observations on the high - field and low - temperature behavior of the eec , as follows : ( i ) the observed @xmath43 dependences do not demonstrate any anisotropy with respect to the field direction , ( ii ) @xmath49 is linear both in @xmath15 and @xmath150 in line with the expected asymptotics ( eq . [ lowtbasympt ] ) , and ( iii ) the observed field and temperature dependences have an anticipated magnitude . in total , the above listed facts prove that the field and temperature dependences of conductivity indeed represent the eec . the prefactor in the temperature dependence of @xmath49 reproduces @xmath147 values that are decreasing with density and are reasonably consistent with earlier data . the prefactor in the @xmath15-dependence does not contradict the microscopic theory and earlier data though it was determined with rather large uncertainty . .summary of interaction parameters found from the eec measurements.[f0summarytable ] [ cols="^,^,^,^,^,^",options="header " , ] a nonmonotonic magnetoresistance in perpendicular field and at low temperatures in si and p - sige has been observed since 1982 @xcite : as field increases , the drop in @xmath0 due to weak localization is followed by the resistivity increase ( see , e.g. fig . [ perpfield]a ) . surprisingly , the effect has not been discussed and understood yet . as can be seen from our data ( fig . [ 15tesla]a ) , field tilting makes this effect even more pronounced , thus pointing to the zeeman nature of the nonmonotonic magnetoresistance . qualitatively , this seems to be transparent : the larger is the zeeman field , the stronger is the positive component of magnetoresistance . in the strictly parallel field orientation there is no weak localization suppression and the magnetoresistance is purely positive . following castellani et al . @xcite , the magnetoresistance in the diffusive regime for 2d systems was often attributed to eec @xcite and used to evaluate interaction parameter @xcite . however , whether the observed parallel field magnetoresistance may be entirely attributed to eec , has not been proven so far . having developed a technique to measure zeeman field dependence of the eec directly , with no adjustable parameters , we can compare now eec with magnetoresistance . figure [ parallel]a shows a set of magnetoresistance curves collected at different temperatures for @xmath99 @xmath31 in the parallel field orientation . high field asymptotics of eec for the same sample and same density @xmath151 is also shown in fig.[parallel]a by dashed line ( the prefactor 0.15 for this particular density is found in the previous section ) . the observed magnetoresistance appears to be ( i ) an order of magnitude larger than eec , and ( ii ) has much stronger temperature dependence ; therefore , it is mainly of a different origin . this is one of the key results of the present study , it brings into a question the validity of usage parallel field magnetoresistance for finding the interaction constant . we have shown above that the multiple scattering ( eec ) approach leads to much smaller magnetoresistance than that observed ; we test below whether or not single impurity scattering processes play dominant role in magnetoresistance . we show in fig . [ parallel]a ( dotted lines ) the magnetocondutivity calculated according to eq . ( 4 ) from ref . @xcite ( two - valley version of theory @xcite ) for a system with @xmath152 , @xmath153 , and @xmath154 . the calculated theory curve seems to demonstrate a reasonable agreement with the same experimental data . this agreement ( in contrast to above sharp disagreement - see the dashed curve in fig . [ parallel]a ) has a simple physical explanation : the calculated curve comprises mainly ( more than 65 % ) single impurity scattering renormalization , and only 35% of the diffusive eec . we believe that by using two different @xmath60 values for ballistic and diffusive contributions @xcite and by introducing finite valley splitting and intervalley scattering rate @xcite ( i.e. about 4 adjustable parameters ) one can achieve a perfect agreement with experiment . this exercise reproduces an apparent successful comparison of the parallel field magnetoresistance data in si - mosfets@xcite with theory@xcite . is this the end of the story and complete understanding of the magnetoresistance in the 2d system ? our answer is no ; in order to argue this point we apply perpendicular field . theoretically @xcite the zeeman field - induced magnetoresistance should remain the same in tilted field . figure [ parallel]b shows magnetoresistance at fixed @xmath155 k and @xmath99 @xmath31 for various tilt angles . in perpendicular field ( 0@xmath110 curve ) , as field increases from zero , weak localization becomes suppressed , leading to the negative magnetoresistance ; further we observe almost flat magnetoresistance until the onset of shubnikov - de haas oscillations . such classically flat magnetoresistance is a signature of a field - independent mobility . in the opposite limit of parallel field ( 90@xmath110 curve ) we observe a huge , 25% magnetoresistance in field of 6 t . this fact itself is remarkable because we did not expect much difference between the parallel and perpendicular configurations in fields @xmath156 within the above simple considerations . although we do not believe in orbital origin of the parallel field magnetoresistance , let us assume for a moment that only parallel component of the field affects mobility . if this is so , then by tilting the field from purely in - plane direction ( @xmath157 ) to slightly out of plane ( 75@xmath110 ) , when parallel component of the field decreases only by 4% , we would expect roughly the same positive magnetoresistance as that in the purely parallel field . surprisingly , after suppression of weak localization , the observed positive upturn in magnetoresistance is quite shallow and is no more than @xmath1586% ! for intermediate tilt angle ( 45@xmath110 ) , the magnetoresistance ( fig . 5b ) only slightly deviates from the perpendicular field data , which further confirms suppression of the parallel field mr by perpendicular component . to summarize our observations , the positive magnetoresistance that is weak in perpendicular and tilted fields increases dramatically when the field turns parallel to the 2d plane . such behavior is counterintuitive and qualitatively different from that in gaas - based structures . in the present study , by using magnetoresitance tensor measurements technique in tilted magnetic field , we extracted eec and explored zeeman effects in conductivity . we demonstrated fruitfulness of this approach and revealed the anticipated @xcite high - field logarithmic asymptotics of the eec . our measurements also reveal several puzzling features of the magnetotransport in si - mosfets . firstly , the low - field drop of the eec ( i.e. , the increase in hall resistance ) , detected in our experiments , lacks an explanation . in order to clarify this issue one requires more precise very - low - field hall measurements with variety of samples ranging in mobility and conductance values , as well as theoretical framework to treat the problem . another important observation is that the parallel field magnetoresistance is not produced by eec solely . this observation poses a question of the applicability of the eec theory to determination of the interaction constant from the parallel field magnetoresistance @xcite . indeed , our data prove that there is another unexplored mechanismthat also contributes to parallel field magnetoresistance ; it s necessary to disentangle different mr mechanisms , before extracting the fermi - liquid constant . finally , our key result is that the parallel field magnetoresistance is suppressed by rather insignificant nonquantizing perpendicular field component . this result directly contradicts to the predictions of the electron - electron interaction theory @xcite . were the parallel field magnetoresistance caused by eec , such behavior could be straightforvardly explained by complete suppression of magnetoresistance already at @xmath159 , according to eq . ( [ aamrtensor ] ) . however , we have shown above , that eec is too small and alone can not be a reason for the parallel field magnetoresistance . according to eq . ( [ aamrtensor ] ) it means that the parallel field mr comes from mobility renormalization . there are several potential mechanisms of such renormalization : ( i ) mobility may drop with parallel field due to finite width effects @xcite . this mechanism , however , is not supported by our data ( compare the 90@xmath110- and 75@xmath110-curves in fig . [ parallel ] that show a factor of 3 diminished magnetoresistance in almost the same parallel field ) , and is not very probable because of thin potential well in si - mosfets ( @xmath160 nm ) compared to magnetic length @xmath161[nm]@xmath162[t])@xmath163 ; ( ii ) mobility may depend on zeeman splitting in agreement with the predictions of the screening theory @xcite . the latter mechanism should produce direction - independent positive magnetoresistance , exactly the behavior observed in heavily - doped multisubband thin si quantum wells@xcite . this however does not reconcile with our data . ( iii ) surface roughness could also affect mr through the weak localization suppression @xcite , however , this mechanism is hardly relevant because it produces a negative mr in contrast to our observations . experimentally , we observe a picture opposite to the common sense arguments : parallel field magnetoresistance is strongly suppressed when sample is rotated in fixed total field . this observation is not a property of particular studied samples , rather , there are numerous examples in literature where strong positive mr in parallel field coexists with almost shallow or even negative mr in perpendicular or tilted field : for p - gaas @xcite , strained si @xcite , si / sige quantum wells @xcite . dramatic delocalizing effect of the perpendicular field was also reported for high mobility si - mosfet samples in the low - density / high resistivity ( @xmath164 ) regime in the vicinity of the metal - insulator transition @xcite . such behavior is not explained at all , and the question why parallel field mr is suppressed by the perpendicular field component remains open . we believe that the parallel field magnetoresistance suppression in perpendicular field is somehow related to the low - field hall feature ( region a in fig . [ perpfield]g ) . the empiric crossover between the low- and high - field regimes @xmath165 points to the localization - related nature of the low - field hall feature . we , therefore , suggest the following model : a small part of electrons is localizedin low binding - energy states and coexist with mobile electrons . we stress that these states are different from conventional tail of localized states located below the mobility edge @xcite . the localized electrons do not contribute to hall effect at @xmath166 because they can be treated as zero mobility carriers . let us assume also that the localized states promote strong parallel field mr . this assumption does not contradict to the empirical observations of the disorder effect in mr @xcite . application of perpendicular field changes the symmetry class of the system , favors delocalization of these electrons and hence decreases the response of a 2d system to parallel magnetic field . such explanation initiates questions about the source of 10 - 30% electrons that are out of the game at @xmath166 and about their influence on the parallel field mr . the tilted field approach suggested in this paper might be applied for similar study of eec with high - mobility si - mosfets where strong metallic conductivity ( @xmath167 ) and metal - insulator transition are considered to be driven by the eec @xcite . this task is however rather challenging experimentally because it requires a combination of millikelvin temperatures ( to insure the deep diffusive limit @xmath10 for an order of magnitude higher mobility samples ) , field sweep and sample rotation . in this paper we applied phenomenological technique of resistivity tensor analysis to check long - standing prediction by lee and ramakrishnan @xcite about zeeman splitting dependence of the electron - electron interaction correction ( eec ) to conductivity in the diffusive regime @xmath168 . our measurements reveal distinctly different behaviors of the magnetotransport in the two domains of perpendicular field , the low @xmath118-field ( lf ) and the high @xmath118-field ( hf ) one . in the lf domain , the 2d system demonstrates a strong and @xmath18-dependent magnetoresistance versus @xmath169 field , whereas the hall angle exceeds the drude value by up to 30% , depends on @xmath118 , and grows as @xmath18 decreases . this hall anomaly obscures determination of the quantum interaction correction from the magnetotransport data in the lf domain . in the hf domain , the hall anomaly is washed - out by the @xmath118 field that enables extracting the interaction quantum correction from experimental data . the eec was found to be field direction independent , and linear in both @xmath150 and @xmath15 , as expected . remarkably , the magnitude of the experimentally determined eec appeared to be more than a factor of 10 smaller than the parallel field magnetoconductance . thus , the parallel field mr observed in si - mosfets is not explained by the eec solely . even more surprisingly , the observed strong parallel field mr quickly diminishes when the perpendicular field component is applied on top of the parallel field . this fact is the direct evidence for non - purely zeeman origin of the parallel field magnetoresistance . in total , our findings point at the incompleteness of the existing theory of magnetotransport in interacting and disordered 2d systems : too strong parallel field magnetoresistance , its suppression by perpendicular magnetic field , and the low - field hall anomaly require an explanation . we believe that these three phenomena are interrelated and originate from a destructive action of the perpendicular field on the localized states . we thank g.m . minkov , a.v . germanenko , a.a . sherstobitov , i.s . burmistrov , and a.m. finkelstein for discussions , and s.i . dorozhkin for stimulating criticism of the preliminary results . the work was supported by rfbr ( grants 12 - 02 - 00579 ) , by russian ministry of education and science ( grant no 8375 ) , and using research equipment of the shared facilities center at lpi . in the original papers @xcite , the interaction correction field dependence , @xmath170 , is expressed via an integral with known low and high - field asymptotics . this integral is inconvenient in handling , therefore , we used its analytical asymptotics eq . ( [ lowtbasympt ] ) instead . in fig . [ asymptotics ] we compare the exact result @xmath170 for several @xmath171 values [ eq . ( 15 ) in ref . @xcite , note that the dimensionless field scale differs by a factor @xmath172 and high - field linear - in-@xmath15 asymptotics , eq . ( [ lowtbasympt ] ) . to exclude interaction constant dependence in the high - field limit all results were divided by @xmath173 . figure [ asymptotics ] shows that for @xmath92 the exact result for different @xmath171 become indistinguishable from the logarithmic high - field asymptotics . at the same time , in the low field regime the curves substantially deviate from each other : the closer @xmath171 is to -1 , the stronger is the magnetoconductance . this fact has simple physical explanation : in the high - field limit , @xmath83 triplets ( out of all @xmath174 triplet terms ) become suppressed , thus excluding any dependence on @xmath171 . in the low field regime , the closer @xmath175 is to -1 ( i.e. to the stoner instability ) , the larger is the effective @xmath3-factor , and the stronger is the response of conductivity to magnetic field . alternative approach used in ref . @xcite is based on the approximation of the crossover function ( eq.(4 ) from ref . @xcite ) . for comparison , we present this approximation also in fig . [ asymptotics ] . from the practical point of view , the condition @xmath92 means that the total magnetic field @xmath50[t ] should exceed @xmath176[k ] . this inequality is valid for our low - temperature high - field data shown in figs . [ 15tesla ] and [ vg21 ] . nonlinearity of the hall resistance in the low field domain affects both @xmath103 and @xmath121 dependences due to the calculation procedure , based on eq . [ aamctensor ] . most of scattering mechanisms and quantum corrections , including low - field weak localization and maki - thompson corrections , renormalize mobility @xmath41 and do not renormalize hall effect @xcite . eec does affect hall resistivity but has no peculiarities close to zero field . the low field hall feature is thus unexpected in theory , though its experimental observation is not surprising . the lack of agreement between low - field hall resistance and quantum correction theory was first pointed by ovadyahu @xcite and reported several times since then @xcite . although there are some theoretically suggested mechanisms @xcite , this phenomenon is still poorly understood . let us briefly summarize our experimental observations of the low - field nonlinearity in the hall resistance in si - mosfets : ( i ) the effect weakens as temperature increases ( see e.g. fig . [ 15tesla ] ) ; ( ii ) the amplitude of the effect may achieve @xmath177 at low temperatures , i.e. variations in the hall angle are as large as 30% ; ( iii ) the @xmath107 dependence is determined by perpendicular field component . to prove this we compare in fig . 7 @xmath107 for perpendicular field orientation , @xmath122 and @xmath107 for @xmath178 ; ( iv ) the field range , where the hall resistance is nonlinear , broadens as density ( and hence conductivity ) decreases . ( v ) the effect is observed in low - mobility si - mosfets ( @xmath179 m@xmath21/vs ) in the temperature range @xmath180k ( @xmath181 ) . for high mobility samples ( @xmath182 m@xmath21/vs ) we did nt observe any nonlinearity in the hall resistance down to 0.05k ( @xmath183 ) . it might be that the low - field hall nonlinearity is somehow related to weak localization , though weak localization itself does not produce correction to hall coefficient . indeed , let us estimate transport field @xmath184 , ( @xmath185 ) , that is the typical value of perpendicular field component where weak localization is suppressed . by substituting the transport mean free path @xmath96 from the drude formula @xmath186 for a two - valley system , we get : @xmath187 , where @xmath188 is the dimensionless drude resistivity , and @xmath189 for simplicity was taken equal to @xmath190 ; this simplification is justified for @xmath191 kohm . for practical use @xmath192= 0.062\times n[10^{12 } { \rm cm}^{-2}]\times ( \rho_d[{\rm kohm}])^2 $ ] . the coincidence of the estimated @xmath193 t value for the most intensively discussed density @xmath99@xmath31 with characteristic field of the hall anomaly suppression at the lowest temperature ( see figs . 1 , 3 , 4 ) points to the relationship between weak localization and the anomalous low - field hall slope . observation ( iv ) is in line with this scenario , because growth of the resistivity pushes @xmath194 to higher fields . a mechanism of the weak - localization - related low - field nonlinearity in the hall resistance was suggested in ref . @xcite , where the effect was interpreted as the second - order correction to conductivity ; 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rev . 54 * , 437 ( 1982 ) . a. houghton , j.r . senna , s.c . ying , phys . rev . b * 25 * , 2196 ( 1982 ) . the density value @xmath29 was determined preliminary with @xmath200 uncertainty from shubnikov - de - haas oscillations , and adjusted more precisely to make @xmath201 in the high - temperature limit . such procedure was substantiated in ref.@xcite . a.m. finkelstein , zh . exp . phys . * 84 * , 168 - 189 ( 1983 ) . a. punnoose , phys . b * 81 * , 035306 ( 2010 ) g. zala , b.n . narozhny , i.l . aleiner , phys . b * 64 * , 214204 ( 2001 ) . original formula from ref.@xcite contains @xmath202 instead of @xmath203 . we omit the term @xmath204 because it is just a constant shift , which enters to mobility only and does not affect conductivity tensor in eq . 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we study diffusive electron - electron interaction correction to conductivity by analyzing simultaneously @xmath0 and @xmath1 for disordered 2d electron systems in si in tilted magnetic field . tilting the field
is shown to be a straightforward tool to disentangle spin and orbital effects .
in particular , by changing the tilt angle we prove experimentally that in the field range @xmath2 the correction depends on modulus of magnetic field rather than on its direction , which is expected for a system with isotropic @xmath3-factor . in the high - field limit
the correction behaves as @xmath4 , as expected theoretically ( lee , ramakrishnan , phys .
rev . b*26 * , 4009 ( 1982 ) ) .
our data prove that the diffusive electron - electron interaction correction to conductivity is not solely responsible for the huge and temperature dependent magnetoresistance in parallel field , typically observed in si - mosfets .
| 18,110 | 243 |
in recent years loop quantum cosmology ( lqc ) has inspired realisation of the cosmological scenario in which the initial singularity is replaced by the bounce . in this picture , the universe is initially in the contracting phase , reaches the minimal , nonzero volume and , thanks to quantum repulsion , evolves toward the expanding phase . such a scenario has been extensively studied with use of the numerical methods @xcite . however , as it was shown for example in @xcite exact solutions for bouncing universe with dust and cosmological constant can be found . the aim of the present paper is to show that analytical solutions can also be obtained for the bouncing models arising from lqc . the main advantage of such exact solutions is that they allow for investigations in whole ranges of the parameter domains . in this paper we consider the flat frw model with a free scalar field and with the cosmological constant . quantum effects are introduced in terms of correction to the classical theory . generally one considers two types of of quantum correction : correction from inverse volume and holonomy corrections . the leading effect of the volume corrections is the appearance of the super - inflationary phase . the effect of holonomy corrections , on the other hand , is the appearance of a bounce instead of singularity . the aim of this paper is to investigate analytically these effects in a flat frw model . that is to say , we neglect corrections from inverse volume , these effects however , has been extensively studied elsewhere . moreover , these two types of corrections are not equally important in the same regimes . the inverse volume corrections are mainly important for small values of the scale factor , whereas holonomy corrections are mainly important for large values of the hubble parameter . in other words , when the minimal scale factor ( during the bounce ) is large enough , the effects of inverse volume corrections can be neglected . the flat frw model in the loop quantum cosmology has been first investigated in the pioneer works of bojowald @xcite and later improved in the works of ashtekar , pawowski and singh @xcite . bojowald s original description of the quantum universe in currently explored in the number of works and regarded as a parallel line of research @xcite . in the present paper , we restrict ourselves to the flat frw models arising in the framework proposed by ashtekar and co - workers . beside the flat models this approach has also been applied to the frw @xmath2 models in @xcite and bianchi i in @xcite . in these models the unambiguity in the choice of the elementary area for the holonomy corrections appear . in the present paper we consider two kind of approaches to this problem : the so called @xmath3scheme and @xmath4scheme ( for a more detailed description see appendix [ appendix1 ] ) . we find analytical solutions for the considered models in these two schemes . the hamiltonian of the considered model is given by @xmath5 ^ 2 + \frac{1}{2 } \frac { p_{\phi}^2 } { { |p|}^{3/2 } } + { |p|}^{3/2}\frac{\lambda}{8\pi g}. \label{model}\ ] ] in appendix [ appendix1 ] we show the derivation of this hamiltonian in the loop quantum gravity setting . the canonical variables for the gravitational field are @xmath6 and for the scalar field @xmath7 . the canonical variables for the gravitational field can be expressed in terms of the standard frw variables @xmath8 . where the factor @xmath9 is called barbero - immirzi parameter and is a constant of the theory , and @xmath10 is the volume of the fiducial cell . the volume @xmath10 is just a scaling factor and can be chosen arbitrarily in the domain @xmath11 . since @xmath12 is the more natural variable than @xmath13 here , we present mostly @xmath14 in the figures . @xmath13 is always the positive square root of @xmath12 so the shape of the graphs would be essentially the same when drawn with @xmath13 . the equations of motions can now be derived with the use of the hamilton equation @xmath15 where the poisson bracket is defined as follows @xmath16 \nonumber \\ & + & \left[\frac{\partial f}{\partial \phi}\frac{\partial g}{\partial p_{\phi } } - \frac{\partial f}{\partial p_{\phi}}\frac{\partial g}{\partial \phi } \right ] . \end{aligned}\ ] ] from this definition we can immediately retrieve the elementary brackets @xmath17 with use of the hamiltonian ( [ model ] ) and equation ( [ hameq ] ) we obtain equations of motion for the canonical variables @xmath18 ^ 2 \right\ } \nonumber \\ & -&\text{sgn}(p)\frac{\kappa \gamma}{4 } \frac{p_{\phi}^2}{{|p|}^{5/2 } } + \text{sgn}(p ) \frac{\lambda \gamma } { 2}\sqrt{|p| } , \nonumber \\ \dot{\phi } & = & { |p|}^{-3/2 } p_{\phi } , \nonumber \\ \dot{p_{\phi } } & = & 0 , \label{equations}\end{aligned}\ ] ] where @xmath19 . the hamiltonian constraint @xmath20 implies @xmath21 ^ 2 = \frac{\kappa } { 3 } \frac{1}{2 } \frac{p_{\phi}^2}{{|p|}^3 } + \frac{\lambda}{3}. \label{constraint}\ ] ] the variable @xmath22 corresponds to the dimensionless length of the edge of the elementary loop and can be written in the general form @xmath23 where @xmath24 and @xmath25 is a constant @xmath26 ( this comes from the fact that @xmath27 is positively defined ) . the choice of @xmath28 and @xmath25 depends on the particular scheme in the holonomy corrections . in particular , boundary values correspond to the cases when @xmath22 is the physical distance ( @xmath29 , @xmath3scheme ) and coordinate distance ( @xmath30 , @xmath4scheme ) . however , the @xmath30 case does not lead to the correct classical limit . when @xmath31 , the classical limit can not be recovered either . only for negative values of @xmath28 is the classical limit @xmath32 correctly recovered @xmath33 strict motivation of the domain of the parameter @xmath28 comes form the investigation of the lattice states @xcite . the number of the lattice blocks is expressed as @xmath34 where @xmath35 is the average length of the lattice edge . this value is connected to the earlier introduced length @xmath22 , namely @xmath36 . during evolution the increase of the total volume is due to the increase of the spin labels on the graph edges or due to the increase of the number of vortices . in this former case the number of the lattice blocks is constant during evolution , @xmath37 . otherwise , when the spin labels do not change , the number of vortices scales with the volume , @xmath38 . the physical evolution correspond to something in the middle , it means the power index lies in the range @xmath39 $ ] . applying the definition of @xmath40 we see that the considered boundary values translate to the domain of @xmath28 introduced earlier , @xmath41 $ ] . more detailed investigation of the considered ambiguities can be found in the papers @xcite and in the appendix [ appendix1 ] . combining equations ( [ constraint ] ) , ( [ correction ] ) and the first one from the set of equations ( [ equations ] ) we obtain @xmath42 where new parameters are defined as follow @xmath43 equation ( [ mainequation ] ) is , in fact , a modified friedmann equation @xmath44 where the effective constants are expressed as follow @xmath45 , \\ \lambda_{\text{eff } } & = & \lambda \left[1 - 2\frac{\rho}{\rho_{\text{c}}}-\frac{\lambda}{\kappa \rho_{\text{c } } } \right],\end{aligned}\ ] ] and @xmath46 we will study the solutions of the equation ( [ mainequation ] ) for both @xmath3scheme and @xmath4scheme . the organisation of the text is the following . in section [ sec : nolambda ] we consider models with @xmath47 . we find solutions of the equations ( [ mainequation ] ) for both @xmath3scheme and @xmath4scheme . next , in section [ sec : lambda ] we add to our considerations a non - vanishing cosmological constant @xmath1 . we carry out an analysis similar to the case without lambda . we find analytical solutions of the equation([mainequation ] ) for @xmath3scheme . then , we study the behaviour of this case in @xmath4scheme . in section [ sec : summary ] we summarise the results . in this section we begin our considerations with the model without @xmath1 . equation ( [ mainequation ] ) is then simplified to the form @xmath48 we solve this equation for both @xmath3scheme and @xmath4scheme in the present section . in the @xmath3scheme , as is explained in the appendix [ appendix1 ] , the @xmath22 is expressed as @xmath50 where @xmath51 is the area gap . so @xmath52 and @xmath29 . to solve ( [ equation1 ] ) in the considered scheme we introduce a new dependent variables @xmath53 in the form @xmath54 with use of the variable @xmath53 the equation ( [ equation1 ] ) takes the form @xmath55 and has a solution in the form of a second order polynomial @xmath56 where @xmath57 is a constant of integration . we can choose now @xmath58 , so that the minimum of @xmath59 occurs for @xmath60 . going back to the canonical variable @xmath12 we obtain a bouncing solution @xmath61^{1/3}. \label{sol1}\ ] ] the main property of this solutions is that @xmath14 never reaches zero value for non vanishing @xmath62 . the minimal value of @xmath14 is given by @xmath63^{1/3}. \label{pmin}\ ] ] we show the solution ( [ sol1 ] ) in the fig . [ solution1 ] . in the @xmath3scheme . the canonical variable @xmath12 is expressed in the @xmath64 $ ] units and time @xmath65 in the @xmath66 $ ] units.,width=226 ] the dynamical behaviour in this model is simple . for negative times we have a contracting pre - big bang universe . for @xmath67 we have a big bang evolution from minimal @xmath68 ( [ pmin ] ) . it is , however , convenient to call this kind of stage big bounce rather than big bang because of initial singularity avoidance . in the @xmath4scheme the @xmath22 is expressed as @xmath70 so @xmath71 and @xmath30 . to solve ( [ equation1 ] ) in this scheme , we change the time @xmath72 and introduce a new variable @xmath73 as follows @xmath74 applying this new parametrisation to the equation ( [ equation1 ] ) leads to the equation in the form @xmath75 which has a solution @xmath76 we can now choose @xmath77 so that the minimal value of @xmath78 occurs for @xmath79 . now we can go back to the initial parameters @xmath12 and @xmath65 , then @xmath80^{1/4 } \label{time1}\end{aligned}\ ] ] introducing the variable @xmath81 we can rewrite integral ( [ time1 ] ) to the simplest form @xmath82^{1/4 } , \end{aligned}\ ] ] and the solution of such integral is given by @xmath83^{1/4 } dy & = & \frac{2}{3 } x\left(1+x^2 \right)^{1/4 } \nonumber \\ & + & \frac{1}{3 } x { _ 2f_1}\left [ \frac{1}{2},\frac{3}{4 } , \frac{3}{2 } ; -x^2 \right ] , \end{aligned}\ ] ] where @xmath84 is the hypergeometric function , defined as @xmath85 = \sum_{k=0}^{\infty } \frac{(a_1)_k ... (a_p)_k}{(b_1)_k ... (b_q)_k}\frac{x^k}{k!},\ ] ] where @xmath86 is the pochhammer symbol defined as follow @xmath87 this solution is very similar to the one in @xmath3scheme . however , the time parametrisation is expressed in a more complex way . we show this solution in the fig . [ solution2 ] . in the @xmath4scheme . the canonical variable @xmath12 is expressed in @xmath64 $ ] units and time @xmath65 in the @xmath66 $ ] units.,width=226 ] in this case minimal value of @xmath14 is expressed as @xmath88 in this section , we investigate the general model with non vanishing cosmological constant . it will be useful to write equation ( [ mainequation ] ) in the form @xmath90\times \nonumber \\ & \times & \left [ 1- \frac{\omega_{\text{iii}}}{\omega_{\text{i}}}{|p|}^{2n-2 } - \frac{\omega_{\text{v}}}{\omega_{\text{ii}}}{|p|}^{2n+1 } \right ] . \label{equation22}\end{aligned}\ ] ] we see that when we perform the multiplication in this equation and define @xmath91 we recover equation ( [ mainequation ] ) . in this and the next section we use equivalently @xmath92 and @xmath93 to simplify notation . in this case equation ( [ equation22 ] ) can be rewritten to the form @xmath94 where the parameters are defined as @xmath95 , \\ \gamma_{\text{ii } } & = & \omega_{\text{i } } - \omega_{\text{iv } } = \frac{2}{3}\kappa p^2_{\phi}\left[1 - 2\frac{\lambda}{3}\gamma^2 \xi^2 \right ] , \\ \gamma_{\text{iii } } & = & \omega_{\text{iii}}= \frac{1}{9 } \kappa^2 \gamma^2 \xi^2 p^4_{\phi } . \end{aligned}\ ] ] to solve equation ( [ equation2 ] ) we re - parametrise the time variable @xmath96 and introduce a new variable @xmath12 as follows @xmath97 this change of variables leads to the equation in the form @xmath98 where @xmath99 } , \\ y_2 & = & -\frac{1}{2 } \frac{\kappa p^2_{\phi}}{\lambda}.\end{aligned}\ ] ] there are three general types of solutions corresponding to the values of the parameters @xmath100 . we summarise these possibilities in the table below . @xmath101 it is important to notice that the product @xmath102 is negative in all cases . this property will be useful to solve equation ( [ equation3 ] ) . in the fig . [ roots1 ] we show values of the roots @xmath103 and @xmath104 as functions of @xmath1 . and @xmath104 as functions of @xmath1 . region 1 corresponds to oscillatory solution , region 2 to bouncing solution . there is no physical solutions in region 3 . the parameter @xmath1 is expressed in units of @xmath105$].,width=226 ] thus , there are two values of cosmological constants where the signs of the roots change , namely @xmath47 and @xmath106 where we have used the value of the @xmath107 calculated in the work @xcite . more recent investigation of the black hole entropy indicate however that the value of the barbero - immirzi parameter is @xmath108 @xcite . in particular , meissner has calculated @xmath109 @xcite . however , this freedom in the choice of @xmath9 does not change the qualitative results . only the region of the parameter space where the particular kind of motion is allowed can be shifted . we now perform a change of variables in equation ( [ equation3 ] ) in the form @xmath110 and @xmath111 we also introduce the parameters @xmath112 and then equation ( [ equation3 ] ) takes the form @xmath113 the equation ( [ equation4 ] ) is the equation of shifted harmonic oscillator and its solution is @xmath114.\ ] ] where we set the integration constant to zero . it is now possible to go back to the initial parameters @xmath12 and @xmath65 which are expressed as follows @xmath115 { \frac{6\alpha}{\lambda_{0,\bar{\mu } } } } } { \sqrt[3 ] { 1 - 2\frac{\lambda}{\lambda_{0,\bar{\mu } } } + \cos \left(3\sqrt{|\upsilon|}u\right ) } } , \label{sol11 } \\ t(u ) & = & \int_0^u du^ { ' } p^3(u^ { ' } ) , \label{sol22}\end{aligned}\ ] ] the integral ( [ sol22 ] ) can be easily solved but solutions depend on the parameters in ( [ sol11 ] ) . we define now an integral @xmath116 so the expression ( [ sol22 ] ) can now be written as @xmath117 solutions of integral ( [ mainintegral1 ] ) depend on the value of parameter @xmath13 and read @xmath118 , \nonumber \\ i(x , a= 1 ) & = & \tan \left(\frac{x}{2 } \right ) , \nonumber \\ i(x,0<|a|<1 ) & = & -\frac{2}{\sqrt{1-a^2 } } \text{arctanh } \left [ \frac{(a-1)\tan\left ( \frac{x}{2 } \right)}{\sqrt{1-a^2 } } \right ] , \nonumber \\ i(x , a=0 ) & = & \ln \left ( \frac{1}{\cos x } - \tan x \right ) . \nonumber\end{aligned}\ ] ] in fig . [ solution4 ] we show the solution for @xmath119 or equivalently with parameter @xmath120 . . oscillating curve ( green ) is @xmath121 and the increasing curve ( red ) is @xmath122 . the canonical variable @xmath12 is expressed in the @xmath64 $ ] units and time @xmath65 in the @xmath66 $ ] units.,width=226 ] with @xmath119 . the canonical variable @xmath12 is expressed in units of @xmath64 $ ] and time @xmath65 in @xmath66$].,width=226 ] in fig . [ solution5 ] we show the solution for @xmath123 which corresponds to @xmath124 . . top curve ( green ) shows @xmath121 and bottom curve ( red ) is @xmath122 . the canonical variable @xmath12 is expressed in units of @xmath64 $ ] and time @xmath65 in @xmath66$].,width=226 ] with @xmath123 . the canonical variable @xmath12 is expressed in units of @xmath64 $ ] and time @xmath65 in @xmath66$].,width=226 ] in this section we study the last case in which @xmath30 and @xmath125 . equation ( [ equation22 ] ) can then be written in the form @xmath126 where @xmath127 and the polynomial s coefficients are expressed as @xmath128 the discriminant of polynomial ( [ polynomial ] ) is @xmath129 inserting the values of parameters @xmath130 listed above we obtain @xmath131.\end{aligned}\ ] ] for @xmath132 , or @xmath133 polynomial ( [ polynomial ] ) has three real roots . when relation ( [ rel ] ) is fulfilled and @xmath134 , oscillatory solutions occur . for @xmath119 equation ( [ equation6 ] ) has bouncing type solutions . this can be seen when we redefine equation ( [ equation6 ] ) to the point particle form . this approach is useful for qualitative analysis and will be fully used in the next section . however , here we will use it to distinguish between different types of solutions . introducing a new time variable @xmath135 we can rewrite equation ( [ equation6 ] ) to the form @xmath136 where the potential function @xmath137 equation ( [ hamconstr ] ) has the form of hamiltonian constraint for a point particle in a potential well . we see that for @xmath119 potential ( [ pot ] ) has only one extremum ( for @xmath138 ) and only a bouncing solution is possible . for @xmath119 potential ( [ pot ] ) has a minimum for @xmath139 . in this case physical solutions correspond to the condition @xmath140 and the energy of the imagined particle in the potential well is greater than the minimum of well . the particle then oscillates between the boundaries of the potential . the condition @xmath140 is equivalent to relation ( [ rel ] ) calculated from the discriminant . upon introducing the parameters @xmath141 and the variable @xmath142 @xmath143 then equation ( [ equation6 ] ) takes on the form of the weierstrass equation @xmath144 detailed analysis and plots of solutions of this equation for different values of parameters can be found in the appendix to the article @xcite . the solution of equation ( [ weierstrass ] ) is the weierstrass @xmath145-function @xmath146 where @xmath147 so the parametric solution for the parameter @xmath12 is @xmath148\ ] ] the time variable can be expressed as the integral @xmath149 in the fig . [ solution9 ] and [ solution9a ] we show an exemplary parametric bounce solution with @xmath119 and a possible oscillatory solution with @xmath134 . with @xmath119 . the canonical variable @xmath12 is expressed in units of @xmath64 $ ] , @xmath53 is dimensionless.,width=226 ] with @xmath134 and relation ( [ rel ] ) satisfied . the canonical variable @xmath12 is expressed in units of @xmath64 $ ] and @xmath53 is dimensionless.,width=302 ] in general the solution is expressible as an explicit function of time , by means of the so called abelian functions . however , we chose not to employ them here as the appropriate formulae are not as clear , and the commonly used numerical packages do not allow for their direct plotting . the fact of existence of such solutions allows us to assume that the above integral is well defined , and so is the solution itself . the main advantage of using qualitative methods of differential equations ( dynamical systems methods ) is the investigation of all solutions for all admissible initial conditions . we demonstrate that dynamics of the model can be reduced to the form of two dimensional autonomous dynamical system . in our case the phase space is @xmath150 @xmath151 . first we can find the solutions corresponding to vanishing of the right hand sides of the system which are called critical points . the information about their stability and character is contained in the linearisation matrix around a given critical point . in the considered case , the dynamical system is of the newtonian type . for such a system , the characteristic equation which determines the eigenvalues of the linearisation matrix at the critical point is of the form @xmath152 where @xmath153 is a potential function and @xmath154 , @xmath13 being the scale factor . as it is well known for dynamical systems of the newtonian type , only two types of critical points are admissible . if the diagram of the potential function is upper convex then eigenvalues are real and of opposite signs , and the corresponding critical point is of the saddle type . in the opposite case , if @xmath155 then the eigenvalues are purely imaginary and conjugate . the corresponding critical point is of the centre type . equation ( [ equation22 ] ) can be written in the form @xmath156 where @xmath157 or equivalently as @xmath158 where we have made the following time reparametrisation @xmath159 @xmath160 now we are able to write the hamiltonian constraint in the form analogous to the particle of the unit mass moving in the one dimensional potential well @xmath161 where the potential function @xmath162 as we can see , the constant @xmath163 plays the role of the total energy of the fictitious particle . the domain admissible for motion in the configuration space is determined by the condition @xmath164 . a dynamical system of the hamiltonian type has the following form @xmath165 where the prime denotes differentiation with respect to a new re - parametrised time variable which is a monotonous function of the original , cosmological time . the structure of the phase plane is organised by the number and location of critical points . in our case , critical points in the finite domain are located only on the line @xmath166 and the second coordinate is determined form the equation @xmath167 which is @xmath168 the number of critical points depends on the value of @xmath1 and @xmath28 . we can distinguish two cases : * for @xmath169 : * for @xmath29 : the full analysis of the behaviour of trajectories requires investigation also at the infinity . to this aim we introduce radial coordinates on the phase plane for compactification of the plane by adjoining the circle at infinity @xmath170 , @xmath171 . the phase portraits for both cases are shown at figs . [ fig:1 ] and [ fig:2 ] . for the general case of @xmath134 we can write the parametric equation of the boundary @xmath172 of the physically admissible region in the phase space , namely @xmath173 where the dot denotes differentiation with respect to cosmological time @xmath65 . this equation greatly simplifies for the special case @xmath29 , and we receive the value of the hubble parameter at the boundary @xmath174 a ) b ) a ) b ) in this special case @xmath172 and @xmath30 we can integrate eq.(1 ) for @xmath134 and choosing the integration constant equal to zero @xmath175 which gives the maximal value o parameter @xmath12 @xmath176 in the same case with @xmath119 and choosing integration constant equal to zero we have @xmath177 for @xmath172 and @xmath178 in the case @xmath29 we obtain the de sitter solution @xmath179 these solutions represent the lines on the boundaries of the physically admissible domains . we have studied dynamics and analytical solutions of the flat friedmann - robertson - walker cosmological model with a free scalar field and the cosmological constant , modified by the holonomy corrections of loop quantum gravity . we performed calculations in two setups called @xmath3scheme and @xmath4scheme , explained in the appendix a. we have explored whole @xmath180 range and whole allowed @xmath181 range . in the case of @xmath3scheme resulting solutions are of oscillating type for @xmath182 and bouncing type for @xmath183 . for @xmath184 there are no physical solutions . in the case of @xmath4scheme for @xmath182 bouncing solutions occur . when both @xmath185 and relation ( [ rel ] ) are fulfilled , oscillatory behaviour occurs . otherwise , bouncing solutions appear . in all considered cases with @xmath186 the initial singularity is avoided . we have investigated the evolutional paths of the model , from the point of view of qualitative methods of dynamical systems of differential equations . we found that in the special case of @xmath29 the boundary trajectory @xmath187 approaches the de sitter state ; and demonstrate that in the case of positive cosmological constant there are two types of dynamical behaviours in the finite domain . for the case of @xmath30 there appear oscillating solutions without the initial and final singularities , and that they change into bouncing for @xmath188 . the results of this paper can give helpful background dynamics to study variety of physical phenomenas during the bounce epoch in loop quantum cosmology . for example , the interesting question of the fluctuations like gravitational waves@xcite or scalar perturbations @xcite during this period . we tried to show that numerical calculations can be `` shifted '' one step further , since the basic model is explicitly solvable , and can be treated as starting ground for more complex problems , like the above , which can not be solved analytically . authors are grateful to orest hrycyna for discussion . this work was supported in part by the marie curie actions transfer of knowledge project cocos ( contract mtkd - ct-2004 - 517186 ) . in this appendix , we derive the form of the hamiltonian ( [ model ] ) considered in the paper . the frw @xmath0 spacetime metric can be written as @xmath189 where @xmath190 is the lapse function and the spatial part of the metric is expressed as @xmath191 in this expression @xmath192 is fiducial metric and @xmath193 are co - triads dual to the triads @xmath194 , @xmath195 where @xmath196 and @xmath197 . from these triads we construct the ashtekar variables @xmath198 where @xmath199 note that the gaussian constraint implies that @xmath200 leads to the same physical results . the factor @xmath9 is called barbero - immirzi parameter . in the definition ( [ a ] ) the spin connection is defined as @xmath201}+\frac{1}{2}e^c_k e^l_a \partial_{[c}e^l_{b ] } ) , \ ] ] and the extrinsic curvature is defined as @xmath202,\ ] ] which corresponds to @xmath203 . the scalar constraint , in ashtekar variables , has the form @xmath204 } \right ] , \label{scalar}\end{aligned}\ ] ] where field strength is expressed as @xmath205 with use of ( [ a]),([e ] ) and ( [ str ] ) the hamiltonian ( [ scalar ] ) assumes the form @xmath206 where we have assumed a gauge of @xmath207 . the constant @xmath208 is the volume of the fiducial cell . this volume can be chosen arbitrarily . it is convenient to absorb the factor @xmath208 by redefinition @xmath209 the holonomy along a curve @xmath210 is defined as follows @xmath211 where @xmath212 and @xmath213 are the pauli matrices . from this definition we can calculate holonomy in the direction @xmath214 and the length @xmath215 @xmath216 where we used definition of the ashtekar variable @xmath217 ( [ a ] ) . from such a particular holonomies we can construct holonomy along the closed curve @xmath218 . this curve is schematically presented on the diagram below . ( 10,10 ) ( 3,3)(1,0)2 ( 5,3)(1,0)2 ( 7,3)(0,1)2 ( 7,5)(0,1)2 ( 7,7)(-1,0)2 ( 5,7)(-1,0)2 ( 3,7)(0,-1)2 ( 3,5)(0,-1)2 ( 4.5,2)(1,0)1 ( 5.5,8)(-1,0)1 ( 8,4.5)(0,1)1 ( 2,5.5)(0,-1)1 ( 4.5,1.2)@xmath214 ( 8.3,5)@xmath219 ( 4.5,8.5)@xmath220 ( 0.2,5)@xmath221 ( 4.5,3.4)@xmath222 ( 5.8,4.6)@xmath223 ( 4.5,6)@xmath224 ( 3.3,4.6)@xmath225 this holonomy can be written as @xmath226 + \mathcal{o}(\mu^3 ) \right ] \nonumber \\ & = & \mathbb{i } + \mu^2 v_0^{2/3 } f^k_{ab } \tau_k { ^oe^a_i } { ^oe^b_j } + \mathcal{o}(\mu^3 ) , \label{deriv}\end{aligned}\ ] ] where we have used baker - campbell - hausdorff formula and the fact that for flat frw , field strength ( [ str ] ) simplifies to the form @xmath227 . now , equation ( [ deriv ] ) can be simply rewritten to the form @xmath228}{\mu^2 v_0^{2/3 } } { ^o\omega^i_a}{^o\omega^j_b}. \label{lim}\ ] ] the trace in this equation can be calculated with use of definition ( [ hol2 ] ) @xmath229 = - \frac{\epsilon_{kij}}{2 } \sin^2\left(\mu c \right ) . \label{tr}\ ] ] in loop quantum gravity the limit @xmath230 in the formula ( [ lim ] ) does not exist because of existence of the area gap . the area gap corresponds to the minimal quantum of area @xmath231 , which arises as the first non - zero eigenvalue of the are operator @xcite . so instead of the limit in equation ( [ lim ] ) we should stop shrinking the loop at the appropriate minimal area @xmath232 . this area corresponds to the area intersected by the loop . for the holonomy in the direction @xmath214 the area is @xmath233 , as explained in the diagram below ( where @xmath234 ) . the limit @xmath239 corresponds to @xmath240 . now , we must connect the area @xmath233 with the length @xmath27 . we can choose that area @xmath233 to correspond to physical area @xmath241 or to the area @xmath242 . so in the case @xmath243 we have the limit @xmath244 this approach we call @xmath3scheme . in the case @xmath245 where @xmath246 correspond to the eigenvalue @xmath247 and taking the limit @xmath248 which we call @xmath4scheme . in the quantum version , we can combine equations ( [ lim ] ) and ( [ tr ] ) , and write @xmath249 whereas in the classical case we have @xmath250 so , from these two equations we see that quantum effects can be introduced by a replacement @xmath251 in the classical expressions . the hamiltonian ( [ hami ] ) with holonomy correction takes the form @xmath5 ^ 2 .\ ] ] p. singh , k. vandersloot and g. v. vereshchagin , phys . d * 74 * ( 2006 ) 043510 [ arxiv : gr - qc/0606032 ] . a. ashtekar , t. pawlowski and p. singh , phys . d * 73 * ( 2006 ) 124038 [ arxiv : gr - qc/0604013 ] . t. stachowiak and m. szydlowski , phys . b * 646 * ( 2007 ) 209 [ arxiv : gr - qc/0610121 ] . a. ashtekar , t. pawlowski and p. singh , phys . rev . lett . * 96 * ( 2006 ) 141301 [ arxiv : gr - qc/0602086 ] . m. bojowald , class . * 19 * ( 2002 ) 2717 [ arxiv : gr - qc/0202077 ] . m. bojowald , class . * 20 * ( 2003 ) 2595 [ arxiv : gr - qc/0303073 ] . a. ashtekar , t. pawlowski and p. singh , phys . d * 74 * ( 2006 ) 084003 [ arxiv : gr - qc/0607039 ] . a. ashtekar , t. pawlowski , p. singh and k. vandersloot , phys . d * 75 * ( 2007 ) 024035 [ arxiv : gr - qc/0612104 ] . m. bojowald , phys . d * 74 * ( 2007 ) 081301 [ arxiv : gr - qc/0608100 ] . m. bojowald , arxiv:0801.4001 [ gr - qc ] . k. vandersloot , phys . d * 75 * ( 2007 ) 023523 [ arxiv : gr - qc/0612070 ] . l. szulc , w. kaminski and j. lewandowski , class . * 24 * ( 2007 ) 2621 [ arxiv : gr - qc/0612101 ] . l. szulc , arxiv:0707.1816 [ gr - qc ] . d. w. chiou , arxiv:0710.0416 [ gr - qc ] . d. w. chiou , arxiv : gr - qc/0703010 . m. bojowald , gen . grav . * 40 * , 639 ( 2008 ) [ arxiv:0705.4398 [ gr - qc ] ] . m. p. dabrowski and t. stachowiak , annals phys . * 321 * ( 2006 ) 771 [ arxiv : hep - th/0411199 ] . a. ashtekar , j. baez , a. corichi and k. krasnov , phys . . lett . * 80 * ( 1998 ) 904 [ arxiv : gr - qc/9710007 ] . m. domagala and j. lewandowski , class . * 21 * ( 2004 ) 5233 [ arxiv : gr - qc/0407051 ] . k. a. meissner , class . * 21 * ( 2004 ) 5245 [ arxiv : gr - qc/0407052 ] . m. bojowald and g. m. hossain , arxiv:0709.2365 [ gr - qc ] . j. mielczarek and m. szydlowski , phys . b * 657 * ( 2007 ) 20 [ arxiv:0705.4449 [ gr - qc ] ] . j. mielczarek and m. szydlowski , arxiv:0710.2742 [ gr - qc ] . m. bojowald , h. h. hernandez , m. kagan , p. singh and a. skirzewski , phys . d * 74 * ( 2006 ) 123512 [ arxiv : gr - qc/0609057 ] . m. bojowald , gen . * 38 * ( 2006 ) 1771 [ arxiv : gr - qc/0609034 ] . a. ashtekar and j. lewandowski , class . * 14 * ( 1997 ) a55 [ arxiv : gr - qc/9602046 ] .
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in this paper we study the flat ( @xmath0 ) cosmological frw model with holonomy corrections of loop quantum gravity . the considered universe contains a massless scalar field and the cosmological constant @xmath1 .
we find analytical solutions for this model in different configurations and investigate its dynamical behaviour in the whole phase space .
we show the explicit influence of @xmath1 on the qualitative and quantitative character of solutions . even in the case of positive @xmath1
the oscillating solutions without the initial and final singularity appear as a generic case for some quantisation schemes .
| 11,083 | 143 |
the cataclysmic variable hu aqr currently consists of a 0.80 white dwarf that accretes from a 0.18 main - sequence companion star . the transfer of mass in the tight @xmath12 orbit is mediate by the emission of gravitational waves and the strong magnetic field of the accreting star . since its discovery , irregularities of the observed - calculated variations have led to a range of explanations , including the presence of circum - binary planets . detailed timing analysis has eventually led to the conclusion that the cv is orbited by two planets @xcite , a 5.7 planet in a @xmath13 orbit with an eccentricity of @xmath14 and a somewhat more massive ( 7.6 ) planet in a wider @xmath15 and eccentric @xmath16 orbit @xcite . although , the two - planet configuration turned out to be dynamically unstable on a 100010,000 year time scale ( * ? ? ? * see also [ sect : stability ] ) , a small fraction of the numerical simulations exhibit long term dynamical stability ( for model b2 in * ? ? ? * see tab.[tab : huaqr ] for the parameters ) . it is peculiar to find a planet orbiting a binary , in particular around a cv . while planets may be a natural consequence of the formation of binaries @xcite , planetary systems orbiting cvs could also be quite common . in particular because of recently timing residual in nn serpentis , dp leonis and qs virgo @xcite were also interpreted is being caused by circum - cv planets . although the verdict on the planets around hu aqr ( and the other cvs ) remains debated ( tom marsh private communication , and * ? ? ? * ) , we here demonstrate how a planet in orbit around a cv , and in particular two planets , can constrain the cv evolution and be used to reconstruct the history of the inner binary . we will use the planets to perform a precision reconstruction of the binary history , and for the remaining paper we assume the planets to be real . because of their catastrophic evolutionary history , cvs seem to be the last place to find planets . the original binary lost probably more than half its mass in the common - envelope phase , which causes the reduction of the binary separation by more than an order of magnitude . it is hard to imagine how a planet ( let alone two ) can survive such turbulent past , but it could be a rather natural consequence of the evolution of cvs , and its survival offers unique diagnostics to constrain the origin and the evolution of the system . after the birth of the binary , the primary star evolved until it overflowed it roche lobe , which initiated a common - envelope phase . the hydrogen envelope of the primary was ejected quite suddenly in this episode @xcite , and the white dwarf still bears the imprint of its progenitor : the mass and composition of the white dwarf limits the mass and evolutionary phase of its progenitor star at the moment of roche - lobe overflow ( rlof ) . for an isolated binary the degeneracy between the donor mass at the moment of rlof ( @xmath17 ) , its radius @xmath18 and the mass of its core @xmath19 can not be broken . the presence of the inner planet in orbit around hu aqr @xcite allows us to break this degeneracy and derive the rate of mass loss in the common - envelope phase . the outer planet allows us to validate this calculation and in addition to determine the conditions under which the cv was born . the requirement that the initial binary must have been dynamically stable further constrains the masses of the two stars and their orbital separation . during the cv phase little mass is lost from the binary system @xmath20constant ( but see * ? ? ? * ) , and the current total binary mass ( @xmath21 ) was not affected by the past ( and current ) cv evolution @xcite . the observed white dwarf mass then provides an upper limit to the mass of the core of the primary star at the moment of roche - lobe contact , and therefore also provides a minimum to the companion mass via @xmath22 . with the mass of the companion not being affected by the common envelope phase , we constrain the orbital parameters at the moment of rlof by calculating stellar evolution tracks to measure the core mass @xmath19 and the corresponding radius @xmath23 for stars with zero - age main - sequence mass @xmath24 . in fig.[fig : amcoreformzams3msun ] we present the evolution of the radius of a 3 star as a function of @xmath19 , which is a measure of time we adopted the henyey stellar evolution code mesa @xcite to calculate evolutionary track of stars from @xmath25 to 8 using amuse @xcite to run mesa and determine the mass of the stellar core . the latter is measured by searching for the mass - shell in the stellar evolution code for which the relative hydrogen fraction @xmath26 . at the moment of rlof the core mass is @xmath19 and the stellar radius @xmath27 . via the relation for the roche radius @xcite , we can now calculate the orbital separation at the moment of rlof @xmath28 as a function of @xmath17 . this separation is slightly larger than the initial ( zero - age ) binary separation @xmath29 due to the mass lost by the primary star since its birth @xmath30 . the long ( main - sequence ) time scale in which this mass is lost guarantees an adiabatic response to the orbital separation , i.e. @xmath31 constant . for each @xmath24 we now have a range of possible solutions for @xmath28 as a function of @xmath19 and @xmath32 . this reflects the assumption that the total mass ( @xmath33 ) in the observed binary with mass @xmath34 is conserved throughout the evolution of the cv . in fig.[fig : amcoreformzams3msun ] we present the corresponding stellar radius @xmath18 and @xmath35 as a function of @xmath19 for @xmath36 . this curve for @xmath28 is interrupted when rlof would already have been initiated earlier for that particular orbital separation . we calculate this curve by first measuring the size of the donor for core mass @xmath19 , and assuming that the primary fills its roche - lobe we calculate the orbital separation at which this happens . during the common envelope phase the primary s mantle is blown away beyond the orbit of the planets . the latter responds to this by migrating from the orbits in which they were born ( semi - major axis @xmath37 and eccentricity @xmath38 , the subscript @xmath39 indicates the inner planet , we adopt a @xmath40 to indicate the outer planet ) to the currently observed orbits . using first order analysis we recognize two regimes of mass loss : fast and slow . in the latter case the orbit expands adiabatically without affecting the eccentricity : the minimum possible expansion of the planet s orbit is achieved when the common envelope is lost adiabatically . fast mass loss leads to an increase in the eccentricity as well and may even cause the planet to escape @xcite . a planet born at the shortest possible orbital separation to be dynamically stable will have @xmath41 @xcite , which is slightly smaller than the distance at which circum binary planets tend to form @xcite . in fig.[fig : amcoreformzams3msun ] we present a minimum to the semi - major axis for a planet that was born at @xmath42 and migrated by the adiabatic loss of the hydrogen envelope from the primary star in the common - envelope phase . the planet can have migrated to a wider orbit , but not to an orbit smaller than the solid black curve ( indicated with @xmath43 ) in fig.[fig : amcoreformzams3msun ] . for the 3 star , presented in fig.[fig : amcoreformzams3msun ] , rlof can successfully result in the migration of the planet to the observed separation in hu aqr for @xmath44 , which occurs for @xmath45 . a core mass @xmath46 would , for a 3 primary star , result in an orbital separation that exceeds that of the inner planet in hu aqr ; in this case the core mass of the primary star must have been smaller than 0.521 . another constraint on the initial binary orbit is provided by the requirement that the mass transfer in the post common - envelope binary should be stable when the companion starts to overfill its roche lobe . to guarantee stable mass transfer we require that @xmath47 . the thick part of the red curve in fig.[fig : amcoreformzams3msun ] indicates the valid range for the initial orbital separation and core - mass for which the observed planet can be explained ; the thin parts indicate where these criteria fail . we repeat the calculation presented in fig.[fig : amcoreformzams3msun ] for a range of masses from @xmath48 to 8 with steps of 0.02 , the results are presented as the shaded region in fig.[fig : am0_distribution_hu ] . the response of the orbit of the planet to the mass loss depends on the total amount of mass lost in the common envelope and the rate at which it is lost . numerical common - envelope studies indicate that for an in - spiraling binary @xmath49 @xcite . at this rate the entire envelope @xmath505.8 is expelled well within one orbital period of the inner planet , which leads to an impulsive response and the possible loss ( for @xmath51 ) of the planet . the fact that the hu aqr is orbited by a planet indicates that at the distance of the planet @xmath52 . the eccentricity of the inner planet in hu aqr ( see tab.[tab : huaqr ] ) can be used to further constrain the rate at which the common - envelope was lost from the planetary orbit . the higher eccentricity of the outer planet indicates a more impulsive response , which is a natural consequence of its wider orbits with the same @xmath53 . this regime between adiabatic and impulsive mass loss is hard to study analytically @xcite . [ cols="<,<,<,<,<,<,<,<",options="header " , ] we calculate the effect of the mass loss on the orbital parameters by numerically integrating the planet orbit . the calculations are started by selecting initial conditions for the zero - age binary hu aqr @xmath24 , @xmath29 and consequently @xmath19 from the available parameter space ( shaded area ) in fig.[fig : am0_distribution_hu ] , and integrate the equations of motion of the inner planet with time . planets ware assumed to be born in a circular orbit ( @xmath54 ) in the binary plane with semi - major axis @xmath37 . the equations of motions are integrated using the high - order symplectic integrator huayno @xcite via the amuse framework . during the integration we adopt a constant mass - loss rate @xmath53 applied at every 1/100th of an orbit , and we continued the calculation until the entire envelope is lost ( see [ sect : ce ] and fig.[fig : am0_distribution_hu ] ) , at which time we measure the final semi - major axis and eccentricity of the planetary orbit . during the integration we allow the energy error to increase up to at most @xmath55 . by repeating this calculation while varying @xmath37 and @xmath56 we iterate ( by bisection ) until the result is within 1% of the observed @xmath57 and @xmath58 of the inner planet observed in hu aqr . the converged results of these simulations are presented in fig.[fig : am0_distribution_hu ] ( circles ) , and these represent the range of consistent values for the inner planet s orbital separation @xmath59752 as a function of @xmath608 and consistently reproduce the observed inner planet when adopting @xmath610.267/yr . the highest value for @xmath53 is reached for @xmath62 at an initial orbital separation of @xmath63 . the orbital solution for the inner planet is insensitive to the semi - major axis of the zero - age binary @xmath29 ( for a fixed @xmath24 ) , and each of these solutions were tested for dynamical stability , which turned out to be the case irrespective of the initial binary semi - major axis ( as discussed in [ sect : stability ] ) . we now adopt the in [ sect : innerplanet ] measured value of @xmath53 to integrate the orbit of the outer planet . the effect of the mass outflow on the planet is proportional to the square of the density in the wind at the location of the planet . we correct for this effect by reducing the mass loss rate in the common envelope that affects the outer planet by a factor @xmath64 . we use the same integrator and assumptions about the initial orbits as in [ sect : innerplanet ] , but we adopt the value of @xmath53 from our reconstruction of the inner planet ( see [ sect : innerplanet ] ) . to reconstruct the initial orbital separation of the outer planet @xmath65 , we vary this value ( by bisection ) until the final semi - major axis is within 1% of the observed orbit ( see tab.[tab : huaqr ] ) . the results are presented in fig.[fig : am0_distribution_hu ] ( triangles ) . the post common - envelope eccentricity of the outer planet then turn out to be @xmath66 . after having reconstructed the initial conditions of the binary system with its two planets we test its dynamical stability by integrating the entire system numerically for 1myr using the huayno integrator @xcite . to test the stability we check the semi - major axis and eccentricity of both planets every 100years . if any of these parameters change by a factor of two compared to the initial values or if the orbits cross we declare the system unstable , otherwise they are considered stable . the calculations are repeated with the 4th order hermite predictor - corrector integrator ph4 @xcite within amuse to verify that the results are robust , which turned out to be the case . we then repeated this calculation ten times with random inital orbital phases and again with a 1% gaussian variation in the initial planetary semi - major axes . in fig.[fig : am0_distribution_hu ] we present the resulting stable systems by coloring them red ( circled ) and blue ( triangles ) , the unstable systems are represented by open symbols . from the wide range of possible systems that can produce hu aqr only a small range around @xmath67 turns out to be dynamically stable . the eccentricity of the outer orbit of the stable systems ( which ware stable for initial conditions within 1% ) @xmath68 , which is somewhat smaller than the observed value for hu aqr ( * ? ? ? * @xmath69 ) . these values are obtained with @xmath70 . the small uncertainty in the derived value of @xmath53 is a direct consequence of its sensitivity to @xmath38 and the small error on @xmath24 from the requirement that the initial system is dynamically stable . we have adopted the suggestive results from the timing analysis of hu aqr , that the cv is orbited by two planets , to reconstruct the evolution of this complex system . a word of caution is well placed in that these observations are not confirmed , and currently under debate ( tom marsh private communication , and comments by the referee ) . however , the predictive power that such an observation would entail is interesting . the possibility to reconstruct the initial conditions of a cv by measuring the orbital parameters of two circum binary planets is a general result that can be applied to other binaries . for cvs in particular it enables us to constrain the value of fundamental parameters in the common - envelope evolution . this in itself makes it interesting to perform this theoretical exercise , irrespective of the uncertainty in the observations . on the other hand , the consistency between the observations and the theoretical analysis give some trust to the correctness of these observations . the presence of one planet in an eccentric orbit around a cv allow us to calculate the rate at which the common - envelope was lost from the inner binary . a single planet provides insufficient information to derive the initial mass of the primary star , but allows us to derive the initial binary separation and planetary orbital separation to within about factor of 5 , and the initial rate of mass loss from the common envelope to about a factor 2 . a second planet can be used to further constrain these parameters to a few per cent accuracy and allows us to make a precision reconstruction of the evolution of the cv . we have used the observed two planets in orbit around the cv hu aqr to reconstruct its evolution , to derived its initial conditions ( primary mass , secondary mass , orbital separation , and the orbital separations of both planets ) and to measure the rate of mass lost in the common - envelope parameters @xmath53 . by comparing the binary parameters at birth with those after the common - envelope phase we subsequently calculate the two parameters @xmath71 and @xmath72 . the measured rate of mass loss for hu aqr of @xmath73 from the inner planetary orbit , which from the binary system itself would entail a mass - loss rate of @xmath74 , when we adopt the initial binary to have a semi - major axis of @xmath75 , which is bracketed by our derived range of @xmath76160 . this is consistent with a mass - loss rate of @xmath77 from numerical common - envelope studies @xcite . by adopting that the binary survives its common envelope at a separation between @xmath78 ( at which separation the secondary star will just fill it s roche - lobe to the white dwarf ) and @xmath79 ( for gravitational wave radiation to drive the binary into roche - lobe overflow within 10gyr ) , we derive the value of @xmath802.0 ( for @xmath75 we arrive at @xmath81 ) . this value is a bit small compared to numerous earlier studies , which tend to suggest @xmath82 . the alternative @xmath72-formalism for common - envelope ejection gives a value of @xmath831.80 ( for @xmath75 we arrive at @xmath84 ) , which is consistent with the determination of @xmath72 in 30 other cvs @xcite . the inner planet in hu aqr formed at @xmath85@xmath86 , with a best value of @xmath87 , which is consistent with the planets found to orbit other binaries , like kepler 16 @xcite and for kepler 34 and 35 @xcite , although these systems have lower primary mass and secondary mass stars . it seems unlikely that more planets were formed inside the orbit of the inner most planet , even though currently there is sufficient parameter space for many more stable planets ; in the zero - age binary there has not been much room for forming additional planets further in . it is however possible that additional planets formed further out and those , we predict , will have even higher eccentricity than those already found . * acknowledgements * it is a pleasure to thank edward p.j . van den heuvel , tom marsh , inti pelupessy , nathan de vries , arjen van elteren and the anonymous referee for comments on the manuscript and discussions . this work was supported by the netherlands research council nwo ( grants # 612.071.305 [ lgm ] , # 639.073.803 [ vici ] and # 614.061.608 [ amuse ] ) and by the netherlands research school for astronomy ( nova ) .
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cataclysmic variables ( cvs ) are binaries in which a compact white dwarf accretes material from a low - mass companion star .
the discovery of two planets in orbit around the cv hu aquarii opens unusual opportunities for understanding the formation and evolution of this system .
in particular the orbital parameters of the planets constrains the past and enables us to reconstruct the evolution of the system through the common - envelope phase . during this dramatic event the entire hydrogen envelope of the primary star
is ejected , passing the two planets on the way .
the observed eccentricities and orbital separations of the planets in hu aqr enable us to limit the common - envelope parameter @xmath0 or @xmath1 and measure the rate at which the common envelope is ejected , which turns out to be copious .
the mass in the common envelope is ejected from the binary system at a rate of @xmath2 .
the reconstruction of the initial conditions for hu aqr indicates that the primary star had a mass of @xmath3 and a @xmath4 companion in a @xmath5160 ( best value @xmath6 ) binary .
the two planets were born with an orbital separation of @xmath7 and @xmath8 respectively . after the common envelope , the primary star turns into a @xmath9 helium white dwarf , which subsequently accreted @xmath10 from its roche - lobe filling companion star , grinding it down to its current observed mass of @xmath11 .
methods : numerical planets and satellites : dynamical evolution and stability planet star interactions planets and satellites :
formation stars : individual : hu aquarius stars : binaries : evolution
| 5,106 | 431 |
the @xmath0 ncsm / rgm was presented in @xcite as a promising technique that is able to treat both structure and reactions in light nuclear systems . this approach combines a microscopic cluster technique with the use of realistic interactions and a consistent @xmath0 description of the nucleon clusters . the method has been introduced in detail for two - body cluster bases and has been shown to work efficiently in different systems @xcite . however , there are many interesting systems that have a three - body cluster structure and therefore can not be successfully studied with a two - body cluster approach . the extension of the ncsm / rgm approach to properly describe three - body cluster states is essential for the study of nuclear systems that present such configuration . this type of systems appear , @xmath3 , in structure problems of two - nucleon halo nuclei such as @xmath1he and @xmath4li , resonant systems like @xmath5h or transfer reactions with three fragments in their final states like @xmath6h(@xmath6h,2n)@xmath2he or @xmath6he(@xmath6he,2p)@xmath2he . recently , we introduced three - body cluster configurations into the method and presented the first results for the @xmath1he ground state @xcite . here we present these results as well as first results for the continuum states of @xmath1he within a @xmath2he+n+n basis . the extension of the ncsm / rgm approach to properly describe three - cluster configurations requires to expand the many - body wave function over a basis @xmath7 of three - body cluster channel states built from the ncsm wave function of each of the three clusters , @xmath8 @xmath9^{(j^{\pi}t ) } \times \frac{\delta(x-\eta_{a_2-a_3})}{x\eta_{a_2-a_3 } } \frac{\delta(y-\eta_{a - a_{23}})}{y\eta_{a - a_{23}}}\ , , \label{eq:3bchannel } \end{aligned}\ ] ] where @xmath10 is the relative vector proportional to the displacement between the center of mass ( c.m . ) of the first cluster and that of the residual two fragments , and @xmath11 is the relative coordinate proportional to the distance between the centers of mass of cluster 2 and 3 . in eq . ( [ eq1 ] ) , @xmath12 are the relative motion wave functions and represent the unknowns of the problem and @xmath13 is the intercluster antisymmetrizer . projecting the microscopic @xmath14-body schrdinger equation onto the basis states @xmath15 , the many - body problem can be mapped onto the system of coupled - channel integral - differential equations @xmath16 g_{\nu}^{j^\pi t}(x , y ) = 0,\label{eq:3beq1 } \end{aligned}\ ] ] where @xmath17 is the total energy of the system in the c.m . frame and @xmath18 are integration kernels given respectively by the hamiltonian and overlap ( or norm ) matrix elements over the antisymmetrized basis states . finally , @xmath19 is the intrinsic @xmath14-body hamiltonian . in order to solve the schrdinger equations ( [ eq:3beq1 ] ) we orthogonalize them and transform to the hyperspherical harmonics ( hh ) basis to obtain a set of non - local integral - differential equations in the hyper - radial coordinate , @xmath20 which is finally solved using the microscopic r - matrix method on a lagrange mesh . the details of the procedure can be found in @xcite . at present , we have completed the development of the formalism for the treatment of three - cluster systems formed by two separate nucleons in relative motion with respect to a nucleus of mass number a@xmath21 . it is well known that @xmath1he is the lightest borromean nucleus @xcite , formed by an @xmath2he core and two halo neutrons . it is , therefore , an ideal first candidate to be studied within this approach . in the present calculations , we describe the @xmath2he core only by its g.s . wave function , ignoring its excited states . this is the only limitation in the model space used . we used similarity - renormalization - group ( srg ) @xcite evolved potentials obtained from the chiral n@xmath6lo nn interaction @xcite with @xmath22 = 1.5 @xmath23 . the set of equations ( [ rgmrho ] ) are solved for different channels using both bound and continuum asymptotic conditions . we find only one bound state , which appears in the @xmath24 channel and corresponds to the @xmath1he ground state . [ [ ground - state ] ] ground state + + + + + + + + + + + + [ tab : a ] lccc approach & & e@xmath25(@xmath2he ) & e@xmath25(@xmath1he ) + ncsm / rgm & ( @xmath26=12 ) & @xmath27 mev & @xmath28 mev + ncsm & ( @xmath26=12 ) & @xmath29 mev & @xmath30 mev + ncsm & ( extrapolated ) & @xmath31 mev & @xmath32 mev + the results for the g.s . energy of @xmath1he within a @xmath2he(g.s.)+n+n cluster basis and @xmath26 = 12 , @xmath33 = 14 mev harmonic oscillator model space are compared to ncsm calculations in table [ tab : a ] . at @xmath34 12 the binding energy calculations are close to convergence in both ncsm / rgm and ncsm approaches . the observed difference of approximately 1 mev is due to the excitations of the @xmath2he core , included only in the ncsm at present . therefore , it gives a measure of the polarization effects of the core . the inclusion of the excitations of the core will be achieved in a future work through the use of the no - core shell model with continuum approach ( ncsmc ) @xcite , which couples the present three - cluster wave functions with ncsm eigenstates of the six - body system . contrary to the ncsm , in the ncsm / rgm the @xmath2he(g.s.)+n+n wave functions present the appropriate asymptotic behavior . the main components of the radial part of the @xmath1he g.s . wave function @xmath35 can be seen in fig . ( [ fig:1 ] ) for different sizes of the model space demostrating large extension of the system . in the left part of the figure , the probability distribution of the main component of the wave function is shown , featuring two characteristic peaks which correspond to the di - neutron and cigar configurations . a thorough study of the converge of the results with respect to different parameters of the calculation was presented in @xcite , showing good convergence and stability . he+@xmath36+@xmath36 relative motion wave function for the @xmath37 ground state , @xmath38 and @xmath39 are the distances between the two neutrons and between the @xmath40 particle and center of mass of the two neutrons , respectively . in the right , the three main components of the radial part of the @xmath1he g.s . wave functions @xmath35 for @xmath26=6,8,10 , and 12 . , title="fig:",height=226 ] he+@xmath36+@xmath36 relative motion wave function for the @xmath37 ground state , @xmath38 and @xmath39 are the distances between the two neutrons and between the @xmath40 particle and center of mass of the two neutrons , respectively . in the right , the three main components of the radial part of the @xmath1he g.s . wave functions @xmath35 for @xmath26=6,8,10 , and 12 . , title="fig:",height=207 ] [ [ continuum - states ] ] continuum states + + + + + + + + + + + + + + + + the use of three - cluster dynamics is essential for describing @xmath1he states in the continuum . therefore , this formalism is ideal for such study . using continuum asymptotic conditions , we solved the set of equations ( [ rgmrho ] ) in order to obtain the low - energy phase shifts for the @xmath41 and @xmath42 channels in the continuum . in our preliminary results , we obtain the experimentally well - known @xmath43 resonance as well as a second low - lying @xmath44 resonance recently measured at ganil @xcite . a resonance is also found in the @xmath45 channel while no low - lying resonances are present in the @xmath46 or @xmath47 channels . in fig . [ fig:2 ] some of the preliminary phase shifts for different channels are shown . results for bigger model spaces and a study of their stability respect to the parameters in the formalism are presently being calculated and will be presented elsewhere . he for different @xmath48 channels.,height=226 ] in this work , we present an extension of the ncsm / rgm which includes three - body dynamics in the formalism . this new feature permits us to study a new range of systems that present three - body configurations . in particular , we presented results for both bound and continuum states of @xmath1he studied within a basis of @xmath2he+n+n . the obtained wave functions feature an appropriate asymptotic behavior , contrary to bound - state @xmath0 methods such as the ncsm . computing support for this work came from the llnl institutional computing grand challenge program and from an incite award on the titan supercomputer of the oak ridge leadership computing facility ( olcf ) at ornl . prepared in part by llnl under contract de - ac52 - 07na27344 . support from the u.s . doe / sc / np ( work proposal no . scw1158 ) and nserc grant no . 401945 - 2011 is acknowledged . triumf receives funding via a contribution through the canadian national research council . 13 quaglioni s , navrtil p ( 2008 ) ab initio many - body calculations of n-3h , n-(4)he , p-(3,4)he , and n-(10)be scattering . phys rev lett 101:092501 quaglioni s , navrtil p ( 2009 ) ab initio many - body calculations of nucleon - nucleus scattering . phys rev c 79:044606 navrtil p , quaglioni s ( 2011 ) ab initio many - body calculations of deuteron - he-4 scattering and li-6 states . phys rev c 83:044609 navrtil p , quaglioni s ( 2012 ) ab initio many - body calculations of the h-3(d , n)he-4 and he-3(d , p)he-4 fusion reactions . phys rev lett 108:042503 quaglioni s , romero - redondo c , navrtil p ( 2013 ) three - cluster dynamics within an ab initio framework . phys rev c 88:034320 tanihata i ( 1996 ) neutron halo nuclei . j phys g 22:157 - 198 tanihata i , hamagaki h , hashimoto o , shida y , yoshikawa n , sugimoto k , yamakawa o , kobayashi t , takahashi n ( 1985 ) measurements of interaction cross sections and nuclear radii in the light p - shell region . phys rev lett 55:2676 - 2679 bogner sk , furnstahl rj , perry rj ( 2007 ) similarity renormalization group for nucleon - nucleon interac- tions . phys rev c 75:061001 roth r , reinhardt s , hergert h ( 2008 ) unitary correlation operator method and similarity renormalization group : connections and differences . phys rev c 77:064003 entem dr , machleidt r ( 2003 ) accurate charge - dependent nucleon - nucleon potential at fourth order of chiral perturbation theory . phys rev c 68:041001 baroni s , navrtil p , quaglioni s ( 2013 ) ab initio description of the exotic unbound @xmath49he nucleus . phys rev lett 110:022505 baroni s , navrtil p , quaglioni s ( 2013 ) unified ab initio approach to bound and unbound states : no - core shell model with continuum and its application to @xmath49he . phys rev c 87:034326 mougeot x , lapoux v , mittig w , alamanos n , auger f , et al ( 2012 ) new excited states in the halo nucleus he-6 . phys lett b 718:441 - 446
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we introduce an extension of the @xmath0 no - core shell model / resonating group method ( ncsm / rgm ) in order to describe three - body cluster states .
we present results for the @xmath1he ground state within a @xmath2he+n+n cluster basis as well as first results for the phase shifts of different channels of the @xmath2he+n+n system which provide information about low - lying resonances of this nucleus .
example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
| 3,679 | 181 |
observations of starburst galaxies can provide vital insights into the processes and spectral characteristics of massive star formation regions . in such regions the physical conditions are similar to those that existed at the time of collapse and formation of galaxies in the early universe , and they can also provide an understanding of early galaxy evolution . the _ infrared astronomical satellite _ ( iras ) made the key discovery of large numbers of infrared luminous galaxies , similar to those found by @xcite . many of these are dominated by intense star formation @xcite in which the luminosity of the young hot stars heats the surrounding dust , producing large amounts of infrared radiation . the theoretical tools required to interpret the spectra of such galaxies are now available . for example , detailed stellar population synthesis models have been developed for both instantaneous and continuous starbursts and using these models , one is able to derive parameters such as the starburst age and metallicity from the continuous spectrum . in such models , the stellar initial mass function ( imf ) , star formation rate ( sfr ) and stellar atmosphere formulations are all adjustable initial parameters . the emission line spectrum of the starburst provides constraints on the physical parameters for the ionized gas and the interstellar medium in general . in particular , the gas density , temperature and pressure can be derived directly from such observations , and the total rates of star formation can be estimated from the luminosity in the balmer lines of hydrogen for the objects without large quantities of dust at least ( _ eg _ @xcite ) . using the ionizing uv radiation fields produced by stellar population synthesis models in conjunction with detailed self - consistent photoionization models such as mappings iii @xcite or cloudy @xcite we can now generate models for any region or starburst . in such models it is vital to include a self - consistent treatment of dust physics and the depletion of various elements out of the gas phase . since the nebular emission line spectrum is very sensitive to the hardness of the ionizing euv radiation , optical line ratio diagnostic diagrams provide an important constraint on the shape of the euv spectrum and these may also be used to estimate the mean ionization parameter and metallicity of the galaxies . such optical diagnostic diagrams were first proposed by @xcite to classify galaxies into starburst or agn type , since agn have a much harder ionizing spectrum than hot stars . the classification scheme was revised by @xcite and @xcite , hereafter vo87 . these revised diagnostics are used here . for both schemes , the line diagnostic tools are based on emission - line intensity ratios which turn out to be particularly sensitive to the hardness of the euv radiation field . in an earlier paper @xcite , we theoretically recalibrated the extragalactic region sequence using these line diagnostic diagrams and others , in order to separate and quantify the effects of abundance , ionization parameter and continuous vs. instantanteous burst models . the theoretical region models were generated by the mappings iii code which uses as input the euv fields predicted by the stellar population synthesis models pegase v2.0 @xcite and starburst99 @xcite . dust photoelectric heating and the gas - phase depletion of the heavy elements were treated in a self - consistent manner . this work found that the high surface brightness isolated extragalactic regions are in general excited by young clusters of ob stars , and that , in this case , the ionizing euv spectra and region emission line spectra predicted by the pegase and starburst99 codes are essentially identical . for starburst galaxies , in which the starburst has a luminosity comparable to the luminosity of the host galaxy , the situation is rather different . in these objects , intense star formation is likely to continue over at least a galactic dynamical timescale , and therefore the assumption of a continuous rather than an instantaneous burst of star formation would be more accurate . as a consequence , the assumptions which go into the theoretical stellar mass loss formulations and evolutionary tracks are likely to play a much more important role in the modeling . furthermore , for starbursts continued for more than a few myr , the wolf - rayet ( w - r ) stars can play an important part in determining both the intensity and shape of the euv spectrum . for the w - r stars , the uncertain assumptions made about the stellar lifetimes , wind mass - loss rates , the velocity law in the stellar wind , and the atmospheric opacities play a critical role in determining the spectral shape and intensity of the emergent euv flux predicted by theory . in this paper , we present new grids of theoretical models ( based on the assumption of continuous star formation ) which again combine region models generated by the mappings iii code with input euv fields given by the stellar population spectral synthesis models pegase 2 and starburst99 . these models are used in conjuction with our large observational data set described in @xcite and @xcite to place new _ observational _ constraints on the shape of the euv ionizing radiation field . since the two stellar population spectral synthesis codes provide a wide choice of stellar mass loss formulations , evolutionary tracks and stellar atmospheric transfer models , they provide strikingly different predictions about the shape and intensity of the euv field as a function of stellar age . in this paper , we use these to separate and quantify the effects of the stellar atmospheric models and the evolutionary tracks used on the optical diagnostic diagrams . in particular , we will show that the models which give the hardest euv spectrum below the ionization limit , but which have relatively few photons above this limit , provide the best empirical fit to the distribution of starburst galaxies on the optical line ratio diagnostic diagrams . these new observational constraints should prove very helpful to theoreticians modeling the late stages of stellar evolution in massive stars . this paper is structured as follows . our observational comparison sample is described in section 2 . the stellar population synthesis models used to calculate the euv ionizing radiation field are presented in section 3 . our theoretical starburst models were produced using the photoionization and shock code mappings iii and are described in section 4 . wolf - rayet emission in our sample of starburst galaxies is discussed in section 5 and the effect of continuum metal opacities on the euv ionizing continuum is discussed in section 6 . a new theoretical classification scheme for starbursts and agn is presented in section 7 and our main conclusions are summarised in section 8 . we have selected a large sample of 285 warm iras galaxies covering a wide range of infrared luminosities . the warm selection criterion ensures that our sample contains galaxies with either concentrated star formation @xcite , agn @xcite , or in many cases , a combination of the two . our sample has been selected from the catalogue by @xcite and consists of all objects south of declination @xmath1 with the additional following criteria ; \1 . flux at 60 @xmath2 with moderate or high quality detections at 25 , 60 and 100 @xmath3 \2 . redshift @xmath4 8000 km s@xmath5 for @xmath6 and @xmath7 30000 km s@xmath5 for @xmath8 \3 . galactic latitude @xmath9 15@xmath10 , and declination @xmath11 + 10@xmath12 \4 . warm fir colours ; @xmath13 and @xmath14 where @xmath15,@xmath16 and @xmath17 are the iras fluxes at 25 , 60 and 100 @xmath3 respectively . these selection criteria ensure that that the galaxies in the sample are well - resolved , and that the sample has a large dynamical range in luminosity , so that luminosity dependent effects can , in principle , be investigated . high resolution ( 50 km s@xmath5 at @xmath18 ) optical spectra with useable s / n ( @xmath19 ) ratios were obtained for 225 of the galaxies in our sample . spectra were obtained in the red and blue wavelength ranges using the double beam spectrograph on the mount stromlo and siding springs 2.3 m telescope . these selection criteria and the full details of our observations are discussed in detail in @xcite and @xcite . all 225 objects were classified into agn , starburst and liner types using new theoretical classification lines on the @xcite ( hereafter vo87 ) diagnostic diagrams . we found 157 starburst galaxies using the vo87 diagrams and semi - empirical classification scheme which provide the primary observational comparison for the starburst spectral modeling presented here . we believe there is little contamination by obscured agn in this starburst sample for two reasons . firstly , in @xcite , we have shown that the optical diagnostic diagrams are extremely sensitive to the presence of an agn . an agn which contributes only 20% to the optical emission increases the line ratios sufficiently that it would be classified as an agn . secondly , we found very few starburst galaxies with warm colors ( @xmath20 ) . since warm colors usually mean that the galaxy is energetically dominated by an agn we can conclude that the fraction of obscured agn in our starburst galaxies must be low . these results are consistent with the infrared studies of @xcite and @xcite , who showed that ultraluminous infrared galaxies classified as starbursts also show a lack of an energetically important agn . the models we have utilized are described in detail in @xcite , and we describe them again briefly here . we have used both the pegase 2 and starburst99 codes to model the starbursts in our sample . the pegase 2 code uses the @xcite grid of atmospheres covering the entire hertzsprung - russell diagram ( hrd ) plus @xcite planetary nebula nuclei ( pnn ) atmospheres for stars with high effective temperatives ( t @xmath21 50000 k ) ( hereafter known as the clegg & middlemass atmospheres ) . the @xcite grid ( hereafter called the lejeune grid ) is derived from three sets of atmosphere calculations , the bulk being the @xcite models with smaller specialized cool star models by @xcite and @xcite . lejeune , cuisinier & buser incorporated observational flux corrections into these models for a range of stellar temperatures , but does not alter the @xmath22k models of kurucz . the starburst 99 code also uses the plane - parallel atmospheric lejeune grid . for stars with strong winds it offers the choice of the lejeune grid or the @xcite extended model atmospheres ( hereafter known as the schmutz atmospheres ) . the prescription for the switch between extended and plane - parallel atmospheres is the same as in @xcite . we ran models for both the `` standard '' mass loss rates and the `` enhanced '' mass loss rates described in @xcite . we found that for the line diagnostic ratios used here , the mass loss prescriptions agree to within 0.03 dex . both codes follow the theoretical stellar tracks from the zero age main sequence ( zams ) to their final stages . these stages include the asymptotic and post - asymptotic giant branch phases for intermediate - mass stars . the padova tracks @xcite are used in pegase 2 and the geneva tracks ( @xcite ) are used in starburst99 . the padova tracks overshoot for masses @xmath23 and use a higher ratio of the overshooting distance to the pressure scale height and down to lower masses than the geneva tracks , which include overshooting only above 1.5 @xmath24 . both tracks use the opal opacities ; @xcite for the padova tracks and @xcite for the geneva tracks . both sets of tracks assume similar mixing lengths . helium contents of 0.28 and 0.30 are used for the padova and geneva tracks respectively . the padova tracks have a higher resolution in mass and time . clear differences between these two sets of tracks ( at solar metallicity ) can be seen by comparing figure ( 5 ) in @xcite and figure ( 7 ) in @xcite . assuming a standard initial mass function , the choices offered by the two spectral synthesis modeling codes allows sufficient flexibility to separately investigate , and to quantify , the effect that either the stellar atmospheres or the stellar evolutionary tracks have upon the theoretical starburst model line intensity ratios as a function of age . for example we can run the starburst99 code with either the lejeune atmospheres or the lejeune plus schmutz atmospheres to investigate the effect of extended atmospheres , or we could compare the pegase 2 and starburst99 codes to quantify the effect of the differences between the padova and the geneva tracks and the effect of the different stellar atmosphere models incorporated into these codes . we are confident that this comparison is valid , since in our earlier paper @xcite , we compared the ionizing euv spectra and region emission spectra predicted by the pegase and starburst99 codes for zero age clusters , and found these to be essentially identical . in the modeling of the starburst emission spectra , we distinguish between a zero - age _ instantaneous _ star formation case , and _ continuous _ starburst models in which a balance between star birth and star death is set up for all stellar masses which contribute significantly to the euv spectrum . the hydrogen - burning lifetime of massive stars is approximately @xmath25 myr , so in practice this condition is satisfied for any starburst which lasts longer than about 6 myr . this gives a dynamical balance between star births and star deaths for all masses greater than about @xmath26 , and is also time enough for the wolf rayet stars to produce their full contribution to the euv spectrum . figure ( 1 ) compares predicted solar metallicity euv spectra produced by the pegase code which uses the lejeune stellar atmospheres plus clegg & middlemass atmospheres for stars with high effective temperatives ( t @xmath21 50000k ) , and the starburst99 codes which use either the lejeune stellar atmospheres grid or the lejeune atmospheres plus schmutz extended model atmospheres . the pegase models used range from ages of @xmath27 myr , and the starburst99 models cover ages of @xmath28 myr . no further evolution in the shape of the euv spectrum is seen after 6 and 8 myr for the pegase and starburst99 models respectively . after a few myr of evolution , quite marked differences in the euv spectrum develop . the comparison of figure ( 1b ) and figure ( 1c ) shows that the schmutz extended atmospheres produce far more ionizing radiation at frequencies above the ionization limit than do the lejeune atmospheres , but the diffences are much less marked at lower energies . the differences between figures ( 1a - c ) are due to a combination of the different evolutionary tracks , and the stellar atmosphere models used . figures ( 2 ) and ( 3 ) show the euv spectra from pegase 2 and starburst99 for metallicities 0.2 and 2 @xmath29 solar . we can observe clear differences in the euv spectra with increasing metallicity . in particular , the euv spectrum becomes harder for lower age models . this is expected if high mass stars are responsible for the euv field in this regime . at higher metallicities , the high mass stars make a larger contribution to the euv radiation field at younger burst ages . the most likely cause of the difference between the euv fields in figures ( 1a - c ) is to be found in the different stellar atmospheric models used for the high mass stars , especially for the wolf - rayet stars . since it was specifically constructed to model starbursts , the starburst99 code uses a more theoretically sophisticated approach to modeling the euv spectrum . in the pegase 2 models , stars with effective temperatures greater than 50000 k ( which includes wolf - rayet stars ) are modeled by the clegg & middlemass planetary nebulae nuclei ( pnn ) atmospheres . these stars have much higher surface gravity than wolf - rayet stars , and so we would expect the atmospheric blanketing to be quite different . in starburst99 code , stars with strong stellar winds ( which includes wolf - rayet stars ) are modeled by the schmutz wolf - rayet atmospheres . these include h and he opacities , but do not include heavy element opacities . the euv spectrum emergent from a w - r model atmosphere depends critically on the fraction of ionizing photons which have been used up to maintain the ionization of the w - r wind region . this is determined by the size of the emission measure of the atmosphere ; @xmath30 this parameter is proportional to the the product @xmath31 , where @xmath32 is the mass - loss rate , @xmath33 the terminal velocity of the wind , and @xmath34 is the photospheric radius of the star . this product is the @xcite density parameter . models with the same density parameter display very similar emission line equivalent widths , but the total scaling in luminosity depends on @xmath35 the density parameter can also be expressed in terms of a `` transformed radius '' , @xmath36 : @xmath37 ^{2/3}\ ] ] where @xmath38 is a ( normalizing ) reference velocity , and @xmath39 is a ( normalizing ) reference mass - loss rate . again , models with similar values of the transformed radius give similar spectra . stars which use a greater fraction of their euv photons in maintaining the photoionization of their extended atmospheres would show a lower - intensity , harder euv spectrum below the ionization edge , and would be expected to exhibit more atmospheric blanketing by heavy elements . to model the starburst spectrum as a function of age of the exciting stars , metallicity and ionization parameter , we input the ionizing spectrum from the both continuous and instantaneous pegase and starburst99 models into the mappings iii code . the photoionization modeling carried out with mappings iii for this analysis is described in @xcite and is described briefly here . we computed plane parallel , isobaric models with electron density of 350 @xmath40 , which is the average electron density of the individual ( frequently unresolved ) regions within the 1 kpc slit aperture extracted for the starbursts in our sample . the electron density was found using the flux in the [ ] @xmath41 and [ ] @xmath42 forbidden lines from our spectra in conjunction with a 5 level model atom using mappings iii . electron densities for the galaxies in our sample can be found in @xcite . the ionization parameter @xmath43 ( cm s@xmath5 ) is defined on the inner boundary of the nebula , that is , the boundary nearest the exciting star . this dimensional ionization parameter can be readily transformed to the more commonly used dimensionless ionization parameter through the identity @xmath44 dust physics is treated explicitly through the absorption of the radiation field on grains , grain charging and photoelectric heating by the grains . we do not yet calculate the re - emission spectrum of dust in the ir but this is currently being implemented . the dust model consists of silicate grains ( 100 @xmath45 1000 ) and small amorphous organic grains ( 10 @xmath45 100 ) , with a size distribution following a @xcite power - law and spherical geometry . the range of sizes is chosen so as to give depletion factors relative to solar similar to those observed by uv interstellar absorption measurements of stars seen through warm diffuse clouds in the local interstellar medium the mappings iii dust model also provides the observed absorption per hydrogen atom for solar metallicity @xcite : @xmath46 . photoelectric yields are found using a more conservative yield curve of the same form as @xcite . photoelectric grain currents are found using @xcite using the dust absorption data from @xcite . collisions with electrons and protons are considered assuming the standard `` sticking '' coefficients , following @xcite and @xcite . more details of the dust physics in mappings iii can be found in @xcite . the undepleted solar abundances are assumed to be those of @xcite ; these abundances and the depletion factors adopted for each element in the starburst modeling are shown in table 1 . for non - solar metallicities we assume that both the dust model and the depletion factors are unchanged , since we have no way of estimating what they may be otherwise . all elements except nitrogen and helium are taken to be primary nucleosynthesis elements . it is known that this assumption may be incorrect in systems where the time history of star formation in the galaxy is different , or where galactic winds are important . for example , the o / fe ratio is different from its solar value in both the lmc and the smc @xcite . again , we are forced use the simplest assumptions possible in the absence of a more detailed understanding of the chemical evolution of starburst galaxies . for helium , we assume a primary nucleosynthesis component in addition to the primordial value derived from @xcite . this primary component is matched empirically to provide the observed abundances at smc , lmc and solar abundances ; @xcite , @xcite . by number , the he / h ratio is : @xmath47 nitrogen is assumed to be a secondary nucleosynthesis element above metallicities of @xmath48 solar , but as a primary nucleosynthesis element at lower metallicities . this is an empirical fit to the observed behaviour of the n / o ratio in regions ( van zee , haynes & salzer , 1997 ) . by number : @xmath49 the abundances and depletion factors for each element used can be found in table [ table1 ] . the ionization parameter was varied from @xmath50 to @xmath51 , and the metallicities varied from 0.01 to 3 solar for pegase and 0.05 to 2 solar for starburst99 . the metallicity values used are restricted by the stellar tracks used by the population synthesis models . figure ( 4 ) shows that , due to the absence of wolf - rayet stars , the use of similar zero - age main sequences , and the use of identical model atmospheres for massive stars , the shape of the euv spectrum for both the pegase 2 and starburst99 models is almost identical for instantaneous burst models . as a consequence , they produce almost identical optical line ratios in the photoionization models . the results from these models are compared with the observational data set on the vo87 line ratio diagnostic diagrams in figures ( 4 ) , ( 5 ) and ( 6 ) . note that the major problem seen with these models is that many starburst galaxies are found in a region lying above and to the right of the `` fold '' in the ionization parameter : metallicity surface . this presents a problem , since these points lie in a `` forbidden zone '' of line ratio space which can not be reached by any combination of metallicity or ionization parameter . the only way to make models which fall into this region of the the diagnostic diagrams is either to mix in another type of excitation _ i.e. _ shocks or a power - law ionizing radiation field or simply to use a harder euv ionizing spectrum , particularly in the 1 - 4 rydberg region . in any event , we should not be too surprised that the instantaneous models do not provide a very good fit to the observed starbursts , since many of these objects are seen in merging pairs of galaxies , and in this case we would theoretically expect star formation to continue over a galactic dynamical timescale . thus massive clusters associated with individual ( usually unresolved ) regions should have a wide variety of ages . direct evidence for continuous star formation , or older regions , is seen in many of our starburst spectra . these show either a low equivalent width in h@xmath52 ( which often indicates dilution by an older stellar population ) or clear evidence of h@xmath53 absorption in the stellar continuum . both of these indicate that star formation has continued over at least several myr @xcite . in @xcite , we showed that the line ratio usually used for measuring the ionization parameter ; [ ] @xmath545007 / [ ] @xmath553726,9 , is indeed a good diagnostic and that the [ ] @xmath54 6584 / [ ] @xmath553726,9 ratio gives the best diagnostic of abundance , as it is monotonic between 0.1 and over 3.0 times solar metallicity . the wavelength range of our spectra was not sufficient for us to observe [ ] , however we present the grids of the instantaneous models for the [ ] @xmath556548,84 / [ ] @xmath563726,9 _ vs. _ [ ] @xmath545007 /h@xmath52 diagram and the [ ] @xmath556548,84/ [ ] @xmath573726,9 _ vs. _ [ ] @xmath545007/ [ ] @xmath553726,9 in figures ( 7 ) and ( 8) for the use of the astronomical community . as we have seen , when star formation continues over several myr , the ionizing spectrum evolves until a dynamic balance between stellar births and stellar deaths has been set up for all intial stellar masses which are important in producing euv photons . as figure ( 1 ) shows , this occurs after 6 and 8 myr for the pegase and starburst99 models respectively , and so these cluster ages were assumed for the continuous star formation models presented in this section . the results for the continuous models shown on the vo87 line ratio diagnostic diagrams are presented in figures ( 9 - 11 ) for the pegase 2 models with ( a ) the lejeune plus clegg & middlemass pnn model atmospheres , ( b ) the starburst99 models with the lejeune plus schmutz model atmospheres and ( c ) the starburst99 models with the lejeune atmospheres respectively . let us first consider the two sets of starburst99 models . the differences seen here simply reflect the differences in the stellar atmospheric models . the lejeune plus schmutz model atmospheres show little change in spectral slope between 1 and 3 ryd as a function of cluster age . it is therefore not surprising that these continuous star formation models give very similar results to the zero age instantaneous models of figures ( 4 - 8) . in these continuous star formation models the w - r stars provide a radiation field of appreciable strength above the ionization limit ( eg . 4 - 8 ryd in figures 1 - 3 ) . in this case , we might expect to detect lines in the optical spectrum . the detection of a nebular @xmath58 line would provide an important observational diagnostic in support of the schmutz extended atmospheric modeling and is discussed in the following section . these models with schmutz extended atmospheres have exactly the same difficulty in explaining the position of the observed points as do the instantaneous models , in that the model grid fails to overlap about half of the observed points , indicating the need for a harder ionizing spectrum than these tracks and atmospheres provide . the second of the starburst99 grids , for continuous starbursts using the geneva tracks along with the lejeune atmospheres provides an even softer radiation field in the @xmath59 ryd energy range . in this case , the theoretical grid falls below and to the left of the majority of the observed points on all three of the vo87 plots . this combination of tracks and atmospheres is therefore excluded with a good degree of certainty . finally , consider the pegase 2 models , which use the padova tracks with the lejeune plus clegg & middlemass pnn extended atmospheres . these models are characterised by the hardest radiation field in the 1 - 4 ryd region , and are the only ones that show the spectrum becoming harder as the w - r stars turn on . this is a direct consequence of the clegg & middlemass pnn atmosphere models used for stars with effective temperatures greater than 50000 k. these are the only set of models which encompass nearly all of the observed starbursts on all three of the vo87 line diagnostic plots . furthermore , individual objects fall into similar regions of the theoretical grid in all three diagnostic plots , indicating that there is some degree of consistency achieved here . in essence , the ionization parameter appears to be limited to the range @xmath60 and the metallicity range seems to cover most of the range ; from about @xmath61 solar up to nearly 3 times solar . however , the low - metallicity objects appear to be rather rare in our sample , and most of the starbursts are consistent with a metallicity 1 - 3 times solar . we present the grids of the continuous models for the [ ] @xmath556548,84 / [ ] @xmath563726,9 _ vs. _ [ ] @xmath545007 /h@xmath52 diagram and the [ ] @xmath556548,84/ [ ] @xmath573726,9 _ vs. _ [ ] @xmath545007/ [ ] @xmath553726,9 in figures ( 12 ) and ( 12 ) for the use of the astronomical community . the starburst models presented here are based on pure photoionization models , and therefore do not incorporate the effect of mechanical luminosity from supernovae shocks . the primary effect of the release of mechanical energy through shocks is to move the theoretical grids upwards , and lightly to the right on the vo87 line ratio diagnostic plots . the size of this effect depends on the relative importance of the shock and photoionization luminosity . the input mechanical energy luminosity @xmath62 produced by supernova events and winds is converted into optical line emission through radiative shocks . the importance of this line emission relative to that produced by photoionization depends on fraction of this energy , @xmath63 , which has been converted to h@xmath53 flux compared to the h@xmath53 flux produced by recombinations in the photoionized plasma : @xmath64 where @xmath65 is the effective recombination coefficient of hydrogen , and @xmath66 is the number of ionizing photons being produced by the hot stars in the cluster . in the case of a supernova remnant , the shock becomes radiative , and the sedov - taylor ( adiabatic ) expansion phase is terminated when its cooling timescale , @xmath67 , becomes shorter than the dynamical expansion timescale of the shell , @xmath68 . from standard sedov - taylor theory , @xmath69 where @xmath70 is the radius of the supernova remnant , @xmath71 is the shock wave velocity , and @xmath72 is the time since the supernova explosion . the radiative shock - wave theory of dopita & sutherland ( 1996 ) shows that the cooling timescale , in years , can be expressed as : @xmath73 where @xmath74 is the shock velocity in units of 100 km s@xmath75 @xmath76 is the metallicity of the plasma relative to solar , and @xmath77 is the pre - shock density . we therefore find that , with @xmath78 @xmath79 , a supernova remnant typically become radiative at a radius of 1 pc , when the shock velocity is 600 km s@xmath80 at this point , the supernova remnant is @xmath81600 years old , and its expansion timescale is @xmath811500 yr . we therefore used a shock model with a velocity of 600 km s@xmath5 in a spherical nebula of solar metallicity and @xmath78 @xmath79 to calculate the possible contribution by supernova remnants to our photoionization models . we assumed a total mechanical luminosity in snr of @xmath82 @xcite . we calculated a star formation rate of @xmath83 from the average @xmath84for the starbursts in our sample , following the prescription given in @xcite ( note that @xmath84 is defined as @xmath85 in @xcite ) . assuming the ir luminosity is distributed evenly throughout the galaxy and a minimum size of 7 kpc , the sfr reduces to @xmath86 in the observed aperture . we note that this is still higher than that found using the h@xmath87 luminosity of our template starburst ( within the 1 kpc aperture ) of @xmath88 . this is to be expected as dust absorption will reduce the h@xmath87 derived sfr compared with that derived using the fir luminosity . we computed a 600 km / s shock model and a spherical ionized precursor with a shocked spherical radius of 1 pc . this model had a mechanical luminosity of @xmath89 erg / s . as we expect the total mechanical luminosity to be @xmath90 erg / s , we expect to obtain on average 11.2 snr within our 1 kpc aperture at any one time . the luminosity produced in the shock and precursor are given in table ( 2 ) . from this snr model , the contribution to h@xmath52 emission in the starbursts due to snrs would be about 16 - 20% . the contribution to the h@xmath52 flux from starbursts is determined more or less directly by the star formation history and the imf . the contribution to h@xmath52 from snrs is determined by the velocity and the total area covered by the snr shocks , which in turn depend on the number of snrs ( determined by the sfr and imf ) and weakly on the density . at low densities , snrs are larger and have lower velocities than at higher densities , and have similar overall luminosities in h@xmath52 . the [ ] emission from the starbursts , on the other hand , is not a function of the sfr , rather it is a measure of the ionization parameter @xmath43 of the radiation field within the starburst which is a function of the density and mass distribution ( ie filaments vs uniform ) of the gas . thus the [ ] in the starburst models varies from bright to negligible levels compared to h@xmath52 . the [ ] mission from the snrs is constrained by the chosen model geometry and is therefore determined by the sfr and the average density , not by the global ionization parameter . the ionization parameter for each snr is determined internally by the snr model without reference to the global starburst value . the observed total @xmath91@xmath92/{\rm h}\beta)$ ] ratio is the sum of the starburst and snr contributions ; @xmath93/{\rm h}\beta ) = \log([{\rm oiii}]_{\rm starb } + [ { \rm oiii}]_{\rm snr } ) - \log({\rm h\beta_{starb}+h\beta_{snr}}).\ ] ] thus , in the limit of @xmath94_{\rm starb}=0 $ ] due to low global @xmath43 in the starbursts , the observed @xmath91@xmath92/{\rm h}\beta)$ ] ratio will reach a lower limit of @xmath93_{\rm snr } ) - \log({\rm h\beta_{starb}+h\beta_{snr } } ) \ge 0.0\ ] ] with a density of 350 @xmath79 and the sfr used here ( @xmath86 in the observed aperture of 1 square kpc ) . this lower limit is not observed , and the actual lower limit is @xmath95/h}\beta ) \approx -1.0 $ ] , a factor of 10 less than the snr model limit . clearly the strong @xmath94_{\rm snr}$ ] contribution modeled here is not compatible with the observations . this incompatibility may be due to either the number of snr we derived , or that the average density in the snr environment is less than 350 @xmath79 . we conclude that the snr contribution to the @xmath91@xmath92/{\rm h}\beta)$ ] ratio is @xmath96% and is probably a factor of 10 less ( @xmath97% ) , and can be neglected for the starburst models derived here . we are currently investigating this snr contribution to starbursts further by considering a range of densities and lower shock velocities to determine the level at which the snrs model is compatible with the observations . we expect to find that velocities in the 200 - 300 km / s range will be more compatible with observations , resulting in a small @xmath94_{\rm snr}$ ] contribution , negligible in all but the starbursts with the lowest ionization parameter . wolf - rayet features were first found in the dwarf emission galaxy he2 - 10 @xcite . as more such features were discovered in galaxies , @xcite defined wolf - rayet ( w - r ) galaxies as those galaxies which contain broad stellar emission lines in their spectra and therefore contain large numbers of w - r stars . the large numbers of w - r stars are thought to be a result of present or very recent star formation . @xcite searched for w - r emission in a sample of blue emission - line galaxies . they found a positive detection in one galaxy and suspected in 14 others and suggested that w - r stars are preferentially detected in low redshift galaxies . the first catalogue of w - r galaxies was presented by @xcite , showing that w - r galaxies can be easily distinguished by their broad @xmath544686 [ ] emission feature , or in some cases a broad line at 4640 due to . high resolution long - slit observations of a sample of wolf - rayet galaxies were carried out recently by @xcite . nearly all the galaxies in their sample show broad w - r emission consisting of an unresolved blend of @xmath544640 , @xmath544650 , [ ] @xmath544658 , and @xmath544686 emission lines . they also found weaker w - r emission lines @xmath544512 and @xmath544565 in some galaxies . the signal - to - noise ratios of our spectra are not sufficient in individual galaxies to detect these signatures of w - r emission . we have therefore constructed a template `` average '' warm infrared starburst spectrum from the 56 starburst galaxies in our sample which have snrs at h@xmath53 of 60 or greater and for which the zero - redshift blue wavelength cut - off is lower than @xmath984620 . we note that this average spectrum is not representative of our sample of warm infrared starbursts because selecting for high snr galaxies may bias the average spectrum towards young and hence more luminous starbursts . however , it is useful simply to assist in the search for w - r features . the average spectrum is shown in figure ( 14 ) and in detail in figure ( 15 ) . the positions of the expected w - r features are marked on figure ( 15 ) . the mean age of the average warm infrared starburst can be estimated from the h@xmath53 absorption equivalent width , bearing in mind that dust absorption may also contribute to the absorption line profile and has not been taken into account . the h@xmath53 absorption equivalent width was found by simultaneously fitting the h@xmath53 absorption and emission lines with gaussians in the iraf task _ ngaussfits_. we find that the h@xmath53 absorption equivalent width is @xmath99 which corresponds to an upper limit age of @xmath100 myr for a continous star formation model at solar metallicity @xcite . however , this is probably an underestimate , since the h@xmath53 absorption equivalent width will be reduced by any underlying old star population . nonetheless , there is likely to be likely a selection affect towards more youthful starbursts , in that the bright starbursts tend to be both younger and intrinsically more luminous . in the `` average '' spectrum of figure ( 15 ) , there appears to be marginal detections of [ ] @xmath984658 , and @xmath984686 at the 2@xmath101 level . these lines do not appear broad , however this may simply be an effect of the low snr for these lines . the @xmath984686 emission in the majority of our galaxies provides an important constraint on the schmutz extended atmospheric modeling . the @xmath102(@xmath103)ratio for our template starburst galaxy is @xmath104 . both the mappings iii models with pegase and starburst99 ( lejeune atmospheres ) stellar ionizing continua produce @xmath102(@xmath103 ) around @xmath105 , while the mappings iii models with starburst99 ( lejeune + schmutz atmospheres ) stellar ionizing continuum produces @xmath102(@xmath103)around @xmath106 , consistent with that observed with our template starburst galaxy . we note again that our template starburst galaxy has been composed of the starbursts in our sample with the highest snrs , and so may be biased towards the most luminous and therefore the youngest starbursts in our sample . this @xmath103ratio for our template starburst may therefore be biased towards those with more w - r stars than the average starburst galaxy in our sample . clearly we require a hard euv field in the 1 - 4 ryd regime to model our starburst galaxies on the optical diagnostic diagrams such as that provided with the pegase 2 model . however , as we would expect the schmutz extended atmospheric modeling to be more appropriate to the modeling of our starburst galaxies , we believe that the inclusion of continuum metal opacity ( or continuum metal blanketing ) in the stellar population synthesis models using the schmutz extended atmospheres is a possible solution . for many years , optical depth estimates in the ism were found using the hydrogen 21 cm line and the assumption of a uniform gas density . as a result of the low optical depths obtained , it was thought that the ism should be opaque to radiation in the euv @xcite . the discovery that the ism is inhomogeneous overturned this conclusion and @xcite established an estimate of the effective absorption cross - section of the ism at euv wavelengths using determinations of cross - sections and abundances . @xcite showed that some euv radiation should be able to be observed over considerable distances . more recent determinations of cross - sections and abundances have been used by @xcite to provide a new estimate of the effective absorption cross - section of the ism at euv wavelengths . the relative difficulty in obtaining euv spectra to compare with theoretical estimates of opacity in the euv continuum described above has decreased significantly over the last decade with the aid of euv telescopes such as euve ( extreme ultraviolet explorer ) , rosat wide field camera , alexis ( array of low energy x - ray imaging sensors ) , fuse ( far ultraviolet spectroscopic explorer ) , and the extreme - ultraviolet imaging telescope ( eit ) . these telescopes have allowed observations of many stellar objects which have greatly advanced the modeling of stellar atmospheres ( * ? ? ? * eg . ) and have allowed the euv study of some seyfert galaxies ( eg . however the euv spectrum in starburst galaxies continues to remain unseen as a result of the weakness of the euv continuum in starburst galaxies due to absorption . it is therefore necessary to rely on theoretical predictions of the euv spectrum from stellar population synthesis codes . although the euv spectrum must be estimated using theoretical modeling , the reprocessing of the euv spectrum into optical emission lines allows limits to be placed on the shape of the euv spectrum in starburst galaxies . as concluded in previous sections , we require a harder euv field between the 1 - 4 ryd regime than is provided by the stellar population synthesis models with schmutz extended atmospheres . currently , only continuous opacities due to helium are included in the models , as continuous metal opacities are relatively unimportant for many studies . we believe that the inclusion of continuum metal opacities in these models would provide a suitable shape in the euv , enabling the models to reproduce the starburst sequence on the standard optical diagnostic diagrams . continuum metal opacities , as with hydrogen and helium opacities , are a result of bound - free transitions , ie photoionization of the metals . continuum metal opacity would allow some fraction of the radiation with energies greater than 4 ryd to be absorbed and re - emitted less than 4 ryd . the fraction of radiation absorbed by metals depends on their individual absorption cross - sections and abundances . the resulting euv continuum would have a softer continuum above the ionization limit , as a result of carbon opacities . the spectrum would also have a harder but fainter euv field between the 1 - 4 ryd regime , as required by the position of our starburst galaxies on the optical diagnostic diagrams . note that if this were the case , we might expect that the models with schmutz extended atmospheres would produce too much flux in the @xmath107 line compared with our observations , contrary to our findings . we conclude that while continuum metal blanketing may be a possible solution to the discrepancy seen between our observations and the models using schmutz extended atmospheres , it may not be the only solution . in order to place a theoretical upper limit for starburst models on the optical diagnostic diagrams , we used the pegase 2.0 grids , since these provide the hardest euv spectrum , and therefore give a theoretical grid which is placed both highest and furthest to the right on the vo87 line ratio diagnostic diagrams . we have shown that , with a realistic range of metallicities ( @xmath108 ) and ionization parameter @xmath43 ( cm / s ) in the range @xmath109 ( or @xmath110 ) , continuous starburst models produced by any modeling procedure always fall below and to the left of an empirical limit on the [ ] /@xmath111 _ vs. _ [ ] /@xmath112 , [ ] /@xmath111 _ vs. _ [ ] /@xmath112 and [ ] /@xmath111 _ vs. _ [ ] /@xmath112 diagrams . this is due to the two - parameter grid of the ionization parameter and metallicity folding back upon itself . therefore , no combination of these parameters can generate a theoretical point above this fold . the lines dividing the theoretical starburst region from objects of other types of excitation are shown in figure ( 16 ) . these can be parametrized by the following simple fitting formulae , which have the shape of rectangular hyperbolae ; @xmath113\,\lambda 5007}}{{\rm h}\beta}\right ) = \frac{0.61 } { \log ( { \rm [ nii]}/{\rm h}\alpha ) -0.47}+1.19\ ] ] @xmath114\,\lambda 5007}{{\rm h}\beta}\right ) = \frac{0.72 } { \log ( { \rm [ sii]\,\lambda \lambda 6717,31}/{\rm h}\alpha ) -0.32 } + 1.30\ ] ] @xmath114\,\lambda 5007}{{\rm h}\beta}\right ) = \frac{0.73 } { \log ( { \rm [ oi]\,\lambda 6300}/{\rm h}\alpha ) + 0.59}+1.33\ ] ] the shape and position of this maximum starburst line has not been previously established from theoretical models , since the shape of the ionizing spectrum of the cluster has not been known to sufficient accuracy . vo87 have attempted to establish both the position and the shape of this boundary in a semi - empirical way , using both observational data from the literature and a combination of models available to them at that time . the theoretical boundaries for starbursts defined by equations ( 1)-(3 ) provide us for the first time with a theoretical ( as opposed to a semi - empirical boundary ) for the region occupied by starbursts in these diagnostic plots . in view of the potential for errors in the modeling which may flow from errors in the assumptions made in the chemical abundances , chemical depletion factors , the slope of the initial mass function , and the evolutionary tracks and the stellar atmosphere models used , we have indicated a `` best guess '' estimate of these errors as dashed lines in figure ( 16 ) . these theoretical upper limits for starburst galaxies have been used in @xcite to classify the galaxies in our sample along with an extreme mixing line produced using our shock modeling to classify galaxies into starburst , liner and agn types . the strength of our theoretical starburst classification line can be seen by observing the number of galaxies which have ` ambiguous ' classifications . these galaxies are those which fall within the starburst region of one or two of the diagnostic diagram(s ) and the agn region of the remaining diagram(s ) . we found that only 6% of our sample have ambiguous classifications using our theoretical extreme starburst line , compared with 16% ambiguous classifications using the standard vo87 classification lines . these results indicate that our theoretical starburst line is a reliable tool for optically classifying galaxies into starburst and agn types , and is more consistent from diagram to diagram than the conventional vo87 method . we have presented a comparison between the stellar population synthesis models pegase 2 and starburst99 using a large sample of 157 warm infrared starburst galaxies . the main differences between the two synthesis models are the stellar tracks and stellar atmosphere prescriptions . pegase and starburst99 were used to generate the spectral energy distribution ( sed ) of the young star clusters . mappings iii was used to compute photoionization models which include a self - consistent treatment of dust physics and chemical depletion . the standard optical diagnostic diagrams are indicators of the hardness of the euv radiation field in the starburst galaxies . these diagnostic diagrams are most sensitive to the spectral index of the ionizing radiation field in the 1 - 4 rydberg region . we find that warm infrared starburst galaxies contain a relatively hard euv field in this region . the pegase ionizing stellar continuum is harder in the 1 - 4 ryd range than that of starburst99 , most likely due to the different stellar atmospheres used for wolf - rayet stars . we have constructed an average spectrum of the high snr warm infrared starbursts in our sample in order to look for wolf - rayet signatures . we find detections of @xmath544658 , and @xmath544686 at the 2@xmath115 level , indicating w - r activity , and constraining the schmutz extended atmospheric modeling . we require a hard euv field in the 1 - 4 ryd regime to model our starburst galaxies on the optical diagnostic diagrams such as that provided with the pegase 2 model . however , as we would expect the schmutz extended atmospheric modeling to be more appropriate to the modeling of our starburst galaxies , we believe that one solution would be to include continuum metal blanketing in the stellar population synthesis models using the schmutz extended atmospheres . continuum metal blanketing would allow much of the radiation with energies greater than 4 ryd to be absorbed and re - emitted at energies less than 4 ryd . the resulting euv continuum would have a softer continuum above the ionization limit and a harder euv field between the 1 - 4 ryd regime , as required by the position of our starburst galaxies on the optical diagnostic diagrams . snrs could also contribute to the hardness of the euv field , although our models and observations suggest that this is likely to be @xmath96% , insufficient to cause the discrepancy between our starburst galaxies and the models using schmutz extended atmospheres . we use the starburst grids produced with the pegase euv ionizing radiation field and our mappings iii models to parametrize an extreme starburst line which is useful in classifying galaxies into starburst or agn types . in a previous paper @xcite , we showed that this theoretical classification scheme produces reliable classifications with less ambiguity than the classical @xcite empirical method . we thank claus leitherer and brigitte rocca - volmerange for helpful discussions and for the use of starburst99 and pegase 2 . we thank the referee for his constructive comments which helped this to be a better paper . we also thank the staff at siding springs observatories for their assistance during our spectroscopy observations . l. kewley gratefully acknowledges support from the australian academy of science young researcher scheme and the french service culturel & scientifique . lrr h & 0.00 & 0.00 + he & -1.01 & 0.00 + c & -3.44 & -0.30 + n & -3.95 & -0.22 + o & -3.07 & -0.22 + ne & -3.91 & 0.00 + mg & -4.42 & -0.70 + si & -4.45 & -1.00 + s & -4.79 & 0.00 + ar & -5.44 & 0.00 + ca & -5.64 & -2.52 + fe & -4.33 & -2.00 +
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we have modeled a large sample of infrared starburst galaxies using both the pegase v2.0 and starburst99 codes to generate the spectral energy distribution ( sed ) of the young star clusters .
pegase utilizes the padova group tracks while starburst99 uses the geneva group tracks , allowing comparison between the two .
we used our mappings iii code to compute photoionization models which include a self - consistent treatment of dust physics and chemical depletion .
we use the standard optical diagnostic diagrams as indicators of the hardness of the euv radiation field in these galaxies .
these diagnostic diagrams are most sensitive to the spectral index of the ionizing radiation field in the 1 - 4 rydberg region .
we find that warm infrared starburst galaxies contain a relatively hard euv field in this region .
the pegase ionizing stellar continuum is harder in the 1 - 4 rydberg range than that of starburst99 .
as the spectrum in this regime is dominated by emission from wolf - rayet ( w - r ) stars , this difference is most likely due to the differences in stellar atmosphere models used for the w - r stars .
the pegase models use the @xcite planetary nebula nuclei ( pnn ) atmosphere models for the wolf - rayet stars whereas the starburst99 models use the @xcite wolf - rayet atmosphere models .
we believe that the @xcite atmospheres are more applicable to the starburst galaxies in our sample , however they do not produce the hard euv field in the 1 - 4 rydberg region required by our observations .
the inclusion of continuum metal blanketing in the models may be one solution .
supernova remnant ( snr ) shock modeling shows that the contribution by mechanical energy from snrs to the photoionization models is @xmath0% .
the models presented here are used to derive a new theoretical classification scheme for starbursts and agn galaxies based on the optical diagnostic diagrams .
| 14,151 | 533 |
in the standard model @xmath3 and @xmath4 are not mass eigenstates . instead we have ( the small cp - violating effects are neglected ) @xmath5 so the time evolution of the @xmath6 states looks like @xmath7 where @xmath8 is the mass eigenvalue and @xmath9 - the corresponding width . it follows from ( 1 ) and ( 2 ) that the probability for @xmath3 meson not to change its flavour after a time @xmath10 from the creation is @xmath11 and the probability to convert into the @xmath4 meson @xmath12where @xmath13 is the average width and @xmath14 . so @xmath15 mass difference between the @xmath16 mass eigenstates defines the oscillation frequency . standard model predicts @xcite that @xmath17 , @xmath18 being the cabibbo - kobayashi - maskawa matrix element . therefore the mixing in the @xmath19 meson system proceeds much more faster than in the @xmath20 system . the total probability @xmath21 that a @xmath3 will oscillate into @xmath4 is @xmath22 in the first @xmath23-mixing experiments @xcite just this time integrated mixing probability was measured . the result @xcite @xmath24 shows that in the @xmath1 system @xmath25 is expected . in fact the allowed range of @xmath26 is estimated to be between @xmath27 and @xmath28 in the standard model @xcite . such a big value of @xmath26 makes impossible time integrated measurements in the @xmath1 system , because @xmath21 in ( 5 ) saturates at @xmath29 for large values of x. although it was thought that unlike the kaon system for the @xmath16 mesons the decay width difference can be neglected @xcite , nowadays people is more inclined to believe the theoretical prediction @xcite that the @xmath30 transition , with final states common to both @xmath1 and @xmath31 , can generate about 20% difference in lifetimes of the short lived and long lived @xmath1-mesons @xcite . but we can see from the ( 3 @xmath32 5 ) formulas that the effect of nonzero @xmath33 is always @xmath34 and so of the order of several percents , because @xmath35 is expected . in the following we will neglect this effect and will take @xmath36 , though in some formulas @xmath33 is kept for reference reason . the development of high precision vertex detectors made it possible to measure @xcite in the @xmath23 system the time dependent asymmetry @xmath37 the same techniques can also be applied to the @xmath38 system . recently the atlas detector sensitivity to the @xmath26 parameter was studied @xcite using @xmath39 decay chain for @xmath1 meson reconstruction . it was shown that @xmath26 up to 40 should be within a reach @xcite . the signal statistics could be increased by using other decay channels , like @xmath40 . the purpose of this note is to study the usefulness of the decay chain @xmath41 for @xmath1 meson reconstruction in the atlas @xmath1-mixing experiments . about 20 000 following b - decays were generated using the pythia monte carlo program @xcite @xmath42 = @xmath43 = @xmath44 + @xmath45 = @xmath46 + @xmath47 + @xmath45 = @xmath48 + @xmath49 the impact parameter was smeared using the following parameterized description of the impact parameter resolution @xmath50 where resolutions are in @xmath51 and @xmath52 is the angle with respect to the beam line . it was shown in @xcite that this parameterized resolution reasonably reproduces the results obtained by using the full simulation and reconstruction programs . for the transverse momentum resolution an usual expression @xcite @xmath53 was assumed . track reconstruction efficiencies for various particles were taken from @xcite . because now we have 6 particles in the final state instead of 4 for the @xmath2 decay channel , we expect some loss in statistics due to track reconstruction inefficiencies , but the effect is not significant because the investigation in @xcite indicates a high reconstruction efficiency of 0.95 . the topology of a considered @xmath19 decay chain is shown schematically in a figure : ( 150,110 ) ( 40,40 ) ( 40,40)(-3,-2)30 ( 12,17)*@xmath54 * ( 40,40)(2,1)40 ( 62,45)*@xmath55 * ( 80,60)(3,-1)15 ( 83,52)*@xmath43 * ( 95,55)(3,1)33 ( 130,65)*@xmath56 * ( 95,55)(3,0)35 ( 133,55)*@xmath57 * ( 95,55)(3,-2)35 ( 130,35)*@xmath58 * ( 80,60)(1,3)10 ( 85,90)*@xmath59 * ( 80,60)(2,3)20 ( 102,90)*@xmath58 * ( 80,60)(3,2)30 ( 112,80)*@xmath59 * the @xmath1 decay vertex reconstruction was done in the following three steps . first of all the @xmath60 was reconstructed by finding three charged particles presumably originated from the @xmath60 decay and fitting their tracks . for this goal all combinations of the properly charged particles were examined in the generated events , assuming that two of them are kaons and one is pion . the resulting invariant mass distribution is shown in fig.1a for signal events . the expected @xmath60 peak is clearly seen along with moderate enough combinatorial background . cuts on @xmath61 , @xmath62 and @xmath63 were selected in order to optimize signal to background ratio . to select one more cut on @xmath64 , the information about the invariant mass resolution is desirable . 2a shows the reconstructed @xmath60 meson from its true decay products . the finite invariant mass resolution is due to applied track smearing and equals approximately to @xmath65 . after @xmath60 meson reconstruction , @xmath66 meson was searched in three particle combinations from the remaining charged particles , each particle in the combination being assumed to be a pion . 1b shows a resulting invariant mass distribution for signal events . because of huge width of @xmath66 , signal to background separation is not so obvious in this case . if @xmath66 is reconstructed from its true decay products as in fig . 2b , its width is correctly reproduced . to draw out @xmath66 from the background , further cuts were applied on @xmath67 , @xmath68 , @xmath69 and @xmath70 . at last @xmath19 decay vertex was fitted , using reconstructed @xmath60 and @xmath66 . almost the same resolution in the @xmath1-decay proper time was reached @xmath71 , as in @xcite . the corresponding resolution in the b - meson decay length in the transverse plane is @xmath72 . the relevant distributions are shown in fig.3 . branching ratios and signal statistics for the @xmath73 channel are summarized in table 1 . note that we use an updated value for br(@xmath74 ) from @xcite . @xmath19 branching ratios are still unknown experimentally . neglecting su(3 ) unitary symmetry breaking effects , we have taken br(@xmath73)@xmath75 br(@xmath76 ) . + branching ratios and signal statistics for @xmath77 . + [ cols="<,^,<",options="header " , ] as we see , about 2065 reconstructed @xmath19 are expected after one year run at @xmath78 luminosity . the corresponding number of events within one standard deviation ( @xmath79 ) from the @xmath19 mass equals 1407 . this last number should be compared to 2650 signal events , as reported in @xcite , when @xmath80 decay channel is used . events which pass the first level muon trigger ( latexmath:[$p_t > 6~ gev / c,~ background can come from other @xmath16 decays of the same or higher charged multiplicity , and from random combinations with some ( or all ) particles originating not from a @xmath16 decay ( combinatorial background ) . the following channels were considered and no significant contributions were found to the background : * @xmath82 . these events do nt pass the analysis cuts , because the @xmath83 mass is shifted from the @xmath60 mass by about 100 mev , and so does the @xmath20 mass compared to the @xmath19 mass . * @xmath84 followed by @xmath85 . taking @xmath86 from @xcite , we see that the expected number of @xmath87 events , originated from this source , is only five times less than the expected number of truly signal events . but the decay topology for this decay chain is drastically different ( 1 + 5 , not 3 + 3 ) and therefore it is unexpected that significant amount of the b - decays will be simulated in this way . + note that even for @xmath88 decay channel the similar background is negligible @xcite , although @xmath89 is about 44 times bigger than @xmath90 . * @xmath91 . about 10 000 such events were generated by pythia and then analyzed . using br(@xmath92 from @xcite and assuming that @xmath93 decay goes through @xmath94 oscillations : @xmath95 , and therefore @xmath96 , we have got fig.4 . it is seen from this figure that because of @xmath97 mass shift , the contribution of this channel to the background proves to be negligible . + note that fig.4 refers to the total number of the @xmath91 events . in fact the distribution of these events with regard to the decay proper time is oscillatory , @xmath98 ( not @xmath26 ) defining the oscillation frequency . so in general this will result in oscillatory dilution factor . the conclusion that this dilution factor is irrelevent relies on the fact that no candidate event was found with invariant mass within one standard deviation from the @xmath1 mass for @xmath99 integrated luminosity . a huge monte - carlo statistics is needed for combinatorial background studies . no candidate event with @xmath100 was found within @xmath101 inclusive @xmath102 events . this indicates that signal / background ratio is expected to be not worse than 1:1 . the observation of the @xmath103 oscillations is complicated by some dilution factors . first of all the decay proper time is measured with some accuracy @xmath104 . from previous discussions we know that in our case @xmath105 is expected . due to this finite time resolution , the observed oscillations are convolutions of the expressions ( 3 ) and ( 4 ) given above with a gaussian distribution . for example @xmath106 } \frac{ds}{\sqrt{2\pi}\sigma } \sim \nonumber \\ & & \frac{1}{2 } e^{-\frac{\gamma t}{\hbar}}\left ( \cosh{\frac{\delta \gamma } { 2\hbar}}(t-\sigma \frac{\sigma}{\tau})-d_{time}\cos { \frac{\delta m}{\hbar}(t-\sigma \frac{\sigma}{\tau } ) } \right ) \ ; \ ; , \label{eq9 } \end{aligned}\ ] ] where @xmath107 } , ~\tau = \frac{\hbar}{\gamma}$ ] . so the main effect of this smearing is the reduction of the oscillation amplitude by @xmath108 . this is quite important in the @xmath1 system where @xmath109 . there is also a time shift @xmath110 in ( 9 ) . this time shift does not really effect the observability of the oscillations and we will neglect it . in fact ( 9 ) is valid only for not too short decay times @xmath111 , because in ( 3 ) and ( 4 ) distributions @xmath112 is assumed . another reduction in the oscillation amplitude is caused by the particle/ antiparticle mistagging at t=0 . in our case particle / antiparticle nature of the @xmath16 meson is tagged by the lepton charge in the semileptonic decay of the associated beauty hadron . mistagging is mainly due to * @xmath113 oscillations : accompanying b - quark can be hadronized as a neutral @xmath16 meson and oscillate into @xmath114 before semileptonic decay . * @xmath115 cascade process , then the lepton is misidentified as having come directly from the @xmath16-meson and associated to the @xmath116 decay . * leptons coming from other decaying particles ( k,@xmath117 , ... ) . * detector error in the lepton charge identification . let @xmath118 be the mistagging probability . if we have tagged @xmath119 @xmath3 mesons , among them only @xmath120 are indeed @xmath3-s and @xmath121 are @xmath4-s misidentified as @xmath3-s . so at the proper time @xmath10 we would observe ( @xmath122 due to cpt invariance ) @xmath123= \frac{n}{2}e^{-\frac{\gamma t}{\hbar } } \left [ \cosh{\frac{\delta \gamma t}{2\hbar}}-(1 - 2\eta ) \cos { \frac{\delta m \ , t}{\hbar } } \right ] \nonumber \end{aligned}\ ] ] decays associated to the @xmath4 meson and therefore @xmath124 \ ; \ ; \ ; .\ ] ] so the dilution factor due to mistagging is @xmath125 . in our studies we have taken @xmath126 , as in @xcite . finally the dilution can emerge from background . suppose that apart from @xmath127 events with @xmath128 oscillations we also have @xmath129 additional background events . half of them will simulate @xmath114 meson and half of them b meson ( assuming asymmetry free background ) . so the observed number of would be @xmath128 oscillations will be @xmath130 and the oscillation amplitude will be reduced by an amount @xmath131 neglecting the proper time dependence of this dilution factor ( that is supposing that the background is mainly due to @xmath16-hadron decays and therefore has approximately the same proper time exponential decay as the signal @xcite),we have taken @xmath132 which corresponds to the 2:1 signal / background ratio . for @xmath133 integrated luminosity the number of reconstructed @xmath19-s would reach @xmath134 from the analyzed channel alone . another @xmath135-s are expected from the @xmath136 channel @xcite . for events in which @xmath19 meson does not oscillate before its decay , the @xmath137 meson and the tagging muon have equal sign charges . if the @xmath19 meson oscillates , opposite charge combination emerges . the corresponding decay time distributions are @xmath138 d is the product of all dilution factors and @xmath119 is the total number of reconstructed @xmath19-s . the unification of samples from @xmath73 and @xmath80 decay channels allows to increase @xmath26 measurement precision . fig.7 and fig.8 show the corresponding @xmath139 asymmetry plots for @xmath140 and 35 . it seems to us that @xmath141 decay channel is almost as good for the @xmath1-mixing exploration as previously studied @xmath142 and enables us to increase signal statistics about 1.5 times . further gain in signal statistics can be reached @xcite by using @xmath143 decay mode and considering other decay channels of @xmath60 . these possibilities are under study . we refrain from giving any particular value of @xmath26 as an attainable upper limit . too many uncertainties are left before a real experiment will start . note , for example , that about two times bigger branching ratios for both @xmath144 and @xmath145 decay channels are predicted in @xcite . @xmath146 as a @xmath147 production cross section can also have significant variation in real life @xcite . so although the results of this investigation strengthen confidence in reaching @xmath26 as high as 40 @xcite , it should be realized that some theoretical predictions about @xmath1-physics and collider operation were involved and according to t.d.lees first law of physicist @xcite `` without experimentalist , theorist tend to drift '' . however maybe it is worthwhile to recall his second law also `` without theorist , experimentalists tend to falter '' . many suggestions of p.eerola strongly influenced this investigation and lead to considerable improvement of the paper . communications with s.gadomski and n.ellis are also appreciated . authors thank n.v . makhaldiani for drawing their attention to t.d.lees paper . 99 a. ali , d. london , j. phys . * g19 * ( 1993 ) , 1069 . see for example : ua1 coll . albajar et al . , phys . lett . * b186 * ( 1987 ) , 247 ; ( 1991 ) , 171 . cleo coll . , j. bartelt et al . , phys . * 71 * ( 1993 ) , 1680 . argus coll . , h. albrecht et al . , z. phys . * c55 * ( 1992 ) , 357 . aleph coll . , d. buskulic et al . , phys . lett . * 284 * ( 1992 ) , 177 . opal coll . acton et al . , phys . lett . * b276 * ( 1992 ) , 379 . , b. adeva et al . , phys . lett . * b288 * ( 1992 ) , 395 . delphi coll . , p. abreu et al . , phys . lett . * b332 * ( 1994 ) , 488 . moser , b - mixing . talk given at the @xmath148 international symposium on heavy flavour physics , montreal , canada , 1993 . cern - ppe/93 - 164 . a. ali , d. london , cp violation and flavour mixing in the standard model , desy-95 - 148 ( hep - ph/9508272 ) . a. ali , d. london , implications of the top quark mass measurement for the ckm parameters , @xmath26 and cp asymmetries . cern - th.7398/94 ( hep - ph/9408332 ) . buras , w. slominsski , h. steger , nucl . b245 * ( 1984 ) , 369 . franzini , phys . * 173 * ( 1989 ) , 1 . voloshin et al . , * 46 * ( 1987 ) , 181 . a. datta , e.a . paschos , u. trke , phys . b196 ( 1987 ) , 382 . bigi , lifetimes of heavy - flavour hadrons - whence and whither ? und - hep-95-big06 ( hep - ph/9507364 ) . i. dunietz , @xmath38 mixing , cp violation and extraction of ckm phases from untagged @xmath1 data samples . fermilab - pub-94/361-t ( hep - ph/9501287 ) . aleph coll . , d. decamp et al . , phys . b313 * ( 1993 ) , 498 . aleph coll . , d. buskulic et al . , cern - ppe/93 - 99 . opal coll . , r. akers et al . , cern - ppe/94 - 43 . p. eerola , s. gadomski , b. murray , @xmath19-mixing measurement in atlas , atlas internal note phys - no-039 , 1994 . atlas technical proposal , cern / lhcc/94 - 43 , 1994 . t.sjstrand , pythia 5.7 and jetset 7.4 : physics and manual , cern - th.7112/93 , 1993 . review of particle properties , phys . rev . * d50 * , 1994 . s. rudaz , m.b . voloshin , phys . lett . * b252 * ( 1990 ) , 443 . p. eerola et al . , asymmetries in @xmath16 decays and their experimental control , atlas internal note phys - no-054 , 1994 . p. camarri , a. nisati , time - dependent analysis of cp - asymmetries in the @xmath149 system , atlas internal note phys - no-065 , 1995 . p. blasii , p. colangelo , g. nandulli , phys . b283(1992 ) , 434 . p. eerola , measurement of cp - violation in b - decays with the atlas experiment , atlas internal note , phys - no-009 , 1992 . t. d. lee , the evolution of weak interactions , talk given at the symposium dedicated to jack steinberger , geneva , 1986 , cern 86 - 07 .
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the usefulness of the @xmath0 decay chain is investigated for the @xmath1-reconstruction in the future atlas @xmath1-mixing experiment .
it is shown that this decay channel is almost as suitable for this purpose as previously studied @xmath2 .
-10 mm
| 5,902 | 70 |
the hubble space telescope and advances in ground based observing have greatly increased our knowledge of the galaxy population in the distant universe . however , the nature of these galaxies and their evolutionary connections to local galaxies remain poorly understood . luminous , compact , star forming galaxies appear to represent a prominent phase in the early history of galaxy formation @xcite . in particular : * the number density of luminous , compact star forming galaxies rises significantly out to z @xmath1 1 @xcite . * the lyman break galaxies at z @xmath12 2 seen in the hubble deep field are characterized by very compact cores and a high surface brightness @xcite . * sub - millimeter imaging has revealed distant galaxies ( z @xmath1 2@xmath134 ) , half of them compact objects , which may be responsible for as much as half of the total star formation rate in the early universe @xcite . however , little is definitively known of their physical properties , or how they are related to subsets of the local galaxy population . a classification for known examples of intermediate redshift ( 0.4 @xmath0 z @xmath0 0.7 ) luminous , blue , compact galaxies , such as blue nucleated galaxies , compact narrow emission line galaxies , and small blue galaxies , has been developed by @xcite in order to be able to choose samples over a wide redshift range . they have found that the bulk of these galaxies , collectively termed luminous compact blue galaxies ( lcbgs ) , can be distinguished quantitatively from local normal galaxies by their blue color , small size , high luminosity , and high surface brightness . ( see 2.1 for more detail . ) from studies at intermediate redshifts , it has been found that lcbgs are a heterogeneous class of vigorously starbursting , high metallicity galaxies with an underlying older stellar population @xcite . while common at intermediate redshifts , they are rare locally @xcite and little is known about the class as a whole , nor their evolutionary connections to other galaxies . lcbgs undergo dramatic evolution : at z @xmath1 1 , they are numerous and have a total star formation rate density equal to that of grand - design spirals at that time . however , by z @xmath1 0 , the number density and star formation rate density of lcbgs has decreased by at least a factor of ten @xcite . since the lcbg population is morphologically and spectroscopically diverse , these galaxies are unlikely to evolve into one homogeneous galaxy class . @xcite and @xcite suggest that a subset of lcbgs at intermediate redshifts may be the progenitors of local low - mass dwarf elliptical galaxies such as ngc 205 . alternatively , @xcite and @xcite suggest that others may be disk galaxies in the process of building a bulge to become local l@xmath2 spiral galaxies . clearly , to determine the most likely evolutionary scenarios for intermediate redshift lcbgs , it is necessary to know their masses and the timescale of their starburst activity . are they comparable to today s massive or low - mass galaxies ? are they small starbursting galaxies which will soon exhaust their gas and eventually fade ? or are they larger galaxies with only moderate amounts of star formation ? only kinematic line widths that truly reflect the masses of these galaxies , as well as measures of their gas content and star formation rates , can answer these questions . using ionized gas emission line widths , @xcite , @xcite , and @xcite , have found that lcbgs have mass - to - light ratios approximately ten times smaller than typical local l@xmath2 galaxies . however , since ionized gas emission lines may originate primarily from the central regions of galaxies , their line widths may underestimate the gravitational potential @xcite . h emission lines provide a better estimate of the total galaxy mass as they measure the gravitational potential out to larger galactic radii . observations of both h and co ( the best tracer of cold h@xmath14 ) , combined with star formation rates , are necessary to estimate the starburst timescales . with current radio instrumentation , h and co can only easily be measured in very nearby lcbgs , at distances @xmath0 150 mpc for h , and @xmath0 70 mpc for co. therefore , to understand the nature and evolutionary possibilities of higher redshift lcbgs , we have undertaken a survey in h 21 cm emission and multiple rotational transitions of co of a sample of 20 local lcbgs , drawn from the sloan digital sky survey @xcite . this work , paper i , reports the optical photometric properties of our sample and the results of the h 21 cm portion of the survey , including dynamical masses and comparisons with local galaxy types . paper ii @xcite will report the results of a survey of the molecular gas conditions . knowledge of the dynamical masses , combined with gas masses and star formation rates , constrains the evolutionary possibilities of these galaxies . nearby blue compact galaxies ( bcgs ) have been studied extensively at radio and optical wavelengths since @xcite originated the term `` compact galaxy '' and @xcite distinguished between `` red '' and `` blue '' compact galaxies . the term bcg typically refers to galaxies with a compact nature , a high mean surface brightness , and emission lines superposed on a blue continuum . however , many different selection criteria have been used , leading to various definitions of bcgs and samples with a range of properties . for example , the term `` dwarf '' has been used to mean bcgs fainter than @xmath1317 ( e.g. thuan & martin 1981 ; kong & cheng 2002 ) or @xmath1318 blue magnitudes ( e.g. taylor et al . 1994 ) , or an optical diameter less than 10 kpc @xcite . the term `` blue '' has been used to mean blue on the palomar sky survey plate ( e.g. gordon & gottesman 1981 ) , or to have emission lines superposed on a blue background ( e.g. thuan & martin 1981 ) . the term `` compact '' has been used to mean smaller than 1 kpc in optical diameter ( e.g. thuan & martin 1981 ) , or qualitatively compact ( e.g. doublier et al . some , for example @xcite , began using the term blue compact dwarf to refer to the common low luminosity , low metallicity bcgs . as high luminosity ( @xmath1l@xmath2 ) , nearby bcgs are rare , many began to use the term blue compact dwarf for all nearby bcgs , regardless of luminosity . there are only a few of the rare local lcbgs ( by the @xcite definition ) in previous bcg surveys . note that @xcite have recently been studying local luminous bcgs . however , their selection criteria are not as stringent as that of @xcite , and they have been focusing on very low metallicity ( less than 15% solar ) galaxies . intermediate redshift lcbgs with rest - frame properties matching the class definition of @xcite have metallicities at least 40% solar @xcite . four of our local lcbgs have metallicities available in the literature , which range from 40 @xmath13 70 % solar @xcite . by using the selection criteria of @xcite , we ensure that our study is of those local lcbgs defined to be analogs to the widely studied higher redshift lcbgs . the observations , including sample selection and data reduction , are described in @xmath152 . the optical photometric properties , h 21 cm spectra , measurements , and derived properties are presented in @xmath153 , and analyzed in @xmath154 . we compare the derived physical properties to local normal galaxies and higher redshift lcbgs in @xmath155 , and conclude in @xmath156 . we assume h@xmath16 = 70 km s@xmath17 mpc@xmath17 throughout . when we compare our results to those of other authors , we scale their results to this value . when @xcite compared intermediate redshift lcbgs with local normal galaxies , they found that lcbgs can be isolated quantitatively on the basis of color , surface brightness , image concentration , and asymmetry . color and surface brightness were found to give the best leverage for separating lcbgs from normal galaxies . specifically , lcbgs can be defined by a region limited to b@xmath13v @xmath18 0.6 , and a b - band surface brightness within the half - light radius , sbe , brighter than 21 b - mag arc sec@xmath19 . this simple definition differs only slightly from the formal definition in @xcite which uses a color - dependent sbe . @xcite also applied a luminosity cut - off of 25% l@xmath20m@xmath21 @xmath18 @xmath1318.5 ) , to distinguish lcbgs from blue compact galaxies . lcbgs are not so extreme that they are completely separated from the continuum of normal galaxies . the sharp borders used to classify them are artificial , but serve the purpose of defining similar objects over a range of redshifts . lcbgs at intermediate redshifts ( z @xmath1 0.6 ) can be studied in deep spectroscopic surveys ( i @xmath1 24 ) , while the brightest lcbgs at high redshifts ( z @xmath1 3 ) require very deep spectroscopic surveys ( i @xmath1 26 ) . therefore , lcbgs selected in this manner are observable over a wide range of redshifts . using these color , surface brightness , and luminosity criteria , we selected our sample of local lcbgs from the sloan digital sky survey ( sdss ) . begun in 2000 , this survey will ultimately image one quarter of the sky using a large format ccd camera on a 2.5 m telescope at apache point observatory in new mexico . images are taken in five broad bands ( u , g , r , i , z ) which range from 3540 ( u ) to 9130 ( z ) . the survey has a limiting magnitude of 22.2 in g ( 4770 ) and r ( 6230 ) , our two bands of interest . after searching through approximately one million galaxies ( @xmath11500 degree@xmath22 on the sky ) , we identified only 16 nearby ( d @xmath0 70 mpc ) lcbgs . this distance cut - off was chosen to ensure our galaxies could be detected quickly in both h and co. we added four markarian galaxies from the literature which fulfilled our selection criteria and were not yet surveyed by sdss , for a total of 20 local lcbgs . the color and surface brightness characteristics of our local sample , as well as higher redshift ( 0.4 @xmath0 z @xmath0 1 ) samples of lcbgs , are compared to other nearby galaxies in figure 1 . we note that while our sample is not complete , it is representative of local lcbgs ( see castander et al . 2004 ) . as at higher redshifts @xcite , our local lcbg sample is a morphologically heterogeneous mixture of galaxies , including sb to sc spirals , s0 galaxies , polar ring galaxies , peculiar galaxies , and h liners and starbursts , as classified by hyperleda and the nasa@xmath23ipac extragalactic database ( ned ) . approximately one quarter of these galaxies are not classified in either database . roughly one third are members of multiple , sometimes interacting or merging , systems . sdss and digitized sky survey images of our sample of 20 galaxies are shown in figures 2 and 3 . the properties of our sample of local lcbgs are described below , and listed in table 1 . _ full sdss galaxy designation of the form sdss jhhmmss.ss@xmath24ddmmss.s , in the j2000 system . in the remainder of the paper , individual galaxies are referred to by an abbreviated sdss name of the form sdssjhhmm@xmath24ddmm . the four non - sdss galaxies are referred to by their markarian ( mrk ) names . _ alternate name . _ mrk and/or ngc designation , if any . _ d@xmath25 . _ hubble distance , in mpc , calculated from the optical redshift ( from sdss , except for the non - sdss galaxies for which we use ned redshifts . ) _ m@xmath21 . _ absolute blue magnitude calculated from the apparent blue magnitude , m@xmath21 , using d@xmath25 . for the sdss galaxies , m@xmath21 = g + 0.30(g@xmath13r ) + 0.18 , b@xmath13v , and r@xmath26(b ) are based on synthetic spectra which fit the observed spectral energy distribution of lcbgs . ] . the r and g magnitudes were calculated from the sdss `` petrosian '' and `` model '' magnitudes : r = r(petro ) and g = r(petro ) @xmath13 [ r(model ) @xmath13 g(model ) ] @xcite . the sdss galactic reddening corrections were applied , but no k corrections were applied as our galaxies are nearby , and many are of unknown spectral type . the k correction is at most 0.015 magnitudes in r and 0.038 magnitudes in g @xcite . no correction was applied for extinction due to inclination , also because of the uncertainty in spectral types . for the non - sdss galaxies , the `` total apparent blue magnitudes '' were used from hyperleda , and corrected for galactic reddening using extinction values from ned . _ b@xmath13v . _ the color transformation is b@xmath13v @xmath27 0.900(g@xmath13r ) + 0.145@xmath28 , where the magnitudes were calculated as above , for the sdss galaxies . the `` total b@xmath13v colors '' from hyperleda were used for the non - sdss galaxies . they were corrected for reddening using extinction values from ned . _ average surface brightness in the b band within the half light or effective radius in b - mag arc sec@xmath19 . for the sdss galaxies , sbe(b ) = m@xmath21 + 2.5 @xmath29 [ 2@xmath30r@xmath31(b ) ] , where r@xmath26(b ) , the effective or half - light radius in the b band , was calculated from the `` petrosian '' effective radii in the g and r bands [ r@xmath26(b ) = 1.30 @xmath32(g ) @xmath13 0.300 @xmath32(r)]@xmath28 . we corrected sbe for cosmological dimming by subtracting 7.5 @xmath29(1 + z ) , where the redshift , z , was calculated using the h velocity ( @xmath153.1 ) . for the non - sdss galaxies , we used sbe from hyperleda ( termed the `` mean effective surface brightness '' ) , and then corrected it for cosmological dimming in the same way . _ ned type . _ morphological type as given by ned , which does not use a homogeneous system . _ hyperleda type . _ morphological type as given by hyperleda , which uses the @xcite system . _ other sources in beam ? _ indicates whether or not each galaxy has other sources at similar velocities within the 9@xmath33.2 beam of the green bank telescope ( used for the h observations ) , as given by ned . we compared hyperleda and sdss magnitudes for our sdss selected galaxies to ensure that there are no systematic offsets between the two . fourteen galaxies had magnitudes measured in both systems . the median difference between sdss and hyperleda magnitudes ( as calculated in @xmath152.1.2 ) is 0.1 magnitudes . since the standard deviation of these differences is @xmath34 0.8 magnitudes , we estimate the uncertainty in the median magnitude difference to be 0.2 magnitudes . therefore , there is no significant difference between the hyperleda and sdss magnitudes . unfortunately , while fourteen galaxies have magnitudes measured in both systems , only two sdss selected galaxies have b@xmath13v and sbe entries in hyperleda ; for these two galaxies there are no significant differences between the hyperleda and sdss values . h 21 cm observations of 19 of the 20 nearby lcbgs in our sample were made with the 100 meter green bank telescope ( gbt ) at the national radio astronomy observatory in green bank , west virginia between 2002 november 30 and december 6 . the h spectrum of sdssj0934@xmath240014 was acquired by @xcite during gbt commissioning on 2001 november 29 . the main beam half power width is 9@xmath33.2 at 21 cm @xcite . both linear and circular polarizations were observed using the l - band receiver ( 1.15 @xmath13 1.73 ghz ) . position - switching was used with an offset of @xmath1318@xmath33 in right ascension . the spectral processor was used with a bandwidth of 20 mhz ( @xmath14000 km s@xmath17 ) for most galaxies , although a few were observed with a bandwidth of 5 mhz ( @xmath11000 km s@xmath17 ) . the sample time was 60 s , with each galaxy observed for between 6 and 30 minutes , for a peak signal - to - noise of at least five . the only exceptions were sdssj0218@xmath130757 and sdssj0222@xmath130830 , marginal detections , which were observed for 52 minutes each . the initial data calibration and reduction was performed using the aips++ single dish analysis environment `` dish . '' the position - switched data were calibrated in the standard way taking the difference of the on and off scans divided by the off scans . the dish package automatically calibrates the data into temperature units using tabulated values for the position of the telescope . the individual scans were then combined and a first order baseline was fit to the line - free regions and removed . our wide bandwidths ensured a sufficient line - free region . the individual polarizations were then combined , except when one polarization was significantly noisier than the other , or was affected by radio interference . finally , each spectrum was smoothed to @xmath112 km s@xmath17 channels using boxcar smoothing . the rest of the reduction and analysis was done using our own procedures written in interactive data language ( idl ) . to convert our data from temperature to flux density units , we observed the radio galaxy 3c295 . it makes an ideal calibration source as it has not varied by more than @xmath11@xmath35 since 1976 for 2.8 cm @xmath36 @xmath37 @xmath36 21 cm @xcite . comparing our observations of 3c295 with those of @xcite , we found the gain of the gbt to be 1.9 k jy@xmath17 and applied this calibration to our data . finally , obvious noise spikes were clipped out of the data . data of sdssj0934@xmath240014 , the galaxy observed during telescope commissioning , were reduced and calibrated by @xcite . the spectrum was then smoothed to @xmath112 km s@xmath17 channels . the central 800 km s@xmath17 of the final h 21 cm spectra of the 20 local lcbgs are shown in figure 4 . each galaxy s velocity calculated from optical redshifts is indicated with a triangle . dashed lines indicate the 20% crossings used to measure the line widths ( @xmath153.1 ) . all 20 galaxies were detected in the 21 cm line of h , although two , sdssj0218@xmath130757 and sdssj0222@xmath130830 , were only detected at the 3 and 4 @xmath38 ( respectively ) level . h measurements and derived optical and h quantities are listed in tables 2 , 3 and 4 and described in the following sections . since the gbt beam is 9@xmath33.2 at 21 cm , the emission from the target galaxies and any other galaxies within the beam and at similar velocities are blended into one spectrum . ( see table 1 for those galaxies with other sources at similar velocities within the gbt beam . ) we find fairly broad h profiles ( 126 km s@xmath17 @xmath18 w@xmath39 @xmath18 362 km s@xmath17 ) and a variety of profile shapes , including double - horned , flat - topped and gaussian . over a third of our profiles appear asymmetric and only half of these are from galaxies known to have other sources within the gbt beam . these asymmetries may indicate asymmetries in the gas density distribution and/or disk kinematics , or unidentified galaxies within the beam @xcite . estimates of the dynamical masses of our local sample of lcbgs constrain their evolutionary possibilities . we estimate the dynamical mass , m@xmath40 , within a radius , r , as @xmath41 the constant c@xmath14 is a geometry - dependent factor which depends on the galactic light profile . for example , the king models of @xcite give 0.9 @xmath0 c@xmath14 @xmath0 1.5 , depending on the ratio of tidal to core radius . as we do not have information on the light profiles of our galaxy sample , we simply choose c@xmath14 = 1 . the line width at 20@xmath35 , corrected for the effects of inclination , w@xmath42 , is used as a measure of twice the maximum rotational velocity , v@xmath43 ( e.g. * ? ? ? w@xmath39 was measured at the points equal to 20% of the peak flux . the first crossing at 20@xmath35 on each side of the emission line was used , once the spectrum became distinguishable from the noise . these crossings are indicated by dashed lines in figure 4 . we find w@xmath39 ranging from 126 @xmath13 362 km s@xmath17 . when @xcite studied lcbgs at intermediate redshifts , they found a similar range of line widths , using optical emission lines . the recessional velocities in the barycentric system , v@xmath44 , were calculated as the midpoint between the 20@xmath35 crossings . the uncertainties listed in table 2 for w@xmath39 , v@xmath44 , and the galaxy distance ( d ) are from uncertainties in measuring the exact w@xmath39 crossings . two galaxies have multiple crossings at the 20% flux level , once the spectrum is distinguishable from the noise . one of these , sdssj0834@xmath240139 , has a wing on the h spectrum which extends over 200 km s@xmath17 . neither sdssj0834@xmath240139 nor its nearby companion have optical velocities coincident with the peak of the h spectrum ; instead , both have velocities in the wing of the spectrum ( see figure 4 ) . for this reason , we used w@xmath39 measured from the last crossing , which is at the edge of the wing . the other galaxy with multiple crossings at 20% is sdssj0904@xmath245136 , which has a low level extension to one side of the spectrum . we used the first crossing at 20% . the uncertainties in these two w@xmath39 measurements are reflected in the large errors associated with w@xmath39 and derived quantities . we have initiated an observing program at the very large array ( vla ) to map these local lcbgs in h to disentangle target galaxy emission from any other galaxies in the field , improving the mass estimates . to correct the line width for the effects of inclination ( _ i _ ) , we divided w@xmath39 by @xmath45(_i _ ) , to give w@xmath42 . each sdss galaxy s inclination was approximated from the sdss data by : @xmath46 the sdss isophotal major and minor axes in the g - band ( 4770 ) were used , except for sdssj0943@xmath130215 where no g band data were available . in that case , r - band ( 6230 ) data were used . the sdss isophotal axes are derived from the ellipticity of the 25 magnitude arc sec@xmath19 isophote in each band . inclinations derived from sdss isophotal axes in r , i ( 7630 ) , and z ( 9130 ) bands agree with inclinations derived from g - band data to within 8@xmath47 . the inclinations from hyperleda were used for the markarian galaxies . thirteen of our sdss galaxies also have inclinations available in hyperleda , which are calculated from the apparent flattening and morphological type of the galaxies . we estimate the dispersion of the differences between sdss and hyperleda inclinations is 12@xmath47 . a further correction can be made to w@xmath42 to account for random motions , giving w@xmath48 . as outlined in @xcite , @xmath49 - 2w_t^2\exp-(w_{20}/w_c)^2\ ] ] and @xmath50 w@xmath51 is the random motion component of the line width ; w@xmath51 = 38 km s@xmath17 . w@xmath52 characterizes the transition region between linear and quadrature summation of rotational and dispersive terms ; w@xmath52 = 120 km s@xmath17 . the formula degenerates to linear summation for giant galaxies , and quadrature summation for dwarf galaxies @xcite . this correction for random motions decreases the line width of the local lcbg sample by 26 to 38 km s@xmath17 , depending on the rotational velocity . it is crucial , when comparing dynamical masses from different studies , that the rotational velocities be calculated in the same way . in general , we calculate the rotational velocities from w@xmath53 . however , when we wish to compare our sample with others ( e.g. * ? ? ? * ) who have not applied this correction for random motions to their line widths , we also do not apply this correction . we indicate dynamical masses calculated without correcting w@xmath42 for random motions as m@xmath54 . unless indicated as such , all dynamical masses are calculated using the line width corrected for random motions , w@xmath53 . note that in all cases , the line widths have been corrected for inclination . it is also crucial , when comparing dynamical masses from different studies , that they be measured within the same radius . typically , the h radius ( r@xmath3 ) is measured at 1 m@xmath7 pc@xmath19 and is used to estimate the total enclosed mass of a galaxy . however , it is only possible to measure r@xmath3 with interferometers in nearby galaxies . by practical necessity then , some other radius must be adopted for our sample and the galaxy s mass is assumed to be spherically distributed in that radius . two different radii are commonly used : r@xmath26 , the effective or half - light radius , and r@xmath10 , the isophotal radius at the limiting surface brightness of 25 b - magnitudes arc sec@xmath19 . we have measurements of both r@xmath26 ( in r and b band from sdss ) and r@xmath10 ( in b - band from hyperleda ) for most galaxies . @xcite found that r@xmath10 = 2.5 r@xmath26 for nearby emission line galaxies measured in r - band . when we compare our values of r@xmath26 to r@xmath10 , we find medians of @xmath55 and @xmath56 for the galaxies with no r@xmath10 ( sdssj0218@xmath130757 , sdssj0222@xmath130830 and sdssj1118@xmath246316 ) or r@xmath26 ( non - sdss galaxies ) measurements available , we estimate these quantities from the above relations . we find our local sample of lcbgs have dynamical masses , measured within r@xmath10 , ranging from 3@xmath410@xmath6 to 1@xmath410@xmath11 m@xmath7 , with a median of 3@xmath410@xmath57 m@xmath7 . the dynamical masses measured within the effective radius ( measured in both the r and b - band ) range from 8@xmath410@xmath5 to 3@xmath410@xmath57 m@xmath7 , with a median of 8@xmath410@xmath58 m@xmath7 . figure 5 compares the dynamical masses measured for local lcbgs to nearby galaxies of hubble type s0a through i m , where `` m '' indicates magellanic , or low luminosity @xcite . note that @xcite did not correct line widths for random motions as we did . therefore , in figure 5 we compare m@xmath54(r @xmath59r@xmath10 ) which have not been corrected for random motions . these dynamical masses are higher than those we list in table 3 , which have been corrected for random motions . as seen in figure 5 , some local lcbgs have dynamical masses as large as l@xmath2 galaxies ( @xmath110@xmath11 m@xmath7 , roberts & haynes 1994 ) . however , at least 75@xmath35 have dynamical masses approximately an order of magnitude smaller than typical local l@xmath2 galaxies , consistent with observations of lcbgs at higher redshifts ( as discussed in @xmath15 1.1 ) . however , our sample includes galaxies down to 0.25 l@xmath2 . galaxies of this luminosity typically have dynamical masses ( within r@xmath10 ) @xmath12@xmath410@xmath57 m@xmath7 @xcite . therefore , it is more appropriate to compare mass - to - light ratios ; see @xmath154.1 . we estimate the random errors associated with the dynamical masses to be approximately 50% . this estimate is made from uncertainties in the inclination ( @xmath34 12@xmath47 , from the earlier comparison of hyperleda and sdss inclinations ) ; uncertainties in radii measurements ( as given by hyperleda and sdss ) ; and uncertainties in measuring w@xmath39 ( as discussed above ) . however , the random errors are overwhelmed by the systematic uncertainties when estimating dynamical masses . we have chosen a structural constant of c@xmath14 = 1 , but this could be as high as 1.5 , as discussed earlier . even more significant is that 45% of our galaxy sample have other galaxies at similar velocities in the beam of the gbt . as the h emission from all sources in the beam and within the bandwidth is blended into one spectrum , we may be overestimating the masses of these galaxies . finally , a fifth of our galaxies have low inclinations ( @xmath18 40@xmath47 ) which lead to large corrections and therefore increasing uncertainties in the rotational velocities and overestimations of the dynamical masses . the dynamical masses we report may be viewed as upper limits we are certainly overestimating the masses in many cases , but not underestimating them . we are pursuing a follow - up program with both the arecibo radio telescope and the vla to address these issues . at 21 cm the arecibo radio telescope has a beam size of 3@xmath33.1 @xmath4 3@xmath33.5 , a third the size of the gbt beam . this decreases the number of target galaxies observed with other galaxies in the beam , decreasing the number of galaxies with overestimates of dynamical masses . we have already observed over 40 lcbgs at arecibo , but results are not yet available . the vla in `` b '' configuration provides the ability to map our galaxies in 21 cm emission to a resolution of 5@xmath33@xmath33 . we have begun a survey of those lcbgs with companions . vla maps allow us to disentangle the emission from each galaxy and more accurately estimate the dynamical mass of the target galaxy . along with dynamical mass , a knowledge of the amount of fuel , i.e. atomic and molecular gas , available for the starburst activity , is critical for narrowing down the evolutionary possibilities for lcbgs . when combined with star formation rates , this gives an estimate of the maximum length of the starburst at the current rate of star formation . we defer a full discussion of this until paper ii @xcite where we present our measurements of the co content of these galaxies , but present the h results here . the h masses are given by : @xmath60 @xcite , where the distance ( measured from h ) is d = v@xmath44 h@xmath61 . the total h flux , @xmath62 , was calculated by numerically integrating under the spectrum where it is distinguishable from the noise . for the two marginal detections , sdssj0218@xmath130757 and sdssj0222@xmath130830 , h masses were calculated using the optical ( sdss ) recessional velocities . we find local lcbgs have h masses ranging from 5@xmath410@xmath5 to 8@xmath410@xmath6 m@xmath7 , with a median of 5@xmath410@xmath6 m@xmath7 . these span the range of h masses in nearby galaxies across the hubble sequence ; the median is that of nearby late - type spiral galaxies @xcite . the uncertainties listed for m@xmath3 ( table 3 ) include uncertainties in the distance and integrated flux density , as listed in table 2 . however , as in the discussion of dynamical mass uncertainties in @xmath153.1 , the h masses for nearly half of our galaxies are most likely overestimates due to the presence of other galaxies within the gbt beam . dynamical mass - to - light ratios indicate whether our local sample of lcbgs are under - massive for their luminosities . we calculated the total blue luminosities , l@xmath21 , of our galaxies from m@xmath21 , assuming m@xmath63 = 5.48 @xcite . we find the median mass - to - light ratio , m@xmath40(r @xmath18 r@xmath10 ) l@xmath8 = 3 m@xmath7 l@xmath64 , with a minimum of 0.6 and a maximum of 9 m@xmath7 l@xmath64 . @xcite find that m@xmath40(r @xmath18 r@xmath10 ) l@xmath8 ranges from 3.6 @xmath13 10 m@xmath7 l@xmath64 in local galaxies across the hubble sequence . the median m@xmath40(r @xmath18 r@xmath10 ) l@xmath8 is fairly constant ranging from 5 to 7 m@xmath7 l@xmath64 . however , as discussed in @xmath153.1 , to compare our mass - to - light ratios to those of local , normal hubble - types in @xcite , we must use dynamical masses which have not been corrected for random motions . figure 6 compares such mass - to - light ratios for our local sample of lcbgs and nearby hubble types . while some local lcbgs have mass - to - light ratios equal to or greater than the median for local hubble types , approximately half the local lcbgs have smaller mass - to - light ratios . many of the galaxies with mass - to - light ratios typical of local hubble types have other galaxies within the beam , possibly leading to an overestimation of the dynamical mass - to - light ratio of the target galaxy . therefore , in general , lcbgs tend to have mass - to - light ratios smaller than local normal galaxies . that is , most are small galaxies undergoing large amounts of star formation and not simply large galaxies with moderate amounts of star formation . we calculated the gas mass fraction , m@xmath3 m@xmath65 ( r @xmath18 r@xmath3 ) for the local sample of lcbgs as a measure of their atomic gas richness . we do not have interferometric observations of these galaxies , so we estimated the hydrogen radius , r@xmath3 , from optical radii . @xcite found for nearby spiral galaxies that r@xmath3 = 2 r@xmath10 , where r@xmath3 is measured at the 1 m@xmath7 pc@xmath19 level . however , @xcite found r@xmath3 ranging from 3 @xmath13 5 r@xmath10 for a small sample of local h galaxies which are similar to lcbgs , but much less luminous ( m@xmath21 @xmath66 @xmath1316 ) . @xcite studied a large sample of nearby galaxies and found that galaxies with smaller optical radii have larger h extensions . @xcite have measured r@xmath3 in one of our lcbgs , mrk 314 , using the vla . they find r@xmath3 = 4 r@xmath10 . therefore , although we estimated r@xmath3 as 2 @xmath4 r@xmath10 following @xcite , we note it may be an underestimate , which would cause us to overestimate the gas mass fraction . we are undertaking interferometric observations of some of our local sample of lcbgs to directly measure r@xmath3 . this will be addressed in a future paper . in local lcbgs , we find the gas mass fraction , m@xmath3 m@xmath65 , ranges from 0.03 to 0.2 , with a median of 0.08 . that is , it ranges from normal hubble type spirals ( @xmath18 0.1 @xcite ) to very gas rich galaxies such as nearby h and irregular galaxies studied by @xcite with gas fractions ranging from 0.2 @xmath13 0.4 . this is similar to the range of 0.01 @xmath13 0.5 found by @xcite for local blue compact galaxies , most of them less luminous than lcbgs . the fraction of hydrogen mass to blue luminosity provides a distance independent alternative measure of the gas richness of galaxies . we find our local lcbgs have m@xmath3 l@xmath8 ranging from 0.09 to 2 m@xmath7 l@xmath67 , with a median of 0.4 m@xmath7 l@xmath67 . this spans the range of nearby galaxies from s0a through i m , the median corresponding to late - type spirals @xcite . @xcite studied a range of blue compact galaxies , ranging from faint blue compact dwarfs to lcbgs , and found the same median value , 0.4 m@xmath7 l@xmath67 . the tully - fisher relationship , a relation between line width and luminosity , is a fundamental scaling relation for non - interacting spiral galaxies . an absolute blue magnitude versus h line width version of the tully - fisher relationship is shown in figure 7 . we have compared our local lcbgs with the tully - fisher relation found for @xmath14500 normal galaxies within @xmath140 mpc @xcite . as seen in figure 7 , the location of local lcbgs is consistent with the @xcite relation , although with a large scatter . @xcite have a 1 @xmath38 scatter of 0.3 magnitudes in m@xmath21 , while the local lcbgs have a 1 @xmath38 scatter of 0.9 magnitudes in m@xmath21 . approximately half the galaxies ( without other sources in the beam ) lie to the left of the tully - fisher relationship , indicating they have lower masses than expected from their luminosities , although this result is not statistically significant . this is consistent with the distribution of lcbg and hubble type galaxy mass - to - light ratios in figure 6 . seven of the galaxies lying to the right of the tully - fisher relationship have other galaxies within the gbt beam , possibly leading to overestimations of the line widths . note that in our comparison of magnitudes , masses , and mass - to - light ratios , as in all other comparisons in this paper , we have adjusted all values to a hubble constant of h@xmath16 = 70 km s@xmath17 mpc@xmath17 we have observed 20 local lcbgs chosen with the same selection criteria as those lcbgs common at higher redshifts . lcbgs can not remain lcbgs for a long period of time : the number density of lcbgs decreases by at least a factor of ten from z @xmath12 0.5 to today . that is , while lcbgs were common in the past , there are very few today . out of approximately a million nearby galaxies observed by the sdss , only about a hundred are lcbgs . ( see castander et al . 2004 for a discussion of the local space density of lcbgs . ) therefore , lcbgs must evolve into some other galaxy type . from studies of intermediate redshift lcbgs , @xcite and @xcite proposed that some may be progenitors of local dwarf elliptical galaxies . alternatively , @xcite and @xcite proposed that some may be disk galaxies in the process of forming a bulge to become present - day l@xmath2 spiral galaxies . two pieces of information are crucial to determining if these possibilities are likely . it is necessary to know the dynamical masses and the duration of the starbursts . we have measured the dynamical masses of our local sample of lcbgs using h . the length of the starburst will not be estimated until we discuss our molecular gas survey in paper ii @xcite . however , we can use our measurements of line width and radius to identify local galaxies comparable to lcbgs , independent of the evolutionary stage of their stellar populations . in figure 8 we plot the effective or half - light radius , r@xmath26 , versus the velocity dispersion , @xmath38 , for our sample of local lcbgs , intermediate redshift ( 0.4 @xmath0 z @xmath0 1 ) lcbgs @xcite , and local samples of elliptical , spiral , magellanic spiral , irregular , and dwarf elliptical galaxies ( guzmn et al . 1996 ; hyperleda ) . we measured the @xmath38 of our sample of galaxies from the h spectra by measuring the moments of the spectra . the measurements were made in the same way as we described in @xmath153.2 for measuring @xmath62 , the zeroth moment . we find similar results if we simply scale w@xmath39 to @xmath38 by assuming the spectra are gaussian , that is , dividing w@xmath39 by 3.6 . the values of @xmath38 for the other galaxy samples are from optical line width measurements , except the measurements for magellanic spirals are from 21 cm h line widths . optical line widths may be smaller than h line widths . for example , @xcite studied a sample of nearby blue compact galaxies and found the ionized gas emission line widths to be systematically smaller than the neutral hydrogen emission line widths . on average , the ratio of w@xmath39 measured from h to w@xmath39 measured from h was 0.66 for their sample of 11 blue compact galaxies . however , the galaxies in @xcite s sample tend to be smaller both in effective radius and h line width than our sample of local lcbgs , suggesting that the difference between the optical and h line widths may be smaller as well @xcite . the r@xmath26 versus @xmath38 plot allows us to compare the dynamical mass properties of our sample of lcbgs with the other galaxy types . these properties are expected to remain constant despite luminosity evolution . we find that while some local lcbgs have @xmath38 consistent with local spiral galaxies , they tend to be too small in r@xmath26 . however , local lcbgs are consistent with the smaller spiral galaxies , magellanic spirals . they are inconsistent with elliptical galaxies , but are consistent with the most massive dwarf elliptical and irregular galaxies . there is much confusion by what various authors mean when discussing dwarf elliptical galaxies ( sometimes called spheroidal galaxies ) . we are referring to galaxies such as ngc 205 and _ not _ the less massive and less luminous galaxies like draco and carina . finally , when we compare our local sample of lcbgs with those lcbgs observed at intermediate redshifts , we find that they occupy the same region of r@xmath26 @xmath13 @xmath38 space , suggesting we are indeed examining a similar mass range in both samples of galaxies . we also compared local lcbgs to other galaxy types in w@xmath39 sin@xmath17(_i _ ) versus r@xmath10 space , where w@xmath39 was measured from h emission for all samples . this allowed us to investigate if the wavelength used to measure the line width , or correcting the line width for inclination , was having an effect on our interpretation of which galaxy types most resemble lcbgs . we could only compare local lcbgs , spirals , magellanic spirals , and irregular galaxies , as h observations of intermediate redshift lcbgs and dwarf ellipticals are rare or non - existent . our findings were entirely consistent with the interpretation from figure 8 . local lcbgs are therefore consistent with the dynamical mass properties of the most massive dwarf ellipticals and irregulars , and lower mass or magellanic spirals . these classes vary in color and magnitude . knowledge of the amount of molecular gas , time scale of starburst and fading , along with the ability of the galaxy to retain its interstellar medium , will allow us to discriminate between these remaining possibilities . we will begin this work in paper ii @xcite . given the diverse nature of lcbgs , it is likely that multiple scenarios apply , each to a different subset of lcbgs . we have performed a single dish 21 cm h survey of 20 local lcbgs chosen to be local analogs to the numerous lcbgs studied at intermediate redshifts ( 0.4 @xmath0 z @xmath0 0.7 ) . our findings have verified results from intermediate redshift lcbg studies . we have found that local lcbgs are a morphologically heterogeneous mixture of galaxies . they are typically gas - rich , with median values of m@xmath3 = 5@xmath410@xmath6 m@xmath7 and m@xmath3 l@xmath8 = 0.4 m@xmath7 l@xmath9 . approximately half have mass - to - light ratios approximately ten times smaller than local galaxies of all hubble types at similar luminosities , confirming that these are indeed small galaxies undergoing vigorous bursts of star formation . this proportion is likely an underestimate , as nearly half our sample of galaxies may have dynamical mass overestimates . by comparing line widths and radii with local galaxy populations , we find that local lcbgs are consistent with magellanic spirals , and the more massive irregulars and dwarf ellipticals . measurements of the length of starburst , amount of fading , and ability of these galaxies to retain their interstellar media will help to constrain the evolutionary possibilities of this galaxy class . we begin to address these issues in paper ii @xcite , where we present the results of a molecular gas survey of these same local lcbgs . _ acknowledgments _ we thank the referee for helpful comments which improved the quality of this paper . we thank rick fisher for providing the h spectrum of sdssj0934 + 0014 . we also thank the operators and staff at the gbt for their help with the observing and reduction , and their hospitality . support for this work was provided by the nsf through award gssp02 - 0001 from the nrao . d. j. p. acknowledges generous support from an nsf mps distinguished international postdoctoral research fellowship , nsf grant ast0104439 . r. g. acknowledges funding from nasa grant ltsa nag5 - 11635 . funding for the creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. we have made extensive use of hyperleda ( http://www-obs.univ-lyon1.fr/hypercat/ ) and the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration ( http://nedwww.ipac.caltech.edu/ ) . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . llcccclcc mrk 297 & ngc 6052 , ngc 6064 & 67 & @xmath1321.0 & 0.4 & 20.6 & & sc & y + mrk 314 & ngc 7468 & 30 & @xmath1318.5 & 0.4 & 20.2 & e3 , pec ( polar ring ? ) & e & n + mrk 325 & ngc 7673 , mrk 325 & 49 & @xmath1320.0 & 0.4 & 20.0 & sac ? , pec , h ii starburst & sc & y + mrk 538 & ngc 7714 & 40 & @xmath1320.1 & 0.4 & 20.2 & sb(s)b , pec , h ii liner & sbb & y + sdss j011932.95 + 145219.0 & ngc 469 & 59 & @xmath1318.9 & 0.4 & 20.3 & & & y + sdss j021808.75@xmath13075718.0 & & 69 & @xmath1318.8 & 0.5 & 20.2 & & & n + sdss j022211.96@xmath13083036.2 & & 67 & @xmath1318.6 & 0.4 & 20.1 & & & n + sdss j072849.75 + 353255.2 & & 56 & @xmath1318.9 & 0.4 & 20.3 & s ? & sbc & n + sdss j083431.70 + 013957.9 & & 59 & @xmath1319.1 & 0.6 & 20.6 & sb(s)b & sbb & y + sdss j090433.53 + 513651.1 & mrk 101 & 68 & @xmath1319.7 & 0.6 & 20.3 & s & sc & n + sdss j091139.74 + 463823.0 & mrk 102 & 61 & @xmath1319.3 & 0.5 & 19.1 & s ? & & n + sdss j093410.52 + 001430.2 & mrk 1233 & 70 & @xmath1319.8 & 0.3 & 19.9 & sb & sbc & y + sdss j093635.36 + 010659.8 & & 71 & @xmath1319.1 & 0.6 & 21.0 & & & y + sdss j094302.60@xmath13021508.9 & & 68 & @xmath1319.3 & 0.5 & 20.4 & s0 & s0 & n + sdss j111836.35 + 631650.4 & mrk 165 & 46 & @xmath1318.6 & 0.4 & 19.4 & compact starburst & & n + sdss j123440.89 + 031925.1 & ngc 4538 & 67 & @xmath1319.2 & 0.6 & 21.0 & s pec & sbc & n + sdss j131949.93 + 520341.1 & & 67 & @xmath1318.7 & 0.2 & 20.1 & & & y + sdss j140203.52 + 095545.6 & ngc 5414 , mrk 800 & 61 & @xmath1319.7 & 0.5 & 19.7 & pec & & y + sdss j150748.33 + 551108.6 & & 48 & @xmath1318.9 & 0.4 & 20.8 & s & sbc & n + sdss j231736.39 + 140004.3 & ngc 7580 , mrk 318 & 63 & @xmath1319.3 & 0.6 & 20.4 & s ? & sbc & n + lllll mrk 297 & 4739 @xmath34 17 & 68 @xmath34 0.2 & 362 @xmath34 17 & 6.5 @xmath34 0.08 + mrk 314 & 2081 @xmath34 17 & 30 @xmath34 0.2 & 188 @xmath34 17 & 12 @xmath34 0.1 + mrk 325 & 3427 @xmath34 17 & 49 @xmath34 0.2 & 202 @xmath34 17 & 11 @xmath34 0.2 + mrk 538 & 2798 @xmath34 17 & 40 @xmath34 0.2 & 240 @xmath34 17 & 20 @xmath34 0.2 + sdssj0119 + 1452 & 4098 @xmath34 17 & 59 @xmath34 0.2 & 266 @xmath34 17 & 2.4 @xmath34 0.1 + sdssj0218@xmath130757 & & & & 0.41 @xmath34 0.07 + sdssj0222@xmath130830 & & & & 0.61 @xmath34 0.09 + sdssj0728 + 3532 & 3953 @xmath34 17 & 56 @xmath34 0.2 & 216 @xmath34 17 & 7.9 @xmath34 0.1 + sdssj0834 + 0139 & 4215 @xmath34 160 & 60 @xmath34 2 & 329 @xmath34 160 & 6.9 @xmath34 0.2 + sdssj0904 + 5136 & 4782 @xmath34 103 & 68 @xmath34 2 & 204 @xmath34 103 & 4.5 @xmath34 0.2 + sdssj0911 + 4636 & 4281 @xmath34 17 & 61 @xmath34 0.2 & 151 @xmath34 17 & 2.0 @xmath34 0.1 + sdssj0934 + 0014 & 4860 @xmath34 17 & 69 @xmath34 0.2 & 332 @xmath34 17 & 4.8 @xmath34 0.2 + sdssj0936 + 0106 & 4920 @xmath34 17 & 70 @xmath34 0.2 & 255 @xmath34 17 & 3.3 @xmath34 0.09 + sdssj0943@xmath130215 & 4823 @xmath34 17 & 69 @xmath34 0.2 & 230 @xmath34 17 & 2.6 @xmath34 0.06 + sdssj1118 + 6316 & 3218 @xmath34 17 & 46 @xmath34 0.2 & 152 @xmath34 17 & 2.2 @xmath34 0.07 + sdssj1234 + 0319 & 4685 @xmath34 17 & 67 @xmath34 0.2 & 269 @xmath34 17 & 4.4 @xmath34 0.1 + sdssj1319 + 5203 & 4619 @xmath34 17 & 66 @xmath34 0.2 & 202 @xmath34 17 & 7.7 @xmath34 0.09 + sdssj1402 + 0955 & 4267 @xmath34 17 & 61 @xmath34 0.2 & 343 @xmath34 17 & 6.8 @xmath34 0.2 + sdssj1507 + 5511 & 3373 @xmath34 17 & 48 @xmath34 0.2 & 126 @xmath34 17 & 3.7 @xmath34 0.1 + sdssj2317 + 1400 & 4413 @xmath34 17 & 63 @xmath34 0.2 & 254 @xmath34 17 & 6.2 @xmath34 0.1 + llccccccc mrk 297 & 7.0 @xmath34 0.1 & 42 & 486 & 243 & 8.0 & 11 & 2.0 & 2.7 + mrk 314 & 2.5 @xmath34 0.05 & 65 & 170 & 85 & 3.8 & 0.63 & 0.94 & 0.16 + mrk 325 & 6.3 @xmath34 0.1 & 43 & 242 & 121 & 9.6 & 3.3 & 2.4 & 0.83 + mrk 538 & 7.6 @xmath34 0.1 & 50 & 264 & 132 & 11 & 4.5 & 2.8 & 1.1 + sdssj0119 + 1452 & 2.0 @xmath34 0.1 & 84 & 230 & 115 & 5.6 & 1.7 & 1.6 & 0.48 + sdssj0218@xmath130757 & 0.47 @xmath34 0.08 & 52 & & & 4.9 & & 1.3 & + sdssj0222@xmath130830 & 0.65 @xmath34 0.1 & 48 & & & 4.2 & & 1.2 & + sdssj0728 + 3532 & 6.0 @xmath34 0.1 & 34 & 321 & 160 & 5.4 & 3.3 & 1.2 & 0.74 + sdssj0834 + 0139 & 5.9 @xmath34 0.5 & 83 & 293 & 147 & 6.5 & 3.3 & 1.6 & 0.82 + sdssj0904 + 5136 & 4.9 @xmath34 0.3 & 34 & 301 & 150 & 7.4 & 3.9 & 2.0 & 1.0 + sdssj0911 + 4636 & 1.7 @xmath34 0.1 & 33 & 221 & 111 & 5.6 & 1.6 & 1.2 & 0.35 + sdssj0934 + 0014 & 5.4 @xmath34 0.2 & 51 & 378 & 189 & 6.8 & 5.7 & 1.9 & 1.5 + sdssj0936 + 0106 & 3.8 @xmath34 0.1 & 48 & 293 & 146 & 7.1 & 3.5 & 2.0 & 1.0 + sdssj0943@xmath130215 & 2.9 @xmath34 0.07 & 84 & 194 & 97 & 4.9 & 1.1 & 1.8 & 0.39 + sdssj1118 + 6316 & 1.1 @xmath34 0.04 & 82 & 123 & 61 & 3.1 & 0.27 & 0.91 & 0.08 + sdssj1234 + 0319 & 4.7 @xmath34 0.1 & 52 & 293 & 147 & 6.7 & 3.4 & 2.1 & 1.1 + sdssj1319 + 5203 & 7.9 @xmath34 0.1 & 44 & 239 & 120 & 5.3 & 1.8 & 1.2 & 0.41 + sdssj1402 + 0955 & 6.0 @xmath34 0.2 & 55 & 372 & 186 & 8.7 & 7.0 & 1.6 & 1.3 + sdssj1507 + 5511 & 2.0 @xmath34 0.07 & 44 & 144 & 72 & 7.9 & 0.95 & 1.8 & 0.21 + sdssj2317 + 1400 & 5.8 @xmath34 0.1 & 31 & 420 & 210 & 7.3 & 7.5 & 1.7 & 1.8 + lccccc mrk 297 & 0.03 & 39 & 0.2 & 3 & 0.7 + mrk 314 & 0.2 & 3.9 & 0.7 & 2 & 0.4 + mrk 325 & 0.1 & 16 & 0.4 & 2 & 0.5 + mrk 538 & 0.08 & 17 & 0.5 & 3 & 0.7 + sdssj0119 + 1452 & 0.06 & 5.6 & 0.4 & 3 & 0.9 + sdssj0218 - 0757 & & 5.2 & 0.09 & & + sdssj0222 - 0830 & & 4.3 & 0.2 & & + sdssj0728 + 3532 & 0.09 & 5.6 & 1 & 6 & 1 + sdssj0834 + 0139 & 0.1 & 6.8 & 0.9 & 5 & 1 + sdssj0904 + 5136 & 0.07 & 12 & 0.4 & 3 & 0.9 + sdssj0911 + 4636 & 0.05 & 8.2 & 0.2 & 2 & 0.4 + sdssj0934 + 0014 & 0.05 & 13 & 0.4 & 4 & 1 + sdssj0936 + 0106 & 0.06 & 6.8 & 0.6 & 5 & 2 + sdssj0943 - 0215 & 0.1 & 8.2 & 0.4 & 1 & 0.5 + sdssj1118 + 6316 & 0.2 & 4.3 & 0.3 & 0.6 & 0.2 + sdssj1234 + 0319 & 0.07 & 7.4 & 0.6 & 5 & 1 + sdssj1319 + 5203 & 0.2 & 4.7 & 2 & 4 & 0.9 + sdssj1402 + 0955 & 0.04 & 12 & 0.5 & 6 & 1 + sdssj1507 + 5511 & 0.1 & 5.6 & 0.4 & 2 & 0.4 + sdssj2317 + 1400 & 0.04 & 8.2 & 0.7 & 9 & 2 + . the spectra have been smoothed to a resolution of @xmath112 km s@xmath17 and only the central 800 km s@xmath17 are shown . the dashed lines indicate the 20% crossings used to measure the line width at 20% . the triangles indicate the recessional velocities calculated from sdss redshifts for the sdss galaxies ; velocities from ned redshifts are shown for the non - sdss galaxies . those galaxies with other sources at similar velocities within the gbt beam are indicated by a star in the upper right corner . [ fig4 ] ] ( r @xmath59r@xmath10 ) , for local lcbgs , as measured from h observations , is shown as a gray histogram . for comparison , the range of dynamical masses for local hubble type galaxies are indicated with black boxes . the `` m '' in `` sm '' and `` i m '' indicates magellanic or low - luminosity spirals and irregulars . note that in order to compare lcbg dynamical masses with @xcite s results for local hubble types , the dynamical masses plotted here do not include a line width correction for random motions . [ fig5 ] ] ) to l@xmath21 ratios for local lcbgs is shown as a gray histogram . for comparison , the range of mass - to - light ratios for local hubble type galaxies ( s0a to i m ) @xcite are shown . as in figure 5 , for accurate comparisons , these mass - to - light ratios do not include a line width correction for random motions . many lcbgs tend to have mass - to - light ratios smaller than local normal galaxies , consistent with findings at intermediate redshifts . note that we may have overestimated the dynamical masses for most of the lcbgs with higher mass - to - light ratios . [ fig6 ] ] ) versus absolute blue magnitude ( m@xmath21 ) . local lcbgs are indicated by filled circles . those galaxies with other sources at similar velocities within the gbt beam are circled ; their line widths may be overestimated . the solid line indicates the tully - fisher relationship from @xcite ; the dotted lines indicate their 1 @xmath38 scatter of 0.3 m@xmath21 . the location of our local lcbgs is consistent with the tully - fisher relationship , but with a higher 1 @xmath38 scatter of 0.9 m@xmath21 . [ fig7 ] ] ) versus the the velocity dispersion ( @xmath38 ) is plotted for our local lcbgs ( filled circles ) and intermediate redshift ( 0.4 @xmath0 z @xmath0 1 ) lcbgs ( open circles ) @xcite . the regions occupied by the bulk of other local galaxy types@xmath13ellipticals ( e ) , spirals ( s ) , magellanic spirals ( sm ) , dwarf ellipticals ( de ) , and irregulars ( irr)@xmath13are indicated ( guzmn et al . 1996 , hyperleda ) . we have also indicated the approximate locations for some representative galaxies : draco , ngc 205 and m 31 ( hyperleda ) . local lcbgs are consistent with higher mass irregulars and dwarf ellipticals , and lower mass or magellanic spirals , as well as intermediate redshift lcbgs .
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we present single - dish h spectra obtained with the green bank telescope , along with optical photometric properties from the sloan digital sky survey , of 20 nearby ( d @xmath0 70 mpc ) luminous compact blue galaxies ( lcbgs ) . these @xmath1l@xmath2 , blue , high surface brightness , starbursting galaxies were selected with the same criteria used to define lcbgs at higher redshifts .
we find these galaxies are gas - rich , with m@xmath3 ranging from 5@xmath410@xmath5 to 8@xmath410@xmath6 m@xmath7 , and m@xmath3 l@xmath8 ranging from 0.2 to 2 m@xmath7 l@xmath9 , consistent with a variety of morphological types of galaxies .
we find the dynamical masses ( measured within r@xmath10 ) span a wide range , from 3@xmath410@xmath6 to 1@xmath410@xmath11 m@xmath7 .
however , at least half have dynamical mass - to - light ratios smaller than nearby galaxies of all hubble types , as found for lcbgs at intermediate redshifts . by comparing line widths and
effective radii with local galaxy populations , we find that lcbgs are consistent with the dynamical mass properties of magellanic ( low luminosity ) spirals , and the more massive irregulars and dwarf ellipticals , such as ngc 205 .
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@xcite found that the galactic cepheids follow a spectral type that is independent of their pulsational periods at maximum light and gets later as the periods increase at minimum light . * hereafter skm ) used radiative hydrodynamical models to explain these observational phenomena as being due to the location of the hydrogen ionization front ( hif ) relative to the photosphere . their results agreed very well with code s observation . skm further used the stefan - boltzmann law applied at the maximum and minimum light , together with the fact that radial variation is small in the optical @xcite , to derive : @xmath3 where @xmath4 are the effective temperature at the maximum / minimum light , respectively . if @xmath5 is independent of the pulsation period @xmath6 ( in days ) , then equation ( 1 ) predicts there is a relation between the @xmath7-band amplitude and the temperature ( or the colour ) at minimum light , and vice versa . in other words , if the period - colour ( pc ) relation at maximum ( or minimum ) light is flat , then there is an amplitude - colour ( ac ) relation at minimum ( or maximum ) light . equation ( 1 ) has shown to be valid theoretically and observationally for the classical cepheids and rr lyrae variables @xcite . for the rr lyrae variables , @xcite and @xcite used linear and non - linear hydrodynamic models of rrab stars in the galaxy to explain why rrab stars follow a flat pc relation at _ minimum _ light . later , @xcite used macho rrab stars in the lmc to prove that lmc rrab stars follow a relation such that higher amplitude stars are driven to cooler temperatures at maximum light . similar studies were also carried out for cepheid variables , as in skm , @xcite , ( * ? ? ? * hereafter paper i ) and ( * ? ? ? * hereafter paper ii ) . in contrast to the rr lyrae variables , cepheids show a flat pc relation at the _ maximum _ light , and there is a ac relation at the minimum light . therefore , the pc relation and the ac relation are intimately connected . all these studies are in accord with the predictions of equation ( 1 ) . in paper i , the galactic , large magellanic cloud ( lmc ) and small magellanic cloud ( smc ) cepheids were analyzed in terms of the pc and ac relations at the phase of maximum , mean and minimum light . one of the motivations for this paper originates from recent studies on the non - linear lmc pc relation ( as well as the period - luminosity , pl , relation . see paper i ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) : the optical data are more consistent with two lines of differing slopes which are continuous or almost continuous at a period close to 10 days . paper i also applied the the @xmath2-test @xcite to the pc and ac relations at maximum , mean and minimum @xmath7-band light for the galactic , lmc and smc cepheids . the @xmath2-test results implied that the lmc pc relations are broken or non - linear , in the sense described above , across a period of 10 days , at mean and minimum light , but only marginally so at maximum light . the results for the galactic and smc cepheids are similar , in a sense that at mean and minimum light the pc relations do not show any non - linearity and the pc(max ) relation exhibited marginal evidence of non - linearity . for the ac relation , cepheids in all three galaxies supported the existence of two ac relations at maximum , mean and minimum light . in addition , the cepheids in these three galaxies also exhibited evidence of the pc - ac connection , as implied by equation ( 1 ) , which give further evidence of the hif - photosphere interactions as outlined in skm . to further investigate the connection between equation ( 1 ) and the hif - photosphere interaction , and also to explain code s observations with modern stellar pulsation codes , galactic cepheid models were constructed in paper ii . in contrast to skm s purely radiative models , the stellar pulsation codes used in paper ii included the treatment of turbulent convection as outlined in @xcite . one of the results from paper ii was that the general forms of the theoretical pc and ac relation matched the observed relations well . the properties of the pc and ac relations for the galactic cepheids with @xmath8 can be explained with the hif - photosphere interaction . this interaction , to a large extent , is independent of the pulsation codes used , the adopted ml relations , and the detailed input physics . the aim of this paper is to extend the investigation of the connections between pc - ac relations and the hif - photosphere interactions in theoretical pulsation models of lmc cepheids , in addition to the galactic models presented in paper ii . in section 2 , we describe the basic physics of the hif - photosphere interaction . the updated observational data , after applying various selection criteria , that used in this paper are described in section 3 . in section 4 , the new empirical pc and ac relations based on the data used are presented . in section 5 , we outline our methods and model calculations , and the results are presented in section 6 . examples of the hif - photosphere interaction in astrophysical applications are given in section 7 . our conclusions & discussion are presented in section 8 . throughout the paper , short and long period cepheid are referred to cepheids with period less and greater than 10 days , respectively . the partial hydrogen ionization zone ( or the hif ) moves in and out in the mass distribution as the star pulsates . it is possible that the hif will interact with the photosphere , defined at optical depth ( @xmath9 ) of 2/3 , at certain phases of pulsation . for example , skm suggested that this happened at maximum light for the galactic cepheids , as the hif is so far out in the mass distribution that the photosphere occurs right at the base of the hif . the sharp rise of the opacity wall ( where the mean free path goes to zero ) due to the existence of hif prevents the photosphere moving further into the mass distribution and hence erases any `` memory '' of global stellar conditions , including the underlying pc relation . this lead to a flat relation between period & temperature , period & colour and period & spectral type at maximum light , as seen in skm and paper ii . at other phases , since the hif does not interact with the photosphere , the temperature of the star ( or the colour ) follows the underlying global pc relation . the hif - photosphere interaction also relies on the properties of the saha ionization equation and the structural properties of the outer envelopes of cepheids . it is well known that the partition functions in the saha ionization equation are formally divergent unless some atomic physics is used to truncate them . in the pulsation codes we used , we approximate the partition functions of various atoms by their ground state statistical weights . the properties of the saha ionization equation in cepheid envelopes are such that hydrogen starts to ionize at a temperature that is almost independent of density , for a certain range of low densities . outside of this range of density , the density dependence increases . thus , when the photosphere is very close to , or engaged with the hif and the density of these regions is reasonably low , the temperature of the photosphere is less dependent on the surrounding density and hence the global stellar parameters . at higher densities , the temperature at which hydrogen ionizes becomes more sensitive to density and hence more sensitive to global stellar parameters . if the photosphere is far from the hif , or disengaged , then the location of the photosphere and hence the temperature of the photosphere , is again strongly dependent on density and hence on global stellar parameters . that is why the photosphere needs to be close to , or engaged with the hif for this effect to take place . moreover , this dependence on density is not sharp so that for `` low '' and `` high '' densities the density dependence of the photospheric temperature is weak and strong respectively . an examination of figure 15.1 in @xcite demonstrates that this is plausible . thus as the star pulsates , the photospheric temperature has a density dependence that can be strong or weak depending on phase . an example where the density dependence is weak are the galactic long period cepheids at maximum light ( skm , paper ii ) : these cepheids display a flat pc relation at maximum light . these properties of the hif - photosphere interaction can , in turn , affect the temperature of the photosphere and hence the colour of the cepheid . here we investigate the idea that lmc cepheids with periods below 10 days are such that the hif and photosphere are engaged through most of the pulsation cycle . at periods greater than 10 days , the photosphere only engages with the hif at maximum light . the transition is sharp because the photosphere is either at the base of the hif or it is not . the transition occurs because as the period increases , the @xmath10 ratio increases and this implies the hif is located further inside in the mass distribution , changing the phase at which it can interact with the photosphere @xcite . the structure of galactic cepheids is such that this interaction only occurs at maximum light , even for cepheids with periods shorter than 10 days . in paper i , we constructed the light curves of fundamental mode cepheids in the lmc by using the extensive photometric dataset in the ogle ( optical gravitational lensing experiment ) database . however , the dataset used in paper i was downloaded in 2002 , prior to the updated version of the dataset that was available after april 24 , 2003 ( ogle website , udalski 2004 [ private communication ] ) . the updated version includes additional @xmath7- and @xmath11-band data for most of the cepheids . in addition , the periods have been refined by the ogle team using the complete set of photometric data . due to these reasons , we decided to repeat the light curve construction @xcite with the updated data and periods . since the cepheids in the ogle database are truncated at @xmath12 , due to the saturation of the ccd detector for the longer period ( hence brighter ) cepheids @xcite , we include some additional lmc cepheid data from @xcite , @xcite and @xcite to extend the period coverage to @xmath13 in our sample . the requirements that govern our choice of the published photometric data are : ( a ) latest observations that use the modern day ccd cameras ; ( b ) high quality data with large number of data points per light curve , which provide uniform phase coverage and small scatter of the light curve ; and ( c ) as homogeneous as possible ( i.e. , from a minimal number of sources ) to avoid any additional systematic errors . these requirements are essential to construct accurate light curves to allow the estimation of colours and magnitudes at maximum , mean and minimum light for our pc and ac study . hence we did not include some of the older photometric data in this study . the photometric data of all cepheids , comprising 771 from ogle database , 14 from @xcite+@xcite and 39 from @xcite , were mainly fit with @xmath14 to @xmath15 fourier expansions ( @xmath16 is the order of fourier expansion ) using the simulated annealing method described in @xcite to the @xmath7- and @xmath11-band photometric data . this is in contrast to paper i that only applied @xmath14 fourier fits . however , for some of the ogle long period cepheids ( @xmath17 days ) , it was found out that the quality of the fitted light curves could be improved by using a higher order fourier expansion , hence we extended the fit to @xmath18 for these long period cepheids . all the fitted light curves were visually inspected and the best - fit light curves from the different orders of the fourier expansions were selected . to the best of our knowledge , this analysis also represents a major improvement in the fourier analysis of the ogle data . the extinction is corrected with the standard procedure , i.e. @xmath19 with @xmath20 and @xmath21 @xcite . the values of @xmath22 for each ogle cepheids are taken from the ogle database @xcite , while for the cepheids in @xcite+@xcite and in @xcite , the values of @xmath22 are adopted from @xcite and/or @xcite . to guard against some `` bad '' cepheids or other contamination in our sample , and select only the good cepheids in both bands , we removed some cepheids in the sample according to the following criteria ( see also * ? ? ? * ) : = 7.5 cm 1 . cepheids without @xmath7- and/or @xmath11-band photometry , or the number of data per light curve ( in either bands or both ) is too low to fit a @xmath14 fourier expansion . cepheids with poorly fitted or unacceptable @xmath7- and/or @xmath11-band light curves in the sample , such as those with a large scatter of data points or with bad - phase coverage ( large gaps between the phased data points ) . most of the magnitudes , as well as the colours , at the maximum and/or minimum light from these fitted light curves are very uncertain . cepheids with possible duplicity in the ogle sample . some of the possible duplicated cepheids were removed in the ogle database by consulting table 4 of @xcite . 4 . cepheids with unusual colour . we first plot out ( as in figure [ figcut][a ] ) the extinction corrected pc relation at mean light . the plot shows that there are number of outliers in the period - colour plane , mostly with @xmath23 . the presence of these outliers is probably due to : ( a ) their extinction is either over- or under - estimated ; ( b ) they have blue or red companions that can not be resolved due to the problems of blending ; or ( c ) other unknown physical reasons . a detailed investigation of these outliers is beyond the scope of this paper , but it is clear that they should be removed from the sample . these outliers are removed with the adopted colour - cut of @xmath24 , a compromise between maximizing the number of cepheids in the sample and excluding the cepheids with unusual colour . cepheids with unusually low ( or high ) amplitude . some cepheids with unusually low @xmath7- and @xmath11-band amplitudes were found in the sample . their amplitudes are typically @xmath25 times smaller as compared to the amplitudes of other cepheids at given period . some examples of the light curves for these low amplitude cepheids are given in @xcite . in addition , most of the light curves for these low amplitude cepheids can be fitted with @xmath14 fourier expansion , while other cepheids with `` normal '' amplitude may require higher order fits . @xcite has briefly discussed some possible physical reasons for these cepheids to have such low amplitudes , e.g. they are just entering or leaving the fundamental mode instability strip @xcite or they have different chemical composition ( see , e.g. , * ? ? ? the detailed investigation of these low amplitudes cepheids is beyond the scope of this paper . here , we apply a conservative amplitude cut of @xmath26 mag . in the @xmath7-band to remove the low amplitude cepheids . besides that , we also remove ogle-286532 ( with unusually low amplitude ) and hv-2883 ( with unusually high amplitude ) as they are clear outliers in the @xmath27-amplitude plot ( not shown , but see * ? ? ? note that @xcite applied a cut of @xmath28 mag . to remove the low amplitude cepheids in ngc 6822 . other examples of removing the low - amplitude cepheids can also be found in @xcite . 6 . cepheids with @xmath29 and @xmath30 . in order to guard against possible contamination from the first overtone cepheids @xcite and to be consistent with the previous studies @xcite , we removed cepheids with @xmath29 ( see further justification in @xcite ) . regarding the removal of cepheids with @xmath30 , a preliminary analysis of the pc relation reveals that few of the longest cepheids should be removed from the sample , because they are clear outliers in the pc plot at maximum light ( see upper panel of figure [ figpcmax ] ) . without these longest period cepheids , the pc(max ) relation for the long period cepheids is flat , which is consistent with the results found in paper i. the hypothesis of the hif - photosphere interaction also suggests the flatness of the pc relation at maximum light for long period cepheids . however , as the period gets longer ( with @xmath31>1.5 $ ] ) , the photosphere disengages from the hif @xcite . these longest period cepheids have biased the slope of the pc(max ) relation by making the slope becomes steeper . these selection criteria are guided mainly by the philosophy that it is better to lose some `` bad '' but real cepheids rather than including those spurious and doubtful cepheids in the sample @xcite , or those with bad fitted light curves that will give inaccurate measurements of the maximum and minimum light . hence , the final sample consists of 641 lmc cepheids that will be considered further . the locations of the outliers from various selection criteria are shown in figure [ figcut ] for the pc(mean ) relation , @xmath7-band pl relation , @xmath32-@xmath27 relation , where @xmath33 are the fourier amplitudes . see @xcite and @xcite for details . ] and the colour - magnitude diagram ( cmd ) . note that some of the outliers are located within the `` good '' cepheids . however they can be eliminated due to various physical reasons as given above , especially those with poorly fit light curves that will give inaccurate measurements at maximum , mean and minimum light . a simple sigma - clipping algorithm ( e.g. , * ? ? ? * ) will not be able to remove these outliers @xcite . to construct the empirical pc & ac relations , we used the following quantities from the fourier fits to the cepheid data as obtained from previous section : * @xmath7-band amplitude : the difference of the numerical maximum and minimum from the fourier expansion , @xmath34 . * @xmath35 : defined as @xmath36 , where @xmath37 is the @xmath11-band magnitude at the same phase as @xmath38 . * @xmath39 : defined as @xmath40 , where @xmath41 is the mean value from the fourier expansion ( see * ? ? ? this is very similar to the conventional definition of the mean colour , @xmath42 , where @xmath43 denotes the intensity mean . * @xmath44 : defined as @xmath45 , where @xmath46 is the @xmath11-band magnitude at the same phase as @xmath47 . @xmath47 is the @xmath7-band magnitude closest to @xmath48 , the mean value from fourier expansion . * @xmath49 : defined as @xmath50 , where @xmath51 is the @xmath11-band magnitude at the same phase as @xmath52 . these quantities have been corrected for extinction as mentioned in previous section . the empirical lmc pc and ac relations at maximum , mean and minimum light for all , long and short period cepheids are summarized in table [ c9tabpc ] & [ c9tabac ] , and the corresponding plots are presented in figure [ c9figpc ] & [ c9figac ] , respectively . lccc phase & @xmath53 & @xmath54 & @xmath55 + + maximum & @xmath56 & @xmath57 & 0.099 + mean & @xmath58 & @xmath59 & 0.075 + phmean & @xmath60 & @xmath61 & 0.081 + minimum & @xmath62 & @xmath63 & 0.075 + + maximum & @xmath64 & @xmath65 & 0.098 + mean & @xmath66 & @xmath67 & 0.075 + phmean & @xmath68 & @xmath69 & 0.092 + minimum & @xmath70 & @xmath71 & 0.081 + + maximum & @xmath72 & @xmath73 & 0.097 + mean & @xmath74 & @xmath75 & 0.074 + phmean & @xmath76 & @xmath77 & 0.078 + minimum & @xmath78 & @xmath79 & 0.073 + lccc phase & @xmath53 & @xmath54 & @xmath55 + + maximum & @xmath80 & @xmath81 & 0.093 + mean & @xmath82 & @xmath83 & 0.092 + phmean & @xmath84 & @xmath85 & 0.099 + minimum & @xmath86 & @xmath87 & 0.098 + + maximum & @xmath88 & @xmath89 & 0.086 + mean & @xmath90 & @xmath91 & 0.085 + phmean & @xmath92 & @xmath93 & 0.103 + minimum & @xmath94 & @xmath95 & 0.082 + + maximum & @xmath96 & @xmath97 & 0.076 + mean & @xmath98 & @xmath99 & 0.074 + phmean & @xmath100 & @xmath101 & 0.078 + minimum & @xmath102 & @xmath103 & 0.079 + to test the non - linearity of the pc and ac relations , or the `` break '' at a period of 10 days , we apply the @xmath2-test as given in paper i and in @xcite . the null hypothesis in the @xmath2-test is single line regression is sufficient , while the alternate hypothesis is that two lines regressions with a discontinuity ( a break ) at 10 days is necessary to fit the data . the probability @xmath104 , under the null hypothesis , can be obtained with the corresponding @xmath2-values and the degrees of freedom . in general , the large value of @xmath2 ( equivalent to the small value of @xmath105 $ ] ) indicates that the null hypothesis can be rejected . for our sample , @xmath106 when @xmath107 ( the 95% confident level ) , therefore the null hypothesis can be rejected if the @xmath2-value is greater than @xmath108 with more than 95% confident level and the data is more consistent with the two - line regression . a glance of table [ c9tabpc ] and figure [ c9figpc ] suggests that the lmc pc relations are broken at maximum , mean and minimum light . these are confirmed with the @xmath2-test results with @xmath109 . similarly , the @xmath2-test results for the ac relation are : @xmath110 . hence , the lmc pc and ac relations are non - linear ( hence broken ) at maximum , means and minimum light . note that the flatness of the long period pc(max ) relation as given in table [ c9tabpc ] ( @xmath64 ) is in good agreement with the slope found in paper i ( @xmath111 ) . recall that equation ( 1 ) predicts that if the pc relation is flat at maximum light , then there is a correlation between the amplitude and the colour at minimum light . this is seen in table [ c9tabac ] ( and in figure [ c9figac ] ) for the long period ac(min ) relation , with a slope of @xmath112 . the stellar pulsation codes we used are both linear @xcite and non - linear @xcite . these codes , which include a 1-d turbulent convection recipe @xcite , are the same as in paper ii . briefly speaking , the codes take the mass ( @xmath113 ) , luminosity ( @xmath114 ) , effective temperature ( @xmath115 ) and chemical composition ( @xmath116 ) as input parameters . the chemical composition is set to be @xmath117 to represent the lmc hydrogen and metallicity abundance ( by mass ) . the mass and luminosity are obtained from the ml relations calculated from evolutionary models . the @xmath115 are chosen to ensure the models oscillate in the fundamental mode and located inside the cepheid instability strip . the pulsation periods for the models are obtained from a linear non - adiabatic analysis @xcite . all other parameters used in the pulsation codes had the same values for the lmc and galactic models ( paper ii ) . this included the @xmath118 parameters that are part of the turbulent convection recipe , though see section 8 . of course , one variable parameter was the metallicity . the only other difference between this study and paper ii , besides the metallicity , is the value set for the artificial viscosity parameter , @xmath119 . in this study , we set @xmath120 for the lmc models to improve the shape of the theoretical light curves , in contrast to the value of @xmath121 used for the galactic models . in paper ii , the ml relations are adopted from @xcite and @xcite . in order to be consistent with previous work , the ml relations used in this paper will also be adopted from these two sources . however , @xcite only provided two ml relations , one for @xmath122 which are used in paper ii , and another one for @xmath123 . hence we have to adopt the second ml relation for the lmc models . even though the lmc metallicity is higher than @xmath123 , the lmc is still considered as a low metallicity system in the literature . hence the @xcite ml relation can be approximately applied for the lmc models . an anonymous referee pointed out that an interpolation of the @xcite ml relations between @xmath122 and @xmath123 should also be used . we have included the interpolated ml relation in our model calculations . in the context of the hif - photosphere interaction , it is the ml relation which dictates at what period and at what phases this will occur . stellar evolutionary theory changes the ml relation as a function of metallicity . hence the coefficients of the ml relation are important in determining the nature of the hif - photosphere interaction ( paper ii ) . in short , the ml relations used are : ccccccc @xmath113 & @xmath124 & @xmath125 & @xmath126 & @xmath127 & @xmath128 & @xmath129 + + 11.0 & 4.375 & 5050 & 46.4155 & 0.124 & 28.98 & -0.118 + 10.0 & 4.236 & 5100 & 35.6727 & 0.091 & 22.92 & -0.093 + 9.50 & 4.161 & 5250 & 28.2406 & 0.094 & 18.68 & -0.046 + 9.10 & 4.099 & 5260 & 25.3960 & 0.082 & 16.92 & -0.042 + 8.75 & 4.042 & 5310 & 22.3804 & 0.076 & 15.07 & -0.027 + 8.40 & 3.982 & 5380 & 19.3886 & 0.071 & 13.20 & -0.008 + 7.95 & 3.902 & 5330 & 17.7750 & 0.055 & 12.09 & -0.027 + 7.00 & 3.717 & 5410 & 12.6085 & 0.035 & 8.722 & -0.018 + 6.55 & 3.620 & 5490 & 10.2940 & 0.031 & 7.183 & -0.006 + 6.40 & 3.587 & 5485 & 9.81474 & 0.027 & 6.853 & -0.010 + 6.00 & 3.493 & 5510 & 8.37226 & 0.020 & 5.866 & -0.014 + 5.90 & 3.468 & 5500 & 8.12498 & 0.017 & 5.691 & -0.018 + 5.80 & 3.443 & 5525 & 7.69466 & 0.017 & 5.400 & -0.015 + 5.70 & 3.418 & 5560 & 7.23505 & 0.017 & 5.090 & -0.009 + 5.30 & 3.312 & 5600 & 6.01283 & 0.012 & 4.244 & -0.009 + + 7.20 & 4.272 & 5380 & 40.2561 & 0.275 & 24.51 & -0.162 + 6.80 & 4.192 & 5380 & 35.4374 & 0.264 & 21.91 & -0.122 + 6.20 & 4.063 & 5410 & 28.2629 & 0.225 & 17.94 & -0.076 + 5.95 & 4.005 & 5420 & 25.6378 & 0.211 & 16.43 & -0.060 + 5.40 & 3.869 & 5510 & 19.4314 & 0.170 & 12.82 & -0.015 + 5.15 & 3.803 & 5510 & 17.5523 & 0.160 & 11.66 & -0.007 + 4.65 & 3.660 & 5490 & 14.3143 & 0.131 & 9.611 & -0.010 + 4.20 & 3.518 & 5510 & 11.3659 & 0.101 & 7.729 & -0.011 + 4.00 & 3.450 & 5545 & 10.0011 & 0.089 & 6.854 & -0.005 + 3.95 & 3.432 & 5540 & 9.77393 & 0.085 & 6.701 & -0.008 + 3.80 & 3.378 & 5550 & 8.94637 & 0.075 & 6.157 & -0.009 + 3.70 & 3.341 & 5575 & 8.31297 & 0.070 & 5.745 & -0.005 + 3.65 & 3.322 & 5570 & 8.10751 & 0.066 & 5.605 & -0.008 + 3.60 & 3.302 & 5530 & 8.09994 & 0.058 & 5.583 & -0.022 + 3.60 & 3.302 & 5600 & 7.71463 & 0.065 & 5.352 & -0.001 + + 6.80 & 4.092 & 5280 & 30.9661 & 0.207 & 19.51 & -0.091 + 5.20 & 3.701 & 5340 & 15.9744 & 0.094 & 10.63 & -0.050 + 4.40 & 3.457 & 5460 & 10.0562 & 0.056 & 6.898 & -0.028 + 4.20 & 3.389 & 5550 & 8.51317 & 0.054 & 5.906 & -0.007 + 3.80 & 3.243 & 5630 & 6.46303 & 0.039 & 4.532 & -0.002 + 1 . ml relation given in @xcite : @xmath130 2 . ml relation given in @xcite : @xmath131 3 . ml relation interpolated between two @xcite relations at @xmath132 and @xmath123 to yield a relation at @xmath133 : @xmath134 the units for both @xmath113 and @xmath114 are in solar units . note that these ml relations cover reasonably broad @xmath10 ratios given in the literature . the input parameters for the lmc models with these ml relations and the periods calculated from linear non - adiabatic analysis are given in table [ tabinput ] . after the full amplitude models are constructed from the pulsation codes , the temperature and the opacity profile can be plotted in terms of the internal mass distribution ( @xmath135 $ ] , where @xmath136 is mass within radius @xmath137 and @xmath113 is the total mass ) at a given phase of pulsation . as in paper ii , the locations of the hif ( sharp rise in the temperature profile ) and photosphere ( at optical depth @xmath138 ) can be identified in the temperature profile . to quantify the hif - photosphere interaction ( if the photosphere is next to the base of the hif or not , see also paper ii ) , we calculate the `` distance '' , @xmath139 , in @xmath140 between the hif and the photosphere from the temperature profile . the definition of @xmath139 can be found in paper ii . a small @xmath139 means there is a hif - photosphere interaction , and vice versa . the theoretical quantities from the models can be compared to the observed quantities using the following prescriptions : ccccc @xmath6 & @xmath141 & @xmath5 & @xmath142 & @xmath143 + + 46.4155 & 27481.28 & 5445.00 & 18329.74 & 4826.00 + 35.6727 & 19742.83 & 5434.01 & 14321.20 & 4962.65 + 28.2406 & 16894.15 & 5502.35 & 12093.99 & 4978.72 + 25.3960 & 14460.62 & 5562.28 & 10570.70 & 5010.85 + 22.3804 & 12772.75 & 5615.34 & 9283.257 & 5069.43 + 19.3886 & 11263.84 & 5687.98 & 8143.460 & 5145.20 + 17.7750 & 9034.392 & 5561.39 & 7044.743 & 5147.04 + 12.6085 & 5637.467 & 5528.15 & 4800.840 & 5278.11 + 10.2940 & 4371.170 & 5537.42 & 3839.001 & 5377.80 + 9.81474 & 4004.569 & 5518.39 & 3580.396 & 5383.61 + 8.37226 & 3217.688 & 5596.34 & 2929.354 & 5437.17 + 8.12498 & 3032.639 & 5581.53 & 2793.441 & 5438.96 + 7.69466 & 2864.745 & 5602.37 & 2640.439 & 5467.54 + 7.23505 & 2705.430 & 5639.03 & 2490.271 & 5504.98 + 6.01283 & 2105.589 & 5669.57 & 1986.586 & 5567.91 + + 40.2561 & 23500.98 & 5821.99 & 11853.38 & 4947.11 + 35.4374 & 19604.60 & 5830.18 & 9898.124 & 4962.26 + 28.2629 & 14661.29 & 5875.98 & 7659.896 & 5057.96 + 25.6378 & 12836.04 & 5884.01 & 6916.299 & 5108.02 + 19.4314 & 9538.226 & 5985.21 & 5555.179 & 5303.60 + 17.5523 & 8035.700 & 5921.07 & 4947.967 & 5142.58 + 14.3143 & 5490.345 & 5783.01 & 3639.964 & 5164.11 + 11.3659 & 3850.971 & 5814.49 & 2722.473 & 5242.47 + 10.0011 & 3286.030 & 5817.00 & 2377.950 & 5300.52 + 9.77393 & 3136.139 & 5801.77 & 2301.751 & 5296.73 + 8.94637 & 2738.003 & 5798.30 & 2071.106 & 5331.19 + 8.31297 & 2499.416 & 5794.49 & 1917.696 & 5365.34 + 8.10751 & 2374.713 & 5780.35 & 1850.048 & 5369.40 + 8.09994 & 2235.292 & 5723.18 & 1796.586 & 5355.41 + 7.71463 & 2271.393 & 5805.49 & 1772.270 & 5408.34 + + 30.9661 & 14958.03 & 5611.77 & 8486.049 & 4963.81 + 15.9744 & 5697.042 & 5665.97 & 4209.487 & 5079.81 + 10.0562 & 3197.451 & 5670.64 & 2560.871 & 5283.85 + 8.51317 & 2723.566 & 5716.48 & 2201.628 & 5382.21 + 6.46303 & 1852.537 & 5684.63 & 1593.714 & 5507.38 + = 7.5 cm 1 . as in paper ii , we use the basel atmosphere database @xcite to construct a fit giving temperature and effective gravity as a function of @xmath144 colour . the effective gravity is obtained at the appropriate phase from the models ( see paper ii ) . these prescriptions are used to convert the temperatures to the @xmath144 colours . the bolometric corrections ( @xmath145 ) are obtained in a similar manner . the anonymous referee has suggested that @xmath144 may not be a good way to convert between temperature and colour unless both of the micro - turbulence and surface gravity are included . as indicated above this is the case , and in any case our results and those of paper ii for galactic models , show good agreement between theory and observations . a number of previous authors have used this method and some authors commented that this colour can be used as an indicator of temperature ( e.g. * ? ? ? * ; * ? ? ? the empirical relations we studied in this series were also mainly in the @xmath144 colour . 2 . in addition to the basel atmosphere , we also use the atmosphere fit from @xcite , referring this as the sbt atmosphere in our paper . the sbt atmosphere does include both of the effective gravity and the micro - turbulence in their table 6 for the temperature and colour conversion . these conversions are tabulated for two micro - turbulence velocities of @xmath146 and @xmath147 , as well as for various metallicities . to apply these conversions to our lmc models , we first interpolated the conversions between @xmath148=0.0 $ ] and @xmath148=-0.5 $ ] to @xmath148=-0.3 $ ] , which is appropriate for the lmc metallicity . the @xmath144 colours at the maximum , mean and minimum light are then obtained from the given effective temperature and the effective gravity for both of the micro - turbulence velocities . we use the prescriptions given in @xcite to convert the observed colours to the temperatures appropriate for the lmc data as follows : @xmath149 + note that these functions are also obtained from the basel atmosphere database . we can compare the colours obtained from the basel and sbt atmosphere for our models constructed in this paper . the results are presented in figure [ figatmos ] . from this figure it can be seen that the colours obtained from both of the atmosphere fits agree within @xmath150mag . the difference is even smaller if the micro - turbulence velocity of @xmath147 is used . this indicates that the @xmath144 colours can be used to indicate the temperature . since the results of our models are qualitatively compared to the observations ( see next section ) and not used to quantitatively derive any theoretical pc and/or ac relations , an accuracy of @xmath151mag . , independent of period , from the atmosphere fit is acceptable and would not cause problems for our results . note that the sbt atmosphere are only defined for @xmath152 and @xmath153 , few of our models either the @xmath115 or @xmath154 or both are beyond these ranges at certain phases , hence no colours can be obtained from the sbt atmosphere ( for example some points are missing at minimum light for few of the long period models , as shown in figure [ figatmos ] ) . due to these reasons , we continue adopt the basel atmosphere fits to convert the temperatures and @xmath144 colours , after taking account of the effective gravity in the fits , as a function of phase . cccccccc @xmath6 & @xmath155 & @xmath156 & @xmath157 & @xmath158 & @xmath159 & @xmath160 ( asc ) & @xmath160 ( des ) + + 46.4155 & 24249.2 & 24423.695 & 5330.06 & 24282.129 & 4882.55 & 5319.42 & 4880.66 + 35.6727 & 17207.7 & 17078.471 & 5293.90 & 17188.051 & 4922.99 & 5305.14 & 4924.47 + 28.2406 & 14484.7 & 14466.869 & 5457.79 & 14529.348 & 5073.65 & 5459.55 & 5069.64 + 25.3960 & 12540.8 & 12461.087 & 5443.66 & 12544.735 & 5096.58 & 5452.77 & 5096.18 + 22.3804 & 10997.1 & 10996.587 & 5493.44 & 11039.123 & 5157.95 & 5493.51 & 5152.87 + 19.3886 & 9587.02 & 9560.1922 & 5545.87 & 9618.0583 & 5232.85 & 5549.58 & 5228.51 + 17.7750 & 7983.35 & 7988.5457 & 5473.94 & 7994.9159 & 5213.83 & 5473.06 & 5211.77 + 12.6085 & 5208.60 & 5208.1564 & 5493.34 & 5202.6440 & 5349.00 & 5493.36 & 5350.84 + 10.2940 & 4170.31 & 4162.1135 & 5560.81 & 4174.0993 & 5450.34 & 5564.43 & 5449.06 + 9.81474 & 3860.04 & 3861.4313 & 5556.01 & 3852.8715 & 5446.51 & 5555.34 & 5449.11 + 8.37226 & 3109.66 & 3106.7337 & 5570.85 & 3110.1806 & 5481.59 & 5572.55 & 5481.37 + 8.12498 & 2939.41 & 2935.0599 & 5553.67 & 2939.6504 & 5476.34 & 5556.38 & 5476.23 + 7.69466 & 2775.73 & 2775.1245 & 5581.66 & 2777.4057 & 5501.67 & 5582.05 & 5500.77 + 7.23505 & 2618.53 & 2620.1965 & 5620.29 & 2618.8674 & 5534.39 & 5619.15 & 5534.20 + 6.01283 & 2051.97 & 2050.8394 & 5646.29 & 2051.3277 & 5583.97 & 5647.27 & 5584.50 + + 40.2561 & 18754.3 & 18580.019 & 5704.60 & 18824.542 & 5144.86 & 5718.84 & 5141.42 + 35.4374 & 15896.6 & 15885.248 & 5749.07 & 15861.179 & 5147.12 & 5750.16 & 5158.63 + 28.2627 & 11157.8 & 11188.594 & 5692.54 & 11143.155 & 5111.26 & 5688.21 & 5112.18 + 25.6378 & 10388.8 & 10366.128 & 5782.33 & 10358.696 & 5184.81 & 5785.59 & 5188.79 + 19.4314 & 7743.89 & 7741.3087 & 5871.14 & 7750.7288 & 5332.20 & 5871.60 & 5330.76 + 17.5523 & 6538.19 & 6537.3819 & 5833.02 & 6532.0823 & 5310.84 & 5833.20 & 5312.28 + 14.3143 & 4564.83 & 4621.4163 & 5750.64 & 4565.7876 & 5280.59 & 5732.80 & 5280.34 + 11.3659 & 3289.17 & 3288.4584 & 5711.02 & 3281.3799 & 5328.36 & 5711.31 & 5331.32 + 10.0011 & 2811.35 & 2823.6619 & 5726.94 & 2800.7972 & 5376.66 & 5721.20 & 5381.50 + 9.77393 & 2702.54 & 2712.0209 & 5715.81 & 2692.4263 & 5378.63 & 5711.29 & 5383.51 + 8.94637 & 2385.34 & 2395.4576 & 5704.28 & 2388.3903 & 5412.32 & 5699.14 & 5410.62 + 8.31297 & 2189.29 & 2193.3351 & 5708.03 & 2188.7842 & 5448.47 & 5705.90 & 5448.80 + 8.10751 & 2094.38 & 2103.6034 & 5698.27 & 2093.0120 & 5449.90 & 5693.36 & 5450.83 + 8.09994 & 2004.90 & 1998.5077 & 5648.26 & 1999.8838 & 5419.02 & 5651.76 & 5422.70 + 7.71463 & 2004.07 & 2009.2157 & 5713.42 & 2004.9007 & 5490.38 & 5710.72 & 5489.80 + + 30.9661 & 12359.6 & 12202.760 & 5566.66 & 12354.904 & 5029.65 & 5585.98 & 5030.07 + 15.9744 & 5011.60 & 5031.4017 & 5550.08 & 5024.6207 & 5173.09 & 5544.24 & 5169.84 + 10.0562 & 2860.25 & 2866.2012 & 5593.28 & 2859.5002 & 5346.96 & 5590.81 & 5347.35 + 8.51317 & 2447.87 & 2441.2040 & 5643.58 & 2453.1647 & 5463.35 & 5646.11 & 5460.12 + 6.46303 & 1749.72 & 1749.6046 & 5719.11 & 1744.6691 & 5577.03 & 5719.24 & 5581.12 + the effective temperatures for the full amplitude models in table [ tabinput ] at the corresponding maximum and minimum light ( or luminosity ) are given in table [ c9tabmaxmin ] . for the effective temperatures at mean light , the temperatures for the mean light at ascending and descending branch of the light ( or luminosity ) curve are not the same ( e.g. , in paper ii ) , hence table [ c9tabmean ] gives the effective temperature at these phases for our lmc models . the layout of table [ c9tabmean ] is the same as table 3 from paper ii . following paper ii , the locations of the photosphere can be identified in the temperature and opacity profiles . these are displayed in figure [ c9bono4]-[c9chiosi13 ] with a @xmath161 , a @xmath162 and a @xmath23 model , respectively . the left and right panels of figure [ c9bono4]-[c9chiosi13 ] are the temperature and opacity profiles respectively . the photospheres are marked as filled circles in these figures . finally , the plots of the @xmath139 , the `` distance '' between the photosphere and the hif from the temperature profiles , as a function of pulsating period for the lmc models are presented in figure [ c9deltalmc ] with the three ml relations used . in paper ii , it is found that the distribution of @xmath139 as a function of period is almost independent of the adopted ml relation . this is also seen in the lmc models as depicted in figure [ c9deltalmc ] . figure [ c9bono4]-[c9chiosi13 ] and figure [ c9deltalmc ] bear witness to the fact that at maximum light , the photosphere lies at the base of the hif for all of the models . although there is a slight deviation for some longer period models , the location of the photosphere is close to the hif within the error bars ( which are defined as the coarseness of the grid points around the location of the hif ) . as in paper ii for the galactic models , the closeness of the photosphere to the base of the hif , for reasonably low densities , results in a flat or almost flat pc relation for the long period lmc cepheids . in the case of minimum light , even though figure [ c9deltalmc ] implies that @xmath163 is nearly constant across the period range and the photosphere is near the base of the hif , as in the case of maximum light , @xmath163 does follow a shallow correlation with period after 10 days . judging from the error bars of @xmath163 and from figure [ c9bono4]-[c9chiosi13 ] , there is tentative evidence that the photosphere is disengaged from the hif for @xmath161 at minimum light . hence the temperatures or the colours at minimum light are more dependent on period for @xmath164-@xmath27 relation may not be correlated with the slopes of the pc relation ] and the global properties . theoretical quantities that can be computed from the models and compared with data include the pulsation periods , the @xmath7-band amplitudes and the fourier parameters , the temperatures and colours at the maximum , mean and minimum light . these are the pc plots , the ac plots , the period - temperature plots and the fourier parameters plots portrayed in figures [ c9modelpc]-[c9lmcfourier ] . the temperatures in table [ c9tabmaxmin ] & [ c9tabmean ] , after conversion to the @xmath144 colours as mentioned in previous section , are superimposed along with the observed lmc pc relations as plotted in figure [ c9modelpc ] . similarly , figure [ c9modelpt ] graphs the same quantities but on the @xmath165-@xmath27 plane with the observed @xmath144 colours converted to temperatures using the prescriptions given in section 5 . the theoretical bolometric light curves are converted to the @xmath7-band light curves with the bolometric corrections obtained from the basel database mentioned previously . from the theoretical @xmath7-band light curves , the amplitudes can be estimated and these are displayed in figure [ c9modelac ] along with the colours from models to compare with the empirical ac relations . the fourier parameters of the theoretical @xmath7-band light curves can also be obtained with ( @xmath166 ) fourier expansion . these fourier parameters are compared with the observational data in figure [ c9lmcfourier ] . several features are noticed from figure [ c9modelpc]-[c9lmcfourier ] : 1 . the general trends of the models qualitatively match the observational data . there are greater discrepancies between the data and short period models , particularly in matching the observed light curve amplitudes . 2 . the models with the ml relation from @xcite , with lower @xmath10 ratio , do better in matching the observations . these models also tend to lie near the envelopes of the pc , ac , @xmath165-@xmath27 and @xmath167-@xmath27 relations defined by the observational data . 3 . the slopes of the period - colour ( or period - temperature ) relations at maximum and minimum light from the models roughly match the observational data , i.e. , the theoretical pc(max ) relation is approximately flat and there is a relation at minimum light . 4 . the temperatures from the models with the @xcite ml relation is cooler ( hence redder ) than the models with the @xcite ml relation and the observed data at maximum light . in contrast , the temperatures ( or the colours ) at minimum light from the models with these two ml relations are consistent with each other and are located near the blue edge of the observed data . the means at the descending branches are in better agreement with the observed data than the means at the ascending branches . this is because the observed means , @xmath168 , are obtained mostly from the descending branches . though previous researchers have noted that temperatures on the ascending and descending branches are not the same at mean light ( as cepheids exhibit loops in cmd ) , what is new here is the way the nature of the hif changes during the pulsation . the behaviors of the models from the interpolated @xcite ml are closer to the models from @xcite ml relation because their slopes are very similar . the amplitudes of the theoretical light curves ( in both of the bolometric and @xmath7-band light curves ) are smaller than the observations at given period , especially for the models with the @xcite ml relation . these can be seen from the ac relations as given in figure [ c9modelac ] and the left panels of figure [ c9lmcfourier ] . overall , some agreements and disagreements are found between the theoretical quantities and the observational data . it is also found out that there are some problems associated with the pulsation codes when the lmc models are constructed : these include the smaller amplitude of the model light curves and the cooler temperatures at the maximum light ( especially with @xcite ml relation ) . note that from equation ( 1 ) , cooler temperatures at maximum light imply that the amplitudes will be lower at given period . varying other parameters in the pulsation codes , including the @xmath118 parameters , does not improve the situation , though perhaps a more detailed and systematic study of the dependence of lmc cepheid pulsation models on the @xmath169 parameters could resolve this situation . however , we believe that the qualitative nature of the photosphere - hif interactions as given in figure [ c9deltalmc ] will still hold even in models which fare better in mimicking observed amplitudes . this is in part because figure [ c9deltalmc ] suggest that the behaviors of @xmath139 as a function of period are nearly , though not completely , independent of amplitudes , as the models with @xcite ml relation have higher amplitudes ( although still smaller than the observations ) than the models with the @xcite ml relation . however , better codes that fix these problems or the 3-d convection codes are needed in the future studies . the temperature profiles from the galactic models given in paper ii and the lmc models are compared in figure [ c9pt ] at maximum and minimum light . the upper panels of figure [ c9pt ] suggest that at maximum light , the photosphere is not far from the base of the hif in both of the galactic and the lmc models . in contrast , the photosphere is further away from the hif in the galactic models than the lmc models at minimum light . the hif is located further out in the mass distribution for the galactic models . the plots of the @xmath139-@xmath27 relation from the galactic and lmc models at maximum and minimum light are also compared in figure [ c9delta ] . it can be seen from the figure that at maximum light , the behavior of both galactic and lmc models is similar , where the photosphere is near the base of the hif . at minimum light , the long period models show that the photosphere is disengaged from the hif , while the behavior of the short period models is different between the galactic and lmc models . the photosphere of the short period lmc models seems to be located closer to the hif at minimum light , but it is not the case for the short period galactic models . this could lead to shallower slopes of the pc(min ) relation seen in the lmc cepheids as compared to the galactic counterparts . in terms of the hif - photosphere interaction , there is some tentative evidence from the models that the lmc long period cepheids behave like the galactic cepheids , while the short period lmc cepheids behave like the rr lyrae stars at minimum light . figure [ c9density ] graphs the density ( defined as @xmath170 , where @xmath7 is the specific volume ) at the photosphere as a function of the period of the model at minimum , maximum and ascending and descending mean light . galactic models generally tend to have the lowest density and , in particular , have significantly lower densities at minimum light than the lmc models . we note that the galactic models always have a photospheric density lower than about @xmath171 whereas the photospheric density for the lmc models only falls below this figure after a period of 10 days . at maximum light , all long period models have a low photospheric density . what we get from this figure is that it provides some evidence that there is a difference in photospheric density between the lmc and galactic models . moreover , this difference appears to be consistent with what is required by our theoretical scenario : short period lmc models have a higher photospheric density than their galactic counterparts . however , for a discussion of some caveats , see section 8 . we now discuss two important applications of the photosphere - hif interaction : reddening corrections and the explanation of the observed non - linear lmc pl ( and pc ) relations . @xcite original interest in the spectral properties of cepheids at maximum light was to estimate reddening . skm used this to correct a number of reddening for galactic cepheids . @xcite used equation ( 1 ) and the theoretical explanation provided in skm to derive a relation linking the colour excess to the colour at maximum light , the @xmath7-band amplitude and the period . such a relation is predicted from equation ( 1 ) . @xcite estimates the error with this method to be comparable to other multi - colour methods . a more interesting application of the hif - photosphere interaction is to explain the recent detected non - linear lmc pl relation as presented in @xcite , paper i , @xcite and @xcite . paper i used the @xmath2-test to provide strong statistical evidence that the optical cepheid pl relation at mean light in the lmc is non - linear around a period close to 10 days . @xcite used the macho and 2mass datasets together with additional long period cepheids from the literature to further support the existence of non - linear lmc pl relation in the optical and near infra - red wave - bands . in contrast , current data indicate that the galactic pl relation is linear at mean light @xcite . non - linearity of the lmc pl relations can be tested using the @xmath2-test with the data given in section 3 . the empirical results of the fitted lmc pl relations at maximum , mean and minimum light using the updated data are presented in table [ c9tabpl ] . the plots of the pl relations at maximum / minimum light and at mean light are shown in figure [ c9plmaxmin ] & [ c9plmean ] , respectively . the @xmath2-test results for these pl relations are : @xmath172 , and @xmath173 . the large @xmath2-values for both @xmath7- and @xmath11-band pl relations at mean and minimum light strongly indicate that the pl relations at these two phases are not linear , and the data is better described with the broken ( i.e , two regressions ) pl relation . however , the small @xmath2-values at maximum light , with corresponding @xmath174-values of @xmath175 and @xmath176 for the @xmath7- and @xmath11-band pl(max ) relations respectively , show that the null hypothesis of the @xmath2-test can not be rejected ( a value of @xmath177 and/or @xmath178 is required for doing this ) . hence there is no observed break seen in the pl(max ) relation and the data is consistent with single line regression . note that the same slopes of the pl(max ) relations for long period cepheids in both bands are consistent of the finding that the pc(max ) relation is flat for these cepheids . lcccccc phase & @xmath179 & @xmath180 & @xmath181 & @xmath182 & @xmath183 & @xmath184 + + maximum & @xmath185 & @xmath186 & 0.260 & @xmath187 & @xmath188 & 0.170 + mean & @xmath189 & @xmath190 & 0.208 & @xmath191 & @xmath192 & 0.141 + minimum & @xmath193 & @xmath194 & 0.204 & @xmath195 & @xmath196 & 0.140 + + maximum & @xmath197 & @xmath198 & 0.257 & @xmath199 & @xmath200 & 0.174 + mean & @xmath201 & @xmath202 & 0.228 & @xmath203 & @xmath204 & 0.158 + minimum & @xmath205 & @xmath206 & 0.243 & @xmath207 & @xmath208 & 0.174 + + maximum & @xmath209 & @xmath210 & 0.260 & @xmath211 & @xmath212 & 0.169 + mean & @xmath213 & @xmath214 & 0.203 & @xmath215 & @xmath216 & 0.138 + minimum & @xmath217 & @xmath218 & 0.194 & @xmath219 & @xmath220 & 0.131 + our tentative theoretical explanation for the non - linear nature of the lmc pl relations across a period of 10 days replies on the hif - photosphere interaction . @xcite and @xcite have established the connection between the pc and pl relations : both these relations arise from the more general plc relation . these relations refer to quantities evaluated at mean light . the existence of such a connection relies on the period - mean density theorem , the instability strip and the stefan - boltzmann law . if we assume the stefan - boltzmann law can be applied at every phase , then it is straightforward to show that a plc relation ( though possibly with different coefficients ) exists at every phase point . thus the standard plc relation and indeed the pc and pl relation expresses at mean light are just the averages of the same relations at different phases points . consequently one way to understand the behavior of plc / pl / pc relations at mean light is to understand their behavior at different phase points . what we try to do in this paper is point out some evidence from our models that shows how the changing behavior of the pc relations at different phases can , in principle , arise from a consideration of the photosphere - hif interaction at these phases . since the mean light pc and pl relation are the average of those at all phases , these properties can affect the pc and , as a consequence , the pl relation ( via the plc relation ) . in fact , the new data with superb phase resolution from such micro - lensing projects such as ogle and macho demands a multiphase analysis . this approach can potentially lead to a deeper understanding of the pulsation and evolution of cepheid variables . for example , @xcite looked at pc relations in the galaxy and lmc as a function of phase . they found that short and long period lmc cepheids have a shallower and steeper slope at most pulsation phases than galactic cepheids respectively . in this paper , we have confronted updated pc and ac relations at maximum , mean and minimum light for lmc cepheids observed by the ogle team , and additional cepheids from the literature , with theoretical , full amplitude pulsation models of lmc cepheids . the observed pc and ac relations provide compelling evidence of a non - linearity or break at a period of 10 days . we also constructed theoretical cepheid pulsation models appropriate for the lmc using the florida pulsation codes @xcite to study the hif - photosphere interaction . the empirical results presented in this paper , as well as in other papers such as @xcite and @xcite , provide strong empirical evidence that the pc and pl relations for the lmc cepheids are non - linear , in the sense described in previous sections . issues such as extinction and a lack of long period cepheids that may cause the non - linear lmc pl and pc relations have been addressed and argued against in paper i , @xcite and @xcite , and will not be repeated here . other arguments against the non - linear lmc pl relation include the results presented in @xcite , as the authors found no evidence for a non - linear pl relation in the lmc at @xmath221-bands . however , @xcite treated the data of @xcite extensively and found , in a statistically rigorous way , that the reason why @xcite found linear @xmath221 pl relations , is due to the small number of short period cepheids ( @xmath222 ) in their sample . @xcite also reduce the number of ogle / macho lmc cepheids and show how the @xmath2-test can produce a non - significant result when the number of short / long period cepheids become small . instead , using the 2mass data that are cross - correlated with macho cepheids , @xcite have found that the lmc @xmath223-band pl relations are non - linear - band than in @xmath224-band , as shown in @xcite . ] and the @xmath225-band pl relation starts to become linear . @xcite also discussed why this is the case . another argument against the non - linear pl relation is that the pl relation should be universal , as found in @xcite . we argue that their results are based on a handful of cepheids ( @xmath226 ) and on short periods cepheids in a cluster whose membership to the lmc is in question . their shallower galactic pl relation based on the revised infra - red surface brightness method also contradicts the steeper galactic pl relation based on independent methods from open cluster main - sequence fitting @xcite . it is worthwhile to point out that our sample selection does not affect the detection on non - linear lmc pl relation at mean light . since the mean magnitudes of a cepheid light curve is less affected by our constrains on selecting the cepheids with good light curves , we can use the published ( reddening corrected ) mean magnitudes to test the non - linear lmc pl relation . the anonymous referee kindly provided a large sample of lmc cepheids that combined the published mean @xmath7-band magnitudes from the ogle , @xcite and @xcite datasets . there are a total of 115 long period cepheids in this sample and the @xmath2-test still return a significant detection of the non - linear lmc pl relation . the ogle+@xcite combined data also give very similar results . similar tests have also been done in @xcite by using the macho data alone and the macho+@xcite combined data . the non - linear lmc pl relation is still present from the @xmath2-test results on these two datasets . therefore we believe our sample selection does not affect the detection of the non - linear lmc pl relation . the detection of non - linear lmc pl relation from totally independent ogle and macho data , using totally independent reddening estimates , suggested that this non - linearity is real and our paper is the first attempt to theoretically explain this non - linearity in terms of the hif - photosphere interaction . due to small number of lmc models , it is impossible to derive the theoretical pc and ac relations with a small error on the slope and compare directly to the empirical relations . however , these lmc models can be qualitatively compared to the observations by converting some physical quantities to the observable quantities and vice versa , such as the temperature - colour conversion . hence we compared our model light curves to the observations in terms of theoretical pc and ac relations at the phases of maximum , mean and minimum light and also in terms of the fourier parameters from theoretical light curves with observations . the theoretical quantities from the models generally agree with the observations , but it was found out that these models tend to have smaller amplitudes and ( hence ) the temperature is cooler at maximum light than the real cepheids . though our models have some drawbacks in this comparison , our main interest is in comparing the interaction of the photosphere and hif as a function of phase with similar results presented in paper ii for galactic cepheid pulsation models . the aim is _ not _ to compare our models rigorously with observations but rather to study models which match observations reasonably well in the context of the theoretical framework described in previous sections and in paper i & ii . nevertheless we argued that the qualitative nature of the photosphere - hif interaction is not seriously affected by these problems . our postulate is that at certain phases , this interaction can affect the pc relation due to the properties of the saha ionization equation : specifically for reasonably low densities in cepheid envelopes , hydrogen ionizes at a temperature that is almost independent of period . consequently , when the photosphere is located at the base of the hif , the photospheric temperature and hence the colour is almost independent of period . however , when this engagement occurs , but the density is greater , then the temperature at which hydrogen ionizes again becomes sensitive to global surroundings and hence on period . when the photosphere is not engaged with the hif in this way , its temperature is again dependent on period and global stellar parameters . for galactic cepheids , this hif - photosphere interaction occurs mainly at maximum light for cepheids with @xmath227 ( paper ii ) . at minimum light , there is a strong correlation between the hif - photosphere distance and period leading to a definite ac relation at minimum light for galactic cepheids ( skm , paper i & ii ) . in this paper , we have found tentative evidence that , for short period lmc models which match observations in the period - color plane , the hif - photosphere interaction occurs at most phases but at densities which are too high to produce a flat pc relation . why would these short period lmc cepheids be different in this regard to short period galactic cepheids ? one possibility could be that this is partly because these lmc cepheids are hotter than their galactic counterparts @xcite . the hif - photosphere are disengaged for most of the pulsation cycle for long period lmc cepheids . this happens because as the period increases , so does the @xmath10 ratio which pushes the hif further inside the mass distribution . when the hif - photosphere are disengaged in this way , the photospheric temperature is more dependent on density and hence on period . the change is sudden because the hif - photosphere are either engaged or they are not . this can lead to a sudden change in the pc relation at 10 days as shown by the observations @xcite . however , at maximum light the hif - photosphere are engaged at low densities for long period lmc cepheids leading to the observed flat pc relation for these stars . taken together with equation ( 1 ) , this theoretical scenario is consistent with the observed pc - ac behavior described in paper i and in this study . the anonymous referee has noted that these suggestions about photospheric density can be tested by spectroscopic means . we now enumerate some caveats to our argument that could be addressed in future papers . 1 . since the smc pc relation at mean light is linear ( e.g. , paper i ) , how do smc ( i.e. , metal - poor ) models fit into the theoretical scenario outlined in this paper and paper ii , if at all ? this is a difficult question and its full answer is beyond the scope of this paper however , as the metallicity decreases , we do note that the smc has a different ml relation to the lmc and galaxy and so does the temperatures associated with the instability strip . these will change the relative location of the hif and photosphere @xcite and possibly alter the phase at which they interact . further the amplitudes for smc cepheids are smaller due to the lower metallicity @xcite . this will also affect the hif - photosphere interaction . one difference which can be consistent with this is the fact that the pc relation at maximum light in the smc is not flat ( see paper i ) but it is the case for the galaxy and lmc pc relations . this indicates that at maximum light , there is less interaction between the hif and photosphere at low densities . this leads to an observed linear pc relation at mean light for the smc cepheids . these will be investigated further in a future paper in this series . 2 . could the well - known hertzsprung progression play any part in causing the observed changes in the galactic and lmc pc relations ? it may also be that higher order overtones becoming unstable or stable , though with the fundamental mode still being dominant , may also have an impact on the pc relation in some as yet unknown way ( paper ii ) . 4 . the behavior of short period lmc cepheids still needs to be understood , for example , what causes the difference between the bottom left panels of figures [ c9deltalmc ] and [ c9delta ] ? that is , why is it that for short period galactic / lmc cepheids , the hif - photosphere are disengaged / engaged ? our experience suggests that constructing short period full amplitude fundamental mode cepheids requires more care than the long period case because the first overtone has a non - negligible growth rate . because of this we feel a thorough study of these short period cepheids merits a separate paper . 5 . would more advanced pulsation codes which , for example , can match the observed amplitudes and which contain a more accurate model of time dependent turbulent convection , yield similar results , especially for figure [ c9delta ] ? could such codes fare better in modeling short period lmc cepheids ? smk acknowledges support from hst - ar-10673.04-a . we thank an anonymous referee for several useful suggestions and providing the data for our testing . we would also like to thank e. antonello , r. buchler & j. kwan for useful discussions , and r. bell & m. marengo for the discussion regarding the atmosphere fits .
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period - colour ( pc ) and amplitude - colour ( ac ) relations are studied for the large magellanic cloud ( lmc ) cepheids under the theoretical framework of the hydrogen ionization front ( hif ) - photosphere interaction .
lmc models are constructed with pulsation codes that include turbulent convection , and the properties of these models are studied at maximum , mean and minimum light . as with galactic models , at maximum light the photosphere is located next to the hif for the lmc models . however very different behavior is found at minimum light . the long period ( @xmath0days )
lmc models imply that the photosphere is disengaged from the hif at minimum light , similar to the galactic models , but there are some indications that the photosphere is located near the hif for the short period ( @xmath1 days ) lmc models .
we also use the updated lmc data to derive empirical pc and ac relations at these phases .
our numerical models are broadly consistent with our theory and the observed data , though we discuss some caveats in the paper .
we apply the idea of the hif - photosphere interaction to explain recent suggestions that the lmc period - luminosity ( pl ) and pc relations are non - linear with a break at a period close to 10 days .
our empirical lmc pc and pl relations are also found to be non - linear with the @xmath2-test .
our explanation relies on the properties of the saha ionization equation , the hif - photosphere interaction and the way this interaction changes with the phase of pulsation and metallicity to produce the observed changes in the lmc pc and pl relations .
cepheids stars : fundamental parameters
| 20,175 | 454 |
it has become clear in the past several years that copper oxide materials are among the most complex systems studied in condensed matter physics , and show many unusual normal - state properties . the complications arise mainly from ( 1 ) strong anisotropy in the properties parallel and perpendicular to the cuo@xmath0 planes which are the key structural element in the whole copper oxide superconducting materials , and ( 2 ) extreme sensitivity of the properties to the compositions ( stoichiometry ) which control the carrier density in the cuo@xmath0 plane @xcite , while the unusual normal - state feature is then closely related to the fact that these copper oxide materials are doped mott insulators , obtained by chemically adding charge carriers to a strongly correlated antiferromagnetic ( af ) insulating state , therefore the physical properties of these systems mainly depend on the extent of dopings , and the regimes have been classified into the underdoped , optimally doped , and overdoped , respectively @xcite . the normal - state properties of copper oxide materials in the underdoped and optimally doped regimes exhibit a number of anomalous properties in the sense that they do not fit in the conventional fermi - liquid theory @xcite , and the mechanism for the superconductivity in copper oxide materials has been widely recognized to be closely associated with the anisotropic normal - state properties @xcite . among the striking features of the normal - state properties in the underdoped and optimally doped regimes , the physical quantity which most evidently displays the anisotropic property in copper oxide materials is the charge dynamics @xcite , which is manifested by the optical conductivity and resistivity . it has been show from the experiments that the in - plane charge dynamics is rather universal within the whole copper oxide materials @xcite . the in - plane optical conductivity for the same doping is nearly materials independent both in the magnitude and energy dependence , and shows the non - drude behavior at low energies and anomalous midinfrared band in the charge - transfer gap , while the in - plane resistivity @xmath1 exhibits a linear behavior in the temperature in the optimally doped regime and a nearly temperature linear dependence with deviations at low temperatures in the underdoped regime @xcite . by contrast , the magnitude of the c - axis charge dynamics in the underdoped and optimally doped regimes is strongly materials dependent , _ i.e. _ , it is dependent on the species of the building blocks in between the cuo@xmath0 planes @xcite . in the underdoped and optimally doped regimes , the experimental results @xcite show that the ratio @xmath2 ranges from @xmath3 to @xmath4 , this large magnitude of the resistivity anisotropy reflects that the c - axis mean free path is shorter than the interlayer distance , and the carriers are tightly confined to the cuo@xmath0 planes , and also is the evidence of the incoherent charge dynamics in the c - axis direction . for the copper oxide materials without the cu - o chains in between the cuo@xmath0 planes @xcite , such as la@xmath5sr@xmath6cuo@xmath7 systems , the transferred weight in the c - axis conductivity forms a band peaked at high energy @xmath8 , and the low - energy spectral weight is quite small and spread over a wide energy range instead of forming a peak at low energies , in this case the behavior of the c - axis temperature dependent resistivity @xmath9 is characterized by a crossover from the high temperature metallic - like to the low temperature semiconducting - like @xcite . however , for these copper oxide materials with the cu - o chains in between the cuo@xmath0 planes @xcite , such as yba@xmath0cu@xmath10o@xmath11 systems , the c - axis conductivity exhibits the non - drude behavior at low energies and weak midinfrared band , moreover , this weak midinfrared band rapidly decrease with reducing dopings or increasing temperatures , while the c - axis resistivity @xmath9 is linear in temperatures in the optimally doped regime , and shows a crossover from the high temperature metallic - like behavior to the low temperature semiconducting - like behavior in the underdoped regime @xcite . therefore there are some subtle differences between the chain and no - chain copper oxide materials . the c - axis charge dynamics of copper oxide materials has been addressed from several theoretical viewpoints @xcite . based on the concept of dynamical dephasing , leggett @xcite thus proposed that the c - axis conduction has to do with scatterings from in - plane thermal fluctuations , and depends on the ratio of the interlayer hopping rate of cuo@xmath0 sheets to the thermal energy . while the theory of tunneling c - axis conductivity in the incoherent regime has been given by many researchers @xcite . based on a highly anisotropic fermi - liquid , some effect from the interlayer static disorder or dynamical one has been discussed @xcite . the similar incoherent conductivity in the coupled fermion chains has been in more detail studied by many authors within the framework of the non - fermi - liquid theory @xcite . moreover , the most reliable result for the c - axis charge dynamics from the model relevant to copper oxide materials has been obtained by the numerical simulation @xcite . it has been argued that the in - plane resistivity deviates from the temperature linear behavior and temperature coefficient of the c - axis resistivity change sign , showing semiconducting - like behavior at low temperatures are associated with the effect of the pseudogap @xcite . to shed light on this issue , we , in this paper , apply the fermion - spin approach @xcite to study the c - axis charge dynamics by considering the interlayer coupling . the paper is organized as follows . the theoretical framework is presented in sec . ii . in the case of the incoherent interlayer hopping , the c - axis current - current correlation function ( then the c - axis optical conductivity ) is calculated in terms of the in - plane single - particle spectral function by using standard formalisms for the tunneling in metal - insulator - metal junctions @xcite . within this theoretical framework , we discuss the c - axis charge dynamics of the chain copper oxide materials in sec . it is shown that the c - axis charge dynamics of the chain copper oxide materials is mainly governed by the scattering from in - plane charged holons due to spinon fluctuations , and the behavior of the c - axis resistivity is the metallic - like in the optimally doped regime and the semiconducting - like in the underdoped regime at low temperatures . in sec . iv , the c - axis charge dynamics of the no - chain copper oxide materials is discussed . our result shows that the scattering from the in - plane fluctuation incorporating with the interlayer disorder dominates the c - axis charge dynamics for the no - chain copper oxide materials . in this case , the c - axis resistivity exhibits the semiconducting - like behavior in the underdoped and optimally doped regimes at low temperatures . v is devoted to a summary and discussions . our results also show that the crossover to the semiconducting - like range in @xmath9 is obviously linked with the crossover from the temperature linear to the nonlinear range in @xmath1 , and the common origin for these crossovers is due to the existence of the holon pseudogap at low temperatures and lower doping levels . among the microscopic models the most simplest for the discussion of doped mott insulators is the @xmath12-@xmath13 model @xcite , which is originally introduced as an effective hamiltonian of the hubbard model in the strong coupling regime , where the electron become strongly correlated to avoid the double occupancy . the interest in the @xmath12-@xmath13 model is stimulated by many researchers suggestions that it may contain the essential physics of copper oxide materials @xcite . on the other hand , there is a lot of evidence from the experiments and numerical simulations in favour of the @xmath12-@xmath13 model as the basic underlying microscopic model @xcite . within each cuo@xmath0 plane , the physics property may be described by the two - dimensional ( 2d ) @xmath12-@xmath13 model , @xmath14 supplemented by the on - site local constraint @xmath15 to avoid the double occupancy , where @xmath16 , @xmath17 is the lattice constant of the square planar lattice , which is set as the unit hereafter , @xmath18 refers to planar sites within the l - th cuo@xmath0 plane , @xmath19 ( @xmath20 ) are the electron creation ( annihilation ) operators , @xmath21 are the spin operators with @xmath22 as the pauli matrices , and @xmath23 is the chemical potential . then the hopping between cuo@xmath0 planes is considered as @xcite @xmath24 where @xmath25 , @xmath26 is the interlayer distance , and has been determined from the experiments @xcite as @xmath27 . as mentioned above , the experimental results show that the c - axis charge dynamics in the underdoped and optimally doped regimes is incoherent , therefore the c - axis momentum can not be defined @xcite . moreover , the absence of the coherent c - axis charge dynamics is a consequence of the weak interlayer hopping matrix element @xmath28 , but also of a strong intralayer scattering , _ i.e. _ , @xmath29 , and therefore the common cuo@xmath0 planes in copper oxide materials clearly dominate the most normal - state properties . in this case , the most relevant for the study of the c - axis charge dynamics is the results on the in - plane conductivity @xmath30 and related single - particle spectral function @xmath31 . since the strong electron correlation in the @xmath12-@xmath13 model manifests itself by the electron single occupancy on - site local constraint , then the crucial requirement is to impose this electron on - site local constraint for a proper understanding of the physics of copper oxide materials . to incorporate this local constraint , the fermion - spin theory based on the charge - spin separation has been proposed @xcite . according to the fermion - spin theory , the constrained electron operators in the @xmath12-@xmath13 model is decomposed as @xcite , @xmath32 with the spinless fermion operator @xmath33 keeps track of the charge ( holon ) , while the pseudospin operator @xmath34 keeps track of the spin ( spinon ) . the main advantage of this approach is that the electron on - site local constraint can be treated exactly in analytical calculations . in this case , the low - energy behavior of the @xmath12-@xmath13 model ( 2 ) in the fermion - spin representation can be written as @xcite , @xmath35 where @xmath36 $ ] , the in - plane holon particle - hole parameter @xmath37 , and @xmath38 and @xmath39 are the pseudospin raising and lowering operators , respectively . as a consequence , the kinetic part in the @xmath12-@xmath13 model has been expressed as the holon - spinon interaction in the fermion - spin representation , which dominates the charge and spin dynamics in copper oxide materials in the underdoped and optimally doped regimes . the spinon and holon may be separated at the mean - field level , but they are strongly coupled beyond mean - field approximation ( mfa ) due to fluctuations . the mean - field theory within the fermion - spin formalism in the underdoped and optimally doped regimes without af long - range - order ( aflro ) has been developed @xcite , and the in - plane mean - field spinon and holon green s functions @xmath40 and @xmath41 have been evaluated @xcite as , @xmath42 @xmath43 respectively , where @xmath44 $ ] , @xmath45 , @xmath46 , @xmath47 , @xmath48 is the number of the nearest neighbor sites at the plane , while the in - plane mean - field spinon spectrum @xmath49-[\alpha c_{z}+{1\over 4z}(1-\alpha)]\right ) ( \epsilon\gamma_{k}-1 ) \nonumber \\ + \lambda^{2}\left ( \alpha\epsilon[{1\over 2}\chi\gamma_{k}+ { 1\over z}\chi_{z}]-{1\over 2}\epsilon[\alpha c+{1\over 2z } ( 1-\alpha)]\right ) ( \gamma_{k}-\epsilon),\end{aligned}\ ] ] and the in - plane mean - field holon spectrum @xmath50 , with the in - plane spinon correlation functions @xmath51 , @xmath52 , @xmath53 , and @xmath54 . in order not to violate the sum rule of the correlation function @xmath55 in the case without aflro , the important decoupling parameter @xmath56 has been introduced in the mean - field calculation @xcite , which can be regarded as the vertex correction . the mean - field order parameter @xmath57 , @xmath58 , @xmath59 , @xmath60 , @xmath61 and chemical potential @xmath23 have been determined @xcite by the self - consistent equations . within the 2d @xmath12-@xmath13 model , the in - plane charge dynamics in copper oxide materials has been discussed @xcite by considering fluctuations around this mean - field solution , and the result exhibits a behavior similar to that seen in the experiments @xcite and numerical simulations @xcite . in the framework of the charge - spin separation , an electron is represented by the product of a holon and a spinon , then the external field can only be coupled to one of them . according to the ioffe - larkin combination rule @xcite , the physical c - axis conductivity @xmath62 is given by , @xmath63 where @xmath64 and @xmath65 are the contributions to the c - axis conductivity from holons and spinons , respectively , and can be expressed @xcite as , @xmath66 with @xmath67 and @xmath68 are the holon and spinon c - axis current - current correlation function , respectively , which are defined as , @xmath69 within the hamiltonian ( 4 ) , the c - axis current densities of spinons and holons are obtained by the time derivation of the polarization operator using heisenberg s equation of motion as , @xmath70 and @xmath71 , respectively , with @xmath72 is the effective interlayer holon hopping matrix element , and the order parameters @xmath73 and @xmath74 are defined @xcite as @xmath75 , and @xmath76 , respectively . as in the previous discussions @xcite , a formal calculation for the spinon part shows that there is no the direct contribution to the current - current correlation from spinons , but the strongly correlation between holons and spinons is considered through the spinon s order parameters entering in the holon part of the contribution to the current - current correlation , therefore the charge dynamics in copper oxide materials is mainly caused by charged holons within the cuo@xmath0 planes , which are strongly renormalized because of the strong interaction with fluctuations of surrounding spinon excitations . in the case of the incoherent charge dynamics in the c - axis direction , _ i.e. _ , the independent electron propagation in each layer , the c - axis holon current - current correlation function is then proportional to the tunneling rate between just two adjacent planes , and can be calculated in terms of the in - plane holon green s function @xmath77 by using standard formalisms for the tunneling in metal - insulator - metal junctions @xcite as , @xmath78 where @xmath79 is the matsubara frequency . therefore the c - axis current - current correlation function is essentially determined by the property of the in - plane full holon propagator @xmath80 , which can be expressed as , @xmath81 where the holon self - energy has been obtained by considering the second - order correction due to the spinon pair bubble as @xcite , @xmath82-n_{b}(\omega_{p+p ' } ) n_{b}(-\omega_{p'})\over i\omega_{n}+\omega_{p+p'}-\omega_{p'}- \xi_{p+k } } \right . \nonumber \\ + { n_{f}(\xi_{p+k})[n_{b}(\omega_{p+p'})-n_{b}(-\omega_{p'})]+ n_{b}(\omega_{p'})n_{b}(\omega_{p+p'})\over i\omega_{n}+\omega_{p ' } + \omega_{p+p'}-\xi_{p+k } } \nonumber \\ \left . -{n_{f}(\xi_{p+k)}[n_{b}(\omega_{p+p ' } ) -n_{b}(-\omega_{p'})]-n_{b}(-\omega_{p'})n_{b}(-\omega_{p+p ' } ) \over i\omega_{n}-\omega_{p+p'}-\omega_{p'}-\xi_{p+k}}\right ) , \end{aligned}\ ] ] with @xmath83 and @xmath84 are the fermi and bose distribution functions , respectively . for the convenience in the following discussions , the above full holon in - plane green s function @xmath80 also can be expressed as frequency integrals in terms of the spectral representation as , @xmath85 with the in - plane holon spectral function @xmath86 . then the c - axis optical conductivity in the present theoretical framework is expressed @xcite as @xmath87 . we firstly consider the chain copper oxide materials . from the experiments testing the c - axis charge dynamics @xcite , it has been shown that the presence of the rather conductive cu - o chains in the underdoped and optimally doped regimes can reduce the blocking effect , and therefore the c - axis charge dynamics in this system is effected by the same electron interaction as that in the in - plane . in this case , we substitute eq . ( 14 ) into eq . ( 11 ) , and evaluate the frequency summations , then the c - axis optical conductivity for the chain copper oxide materials can be obtained as @xcite , @xmath88 where the in - plane momentum is conserved . this c - axis conductivity @xmath89 has been calculated numerically , and the result at the doping @xmath90 ( solid line ) , @xmath91 ( dashed line ) , and @xmath92 ( dot - dashed line ) for the parameters @xmath93 , @xmath94 , and @xmath95 at the temperature @xmath96=0 is shown in fig . 1 , where the charge @xmath97 has been set as the unit . from fig . 1 , we find that there is two bands in @xmath62 separated at @xmath98 , the higher - energy band , corresponding to the `` midinfrared band '' in the in - plane optical conductivity @xmath30 @xcite , shows a broad peak at @xmath99 , in particular , the weight of this band is strongly doping dependent , and decreasing rapidly with dopings , but the peak position does not appreciably shift to higher energies . on the other hand , the transferred weight of the lower - energy band forms a sharp peak at @xmath100 , which can be described formally by the non - drude formula , and our analysis indicates that this peak decay is @xmath101 at low energies as in the case of @xmath30 @xcite . in comparison with @xmath30 @xcite , the present result also shows that the values of @xmath89 are by @xmath102 orders of magnitude smaller than those of @xmath30 in the corresponding energy range . the finite temperature behavior of @xmath89 also has been discussed , and the result shows that @xmath89 is temperature dependent , the higher - energy band is severely suppressed with increasing temperatures , and vanishes at higher temperatures . these results are qualitatively consistent with the experimental results @xcite of the chain copper oxide materials and numerical simulations @xcite . with the help of the c - axis conductivity , the c - axis resistivity can be obtained as @xmath103 , and the numerical result at the doping @xmath90 and @xmath92 for the parameters @xmath93 , @xmath104 , and @xmath95 is shown in fig . 2(a ) and fig . 2(b ) , respectively . in the underdoped regime , the behavior of the temperature dependence of @xmath9 shows a crossover from the high temperature metallic - like ( @xmath105 ) to the low temperature semiconducting - like ( @xmath106 ) , but the metallic - like temperature dependence dominates over a wide temperature range . therefore in this case , there is a general trend that the chain copper oxide materials show nonmetallic @xmath9 in the underdoped regime at low temperatures . while in the optimally doped regime , @xmath9 is linear in temperatures , and shows the metallic - like behavior for all temperatures . these results are also qualitatively consistent with the experimental results of the chain copper oxide materials @xcite and numerical simulation @xcite . now we turn to discuss the c - axis charge dynamics of the no - chain copper oxide materials . it has been indicated from the experiments @xcite that for the no - chain copper oxide materials the doped holes may introduce a disorder in between the cuo@xmath0 planes , contrary to the case of the chain copper oxide materials @xcite , where the increasing doping reduces the disorder in between the cuo@xmath0 planes due to the effect of the cu - o chains . therefore for the no - chain copper oxide materials , the disorder introduced by doped holes residing between the cuo@xmath0 planes modifies the interlayer hopping elements as the random matrix elements . in this case , only the in - plane holon density of states ( dos ) @xmath107 enters the holon current - current correlation function as in disordered systems @xcite , and after a similar discussion as in sec . ii , we find that the corresponding momentum - nonconserving expression of the c - axis conductivity @xmath108 for the no - chain copper oxide materials is obtained by the replacement of the in - plane holon spectral function @xmath109 in eq . ( 15 ) with the in - plane holon dos @xmath110 as @xcite , @xmath111 where the @xmath112 is some average of the random interlayer hopping matrix elements @xmath113 . we have performed a numerical calculation for this c - axis conductivity @xmath114 , and the result at the doping @xmath115 ( solid line ) , @xmath116 ( dashed line ) , and @xmath92 ( dash - dotted line ) for the parameters @xmath93 , @xmath117 with the temperature @xmath118 is plotted in fig . this result shows that the c - axis conductivity @xmath108 contains two bands , the higher - energy band , corresponding to the midinfrared band in the in - plane conductivity @xmath30 @xcite , shows a broad peak at @xmath119 . the weight of this band is increased with dopings , but the peak position does not appreciably shift to lower energies . as a consequence of this pinning of the transferred spectral weight , the weight of the lower - energy band , corresponding to the non - drude peak in @xmath30 @xcite , is quite small and does not form a well - defined peak at low energies in the underdoped and optimally doped regimes . in this case , the conductivity @xmath108 at low energies can not be described by the over - damped drude - like formula even an @xmath120-dependence of the in - plane holon scattering rate has been taken into consideration . in comparison with the momentum - conserving @xmath89 , our result also shows that at low energies the suppression of the momentum - nonconserving @xmath114 is due to the disordered effect introduced by doped holes residing between the cuo@xmath0 planes . these results are qualitatively consistent with the experimental results of the no - chain copper oxide materials @xcite . for the further understanding the transport property of the no - chain copper oxide materials , we have also performed the numerical calculation for the c - axis resistivity @xmath121 , and the results at the doping @xmath115 and @xmath92 for the parameters @xmath93 , @xmath117 , and @xmath122 are shown in fig . 4(a ) and fig . 4(b ) , respectively . in accordance with the c - axis conductivity @xmath108 , the behavior of the c - axis resistivity @xmath123 in the underdoped and optimally doped regimes is the semiconducting - like at low temperatures , and metallic - like at higher temperatures . in comparison with the in - plane resistivity @xmath1 @xcite , it is shown that the values of the c - axis resistivity @xmath123 for the no - chain copper oxide materials are by @xmath124 orders of magnitude larger than these of the in - plane resistivity @xmath1 in the corresponding energy range , which are also qualitatively consistent with the experimental results of the no - chain copper oxide materials @xcite . in the above discussions , the central concerns of the c - axis charge dynamics in copper oxide materials are the two dimensionality of the electron state and incoherent hopping between the cuo@xmath0 planes , and therefore the c - axis charge dynamics in the present fermion - spin picture based on the charge - spin separation is mainly determined by the in - plane charged holon fluctuation for the chain copper oxide materials and the in - plane charged holon fluctuation incorporating with the interlayer disorder for the no - chain copper oxide materials . in comparison with the in - plane resistivity @xmath1 @xcite , it is shown that the crossover to the semiconducting - like range in @xmath9 is obviously linked with the crossover from the temperature linear to the nonlinear range in @xmath1 , _ i.e. _ , they should have a common origin . in the fermion - spin theory @xcite , the charge and spin degrees of freedom of the physical electron are separated as the holon and spinon , respectively . although both holons and spinons contributed to the charge and spin dynamics , it has been shown that the scattering of spinons dominates the spin dynamics @xcite , while the results of the in - plane charge dynamics @xcite and present c - axis charge dynamics show that scattering of holons dominates the charge dynamics , the two rates observed in the experiments @xcite are attributed to the scattering of two distinct excitations , spinons and holons . it has been shown that an remarkable point of the pseudogap is that it appears in both of spinon and holon excitations @xcite . the present study indicates that the observed crossovers of @xmath125 and @xmath126 for copper oxide materials seem to be connected with the pseudogap in the in - plane charge holon excitations , which can be understood from the physical property of the in - plane holon dos . the numerical result of the in - plane holon dos @xmath110 at the doping @xmath92 , @xmath90 , and @xmath127 for the parameter @xmath93 at the temperature @xmath96=0 is shown in fig . 5(a ) , fig . 5(b ) , and fig . 5(c ) , respectively . for comparison , the corresponding mean - field result ( dashed line ) is also shown in fig . 5 . while the in - plane holon density of states @xmath110 in the underdoping @xmath92 as a function of energy for the temperature ( a ) @xmath118 , ( b ) @xmath128 , and ( c ) @xmath129 is plotted in fig . 6 . from fig . 5 and fig . 6 , we therefore find that the in - plane holon dos in mfa consists of the central part only , which comes from the noninteracting particles as pointed in ref . @xcite . after including fluctuations the central part is renormalized and two side bands @xcite and a v - shape holon pseudogap near the chemical potential @xmath23 in the underdoped regime appear . but these two side bands are almost doping and temperature independent , while the v - shape holon pseudogap is doping and temperature dependent , and grows monotonously as the doping @xmath130 decreases , and disappear in the overdoped regime . moreover , this holon pseudogap also decreases with increasing temperatures , and vanishes at higher temperatures . since the full holon green s function ( then the holon spectral function and dos ) is obtained by considering the second - order correction due to the spinon pair bubble , then the holon pseudogap is closely related to the spinon fluctuation . for small dopings and lower temperatures , the holon kinetic energy is much smaller than the magnetic energy , _ i.e. _ , @xmath131 , in this case the magnetic fluctuation is strong enough to lead to the holon pseudogap . this holon pseudogap would reduce the in - plane holon scattering and thus is responsible for the metallic to semiconducting crossover in the c - axis resistivity @xmath126 and the deviation from the temperature linear behavior in the in - plane resistivity @xmath125 @xcite . this holon pseudogap will also lead to form the normal - state gap in the system , and the similar result has been obtained from the doped _ kagom _ and triangular antiferromagnets @xcite , where the strong quantum fluctuation of spinons due to the geometric frustration leads to the normal - state gap . with increasing temperatures or dopings , the holon kinetic energy is increased , while the spinon magnetic energy is decreased . in the region where the holon pseudogap closes , at high temperatures or at higher doping levels , the charged holon scattering would give rise to the temperature linear in - plane resistivity as well as the metallic temperature dependence of the c - axis resistivity . our results also show that @xmath1 is only slightly affected by this holon pseudogap @xcite , while @xmath9 is more sensitive to the underlying mechanism . in summary , we have studied the c - axis charge dynamics of copper oxide materials in the underdoped and optimally doped regimes within the @xmath12-@xmath13 model by considering the incoherent interlayer hopping . our result shows the c - axis charge dynamics for the chain copper oxide materials is mainly governed by the scattering from the in - plane fluctuation , and the c - axis charge dynamics for the no - chain copper oxide materials is dominated by the scattering from the in - plane fluctuation incorporating with the interlayer disorder , which would be suppressed when the holon pseudogap opens at low temperatures and lower doping levels , leading to the crossovers to the semiconducting - like range in the c - axis resistivity @xmath9 and the temperature linear to the nonlinear range in the in - plane resistivity @xmath1 . because copper oxide materials are very complex systems , it is also possible that the actual c - axis conductivity may be a linear combination of the momentum - conserving @xmath89 and momentum - nonconserving @xmath108 , but we believe that @xmath89 should be the major part of the c - axis conductivity in the chain copper oxide materials , while @xmath132 should be the major part of the c - axis conductivity in the no - chain copper oxide materials . the authors would like to thank professor ru - shan han , professor h. q. lin , and professor t. xiang for helpful discussions . this work was supported by the national natural science foundation under grant no . 19774014 and the state education department of china through the foundation of doctoral training . s. l. cooper _ et al . b*47 * , 8233 ( 1993 ) ; s. l. cooper , _ et al . _ , lett . * 70 * , 1533 ( 1993 ) ; j. schtzmann _ et al . * 73 * , 174 ( 1994 ) ; a. v. puchkov _ et al . _ , 77 * , 1853 ( 1996 ) ; c. c. homes _ et al . * 71 * , 1645 ( 1993 ) ;
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the c - axis charge dynamics of copper oxide materials in the underdoped and optimally doped regimes has been studied by considering the incoherent interlayer hopping .
it is shown that the c - axis charge dynamics for the chain copper oxide materials is mainly governed by the scattering from the in - plane fluctuation , and the c - axis charge dynamics for the no - chain copper oxide materials is dominated by the scattering from the in - plane fluctuation incorporating with the interlayer disorder , which would be suppressed when the holon pseudogap opens at low temperatures and lower doping levels , leading to the crossovers to the semiconducting - like range in the c - axis resistivity and the temperature linear to the nonlinear range in the in - plane resistivity .
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the study of cellular processes at the single - molecule level is a flourishing field of research in biology . individual molecules labeled with sub - micron markers can now be tracked in a cellular environment , and quantitative information about their dynamics can be obtained by reconstructing their trajectory . one of the most used techniques for this purpose is single - molecule fluorescence microscopy ( smfm ) , which relies on a labeling with nanometer - sized fluorescent markers such as organic dyes or quantum dots . but standard smfm provides no information on the axial position of the marker , limiting this technique to 2d tracking . recent improvements of smfm such as astigmatism optic@xcite , @xmath1pi microscopy @xcite , double - helix psf @xcite , or multi - planes detection @xcite have made possible 3d tracking . since the depth of field of these techniques is limited to a few microns , 3d tracking of molecules that explore larger distances in the thickness of a sample requires to continuously adjust the position of the focal plane of the microscope objective , which strongly limits the time resolution . digital holographic microscopy ( dhm ) @xcite circumvents this drawback . in dhm , a ccd camera records the interference pattern between the light scattered by the sample and a reference wave , and a single shot is sufficient to determine the 3d positions of scatterers embedded in a non - diffusing environment , over a depth of typically a hundred of microns . as the scattering cross section of a particle scales as the sixth power of its radius @xcite , how easily and accurately a particle can be detected strongly depends on its size . several publications demonstrate the tracking of micron - sized colloids by using in - line holography @xcite , with a localization accuracy in the nanometer range through the use of high numerical aperture ( na ) microscope objectives . for example , with a @xmath2 na 1.4 oil immersion objective , cheong et al . @xcite reported lateral and axial localization accuracies of 4 and 20 nm respectively . this result was obtained with polystyrene spheres of diameter @xmath3 m , whose scattering cross section is quite large . the tracking of @xmath4 nm particles , whose scattering cross section is extremely low , is much more difficult , and , as far as we know , has not been demonstrated using in - line holography . yet the detection of such small particles is possible using dhm in a off - axis geometry @xcite with a noise level as low as possible @xcite and using good light scatterers , such as gold nanobeads @xcite . by this way , @xmath5 to @xmath6 nm gold particles embedded in an agarose gel have been detected and localized @xcite . since gold particles are not toxic for cells , they can be used as markers in biology @xcite , and d = 40 nm gold nanobeads fixed on the membrane receptors of a living cell have been localized @xcite . more recently , 3d tracking of batio3 particles with second harmonic generation dhm was also demonstrated @xcite . here , the main advantage of dhm , with respect to in - line holography , is the possibility to independently adjust the intensity of the illumination and reference beams in order to get the best detection sensitivity , by adjusting the reference beam intensity @xcite , and the largest signal , by adjusting the illumination beam intensity . combining dhm with dark field illumination allows then to detect nanometer - sized particles , as the sample can be illuminated with an intensity as large as possible , while avoiding a saturation of the camera @xcite . for example , atlan et al . and warnasooriya et al . uses total internal reflection ( tir ) to detect and localize @xmath5 nm and @xmath7 nm particles @xcite . but the tir configuration used in these experiments yield a standing wave which does not allow to track moving particles : when a moving particle crosses a node , the illumination ( and thus the signal ) goes down to zero , and the particle is lost . dubois et al . uses another dark field illumination configuration that focuses the illumination on a mask @xcite . since the illumination is parallel to the optical axis , no standing wave can appear , but since the illumination passes through the microscope objective , one expects parasitic reflections of the illumination beam . in this paper , we present a digital holographic microscopy technique which makes possible to track @xmath8 nm gold particles 3d diffusing in water . the illumination is parallel to the optical axis to prevent the formation of standing waves , and the holographic signal is collected by a na=0.5 dark field reflecting microscope objective . with such objective , the illumination beam is masked before the microscope objective and parasitic reflections are avoided . this yields high dynamic dark field illumination , which makes possible to detect , localize and track @xmath8 nm particles . first we describe the setup , which combines dark - field microscopy and off - axis holography . then we present the algorithm of reconstruction , our procedure to localize the beads , and describe how we can reach a real - time localization by performing calculations on a graphic card . finally we show that our setup allows us to track gold nanoparticles in motion with a lateral ( @xmath9 ) resolution of @xmath10 nm and an axial ( @xmath11 ) resolution of @xmath12 . since na=0.5 , the resolution ( especially in @xmath11 ) is lower than with na=1.4 in - line holography @xcite . we also show that the depth of field of our holographic microscope is made two orders of magnitude larger than in optical microscopy . our dhm experimental setup , depicted in fig . [ fig : setup ] , is designed to investigate the brownian motion of @xmath13 diameter gold particles diffusing in a @xmath14 ( length @xmath15 width @xmath15 height ) cell chamber ( ibidi @xmath16 ) filled with water . the concentration of nanoparticles is adjusted to @xmath17 particles / mm@xmath18 to have a few particles per field of view . the light source is a @xmath19 diode pump solid state laser ( crystal laser ) with a short coherence length ( @xmath20 ) to avoid parasitic interferences raising from reflexions between the optical elements . the laser beam is split into two beams by a polarizing beam splitter ( pbs ) , a half - wave plate before the pbs setting the ratio of energy between the emerging beams . the reference beam passes through a dove prism fixed on a micrometer translation stage to adjust the length of the optical path . this beam is spatially filtered through a @xmath21 m diameter pinhole and then expanded as to uniformly cover the ccd chip of the camera ( 512 @xmath15 512 pixels , andor luca r ) . the illumination beam is focalized on the sample with a plano - convex lens of focal length @xmath22 cm ( waist diameter @xmath23 , laser intensity @xmath24 w/@xmath25 ) . the light scattered by the beads is collected in transmission with a dark - field reflecting objective ( edmund optics , reflx series ) of @xmath26 and @xmath27 magnification . a small mask on the input of the objective limits the collection of light between @xmath28 and @xmath26 , so the illumination beam is totally blocked after passing through the sample . this dark - field configuration prevents the saturation of the ccd chip . a non - polarizing @xmath29 beam splitter behind the objective combines the scattered light with the reference beam . the ccd camera records the interference pattern on @xmath30 frames with a @xmath31 rate . the last beam - splitter is tilted by few degrees to be in off - axis configuration . in digital holography , the ccd sensor records an intensity @xmath32 which is the interference of the reference beam with the light scattered by the nano - objects . @xmath32 is thus a sum of 4 terms : @xmath33 where @xmath34 is the intensity of the reference beam , @xmath35 the intensity of the scattered light and @xmath36 are the electric fields for the reference beam and the scattered light respectively . as the scattering cross section of a @xmath0 diameter gold particle calculated with the rayleigh - mie scattering model is about @xmath37 at @xmath19 @xcite , the integration over the collection solid angle of the objective ( @xmath38 ) gives @xmath39 for a single particle . thus @xmath35 can be neglected compared to the other terms of eq.([eq : intensity ] ) . the field @xmath40 , the amplitude of which is about @xmath41 , contains the phase of the scattered light necessary to a 3d localization of the particle . in the fourier space , the tilted beam splitter adds a spatial frequency on the two conjugates terms of interference , and thus the different terms of @xmath32 are spatially separated , i.e. @xmath34 remains centered on the zero frequency of the fourier plane while the two cross terms of interference are centered on the spatial frequency induced by the off - axis geometry . since 3d reconstruction is a time consuming task even using recent multi - cores processors , we developed parallel calculations on a graphic card ( nvidia geforce gt470 , 448 cores ) @xcite . our ` c++ ` based algorithm uses the nvidia cuda library to decompose the calculations on the gpus of the card . among the existing methods to reconstruct holograms , the most common is the convolution method described by schnars et al . @xcite . the main drawback of this method is that the pixel size of the reconstructed image depends on the distance of reconstruction , so that the image of a thick sample is distorted . therefore this method is not convenient for 3d tracking , as the lateral scale depends on the depth of reconstruction considered . here we chose to reconstruct the holograms with the angular spectrum method @xcite , which , by compensating the sphericity of the signal wave , allows to reconstruct the hologram without distortion . our algorithm can be decomposed in 6 steps : i. _ subtraction of the background : _ : : in order to increase the signal - to - noise ratio , we subtract from the last recorded frame the average of the ten previous frames . phase shifting holography @xcite is also an effective technique for reducing noise , but the minimal delay @xmath42 ms between two frames , which is driven by our camera , is too long to use this technique for nanoparticle tracking . this step of calculation is performed only for particles in motion ( i.e. we skipped this step for the results presented in [ fixed ] ) . numerical correction of the signal wave sphericity : _ : : the hologram @xmath32 is multiplied by a complex phase matrix @xmath43 to compensate the sphericity induced by the microscope objective on the signal wave : @xmath44 where @xmath45 is the local radius of curvature of the wave on the ccd plane . if the reference wave is a plane wave , this distance @xmath45 is also the distance between the ccd chip and the back focal plane of the objective . _ first fft : _ : : the direct fourier transform of the corrected hologram is calculated using the cuda cufft library : @xmath46.\ ] ] fig . [ fig : recons](a ) shows the intensity @xmath47 in the _ k - space _ , in logarithmic scale in the case where the background is not removed ( step ( i ) skipped ) . in the middle of fig . [ fig : recons](a ) , the zero - order appears as a square because of the multiplication by the matrix @xmath43 . the term related to @xmath40 is in the down - right corner , centered on the spatial frequency induced by the off - axis geometry . the term related to @xmath48 is centered on the conjugate frequency ( top - left corner ) . at this step , the calculation is equivalent of the reconstruction in one fft ( convolution method ) of the hologram at the distance @xmath45 described in the previous step . since the back focal plane of our microscope objective coincides with the output pupil plane , which is common for high magnification objective , we see on the down - right corner a sharp reconstruction of the output pupil plane . if we change in step ( i ) the parameter @xmath45 to @xmath49 , the image of the output pupil would be sharp in the top - left corner , while the term related to @xmath40 would be blurred . [ fig : recons](b ) shows @xmath47 in the _ k - space _ when the background is removed ( step ( i ) performed ) . the zero - order term in the middle of the _ k - space _ is largely removed compare to fig . [ fig : recons](a ) , which reduces the recovery between the zero - order and @xmath40 . _ spatial filtering and centering : _ : : to remove the zero - order term and replace the term related to @xmath40 in the middle of the fourier plane , a round numerical filter which matches with the output pupil of the objective is applied . since the shape of the pupil is sharp in the k - space , we can isolate precisely the pixels containing the signal , minimizing the loss of information . to more precisely calibrate the radius and the center of the filter , we used a diffusive paper as a sample before performing experiments . since the paper scatters light uniformly , all the spatial frequencies that the microscope objective can collect are recorded . [ fig : recons](d ) shows the intensity @xmath47 in logarithmic scale when the sample is replaced by a diffusing paper . we clearly see the shape of the output pupil of the objective and the dark - field mask in the center . we set the filter mask to match with the shape of the output pupil ( white dotted circle in fig . [ fig : recons](d ) ) . the filtered part is then translated into the middle of a @xmath50 calculation grid in order to compensate the off - axis shift . v. _ propagation : _ : : the matrix obtained is multiplied by a propagation matrix @xmath51 , which is the exact form of the matrix propagation as given by kim et al . @xcite : @xmath52 where @xmath53 to propagate the hologram by a distance @xmath11 in the axial direction . @xmath54 is the magnified pixel size . these equations are suited for holograms of @xmath50 pixels . _ second fft : _ : : finally the inverse fft is calculated . for each hologram , the steps ( v ) and ( vi ) are repeated in order to get a stack of the scattered field at different depths , with a propagation step @xmath55 : @xmath56,\ ] ] where @xmath57 is an integer . in logarithmic scale . the zero - order term appears as a red square distorted by the multiplication with the phase matrix @xmath43 . the term of interest is located in the down - right corner . b : @xmath58 when the average of the ten previous holograms is subtracted before calculating the fft . the zero - order term is largely removed , so the recovery between this term and the region of interest in the down - right corner is reduced . c : two - dimensional reconstruction at a fixed depth of the sample . a gold nanobead ( 1 ) is localized in this plan and the shape of the intensity of the field scattered by other beads at other depths ( 2 and 3 ) is visible . d : intensity @xmath58 in logarithmic scale when the sample is replaced by a diffusive paper . the area of the output pupil of the objective is sharply defined . the white - dotted circle shows the mask of the numerical filter used for the reconstruction.,height=10 ] an example of reconstruction is shown in fig . [ fig : recons](c ) . we can see a nanoparticle ( 1 ) localized in the considered plane and the intensity of the field scattered by two particles at different depths ( 2 and 3 ) . the reconstruction of a @xmath59 thick volume requires to calculate one fft to reconstruct the hologram in the output pupil plane , then to calculate 500 inverse ffts for the slices of the stack . these 501 ffts typically require one minute when the calculation is performed on the cpu , even with a recent multi - cores computer . we reduce this time by a factor of 30 when the calculation is parallelized on the graphic card . once the three dimensional map of the scattered field is calculated , the beads are localized by pointing the local maxima of the field s intensity . [ fig : psf ] shows the field s intensity in the @xmath60,@xmath61 and @xmath62 directions . the full width at half maximum ( fwhm ) is about @xmath63 in lateral direction , and @xmath64 in the axial direction as expected for the point spread function associated with our microscope objective@xcite . in order to reach a sub - pixel size resolution , the intensity is fitted by a gaussian curve in @xmath60 and @xmath61 using the pixel for which the intensity is maximum and the two adjacent pixels . as the intensity profile in @xmath62 is not gaussian and 10 times larger than in @xmath60 and @xmath61 , we fit the maximum of the peak with a parabola using the pixel @xmath65 for which the intensity is maximum and the two adjacent pixels @xmath66 and @xmath67 ( fig . [ fig : psf](d ) ) . we chose this simple localization method because programming a more elaborate fit , as t - matrix theory based computation @xcite , with cuda ( programming language of the graphic card ) is more complicated and would considerably slow down the process . our method shows good performance ( see fig . [ fig : reso ] and fig . [ fig : error ] ) , but a better resolution may be achieved using t - matrix theory . to evaluate the lateral and axial localization accuracy of our setup , we localize a single @xmath0 diameter gold particle embedded in a 1% agarose gel . the localization accuracy is evaluated by calculating the standard deviation of 200 positions of the bead obtained from successive frames , with an exposure time for each frame @xmath68 . for a particle in the focal plane of the objective , we found a lateral localization accuracy of @xmath69 and an axial localization accuracy of @xmath70 ( fig . [ fig : reso](a ) and [ fig : reso](b ) ) . then the distance between the particle and the focal plane was increased by steps of @xmath71 . for each step we recorded 200 holograms and determined the mean position of the bead as well as the lateral and axial accuracy . [ fig : error ] compares the mean axial position of the particle with the mechanical displacement along @xmath62 of the sample , which are in excellent agreement . the localization accuracy in @xmath60,@xmath61 and @xmath62 as a function of the axial position of the bead is shown in fig . [ fig : reso](a ) and [ fig : reso](b ) . while the lateral accuracy is constant ( @xmath69 ) for @xmath72 nm , the axial accuracy slightly depends on @xmath62 . it is about @xmath70 around @xmath73 , then decreases to @xmath74 for @xmath72 nm . this accuracy strongly increases for @xmath75 , and for @xmath76 , the localization of the particle is not possible because the scattered signal level reaches the noise level . the local maximum at @xmath73 observed on the axial accuracy curve ( fig . [ fig : reso](b ) ) shows that the localization is not optimal when the gold particle is the focal plane of the objective . in this case the particle is imaged on a small area of the ccd chip , so that the interference pattern spreads on a small number of pixels , which degrades the quality of the reconstruction @xcite . in the case of particle tracking , the position of the focal plane in the sample has to be fixed . to minimize the spherical aberrations due to the presence of the coverslip , the focal plane should coincide with the sample - coverslip interface . in this configuration , a moving particle can not cross the focal plane , so that the localization accuracy remains optimal and constant over a depth of @xmath77 , that corresponds to the part between abscissa 0 and abscissa 250 in fig . [ fig : reso](a ) and [ fig : reso](b ) . yet tracking particles remains possible when the focal plane is set above the sample - coverslip interface as demonstrated below , as the cost of a slightly worse axial localization accuracy . we now consider gold particles in brownian motion . according to the stokes - einstein equation , the diffusion coefficient @xmath78 of the particles is given by : @xmath79 where @xmath80 is the boltzmann constant , @xmath81 the room temperature ( @xmath82 ) , @xmath83 the viscosity of water ( 1.0mpa.s at @xmath82 ) and @xmath84 nm the radius of the nanobead ( size dispersity given by the provider bbinternational ) . the exposure time is @xmath68 and the time between two frames @xmath85 . the mean distance traveled along one direction by a brownian particle during @xmath86 is @xmath87 , which is smaller than the lateral size of the magnified pixels ( @xmath88 ) . consequently , the signal from a particle is not blurred over several pixels during @xmath86 . the mean distance covered along a given axis during @xmath89 is @xmath90 , which corresponds to approximately @xmath91 pixels . as shown in fig . [ fig : motion ] , our method allows us to simultaneously track several particles . a volume of @xmath92 ( @xmath93 ) , i.e. @xmath94 pixels , can be reconstructed from a single hologram , and the localization method described in the section _ [ method ] _ can be performed to localize several beads with a sub - pixel accuracy . by repeating the algorithm for successive frames , we could for instance reconstruct the trajectories of 3 gold particles diffusing in water ( fig . [ fig : motion ] ) . we were able to track particles during up to 10 s ( @xmath95 frames ) . since the time needed to reconstruct a volume of @xmath94 pixels is about 0.5 s ( i.e. much larger than @xmath96 ms ) , reconstruction is necessarily a post - processing process in this case . in contrast , when a single particle is tracked , 3d localization in a given frame requires to reconstruct only a few slices around the position of the particle in the previous frame . for a @xmath0 particle , the mean distance traveled along @xmath62 during @xmath89 is @xmath97 , thus only 24 reconstructions ( 12 reconstructions above / below the previous position of the particle ) are sufficient to find the new position of the particle . the calculation of these 24 slices from the hologram requires @xmath98 , which is smaller than @xmath96 ms . real - time tracking is thus possible for a single particle under the condition that reconstruction is performed fast enough , for instance by using a graphic card unit as described above . to evidence the brownian motion , we calculated the 3d mean square displacement ( msd ) of a bead from the red trajectory plotted in fig . [ fig : motion ] : @xmath99 where @xmath100 is the total number of positions , as well as the 1d msd along the directions @xmath60 , @xmath61 , and @xmath62 . for a brownian motion with a diffusion constant @xmath78 , the msd curve depends linearly on time , and the slope of the curve is @xmath101 , where @xmath102 for the 3d msd ( fig . [ fig : msd](a ) ) and @xmath103 for 1d msd ( fig . [ fig : msd](b ) , ( c ) and ( d ) ) . as expected for a brownian motion , experimental 3d and 1d msd depend linearly on time , and a fit of the first six points of the curves gives @xmath104 along @xmath60 , @xmath105 along @xmath61 , @xmath106 along z and @xmath107 for the 3d motion . these values are in agreement with the theoretical value @xmath108 predicted by eq . ( [ stokes ] ) . in this paper , we show that digital holographic microscopy can be used to track @xmath109 gold particles diffusing in water . as the intensity of the light scattered by such nanoparticles is height orders of magnitude smaller than that of the excitation light , we combined holography to standing wave free dark - field microscopy to completely block the illumination beam , thereby preventing a saturation of the ccd chip of the camera . a single hologram , recorded with an exposure time of only 1 ms , is sufficient to localize several particles in a 250 @xmath110 m thick sample , with a lateral ( @xmath9 ) localization accuracy of @xmath69 and an axial ( @xmath11 ) localization accuracy of @xmath74 . as our dark field microscope involves a na=0.5 reflecting microscope objective , the resolution , especially in @xmath11 , obtained here with nanoparticles ( @xmath8 nm ) , is lower than the resolution reached in in - line holography using a na=1.4 objective and micron - sized objects ( @xmath3 m ) @xcite . this is the cost to pay for the detection of nanometer - sized particles . we were able to reconstruct particle trajectories , evidence the brownian nature of the motion and determine the related diffusion coefficient . the accuracy achieved by our setup is comparable with that reached with super - resolution microscopy : a localization accuracy of @xmath111 nm in x and y , and @xmath112 nm in z has been reported for fluorescence dye or quantum dots imaged with storm microscopy using optical astigmatism @xcite , and an accuracy of @xmath113 nm over a depth of @xmath114 m has been reached using a double helix psf @xcite . severals results of 3d tracking of quantum dots using palm microscopy report an axial accuracy between @xmath113 and @xmath115 nm , over a depth of typically @xmath116 m @xcite . palm and storm microscopy reach higher 3d localization accuracies than our dhm setup , but over a depth two order of magnitude lower . this ability to track nanoparticles up to @xmath117 m from the focal plane with a constant localization accuracy is the strength of dhm compare to super - resolution techniques . our results pave the way for the use of gold nanobeads as markers in more complex media such as cellular environment . this work was supported by funds from the french national research agency ( anr simi 10 and anr 3d brom ) , centre national de la recherche scientifique ( cnrs ) and cole normale suprieure ( ens ) . the authors thank mathieu coppey , fred etoc and jasmina dikic for their suggestions and careful reading .
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we present a new technique that combines off - axis digital holography and dark field microscopy to track @xmath0 gold particles diffusing in water .
we show that a single hologram is sufficient to localize several particles in a thick sample with a localization accuracy independent of the particle position . from our measurements
we reconstruct the trajectories of the particles and derive their 3d diffusion coefficient .
our results pave the way for quantitative studies of the motion of single nanoparticle in complex media .
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soon after the fermi - dirac form ( in 1926 ) of statistical mechanics was proposed for particles which obey pauli s exclusion principle ( in 1925 ) , fowler ( 1926 ) realized that the electron degeneracy pressure can balance for those stars , called as white dwarfs , discovered by astronomers in 1914 . by a numerical calculation ( 1931 ) for a polytropic gas of extremely relativistic electrons , chandrasekhar found a unique mass , which was interpreted as a mass limit of white dwarfs . landau ( 1932 ) presented an elementary explanation of the chandrasekhar limit by considering the lowest total energy of stars , and recognized that increasing density favors energetically the formation of neutrons , discovered only several months before by chadwick , through the action @xmath0 . a very massive object with much high density may have almost neutrons in the chemical equilibrium , which was then called as _ neutron stars _ ( nss ) . detailed calculations of ns structures showed ( e.g. , oppenheimer & volkoff 1939 ) that an ns can have a mass of @xmath1 , but is only @xmath2 km in radius , which makes it hard to be observed by astronomers . however , on one hand , a few authors do investigate possible astrophysical implications of nss . for example , baade & zwicky ( 1934 ) combined the researches of supernovae , cosmic rays , and nss , and suggested that nss may form after supernovae ; pacini ( 1967 ) even proposed that the stored energy in rotational form of an ns could be pumped into the supernova remnant by emitting electromagnetic waves . on the other hand , ns models were developed with improved treatments of equation of states , involving not only \{@xmath3 } , but also mesons and hyperons . the cooling behavior of nss was also initiated in 1960s due to the discovery of x - ray sources which were at first though mistakenly to be the thermal emission from nss . the discovery of _ radio pulsars _ by hewish & bell ( and their coauthors 1968 ) is a breakthrough in the study , and this kind of stars were soon identified as spinning nss by gold ( 1968 ) . since then more and more discoveries in other wave bands broaden greatly our knowledge about these pulsar - like compact stars ( plcss ) , including x - ray pulsars , x - ray bursts , anomalous x - ray pulsars , soft @xmath4-ray repeaters , and rosat - discovered `` isolated neutron stars '' . it is still a current concept among astrophysicists that such stars are really nss . ns studies are therefore in two major directions : 1 , the emission mechanisms for the stars , both rotation - powered and accretion - powered ; 2 , the ns interior physics . however , neutrons and protons are in fact _ not _ structureless points although they were thought to be elementary particles in 1930s ; they ( and other hadrons ) are composed of _ quarks _ proposed by gell - mann and zweig , respectively , in 1964 . the quark model for hadrons developed effectively in 1960s , and ivanenko & kurdgelaidze ( 1969 ) began to suggest a quark core in massive compact stars . itoh ( 1970 ) speculated about the exist of 3-flavor _ full _ quark stars ( since only @xmath5 , @xmath6 and @xmath7 quarks were recognized at that time ) , and even calculated the hydrostatic equilibrium of such quark stars which are now called as _ strange stars _ ( sss ) . is it possible that strange stars really exist in nature ? the possibility increases greatly if the bodmer - witten s conjecture is correct : bodmer ( 1971 ) initiated the discussion of quark matter with lower energy per baryon than normal nucleus , whereas witten ( 1984 ) considered an assumption of stable 3-flavor quark matter in details and discussion extensively three aspects related ( 1 , dark baryon and primordial gravitational wave due to the cosmic separation in the qcd epoch ; 2 , strange quark stars ; 3 , cosmic rays ) . farhi & jaffe s ( 1984 ) calculation in the mit bag model showed that the energy per baryon of strange matter is lower than that of nucleus for qcd parameters within rather wide range although we can hardly prove weather the bodmer - witten s conjecture is correct or not from first principles . haensel , zdunik & schaeffer ( 1986 ) and alcock , farhi & olinto ( 1986 ) then modelled sss , and found that sss can also have typical mass ( of @xmath8 ) and radius ( of @xmath2 km ) , which mean that _ the pulsar - like compact stars believed previously to be nss might actually be sss_. yet the most important and essential thing in the study is : how to distinguish sss from nss observationally ? more and more ss candidates appeared recently in literatures ( e.g. , bombaci 2002 , xu 2002 ) . it is generally suggested that sss as radio pulsars , the most popular ones of plcss , should have crusts ( with mass @xmath9 ) being similar to the outer crusts of nss ( witten 1984 , alcock et al . but this view was criticized by xu & qiao ( 1998 ) , who addressed that _ bare _ strange stars ( bsss , i.e. , sss without crusts ) being chosen as the interior of radio pulsars have three advantages : 1 , the spectral features ; 2 , the bounding energy ; and 3 , the core collapse process during supernova . it is thus a new window to distinguish bsss from nss via their magnetosphere and surface radiation according to the striking differences between the exotic quark surfaces of bsss and the normal matter surfaces of nss . with regard to the possible methods of finding strange stars in literatures , hard evidence to identify a strange star may be found by studying only the surface conditions since the other avenues are subject to many complex nuclear and/or particle physics processes that are poorly known . thanks to those advanced x - ray missions , it may be a very time for us to identify real strange stars in the universe . it is worth mentioning that , though some authors may name a general term `` _ neutron star _ '' , regardless of that the stars are `` neutron '' or `` strange '' , it is actually not suitable to call a real ss to be an ns since no _ neutron _ in an ss . one of the most great achievements in the last century is the construction of the standard model in particle physics ( e.g. , cottingham & greenwood 1998 ) , which asserts that the material in the universe is made up of elementary fermions ( divided into quarks and leptons ) interacting though gauge bosons : photon ( electromagnetic ) , w@xmath10 and z@xmath11 ( weak ) , 8 types of gluons ( strong ) , and graviton ( gravitational ) . there are totally 62 types of `` building blocks '' in the model . besides the 13 types of gauge bosons , there are three generations of fermions ( 1st : \{@xmath12 , e ; u , d } , 2nd : \{@xmath13 , @xmath14 ; c , s } , and 3rd : \{@xmath15 , @xmath16 ; t , b}. note that each types of quarks has three colors ) and their antiparticles . the final one , which is still not discovered , is the higgs particle that is responsible to the origin of mass . it is a first principle in the yang - mills theory that an interaction satisfies a corresponding local gauge symmetry , sometimes being broken spontaneously due to vacuum phase transition . the gauge theory for electromagnetic and weak interactions is very successful , with a high precision in calculation ; whereas one can treat gravity using einstein s general relativity theory if the length scale is not as small as the plank scale ( @xmath17 cm ) although a gauge theory of gravity is still not successful . as for the gauge theory for strong interaction , the quantum chromodynamics ( qcd ) , however , is still developing , into which many particle physicists are trying to make efforts . nonetheless , qcd has two general properties . for strong interaction in small scale ( @xmath18 fm ) , i.e. , in the high energy limit , the interacting particles can be treated as being _ asymptotically free _ ; a perturbation theory of qcd ( pqcd ) is possible in this case . whereas in larger scale ( @xmath19 fm ) , i.e. , in the low energy limit , the interaction is very strong , which results in _ color confinement_. the pqcd is not applicable in this scale ( many non - perturbative effects appear then ) , and a strong interaction system can be treated as a system of hadrons in which quarks and gluons are confined . in this limit , we still have effective means to study color interaction : 1 , the lattice formulation ( lqcd ) , with the discretization of space - time and on the base of qcd , provide a non - perturbative framework to compute numerically relations between parameters in the standard model and experiments by first principles ; 2 , phenomenological models , which rely on experimental date available at low energy density , are advanced for superdense hadronic and/or quark matter . these two properties result in two distinct phases ( rho 2001 ) of hadronic matter , depicted in the qcd phase diagram in terms of temperature @xmath20 vs. baryon chemical potential @xmath21 ( or baryon number density ) . hadron gas phase locates at the low energy - density limit where both @xmath20 and @xmath21 are relatively low , while a new phase called _ quark gluon plasma _ ( qgp ) or _ quark matter _ appears in the other limit when @xmath20 _ or _ @xmath21 is high although this new state of matter is still not found with certain yet . it is therefore expected that there is a kind of phase transition from hadron gas to qgp ( or reverse ) at critical values of @xmath20 and @xmath21 . actually a deconfinement transition is observed in numerical simulations of lqcd for zero chemical potential @xmath22 , when @xmath23 mev . can we find real quark matter ? certainly we may improve very much the knowledge about the strong interaction by studying the matter s various properties if a qgp state is identified in hand . one way is to create high energy - density fireball in laboratory through the collisions of relativistic heavy ions in accelerators . quark matter is expected if the center of the fireball reaches a temperature of @xmath24 , but the qgp is hadronized soon and there are final states of hadrons detected . it is thus a challenge to find clear signatures of qgp without ambiguousness in these experiments . another way is to search celestial bodies which contain quark matter via cosmological or stellar processes . strange stars ( 3 ) are very probably such kind of objects , which are a kind of bulk qgp with mass @xmath25 , composed of nearly equal number of up , down , and strange quarks ( called as strange quark matter , or strange matter ) . some possible observational signatures of sss are also addressed in literatures . the first way to detect this new form of matter is in terrestrial laboratory physics ( lab - physics ) , whereas the second is in astrophysics . they compensate each other . up to date , the research both in lab - physics and in astrophysics faces a general difficulty in finding quark matter : to search a _ clear _ signature for its existence . in fact , the investigations in lab - physics and astrophysics are in two _ different _ regions in the qcd phase diagram . available monte carlo simulations of lqcd are only applicable for cases with @xmath22 , and thus give valuable guidance for lab - physics . however , it is a very different story with sss , where the density effects dominate ( @xmath26 ) . it turns out , for technical reasons , to be extremely difficult to study strange matter by lqcd , and we have to rely on phenomenological models to speculate on the properties of sss by extrapolating our knowledge at nuclear matter density . for an ss , with high @xmath21 but low @xmath20 , one can not naively think that it is a simple qgp ; in fact many interesting phenomena , e.g. , color superconductivity ( alford et al . 2001 ) of strange matter , are discussed . in this meaning , sss provide only examples to study such hadronic system at high density , and one thus has to learn `` experimental date '' from astrophysics . that strange stars ( rather than neutron stars ) are residual after core - collapse type supernova explosion depends upon 1 , quark deconfinement occurs , 2 , strange matter in bulk is absolutely stable ( bodmer - witten s conjecture ) . unfortunately no certain answer to these issues is obtained theoretically ( 2 ) . nevertheless , we still can obtain general concepts about those two requirement for forming sss by following simple arguments . in the view of bag model ( called as the `` bag constant '' ) . ] , the vacuum in and out hadrons is different ; in - hadron is of `` pqcd vacuum '' where the strong interaction is weak and pqcd is applicable , but out - hadron is of `` qcd vacuum '' where the nonlinear strong coupling between quarks or gluons is in control . as baryon number density @xmath27 gets higher and higher , the pqcd vacuum in nucleons becomes more and more dominated ; if nucleon keeps a radius @xmath28 fm , the qcd vacuum disappears when @xmath29 ( @xmath30 is the density of ordinary nuclear matter ) . depending on rotation frequency and stellar mass , such density are easily surpassed in the cores of neutron stars ( weber 1999 ) . therefore the first requirement quark deconfinement may be satisfied . in case of @xmath31 , @xmath5 and @xmath6 quarks deconfined from nucleons could have a fermi energy quark and 2 @xmath6 quarks , and the fermi energy @xmath32 ( @xmath33 the number density of quarks ) . if @xmath34 , @xmath35 mev for @xmath5 quark and @xmath36 mev for @xmath6 quark . ] @xmath37 mev , which is much larger than the mass of @xmath7 quark @xmath38 mev ; the system should be energetically favorable by opening of a third flavor degree of freedom ( @xmath7 quark ) since high - energy @xmath5 and @xmath6 quarks are expected to decay into @xmath7 quarks via weak interactions . the second requirement may thus also be reasonable . it is worth addressing that , if only the first requirement is satisfied while strange matter is not absolutely stable but only metastable , a mixed phase where bulk quark and nuclear matter could coexist over macroscopical distances may appear in ns cores ( such nss are called as `` mixed stars '' , see , e.g. , heiselberg & hjorth - jensen 2000 , for a review ) . these two requirements may lead to the existence of strange matter in nature . in a simplified version of the bag model , assuming quarks are massless and noninteracting , we then have quark pressure @xmath39 ( @xmath40 is the quark energy density ) ; the total energy density is @xmath41 but the total pressure is @xmath42 . one therefore have the equation of state for strange matter , @xmath43 ( actually strange matter may have baryons from several hundreds , called as strangelets , to about that of our sun , called as strange stars . ) one can obtain the mass @xmath44 and radius @xmath45 of an ss by integrating numerically the tov equation , with a strange matter equation of state [ e.g. , eq.(1 ) ] , assuming a certain central density ( or pressure ) . the mass - radius relations of sss calculated with various strange matter equations of states are very different from that of nss , which may provide a possible way to differentiate sss from nss ( 3.2 ) . but an ss usually can have a mass of @xmath46 , and correspondent radius @xmath2 km . it is worth noting that the apparent temperature @xmath47 and radius @xmath48 ( also called as radiation radius , defined by @xmath49 , where @xmath50 is the apparent luminosity at infinity and @xmath51 is the stefan - boltzmann constant ) , i.e. , the values observed at infinity , is not that observed at the stellar surface ( @xmath45 is also the schwartzschild radius coordinate at the surface , defined by @xmath52 ) . for a compact star with mass @xmath44 and radius @xmath45 , the relations are ( haensel 2001 ) @xmath53 , @xmath54 , where @xmath55 is the schwartzschild radius . as the strange quarks are more massive ( @xmath38 mev ) than the up ( @xmath56 mev ) and down ( @xmath57 mev ) quarks , some electrons are required to keep the chemical equilibrium of an ss . this brings some interesting properties near the bare quark surface . since quarks can be bound through strong interaction , whereas electrons by much weaker interaction ( only the electromagnetism ) , the electrons can thus spread out the quark surface and be distributed in such a way that a strong outward static electric field is formed . adopting a simple thomas - fermi model , one can deduce analytical expressions for the electron number density @xmath58 and the electric field @xmath59 above the surface ( xu & qiao 1999 ) , @xmath60 where @xmath61 is a measured height above the quark surface , @xmath62 cm ) . very strong electric field , @xmath63 v / cm from eq.(2 ) , should be near the quark surface , which makes it possible to support a normal - matter crust with mass @xmath64 ( alcock et al . sss covered by such crusts are called as crusted strange stars ( csss ) , while sss without crusts as bare strange stars ( bsss ) . can radio pulsars , the largest population of plcss , be bsss ? the answer was no in alcock et al s ( 1986 ) paper : `` pulsar emission mechanisms which depend on the stellar surface as a source of plasma will not work if there is a bare quark surface '' and `` the universe is a dirty environment and a bare strange star may readily accrete some ambient material '' . their first point is certainly incomplete , because @xmath65 pairs produced rapidly in strong electro - magnetic fields [ as expressed in eq.(2 ) , or induced by the unipolar effect ] should create a corotating magnetosphere although no charged particles can be pulled out into the magnetosphere ( xu & qiao 1998 ) . a nascent protostrange star should be bare because of strong mass ejection and high temperature ( usov 1998 ) after the supernova detonation flame ; an ss can hardly accrete due to rapid rotating and strong magnetic field ( xu et al . 2001 ) ; even accretion is possible , a crust still can not form as long as the accretion rate is not much larger than the eddinton one ( xu 2002 ) . their second point is thus also not reasonable . therefore radio pulsars could be bsss . furthermore , it is found that bsss may be better for explaining the observations of radio pulsars as well as other plcss ( 3.3 ) . although the ss idea is not new , ss identification becomes a hot topic only in recent years because of advanced techniques in space . since sss may have similar masses and radii , which are conventional quantities observable in astronomy , to that of nss , it is very difficult to find observational signals of quark matter in the plcss . it was argued that the ss and ns cooling behaviors could be distinguishable , since sss may cool much faster than nss ( e.g. pizzochero 1991 ) . however , recent more complete analyses on this issue indicate that this may be impossible except for the first @xmath6630 years after their births ( schaab et al . nevertheless , the author thinks there may still be three effective ways to do . the minimum rotation periods of sss are smaller than that of nss . rotating stars composted of ideal fluid are subject to rotation - mode instability , which leads to the loss of rotation energy by gravitational radiation and results in substantial spindown . however the matter of a real star is not ideal but have viscosity ; the calculated bulk viscosity , based on the work of wang & lu ( 1984 ) , of strange matter is much higher than that of neutron matter although their shear viscosities are similar ; therefore sss could have smaller periods at which their higher viscosity can prevent them from developing the instability ( madsen 1998 , 2000 ) . the 2.14 ms optical source in sn 1987a ( middleditch et al 2000 ) should be an ss if being confirmed in further observations . the approximate mass - radius ( @xmath67 ) relations of sss ( @xmath68 ) are in surprising contrast to that of nss ( @xmath69 ) , and sss can have much small radii . comparisons of observation - determined relations in x - ray binaries with modelled ones may thus tell if an object is an ns or an ss ( li et al . also , a plcs with radius @xmath70 km could be an ss ( drake et al . 2002 ) . there are striking differences between the surfaces of bsss and that of nss . the very properties of the quark surface , e.g. , strong bounding of particles , abrupt density change from @xmath71g/@xmath72 to @xmath73 [ eq.(1 ) ] in @xmath19 fm , and strong electric fields , may eventually help us to identify of a bss ( 3.3 ) . three parts of possible evidence for bsss are discussed below . although pulsar emission mechanism is not well understood , the rs - type ( ruderman & sutherland 1975 ) sparking model is still the popular one to connect magnetospheric dynamics with general observations , with an `` user friendly '' nature . maybe the strongest support to the rs - type vacuum gap model is the drifting subpulses observed from some pulsars . however rs model faces at least two difficulties for nss : the binding energy problem and the antipulsar issue , which can be solved completely if radio pulsars are bsss ( xu et al . 1999 ) . the soft @xmath4-ray burst of sgr 0526 - 66 , with peak luminosity @xmath74 , needs ultra - strong field ( @xmath63 g ) to constrain the fireball . an alternative bounding is through the quark surface ; and it may be natural to explain the bursting energy and the light curves in a framework that a comet - like object falls to a bss ( zhang et al . 2000 , usov 2001 ) . bsss are expected to have featureless spectra ( of both surface thermal and magnetospheric non - thermal components ) since no ion is above the quark surface or in the magnetosphere , except for electron cyclotron lines due to the landau levels appeared in strong fields ( xu & qiao 1998 , xu 2002 ) . recent observations known hitherto of several plcss actually show featureless spectra except for two sources 1e1207 and sgr1806 , the lines of which may originate from landau level transitions in suitable field strength for the space facilities ( astro - ph/0207079 ) . the theoretical bases of sss are , to some extent , solid in physics and the formation of strange matter stars is possible in astrophysics ; sss could thus exist . although each of the observed phenomena from plcss may be interpreted under the ns regime with unusual or artificial physical properties , it could be a natural way to understand the observations by updating nss with sss .
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a pedagogical overview of strange quark matter and strange stars is presented . after a historical notation of the research and an introduction to quark matter ,
a major part is devoted to the physics and astrophysics of strange stars , with attention being paid to the possible ways by which neutron stars and strange stars can be distinguished in astrophysics .
recent possible evidence for _ bare _ strange stars is also discussed .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in
| 6,671 | 130 |
numerous precision tests of the standard model ( sm ) and searches for its possible violation have been performed in the last few decades , serving as an invaluable tool to test the theory at the quantum level . they have also provided stringent constraints on many `` new physics '' ( np ) scenarios . a typical example is given by the measurements of the anomalous magnetic moment of the electron and the muon , where recent experiments reached the fabulous relative precision of 0.7 ppb @xcite and 0.5 ppm,@xcite respectively . these experiments measure the so - called gyromagnetic factor @xmath1 , defined by the relation between the particle s spin @xmath5 and its magnetic moment @xmath6 , = g , where @xmath7 and @xmath8 are the charge and mass of the particle . in the dirac theory of a charged point - like spin-@xmath9 particle , @xmath10 . quantum electrodynamics ( qed ) predicts deviations from dirac s value , as the charged particle can emit and reabsorb virtual photons . these qed effects slightly increase the @xmath1 value . it is conventional to express the difference of @xmath1 from 2 in terms of the value of the so - called anomalous magnetic moment , a dimensionless quantity defined as @xmath11 . the anomalous magnetic moment of the electron , @xmath12 , is rather insensitive to strong and weak interactions , hence providing a stringent test of qed and leading to the most precise determination of the fine - structure constant @xmath13 to date.@xcite on the other hand , the @xmath1@xmath2@xmath3 of the muon , @xmath14 , allows to test the entire sm , as each of its sectors contributes in a significant way to the total prediction . compared with @xmath12 , @xmath14 is also much better suited to unveil or constrain np effects . indeed , for a lepton @xmath15 , their contribution to @xmath16 is generally expected to be proportional to @xmath17 , where @xmath18 is the mass of the lepton and @xmath19 is the scale of np , thus leading to an @xmath20 relative enhancement of the sensitivity of the muon versus the electron anomalous magnetic moment . this more than compensates the much higher accuracy with which the @xmath1 factor of the latter is known . the anomalous magnetic moment of the @xmath0 lepton , @xmath21 , would suit even better ; however , its direct experimental measurement is prevented by the relatively short lifetime of this lepton , at least at present . the existing limits are based on the precise measurements of the total and differential cross sections of the reactions @xmath22 and @xmath23 at lep energies . the most stringent limit , @xmath24 at 95% confidence level , was set by the delphi collaboration,@xcite and is still more than an order of magnitude worse than that required to determine @xmath21 . in the 1990s it became clear that the accuracy of the theoretical prediction of the muon @xmath1@xmath2@xmath3 , challenged by the e821 experiment underway at brookhaven,@xcite was going to be restricted by our knowledge of its hadronic contribution . this problem has been solved by the impressive experiments at low - energy @xmath25 colliders , where the total hadronic cross section ( as well as exclusive ones ) were measured with high precision , allowing a significant improvement of the uncertainty of the leading - order hadronic contribution.@xcite as a result , the accuracy of the sm prediction for @xmath14 now matches that of its measurement . in parallel to these efforts , very many improvements of all other sectors of the sm prediction were carried on by a large number of theorists ( see refs . for reviews ) . all these experimental and theoretical developments allow to significantly improve the theoretical prediction for the anomalous magnetic moment of @xmath0 lepton as well . in this article we review and update the sm prediction of @xmath21 , analyzing in detail the three contributions into which it is usually split : qed , electroweak ( ew ) and hadronic . updated qed and ew contributions are presented in secs . [ sec : qed ] and [ sec : ew ] ; new values of the leading - order hadronic term , based on the recent low energy @xmath4 data from babar , cmd-2 , kloe and snd , and of the hadronic light - by - light contribution are presented in sec . [ sec : had ] . the total sm prediction is confronted to the available experimental bounds on the @xmath0 lepton @xmath1@xmath2@xmath3 in sec . [ sec : sm ] , and prospects for its future measurements are briefly discussed in sec . [ sec : conc ] , where conclusions are drawn . the qed part of the anomalous magnetic moment of the @xmath0 lepton arises from the subset of sm diagrams containing only leptons and photons . this dimensionless quantity can be cast in the general form:@xcite a_^qed = a_1 + a_2 ( ) + a_2 ( ) + a_3 ( , ) , [ eq : atauqedgeneral ] where @xmath27 , @xmath28 and @xmath29 are the electron , muon and @xmath0 lepton masses , respectively . the term @xmath30 , arising from diagrams containing only photons and @xmath0 leptons , is mass and flavor independent . in contrast , the terms @xmath31 and @xmath32 are functions of the indicated mass ratios , and are generated by graphs containing also electrons and/or muons . the functions @xmath33 ( @xmath34 ) can be expanded as power series in @xmath35 and computed order - by - order a_i = a_i^(2 ) ( ) + a_i^(4 ) ( ) ^2 + a_i^(6 ) ( ) ^3 + a_i^(8 ) ( ) ^4 + . only one diagram is involved in the evaluation of the lowest - order ( first - order in @xmath13 , second - order in the electric charge ) contribution it provides the famous result by schwinger @xmath36.@xcite the mass - dependent coefficients @xmath31 and @xmath32 , discussed below , are of higher order . they were derived using the latest codata@xcite recommended mass ratios : m_/m_e & = & 3477.48 ( 57 ) [ eq : rte ] + m_/m _ & = & 16.8183 ( 27 ) . [ eq : rtm ] the value for @xmath29 adopted by codata in ref . , @xmath37 mev , is based on the pdg 2002 result.@xcite it remained unchanged until very recently ( see refs . ) , when preliminary results of two new measurements ( from the belle@xcite and kedr@xcite detectors ) were reported . the central values of the new mass values are slightly lower than the current world average value , but agree with it within the uncertainties , which are approaching that of the world average value ( used in this work ) . seven diagrams contribute to the fourth - order coefficient @xmath38 , one to @xmath39 and one to @xmath40 . they are depicted in fig . [ fig : qed2 ] . as there are no two - loop diagrams contributing to @xmath41 that contain both virtual electrons and muons , @xmath42 . the mass - independent coefficient has been known for almost fifty years:@xcite a_1^(4 ) & = & + + ( 3 ) - 2 + & = & -0.328 478 965 579 193 78 , [ eq : a14 ] where @xmath43 is the riemann zeta function of argument @xmath44 . for @xmath45 , @xmath46 or @xmath0 , the coefficient of the two - loop mass - dependent contribution to @xmath47 , @xmath48 , with @xmath49 , is generated by the diagram with a vacuum polarization subgraph containing the virtual lepton @xmath50 . this coefficient was first computed in the late 1950s for the muon @xmath1@xmath2@xmath3 with @xmath51 , neglecting terms of @xmath52.@xcite the exact expression for @xmath53 was reported by elend in 1966.@xcite however , its numerical evaluation was considered tricky because of large cancellations and difficulties in the estimate of the accuracy of the results , so that common practice was to use series expansions instead.@xcite taking advantage of the properties of the dilogarithm @xmath54,@xcite the exact result was cast in ref . in a very simple and compact analytic form , valid , contrary to the one in ref . , also for @xmath55 ( the case relevant to @xmath56 and part of @xmath57 ) : a_2^(4)(1/x ) & = & - - + x^2 ( 4 + 3x ) + ( 1 - 5 x^2 ) + & & + + & & + x^4 . [ eq : ea24 ] for @xmath58 , gives @xmath59 ; of course , this contribution is already part of @xmath38 in . numerical evaluation of with the mass ratios given in eqs . ( [ eq : rte])([eq : rtm ] ) yields the two - loop mass - dependent qed contributions to the anomalous magnetic moment of the @xmath0 lepton,@xcite a_2^(4)(m_/m_e ) & = & 2.024 284 ( 55 ) , [ eq : ta24e ] + a_2^(4)(m_/m _ ) & = & 0.361 652 ( 38 ) . [ eq : ta24 m ] these two values are very similar to those computed via a dispersive integral in ref . ( which , however , contain no estimates of the uncertainties ) . equations ( [ eq : ta24e ] ) and ( [ eq : ta24 m ] ) are also in agreement ( but more accurate ) with those of ref . . adding up eqs . ( [ eq : a14 ] ) , ( [ eq : ta24e ] ) and ( [ eq : ta24 m ] ) one gets:@xcite c_^(4 ) = 2.057 457 ( 93 ) [ eq : tc2 ] ( note that the uncertainties in @xmath60 and @xmath61 are correlated ) . the resulting error @xmath62 leads to a @xmath63 uncertainty in @xmath41 . more than one hundred diagrams are involved in the evaluation of the three - loop ( sixth - order ) qed contribution . their analytic computation required approximately three decades , ending in the late 1990s . the coefficient @xmath64 arises from 72 diagrams . its exact expression , mainly due to remiddi and his collaborators , reads:@xcite a_1^(6 ) & = & ^2 ( 3 ) - ( 5 ) - ^4 + + + & & + ( 3 ) - ^2 2 + ^2 + + & & + + & = & 1.181 241 456 587 . [ eq : a16 ] this value is in very good agreement with previous results obtained with numerical methods.@xcite the calculation of the exact expression for the coefficient @xmath65 for arbitrary values of the mass ratio @xmath66 was completed in 1993 by laporta and remiddi @xcite ( earlier works include refs . ) . let us focus on @xmath67 ( @xmath68 , @xmath69 ) . this coefficient can be further split into two parts : the first one , @xmath70 , receives contributions from 36 diagrams containing either electron or muon vacuum polarization loops,@xcite whereas the second one , @xmath71 , is due to 12 light - by - light scattering diagrams with either electron or muon loops.@xcite the exact expressions for these coefficients are rather complicated , containing hundreds of polylogarithmic functions up to fifth degree ( for the light - by - light diagrams ) and complex arguments ( for the vacuum polarization ones ) they also involve harmonic polylogarithms.@xcite series expansions were provided in ref . for the cases of physical relevance . using the recommended mass ratios given in eqs . ( [ eq : rte ] ) and ( [ eq : rtm ] ) , the following values were recently computed from the full analytic expressions:@xcite a_2^(6)(m_/m_e , ) & = & 7.256 99 ( 41 ) [ eq : ta26evac ] + a_2^(6)(m_/m_e , ) & = & 39.1351 ( 11 ) [ eq : ta26elbl ] + a_2^(6)(m_/m_,)&= & -0.023 554 ( 51 ) [ eq : ta26mvac ] + a_2^(6)(m_/m_,)&= & 7.033 76 ( 71 ) . [ eq : ta26mlbl ] almost identical values were obtained employing the approximate series expansions of ref . : 7.25699(41 ) , 39.1351(11 ) , @xmath20.023564(51 ) , 7.03375(71).@xcite the previous estimates of ref . were different : 10.0002 , 39.5217 , 2.9340 , and 4.4412 ( no error estimates were provided ) , respectively ; they are superseded by the results in eqs . ( [ eq : ta26evac])([eq : ta26mlbl ] ) , derived from the exact analytic expressions . the estimates of ref . compare slightly better : 7.2670 , 39.6 , @xmath72 , 4.47 ( no errors provided ) . in the specific case of @xmath73 , the values of refs . and differ from because their derivations did not include terms of @xmath74 , which turn out to be unexpectedly large . the sums of eqs . ( [ eq : ta26evac])([eq : ta26elbl ] ) and ( [ eq : ta26mvac])([eq : ta26mlbl ] ) are a_2^(6)(m_/m_e ) & = & 46.3921 ( 15 ) , [ eq : ta26e ] + a_2^(6)(m_/m_)&= & 7.010 21 ( 76 ) . [ eq : ta26 m ] the contribution of the three - loop diagrams with both electron- and muon - loop insertions in the photon propagator was calculated numerically from the integral expressions of ref . , obtaining:@xcite a_3^(6)(m_/m_e , m_/m _ ) = 3.347 97 ( 41 ) . [ eq : ta36 ] this value disagrees with the results of refs . ( 1.679 ) and ( 2.75316 ) . combining the three - loop results of eqs . ( [ eq : a16 ] ) , ( [ eq : ta26e ] ) , ( [ eq : ta26 m ] ) and ( [ eq : ta36 ] ) one finds the sixth - order qed coefficient,@xcite c_^(6 ) = 57.9315 ( 27 ) . [ eq : tc3 ] the error @xmath75 induces a @xmath76 uncertainty in @xmath41 . the order of magnitude of the three - loop contribution to @xmath41 , dominated by the mass - dependent terms , is comparable to that of ew and hadronic effects ( see later ) . contrary to the case of the electron and muon @xmath1@xmath2@xmath3 , qed contributions of order higher than three are not known.@xcite ( an exception is the mass- and flavor - independent term @xmath77,@xcite which is however expected to be a very small part of the complete four - loop contribution . ) adding up all the above contributions and using the new value of @xmath13 derived in refs . and , @xmath78 , one obtains the total qed contribution to the @xmath1@xmath2@xmath3 of the @xmath0 lepton,@xcite a_^qed = 117 324 ( 2 ) 10 ^ -8 . [ eq : tqed ] the error @xmath79 is the uncertainty @xmath80 assigned to @xmath41 for uncalculated four - loop contributions . as we mentioned earlier , the errors due to the uncertainties of the @xmath81 and @xmath82 terms are negligible . the error induced by the uncertainty of @xmath13 is only @xmath83 ( and thus totally negligible ) . with respect to schwinger s contribution , the ew correction to the anomalous magnetic moment of the @xmath0 lepton is suppressed by the ratio @xmath84 , where @xmath85 is the mass of the @xmath86 boson . numerically , this contribution is of the same order of magnitude as the three - loop qed one . the analytic expression for the one - loop ew contribution to @xmath26 , due to the diagrams in fig . [ fig : ew1 ] , is:@xcite ^ew ( ) = , [ eq : ewoneloop ] where @xmath87 is the fermi coupling constant,@xcite @xmath88 , @xmath85 and @xmath89 are the masses of the @xmath90 , @xmath86 and higgs bosons , and @xmath91 is the weak mixing angle . closed analytic expressions for @xmath92 taking exactly into account the @xmath93 dependence ( @xmath94 higgs , or other hypothetical bosons ) can be found in refs . . following ref . , we employ for @xmath95 the on - shell definition,@xcite @xmath96 , where @xmath97,@xcite and @xmath85 is the theoretical sm prediction of the @xmath86 mass . the latter can be easily derived from the simple analytic formulae of ref . ( see also refs . ) , = , [ eq : fops ] ( on - shell scheme ii with @xmath98,@xcite @xmath99,@xcite and @xmath100 gev,@xcite ) leading to @xmath101 for @xmath102 . this result should be compared with the direct experimental value @xmath103,@xcite which corresponds to a very small @xmath89 . in any case , these shifts in the @xmath85 prediction induced by the variation of @xmath89 from 114.4 gev , the current lower bound at 95% confidence level,@xcite up to a few hundred gev , only change @xmath92 by amounts of @xmath104 . from , including the tiny @xmath105 corrections of refs . , for @xmath102 we get ^ew ( ) = 55.1(1 ) 10 ^ -8 . [ eq : ewoneloopn ] the uncertainty encompasses the shifts induced by variations of @xmath89 from 114 gev up to a few hundred gev , and the tiny uncertainty due to the error in @xmath29 . the estimate of the ew contribution in ref . , @xmath106 , obtained from the one - loop formula ( without the small corrections of order @xmath107 ) , is similar to our value in . however , its uncertainty ( @xmath108 ) is too small , and it does nt contain the two - loop contribution which , as we ll discuss in the next section , is not negligible . the two - loop ew contributions to @xmath16 ( @xmath45 , @xmath46 or @xmath0 ) were computed in 1995 by czarnecki , krause and marciano.@xcite this remarkable calculation leads to a significant reduction of the one - loop prediction . navely one would expect the two - loop ew contribution @xmath109 to be of order @xmath110 , but this turns out not to be so . as first noticed in the early 1990s,@xcite @xmath111 is actually quite substantial because of the appearance of terms enhanced by a factor of @xmath112 , where @xmath113 is a fermion mass scale much smaller than @xmath85 . the two - loop contribution to @xmath114 involves 1678 diagrams in the linear t hooft - feynman gauge@xcite ( as a check , the authors of refs . and employed both this gauge and a nonlinear one in which the vertex of the photon , the @xmath86 and the unphysical charged scalar vanishes ) . it can be divided into fermionic and bosonic parts ; the former , @xmath115 , includes all two - loop ew corrections containing closed fermion loops , whereas all other contributions are grouped into the latter , @xmath116 . the expressions of ref . for the bosonic part were obtained in the approximation @xmath117 , computing the first two terms in the expansion in @xmath118 , and expanding in @xmath95 , keeping the first four terms in this expansion ( this number of powers is sufficient to obtain an exact coefficient of the large logarithms @xmath119 ) . recent analyses of the ew bosonic corrections of the @xmath1@xmath2@xmath3 of the muon@xcite relaxed these approximations , providing analytic results valid also for a light higgs . considering the present @xmath120 gev lower bound,@xcite we can safely employ the results of ref . , obtaining , for @xmath102 @xmath121 . the neglected terms are of @xmath122 . the fermionic part of @xmath123 contains the contribution of diagrams with light quarks ; they involve long - distance qcd for which perturbation theory can not be employed . in particular , these hadronic uncertainties arise from two types of two - loop diagrams : the hadronic photon@xmath90 mixing , and quark triangle loops with the external photon , a virtual photon and a @xmath90 attached to them ( see fig . [ fig : ew2 ] ) . the hadronic uncertainties mainly arise from the latter ones . two approaches were suggested for their study : in ref . the nonperturbative effects where modeled introducing effective quark masses as a simple way to account for strong interactions . in view of the high experimental precision of the @xmath1@xmath2@xmath3 of the muon , a more realistic treatment of the relevant hadronic dynamics was introduced in ref . within a low - energy effective field theory approach , later on developed in the detailed analyses of refs . . however , from a numerical point of view , the discrepancy between the results provided by these two different approaches turns out to be irrelevant for the present interpretation of the experimental result of the muon @xmath1@xmath2@xmath3 , in spite of its precision . the use of effective quark masses for the study of the @xmath1@xmath2@xmath3 of the @xmath0 whose experimental precision is a far cry from that of the muon ! thus appears to be sufficient at present . the tiny hadronic @xmath124@xmath90 mixing terms can be evaluated either in the free quark approximation or via a dispersion relation using data from @xmath25 annihilation into hadrons ; the difference was shown to be numerically insignificant.@xcite references and contain simple approximate expressions for the contributions of the diagrams with fermion triangle loops shown in fig . [ fig : ew2 ] ( right ) . in general , for a lepton @xmath45 , @xmath46 or @xmath0 , neglecting small mass - ratios , they are a_l^ew ( ) _ f , [ eq : ew2lfd ] where @xmath125 is the third component of the weak isospin of the fermion @xmath126 in the loop , @xmath127 is its charge , @xmath128 its number of colors ( 3 for quarks , 1 for leptons ) , and c(f ) \ { ll ( ^2/m_l^2 ) + 5/6 & + ( ^2/m_l^2 ) - 8 ^2/27 + 11/18 & + ( ^2/m_f^2 ) -2 & + ( m_top^2/^2 ) + + + ( 5/18)(^2/m_top^2 ) -4/3 & . [ eq : ew2lfdlogs ] the contribution of the top - quark triangle loop diagram of fig . [ fig : ew2 ] ( right ) with the @xmath90 boson replaced by the neutral goldstone boson ( @xmath129 ) has also been included in this expression.@xcite it is clear from eqs . ( [ eq : ew2lfd])([eq : ew2lfdlogs ] ) that the logarithms @xmath130 cancel in sums over all fermions of a given generation , as long as @xmath131 , due to the no - anomaly condition @xmath132 valid within every generation.@xcite this does not occur for the third generation due to the large mass of the top quark . note that this short - distance cancellation does not get modified by strong interaction effects on the quark triangle diagrams.@xcite contrary to the case of the muon @xmath1@xmath2@xmath3 , where all fermion masses , with the exception of @xmath27 , enter in @xmath133 , the approximate expressions in show that this is not the case for the @xmath1@xmath2@xmath3 of the @xmath0 lepton . indeed , due to the high infrared cut - off set by @xmath29 , for @xmath134 does not depend on any fermion mass lighter than @xmath29 ; apart from @xmath29 , it only depends on @xmath135 and @xmath136 , the masses of the top and bottom quarks ( assuming @xmath137 ) . the charm contribution requires some care , as the crude approximation provided by for a charm lighter than the @xmath0 lepton is valid only if @xmath138 . clearly , this is not a good approximation , and the spurious shift induced by when @xmath139 is varied across the @xmath29 threshold is of @xmath140 . one possibility is to use with @xmath139 equal to @xmath29.@xcite better still , we numerically integrated the exact expressions for @xmath141 provided in ref . for arbitrary values of @xmath113 , obtaining a smooth dependence on the value of @xmath139 . for completeness we repeated this detailed analysis for all light fermions . as expected , the result depends very mildly on the values chosen for their masses . employing the values @xmath142 gev , @xmath143 gev , @xmath144 gev and @xmath145 gev , and adding to the contribution of the remaining fermionic two - loop diagrams studied in ref . , for @xmath146 we obtain @xmath147 . in this evaluation we also included the tiny @xmath122 contribution of the @xmath124@xmath90 mixing diagrams , suppressed by ( @xmath148 for quarks and ( @xmath149 for leptons , via the explicit formulae of ref . . the sum of the fermionic and bosonic two - loop ew contributions described above gives @xmath150 , a 14% reduction of the one - loop result . the leading - logarithm three - loop ew contributions to the muon @xmath1@xmath2@xmath3 were determined to be extremely small via renormalization - group analyses.@xcite we assigned to our @xmath0 lepton @xmath1@xmath2@xmath3 ew result an additional uncertainty of @xmath151 \!\sim\ ! o(10^{-9})$ ] to account for these neglected three - loop effects . adding @xmath152 to the one - loop value of we get our total ew correction ( for @xmath146 ) ^ew = 47.4 ( 5 ) 10 ^ -8 . [ eq : tew ] the uncertainty allows @xmath89 to range from 114 gev up to @xmath153 gev , and reflects the estimated errors induced by hadronic loop effects ( @xmath154 and @xmath155 can vary between 70 mev and 400 mev ) , neglected two - loop bosonic terms , and the missing three - loop contribution . it also includes the small errors due to the uncertainties in @xmath135 and @xmath29 . the value in is in agreement with the prediction @xmath156,@xcite with a reduced uncertainty . as we mentioned in sec . [ subsec : ew1 ] , the ew estimate of ref . , @xmath106 , mainly differs from in that it does nt include the two - loop corrections . in this section we will analyze @xmath157 , the contribution to the @xmath0 anomalous magnetic moment arising from qed diagrams involving hadrons . hadronic effects in ( two - loop ) ew contributions are already included in @xmath114 ( see the previous section ) . similarly to the case of the muon @xmath1@xmath2@xmath3 , the leading - order hadronic contribution to the @xmath0 lepton anomalous magnetic moment is given by the dispersion integral:@xcite @xmath158 where the kernel @xmath159 is a bounded function of energy monotonously increasing to unity at @xmath160 , and @xmath161 is the total hadronic cross section of the @xmath25 annihilation in the born approximation . in fig . [ fig : rat ] we plot the ratio of the kernels in the @xmath0 lepton and muon case . clearly , although the role of the low energies is still very important , the different structure of @xmath162 compared to @xmath163 , induced by the higher mass of the @xmath0 , results in a relatively higher role of the larger energies . the history of these calculations is not as rich as that of the muon . the first calculation performed in 1978 in ref . was based on experimental data available at that time below 7.4 gev , whereas at higher energies the asymptotic qcd prediction was used . ten years later , a rough estimate was made in ref . based on low energy @xmath25 data . in ref . the contribution of the @xmath164 meson was estimated by integrating the approximation obtained using the breit - wigner curve , while other contributions used the data . the accuracy of the calculation was considerably improved in refs . where , below 40 gev , only data were used . in ref . , data were only used below 3 gev ( together with the experimental parameters of the @xmath165 and @xmath166 family states ) . in our opinion this can significantly underestimate the resulting uncertainty . in addition , in the same reference , data from @xmath0 lepton decays were extensively used ; as it is known today , this leads to higher spectral functions than in @xmath25 case,@xcite and can therefore overestimate the result . the results of these calculations are summarized in table [ tab : atau ] . [ tab : atau ] for completeness , in the second part of table [ tab : atau ] we also show purely theoretical estimates . the analysis based on qcd sum rules performed in ref . gives results which strongly depend on the choice of quark and gluon condensates . qcd sum rules are also used in ref . . in ref . the authors use a nonlocal constituent quark model for the description of the photon vacuum polarization function @xmath167 at space - like momenta and obtain @xmath168 , close to the estimates based on the experimental data . they also show that a simpler model with constituent quark masses independent of momentum is strongly dependent on the values chosen for the quark masses . for example , with @xmath169 mev and @xmath170 550 mev their result is @xmath171 , i.e. , significantly smaller than the previous estimate . they could reproduce the value @xmath172 using @xmath173 mev . in a recent analysis using the instanton liquid model the author obtains @xmath174.@xcite all these estimates somewhat undervalue the hadronic contribution and have rather large uncertainties . we updated the calculation of the leading - order contribution using the whole bulk of experimental data below 12 gev , which include old data compiled in refs . , as well as the recent datasets from the cmd-2@xcite and snd@xcite experiments in novosibirsk , and from the radiative return studies at kloe in frascati@xcite and babar at slac.@xcite the improvement is particularly visible in the channel @xmath175 , where four new independent measurements exist in the most important @xmath164 meson region : cmd-2,@xcite snd,@xcite and kloe@xcite ( see fig . [ fig : pi ] ) . our result is a_^hlo = 337.5 ( 3.7 ) 10 ^ -8 [ eq : thlo ] ( we recently presented a preliminary estimate of this value in ref . ) . the breakdown of the contributions of different energy regions as well as their relative fractions in the total leading - order contribution are given in table [ tab : at ] . the contribution of the @xmath164 meson energy range is still important , but its relative weight is smaller than in the case of the muon anomaly , 51.3% compared to about 72%.@xcite the contributions of the narrow resonances ( @xmath165 and @xmath166 families ) are included in the corresponding energy regions . it is worth noting that uncertainties of the contributions from the hadronic continuum are larger than that of the very precise @xmath176 one . the overall uncertainty is 2.5 times smaller than that of the previous data - based prediction.@xcite [ tab : at ] the hadronic higher - order @xmath177 contribution @xmath178 can be divided into two parts : @xmath179 the first one is the @xmath82 contribution of diagrams containing hadronic self - energy insertions in the photon propagators . it was determined by krause in 1996:@xcite a_^hho()= 7.6 ( 2 ) 10 ^ -8 . [ eq : thhovac ] note that navely rescaling the muon result by the factor @xmath180 ( as it was done in ref . ) leads to the totally incorrect estimate @xmath181 ( the @xmath182 value is from ref . ) ; even the sign is wrong ! the second term , also of @xmath82 , is the hadronic light - by - light contribution . similarly to the case of the muon @xmath1@xmath2@xmath3 , this term can not be directly determined via a dispersion relation approach using data ( unlike the leading - order hadronic contribution ) , and its evaluation therefore relies on specific models of low - energy hadronic interactions with electromagnetic currents . actually , very few estimates of @xmath183 exist in the literature,@xcite and all of them were obtained simply rescaling the muon results @xmath184 by a factor @xmath180 . following this very nave procedure , the @xmath185 estimate varies between @xmath186\times ( m_{\tau}^2/m_{\mu}^2)= 23(11)\times 10^{-8 } $ ] , and @xmath187\times ( m_{\tau}^2/m_{\mu}^2)= 38(7)\times 10^{-8 } $ ] , according to the values chosen for @xmath188 from refs . and , respectively . these very nave estimates fall short of what is needed . consider the function @xmath71 , the three - loop qed contribution to the @xmath1@xmath2@xmath3 of a lepton of mass @xmath18 due to light - by - light diagrams involving loops of a fermion of mass @xmath189 ( see sec . [ subsec : qed3 ] ) . the exact expression of this function , computed in ref . for arbitrary values of the mass ratio @xmath66 , is rather complicated , but series expansions were provided in the same article for the cases of physical relevance . in particular , if @xmath190 , then @xmath191 . this implies that , for example , the ( negligible ) part of @xmath192 due to diagrams with a top - quark loop can be reasonably estimated simply rescaling the corresponding part of @xmath188 by a factor @xmath180 . on the other hand , to compute the dominant contributions to @xmath192 , i.e.those induced by the light quarks , we need the opposite case : @xmath193 . in this limit , @xmath71 does not scale as @xmath194 , and a nave rescaling of @xmath184 by @xmath180 to derive @xmath195 leads to an incorrect estimate . we therefore decided to perform a parton - level estimate of @xmath192 based on the exact expression for @xmath71 using the quark masses recently proposed in ref . for the determination of @xmath184 : @xmath196 mev , @xmath197 mev , @xmath198 gev and @xmath199 gev ( note that with these values the authors of ref . obtain @xmath200 , in perfect agreement with the value in ref . see also ref . for a similar earlier determination ) . we obtain a_^hho()= 5 ( 3 ) 10 ^ -8 . [ eq : thholbl ] this value is much lower than those obtained by simple rescaling of @xmath188 by @xmath180 . the up - quark provides the dominating contribution ; the uncertainty @xmath201 allows @xmath154 to range from 70 mev up to 400 mev . further independent studies ( following the approach of ref . , for example ) would provide an important check of this result . the total hadronic contribution to the anomalous magnetic moment of the @xmath0 lepton can be immediately derived adding the values in eqs . ( [ eq : thlo ] ) , ( [ eq : thhovac ] ) and ( [ eq : thholbl ] ) , a_^had = a_^hlo+ a_^hho()+ a_^hho ( ) = 350.1 ( 4.8 ) 10 ^ -8 . [ eq : thad ] errors were added in quadrature . we can now add up all the contributions discussed in the previous sections to derive the sm prediction for @xmath21 : a_^sm = a_^qed + a_^ew + a_^hlo + a_^hho ( ) + a_^hho ( ) , [ eq : sm ] where @xmath202 ( the sum of the hadronic contributions is given in ) . adding errors in quadrature , our final result is a_^sm = 117 721 ( 5 ) 10 ^ -8 . [ eq : nsm ] the present pdg limit on the anomalous magnetic moment of the @xmath0 lepton was derived in 2004 by the delphi collaboration from @xmath22 total cross section measurements at @xmath203 between 183 and 208 gev at lep2:@xcite -0.052 < a _ < 0.013 [ eq : exp_delphi1 ] at 95% confidence level . the authors of ref . also quote their result in the form of central value and error : a _ = -0.018 ( 17 ) . [ eq : exp_delphi2 ] comparing this result with ( their difference is roughly one standard deviation ) , it is clear that the sensitivity of the best existing measurements is still more than an order of magnitude worse than needed . a reanalysis of various measurements of the cross section of the process @xmath204 , the transverse @xmath0 polarization and asymmetry at lep and sld , as well as of the decay width @xmath205 at lep and tevatron , allowed to set a stronger model - independent limit:@xcite -0.007 < a _ 0.005 . other limits on @xmath21 can be found in refs . . in this article we reviewed and updated the sm prediction of the @xmath0 lepton @xmath1@xmath2@xmath3 . updated qed and electroweak contributions were presented , together with new values of the leading - order hadronic term , based on the recent low energy @xmath4 data from babar , cmd-2 , kloe and snd , and of the hadronic light - by - light contribution . these results were confronted in sec . [ sec : sm ] to the available experimental bounds on the @xmath0 lepton anomaly . as we already mentioned in the introduction , quite generally , np associated with a scale @xmath19 is expected to modify the sm prediction of the anomalous magnetic moment of a lepton @xmath15 of mass @xmath18 by a contribution @xmath206 . therefore , given the large factor @xmath207 , the @xmath1@xmath2@xmath3 of the @xmath0 lepton is much more sensitive than the muon one to ew and np loop effects that give contributions @xmath208 , making its measurement an excellent opportunity to unveil ( or just constrain ) np effects . another interesting feature can be observed comparing the magnitude of the ew and hadronic contributions to the muon and @xmath0 lepton @xmath1@xmath2@xmath3 . the ew contribution to the @xmath1@xmath2@xmath3 of the @xmath0 is only a factor of seven smaller than the hadronic one , compared to a factor of 45 for the @xmath1@xmath2@xmath3 of the muon . also , while the ew contribution to @xmath209 is only a factor of three larger than the present uncertainty of the hadronic contribution , this factor raises to 10 for the @xmath0 lepton . if a np contribution were of the same order of magnitude as the ew one , from a purely theoretical point of view , the @xmath1@xmath2@xmath3 of the @xmath0 would provide a much cleaner test of the presence ( or absence ) of such np effects than the muon one . indeed , if this were the case , such a np contribution to the @xmath0 lepton @xmath1@xmath2@xmath3 would be much larger than the hadronic uncertainty , which is currently the limiting factor of the sm prediction . unfortunately , the very short lifetime of the @xmath0 lepton makes it very difficult to determine its anomalous magnetic moment by measuring its spin precession in the magnetic field , like in the muon @xmath1@xmath2@xmath3 experiment.@xcite instead , experiments focused on high - precision measurements of the @xmath0 lepton pair production in various high - energy processes , comparing the measured cross sections with the qed predictions.@xcite as we can see from , the sensitivity of the best existing measurements is still more than an order of magnitude worse than that required to determine @xmath21 . nonetheless , the possibility to improve such a measurement is certainly not excluded . for example , it was suggested to determine the @xmath0 lepton @xmath1 factor taking advantage of the radiation amplitude zero which occurs at the high - energy end of the lepton distribution in radiative @xmath0 decays.@xcite this method requires a very good energy resolution and could perhaps be employed at a @xmath0-charm or @xmath210 factory also benefiting from the possibility to collect very high statistics . it is not clear whether the huge data samples at @xmath210 factories will result in a corresponding gain for the limits on @xmath21 . indeed , lep measurements were rather limited by systematic uncertainties , which were of the order of 2 - 3% for the discussed processes and , until now , experiments at @xmath210 factories have not yet reached such a level of accuracy in the absolute measurements of the total cross sections . however , a search for the @xmath0 lepton electric dipole moment at belle@xcite showed that with the appropriate choice of observables , using full information about events , the improvement in sensitivity can be proportional to the square root of luminosity , i.e. , determined mainly by statistics . one can hope that this is also the case with the determination of @xmath21 . a similar method to study @xmath21 using radiative @xmath86 decays and potentially very high data samples at lhc was suggested in ref . . yet another method would use the channeling in a bent crystal similarly to the suggestion for the measurement of magnetic moments of short - living baryons.@xcite this method has been successfully tested by the e761 collaboration at fermilab , which measured the magnetic moment of the @xmath211 hyperon.@xcite in the case of the @xmath0 lepton , it was suggested to use the decay @xmath212 , which would produce polarized @xmath0 leptons.@xcite in 1991 , when this suggestion was published , the idea seemed completely unlikely . however , in the era of @xmath210 factories , when the decay @xmath212 is already observed by the belle collaboration,@xcite and the possibility of a super-@xmath210 factory is actively discussed , this is no longer a dream . even more promising could be the realization of this idea in a dedicated experiment at a hadron collider with its huge number of @xmath210 mesons produced and a more suitable geometry . we believe that a detailed feasibility study of such an experiment , as well as further attempts to improve the accuracy of the theoretical prediction for @xmath21 , are quite timely . we would like to thank m. giacomini and f.v . ignatov for many valuable comments and collaborations on topics presented in this manuscript . we are greatly indebted to a. vainshtein for communications concerning the hadronic light - by - light contribution to @xmath21 and to f. jegerlehner for many fruitful discussions . we are also grateful to k. inami for an interesting discussion on the feasibility of @xmath21 measurements at @xmath210 factories . thanks the dipartimento di fisica , universit di padova and infn , sezione di padova , where part of this work was done , for its hospitality . the work of s.e . was supported in part by the grants of rfbr 06 - 02 - 04018 and 06 - 02 - 16156 as well as by the grant of dfg gz : 436 rus 113/769/0 - 2 . is grateful to the instituto de fsica da universidade federal da bahia , brasil , for the hospitality during a visit when this manuscript was finalized . the work of m.p . was supported in part by the european community s marie curie research training networks under contracts mrtn - ct-2004 - 503369 and mrtn - ct-2006 - 035505 . all diagrams were drawn with jaxodraw.@xcite g.w . bennett _ et al . _ , phys . d * 73 * ( 2006 ) 072003 ; phys . * 92 * ( 2004 ) 161802 ; phys . * 89 * ( 2002 ) 101804 ; _ ibid . _ * 89 * ( 2002 ) 129903 ; h.n . brown _ et al . _ , phys . rev . lett . * 86 * ( 2001 ) 2227 . g. gabrielse , d. hanneke , t. kinoshita , m. nio , and b. odom , phys . rev . lett . * 97 * ( 2006 ) 030802 . k. melnikov and a. vainshtein , _ theory of the muon anomalous magnetic moment _ , springer , 2006 ; m. passera , nucl . * 155 * ( 2006 ) 365 ; m. davier and w.j . marciano , annu . nucl . part . * 54 * ( 2004 ) 115 ; m. knecht , arxiv : hep - 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this article reviews and updates the standard model prediction of the @xmath0 lepton @xmath1@xmath2@xmath3 . updated qed and electroweak contributions
are presented , together with new values of the leading - order hadronic term , based on the recent low energy @xmath4 data from babar , cmd-2 , kloe and snd , and of the hadronic light - by - light contribution .
the total prediction is confronted to the available experimental bounds on the @xmath0 lepton anomaly , and prospects for its future measurements are briefly discussed .
| 17,582 | 153 |
intensity modulated radiation therapy ( imrt ) is usually used for head and neck cancer patients because it delivers highly conformal radiation doses to the target with reduction of toxicity to normal organs , as compared with conventional radiation therapy techniques @xcite . volumetric modulated arc therapy ( vmat ) is a novel imrt technique . vmat has less mu , less treatment time , high quality planning and more efficiency than static gantry angle imrt @xcite . during vmat the linear accelerator ( linac ) control system changes the dose rate and the multi leaf collimator ( mlc ) positions while gantry is rotating around the patient . collimator angle is usually rotated in the plans of vmat to reduce radiation leakage between mlc leaves . at a zero angle , the leakage between mlc leaves accumulates during the gantry rotation and the summed leakage results in unwanted dose distributions , which can not be controlled by optimization . at different collimator angles , the unwanted doses can be controlled by dose constraints in the optimization procedure so that we can reduce the unwanted doses . the optimal collimator angle for vmat plan is thus required to be determined . there are several factors for consideration in the choice of the collimator angle of the vmat plan . among them we concentrated on the accuracy of the vmat delivery . we studied the effect of the collimator angle on the results of dosimetric verifications of the vmat plan for nasopharyngeal cancer ( npc ) . ten patients with late - stage nasopharyngeal cancer were treated with concurrent chemo radiation therapy ( ccrt ) . eight patients had stage iii disease and 2 patients had stage iv disease according to american joint committee on cancer staging system 7 . nine patients were male and 1 patient was female . one radiation oncologist delineated radiation targets and organs at risk ( oars ) . the clinical target volume ( ctv ) included the primary nasopharyngeal tumor , neck nodal region and subclinical disease . considering the setup uncertainty , margins ranging from 3 - 10 mm were added to each ctv to create a planning target volume ( ptv ) . reduced - field techniques were used for delivery of the 66 - 70 gy total dose . the treatment plan course for each patient consisted of several sub - plans . in this study , we selected the first plan with prescribed doses of 50 - 60 gy in 25 - 30 fractions to study the effect of the collimator angles on dosimetric verifications of the vmat . the radiation treatment planning system eclipse v10.0.42 ( varian medical systems , usa ) was used to generate vmat plans . the vmat ( rapidarc : varian ) plans were generated for clinac ix linear accelerator using 6 mv photons . the clinac ix is equipped with a millennium 120 mlc that has spatial resolution of 5 mm at the isocenter for the central 20 cm region and of 10 mm in the outer 2@xmath110 cm region . the maximum mlc leaf speed is 2.5 cm / s and leaf transmission is 1.8% . dosimetric leaf gap of the mlc was measured using the procedure recommended by varian medical systems . the value of the dosimetric leaf gap was 1.427 mm for 6 mv photons . for volume dose calculation , grid size of 2.5 mm , inhomogeneiy correction , the anisotropic analytical algorithm ( aaa ) v10.0.28 and the progressive resolution optimizer ( pro ) v10.0.28 were used in all plans . vmat plans for npc patients were composed of 2 coplanar full arcs in 181 - 179 degree clockwise and 179 - 181 degree counterclockwise directions . the 2 full - arc delivery was expected to achieve better target coverage and conformity than the single arc @xcite . we generated 10 vmat plans ( plan set a ) with different collimator angles for each patient . ten collimator angles for the first arc were 0 , 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 and 45 degrees . for the second arc , the collimator angle was selected explementary to the collimator angle of the first arc in the same plan , i.e. , the 2 collimator angles added up to 360 degree . the average field size of vmat plans was 22 @xmath1 22 @xmath2 . we used the same dose constraints for all the 10 vmat plans and optimization was conducted for each plan . the maximum dose rate was 600 mu / min . the target coverage was aimed to achieve a 100% volume covered by 95% of prescribed dose . optimization of each plan resulted in different fluences and different mlc motions for each plan . therefore we had 2 variables , i.e. , the collimator angle and mlc motions . to simplify the analysis we generated another set of 10 plans ( plan set b ) with the same mlc motions and different collimator angles for each patient . the mlc motions were those of the plan with 30 degree collimator angle . the plans in this set had different dose distributions and usually can not be used for treatment purposes excepting the plan with a 30 degree collimator angle . we performed patient specific quality assurances ( qa ) of 2 sets of 10 vmat plans for each patient . the measurements were made by the 2-dimensional ion chamber array matrixx ( iba dosimetry , germany ) @xcite . the matrixx has 1020 pixel ion chambers arranged in a 32@xmath132 matrix covering 24.4@xmath124.4 @xmath2 . each ion chamber has the following dimensions : 4.5 mm in diameter , 5 mm in height and a sensitive volume of 0.08 @xmath3 . the distance between chambers is 7.619 mm . the matrixx has an intrinsic buildup and backscatter thicknesses of 0.3 mm and 3.5 mm , respectively . the matrixx was placed between solid water phantoms multicube ( iba dosimetry , germany ) ( figure [ fig1 ] ) so that thickness of total buildup and backscatter was 5 cm ( figure [ fig2 ] ) . the source to surface distance was 95 cm with the measurement plane of the matrixx at the isocenter of the linac . measurement was done for each arc in the plan ; therefore , we conducted 40 measurements for each patient and the total number of measurements was 400 . the angular dependence of the matrixx was corrected after the measurements using the gantry angle sensor @xcite ( iba dosimetry , germany ) . the comparison between the calculations and the measurements were made by @xmath0-index ( 2%/2 mm , 3%/3 mm ) analysis @xcite using omnipro imrt v1.7b ( iba dosimetry , germany ) . the @xmath0-index was calculated only for the regions that have dose values above 10% @xcite in the measured area . average @xmath0-index passing rates of patient specific qas were given in table [ table1 ] . the results were averaged over the 2 arcs and 10 patients . because the 2 arcs in each vmat plan rotated almost 360 degrees and the measurement set - up is mirror symmetric about the measurement plane ( @xmath4 plane in figure [ fig2 ] ) of the matrixx detector and a vertical plane passing through the isocenter ( @xmath5 plane in figure [ fig2 ] ) the arc with collimator angle @xmath6 is symmetric to the arc with collimator angle @xmath7 . therefore we regarded the collimator angle of the second arc , which was equal to 360 minus the collimator angle of the first arc , as the collimator angle of the first arc in the analysis . .@xmath0-index passing rates of the patient specific qas as a function of the collimator angle [ cols="^,^,^,^,^ " , ] [ table1 ] maximum difference between @xmath0-index ( 2%/2 mm ) passing rates of plans in plan set a for each patient ranged from 2.83% to 14.32% and the average value was 8.44@xmath84.24% . using the 3%/3 mm criteria the maximum difference ranged from 1.46% to 5.60% and the average value was 3.67@xmath81.29% . maximum difference between @xmath0-index ( 2%/2 mm ) passing rates of plans in plan set b for each patient ranged from 3.71% to 10.44% and the average value was 7.97@xmath82.17% . using the 3%/3 mm criteria the maximum difference ranged from 1.46% to 7.23% and the average value was 4.69@xmath82.51% . 2-dimensional dose distributions calculated by the eclipse treatment planning system , dose distributions measured by the matrixx detector and @xmath0-index ( 3%/3 mm ) distributions of 1 patient plans in the plan set a for collimator angle 5 and 35 degree were shown in figure [ fig3 - 1 ] and [ fig3 - 2 ] , respectively . the passing rate for the 35 degree collimator angle was less than the passing rate for the 5 degree collimator angle . -index ( 3%/3 mm ) distributions . in the @xmath0-index distributions red color indicates the region where the 3%/3 mm criteria failed . , width=566 ] ) plan in the plan set a for collimator angle 35 . the first figure is the 2-dimensional dose distribution calculated by the eclipse treatment planning system . the second one is the dose distribution measured by the matrixx detector . the last one is the @xmath0-index ( 3%/3 mm ) distributions . in the @xmath0-index distributions red color indicates the region where the 3%/3 mm criteria failed . , width=566 ] the increase in collimator angle resulted in decreased @xmath0-index passing rates , as shown in figure [ fig4 ] . in the figure , passing rates were normalized to the value of 0 degree . black and white squares indicated @xmath0-index ( 2%/2 mm ) and ( 3%/3 mm ) passing rates , respectively , averaged over plan set a. black and white triangles indicated @xmath0-index ( 2%/2 mm ) and ( 3%/3 mm ) passing rates , respectively , averaged over plan set b. -index passing rates ( 2%/ 2 mm ) of patient specific delivery qas as a function of the collimator angle . , width=377 ] there were statistically significant negative correlations between the collimator angle and the @xmath0-index passing rates . pearson correlation coefficients for pair - wise ratings of the @xmath0-index ( 2%/2 mm ) and ( 3%/3 mm ) passing rates of plans in the plan set a and b were -0.524 and -0.412 , respectively with p - values @xmath9 0.001 . for accuracy of vmat a smaller collimator angle is better , and for mlc leakage a larger collimator angle is better , we were thus required to make a compromise . based on this study , in our hospital the collimator angles of the vmat plans for head and neck patients range between 15 - 25 degrees because the average @xmath0-index passing rates were above or near to 90% for the 2%/2 mm criteria and 97% for the 3%/3 mm criteria , as shown in the results of the passing rates for the plan set a ( table [ table1 ] ) . in other hospitals these results can be somewhat different because they have different vmat delivery systems and diffterent vmat planning systems . we think that they can find optimal collimator angles by conducting the similar measurements described in this article . although not included in this article , we performed the patient specific qas for other treatment sites with smaller field sizes that are @xmath9 13 @xmath1 13 @xmath2 . maximum difference of the passing rates for vmat plans with various collimator angles was @xmath9 1.5% . collimator angle does not affect the accuracy of the vmat delivery with small field sizes . the accuracy of radiation delivery by the linac depends on geometrical accuracies such as gantry isocentricity , collimator isocentricity and mlc position . it was reported that leaf limiting velocity , mlc position and mechanical isocenter varied at different collimator and gantry angles @xcite . this may explain the @xmath0-index passing rates dependence on the collimator angle . further study is needed to investigate the origin of the collimator angle dependence of the accuracy of vmat delivery . the quality of the plan itself is another factor for consideration in the choice of the collimator angle of the vmat plan . optimized dose distributions with the same dose constraints can vary according to the collimator angle of the vmat plan . further study is needed to evaluate the quality of vmat plans with different collimator angles . we found that the results of the patient specific qas for vmat plans using the 2-dimensional ion chamber array matrixx are dependent on the collimator angle of the vmat plans . the @xmath0-index ( 2%/2 mm ) and ( 3%/3 mm ) passing rates were negatively correlated with the collimator angle . we showed that collimator angles of the vmat plans for head and neck cancer patients range between 15 - 25 degrees resulting in the average @xmath0-index passing rates above or near to 90% for the 2%/2 mm criteria and 97% for the 3%/3 mm criteria . f. k. lee et al . , med . 39 , 44 ( 2014 ) . m w. k kan et al . , j. appl . 13 , 6 ( 2012 ) . s. a. syam kumar et al . , rep . radiother . 8 , 87 ( 2013 ) . a. holt et al . , radiat . 8 , 26 ( 2013 ) . j. alvarez - moret et al . , raiat . oncol . 5 , 110 ( 2010 ) . t. lee et al . , j. appl . 12 , 4 ( 2011 ) . x. jin et al . , med . 38 , 418 ( 2013 ) . j. herzen , et al . 52 , 1197 ( 2007 ) . l. d. wolfsberger et al . , j. appl . 11 , 1 ( 2010 ) . m. rao et al . , med . 37 , 1350 ( 2010 ) s. korreman et al . , acta oncologica . 45 , 185 ( 2009 ) m. stasi et al . 39 , 7626 ( 2012 ) . g. a. ezzell et al . , med . 36 , 5359 ( 2009 ) . c. c. ling et al . j. radiat . 72(2 ) , 575 ( 2008 ) . m. okumura et al . , phys . 55 , 3101 ( 2010 ) .
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collimator angle is usually rotated when planning volumetric modulated arc therapy ( vmat ) due to the leakage of radiation between multi - leaf collimator ( mlc ) leaves .
we studied the effect of the collimator angles on the results of dosimetric verification of the vmat plans for head and neck patients .
we studied vmat plans for 10 head and neck patients .
we made 2 sets of vmat plans for each patient .
each set was composed of 10 plans with collimator angles of 0 , 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 , 45 degrees .
plans in the first set were optimized individually and plans in the second set shared the 30 degree collimator angle optimization .
two sets of plans were verified using the 2-dimensional ion chamber array matrixx ( iba dosimetry , germany ) .
the comparison between the calculation and measurements were made by the @xmath0-index analysis .
the @xmath0-index ( 2%/2 mm ) and ( 3%/3 mm ) passing rates had negative correlations with the collimator angle .
maximum difference between @xmath0-index ( 3%/3 mm ) passing rates of different collimator angles for each patient ranged from 1.46% to 5.60% with an average of 3.67% .
there were significant differences ( maximum 5.6% ) in the passing rates of different collimator angles .
the results suggested that the accuracy of the delivered dose depends on the collimator angle .
these findings are informative when choosing a collimator angle in vmat plans .
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understanding the growth history of supermassive black holes ( smbhs ) is one of the fundamental issues in studies of galaxy formation and evolution . the intimate connection between smbhs and host galaxies is evidenced through empirical correlations between the masses of smbhs ( m@xmath8 ) and the overall properties of the host galaxy spheroids ( e.g. , magorrian et al . 1998 ; ferraresse & merritt 2000 ; gebhardt et al . the cosmic evolution of these scaling relationships has been investigated in the literature , where a tentative evolution has been reported utilizing observational approaches ( e.g. , peng et al . 2006 ; woo et al . 2006 , 2008 ; treu et al . 2007 ; merloni et al . 2010 ; bennert et al . 2010 , 2011 ; hiner et al . 2012 ; canalizo et al . 2012 ) . in order to provide better empirical constraints on the cosmic growth of smbhs and its connection to galaxy evolution , reliable m@xmath8 estimation at low and high redshifts is of paramount importance . the m@xmath8 can be determined for type 1 agn with the reverberation mapping ( rm , peterson 1993 ) method or the single - epoch ( se , wandel et al . 1999 ) method under the virial assumption : @xmath9 , where @xmath10 is the gravitational constant . the size of the broad - line region ( blr ) , @xmath11 , can be directly measured from rm analysis ( e.g. , peterson et al 2004 ; bentz et al . 2009 ; denney et al . 2010 ; barth et al . 2011b ; grier et al . 2012 ) or indirectly estimated from the monochromatic agn luminosity measured from se spectra based on the empirical blr size - luminosity relation ( kaspi et al . 2000 , 2005 ; bentz et al . 2006 , 2009 , 2013 ) . the line - of - sight velocity dispersion , @xmath12 , of blr gas can be measured either from the broad emission line width in the rms spectrum ( e.g. , peterson et al . 2004 ) obtained from multi - epoch rm data or in the se spectra ( e.g. , park et al . 2012b ) , while the virial factor , @xmath13 , is the dimensionless scale factor of order unity that depends on the geometry and kinematics of the blr . currently , an ensemble average , @xmath14 , is determined empirically under the assumption that local active and inactive galaxies have the same @xmath15 relationship ( e.g. , onken et al . 2004 ; woo et al . 2010 ; graham et al . 2011 ; park et al . 2012a ; woo et al . 2013 ) and recalibrated to correct for the systematic difference of line widths in between the se and rms spectra ( e.g. , collin et al . 2006 ; park et al . 2012b ) . the rm method has been applied to a limited sample ( @xmath16 ) to date , due to the practical difficulty of the extensive photometric and spectroscopic monitoring observations and the intrinsic difficulty of tracing the weak variability signal across very long time - lags for high - z , high - luminosity qsos . in contrast , the se method can be applied to any agn if a single spectrum is available , although this method is subject to various random and systematic uncertainties ( see , e.g. , vestergaard & peterson 2006 , collin et al . 2006 ; mcgill et al . 2008 ; shen et al . 2008 ; denney et al . 2009 , 2012 ; richards et al . 2011 ; park et al . 2012b ) . in the local universe , the se mass estimators based on the h@xmath4 line are well calibrated against the direct h@xmath4 rm results ( e.g. , mclure & jarvis 2002 ; vestergaard 2002 ; vestergaard & peterson 2006 ; collin et al . 2006 ; park et al . 2012b ) . for agns at higher redshift ( @xmath17 ) , rest - frame uv lines , i.e. , or , are frequently used for m@xmath8 estimation since they are visible in the optical wavelength range . unfortunately the kinds of accurate calibration applied to h@xmath4-based se bh masses are difficult to achieve for the mass estimators based on the and lines , since the corresponding direct rm results are very few ( see peterson et al . 2005 ; metzroth et al . 2006 ; kaspi et al . 2007 ) . instead , se m@xmath8 based on these lines can be calibrated indirectly against either the most reliable h@xmath4 rm based masses ( e.g. , vestergaard & peterson 2006 ; wang et al . 2009 ; rafiee & hall 2011a ) or the best calibrated h@xmath4 se masses ( mcgill et al . 2008 ; shen & liu 2012 , sl12 hereafter ) under the assumption that the inferred m@xmath8 is the same whichever line is used for the estimation . while several studies demonstrated the consistency between based and h@xmath4 based masses ( e.g. , mclure & dunlop 2004 ; salviander et al . 2007 ; mcgill et al . 2008 ; shen et al . 2008 ; wang et al . 2009 ; rafiee & hall 2011a ; sl12 ) , the reliability of utilizing the line is still controversial , since can be severely affected by non - virial motions , i.e. , outflows and winds , and strong absorption ( e.g , leighly & moore 2004 ; shen et al . 2008 ; richards et al . 2011 ; denney 2012 ) . other related concerns for the line include the baldwin effect , the strong blueshift or asymmetry of the line profile , broad absorption features , and the possible presence of a narrow line component ( see denney 2012 for discussions and interpretations of the issues ) . several studies have reported a poor correlation between and h@xmath4 line widths and a large scatter between and h@xmath4 based masses ( e.g. , baskin & laor 2005 ; netzer et al . 2007 ; sulentic et al . 2007 ; sl12 ; ho et al . 2012 ; trakhtenbrot & netzer 2012 ) . on the other hand , other studies have shown a consistency between them and/or suggested additional calibrations for bringing and h@xmath4 based masses further into agreement . ( e.g. , vestergaard & peterson 2006 ; kelly & bechtold 2007 ; dietrich et al . 2009 ; greene et al . 2010 ; assef et al . 2011 ; denney 2012 ) . given the practical importance of the line , which can be observed with optical spectrographs over a wide range of redshifts ( @xmath18 ) , in studying high - z agns , it is important and useful to calibrate the based m@xmath8 estimators . vestergaard & peterson 2006 ( vp06 hereafter ) have previously calibrated mass estimators against h@xmath4 rm masses , providing widely used m@xmath8 recipes . since then , however , the h@xmath4 rm sample has been expanded and many of rm masses have been updated based on various recent rm campaigns ( e.g. , bentz et al . 2009 ; denney et al . 2010 ; barth 2011a , b ; grier et al . 2012 ) . at the same time , new uv data became available for the rm sample , substantially increasing the quality and quantity of available uv spectra for the rm sample . in this paper we present the new calibration of the based m@xmath8 estimators utilizing the highest quality uv spectra and the most updated rm sample . in section 2 we describe the sample of h@xmath4 reverberation mapped agn having available uv spectra . section 3 describes our detailed spectral analysis of the emission line complex to obtain the relevant luminosity and line width measurements necessary for estimating se m@xmath8 . we provide the updated se m@xmath8 calibration in section 4 and conclude with a discussion and summary in section 5 . we adopt the following cosmological parameters to calculate distances in this work : @xmath19 km s@xmath20 mpc@xmath20 , @xmath21 , and @xmath22 . for our analysis , we start with the reverberation mapped agn sample , which is considered as a calibration base with reliable mass estimates . to date , there are @xmath23 agns , for which h@xmath4 reverberation based masses are available ( peterson et al . 2004 ; denney et al . 2006 ; bentz et al . 2009 ; denney et al . 2010 ; barth et al . 2011a , b ; grier et al . 2012 ) . of those @xmath23 objects , we selected @xmath24 agns whose archival uv spectra are available from _ international ultraviolet explorer _ ( _ iue _ ) or _ hubble space telescope _ ( _ hst _ ) data archives . first , we collected all available uv spectra covering the spectral region from the public archives . if there were multiple spectra for a given individual object , either multiple epochs taken with the same instrument or from multiple instruments , we combined the spectra for each instrument by using a standard weighted average method to get the better signal - to - noise ( s / n ) ratio . at the same time , we tried to keep contemporaneity as far as possible . then we selected the best quality spectra for each object based on visual inspection and by setting a limiting s / n ratio of @xmath25 per pixel , which was measured in an emission - line free region of the continuum near 1450 or 1700 ( see denney et al . 2013 for the s / n related issues ) . among these @xmath24 objects , we excluded four agns ( i.e. , ngc 3227 , ngc 4151 , pg 1411 + 442 , pg 1700 + 518 ) because they are severely contaminated with absorption features . other 9 agns ( i.e. , mrk 79 , mrk 110 , mrk 142 , mrk 1501 , ngc 4253 , ngc 4748 , ngc 6814 , pg 0844 + 349 , pg 1617 + 175 ) were also excluded due to the low quality and unreliability of uv spectra . thus , our sample contains @xmath26 agns . table [ tab : optdata ] lists the agns in the sample and their properties . note that we adopt the updated virial factor , @xmath27 ( park et al . 2012a ; woo et al . 2013 ) . compared to the previous sample of vp06 , one object , mrk 290 , is newly included and seven objects ( i.e. , 3c 120 , mrk 335 , ngc 3516 , ngc 4051 , ngc 4593 , ngc 5548 , pg 2130 + 099 ) have updated reverberation m@xmath8 ( denney et al . 2006 , bentz et al . 2009 ; denney et al . 2010 ; grier et al . one object , pg 0804 + 761 , that was excluded by vp06 , is included since it has a new high - quality uv spectrum from the _ cosmic origins spectrograph _ ( cos ) aboard _ contrary to vp06 , ngc 4151 is omitted in this work due to the strong absorption features near the line center ( see section 3 ) . in summary , @xmath28 agns have recent high - quality uv spectra from _ hst _ cos , @xmath29 km s@xmath30 ) by smoothing and re - binning by 7-pixels . ] compared to vp06 . for the remaining objects , uv spectra were obtained from the _ space telescope imaging spectrograph _ ( stis ) aboard _ hst _ for one object , from the _ faint object spectrograph _ ( fos ) aboard _ hst _ for eight objects , and from _ short - wavelength prime _ ( swp ) camera aboard _ iue _ for four objects as listed in table [ tab : uvdata ] . we corrected the galactic extinction using the values of @xmath31 from the recalibration of schlafly & finkbeiner ( 2011 ) listed in the nasa / ipac extragalactic database ( ned ) and the reddening curve of fitzpatrick ( 1999 ) . in order to calibrate the m@xmath8 estimator , we measured the line width of and the continuum luminosity at 1350 , following the multi - component fitting procedure developed by park et al . ( 2012b ) with a modification for the region . we first fitted a single power - law continuum using the typical emission - line - free windows in both sides of ( i.e. , @xmath32 or @xmath33 and @xmath34 ) , which were slightly adjusted for each spectrum to avoid the contaminating absorption and emission features . we did not subtract the iron emission , since it is generally too weak to constrain at least in our data sets , although we indeed tested the pseudocontinuum model by including the uv template from vestergaard & wilkes ( 2001 ) . after subtracting the best - fit power - law continuum , we simultaneously fitted the complex region with the multi - component model consisting of a gaussian function for the ] @xmath01486 , a gaussian function for the @xmath01531whenever clearly seen , a gaussian function + a sixth - order gauss - hermite series for the @xmath01549 , two gaussian functions for the @xmath01640 , and a gaussian function for the ] @xmath01663 . note that we fitted the 1600 feature , which is contaminating the red wing of , with a broad component ( cf . appendix a. in fine et al . 2010 ; marziani et al . 2010 ) . in the fitting process , the centers of , ] , ] , and emission line components were fixed to be laboratory wavelengths . we suppressed some components in , ] , ] , and lines based on empirical tests with and without such components . narrow absorption features were excluded automatically in the calculation of @xmath35 statistics by masking out the 3 sigma outliers below the smoothed spectrum ( cf . shen et al . 2011 ) . strong broad absorption features around the line center were also masked out manually by setting exclusion windows from visual inspection . although it is still controversial whether or not to remove a narrow emission - line component from before measuring the width , we use the full line profile of , i.e. , without subtracting a narrow emission - line component , in order to be consistent with other studies ( vp06 , shen et al . 2011 , assef et al . 2011 ; ho et al . we measured the continuum luminosities at @xmath36 and @xmath37 from the power - law model and measured the line widths ( fwhm and @xmath38 ) from the best - fit model ( i.e. , a gaussian function + a sixth - order gauss - hermite series ) as shown in figure [ fig : all_spec_hst ] . the measured line widths were corrected for the instrumental resolution following the standard practice by subtracting the instrumental resolution from the measured velocity in quadrature . in figure 2 , we explicitly show spectra and best - fit models for the objects showing absorption features at the center of . note that the fitting results are uncertain for these objects , in particular ngc 4051 ( see section 4.1 ) . to assess measurement uncertainties of the line width and continuum luminosity , we applied the monte carlo flux randomization method used by park et al . ( 2012b ; see also shen et al . 2011 ) . using the @xmath39 realizations of resampled spectra made by randomly scattering flux values based on the flux errors , we fitted and measured the line width and continuum flux , and adopted the standard deviation of the distribution as the measurement uncertainties for individual objects as listed in table [ tab : uvdata ] . and @xmath37 , respectively . the bottom panel compares the fwhm and line dispersion ( @xmath38 ) , where both were measured from the full line profile . the ratio between fwhm and @xmath38 is close to one ( dashed line ) , indicating the line profile is more peaky than gaussian ( dotted line ) . objects with new uv spectra from the _ hst _ cos is denoted with red filled circles . average offset and 1@xmath38 scatter are given in the lower right corner in each panel . , title="fig:",scaledwidth=45.0% ] + and @xmath37 , respectively . the bottom panel compares the fwhm and line dispersion ( @xmath38 ) , where both were measured from the full line profile . the ratio between fwhm and @xmath38 is close to one ( dashed line ) , indicating the line profile is more peaky than gaussian ( dotted line ) . objects with new uv spectra from the _ hst _ cos is denoted with red filled circles . average offset and 1@xmath38 scatter are given in the lower right corner in each panel . , title="fig:",scaledwidth=45.0% ] figure [ fig : uv_measurements ] presents the continuum luminosities measured at @xmath36 and @xmath37 , respectively , which are commonly adopted for the m@xmath8 estimator . since they are almost identical , we choose to use @xmath3 for the mass estimator . the comparison between fwhm and @xmath38 of is plotted in the bottom panel of figure [ fig : uv_measurements ] . it shows on average , a one - to - one relation between fwhm and @xmath38 , indicating that the profile is more peaky than a gaussian profile , although there is large scatter . scatter are given in the lower right corner in each panel . , title="fig:",scaledwidth=43.0% ] + scatter are given in the lower right corner in each panel . , title="fig:",scaledwidth=43.0% ] + scatter are given in the lower right corner in each panel . , title="fig:",scaledwidth=43.0% ] we compare our measurements with those in vp06 in figure [ fig : comapre_uv_measure_vp06 ] , using the common sample ( 23 out of 27 objects given in their table 2 , except for mrk 79 , mrk 110 , ngc 4151 , and pg 1617 + 175 ) . since there are multiple measurements in vp06 , we here show weighted average values of vp06 measurements for the purpose of comparison . for the comparison of @xmath3 , there is 0.24 dex scatter , which may stem from a combination of the differences , e.g. , adopted spectra and the galactic extinction correction , between our study and vp06 . we used the combined single spectra with the best quality while vp06 used all available se spectra for each object . especially for the objects with the new _ hst _ cos spectra ( red filled circles ) observed in different epochs , there could be an intrinsic difference due to the variability . in the case of the galactic extinction correction , we utilized the recent values of @xmath31 listed in the ned taken from the schlafly & finkbeiner ( 2011 ) recalibration , while vp06 used the original values from schlegel et al . ( 1998 ) . when comparing our @xmath40 measurements to those of vp06 , a slight positive systematic trend seems to be present ( middle panel of figure [ fig : comapre_uv_measure_vp06 ] ) . the most likely origin of this trend is the difference in the adopted line width measurement methods between vp06 and this work . based on the investigation by fine et al . ( 2010 ) , we modeled the complex region with multiple components and measured line dispersion from the de - blended line model profile , whereas vp06 measured line dispersion from the data without functional fits by limiting the line profile range to @xmath41 km s@xmath30 of the line center , regardless of the intrinsic line width of each civ profile . thus , the line dispersion measured by vp06 will be biased if line wings are extended much further than the fixed line limit ( i.e. , underestimation ) or the wings are smaller than the fixed line limit ( i.e. , overestimation by including other features ) . we avoid these biases by de - blending the line from other lines using the multi - component fitting analysis and measuring the line widths from the best - fit models . since the line dispersion is more sensitive to the line wings than the line core , the decomposition and thus recovery of the line wing profile from contaminating lines is essential . the bottom panel of figure [ fig : comapre_uv_measure_vp06 ] compares fwhm@xmath42 , indicating on average consistency between vp06 and this work , except for a few outliers . this is because fwhm is less sensitive to the line wings than @xmath43 , hence the difference in the measuring method does not generate significant difference in measurements . note that although fwhm is sensitive to the narrow - line component , both vp06 and our study used the full line profile without decomposing the broad and narrow components . instead , another source of discrepancy comes from the fact that vp06 measured the line width directly from the data while in this study the best - fit models were used for line width measurements . thus , there may be object - specific differences depending on how the absorptions above the `` half - maximum '' flux level were dealt with by vp06 , and how well the functional fits represent the peak of the profile in our study . -based se bh masses ( top ) and fwhm - based se bh masses ( bottom ) to the h@xmath4 rm - based bh masses . the new sample from the recent rm results is marked with a blue open square . the regressed parameters ( @xmath45 ) with the uncertainty estimates are given in the upper left part in each panel . , title="fig:",scaledwidth=45.0% ] + -based se bh masses ( top ) and fwhm - based se bh masses ( bottom ) to the h@xmath4 rm - based bh masses . the new sample from the recent rm results is marked with a blue open square . the regressed parameters ( @xmath45 ) with the uncertainty estimates are given in the upper left part in each panel . , title="fig:",scaledwidth=45.0% ] by adopting h@xmath4 rm - based masses as true m@xmath8 ( see table 1 ) , we calibrate the mass estimators by fitting @xmath46 ~=~ \alpha & + & \beta ~ \log\left(\frac{l_{1350}}{10^{44}~\rm erg~s^{-1}}\right ) \nonumber\\ & ~+~ & \gamma ~ \log \left[\frac{\vardelta v(\textrm{{\ion{c}{4}}})}{1000~ \rm km~s^{-1}}\right]~,\end{aligned}\ ] ] where @xmath3 is the monochromatic continuum luminosity at 1350 and @xmath47 is the line width of , either fwhm@xmath1 or @xmath2 . we regress equation [ eq : calibration ] to determine the free parameters ( @xmath45 ) using the ` fitexy ` estimator implemented in park et al . ( 2012a ) . note that this approach is different from that used by vp06 , who adopted the luminosity slope from the size - luminosity relation and fixed the velocity slope to @xmath48 . instead , this method is consistent with the recent approaches described by wang et al . ( 2009 ) , rafiee & hall ( 2011b ) , and shen & liu ( 2012 ) . because a non - linear dependence is often observed between the line widths of h@xmath4 and line ( especially based on the fwhm ; see denney 2012 for a likely physical interpretation ) , leaving @xmath49 , @xmath4 , and @xmath50 as free parameters arguably results in a better statistical regression by accommodating the possible covariance between luminosity and line width . in figure [ fig : recali3 ] , we present the final best - fit calibration results for -based mass estimators by directly comparing the se masses with the h@xmath4 rm - based masses , using equation [ eq : calibration ] . the regression results with various conditions and the previous calibrations from the literature are listed in table [ tab : calibration ] . we adopt the regression results without ngc 4051 , which is subject to the largest measurement errors among our sample since modeling the line of this object is highly uncertain due to the strong absorption at the center ( see fig . [ fig : obj_strabs ] ) . in addition , the variability of ngc 4051 is expected to have a large amplitude since it is the lowest - luminosity object in our sample . thus , ngc 4051 can add large scatter to the regression and potentially skew the slope because there is only a single object at the low - mass regime . thus , excluding this object possibly will lead to less biased results in terms of sample selection and measurement uncertainties . we will present the results without ngc 4051 hereafter unless explicitly stated . the slope of the velocity term , when it is treated as a free parameter , is closer to the virial assumption ( i.e. , 2 ) for the @xmath43-based estimator , i.e. , @xmath51 ( @xmath52 if ngc 4051 is included ) than for the fwhm@xmath42-based estimator , i.e. , @xmath53 ( @xmath54 if ngc 4051 is included ) . this reinforces the use of @xmath38 for characterizing the line width of , as suggested by denney ( 2012 , 2013 ; see also peterson et al . 2004 and park et al . 2012b for the case of h@xmath4 ) . the slope of the luminosity term ( i.e. , @xmath55 for @xmath38 ; @xmath56 for fwhm ) is almost consistent to that of photoionization expectation ( i.e. , 0.5 ; bentz et al . 2006 ) within the uncertainty . this may indicates that asynchronism between h@xmath4 and measurements does not introduce a significant overall difference for our high - luminosity , high - quality calibration sample . in this calibration , we treat @xmath4 and @xmath50 as free parameters in addition to @xmath49 . letting @xmath4 be a free parameter is required to reduce luminosity dependent systematics since we are dealing with non - contemporaneous h@xmath4 and measurements , which is expected to be not necessarily linear . in addition , it is currently questionable to directly adopt the size - luminosity relation ( e.g. , kaspi et al . 2007 ) for the estimator since it is based on such a small sample . relaxing the constraint of @xmath57 for fwhm@xmath42 can be corroborated by the investigation by denney ( 2012 ) , which shows that there are severe biases in measuring fwhm@xmath42 due to the non - variable component and dependence on the line shape . these systematic uncertainties may be properly calibrated out by taking @xmath50 as a free parameter . in the case of @xmath43 , however , a similar systematic does not seem to be present for @xmath43 ( see denney 2012 ) . even if we allow @xmath50 to be free , the regression slope for @xmath43 is consistent to the virial expectation ( i.e. , 2 ) within @xmath58 uncertainty , thus we opt to fixing @xmath50 to a value of 2 , avoiding systematic uncertainties due to small number statistics or sample - specific systematics . thus , here we provide the best estimator for the -based m@xmath8 ( see also , fig . [ fig : recali3 ] ) as @xmath59 & ~=~ & ( 6.71\pm0.07 ) \nonumber\\ & ~+~ & ( 0.50\pm0.07 ) ~ \log\left(\frac{l_{1350}}{10^{44}~\rm erg~s^{-1}}\right ) \nonumber\\ & ~+~ & 2 ~ \log \left[\frac{\sigma ( \textrm{{\ion{c}{4}}})}{1000~ \rm km~s^{-1}}\right]\end{aligned}\ ] ] with the statistical scatter against rm masses of @xmath60 dex and @xmath61 & ~=~ & ( 7.48\pm0.24 ) \nonumber\\ & ~+~ & ( 0.52\pm0.09 ) ~ \log\left(\frac{l_{1350}}{10^{44}~\rm erg~s^{-1}}\right ) \nonumber\\ & ~+~ & ( 0.56\pm0.48 ) ~ \log \left[\frac{\rm fwhm ( \textrm{{\ion{c}{4}}})}{1000~ \rm km~s^{-1}}\right]\end{aligned}\ ] ] with the statistical scatter against rm masses of @xmath62 dex . apart from the interpretation of values of zero point and slopes , it is worth noting that these estimators are the best calibrated ones to reproduce h@xmath4 rm masses as closely as possible for the current sample and data sets . -based ( top ) and fwhm - based ( middle ) m@xmath8 respectively calculated using the estimators in this study and the estimators of vp06 . in the bottom panel , m@xmath8 calculated with the fwhm - based estimator from sl12 is compared to our mass estimates . average offset and 1@xmath38 scatter are given in the lower right corner in each panel . , title="fig:",scaledwidth=43.0% ] + -based ( top ) and fwhm - based ( middle ) m@xmath8 respectively calculated using the estimators in this study and the estimators of vp06 . in the bottom panel , m@xmath8 calculated with the fwhm - based estimator from sl12 is compared to our mass estimates . average offset and 1@xmath38 scatter are given in the lower right corner in each panel . , title="fig:",scaledwidth=43.0% ] + -based ( top ) and fwhm - based ( middle ) m@xmath8 respectively calculated using the estimators in this study and the estimators of vp06 . in the bottom panel , m@xmath8 calculated with the fwhm - based estimator from sl12 is compared to our mass estimates . average offset and 1@xmath38 scatter are given in the lower right corner in each panel . , title="fig:",scaledwidth=43.0% ] in figure [ fig : comapre_vp06_sl12 ] , we present the systematic difference of m@xmath8 estimates based on our new estimators ( equations [ eq : final_cal_sigma ] and [ eq : final_cal_fwhm ] ) compared to the previous estimators from vp06 and sl12 , respectively , using line width and l@xmath63 measurements . the @xmath43-based m@xmath8 estimates show almost consistent results to vp06 with a slight offset of @xmath64 , which is expected from a difference in the adopted values of the virial factor ( i.e. , @xmath27 here versus @xmath65 in vp06 ) . in contrast , the comparison of fwhm@xmath42-based masses , respectively estimated with our recipe and with that of vp06 , shows large scatter and a systematic trend that the vp06 recipe underestimates m@xmath8 in the low - mass regime and overestimates m@xmath8 in the high - mass regime , compared to our recipe . the bottom panel of figure [ fig : comapre_vp06_sl12 ] shows that the sl12 recipe systematically overestimates m@xmath8 over the whole dynamic range of the sample ( i.e. , @xmath66 @xmath67 ) . this is understandable because sl12 used the fwhm@xmath68-based m@xmath8 in vp06 as a fiducial one and recalibrate the estimator using their high - mass ( @xmath69 @xmath67 ) qsos . thus , the calibration performed by sl12 in their limited dynamical range inherits the overestimation behavior of the vp06 recipe with respect to our recipe , and propagates it into the low mass regime with larger effect . in addition , sl12 subtracted a narrow component before measuring fwhm of , leading to an overestimated fwhm@xmath42 , compared to vp06 and our methods . we note that a large dynamic range is necessary for better calibration and investigation of the biases , as pointed out by sl12 . in order to explicitly compare the calibration methods used in here and vp06 , we regress equation [ eq : calibration ] by fixing @xmath4 and/or @xmath50 with the adopted values in vp06 as listed in table [ tab : calibration ] . for the @xmath38-based mass estimator , we obtain almost same calibration result ( @xmath70 ) to that of vp06 ( @xmath71 ) using the sample including ngc 4051 . when ngc 4051 is exclude , the zero points reduces slightly ( @xmath72 ) and intrinsic scatter becomes smaller . it is interesting to see the consistency of the @xmath38-based calibration between our study and vp06 , despite the systematic bias in @xmath40 measurements of vp06 as shown in section [ compare_vp06 ] . we interpret this as follows . although there is a bias in the vp06 measurement method for @xmath38 , due to their choice of line limits ( i.e. , @xmath41 km s@xmath30 ) , their @xmath38-based m@xmath8 measurements serendipitously scatter evenly below and above the central point of the mass scale of the sample , consequently resulting in a similar zero point regardless of the bias in @xmath38 measurements . in the end the calibrations are very similar , however , the intrinsic scatter of our calibration is smaller than that of vp06 , which demonstrates a general increase in accuracy of our @xmath38-based masses over those of vp06 , advocating for our @xmath38 measurement prescription . to demonstrate the effect of our new estimators on m@xmath8 studies , we present in figure [ fig : sdss_dr7_quasar ] the m@xmath8 distribution of the sdss qso sample as a function of redshift , based on various mass estimators . these masses are calculated using the fwhm@xmath42 measurements from shen et al . ( 2011 ) , who provides only fwhm measurements using sdss dr7 spectra . note that fwhm@xmath42-based m@xmath8 determined with our new estimator is on average smaller by @xmath73 dex than that calculated with the previous estimator by vp06 , since the vp06 recipe tends to overestimate m@xmath8 in the high - mass regime as explained in section 4.2 . in contrast , there is a smooth transition between -based masses estimated from the recipe of wang et al . ( 2009 ) and -based masses from our new calibration since both estimators are based on the same calibration scheme ( see section [ sec : calibration ] ) . kelly & shen ( 2013 ) derived a predicted maximum m@xmath8 of broad - line qsos as a function of redshift ( their figure 7 ) , showing a slight trend that the maximum m@xmath8 was larger at higher redshift . however , this subtle trend may simply be a result of systematic overestimation of masses at high redshift because their mass determination was based on the vp06 recipe . adopting our new mass estimator may eliminate such a trend . we investigated the calibration of m@xmath8 estimators based on the updated sample of @xmath26 agns , for which both h@xmath4 reverberation masses and uv archival spectra were available . the sample of agns with rm masses as well as the uv spectra including have been expanded and updated since the calibrations performed by vp06 ; it is therefore useful to revisit the calibration of the -based mass estimators to provide the most consistent virial m@xmath8 estimates using the emission line . major differences of the calibration method between vp06 and the current study is twofold . first , we derived line widths ( i.e. , line dispersion and fwhm ) from the spectral fits by performing multi - component fitting on the complex region to accurately de - blend from other contaminating lines while vp06 measured the line width of civ directly from the spectra . when `` applying '' a se scaling relationship to calculate m@xmath8 , it is important to use the same fitting and line width measurement prescriptions that were used in `` calibrating '' the scaling relationship because significant systematic differences can arise in m@xmath8estimates if different analysis and measurement techniques are utilized ( e.g. , assef et al . 2011 , denney 2012 , sl12 , park et al . second , we treated the slope parameters ( i.e. , @xmath4 and @xmath50 ) in the virial m@xmath8 equation ( i.e. , eq . [ eq : calibration ] ) as free parameters as in wang et al . ( 2009 ) , which is particularly important for fwhm . we provided the best - fit calibrations for both @xmath43- and fwhm@xmath42-based mass estimators . while we presented a consistent estimator for the @xmath38-based masses to that of vp06 , we obtained significantly different m@xmath8 estimator for the fwhm - based masses , presumably due to relaxing the constraint of the virial expectation ( i.e. , @xmath74 ) to mitigate the fwhm - dependent biases . we generally recommend to use the @xmath38-based mass estimator if the @xmath38 measurement is available , as it shows the better consistency with the virial relation , and @xmath38-based masses show a smaller scatter than the fwhm - based masses when compared to h@xmath4 rm - based masses . using @xmath43-based estimator is also preferred by denney ( 2012 ) , who showed that the @xmath43 measured from mean spectra is the better tracer of the broad - line velocity field than the fwhm@xmath42 since fwhm of is much more affected by the non - variable core component . compared to the previously calibrated fwhm@xmath42 estimator by vp06 , our new estimator shows a systematic trend as a function of mass . the vp06 recipe overestimates m@xmath8 in the high - mass regime ( i.e. , @xmath75 ) while it underestimate m@xmath8 in the low - mass regime ( i.e. , @xmath76 ) , compared to masses based on our new estimator . this systematic discrepancy is due to a combination of effects , including difference in the rm sample and updated rm masses , newly available uv spectra , emission - line fitting method , and calibration method . for the sdss quasar sample ( shen et al . 2011 ) , we find that m@xmath8 estimates based on our new estimator are systematically smaller by @xmath73 dex than those based on the previous recipe of vp06 . one of the main differences in calibrating the fwhm@xmath42-based mass estimator is that we fit the exponent of velocity ( @xmath77 ) as in eq . 3 , instead of adopting @xmath78 as in vp06 . this provides effectively the same effect as adopting a varying virial factor . if a constant virial factor is used for mass determination , then fwhm - based masses will show systematic difference compared to @xmath38-based masses , since the fwhm/@xmath38 ratio has a broad distribution , while the fiducial rm masses are derived with @xmath38 measurements from rms spectra . to resolve this issue , collin et al . ( 2006 ) , for example , introduced varying virial factors for the fwhm@xmath68-based mass estimator depending on the range of the line widths . in our case , relaxing the virial ( fwhm@xmath79 ) requirement in calibrating fwhm@xmath42-based masses against @xmath80-based rm masses provides virtually the same effect as adopting a varying virial factor , thus resulting in better consistency with @xmath38-based masses . it also mitigates the bias caused by the contamination of the non - variable emission component , where fwhm@xmath42-based masses in objects with peaky ( boxy ) profiles are under- ( over- ) estimated with previous fwhm@xmath42-based mass estimators . the calibration of mass estimators provided in this study still suffers from a sample bias as in the case of vp06 . the incompleteness or lack of low - mass objects ( i.e. , @xmath81 ) in the current sample will be resolved when new _ hst _ stis observations become available for six low - mass reverberation - mapped agns ( go-12922 , pi : woo ) . however , the extrapolation of this calibration to high - luminosity , higher - redshift agns more similar to the sdss sample can only be realized with the extension of the rm sample to this regime an endeavor that we strongly advocate . apart from the calibration analysis performed in this and previous studies , several schemes to correct for the -based masses have been suggested in the literature to reduce the large scatter between the -based masses and the h@xmath4-based masses . for example , assef et al . ( 2011 ) suggested a prescription to reduce mass residuals using the ratio of the rest - frame uv to optical continuum luminosities based on a sample of 12 lensed quasars . shen & liu ( 2012 ) reported a poor correlation with large scatter between fwhm@xmath68 and fwhm@xmath1 using a sample of 60 high - luminosity qsos , showing that some part of the scatter correlated with the blueshift of with respect to h@xmath4 . they suggested a correction for the fwhm of and ] lines as a function of the blueshift . recently , denney ( 2012 ) showed that the line profile consists of both non - variable and variable components based on the sample of seven agns with reverberation data , and concluded that this non - variable component is a main source of the large scatter of the se m@xmath8 . they provided an empirical correction for the fwhm - based mass depending on the line shape as parameterized as the ratio of fwhm to the line dispersion . since the line region is more likely to be affected by non - virial motions such as outflows and winds than the lower ionization line region , such as h@xmath4 ( e.g. , shen et al . 2008 ; richards et al . 2011 ) , aforementioned corrections are also important and worth investigating further with a larger sample with enlarged dynamic range . in general , correcting for possible systematic biases and providing accurate m@xmath8 estimates is crucial for studies of the cosmic evolution of bh population , particularly at high - redshifts ( e.g. , fine et al . 2006 ; shen et al . 2008 , 2011 , shen & kelly 2012 ; kelly & shen 2013 ) . thus , it is important to ensure a reliable calibration at the high - mass end ( @xmath82 @xmath67 ) since the mass estimators are most applicable to high - mass agns at high - redshift utilizing optical spectroscopic data from large agn surveys . note that the current rm sample used for calibrating mass estimators still suffers from the lack of high - luminosity agns , suggesting that the rm sample may not best represent the high - luminosity qsos at high - redshifts , i.e. , sdss qsos . thus , obtaining direct reverberation mapping results for high - mass qsos will be even more useful ( see kaspi et al . 2007 for tentative results ) , despite the practical observational challenges . such measurements will be used to better determine a reliable size - luminosity relation and to directly investigate non - varying component of . we thank the anonymous referee for constructive suggestions and charles danforth for helpful comments for the _ hst _ cos data co - addition . this work was supported by the national research foundation of korea ( nrf ) grant funded by the korea government ( mest ) ( no . 2012 - 006087 ) . has received funding from the people programme ( marie curie actions ) of the european union s seventh framework programme fp7/2007 - 2013/ under rea grant agreement no . 300553 . lcccccc 3c120 & @xmath83 & @xmath84 & @xmath85 & @xmath86 & @xmath87 & 6 + 3c390.3 & @xmath88 & @xmath89 & @xmath90 & @xmath91 & @xmath92 & 1 + ark120 & @xmath93 & @xmath94 & @xmath95 & @xmath96 & @xmath97 & 1 + fairall9 & @xmath98 & @xmath99 & @xmath100 & @xmath101 & @xmath102 & 1 + mrk279 & @xmath103 & @xmath104 & @xmath105 & @xmath106 & @xmath107 & 1 + mrk290 & @xmath108 & @xmath109 & @xmath110 & @xmath111 & @xmath112 & 4 + mrk335 & @xmath113 & @xmath114 & @xmath115 & @xmath116 & @xmath117 & 6 + mrk509 & @xmath118 & @xmath119 & @xmath120 & @xmath121 & @xmath122 & 1 + mrk590 & @xmath123 & @xmath124 & @xmath125 & @xmath126 & @xmath127 & 1 + mrk817 & @xmath128 & @xmath129 & @xmath130 & @xmath131 & @xmath132 & 1 + ngc3516 & @xmath133 & @xmath134 & @xmath135 & @xmath136 & @xmath137 & 4 + ngc3783 & @xmath138 & @xmath139 & @xmath140 & @xmath141 & @xmath142 & 1 + ngc4051 & @xmath143 & @xmath144 & @xmath145 & @xmath146 & @xmath147 & 4 + ngc4593 & @xmath148 & @xmath149 & @xmath150 & @xmath151 & @xmath152 & 2 + ngc5548 & @xmath153 & @xmath154 & @xmath155 & @xmath156 & @xmath157 & 3 , 5 + ngc7469 & @xmath158 & @xmath159 & @xmath160 & @xmath161 & @xmath162 & 1 + pg0026 + 129 & @xmath163 & @xmath164 & @xmath165 & @xmath166 & @xmath167 & 1 + pg0052 + 251 & @xmath168 & @xmath169 & @xmath170 & @xmath171 & @xmath172 & 1 + pg0804 + 761 & @xmath173 & @xmath174 & @xmath175 & @xmath176 & @xmath177 & 1 + pg0953 + 414 & @xmath178 & @xmath179 & @xmath180 & @xmath181 & @xmath182 & 1 + pg1226 + 023 & @xmath183 & @xmath184 & @xmath185 & @xmath186 & @xmath187 & 1 + pg1229 + 204 & @xmath188 & @xmath189 & @xmath190 & @xmath191 & @xmath192 & 1 + pg1307 + 085 & @xmath168 & @xmath193 & @xmath194 & @xmath195 & @xmath196 & 1 + pg1426 + 015 & @xmath197 & @xmath198 & @xmath199 & @xmath200 & @xmath201 & 1 + pg1613 + 658 & @xmath202 & @xmath203 & @xmath204 & @xmath205 & @xmath206 & 1 + pg2130 + 099 & @xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & 6 [ tab : optdata ] lclcccccccc 3c120 & _ iue_/swp & 1994 - 02 - 19,27;1994 - 03 - 11 & @xmath212 & 0.263 & @xmath213 & @xmath214 & @xmath215 & @xmath216 & @xmath217 & + 3c390.3 & _ hst_/fos & 1996 - 03 - 31 & @xmath218 & 0.063 & @xmath219 & @xmath220 & @xmath221 & @xmath222 & @xmath223 & + ark120 & _ hst_/fos & 1995 - 07 - 29 & @xmath224 & 0.114 & @xmath225 & @xmath226 & @xmath227 & @xmath228 & @xmath229 & + fairall9 & _ hst_/fos & 1993 - 01 - 22 & @xmath230 & 0.023 & @xmath231 & @xmath232 & @xmath233 & @xmath234 & @xmath235 & + mrk279 & _ hst_/cos & 2011 - 06 - 27 & @xmath236 & 0.014 & @xmath237 & @xmath238 & @xmath239 & @xmath240 & @xmath241 & + mrk290 & _ hst_/cos & 2009 - 10 - 28 & @xmath230 & 0.014 & @xmath242 & @xmath243 & @xmath244 & @xmath245 & @xmath246 & + mrk335 & _ hst_/cos & 2009 - 10 - 31;2010 - 02 - 08 & @xmath247 & 0.032 & @xmath248 & @xmath249 & @xmath250 & @xmath251 & @xmath252 & + mrk509 & _ hst_/cos & 2009 - 12 - 10,11 & @xmath253 & 0.051 & @xmath254 & @xmath255 & @xmath256 & @xmath257 & @xmath258 & + mrk590 & _ iue_/swp & 1991 - 01 - 14 & @xmath224 & 0.033 & @xmath259 & @xmath260 & @xmath261 & @xmath262 & @xmath263 & + mrk817 & _ hst_/cos & 2009 - 08 - 04;2009 - 12 - 28 & @xmath264 & 0.006 & @xmath265 & @xmath266 & @xmath267 & @xmath268 & @xmath269 & + ngc3516 & _ hst_/cos & 2010 - 10 - 04;2011 - 01 - 22 & @xmath270 & 0.038 & @xmath271 & @xmath272 & @xmath273 & @xmath274 & @xmath275 & abs + ngc3783 & _ hst_/cos & 2011 - 05 - 26 & @xmath247 & 0.105 & @xmath276 & @xmath277 & @xmath278 & @xmath279 & @xmath280 & + ngc4051 & _ hst_/cos & 2009 - 12 - 11 & @xmath281 & 0.011 & @xmath282 & @xmath283 & @xmath284 & @xmath285 & @xmath286 & abs + ngc4593 & _ hst_/stis & 2002 - 06 - 23,24 & @xmath287 & 0.022 & @xmath288 & @xmath289 & @xmath290 & @xmath291 & @xmath292 & + ngc5548 & _ hst_/cos & 2011 - 06 - 16,17 & @xmath293 & 0.018 & @xmath294 & @xmath295 & @xmath296 & @xmath297 & @xmath298 & abs + ngc7469 & _ hst_/cos & 2010 - 10 - 16 & @xmath299 & 0.061 & @xmath300 & @xmath301 & @xmath302 & @xmath303 & @xmath304 & + pg0026 + 129 & _ hst_/fos & 1994 - 11 - 27 & @xmath305 & 0.063 & @xmath306 & @xmath307 & @xmath308 & @xmath309 & @xmath310 & + pg0052 + 251 & _ hst_/fos & 1993 - 07 - 22 & @xmath311 & 0.042 & @xmath312 & @xmath313 & @xmath314 & @xmath315 & @xmath316 & + pg0804 + 761 & _ hst_/cos & 2010 - 06 - 12 & @xmath317 & 0.031 & @xmath318 & @xmath319 & @xmath320 & @xmath321 & @xmath316 & + pg0953 + 414 & _ hst_/fos & 1991 - 06 - 18 & @xmath218 & 0.012 & @xmath322 & @xmath323 & @xmath324 & @xmath325 & @xmath325 & + pg1226 + 023 & _ hst_/fos & 1991 - 01 - 14,15 & @xmath326 & 0.018 & @xmath327 & @xmath328 & @xmath329 & @xmath330 & @xmath331 & + pg1229 + 204 & _ iue_/swp & 1982 - 05 - 01,02 & @xmath332 & 0.024 & @xmath333 & @xmath334 & @xmath335 & @xmath336 & @xmath337 & + pg1307 + 085 & _ hst_/fos & 1993 - 07 - 21 & @xmath338 & 0.030 & @xmath339 & @xmath340 & @xmath341 & @xmath342 & @xmath343 & + pg1426 + 015 & _ iue_/swp & 1985 - 03 - 01,02 & @xmath344 & 0.028 & @xmath345 & @xmath346 & @xmath347 & @xmath348 & @xmath349 & + pg1613 + 658 & _ hst_/cos & 2010 - 04 - 08,09,10 & @xmath350 & 0.023 & @xmath351 & @xmath352 & @xmath353 & @xmath354 & @xmath355 & + pg2130 + 099 & _ hst_/cos & 2010 - 10 - 28 & @xmath356 & 0.039 & @xmath357 & @xmath358 & @xmath359 & @xmath360 & @xmath361 & [ tab : uvdata ] lccccccc + + @xmath362 & @xmath363 & @xmath364 & @xmath48 & @xmath365 & & & vp06 + @xmath362 & @xmath366 & @xmath364 & @xmath48 & @xmath367 & & & vp06 + fwhm & @xmath368 & @xmath364 & @xmath48 & @xmath369 & & & vp06 + fwhm & @xmath370 & @xmath364 & @xmath48 & @xmath371 & & & vp06 + fwhm & @xmath372 & @xmath373 & @xmath374 & & @xmath375 & @xmath376 & sl12 + + + @xmath362 & @xmath377 & @xmath378 & @xmath379 & @xmath380 & @xmath381 & @xmath382 & + fwhm & @xmath383 & @xmath384 & @xmath385 & @xmath386 & @xmath381 & @xmath387 & + + + @xmath362 & @xmath388 & @xmath389 & @xmath48 & @xmath390 & @xmath381 & @xmath391 & + fwhm & @xmath392 & @xmath393 & @xmath48 & @xmath394 & @xmath381 & @xmath395 & + + + @xmath362 & @xmath396 & @xmath397 & @xmath398 & @xmath399 & @xmath381 & @xmath400 & + fwhm & @xmath401 & @xmath397 & @xmath402 & @xmath403 & @xmath381 & @xmath387 & + + + @xmath362 & @xmath404 & @xmath397 & @xmath48 & @xmath380 & @xmath381 & @xmath405 & + fwhm & @xmath406 & @xmath397 & @xmath48 & @xmath394 & @xmath381 & @xmath407 & + + + @xmath362 & @xmath408 & @xmath364 & @xmath48 & @xmath409 & @xmath381 & @xmath410 & + fwhm & @xmath411 & @xmath364 & @xmath48 & @xmath412 & @xmath381 & @xmath413 & + + + @xmath362 & @xmath414 & @xmath384 & @xmath415 & @xmath399 & @xmath381 & @xmath416 & + fwhm & @xmath417 & @xmath418 & @xmath419 & @xmath420 & @xmath381 & @xmath421 & best - fit + + + @xmath362 & @xmath422 & @xmath423 & @xmath424 & @xmath425 & @xmath381 & @xmath426 & best - fit + fwhm & @xmath427 & @xmath428 & @xmath48 & @xmath394 & @xmath381 & @xmath429 & + + + @xmath362 & @xmath430 & @xmath397 & @xmath431 & @xmath425 & @xmath381 & @xmath416 & + fwhm & @xmath432 & @xmath397 & @xmath433 & @xmath434 & @xmath381 & @xmath421 & + + + @xmath362 & @xmath435 & @xmath397 & @xmath48 & @xmath425 & @xmath381 & @xmath426 & + fwhm & @xmath436 & @xmath397 & @xmath48 & @xmath437 & @xmath381 & @xmath438 & + + + @xmath362 & @xmath439 & @xmath364 & @xmath48 & @xmath425 & @xmath381 & @xmath440 & + fwhm & @xmath441 & @xmath364 & @xmath48 & @xmath394 & @xmath381 & @xmath442 & [ tab : calibration ]
|
we present the single - epoch black hole mass estimators based on the @xmath01549 broad emission line , using the updated sample of the reverberation - mapped agns and high - quality uv spectra . by performing multi - component spectral fitting analysis
, we measure the line widths ( fwhm@xmath1 and line dispersion , @xmath2 ) and the continuum luminosity at 1350 ( @xmath3 ) to calibrate the -based mass estimators . by comparing with the h@xmath4 reverberation - based masses ,
we provide new mass estimators with the best - fit relationships , i.e. , @xmath5 and @xmath6 .
the new -based mass estimators show significant mass - dependent systematic difference compared to the estimators commonly used in the literature . using the published sloan digital sky survey qso catalog
, we show that the black hole mass of high - redshift qsos decreases on average by @xmath7 dex if our recipe is adopted .
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we begin with the definitions of the essential concepts of our model . * random geometric graph * consists of @xmath10 agents randomly distributed in a unit square @xmath11 . each agent has an interaction range defined by @xmath12 , where @xmath5 is the local interaction radius . two agents are connected if they fall in each others interaction range . the choice of network topology , denoted as @xmath13 , impacts the boundary conditions . some studies , like @xcite , choose the natural topology of the unit square which leads to the free boundary condition . in this paper , we assume that @xmath13 is a torus , imposing the periodic boundary condition . consequently , the opinion dynamics is free of boundary effects until the correlation length of the opinions grows comparable to the length scale of @xmath13 . * microstate * of a network is given by a spin vector @xmath14 where @xmath15 represents the opinion of the @xmath16 individual . in the ng , the spin value is assigned as follows : @xmath17 the evolution of microstate is given by spin updating rules : at each time step , two neighboring agents , a speaker @xmath18 and a listener @xmath19 are randomly selected , only the listener s state is changed ( lo - ng ) . the word sent by the speaker @xmath18 is represented by @xmath20 , @xmath21 if the word is a and @xmath22 if the word is b. @xmath20 is a random variable depending on @xmath15 . the updating rule of the ng can be written as : @xmath23 * macrostate * is given by @xmath24 , @xmath25 and @xmath26 , the concentrations of agents at the location @xmath27 with opinion a , b and ab , respectively , that satisfy the normalization condition @xmath28 . we define @xmath29 as the local order parameter ( analogous to magnetization " ) , and @xmath30 as the local mean field @xmath31 finally , @xmath32 denotes the probability for an agent to receive a word a if it is located at @xmath27 . through the geographic coarsening approach discussed in more detail in * methods * , we obtain the mean - field equation describing the evolution of macrostate @xmath33 while the macrostate itself is defined as @xmath34 there are two characteristic length scales in this system , one is the system size ( which is set to @xmath35 ) , the other is the local interaction radius @xmath5 . so regarding the correlation length or the typical scale of opinion domains @xmath36 , the dynamics can be divided into two stages : ( 1 ) @xmath36 is smaller or comparable to @xmath5 ; ( 2 ) @xmath37 . in the second stage , the consensus is achieved when @xmath36 grows up to 1 . figure [ meanfield ] present snapshots of solution of the mean - field equation . they illustrate the formation of opinion domains and the coarsening of the spatial structure . ) . snapshots are taken at @xmath38 , the scale of opinion domains are much bigger than @xmath39 . black stands for opinion a , white stands for opinion b and gray stands for the coexistence of two types of opinions . the consensus is achieved at @xmath40.,scaledwidth=95.0% ] to study the spatial coarsening , we consider the pair correlation function @xmath41 defined by the conditional expectation of spin correlation . @xmath42.\ ] ] figure [ meanfield ] implies there exists a single characteristic length scale @xmath43 so that the pair correlation function has a scaling form @xmath44 , where the scaling function @xmath45 does not depend on time explicitly . for coarsening in most systems with non - conserved order parameter such as the opinion dynamics on a d - dimensional lattice , the characteristic length scale is @xmath46@xcite . according to the numerical results in fig . [ scaling ] , the length scale for opinion dynamics on rgg at the early stage ( t=30,50 ) is also @xmath8 , but at the late stage ( t=100,200,400 ) , the length scale @xmath47 fits more precisely simulation results than the previous one . for the pair correlation function at times @xmath48 . overlapped curves indicate correct scaling of @xmath49 . simulations are done for the case @xmath50 , @xmath39 , @xmath51 . @xmath49 is normalized by the length scale ( a ) @xmath8 and ( b ) @xmath52.,scaledwidth=95.0% ] here , we find all the possible stationary solutions of the mean - field equation eq . ( [ eqn_macro ] ) . taking @xmath53 , we obtain @xmath54 the eigenvalues of the linear dynamical system @xmath55 are both negative , so @xmath56 is stable . applying the definition of @xmath57 and @xmath30 , we have @xmath58 @xmath59 once we solve the above integral equation , we can retrieve the stationary macrostate @xmath60 by eq . ( [ local_equil ] ) . taking @xmath30 as a constant , we find three solutions @xmath61 or @xmath62 . @xmath61 are both asymptotically stable , while @xmath63 is unstable . another class of solution is obtained by taking @xmath64 ( or similarly @xmath65 ) . the solution consists of an even number of stripe - like opinion domains demarcated by two types of straight intermediate layers parallel to one side of the unit square @xmath13 as shown in fig . [ stationary_solution](b ) . with the boundary condition @xmath66 or vice versa , we solve the two types of intermediate layers @xmath67 as shown in fig . [ stationary_solution ] ( a ) . the intermediate layers are of the scale @xmath5 and can be placed at arbitrary @xmath68 . this class of solution is neutrally stable . finally , there is another class of solution shown in fig . [ stationary_solution](c ) with intermediate layers both in @xmath69 and @xmath70 directions and opinion domains assigned as a checker board . this type of solution is unstable at the intersections of two types of intermediate layers . the latter two classes of solutions can be easily generalized to the cases when the intermediate layers are not parallel to x or y axis . later we will show that in stationary solutions all the curvature of the opinion domain boundary has to be 0 , so the solutions mentioned above are the only possible stationary solutions . . @xmath71 is the location of the intermediate layer . the slope of the intermediate layer at @xmath72 is about @xmath73 . ( b ) stripe - like stationary solution , neutrally stable . ( c ) checker - board - like stationary solution , unstable . ] in conclusion , considering the stability , the final state of the macrostate dynamics can be : ( 1 ) all a or all b consensus states which are both asymptotically stable , ( 2 ) stripe - like solution . the probability for the dynamics stuck in the stripe - like state before achieving full consensus is roughly @xmath74 in analogy to similar cases in continuum percolation and spin dynamics @xcite . one important observation regarding the macrostate dynamics is that the change of local mean field is usually much slower than the convergence of local macrostate @xmath75 to its local equilibrium @xmath56 . let @xmath76 , @xmath77 , @xmath78 . the following equation shows the exponential rate of convergence of @xmath75 : @xmath79 the largest eigenvalue is @xmath80 . so the typical time scale @xmath81 of the convergence of the local macrostate is @xmath82 which is independent of time and system size . the typical time scale @xmath83 of the change of the local mean field is inversely proportional to the propagation speed @xmath84 of opinion domain boundaries , and as we will show later , is of the order @xmath85 where @xmath86 is the curvature of the opinion domain boundary . therefore , @xmath87 for both long time ( @xmath86 grows to infinity along with the time @xmath6 ) and big systems ( in the sense that @xmath88 ) . [ adiabatic ] shows the significant two - time - scale separation observed in numerical results . the equilibrium value of the local order parameter , @xmath89 , can be predicted by the local mean field , @xmath90 . in fig . [ adiabatic ] , we present the empirical local order parameter @xmath91 for different local mean field values @xmath92 and show that it is very close to its local equilibrium @xmath89 . from numerical simulation and the equilibrium @xmath89 for different local mean fields @xmath92 s . the solid line stands for @xmath89 . the error bars present the means and standard deviations of @xmath91 that is the empirical local order parameter for the given @xmath92 in numerical simulations.,title="fig:",scaledwidth=75.0% ] + since @xmath93 , we have @xmath94 \label{eq_s}.\ ] ] this ode is quite relevant to reaction - diffusion systems . on the right hand side , the coefficient @xmath95 is easy to get rid of by scaling the time @xmath96 . after the scaling , the first term is diffusive since @xmath97 is the continuum approximation of the laplace operator on rgg network acting on @xmath91 . the second term @xmath98 is the local reaction term . though classified rigorously , it is non - local as defined in reaction - diffusion system , it represents a reaction in local neighborhood @xmath99 . the adiabaticity of the dynamics implies that the diffusion is much slower than the local reaction . we can obtain an approximated ode for slow time scale dynamics in a closed form by estimating @xmath100 by its local equilibrium @xmath101 . @xmath102 the qualitative behavior of the reaction - diffusion system is determined by the linear stability of the reaction term @xcite . in this sense , ( [ slow ] ) provides clear differentiation between dynamics in our model and in the voter model , the majority game and the glauber ordering . taking a similar approach , we find that the voter model on rgg is purely diffusive , i.e. the reaction term is 0 . for the glauber ordering , the reaction term is @xmath103 in which @xmath104 is the inverse of temperature and @xmath105 is the interaction intensity . [ reaction_term ] shows the reaction term @xmath106 for the voter , ng , and glauber ordering ( go ) at different temperatures . the majority game , ng , and glauber ordering at zero temperature have reaction terms with the same equilibria and stability ( @xmath107 stable , @xmath62 unstable ) . thus , the mean - field solutions of these models behave similarly . however , at the level of the discrete model , the ng on rgg will always go to a microstate corresponding to some stationary mean - field solution , while glauber ordering at zero temperature on rgg may get stuck in one of many local minima of its hamiltonian . for voter , ng and glauber ordering ( go ) at different temperatures.,title="fig:",scaledwidth=75.0% ] + the evolution of the opinion domains is governed by a very simple rule . the boundary of opinion domains propagate at the speed @xmath84 that is proportional to its curvature @xmath108 , i.e. @xmath109 . here , @xmath110 is a constant defined by the average degree @xmath9 . in * methods * , we provide a heuristic argument and using the perturbation method prove this relation for the mean - field equation , i.e. the case when @xmath111 . this behavior is common for many reaction - diffusion systems and it is qualitatively the same as the behavior of glauber ordering at zero temperature @xcite . following the rule of boundary evolution minimizes the length of the domain boundary . a direct consequence of this fact is that if any stationary solution exists , its boundaries must be all straight ( geodesic ) , confirming our conclusion about the stationary solutions found in the previous paragraph . since global topology is irrelevant to our derivation , this relation applies also to other two - dimensional manifolds . the manifold considered here is the torus embedded in 2d euclidean space . however , for the standard torus embedded in 3d euclidean space , the topology is the same but metrics are not , hence the geodesics are different . therefore , there are quite different and more complicated stationary solutions there . another example is the sphere in 3d space . on the sphere , the only inhomogeneous stationary solution consists of two hemispherical opinion domains , because the great circle is the only closed geodesic on a sphere . the numerical result presented in fig . [ propagation_speed ] confirms this relation . in fig . [ propagation_speed ] , we gather @xmath112 data points from numerical solutions of the macrostate equation , using different initial conditions , taking snapshots at different times , tracking different points on the boundary and calculating the local curvature radius @xmath86 and boundary propagation speed @xmath84 . these data points in the double - log plot are aligned well with the straight line with slope @xmath113 . the curve formed by data points is jiggling with some period because we implemented the numerical method on a square lattice , so the numerical propagation speed is slightly anisotropic . of the boundary of opinion domains vs. curvature radius @xmath86 . data points are gathered from 100 runs of macrostate equation with different initial conditions . in each run , propagation speed and curvature radius are calculated for 10000 points on the boundary.,title="fig:",scaledwidth=80.0% ] + another way to confirm this rule is to consider a round opinion domain with initial radius @xmath114 . given @xmath115 , the size of this opinion domain decreases as @xmath116 when @xmath117 . this relation is shown in fig . [ shrink ] . in simulations , the size of opinion domain @xmath118 is evaluated by @xmath119 where @xmath120,@xmath121 are the total numbers of a , ab nodes , respectively . we also observe from this plot that @xmath110 increases with average degree @xmath122 and converges to its upper bound when @xmath123 . as a function of time with @xmath124 . the initial radius of the opinion domain is @xmath125 . the straight line is for the numerical solution of mean - field equation . the dotted line and dash line are for simulation of discrete model with @xmath126 and @xmath127 , as well as with @xmath128 and @xmath129 , respectively.,title="fig:",scaledwidth=80.0% ] + we now consider influencing the consensus by committed agents . in sociological interpretation , a committed agent is one who keeps its opinion unchanged forever regardless of its interactions with other agents . the effect of committed agents in the ng has been studied in @xcite . a critical fraction of committed agent @xmath130 is found for ng - lo on complete graph which is also relevant here . in our setting , a fraction @xmath131 of agents are committed to opinion a , and all other agents are uncommitted and hold opinion b. the macrostate with committed agents is still defined by eq . ( [ eqn_macro ] ) , but the definition of local mean field @xmath92 is replaced by @xmath132 generally , @xmath131 can vary on the x - y plane , but we only consider the case that @xmath131 is a constant , i.e. the committed agents are uniformly distributed . then we reanalyze the stationary solution . firstly there is a critical committed fraction @xmath133 which is exactly the same as that on complete graph . when @xmath134 , the only stationary solution is @xmath135 and it is stable . when @xmath136 , there are three solutions , of which two @xmath135 , @xmath137 are stable , and the third @xmath138 is unstable . besides there is a class of stationary solutions when the committed fraction is below the critical . they are analogues of the stripe - like solutions in the non - committed agent case . the evolution of the boundary can be interpreted as a mean curvature flow . in such view , the fraction of agents committed to a opinion exert a constant pressure on the boundary surface from the side of a opinion domain . similarly , agents holding opinion b exert a pressure from the side of b opinion domain . so the stationary solution will contain opinion domains in the form of disks with critical radius @xmath139 for which the pressure arising from the curvature offsets the pressure from the committed fraction ; thus we have @xmath140 . this type of stationary solutions are unstable , the round disk of the opinion domain will grow when @xmath141 and will shrink when @xmath142 . in the first case , when the typical length scale of opinion domains grows beyond @xmath143 , the system will achieve consensus very quickly . on the basis of the above stability analysis , we then analyze the dependence of the consensus time on the system size @xmath10 , the committed fraction @xmath131 , and the average degree @xmath122 , and show our conclusions are consistent with the numerical results in fig . [ committed2 ] which for a given fraction @xmath110 ( @xmath144 in the figure ) depicts the time for @xmath110-consensus in which at least fraction @xmath110 of agents hold the same opinion . the time to achieve @xmath145consensus is independent of @xmath10 both according to the mean field prediction and numerical plots . when @xmath134 the dynamics will converge to its unique local equilibrium @xmath146 at all locations simultaneously . the consensus in this case is close to that on the complete network , especially when @xmath122 tends to infinity . in the opposite case , when @xmath136 , the process to consensus is twofold - before and after the a opinion domain achieves the critical size @xmath143 . after this criticality , the process is just the extension of the opinion domain driven by the mean field eq . ( [ eqn_macro ] ) . this stage is relatively fast and the consensus time is dominated by the duration of the other stage , the one before the criticality , in which the dynamic behavior is a joint effect of the mean field and the random fluctuation we neglect in mean field analysis . assuming the dynamics was purely driven by the random fluctuation , the typical length scale of opinion domains would have the scaling @xmath147 at the early stage , hence the time scale of this stage would be @xmath148 , i.e. @xmath149 . if the dynamics was purely driven by the mean field , the a opinion domain would never achieve the critical size . the actual dynamics behavior is in between the two extreme cases . when @xmath122 decreases , the fluctuation level relative to the mean field becomes higher , leading to faster consensus . in fig . [ committed2 ] , linear regression for the data points @xmath150 gives @xmath151 in which @xmath152 for @xmath153 respectively , where @xmath154 is the value corresponding to the purely random extreme case . two observations here may have meaningful sociological interpretation : \(1 ) when @xmath123 , for both @xmath136 and @xmath134 , the dynamics behavior converges to that on the complete networks , though the rgg itself may be far from the complete network ( with @xmath5 kept constant , the diameter of the rgg network is much larger than 1 ) . \(2 ) when @xmath136 , committed agents are more powerful in changing the prevailing social opinion on rggs with low average degree . it is similar to the result of the previous study@xcite on the social dynamics on sparse random networks but the `` more powerful '' is in a different sense meaning not the smaller tipping fraction ( with longer consensus time ) , but the shorter time to consensus . for @xmath155-consensus for ng on rgg with different fractions ( @xmath131 ) of committed agents with direct simulation on networks with average degrees @xmath156 , and network sizes @xmath157 . the solid black curve shows the mean field prediction of ng on the complete network(cn ) as the limit case when @xmath123 . the slope of the red data points near @xmath133 is -2.19.,title="fig:",scaledwidth=80.0% ] + on rggs , the average degree of an agent @xmath158 is an important structural parameter which also strongly impacts the local dynamic behavior . there are two critical values of @xmath122 : one is for the emergence of the giant component , @xmath159 @xcite ; the other one , @xmath160 , only applicable for finite - size networks , is for the emergence of the giant component with all vertices belonging to it . in this paper , we only considered the case when @xmath122 is above the critical value @xmath161 so that the network is connected @xcite . we mainly focused on analyzing the mean - field equation of the ng dynamics on rgg . we predicted a number of behaviors , including the existence of metastable states , the two - time - scale separation , and the dependence of the boundary propagation speed on the boundary curvature . we demonstrated in detail that the evolution of the spatial domains for the two - word lo - ng is governed by coarsening dynamics , similar to the broader family of generalized voter - like models with intermediate states @xcite . however , there are still some behaviors that can not be explained by the mean - field equation , such as : ( _ i _ ) in the large @xmath6 limit , the scaling of correlation length is not @xmath8 as on the 2-@xmath162 regular lattice , but @xmath163 ; ( _ ii _ ) the propagation speed increases along with the average degree @xmath122 and its upper - bound is the mean field prediction , i.e. , the @xmath164 case . so the major limitation of the mean - field equation derived from the geographic coarsening approach is that it neglects the fluctuation among replicas ( see * methods * ) and loses the information about @xmath122 . the dependence of the dynamics on @xmath122 is left for further study . first , we provide the equation for the evolution of microstate @xmath165 . denote the probabilities for @xmath15 taking values @xmath35 , @xmath62 , @xmath113 as @xmath166 , @xmath167 and @xmath168 , respectively . the master equation for spin @xmath15 is given by @xmath169 where @xmath170 is the probability for the @xmath16 agent receiving a signal @xmath171 , while @xmath172 is the local mean field defined as the average of the neighboring spins , @xmath173 where @xmath174 is the coordinate of the @xmath16 agent and @xmath175 is the degree of the @xmath16 agent . master equations for all spins together describe the evolution of microstate . the motivation for geographic coarsening comes from the fact that rgg is embedded in a geographic space , so we may skip the level of agents and relate the opinion states directly to the geographic coordinates . instead of taking into account the opinion of every agent , in geographic coarsening , we consider the concentration of agents with different opinions at a specific location . @xmath176 , @xmath57 , @xmath177 are continuously differentiable w.r.t . @xmath69 , @xmath70 and @xmath6 . we derive the equation of @xmath176 from eq . ( [ eqn_micro ] ) by taking limits , @xmath178 the limit is done as follows @xmath179 \;.\ ] ] the coarsening based on the above limit is valid under either of the following two assumptions . the first is when the rgg is very dense ( @xmath180 ) and @xmath181 keeps constant . the second is when we consider @xmath182 replicas of rgg with the same set of parameters , and the summation above is over all replicas . in addition , @xmath183 keeps constant . our derivation is actually based on the second assumption . however under the first assumption , the fluctuation is vanishing , and the dynamics behavior of a single run converges to the mean field result . we show here the qualitative property of the boundary evolution by a heuristic argument . we consider a solution with the form @xmath184 where @xmath185 and with boundary conditions @xmath186 and @xmath187 . @xmath188 has an intermediate layer at @xmath86 as shown in fig . [ stationary_solution](a ) , so near @xmath86 , @xmath189 . when @xmath190 , using moving coordinate @xmath191 where @xmath192 is unit wave vector , @xmath193 is spatial coordinate and @xmath84 is the wave speed , eq . ( [ eq_s ] ) becomes @xmath194 here , @xmath92 can be approximated by @xmath195 then we make perturbation on eq . ( [ traveling_wave ] ) @xmath196 , @xmath197 , @xmath198 and so on , requiring @xmath199 , and obtain the equation for @xmath200 @xmath201 at @xmath202 , @xmath203 , @xmath204 , @xmath205 and @xmath206 is @xmath207 , the above equation becomes @xmath208 , so @xmath209 this work was supported in part by the army research laboratory under cooperative agreement number w911nf-09 - 2 - 0053 , by the army research office grant nos w911nf-09 - 1 - 0254 and w911nf-12 - 1 - 0546 , and by the office of naval research grant no . n00014 - 09 - 1 - 0607 . the views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies either expressed or implied of the army research laboratory or the u.s . government . z.w . , c.l . , g.k . and b.k.s . designed the research ; z.w implemented and performed numerical experiments and simulations ; z.w . , c.l . , g.k . and b.k.s . analyzed data and discussed results ; z.w . , g.k . and b.k.s . wrote and reviewed the manuscript . the authors declare no competing financial interests . lu , q. , korniss , g. & szymanski , b.k . naming games in spatially - embedded random networks . in _ proceedings of the 2006 american association for artificial intelligence fall symposium series , interaction and emergent phenomena in societies of agents _ ( aaai press , menlo park , ca 2006 ) , pp . 148155 ; arxiv : cs/0604075v3 .
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we investigate the two - word naming game on two - dimensional random geometric graphs . studying this model advances our understanding of the spatial distribution and propagation of opinions in social dynamics .
a main feature of this model is the spontaneous emergence of spatial structures called opinion domains which are geographic regions with clear boundaries within which all individuals share the same opinion .
we provide the mean - field equation for the underlying dynamics and discuss several properties of the equation such as the stationary solutions and two - time - scale separation . for the evolution of the opinion
domains we find that the opinion domain boundary propagates at a speed proportional to its curvature .
finally we investigate the impact of committed agents on opinion domains and find the scaling of consensus time .
@xmath0 department of mathematical sciences , rensselaer polytechnic institute , 110 8@xmath1 street , troy , ny , 12180 - 3590 usa + @xmath2 department of physics , applied physics and astronomy , rensselaer polytechnic institute , 110 8@xmath1 street , troy , ny , 12180 - 3590 usa + @xmath3 department of computer science , rensselaer polytechnic institute , 110 8@xmath1 street , troy , ny , 12180 - 3590 usa + @xmath4 e - mail : zhangw14@rpi.edu relevant features of social and opinion dynamics @xcite can be investigated by prototypical agent - based models such as the voter model @xcite , the naming game @xcite , or the majority model @xcite .
these models typically include a large number of individuals , each of which is assigned a state defined by the social opinions that it accepts and updates its state by interacting with its neighbors .
opinion dynamics driven by local communication on geographically embedded networks is of great interest to understanding the spatial distribution and propagation of opinions . in this paper
we investigate the naming game ( ng ) on random geometric graphs as a minimum model of this type .
we focus on the ng but will also compare it with other models of opinion dynamics .
the ng @xcite was originally introduced in the context of linguistics and spontaneous emergence of shared vocabulary among artificial agents @xcite to demonstrate how autonomous agents can achieve global agreement through pair - wise communications without a central coordinator . here
, we employ a special version of the ng , called the two - word @xcite listener - only naming game ( lo - ng ) @xcite . in this version of the ng
, each agent can either adopt one of the two different opinions a , b , or take the neutral stand represented by their union , ab . in each communication , a pair of neighboring agents are randomly chosen , the first one as the speaker and the second one as the listener .
the speaker holding a or b opinion will transmit its own opinion and the neutral speaker will transmit either a or b opinion with equal probability . the listener holding a or b opinion will become neutral when it hears an opinion different from its own and the neutral listener will adopt whatever it hears .
detailed instances are shown in supplementary table .
consensus formation in the original ng ( and its variations ) on various regular and complex networks have been studied @xcite . in particular , the spatial and temporal scaling properties
have been analyzed by direct simulations and scaling arguments in spatially - embedded regular and random ( rgg ) graphs @xcite .
these results indicated @xcite that the consensus formation in these systems is analogous to coarsening @xcite . in this paper , we further elucidate on the emerging coarsening dynamics in the two - word lo - ng on rgg by developing mean - field ( or coarse - grained ) equations for the evolution of opinions .
our method of relating microscopic dynamics to macroscopic behavior shares similar features with those leading to effective fokker - planck and langevin equations in a large class of opinion dynamic models ( including generalized voter models with intermediate states ) @xcite . a random geometric graph ( rgg ) , also referred to as a spatial poisson or boolean graph , models spatial effects explicitly and therefore is of both technological and intellectual importance @xcite . in this model , each node is randomly assigned geographic coordinates and two nodes are connected if the distance between them is within the interaction radius @xmath5 .
another type of network with geographic information is the regular lattice .
fundamental models for opinion dynamics on regular lattice has been intensively studied @xcite . in many aspects ,
opinion dynamics behaves similarly on rggs and regular lattices with the same dimensionality , but in our study , we also observe several differences .
for example , the length scale of spatial coarsening for large @xmath6 is @xmath7 on rgg while it is @xmath8 on regular lattice .
more generally , concerning the spatial propagation of opinions in social systems or agreement dynamics in networks of artificial agents , random geometric graphs are more realistic for a number of reasons : ( _ i _ ) rgg is isotropic ( on average ) while regular lattice is not ; ( _ ii _ ) the average degree @xmath9 for an rgg can be set to an arbitrary positive number , instead of a small fixed number for the lattice ; ( _ iii _ ) rggs closely capture the topology of random networks of short - range - connected spatially - embedded artificial agents , such as sensor networks .
an important feature of the ng which makes it different from other models of opinion dynamics , e.g. , voter model , is the spontaneous emergence of clusters sharing the same opinion .
generally these opinion clusters are communities closely connected within the network .
this feature of the ng can be used to detect communities of social networks @xcite . for the ng on random geometric graph concerned in this paper , the clusters form a spatial structure to which we refer to as _ opinion domains _ and which are geographic regions in which all nodes share the same opinion .
a number of relevant properties of the ng on random geometric graph have been studied by direct individual - based simulations and discussed in @xcite , such as the scaling behavior of the consensus time and the distribution of opinion domain size .
in contrast to these previous works , here we develop a coarse - grained approach and focus on the spatial structure of the two - word lo - ng , such as the correlation length , shape , and propagation of the opinion domains . in this paper
, we provide the mean - field ( or coarse - grained ) equation for the ng dynamics on rgg . by analyzing the mean - field equation
, we list all possible stationary solutions and find that the ng may get stuck in stripe - like metastable states rather than achieve total consensus .
we find significant two - time - scale separation of the dynamics and retrieve the slow process governed by reaction - diffusion system . using this framework , we identify similarities and differences between ng and other relevant models of opinion dynamics , such as voter model , majority game and glauber ordering .
next , we present the governing rule of the opinion domain evolution , that in the late stage of dynamics , the propagation speed of the opinion domain boundary is proportional to its curvature .
thus , an opinion domain can be considered as a mean curvature flow making many results of the previous works applicable here @xcite .
finally we investigate the impact of committed agents . the critical fraction of committed agents found in the case of the ng on complete graph is also present here .
we discuss the dependence of the consensus time on the system size , the committed fraction and the average degree .
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lipid bilayer membranes constitute one of the most fundamental components of all living cells . apart from their obvious _ structural _ role in organizing distinct biochemical compartments , their contributions to essential _ functions _ such as protein organization , sorting , or signalling are now well documented @xcite . in fact , their tasks significantly exceed mere passive separation or solubilization of proteins , since often _ mechanical _ membrane properties are intricately linked to these biological functions , most visibly in all cases which go along with distinct membrane deformations , such as exo- and endocytosis @xcite , vesiculation @xcite , viral budding @xcite , cytoskeleton interaction @xcite , and cytokinesis @xcite . consequently , a quantitative knowledge of the material parameters which characterize a membrane s elastic response most notably the bending modulus @xmath0 is also biologically desirable . several methods for the experimental determination of @xmath0 have been proposed , such as monitoring the spectrum of thermal undulations via light microscopy @xcite , analyzing the relative area change of vesicles under micropipette aspiration @xcite , or measuring the force required to pull thin membrane tethers @xcite . with the possible exception of the tether experiments , these techniques are _ global _ in nature , _ _ i.__@xmath1_e . _ , they supply information averaged over millions of lipids , if not over entire vesicles or cells . yet , in a biological context this may be insufficient @xcite . for instance , membrane properties such as their lipid composition or bilayer phase ( and thus mechanical rigidity ) have been proposed to vary on submicroscopic length scales @xcite . despite being biologically enticing , this suggestion , known as the `` raft hypothesis '' , has repeatedly come under critical scrutiny @xcite , precisely because the existence of such small domains is extremely hard to prove . an obvious tool to obtain mechanical information for small samples is the atomic force microscope ( afm ) @xcite , and it has indeed been used to probe cell elastic properties ( such as for instance their young modulus ) @xcite . yet , obtaining truly _ local _ information still poses a formidable challenge . apart from several complications associated with the inhomogeneous cell surface and intra - cellular structures beneath the lipid bilayer , one particularly notable difficulty is that the basically unknown boundary conditions of the cell membrane away from the spot where the afm tip indents it preclude a quantitative interpretation of the measured force , _ _ i.__@xmath1_e . _ a clean way to translate this force into ( local ) material properties . to overcome this problem steltenkamp _ _ et__@xmath1_al . _ have recently suggested to spread the cell membrane over an adhesive substrate which features circular pores of well - defined radius @xcite . poking the resulting `` nanodrums '' would then constitute an elasto - mechanical experiment with precisely defined geometry . using simple model membranes , the authors could in fact show that a quantitative description of such measurements is possible using the standard continuum curvature - elastic membrane model due to canham @xcite and helfrich @xcite . spreading a cellular membrane without erasing interesting local lipid structures obviously poses an experimental challenge ; but the setup also faces another problem which has its origin in an `` elastic curiosity '' : even significant indentations , which require the full _ nonlinear _ version of the helfrich shape equations for their correct description , end up displaying force - distance - curves which are more or less _ linear_a finding in accord with the initial regime of membrane tether pulling @xcite . yet , this simple functional form makes a unique extraction of the two main mechanical properties , tension and bending modulus , difficult . is the nanodrum setup thus futile ? in the present work we develop the theoretical basis for a slight extension of the nanodrum experiment that will help to overcome this impasse . we will show that an additional _ adhesion _ between the afm tip and the pore - spanning membrane will change the situation very significantly quantitatively and qualitatively . force - distance - curves cease to be linear , hysteresis , nonzero detachment forces and membrane overhangs can show up , and various new stable and unstable equilibrium branches emerge . the magnitude and characteristics of all these new effects can be quantitatively predicted using well established techniques which have previously been used successfully to study vesicle shapes @xcite , vesicle adhesion @xcite , colloidal wrapping @xcite or tether pulling @xcite . indents a pore - spanning membrane with a force @xmath2 to a certain depth @xmath3 . the radius of the pore is @xmath4 . the membrane detaches from the tip at a radial distance @xmath5 . the two possible parametrizations @xmath6 and @xmath7 are explained in the beginning of chapter [ sec : shapeeqn ] . ] the key `` ingredient '' underlying most of the new physics is the fact that the membrane can _ choose _ its point of detachment from the afm tip . unlike in the existing point force descriptions @xcite , in which a certain ( pushing or pulling ) force is applied at one point of the membrane , our description accounts for the fact that the generally nonvanishing interaction energy per unit area between tip and membrane co - determines their contact area over which they transmit forces , and thereby influence the entire force - distance - curve . what may at first seem like a minor modification of boundary conditions quickly proves to open a very rich and partly also complicated scenario , whose various facets may subsequently be used to extract information about the membrane . in fact , smith _ _ et__@xmath1_al._@xcite have demonstrated in a related situation that the competition between adhesion and tether pulling for substrate - bound vesicles gives rise to various first- and second - order transitions , details of which depend in a predictable way on the experimental setup . in our case we will for instance find snap - on and snap - off events between tip and membrane , which rest on the fact that binding is _ not _ pre - determined , and whose correct description is very important for reliably interpreting any afm force experiments . moreover , we will also see that the very occurrence of tethers is a much more subtle phenomenon , since an adhering membrane pulled upwards may in fact prefer to _ detach _ rather than being pulled into a tether a question treated previously ( and on the linear level ) by boulbitch @xcite . our paper is organized as follows : in chapter [ sec : themodel ] we introduce the model of our system and discuss the relevant energies . in chapter [ sec : shapeeqn ] we present the equations that have to be solved in order to find membrane profiles , force - indentation curves and detachment forces . this will also include a treatment of the nonlinear case which was only mentioned very briefly in ref . @xcite . in chapter [ sec : results ] the results of our calculations are summarized and compared to existing @xcite measurements . we end in chapter [ sec : discussion ] with a discussion how the predictions for indentation and adhesion characteristics can be used to extract material properties in future experiments . we consider a flat solid substrate with a circular pore of radius @xmath4 . a lipid bilayer membrane is adsorbed onto the substrate and spans the pore . in the situation we want to analyze an afm tip is used to probe the properties of the free pore - spanning membrane . we assume that the tip has a parabolic shape with curvature radius @xmath8 at its apex . furthermore , we restrict ourselves to the static axisymmetric situation in which the tip pokes the free - standing membrane exactly in the middle of the pore ( see fig . [ fig : poregeometry ] ) . for a certain downward force @xmath9 the membrane is indented to a corresponding depth @xmath10 which is measured from the plane of the substrate to the depth of the apex of the tip . note that it is also possible to pull the membrane up with a force @xmath11 in the opposite direction if attractive interactions attach the membrane to the tip . in the following , we will model the bilayer as a two - dimensional surface . this is a valid approach provided the thickness of the membrane ( approx . 5 nm ) is much smaller than ( i ) the membrane s lateral extension as well as ( ii ) length scales of interest such as local radii of curvature . with this geometric setup in mind , let us now consider the different energy contributions we want to include in our model . the total energy of the system `` pore - tip '' comprises different contributions : the membrane is under a _ lateral tension _ @xmath12 . to pull excess area into the pore , work has to be done against the adhesion between membrane and flat substrate @xcite . it is given by @xmath12 times the excess area @xcite . additionally , a _ curvature energy _ is associated with the membrane . according to canham @xcite and helfrich @xcite the hamiltonian for an up - down symmetric membrane is then @xmath13 where @xmath14 denotes the surface of the membrane part which spans the pore . the proportionality constants @xmath0 and @xmath15 are called bending rigidity and saddle - splay modulus , respectively . the gaussian curvature @xmath16 is the product of the two principal curvatures whereas @xmath17 is their sum @xcite . note that the last term of energy ( [ eq : helfrich ] ) yields zero in our specific problem @xcite . with the help of the two material constants @xmath12 and @xmath0 one can define a characteristic lengthscale @xmath18 which does not depend on geometric boundary conditions such as the radius of the tip or the pore but only on properties of the membrane . on scales larger than @xmath19 tension is the more important energy contributions ; on smaller scales bending dominates . apart from tension and bending , an _ adhesion between tip and membrane _ may contribute to the total energy . we assume that it is proportional to the contact area @xmath20 between tip and membrane with a proportionality constant @xmath21 , the adhesion energy per area . if the indentation @xmath3 is given and one wants to determine the force @xmath2 , the total energy can thus be written as @xmath22 under certain circumstances , however , it is more convenient to consider the problem for a given force @xmath2 . both ensembles ( `` constant indentation '' _ vs. _ `` constant force '' ) are connected via a legendre transformation @xcite , @xmath23 . while the ground states one obtains for the two ensembles will be the same , questions of stability depend on the ensemble : a profile found to be stable under constant height conditions is not necessarily stable under constant force conditions . the route we want to follow here in order to find force - indentation curves is to determine the equilibrium shapes of the non - bound section of the membrane via a functional minimization . the energy contributions caused by the bounded section of the membrane enter via the appropriate boundary conditions ( see chapter [ sec : shapeeqn ] and appendix [ app : boundaryconditions ] ) . these imply that the contact point @xmath24 is not known a priori but has to be determined as well ( `` moving boundary problem '' ) . in the next section we will show how one can set up the appropriate mathematical formulation of the problem to get membrane profiles and force - indentation curves . to describe the shape of the membrane we use two different kinds of parametrization ( see fig . [ fig : poregeometry ] ) : for the linear approximation it is sufficient to use `` monge '' gauge where the position of the membrane is given by a height @xmath6 above ( or below ) the underlying reference plane . the disadvantage of this parametrization is that it does not allow for `` overhangs '' . since these may be present in the full nonlinear problem , we will use the `` angle - arclength '' parametrization in the exact calculations : the angle @xmath7 with respect to the horizontal substrate as a function of arclength @xmath25 fully describes the shape . to get the profile of the free membrane one has to solve the appropriate euler - langrange ( `` shape '' ) equation . this equation is typically a fourth order nonlinear partial differential equation and thus in most cases impossible to solve analytically . one may , however , consider cases where the membrane is indented only a little and gradients are small . in that case one may linearize the energy functional . in the constant indentation ensemble one gets for the free part @xmath26 \ ; , \label{eq : energyfunctionalconstantheight}\ ] ] where @xmath27 is the area element on the flat reference plane and @xmath28 is the projected surface of the free pore - spanning membrane . the symbol @xmath29 denotes the two - dimensional nabla operator in the reference plane . the appropriate shape equation can be derived by setting the first variation of energy ( [ eq : energyfunctionalconstantheight ] ) to zero , yielding @xmath30 the solution to this equation is a linear combination of the eigenfunctions of the laplacian corresponding to the eigenvalues 0 and @xmath31 . for axial symmetry it is given by @xmath32 , where @xmath33 and @xmath34 are the modified bessel functions of the first and the second kind , respectively @xcite . the constants @xmath35 are determined from the appropriate boundary conditions ( see appendix [ app : boundaryconditions ] ) : [ eq : boundaryequationslinear ] @xmath36 where a dash denotes a derivative with respect to @xmath37 . even though the differential equation is of fourth order , _ five _ conditions are required due to its moving boundary nature , _ _ i.__@xmath1_e . _ , @xmath24 is to be determined from an adhesion balance which is in fact the origin of the fifth condition ( [ eq : boundeqlin3 ] ) ( see appendix [ app : boundaryconditions ] ) . the solution of the boundary value problem ( [ eq : shapeequationlinear],[eq : boundaryequationslinear ] ) can be used in two ways to calculate the force for a prescribed indentation : first , one can insert the profile back into the functional ( [ eq : energyfunctionalconstantheight ] ) to obtain the energy of the equilibrium solution , which will then parametrically depend on the indentation @xmath3 . its derivative with respect to @xmath3 yields the force @xmath2 . second , one can also consider stresses : in analogy to elasticity theory @xcite @xmath2 is given by the integral of the flux of surface stress @xcite through a closed contour around the tip . the second approach is used in the present work ; it has the advantage that the final expression for the force can be written in a closed form @xcite ( see also appendix [ app : diffgeostress ] ) : @xmath38 this equation is _ exact_. inserting the solution @xmath6 of the boundary value problem ( [ eq : shapeequationlinear],[eq : boundaryequationslinear ] ) into ( [ eq : forcelinear ] ) yields the value of the force in the linear regime . a little warning might be due here : expression ( [ eq : forcelinear ] ) is evaluated at the rim of the pore where the profile is flat even for high indentations . one might thus wonder whether inserting the small gradient solution would actually lead to an exact result . this is , however , not the case , because the membrane shape at the rim predicted by the linear calculation is not identical with the prediction from the full nonlinear theory except for its flatness , which is enforced by the boundary conditions . there is no magical way to avoid solving the nonlinear shape equation if one wants the exact answer . let us now shift to the angle - arclength parametrization and consider the full nonlinear problem . in principle , the constant height ensemble could be used here as well . it is , however , technically much easier to fix @xmath2 instead in order to reduce the number of boundary conditions one has to fulfill at the rim of the pore ( see below and appendix [ app : numericalmethod ] ) . in this paragraph all variables with a tilde are scaled with @xmath39 , _ _ i.__@xmath1_e . _ : @xmath40 , @xmath41 , etc . the energy functional of the free membrane can then be written as @xcite : @xmath42 where @xmath43 is the arclength at the contact point @xmath24 and @xmath44 the arclength at @xmath4 . the dot denotes the derivative with respect to @xmath25 . the langrange multiplier functions @xmath45 and @xmath46 ensure that the geometric conditions @xmath47 and @xmath48 are fulfilled everywhere . in order to make the numerical integration easier let us rewrite the problem in a hamiltonian formulation @xcite : the conjugate momenta are @xmath49 $ ] , @xmath50 , and @xmath51 . the ( scaled ) hamiltonian is then given by @xmath52 note that @xmath53 is not explicitly dependent on @xmath25 and is thus a conserved quantity . instead of one fourth order one then has six first order ordinary differential equations , the hamilton equations : [ eq : shapeequationsnonlinear ] @xmath54 \cos{\psi } + p_{\rho } \sin{\psi } \label{eq : hamiltonequation3 } \\ \dot{p}_{\rho } & \ ; = & -\frac{\partial \tilde{h}}{\partial \rho } & \ ; = \ ; \frac{p_{\psi}}{\rho } \big(\frac{p_{\psi}}{4 \rho } - \frac{\sin{\psi}}{\rho } \big ) + \frac{2}{\lambda^{2 } } \\ \dot{p}_{z } & \ ; = & -\frac{\partial \tilde{h}}{\partial z } & \ ; = \ ; 0 \label{eq : hamiltonequation4a } \ ; .\end{aligned}\ ] ] according to the last equation , @xmath55 has to be constant along the profile . its value can be found by considering the integral over the flux of surface stress which has to equal the applied force . this implies that @xmath55 vanishes everywhere ( see appendix [ app : diffgeostress ] ) . equations ( [ eq : shapeequationsnonlinear ] ) can be solved numerically subject to the boundary conditions ( see also appendices [ app : boundaryconditions ] and [ app : numericalmethod ] ) : [ eq : boundaryequationsnonlinear ] @xmath56 where contact radius @xmath24 and contact angle @xmath57 are connected via @xmath58 . the solution to ( [ eq : shapeequationsnonlinear ] , [ eq : boundaryequationsnonlinear ] ) gives the indentation @xmath3 for some prescribed force @xmath59 . this chapter will summarize the characteristic features of the solution to the boundary value problems ( [ eq : shapeequationlinear ] , [ eq : boundaryequationslinear ] ) and ( [ eq : shapeequationsnonlinear ] , [ eq : boundaryequationsnonlinear ] ) . in addition , the theory will be shown to be in accord with available experimental results . we will introduce some additional variable rescaling in order to make generalizations of the results easier : lengths will be scaled with @xmath8 . we also define @xmath60 in a typical experiment the curvature of the tip is of the order of ten nanometer ( 540 nm ) and pore radii may lie between 30 and 200 nm @xcite . the bending rigidity of a fluid membrane may vary between one and a hundred @xmath61 @xcite . one expects a maximum surface tension of the order of a few mn / m , which is approximately the rupture tension for a fluid phospholipid bilayer @xcite . a maximum value of the adhesion can be found by assuming that a few @xmath61 per lipid is stored if membrane and tip are in contact . one arrives at @xmath62 . for the continuum theory to be valid eqns . ( [ eq : boundeqlin3 ] , [ eq : boundeqnonlin2 ] ) imply that @xmath63 , where @xmath64 is the bilayer thickness . this estimate yields approx . the same maximum value for @xmath65 as before since @xmath0 is at most 100 @xmath61 . thus , @xmath66 and @xmath67 can in principal vary between 0 and @xmath68 . realistically , if we set @xmath69 and consider a typical fluid phospholipid bilayer with @xmath70 , @xmath66 and @xmath67 are of the order of 1 . furthermore , we will focus on a pore radius of @xmath71 in the following . in order to understand , how adhesion energy modifies the force - distance behavior , let us first briefly revisit the case where there is no adhesion between tip and membrane ( @xmath72 ) . in fig . [ fig : profilesh0 ] the shapes of the membrane for different values of indentation are presented in scaled units . the linear calculations are dotted whereas the exact result is plotted with solid lines . for small indentations the two solutions overlap ; for increasing @xmath73 , however , the deviations become larger just as one expects for a small gradient approximation ( see also ref . @xcite for another example ) . while the differences are noticeable , they appear fairly benign , such that one would maybe not expect big changes in the force - distance behavior . we will soon find out that these hopes will not be fulfilled . , all for @xmath74 ( solid lines : nonlinear calulations , dashed lines : linear approximation , grey shades : afm tips ) . the corresponding forces @xmath75 for the three different indentations are ( nonlinear calculations ) : @xmath76 , @xmath77 , @xmath78 . ] a deeper indentation also means that the tip has to exert a higher force . in figs . [ fig:1forcedistancew=0 ] and [ fig:2forcedistancew=0 ] log - log plots of force - distance curves for different values of @xmath66 are shown . the dashed line marks the maximum indentation @xmath79 which is allowed by the geometry of tip and pore . in the limit of high forces all curves converge and approach @xmath80 ; for small forces the curves are linear in @xmath81 . let us quantify the indentation response by defining the ( scaled ) apparent spring constant @xmath82 of the nanodrum - afm system via @xmath83 a linear force - distance - curve has a constant @xmath82 and thus follows an apparent hookean behavior @xmath84 . in unscaled units , the spring constant is given by @xmath85 . for typical values @xmath86 and @xmath87 this implies @xmath88 . and @xmath89 and @xmath90 ( @xmath12 increasing from left to right ) . the curve for @xmath91 is dashed - dotted . the inset shows the corresponding scaled apparent spring constant @xmath82 ( see eqn . ( [ eq : calk ] ) ) in the small force limit , illustrating its two different regimes of small and large tension with a crossover around @xmath92 . ] and @xmath89 and @xmath90 ( @xmath12 increasing from right to left ) . the solution for @xmath93 in the linear regime is dashed - dotted . nonlinear results are plotted with solid lines , the linear approximation is dotted . ] the smaller @xmath66 , the less force has to be applied to reach the same indentation ( see fig . [ fig:1forcedistancew=0 ] ) . for decreasing @xmath66 the force - distance curves converge to the limiting curve of the pure bending case , for which @xmath91 ; this is plotted dashed - dotted in fig . [ fig:1forcedistancew=0 ] . in the opposite pure tension limit ( @xmath94 or @xmath93 ) the curves become essentially linear in @xmath66 , as can be seen clearly after scaling out the tension ( see fig . [ fig:2forcedistancew=0 ] ) . it is possible to calculate this second limiting curve in the linear regime : the linearized euler lagrange equation reduces to the laplace equation , @xmath95 , which is solved by @xmath96 in the present axial symmetry . the constants @xmath97 and @xmath98 can be determined with the help of the two boundary conditions @xmath99 and @xmath100 . the contact point @xmath24 is then determined by a straightforward energy minimization . the final result for the indentation depth is : @xmath101 \ ; , \ ] ] which is plotted dashed - dotted in fig . [ fig:2forcedistancew=0 ] . at any given penetration the force is now strictly proportional to the tension . notice also the remarkably weak ( logarithmic ) dependency of penetration on pore size . , @xmath102 , @xmath103 , @xmath104 . ] all force - distance curves presented in this section exhibit a linear behavior for small forces . in this limit the scaled spring constant for the systems just discussed is well described by the empirical relation @xmath105 ( see inset in fig . [ fig:1forcedistancew=0 ] ) . combining this with our observation that for typical system parameters @xmath106 , we see that a nanodrum s stiffness can be very well matched by available ( soft ) afm cantilevers , showing that the suggested experiments are indeed feasible . in fact , fig . [ fig : experiment ] shows the results of such an indentation experiment ( solid grey line ) . here , a fluid dotap ( 1,2-dioleoyl-3-trimethylammonium - propane chloride ) membrane was suspended over a pore of radius @xmath107 and subsequently probed with a tip of radius @xmath102 @xcite . the apparent spring constant is found to be @xmath108 . to fit the data we optimized the material parameters @xmath12 and @xmath0 . the linear approximation ( asymptotically ) matches the curve down to an indentation depth of about 40 nm as one can see in fig . [ fig : experiment ] ( dashed line ) . for larger indentations the small gradient assumption breaks down . the nonlinear calculation ( solid black line ) describes the data correctly down to a much deeper penetration depth of 150 nm but diverges for larger values . this deviation is most likely _ not _ a failure of the elastic model but a consequence of our simplified assumptions for the tip geometry . as shown in fig . 1b of the supplementary information to ref . @xcite the tip is parabolic at its apex , but further up it narrows quicker and assumes a more cylindrical shape . it therefore can penetrate the pore much deeper than one would expect if the parabolic shape were correct for the entire tip . apart from this difficulty , theory and experiment are in good agreement . there is , however , a catch . since we can not trust the force - distance behavior close to the depth - saturation ( due to its displeasingly strong dependence on the actual tip shape ) , the remaining interpretable part of the force - distance - curve is linear , and its slope is the only parameter that can be extracted from the data @xcite . for the theoretical calculation one needs two parameters , @xmath12 and @xmath0 . fitting both to a line is not possible . in ref . @xcite this obstacle was overcome by estimating @xmath0 from other measurements to be about @xmath109 . the surface tension @xmath12 could then be adjusted to @xmath110 to match the data which , reassuringly , is a very meaningful value . alternatively , one may proceed in a different manner . in the experiment a small snap - off peak could be observed upon retraction of the afm tip which was due to the attraction between tip and membrane . although this could be neglected in the interpretation of the measurements of ref . @xcite , one may think of deliberately increasing the adhesion between membrane and tip in a follow - up experiment by chemically functionalizing the tip . with this additional tuning parameter one may get further information on the values of the material parameters in question . in the following , we will also allow for adhesion between tip and membrane , _ _ i.__@xmath1_e . _ @xmath67 is not necessarily equal to zero . this will change the qualitative behavior of the force - distance curves dramatically : for fixed @xmath66 and @xmath67 different solution branches can be found . a hysteresis may occur as well , as we will see in the next section . additionally , stable membrane profiles exist even if the tip is pulled upwards . it is therefore possible to calculate the maximum pulling force that can be applied before the tip detaches from the membrane and relate it to the value of the adhesion between tip and membrane . in this section , we will first investigate the case of weak adhesion , @xmath111 . the scaled surface tension @xmath66 will be fixed to 1 . it turns out that once the tip is adhesive , `` overhang '' profiles may occur , _ _ i.__@xmath1_e . _ , shapes where at some point @xmath112 . we will first ignore these solution branches and come back to them later . and @xmath113 ( from right to left ) . the region of hysteresis in the curve for @xmath114 is magnified in the inset . in this case the energy barrier at @xmath115 is approx . overhang branches are omitted . ] [ fig : forcedistancew1 ] illustrates force - distance - curves for @xmath117 . compared to the nonadhesive case , for which an essentially linear behavior levels off towards maximum penetration , adhesive tips behave quite differently . already for @xmath118 an initial hookean response at small forces is soon followed by a regime in which the system displays a much greater sensitivity towards an externally applied stress , _ _ i.__@xmath1_e . _ , where the scaled spring constant @xmath82 drops at intermediate penetrations . physically this of course originates from the fact that adhesion _ helps _ to achieve higher penetrations , because the tip is pulled towards the membrane , but notice that this does not lead to a uniform reduction of @xmath82 : softening only sets in beyond a certain indentation . shortly beyond @xmath118 a point is reached where the force - distance - curve displays a vertical slope at which the apparent spring constant @xmath82 vanishes . for even larger values of adhesion a hysteresis loop opens , featuring a locally unstable region with @xmath119 . this is the case for @xmath114 , and the region around the instability is magnified in the inset of fig . [ fig : forcedistancew1 ] . notice that the dotted branch corresponding to @xmath119 still belongs to solutions for which the functional ( [ eq : etotal ] ) is stationary , yet the energy plotted against penetration @xmath120 ( or , alternatively , contact angle @xmath57 ) has a local _ maximum _ , confirming that these solutions are unstable against contact point variations . the two dashed branches in the inset of fig . [ fig : forcedistancew1 ] have a positive @xmath82 and correspond to local minima in the energy , however , they are _ globally _ unstable against the alternative minimum of larger or smaller @xmath120 . the true global minimum is indicated by the bold solid curve , which exhibits a discontinuity at @xmath115 . depending on the current scanning direction this hysteretic force - distance - curve manifests itself in a snap - on or snap - off event . such a behavior is reminiscent of a _ buckling _ transition ( such as for instance euler buckling of a rod under compression @xcite)with two caveats : first , notice that the membrane does _ not _ stay flat up to a critical buckling force at which it suddenly yields ; rather , the system starts off with a linear stress - strain relation and only later undergoes an adhesion - driven discontinuity . appreciating this point is quite important for the interpretation of measured force - distance curves : upon approach of tip and membrane the snap - on will occur _ neither _ at zero force _ nor _ at zero penetration . second , one should not forget that hysteresis is ultimately a consequence of the energy barrier which goes along with such discontinuities . for macroscopic systems this barrier is typically so big that the transition actually happens at either of the two end - points of the s - shaped hysteresis curve , where the barrier vanishes ( the `` spinodal points '' ) . however , for nano - systems barriers are much smaller , comparable to thermal energy @xmath61 , such that thermal fluctuations can assist the barrier - crossing event . in the present case the barrier at the equilibrium transition point is about @xmath116 , _ _ i.__@xmath1_e . _ , about @xmath121 for typical bilayers . however , already at @xmath122 its magnitude has decreased by about 20% . this shows that we have to expect a narrowing - down of the hysteresis amplitude compared to an athermal buckling scenario . upon increasing the adhesion @xmath67 even further , one will reach a critical value @xmath123 at which the `` back - bending branch '' of the force - distance - curve touches the vertical line @xmath124 . at this point the tip is being pulled into the pore even if there is no force at all . conversely , neglecting barrier complications , this also implies that at the critical adhesion energy @xmath125 an _ infinitesimal _ pulling force will suffice to unbind tip and membrane , _ even though _ the adhesion between tip and membrane is greater than zero . it is very important to keep this fact in mind if one wants to use afm measurements for the determination of adhesion energies . for @xmath126 one obtains stable solutions even when pulling the tip upwards ( where @xmath127 ) @xcite . the maximum possible force before detachment , @xmath128 , again corresponds to the leftmost point of the back - bend , and it increases with increasing @xmath67 ; we will come back to this later . notice that detachment always happens for values of @xmath120 which are _ positive _ , _ _ i.__@xmath1_e . _ , when the afm tip is still _ below _ the substrate level . contrary to what one might have expected , pulling will in this case _ not _ draw the membrane upwards into a tubular lipid bilayer structure ( a `` tether '' ) , which at some specific elongation will fall off from the tip and snap back . rather , the strong adhesion pulls the tip far into the pore , and while pulling on it indeed lifts it up , unbinding still happens below pore rim level . at even larger adhesion energy entirely new stationary solution branches emerge , as fig . [ fig : forcedistancew2 ] illustrates for @xmath129 and @xmath74 . we first recognize the well - known hysteretic branch , already seen in fig . [ fig : forcedistancew1 ] , which for increasing @xmath67 extends to much larger negative forces , even though the snap - off height @xmath120 does only change marginally . the shapes of two typical profiles are illustrated in the insets _ a _ and _ b_. notice that this branch is always connected to the origin , but for larger values of @xmath67 it starts off into the third quadrant ( negative values for @xmath81 and @xmath120 ) . at first sight it seems that we finally get solutions which correspond to a pulled - up membrane ; however , this region close to the origin corresponds to a maximum and is thus unstable . [ [ overhang - branches . ] ] overhang branches . + + + + + + + + + + + + + + + + + + contrary to the hysteretic branches , the new branches depicted in fig . [ fig : forcedistancew2 ] do _ not _ connect to the origin . this classifies them as a genuinely nonlinear phenomenon , since they can not be obtained as a small perturbation around the state @xmath130 . in the first quadrant ( @xmath131 ) they all correspond to profiles which show overhangs ( see inset @xmath24 and @xmath132 ) . these branches had been omitted in fig . [ fig : forcedistancew1 ] , since for weak adhesion they always correspond to maxima and are thus irrelevant . this changes for stronger adhesion , though , where they become stable in certain regions ( for instance , inset @xmath24 is locally stable ) . the details by which this happens are complicated and will be discussed in more detail below . following the new branches to negative forces we see that the one for @xmath133 loses its overhang around @xmath134 . that this can happen continuously is not surprising , since within angle - arc - length parametrization there is nothing special about the point where @xmath135 ( only the shooting method might use occurrences of @xmath136 as a potential termination criterion for integration ) . [ [ branch - splitting . ] ] branch splitting . + + + + + + + + + + + + + + + + + we also see that ( for sufficiently large @xmath67 ) there is a point where the hysteresis branch _ intersects _ the new nonlinear branch . there the values of @xmath81 and @xmath120 coincide for both branches , but the detachment angle @xmath57 and the total energy of the profile are generally different . however , the difference in energy at the intersection decreases with increasing @xmath67 , and around @xmath137 it finally vanishes . at this degenerate point a _ branch splitting _ occurs , where the connectivity of the two branches re - bridges , as illustrated in the lower left inset in fig . [ fig : forcedistancew2 ] . rather than connecting to the origin , the wide loop of the original hysteresis branch now joins into the overhang branch of the first quadrant , while its bit that was connected to the origin now joins into the overhang branch in the third quadrant . [ [ cusps . ] ] cusps . + + + + + + figure [ fig : forcedistancew3 ] shows the force - distance curve branches for the even larger adhesion energy @xmath138 . this depicts a situation well after the branch - splitting , so we recognize the old hysteretic branch connecting with overhangs , and the branch connecting to the origin extending exclusively in the third quadrant . in contrast to fig . [ fig : forcedistancew2 ] , the line styles in fig . [ fig : forcedistancew3 ] are chosen to illustrate local minima ( solid ) or maxima ( dotted ) . what immediately strikes one as surprising is that the profiles at @xmath139 belonging to the insets @xmath140 and @xmath141 _ both _ correspond to maxima , even though they sit on both sides of a back - bending branch , close to its end ( compare this to the `` usual '' scenario at ( @xmath142 , @xmath143 ) . moreover , the solution belonging to inset @xmath140 turns into a local minimum for slightly more negative forces , _ without _ any noticeable features of the branch . how can this happen ? the explanation is illustrated in the lower left inset in fig . [ fig : forcedistancew3 ] , which shows the total energy as a function of detachment angle @xmath57 . recall that extrema in this plot correspond to stationary solutions . as can be seen , the energy is _ multivalued _ , meaning that there exists more than one solution at a given detachment angle ( these would then also differ in their value of their penetration @xmath120 ) . but more excitingly , this graph exhibits a _ boundary extremum _ at a lowest possible nonzero value of @xmath57 in the form of a _ cusp_. this is how one can have two successive maxima on a curve without an intervening minimum the minimum is simply not differentiable . hence , there is a _ third _ solution branch , corresponding to the cusp , at which the contact curvature condition from eqn . ( [ eq : boundeqnonlin2 ] ) is _ not _ satisfied , because this condition is blind to the possibility of having non - differentiable extrema . plotting this cusp branch also into fig . [ fig : forcedistancew3 ] , we finally understand how the switching of a maximum into a minimum happens : it occurs at the point of intersection with the cusp branch . as the lower left inset in fig . [ fig : forcedistancew3 ] illustrates , the maximum belonging to the solution @xmath140 joins the cusp - minimum ( belonging to solution @xmath144 ) in a _ boundary flat point _ , roughly at force @xmath145 . for more negative forces this flat point turns up , leaving a boundary cusp _ maximum _ and a new differentiable minimum . notice that a similar exchange happens once more at ( @xmath146 , @xmath147 ) . incidentally , since at the cusp the contact curvature condition is not satisfied , and since this is the only point where the adhesion energy @xmath67 enters , the location and form of the cusp branch is _ independent _ of the value of @xmath67 . the existence of the cusp branch poses the question , whether the solutions corresponding to it are physically relevant ( at least the ones which are minima ) . it is not so much the lack of differentiability at a cusp minimum which causes concern , but rather the fact that it is located at a _ boundary_. take for instance the @xmath148 curve in the lower left inset of fig . [ fig : forcedistancew3 ] corresponding to @xmath139 . now consider a ( nonequilibrium ) solution which sits on the upper branch , somewhere between the solutions @xmath144 and @xmath141 . to lower the energy , this solution will reduce the detachment angle @xmath57 , thereby approaching the minimum at @xmath144 . but once @xmath144 has been reached , no further reduction in @xmath57 seems possible , since for smaller values no equilibrium solution exists . the crucial point is that our present theory is insufficient to answer what _ else _ would be going on for smaller @xmath57 . it could for instance be that there are indeed solutions , _ but they are not time - independent_. this might be analogous to the well - known situation of a soap film spanned in the form of a catenoidal minimal surface between two coaxial circular rings of equal radius @xmath149 . it is easy to show that for a ring separation exceeding @xmath150 no more stationary solution exists , even though the limiting profile is in no way singular @xcite . however , when slowly pulling the two rings beyond this critical separation , the soap film does not suddenly rupture . rather , it becomes _ dynamically _ unstable and begins to collapse . in the case we are studying here , the system drives itself to the singular boundary point , and without a truly dynamical treatment it is not possible to conclude whether it would remain there or start to dynamically approach a different solution . for this reason we do not want to overrate the significance of the cusp branch ; yet , its existence is still important in order to explain the behavior of the other `` regular '' branches , for instance their metamorphosis from maximum - branches into minimum - branches or vice versa . [ [ detachment - forces . ] ] detachment forces . + + + + + + + + + + + + + + + + + + a measurable quantity in the experiment is the detachment force between tip and membrane , which is the maximum applicable pulling force @xmath151 before the tip detaches from the membrane . in fig . [ fig : detachmentforce ] this force is plotted as a function of adhesion energy @xmath67 for different values of the scaled tension @xmath66 . starting from a certain threshold adhesion @xmath152 , below which no hysteresis occurs , @xmath151 decreases with increasing @xmath67 and exhibits a linear behavior for higher adhesions . increasing @xmath66 also increases the threshold adhesion ( _ _ e.__@xmath1_g . _ @xmath153 compared to @xmath154 ) . in the large-@xmath67-limit @xmath155 finally approaches a limit which is independent of @xmath0 and @xmath12 and only depends on the geometry . the elasticity of the membrane no longer influences the measurement of the adhesion energy not because the membrane is not deformed , but rather because its deformation energy is subdominant to adhesion . but for more realistic smaller values of @xmath67 this decoupling does not happen , and adhesion energies can only be inferred from the detachment force when a full profile calculation is performed . at higher values of @xmath66 also other qualitative features ( such as additional instabilities ) occur . however , these ramifications will not be discussed in the present work . as a function of scaled adhesion energy @xmath67 for four values of the scaled tension , @xmath156 , @xmath157 , @xmath158 and @xmath159 ( @xmath66 increasing from left to right ) . ] [ [ tethers . ] ] tethers . + + + + + + + + characteristically , the detachment happens at deep indentations ( @xmath73 close to the maximum indentation possible ) . long pulled - out membrane tubes ( `` tethers '' ) , as they have been studied in the literature @xcite , are not observed . even though in our calculations we find profiles with @xmath160 , these solutions either correspond to energetic maxima , or they are only _ local _ minima with the global minimum at @xmath161 corresponding to a significantly lower energy . this is a consequence of the adhesion balance present in our situation : upon pulling upwards , it is more favorable for the tip either to be `` sucked in '' completely or to detach from the membrane , rather than forming a long tether . as fig . [ fig : forcedistancew3 ] shows , there is a very small `` window of opportunity '' at @xmath162 where ( locally ) stable solutions pulled above the surface exist . yet , their profiles look essentially like the ones of inset @xmath140 or @xmath144 and show no resemblance to real long tethers . upon increasing the force they become unstable , such that the tip either falls of the membrane , or is drawn below the membrane plane ( notice that there exist two minima at @xmath81 slightly smaller than @xmath163 , but both at positive indentation ) . this analysis shows that it appears impossible to pull tethers using a probe with a certain binding energy , despite existing experiments in which tethers of micrometer size were generated @xcite . consequently , the assumption of an adhesion balance does not seem to be correct in these cases . indeed , in these studies the experimental setup was different ( membrane - covered micron - sized beads @xcite and afm tips covered by lipid multilayers @xcite ) . in the present situation tethers are also observed @xcite , but these events are not very reproducible , and based on the above calculations we would tentatively attribute them to a pinning of the membrane at some irregularity of the tip . in the previous sections we have discussed the indentation of a pore - spanning bilayer by an afm tip . we have seen that the force - distance curves show a linear behavior for small forces in a broad parameter range if the adhesion between tip and membrane vanishes . even though this is in agreement with recent experiments ( @xcite , see also fig . [ fig : experiment ] ) , such a linear behavior is unfortunately too featureless to reveal the values of both elastic material constants , @xmath12 and @xmath0 . one way out of this apparent cul - de - sac would be to repeat the experiment for different pore radii @xmath4 while keeping all other parameters fixed . since @xmath12 and @xmath0 are the same for all pore sizes in that case , it should be possible to extract their value from the measured force - distance curves . note that one does not have to fit both parameters simultaneously if one at first considers a pore where the radius is much larger than the characteristic lengthscale @xmath19 ( see eqn . ( [ eq : characteristiclengthscale ] ) ) . the corresponding system is in the high tension regime and the force - distance relation only depends on the surface tension ( see section [ subsec : noad ] ) . after determining @xmath12 from the resulting curve , one may subsequently extract the value of @xmath0 from a measurement of a system with smaller pore size . the elastic constants can also be obtained by considering systems where the adhesion @xmath21 between tip and membrane has been increased experimentally . as we have seen , the curves change their behavior dramatically for @xmath164 . it should thus be possible to fit two parameters to the resulting curves which would yield a local @xmath0 and @xmath12 in one fell swoop whereas @xmath21 can simultaneously be determined from the snap - on of the tip upon approach to the bilayer . the experimentalist , however , has to make sure in that case that the line of contact between tip and membrane is really due to a force balance as described in this paper and not due to other effects such as pinning of the membrane to single spots on the tip . in practice , this is rather difficult and will be a challenge for future experiments . one also has to keep in mind that the assumption of a perfect parabolic tip is quite simplistic compared to the experimental situation . it is probably valid in the vicinity of the apex but generally fails further up . since the force - distance behavior close to the depth - saturation depends strongly on the actual tip shape , one can only use that part of the force - distance curve for data interpretation where the indentation is still small . to predict the whole behavior the exact indenter shape has to be known : as long as the situation stays axisymmetric one may , in principle , redo the calcutions of this publication with the new shape . this is , however , rather tedious and therefore inexpedient in practice . our theorical approach does not account for hydrodynamic effects although the whole setup is in water and the afm tip is moved with a certain velocity . first measurements have shown , however , that it is possible to increase the velocity of the tip up to 60@xmath165@xmath166 without altering the force - distance curves dramatically @xcite . one can understand this result with the help of the following simple estimate : assume that the tip is a sphere of radius @xmath8 moving with the velocity @xmath167 in water . when indenting the membrane to a distance @xmath132 it will also have to overcome a dissipative hydrodynamic force @xmath168 in addition to the elastic resistance of the membrane . the energy dissipated in this process , @xmath169 , is of the order of the thermal energy if typical values are inserted ( @xmath170 , @xmath86 , @xmath171 60@xmath165@xmath166 , @xmath172 ) . this is substantially smaller than the corresponding elastic energy @xmath173 . complications arising from a correct hydrodynamical treatment were thus omitted here . including adhesion , the velocity of the measurement should nevertheless be as slow as possible to ensure that the line of contact equilibrates due to the force balance . if this is guaranteed , one can also check whether the predicted linear behavior between detachment force and adhesion is actually valid . we thank siegfried steltenkamp and andreas janshoff for providing the experimental data ( see fig . [ fig : experiment ] ) . we have greatly benefitted from discussions with them and with jemal guven . md acknowledges financial support by the german science foundation through grant de775/1 - 3 . in this appendix we will explain the origin of the boundary conditions ( [ eq : boundaryequationslinear ] ) and ( [ eq : boundaryequationsnonlinear ] ) : eqns . ( [ eq : boundeqlin1 ] ) follow simply from the requirement of continuity at the pore rim and the point where the membrane leaves the tip . asking for a membrane that has no kinks and thus no diverging bending energy gives eqns . ( [ eq : boundeqlin2 ] ) and ( [ eq : boundeqnonlin1 ] ) . if the membrane is free to choose its point of detachment as it is assumed here , an adhesion balance at the tip yields another boundary condition for the contact curvatures ( [ eq : boundeqlin3]/[eq : boundeqnonlin2 ] ) ( see ( * ? ? ? 12 , problem 6 ) , @xcite , and @xcite ) . in ref . @xcite a quick derivation can be found for the axisymmetric case in the constant height ensemble : varying the point of contact changes the energy of the free profile but also the energy due to the part at the tip . by setting the total variation to zero one obtains the well - known contact curvature condition ( eqn . ( [ eq : boundeqlin3 ] ) in monge gauge ) . observe that this assumes differentiability of the energy as a function of contact point position . in the force ensemble an extra term @xmath174 has to be added to the variation of the bound membrane . a term that is equal and opposite , however , enters the variation of the free membrane via the hamiltonian ( [ eq : hamiltonian ] ) . in total , both terms cancel and one again obtains the same condition ( eqn . ( [ eq : boundeqnonlin2 ] ) in angle - arclength parametrization ) . the remaining condition ( [ eq : boundeqnonlin3 ] ) stems from the fact that the total arclength is not a conserved quantity , which it would be if we used a fixed interval of integration . relaxing this unphysical constraint requires the hamiltonian to vanish @xcite . if the shape of the free membrane is known , the stress tensor @xmath175 ( @xmath176 ) can be evaluated at every point of the surface @xmath28 . the integral of its flux through an arbitrary contour @xmath177 which encloses the tip gives the force @xcite @xmath178 \ , { { \boldsymbol{l}}}- \kappa ( \nabla_{\perp}k ) \ , { { \boldsymbol{n}}}\big\ } \ ; . \label{eq : forceviastresstensor}\ ] ] the normal vectors @xmath179 and @xmath180 are perpendicular to @xmath177 and to each other in every point of the curve . in addition , @xmath179 is tangential to the surface whereas @xmath180 is normal to it . @xmath181 and @xmath182 are the curvatures perpendicular ( in direction of @xmath179 ) and tangential to the curve . the symbol @xmath183 denotes the derivative along @xmath179 . in angle - arclength parametrization , the curvatures are given by : @xmath184 , @xmath185 , and @xmath186 . ( [ eq : forceviastresstensor ] ) can then be written as @xmath187 \sin{\psi } \nonumber \\ & & \qquad\qquad\quad + \ ; \frac{1}{\rho } \big ( \dot{p}_{\psi } - \frac{p_{\psi}}{\rho}\dot{\rho } \big ) \cos{\psi } \big\ } \ ; .\end{aligned}\ ] ] the integrand can be evaluated further by inserting the hamilton equations ( [ eq : shapeequationsnonlinear ] ) and making use of the fact that the hamiltonian ( [ eq : hamiltonian ] ) is zero . one obtains @xmath188 if we now exploit axial symmetry by integrating around a circle of radius @xmath189 , we finally get @xmath190 ; the momentum @xmath55 conjugate to @xmath191 has to vanish identically which implies that the lagrange multiplier function @xmath46 is equal to @xmath59 . this at first maybe surprising result is no coincidence at all . in fact , in ref . @xcite it was shown that the lagrange multiplier functions which fix the geometrical constraints are closely related to the external forces via the conservation of stresses . expression ( [ eq : forceviastresstensor ] ) can also be translated into `` monge gauge '' . if we again exploit axial symmetry and integrate around a circle of radius @xmath189 , it reads @xmath192 \frac{h'(\rho)}{\sqrt{g } } \nonumber \\ & & \qquad \qquad + \ ; \kappa \big ( \frac{h''(\rho)}{\sqrt{g}^{3 } } + \frac{h'(\rho)}{\rho \sqrt{g } } \big ) ' \ ; \frac{1}{g } \bigg\}\bigg|_{\rho={r_{\text{int } } } } \!\ ! , \label{eq : forceviastresstensormonge}\end{aligned}\ ] ] where @xmath193 . note that the dash denotes derivatives with respect to @xmath37 . if in particular we choose to evaluate the force at @xmath194 , the expression ( [ eq : forceviastresstensormonge ] ) simplifies considerably to eqn . ( [ eq : forcelinear ] ) . the hamilton equations ( [ eq : shapeequationsnonlinear ] ) were solved by using a shooting method @xcite : for a trial contact point @xmath24 eqns . ( [ eq : shapeequationsnonlinear ] ) were integrated with a fourth - order runge - kutta method . the value of @xmath24 determined the contact angle @xmath57 and with it @xmath195 , @xmath37 , @xmath196 , and @xmath197 at @xmath198 via the boundary conditions ( [ eq : boundaryequationsnonlinear ] ) . the integration was stopped as soon as @xmath37 was equal or greater than @xmath4 . to reach @xmath4 exactly one extra integration with the correct stepsize backwards was performed . finally , the value(s ) of @xmath24 for which @xmath199 at @xmath4 were identified for fixed parameters @xmath2 , @xmath12 , @xmath21 , etc . if the calculation had been done in the constant height ensemble , one would additionally have to check whether the correct indentation @xmath3 was reached at @xmath200 after shooting . in the constant force ensemble this complication of meeting a second condition is avoided which is why we chose to use it for the nonlinear calculations . the gauss - bonnet theorem states that : @xmath201 for a simply connected surface @xcite . in our case the boundary @xmath202 of the surface @xmath14 is a circle of radius @xmath4 . its geodesic curvature @xmath203 is equal to @xmath204 , such that the second integral yields @xmath205 . thus , the integral over the gaussian curvature @xmath16 is zero as long as no topological changes occur .
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measurements with an atomic force microscope ( afm ) offer a direct way to probe elastic properties of lipid bilayer membranes locally : provided the underlying stress - strain relation is known , material parameters such as surface tension or bending rigidity may be deduced . in a recent experiment
a pore - spanning membrane was poked with an afm tip , yielding a linear behavior of the force - indentation curves .
a theoretical model for this case is presented here which describes these curves in the framework of helfrich theory .
the linear behavior of the measurements is reproduced if one neglects the influence of adhesion between tip and membrane . including it via an adhesion balance changes the situation significantly : force - distance curves cease to be linear , hysteresis and nonzero detachment forces can show up .
the characteristics of this rich scenario are discussed in detail in this article .
| 16,130 | 223 |
of the 77 extrasolar planets currently listed by the iau working group on extrasolar planets ] ( including planet candidates published in a refereed journals with @xmath0@xmath2 10 m@xmath1 ) , only three systems have been found to harbor planets in circular orbits ( e @xmath2 0.1 ) orbits beyond 0.5 au 47 uma ( fischer et al . 2002 ; butler & marcy 1996 ) , hd 27442 ( butler et al . 2001 ) , and hd 4208 ( vogt et al . 2002 ) . with 13 `` 51 peg type '' planets ( p @xmath2 5 d ) , and @xmath360 eccentric planets ( e @xmath4 0.1 ) , the long period circular orbits are the rarest of the three types of planetary systems to emerge over the last 8 years . with one exception , all the iau working group list planets orbit within 4 au of their host stars . as all these planets have been discovered via the precision doppler technique , there is a strong selection bias toward discovering systems with small semimajor axes . unsurprisingly , the only extrasolar planet so far found to orbit beyond 4 au was detected by the precision doppler survey that has been gathering data the longest ( marcy et al . 2002 ) . perhaps the most critical question facing the field of extrasolar planetary science is `` are solar system analogs ( ie . systems with giants planets in circular orbits beyond 4 au and small rocky planets orbiting in the inner few au ) ubiquitous , or rare ? '' existing precision doppler surveys will become sensitive to giant planets orbiting beyond 4 au by the end of this decade , though only those programs with long term precision of 3 or better will be able to determine if the orbits of such planets are eccentric or circular ( butler et al . 2001 , figure 11 ) . we report here a new extrasolar planet in an approximately circular orbit beyond 3 au , discovered with the 3.9 m anglo australian telescope ( aat ) . the anglo - australian planet search program is described in section 2 . the characteristics of the host star and the precision doppler measurements are presented in section 3 . a discussion follows . the anglo - australian planet search began in 1998 january , and is currently surveying 250 stars . fourteen planet candidates with @xmath0ranging from 0.2 to 10 m@xmath1 have first been published with aat data ( tinney et al . 2001 ; butler et al . 2001 ; tinney et al . 2002a ; jones et al . 2002a ; butler et al . 2002 ; jones et al . 2002b ; tinney et al . 2003a ; jones et al . 2003 ) , and an additional four planet candidates have been confirmed with aat data ( butler et al . 2001 ) . precision doppler measurements are made with the university college london echelle spectrograph ( ucles ) ( diego et al . 1990 ) . an iodine absorption cell ( marcy & butler 1992 ) provides wavelength calibration from 5000 to 6000 . the spectrograph psf and wavelength calibration are derived from the embedded iodine lines ( valenti et al . 1995 ; butler et al . this system has demonstrated long term precision of 3 ( butler et al . 2001 ) , similar to ( if not exceeding ) the iodine systems on the lick 3-m ( butler et al . 1996 ; 1997 ) and the keck 10-m ( vogt et al . hd 70642 ( hip 40952 , sao 199126 ) is a nearby g5 dwarf , at a distance of 28.8 pc ( perryman et al . 1997 ) , a @xmath5 magnitude of 7.17 , and an absolute magnitude of @xmath6 = 4.87 . the star is photometrically stable within hipparcos measurement error ( 0.01 magnitudes ) . the star is chromospherically inactive , with log@xmath7(hk ) @xmath8 @xmath94.90 @xmath100.06 , determined from aat / ucles spectra of the ca ii h&k lines ( tinney et al . 2003b ; tinney et al . figure 1 shows the h line compared to the sun . the chromospherically inferred age of hd 70642 is @xmath34 gyr . spectral synthesis ( lte ) of our aat / ucles spectrum of hd 70642 yields t@xmath11 @xmath85670 @xmath1020 k and @xmath12@xmath8 2.4 @xmath101 consistent with its status as a middle aged g5 dwarf . like most planet bearing stars , hd 70642 is metal rich relative to the sun . we estimate [ fe / h ] @xmath8 @xmath130.16 @xmath100.02 from spectral synthesis , in excellent agreement with the photometric determination of eggen ( 1998 ) . while ni tracks fe for most g & k dwarfs , the [ ni / h ] @xmath8 @xmath130.22 @xmath100.03 appears slightly high for hd 70642 . the mass of hd 70642 estimated from @xmath14@xmath5 , m@xmath15 , and [ fe / h ] is 1.0 @xmath100.05 m@xmath16 . a total of 21 precision doppler measurements of hd 70642 spanning more than 5 years are listed in table 1 and shown in figure 2 . the solid line in figure 2 is the best fit keplerian . the keplerian parameters are listed in table 2 . the reduced @xmath17 of the keplerian fit is 1.4 . figure 3 is a plot of orbital eccentricity vs. semimajor axis for the planet orbiting hd70642 , for extrasolar planets listed by the iau working group on extrasolar planets , and solar system planets out to jupiter . hd 70642b joins 47 uma c ( fischer et al . 2002 ) as the only planets yet found in an approximately circular ( e @xmath18 0.1 ) orbit beyond 3 au . prior to the discovery of extrasolar planets , planetary systems were predicted to be architecturally similar to the solar system ( lissauer 1995 ; boss 1995 ) , with giant planets orbiting beyond 4 au in circular orbits , and terrestrial mass planets inhabiting the inner few au . the landscape revealed by the first @xmath380 extrasolar planets is quite different . extrasolar planetary systems have proven to be much more diverse than imagined , as predicted by lissauer ( 1995 ) , `` the variety of planets and planetary systems in our galaxy must be immense and even more difficult to imagine and predict than was the diversity of the outer planet satellites prior to the voyager mission . '' the discovery here of a jupiter mass planet in a circular orbit highlights the existence , but also the rarity , of giant planets that seem similar to the original theoretical predictions . review of all the known extrasolar planets , both those described in refereed , published journals ( @xmath19 ) and those in the larger list of claimed extrasolar planets ( @xmath20 ) shows that @xmath37% of extrasolar planets orbiting beyond 0.5 au reside in circular orbits ( @xmath210.1 ) . further detections of planets beyond 1 au are needed to determine if circular orbits are more common for planets that orbit farther from the host star . over the next decade precision doppler programs will continue to be the primary means of detecting planets orbiting stars within 50 parsecs . by the end of this decade , doppler programs carried out at precisions of 3 or better by our group , and by others ( e.g. , mayor & santos 2002 ) , will be sensitive to jupiter and saturn mass planets orbiting beyond 4 au . the central looming question is whether these planets will commonly be found in circular orbits , or if the architecture of the solar system is rare . of the greatest anthropocentric interest are planets in intrinsically circular orbits , as opposed to the short period planets in tidally circularized orbits . nasa and esa have made plans for new telescopes to detect terrestrial mass planets . transit missions such as corot , kepler and eddington may be sensitive to terrestrial mass planets orbiting near 1 au , providing valuable information on the incidence of such planets . as terrestrial planets make photometric signatures of 1 part in 10,000 , these missions may be subject to interpretive difficulties that already challenge current ground based transit searches for jupiter sized planets . transit missions can not determine orbital eccentricity , and thus can not determine if planets are solar system analogs . these space based transit missions are targeting stars at several hundred parsecs , making follow up by other techniques difficult . ground based interferometric astrometry programs at keck and vlt are projected to begin taking data by the end of this decade . these programs are complementary to existing precision doppler velocity programs in that they are most sensitive to planets in distant orbits . like doppler velocities , astrometry needs to observe one or more complete orbits to make a secure detection and solve for orbital parameters . it is thus likely that the first significant crop of ground based astrometry planets will emerge after 2015 . the nasa space interferometry mission ( sim ) is scheduled to launch in 2009 . a key objective of the sim mission is the detection of planets as small as 3 earth masses in 1 au orbits around the nearest stars . sim offers the promise of determining actual masses of terrestrial planets , thereby securing their status unambiguously . simulations based on the sim measurement specifications , along with the proposed target stars , the 5 year mission lifetime , and a planet mass function extrapolated to the earth mass regime , yield predictions that as many as @xmath35 terrestrial planets could be found ( ford & tremaine 2003 ) . giant planets orbiting 25 au from the host stars are also detectable with sim , though the orbital parameters will not be not well determined in a 5 year mission . a 10 year sim mission yields significantly better orbital determination for jupiter analogs . overall the detection capabilities of sim are similar to existing precision doppler programs with a precision of 3 ( ford & tremaine 2003 ) , thereby providing confirmation of known planets and unambiguous masses . direct imaging missions such as the nasa terrestrial planet finder ( tpf ) and the esa darwin mission have the primary goal of detecting earth like planets and obtaining low resolution spectra that might reveal biomarkers . such missions will not return dynamical information and hence will not directly reveal the masses of detected planets . current plans call for the launching of such missions around 2015 , perhaps optimistically . we expect that continued doppler measurements , as well as future astrometric missions , will contribute significantly to the interpretation of the unresolved companions detected by tpf / darwin . we acknowledge support by nsf grant ast-9988087 , nasa grant nag5 - 12182 , and travel support from the carnegie institution of washington ( to rpb ) , nasa grant nag5 - 8299 and nsf grant ast95 - 20443 ( to gwm ) , and by sun microsystems . we also wish to acknowledge the support of the nasa astrobiology institute . we thank the australian ( atac ) and uk ( patt ) telescope assignment committees for allocations of aat time . we are grateful for the extraordinary support we have received from the aat technical staff e. penny , r. paterson , d. stafford , f. freeman , s. lee , j. pogson , and g. schaffer . we would especially like to express our gratitude to the aao director , brian boyle . the aat planet search program was created and thrived under brian s tenure as director in large measure because of his critical support and encouragement . the australia telescope national facility ( atnf ) is fortunate to have brian as their new director . butler , r. p. & marcy , g. w. 1997 , `` the near term future of extrasolar planet searches '' , brown dwarfs and extrasolar planets , held on tenerife , 17 - 21 march 1997 , ed . r. rebolo , e.l . martin , & m.r . zapatero osorio , asp conference series , vol . 134 , 162 rrr [ vel70642 ] 830.1082 & -25.8 & 4.2 + 1157.2263 & -36.4 & 4.4 + 1213.1051 & -42.6 & 4.3 + 1236.0850 & -37.5 & 5.2 + 1630.0095 & -15.9 & 3.6 + 1717.8810 & -9.4 & 3.9 + 1920.1348 & 15.0 & 4.8 + 1983.9687 & 13.7 & 5.8 + 2009.0210 & 13.2 & 4.6 + 2060.8744 & 27.7 & 3.4 + 2420.9072 & 12.7 & 3.1 + 2424.8981 & 11.4 & 3.1 + 2455.8416 & 20.0 & 2.8 + 2592.2229 & 12.7 & 2.9 + 2595.2255 & 15.2 & 3.4 + 2710.0700 & 0.8 & 3.0 + 2744.9571 & 0.3 & 3.1 + 2747.9155 & -4.2 & 2.7 + 2749.9755 & -7.4 & 2.2 + 2751.9384 & -5.6 & 2.4 + 2785.9082 & -3.4 & 2.4 + lcc [ orbit ] orbital period @xmath22 ( days ) & 2231 & 400 + velocity semiamplitude @xmath23 ( ms@xmath24 ) & 32 & 5 + eccentricity @xmath25 & 0.10 & 0.06 + periastron date ( julian date ) & 2451749 & 300 + @xmath26 ( degrees ) & 277 & 75 + m@xmath27 ( m@xmath1 ) & 2.0 + semimajor axis ( au ) & 3.3 + n@xmath28 & 21 + rms ( ms@xmath24 ) & 4.0 +
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precision doppler velocity measurements from the 3.9m anglo - australian telescope reveal a planet with a 6 year period orbiting the g5 dwarf hd 70642 .
the a = 3.3 au orbit has a low eccentricity ( e = 0.1 ) , and the minimum ( @xmath0 ) mass of the planet is 2.0 m@xmath1 .
the host star is metal rich relative to the sun , similar to most stars with known planets .
the distant and approximately circular orbit of this planet makes it a member of a rare group to emerge from precision doppler surveys .
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peculiar velocity surveys covering a fair fraction of the sky are now reaching to 6000 and beyond ( @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) and are being interpreted as evidence for substantial flows on these scales ( @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) . however , the amplitude , direction , and scale of these flows remain very much in contention , with resulting uncertainties in the theoretical interpretation and implications of these measurements ( @xcite , @xcite ) . indeed , recently published conflicting results suggest that the motion of the lg is either due , or is not due , to material within 6000 , and that _ iras _ galaxies either trace , or do not trace , the dark matter which gives rise to the observed peculiar velocities . the most recent potent reconstruction of the markiii velocities ( @xcite ) shows that the bulk velocity can be decomposed into two components arising from the mass fluctuation field within the sphere of radius @xmath3 about the lg and a component dominated by the mass distribution outside that volume . for convenience , we refer to this boundary at @xmath3 as the `` supergalactic shell '' since it includes the main local attractors in the supergalactic plane , the great attractor and perseus - pisces . this new analysis shows dominant infall patterns by the ga and pp but very little bulk flow within the supergalactic shell . the tidal component inside this volume is dominated by a flow of amplitude @xmath4 in the supergalactic direction @xmath5 , which is likely generated by the external mass distribution on very large scales ( see also @xcite , @xcite ) . this interpretation is also supported by an increasingly large number of tf / fp investigations ( based on the distribution and motion of abell clusters ) which report the detection of streaming motions of amplitudes greater than 700 beyond @xmath6 and away from the cmb dipole ( @xcite , @xcite , @xcite , @xcite ) . other investigations using nearly homogeneous samples of galaxies within and outside the supergalactic shell find motion consistent with the amplitude and direction of the cmb dipole @xcite . this suggests that the reflex motion of the local group could be explained by material contained within the supergalactic shell . this confusion stems , in large part , in our inability to perfectly match the many heterogeneous samples for flow studies into one self - consistent homogeneous catalogue . much of the problem lies in the fact that , with the exception of a few surveys beyond @xmath7 ( @xcite , @xcite , @xcite ) , none of the surveys within the supergalactic sphere sample the _ entire _ sky uniformly . in an attempt to overcome this problem , two of us ( jw & sc @xmath8 collaborators ) have recently combined the major distance - redshift surveys from both hemispheres ( published before 1994 ) into a catalog of 3100 galaxies ( @xcite ) , but showed that full homogenization at the @xmath9% level , the minimum required for a @xmath10 bulk flow detection at 6000 , can not be achieved . due to subjective reduction techniques and varying selection criteria , fundamental uncertainties remain when trying to match greatly disparate tf datasets ( @xcite ) . furthermore , a revised calibration of the markiii tf zero - points based on maximal agreement with the peculiar velocities predicted by the iras 1.2jy redshift survey suggests a possible source of systematic error for the data sets which cover the pp cone ( @xcite ) . this uncertainty has not seriously affected mass density reconstructions within the supergalactic shell ( @xcite ) but it could lead to spurious estimates of the bulk flows on larger scales . a newer calibration of the courteau / faber catalogue of northern spirals , not included in markiii , has been published ( @xcite , @xcite ) but a revision of the markiii catalogue is in progress ( @xcite ) . the need to tie all existing data bases for cosmic flow studies in an unambiguous fashion is clear . to that effect , we initiated a new survey in 1996 using noao facilities to measure tf distances for a complete , full - sky sample of sb@xmath0sc galaxies in the supergalactic shell for which we will obtain _ precise _ and _ uniform _ photometric and spectroscopic data . this will be the first well - defined full - sky survey to sample this scale , free of uncertainties from matching heterogeneous data sets . the sfi survey of giovanelli @xcite resembles ours in its scope and sky coverage , but it relies on a separate dataset ( @xcite ) for coverage of the southern sky and thus can not attain full - sky homogeneity . our survey , on the other hand , is designed from the outset to be homogeneous to the minimum level required for unambiguous bulk flow detection at the supergalactic shell . because of the overlap with existing surveys at comparable depth ( markiii + sfi ) , this new compilation will be of fundamental importance in tying the majority of existing data sets together in a uniform way , which will greatly increase their usefulness for global analyses of mass fluctuations in the universe . our sample is selected from the optical redshift survey ( @xcite ) , consisting of galaxies over the whole sky with m@xmath11 and @xmath12 from the ugc , eso , and esgc ( @xcite ) . it includes all non - interacting sb and sc galaxies with redshifts between 4500 and 7000 from the local group and inclinations between @xmath13 and @xmath14 , in regions where burstein - heiles extinction is less than 03 . this yields an all - sky catalog of 297 galaxies . following the approach of @xcite , we use the sample itself to calibrate the distance indicator relation ; this mitigates the need to tie the sample to external tf calibrators such as clusters ( although it precludes measurement of a monopole term in the velocity field ) . given a tf fractional distance error of 20% , the statistical uncertainty on a bulk flow from @xmath15 galaxies at common distance @xmath16 is @xmath17 . as the measured ( and much contested ) bulk motions on these scales are of the order of 300 , a detection of high statistical significance is well within reach . data taking and reduction techniques follow the basic guidelines of previous optical tf surveys ( @xcite , @xcite , @xcite , @xcite ) . our survey is now complete , which is essential to achieve our statistical requirements and ensure a rigorous analysis . the spectroscopy relies on measurement of h@xmath18 rotation velocities at 2.2 disk scale lengths for the tightest tf calibration and best match to analogous 21 cm line widths ( @xcite , @xcite ) . the photometry is based on the kron - cousins @xmath19 and @xmath20 systems which will allow direct matching with two largest tf field samples to date ( @xcite,@xcite ) . one of the key features of this study is not only its all - sky sample selection but the independent duplication of all data reductions ( by at least 2 , if not 3 , of us ) . these reductions and a first flow analysis based on the shellflow sample alone should be published soon ( @xcite ) . we also plan a more extensive analysis using the recalibrated markiii combined with other new catalogs not included in the original markiii . 99 corwin , h. g , & skiff , b. a. 1994 , extension to the southern galaxies catalogue , in preparation clutton - brock , m . , & peebles , p.j.e . 1981 , aj , 86 , 1115 courteau , s. 1992 , phd . thesis , uc santa cruz courteau , s. , faber , s.m . , dressler , a. , & willick , j.a . 1993 , apj , 412 , l51 courteau , s. 1996 , apjs , 103 , 363 courteau , s. 1997 , aj , 114 , 2402 courteau , s. ( + shellflow team ) 1999 ( in preparation ) da costa , l. n. , freudling , w. , wegner , g. , giovanelli , r. , haynes , m.p . , & salzer , j.j . 1996 , apj , 468 , l5 dekel , a. , eldar , a. , kolatt , t. , yahil , a. , willick , j. a. , faber , s. m. , courteau , s. , & burstein , d. 1999 , apj ( submitted ) giovanelli , r. , haynes , m.p . , freudling , w. , da costa , l. n. , salzer , j.j . , & wegner , g. 1998 , apjl , in print , astro - ph/9807274 han , m .- s . , & mould , j. r. 1992 , apj , 396 , 453 hudson , m. ( + smac team ) 1998 ( in preparation ) lauer , t. r. , & postman , m. 1994 , apj , 425 , 418 mathewson , d. s. , & ford , v. l. 1994 , apj , 434 , l39 mathewson , d. s. , ford , v. l , & buchhorn , m. 1992 , apjs , 81 , 413 [ m92 ] postman , m. 1995 , in _ dark matter _ , aip conf . series 336 , 371 postman , m. , & lauer , t. r. 1995 , apj , 440 , 28 riess , a. , press , w. , & kirshner , r. p. 1995 , apj , 445 , l91 santiago , b. x. , strauss , m. a. , lahav , o. , davis , m. , & huchra , j. p. 1995 , apj , 446 , 457 scaramella , r. , 1989 , nature , 338 , 562 schlegel , d. 1996 , phd . thesis , uc berkeley strauss , m.a . 1996 , in _ critical dialogues in cosmology _ , ed . neil turok ( singapore : world scientific ) strauss , m.a . , & willick , j.a . 1995 , physics reports , 261 , 271 willick , j. a. , courteau , s. , faber , s. m. , burstein , d. , dekel , a. , & strauss , m. a. 1997 , apjs , 109 , 333 willick , j. a. & strauss , m. s. 1998 , apj , in press ( astro - ph/9801307 ) willick , j.a . 1998 , apj ( submitted ) willick , j.a . 1999 ( in preparation )
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we present a new optical tully - fisher ( tf ) investigation for a complete , full - sky sample of 297 sb@xmath0sc spirals with redshifts between 4500 and 7000 .
the survey was specifically designed to provide _ uniform , well - calibrated _ data over both hemispheres . all previous tf surveys within the supergalactic shell ( @xmath1 ) have relied on matching separate data sets in the northern and southern hemispheres and thus can not attain full - sky homogeneity .
analyses of the cosmological dipole and peculiar velocities based on these studies have produced contradictory claims for the amplitude of the bulk flow and whether it is generated by internal or external mass fluctuations .
with shellflow , and further zero - point calibration of existing tf data sets , we expect a high - accuracy detection of the bulk flow amplitude and an unambiguous characterization of the tidal field at 6000 .
# 1#1@xmath2
| 3,188 | 255 |
the experimental evidence of the theoretically predicted skyrmions in non - centrosymmetric compounds with dzyloshinskii - moriya interaction has intrigued many scientists over the last years.@xcite recently , the preparation of thin films of b20 type mnsi on silicon substrates @xcite has offered promising prospects with regard to possible applications in future spintronic devices . on the one hand mnsi films offer a variety of interesting magnetic phases and on the other hand they are easy to integrate into devices due to the use of silicon as substrate material being well established in technology . the benefit of thin films compared to bulk material is the existence of the skyrmion phase in an extended region of the magnetic phase diagram due to the uniaxial anisotropy.@xcite this pioneers new opportunities for data storage devices . the drawback using mnsi films is the low magnetic ordering temperature , which is considerably below liquid nitrogen temperature . therefore , it is the aim to find compounds with similar spin order at higher temperatures . a suitable candidate is the b20 compound mnge ( bulk lattice constant of 4.795 ) with a magnetic ordering temperature @xmath0 of 170k.@xcite the magnetic ground state of mnge is a helical spin structure with a helix length between 3 nm at lowest temperatures and 6 nm near @xmath0.@xcite the helix axis is due to magnetic anisotropy pinned along @xmath1001@xmath2,@xcite but rotates into field direction in an applied field . recently , a large topological hall effect exceeding by 40 times that of mnsi was observed in mnge which was attributed to a skyrmion phase in analogy to mnsi.@xcite . further evidence for the existence of skyrmions was given by small angle neutron scattering experiments.@xcite unfortunately , the synthesis of mnge is considerably laborious , since it forms only under high pressure and temperatures between 600 and 1000@xmath3c.@xcite however , molecular beam epitaxy ( mbe ) allows for thin film growth under strong non - equilibrium conditions . nevertheless , there has been no successfull attempt to grow mnge on ge , since mn and ge tends to form mn@xmath4ge@xmath5.@xcite the use of si(111 ) as substrate offers the opportunity to prepare a seedlayer of mnsi , which realizes the b20 crystal structure for mnge growth . the lattice constant of mnge within the ( 111 ) plane matches that of si with a misfit of only 2@xmath6 , thus , compressively strained mnge films may be grown on si(111 ) substrates . in this paper we show a preparation method for mnge thin films on si substrates with the aid of a mnsi seedlayer . the structure and morphology of the films have been investigated by reflection high - energy electron diffraction ( rheed ) , atomic force microscopy ( afm ) and x - ray diffraction ( xrd ) . to determine the physical properties of the samples magnetization and magnetoresistance measurements have been performed . for the growth of mnge thin films p - doped si(111 ) substrates were used , which possess a resistivity between 1 and 10 @xmath7 cm at room temperature . prior to film deposition the substrates were heated to 1100@xmath3c under uhv conditions in order to remove the oxide layer and to achieve a clean and flat surface with 7@xmath87-reconstruction , which was verified by in - situ rheed investigations . the depostion of mn and ge directly on the si(111 ) surface does not produce b20 mnge films but results in a mn@xmath4ge@xmath5 layer . in order to establish the b20 crystal structure a 5mn layer was deposited onto the si surface and heated to 300@xmath3c subsequently . this procedure provides for the formation of a thin mnsi seedlayer with a thickness of 10 . in a second step , mnge is codeposited by simultanoeus evaporation of mn and ge from an effusion cell and an electron beam evaporator , respectively . during film growth with a rate of 0.15 / s the substrate is held at a temperature of 250@xmath3c . $ ] crystal direction and b ) line scans across the rheed streaks for the mnge film in comparison with the si substrate . the scans were taken parallel to the shadow edge.,scaledwidth=35.0% ] the mnge films have been investigated by in - situ rheed in order to determine their structure and morphology . the rheed pattern of a 135 mnge film observed along the @xmath9 $ ] direction of the si substrate indicates two - dimensional film growth [ fig . [ fig : rheed](a ) ] . the arrangement of the streaks is very similar to the pattern of mnsi thin films,@xcite and suggests that mnge sustains the b20 crystal structure provided by mnsi seedlayer . the uniformity of the intensity of the detected streaks implies a flat surface of a size comparable to the area contributing to the rheed pattern of around 100 nm in diameter . line scans across rheed patterns of a 135 mnge film [ fig . [ fig : rheed](b ) ] compared to the si substrate reveal a nearly pseudomorphic growth of the mnge layer . however , a small deviance of the mnge streaks from the corresponding si reflections indicates that the mnge lattice has at least partly relaxed from the compressive strain imposed by the substrate . + afm images of films with thicknesses of 45 , 90 and 135 give evidence that island growth of vollmer - weber type is the predominant growth mode [ fig . [ fig : afm ] ] . the thinnest film of 45 thickness [ fig . [ fig : afm](a ) ] consists of islands with a typical diameter of 100 nm separated by valleys of similar size . with increasing film thickness the islands are enlarged and gradually fill the space between them . for the 135 film only very thin valleys of a few nm can be observed [ fig . [ fig : afm](c ) ] , and the morphology has transformed into elongated islands with a length of up to 2@xmath10 m and a width of around 200 nm . $ ] and @xmath11 $ ] crystal directions . inset : intensity plot along the @xmath12 $ ] direction.,scaledwidth=40.0% ] x - ray diffraction measurements were performed using synchrotron radiation with @xmath13 at the swiss - norwegian beamline bm1a of the esrf ( grenoble , france ) with the pilatus@snbl diffractometer . the investigation of the 135 film confirms the b20 crystal structure of the mnge . in fig . [ fig : xray ] the ( 111 ) and ( 333 ) peaks of the si substrate and the mnge thin film are clearly resolved as single crystal peaks . the inset shows an integrated diffraction pattern along the [ 111 ] direction . from the position of the mnge(111 ) peak the lattice parameter of the mnge film of ( 4.750 @xmath14 0.004) is obtained , which is 1% smaller than the value for bulk mnge due to compressive strain . the magnetic characterization of the mnge films was carried out using a quantum design mpms-5s squid magnetometer . for films of different thickness the temperature dependence of the magnetic susceptibility has been measured in the range from 5k to 300k in an applied magnetic field of 10mt [ fig . [ fig : susceptibility ] ] . below 40k the susceptibility slightly increases due to the mnsi seedlayer , that orders magnetically in this temperature range . the measurements were normalized with respect to the saturation magnetization of the mnsi seedlayer , because this layer is the same for all three films . the susceptibility of mnge films exhibits an ordering temperature of @xmath15k indicated by a broad peak . regarding mnge bulk material , the susceptibility shows a qualitatively similar behavior with a lower @xmath16k.@xcite an enhancement of the ordering temperature has also been observed for mnsi thin films.@xcite in contrast to mnsi thin films , no thickness dependence of @xmath0 can be detected for films between 45 and 135 . possibly , the spin - spin correlation length is shorter than the value for mnsi films ( 7 monolayers ) @xcite , i.e. the thickness dependence may occur for mnge when the films are thinner than investigated in this work . both materials belong to the b20 compounds , which possess a helical spin structure , since their magnetic properties are governed by the interplay of ferromagnetic exchange and dzyaloshinskii - moriya interactions . nevertheless , the susceptibility of mnge shows a behavior which is typical for antiferromagnetic order , whereas for mnsi an increase to a constant value of magnetization towards low temperature is observed . the helix length in mnsi is very long ( 18 nm ) and , thus , the local magnetic structure is related to ferromagnetism . a small field easily deforms the soft helix and induces a net magnetization . in the case of mnge the helix is more rigid . therefore , at low temperature no net magnetic moment is induced by a field as small as 10mt . the helix wavelength is connected to the dzyaloshinskii constant @xmath17 via @xmath18 , where @xmath19 is the magnetic stiffness.@xcite since the helix in mnge is extremely short ( @xmath20nm)@xcite the dzyaloshinskii constant is expected to be large , and the magnetic structure is very close to an antiferromagnet . field dependent magnetization measurements at @xmath21k were carried out on the same three samples as in fig . [ fig : susceptibility ] . for all samples the magnetization increases in fields up to 1 t [ fig . [ fig : magnetization ] ] . the inset of fig . [ fig : magnetization ] shows a magnetization measurement on the 135 film in fields up to 5 t , which reveals that saturation is reached around 1 t . this is in agreement with measurements performed on bulk mnge at temperatures close to @xmath0.@xcite since the helix length is much shorter than the size of the mnge islands , the magnetization behavior is not expected to be different from the bulk . the magnetic moment per mn atom was calculated assuming that the complete amount of mn deposited during growth has reacted to mnge . however , since the magnetic moment is only half of the bulk value , some part of the deposited mn did not form mnge . furthermore , we observe an apparently larger magnetic moment for thicker films . this can be explained by the fact that especially in the beginning of mnge growth not every mn atom is incorporated into the mnge crystal . evidence for this is also given by magnetization measurements at 5k , where mainly the ordering of the mnsi seedlayer is observed , since mnge only gives a linear contribution in the considered field range . we observe a moment of the mnsi layer that is about twice as large as expected for 10 mnsi . thereby , we conclude that some part of the deposited mn has reacted with si from the substrate to form mnsi . + resistivity measurements were performed on the 135 film using the van - der - pauw method . the sample was found to be metallic , and the residual resistivity at 3k was determined as 83 @xmath22 cm . the field dependence of the resistivity was measured in magnetic fields up to 5 t for several temperatures . in fig . [ fig : mr ] three curves are depicted , which represent the magnetoresistivity defined by @xmath23 . the data were obtained at temperatures between 60k and 100k , where the sample is in a magnetically ordered state . the mr effect is negative for all temperatures and fields and exhibits no remarkable features in the investigated field range . for comparison the equivalent data for a 19 nm film mnsi is shown in the inset of fig . [ fig : mr ] . in the case of mnsi the critical magnetic field , where the spins align ferromagnetically , occurs around 1 t . at this field a clear kink accompanied by a change in curvature is observed . furthermore the size of the mr effect is considerably larger for mnsi . regarding mnge the absence of magnetic phase transitions in moderate magnetic fields and the smallness of the magnetoresistivity evidences that the helical structure is more rigid than in mnsi . as discussed in the previous paragraph this can be ascribed to a stronger dzyaloshinskii - moriya interaction . in this work we have proved that we succeeded in growing crystalline mnge as a thin film on a si(111 ) substrate . the film adopts the b20 structure from a thin seedlayer of mnsi prepared prior to mnge growth . morphological investigations using rheed and afm give evidence that the mnge thin films consist of islands with a flat surface , which enlarge during growth . the b20 structure was confirmed by xrd and the lattice parameter was determined to be 1% smaller than in bulk mnge due to compressive strain imposed by the si substrate . + although the magnetic properties of mnge thin films are found to be qualitatively similar to bulk , the ordering temperature is enhanced to 200k . in magnetoresistivity measurements no critical fields were observed up to 5 t . compared to mnsi , the helix in mnge is shorter and more rigid than in mnsi . therefore , the magnetic structure is related to antiferromagnetism rather than to ferromagnetism . we would like to thank dmitry chernyshov for his support with the x - ray measurements at the european synchrotron radiation facility . the afm measurements were performed at the institute of semiconductor technology in braunschweig . we thank alexander wagner for his help with the equipment . 99 s. mhlbauer , b. binz , f. jonietz , c. pfleiderer , a. rosch , a. neubauer , r. georgii , and p. bni , science * 323 * , 915 ( 2009 ) . a. neubauer , c. pfleiderer , b. binz , a. rosch , r. ritz , p. g. niklowitz , and p. bni , phys . * 102 * , 186602 ( 2009 ) . u. k. rler , a. n. bogdanov , and c. pfleiderer , nature * 442 * , 797 ( 2006 ) . j. engelke , t. reimann , l. hoffmann , s. gass , d. menzel , and s. sllow , j. phys . . jpn . * 81 * , 124709(2012 ) . e. a. karhu , u. k. rler , a. n. bogdanov , s. kahwaji , b. j. kirby , h. fritzsche , m. d. robertson , c. f. majkrzak , and t. l. monchesky , phys . b * 85 * , 094429 ( 2012 ) . m. n. wilson , e. a. karhu , a. s. quingley , u. k. rler , a. b. butenko , a. n. bogdanov , m. d. robertson , and t. l. monchesky , phys . b * 86 * , 144420 ( 2012 ) . a. b. butenko , a. a. leonov , u. k. rler , and a. n. bogdanov , phys . rev . b * 82 * , 052403 ( 2010 ) . y. li , n. kanazawa , x. z. yu , a. tsukazaki , m. kawasaki , m. ichikawa , x. f. jin , f. kagawa , and y. tokura , phys . * 110 * , 117202 ( 2013 ) . h. takizawa , t. sato , t. endo , and m. shimada , j. solid state chem . * 73 * , 40 ( 1988 ) . n. kanazawa , y. onose , t. arima , d. okuyama , k. ohoyama , s. wakimoto , k. kakurai , s. ishiwata , and y. tokura , phys . lett . * 106 * , 156603 ( 2011 ) . o. l. makarova , a. v. tsvyashchenko , g. andre , f. porcher , l. n. fomicheva , n. rey , and i. mirebeau , phys . rev . b. * 85 * , 205205 ( 2012 ) . n. kanazawa , j .- h . kim , d. s. inosov , j. s. white , n. egetenmeyer , j. l. gavilano , s. ishiwata , y. onose , t. arima , b. keimer , and y. tokura , phys . b * 86 * , 134425 ( 2012 ) . s. olive - mendez , a. spiesser , l. a. michez , v. le thanh , a. glachant , j. derrien , t. devillers , a. barski , and m. jamet , thin solid films * 517 * , 191 ( 2008 ) . r. gunnella , l. morresi , n. pinto , r. murri , l. ottaviano , m. passacantando , f. dorazio , and f. lucari , surf . sci . * 577 * , 22 ( 2005 ) . e. karhu , s. kahwaji , and t. l. monchesky , phys . rev . b * 82 * , 184417 ( 2010 ) .
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mnge has been grown as a thin film on si(111 ) substrates by molecular beam epitaxy .
a 10 layer of mnsi was used as seedlayer in order to establish the b20 crystal structure .
films of a thickness between 45 and 135 have been prepared and structually characterized by rheed , afm and xrd .
these techniques give evidence that mnge forms in the cubic b20 crystal structure as islands exhibiting a very smooth surface .
the islands become larger with increasing film thickness .
a magnetic characterization reveals that the ordering temperature of mnge thin films is enhanced compared to bulk material .
the properties of the helical magnetic structure obtained from magnetization and magnetoresistivity measurements are compared with films of the related compound mnsi . the much larger dzyaloshinskii - moriya interaction in mnge results in a higher rigidness of the spin helix .
| 5,100 | 232 |
piecewise isometries have rich dynamical phenomena and they sometimes produce fractal - like pictures . to define these maps , let @xmath5 be a subset of @xmath6 with a finite partition @xmath7 . a piecewise isometry @xmath8 is a map such that the restriction of @xmath9 to each @xmath10 is a euclidean isometry . the map is not defined on the boundaries @xmath11 for @xmath12 . in this paper , we introduce a one - parameter family of piecewise isometries called the _ tetrahedral twists_. the intuitive definition is the following : let @xmath13 be the surface of a regular tetrahedron of side length 1 . pick one pair of the opposite edges of @xmath13 and cut them open , then @xmath13 becomes a cylinder intrinsically . rotate the cylinder by amount @xmath1 counterclockwise . glue the opposite edges so that @xmath13 becomes the surface of a tetrahedron again . apply this procedure on the other two pairs of opposite edges of @xmath13 . the entire process defines a piecewise isometry on @xmath13 which is called the tetrahedral twist . a polytope exchange transformation ( pet ) is a piecewise isometry @xmath8 on a polytope @xmath5 with two conditions : 1 . the restriction on each @xmath14 is a translation . the image @xmath15 has the full area in @xmath5 . the tetrahedral twist maps are not piecewise translations . however , there exist double covers @xmath16 of @xmath13 such that the liftings of the tetrahedral twists produce pets which we call the tetrahedral pets . we will discuss this construction in section 2.1 . let @xmath17 be a subset of @xmath5 . given a map @xmath18 , the _ first return _ @xmath19 is a map assigns every point @xmath20 to the first point in the forward orbit of @xmath21 lies in @xmath17 under @xmath22 , i.e. @xmath23 a _ renormalization _ of a pet @xmath8 is the choice of a subset @xmath17 of x such that the first return map @xmath24 is also a pet . the existence of renormalization scheme in a dynamical system allows us to study the acceleration of the orbits . if a renormalization scheme exists , we say that the system is _ renormalizable_. the family of tetrahedral pets is renormalizable in the parameter space @xmath2 . out main goal is to show the following theorem : the family of tetrahedral pets has a renormalization scheme when the input parameter lies in the subintervals @xmath25 and @xmath26 $ ] . the subinterval @xmath27 of @xmath2 is a neighborhood of the irrational number @xmath28 . the interval exchange transformations ( iets ) are the examples of piecewise isometries in dimension 1 , see @xcite , @xcite for surveys . the paper @xcite introduces rectangle exchanges which are the products of iets . another important class of piecewise isometries is piecewise rotation on polygons , which is studied in papers such as @xcite , @xcite . we know that piecewise rotations are closely related to the study of pets in the following sense : let @xmath29 be a polygon together with a finite partition @xmath30 . if the piecewise rotation map @xmath9 performs a translation or rotation by a rational multiple of @xmath31 restricted on each element of @xmath30 , then there is a pet @xmath32 conjugate to @xmath9 by a covering map @xmath33 . the outer billiard maps on a convex polygon @xmath34 also give rise to piecewise isometries , see @xcite for reference . the square of the outer billiards map is a piecewise translation outside @xmath34 . in the paper @xcite , @xcite , a higher dimensional pet is constructed from the compactification of the outer billiard outside a kite . for work concerning renormalization of piecewise isometries , the rauzy induction @xcite introduces a renormalization theory for iets . in the paper @xcite , a general theory of renormalization of piecewise rotations is developed . the paper @xcite shows that the renormalization scheme exists for pets arising from the outer billiards on penrose kites . the tetrahedral twists are very similar to the pets described in @xcite , @xcite by hooper . in @xcite , the map is defined on four copies of torus , which we denote by @xmath5 . for every point @xmath35 , the map performs a translation in either horizontal direction parametrized by @xmath36 or in the vertical direction parametrized by @xmath37 , respectively . this is the first example of pets in 2-dimensional parameter space which is invariant under renormalization . hooper describes the renormalization procedure in terms of the renormalization of the truchet tilings , see @xcite . let @xmath38 be reflections about the points @xmath39 , @xmath40 and @xmath41 , respectively . let @xmath42 be the group generated by @xmath38 . define the space @xmath43 . a fundamental domain for the action of @xmath42 by reflection is the union of four equilateral triangles @xmath44 of side length 1 where @xmath45 has the vertices @xmath46 and @xmath47 is the reflection of @xmath45 by the line connecting the points @xmath48 and @xmath49 for @xmath50 . in fact , these four triangles are the faces of a regular tetrahedron . for the action of @xmath42.,width=134 ] fix three parameters @xmath51 . let @xmath52 , @xmath53 , @xmath54 be the unit vectors in the directions of cube roots of unity . for each @xmath55 or @xmath56 , we define the map @xmath57 as follows : @xmath58 where @xmath59 and @xmath60 from top to bottom . ] the maps @xmath61 are illustrated in figure @xmath56 . now , suppose @xmath62 for @xmath63 . a family of tetrahedral twists @xmath64 is defined as @xmath65 let @xmath66 be the lattice generated by two vectors @xmath67 and @xmath68 and @xmath69 be the torus @xmath70 . let @xmath71 be the projection given by @xmath72 and @xmath16 is a double cover of @xmath13 . let @xmath73 be the reflection about the origin . define @xmath74 for @xmath75 , so @xmath76 for each @xmath77 . for fixed parameters @xmath78 , the lifting @xmath79 of the map @xmath80 is given by the following equation : @xmath81 where @xmath82 , @xmath83 and @xmath60 ] on @xmath69 . for each @xmath85 , @xmath69 is divided into halves by a line @xmath86 through the origin in direction @xmath87 . on one half of @xmath69 , the map @xmath79 translates every point @xmath88 by amount @xmath89 ( mod @xmath90 ) in direction @xmath87 . in the other half , every point is translated by the same amount but in the opposite direction @xmath91 . therefore , @xmath79 is a pet , for each @xmath63 . as mentioned in previous section , we set @xmath62 for all @xmath63 . the composition @xmath92 is defined as @xmath93 for @xmath94 , the map @xmath95 is a pet . we call @xmath95 a _ tetrahedral pet_. more precisely , for every point @xmath96 , there is some translation vector @xmath97 such that @xmath98 where @xmath97 is in the form of @xmath99 for some @xmath100 . fix @xmath101 . the partition @xmath102 of @xmath69 associated to the tetrahedral pet @xmath95 is obtained by the following fact : suppose that @xmath103 and @xmath104 are pets and @xmath105 , @xmath106 are the partitions of @xmath107 determined by the maps @xmath108 and @xmath109 , respectively . then , @xmath110 is a finer partition of @xmath5 determined by the pet @xmath111 where @xmath112 the following figures show an example of the tetrahedral pet @xmath95 when @xmath113 . the figure on the left shows the partition @xmath102 determined by a tetrahedral pet @xmath95 . the figure on the right shows the image of every element in @xmath102 under @xmath95 . to be clear , we assign a number to each element in the partition @xmath102 in the left figure whose image is the shape with the same number in the right figure . for @xmath113 ] let @xmath115 be a periodic point with period @xmath116 of the map @xmath95 . a periodic tile @xmath117 of @xmath95 is a maximal subset containing @xmath118 such that @xmath95 is entirely defined on @xmath117 and all points in @xmath117 have the same period as @xmath118 . for a given point @xmath119 , we provide a pseudo - code algorithm to produce a periodic tile @xmath117 containing @xmath118 of @xmath95 . 1 . let @xmath120 be a polygon in the partition @xmath102 such that @xmath121 . 2 . if @xmath122 , then let @xmath123 where @xmath124 is some element in the partition @xmath102 and @xmath125 . set @xmath126 3 . else , return @xmath127 . by construction , every periodic tile @xmath117 is convex since it is the intersection of convex polygons . a periodic tiling @xmath128 is the union of all periodic tiles @xmath117 for all @xmath119 . for @xmath129,width=268 ] for @xmath130,width=268 ] define the renormalization map @xmath131 by the formula . @xmath132 for any subset @xmath133 , we write @xmath134 as the first return map of @xmath95 on @xmath135 . let @xmath101 and @xmath136 be the reflection about the line @xmath137 . define @xmath138 as the semi - regular hexagon with vertices : @xmath139 @xmath140 figure 6 shows an example of @xmath141 for @xmath142 . define the subsets @xmath143 of @xmath69 as follows : @xmath144 suppose @xmath145 $ ] and @xmath146 . there exists a set @xmath147 such that @xmath148 where @xmath149 is a similarity with the scale factor @xmath150 . let @xmath151 be the upper half plane in of @xmath152 and @xmath153 be the lower half . define @xmath154 if @xmath155 , then @xmath95 is conjugate to @xmath156 by a piecewise translation map @xmath149 where @xmath157 the theorem above says that the periodic tiling of @xmath158 and @xmath156 are same up to the interchange of @xmath159 and @xmath160 . for any @xmath161 , there exists a set @xmath147 such that @xmath162 is conjugate to @xmath163 via a similarity @xmath149 with the scale @xmath164 . a space @xmath5 has a mostly self - similar structure if there is a disjoint union @xmath165 such that each @xmath166 is self - similar . let @xmath167 $ ] that is a fixed point under the renormalization map @xmath168 . the periodic tiling @xmath128 is mostly self - similar . the proofs will be provided in section 5 . on the left and @xmath169 on the right ] on the left and @xmath169 on the right , title="fig:",width=264 ] on the left and @xmath169 on the right , title="fig:",width=275 ] for @xmath170,title="fig:",width=288 ] for @xmath170,title="fig:",width=249 ] in this section , we explore the properties of the renormalization map @xmath168 and its connection to continued fraction expansions . recall that @xmath131 is given by the formula : @xmath171 each fixed point @xmath172 of @xmath168 in @xmath173 is in the form of @xmath174 moreover , all fixed points @xmath172 have continued fractions expansion in the following form : @xmath175 let @xmath176 in @xmath2 . there exists some integer @xmath177 such that @xmath178 . we have @xmath179 write @xmath180 . when we apply the square map @xmath181 , the denominators have the fact that @xmath182 . thus , the @xmath183 drops to a value of @xmath184 or @xmath56 for some @xmath177 . it means that @xmath185 must be @xmath186 or @xmath187 for some @xmath177 . for every @xmath188 , we can define a coding map @xmath189 as follows : @xmath190 let @xmath101 . a coding sequence @xmath191 for @xmath172 is a sequence of finite or infinite length such that every element @xmath192 in the sequence is given by the formula @xmath193 for @xmath177 . if @xmath194 , then the sequence terminates at step at step @xmath195 . if @xmath196 , then the sequence terminates at step @xmath197 . by lemma 3.1 , the coding sequence for a rational number terminates after a finite number of steps . for example , the coding sequence of @xmath198 is @xmath199 now , we define @xmath200 as the set of all sequence @xmath201 of finite or infinite length satisfying the following condition : 1 . @xmath202 can not appear consecutively in the sequence 2 . for a finite sequence @xmath203 , the last element @xmath204 . let @xmath205 be a sequence in @xmath200 . 1 . if @xmath205 is infinite , there is a unique @xmath1 determined via the formula @xmath206 moreover , the coding sequence of @xmath172 is @xmath207 . if the sequence @xmath208 is finite of length @xmath209 and @xmath210 , then there is a unique @xmath1 determined by @xmath211 3 . if @xmath212 is finite and @xmath213 , then there is a unique @xmath101 determined by the formula ( * ) but without @xmath214 . let @xmath215 be a sequence of elements in @xmath200 and @xmath1 is determined by the formula ( * ) . we want to show that @xmath172 has the coding sequence @xmath77 . 1 . suppose the first element in the sequence @xmath216 is @xmath217 for @xmath218 . write @xmath146 i.e. , @xmath219 by computation , we have @xmath220 2 . suppose @xmath221 . similarly , we set @xmath146 so that @xmath222 . therefore , @xmath223 repeat this argument by substituting @xmath224 . if there exists some element @xmath225 or @xmath226 in the sequence , then the sequence terminates at the length @xmath197 . we obtain the desired statement . let @xmath101 and @xmath227 be the coding sequence of @xmath172 . a splitted expansion of @xmath172 is defined as follows : * @xmath228 , * @xmath229 for @xmath230 . * if @xmath172 is rational and the coding sequence of @xmath172 has length @xmath231 , then the splitted expansion terminates at @xmath232 if @xmath233 or at @xmath234 if @xmath235 . for example , the splitted sequence for @xmath236 is @xmath237 here are several observations of the splitted expansion : * if @xmath172 is a fixed point by @xmath168 , then the splitted expansion is same as the continued fraction expansion of @xmath172 which is in the form of @xmath238 * the splitted expansion is shifted by 2 digits to the left under the renormalization map @xmath168 . * to translate between the splitted expansion and the signed continued fraction expansion , we have to replace the fragment @xmath239 with @xmath240 . + for example , @xmath241 has the splitted fraction expansion @xmath242 and its signed continued fraction expansion is @xmath243 . suppose @xmath172 has a splitted expansion @xmath244 of inifite length . we set the @xmath197th convergent @xmath245 of @xmath172 as @xmath246 the recurrent formulas for @xmath247 and @xmath183 are same to the ones of the continued fraction expansion , i.e. * @xmath248 . * if @xmath249 , then we set @xmath250 and @xmath251 * if @xmath252 , then @xmath253 . we can set @xmath254 and @xmath255 . then @xmath256 and @xmath257 are obtained by the same formula as above for all integer @xmath258 . the theorem below says that the splitted expansion gives us a good approximation of irrationals . let @xmath1 be irrational with infinite splitted expansion @xmath259 . if @xmath260 as @xmath261 , then @xmath262 . the proof is same as theorem 11.4 in @xcite by passing to the signed continued fraction expansion of @xmath172 . the motivation of this section is to construct convex polyhedra and reduce all the calculations to the polyhedra , which is very similar to schwartz s construction in @xcite . recall that @xmath69 obtained by gluing the parallelogram with vertices @xmath263 we define @xmath264\}.\ ] ] as a fiber bundle over @xmath265 $ ] . the fiber above @xmath172 is the parallelogram @xmath69 . define the fiber bundle map @xmath266 as @xmath267 define @xmath268 as the set @xmath269 it is useful to split the fiber bundle @xmath270 as @xmath271 \cup \mathcal x[1/2 , 1].\ ] ] a maximal domain of @xmath273 is a maximal subset of @xmath273 such that the bundle map @xmath274 is entirely defined and continuous . for @xmath275 , every cross section of the union of maximal domains in @xmath273 at the plane @xmath276 is the partition of @xmath69 determined by the tetrahedral pet @xmath95 . by the assistant of computer , we know that @xmath273 is partitioned into 22 maximal domains . each maximal domain is a convex polyhedron which has rational vertices . experimentally , we obtain the fact that every maximal domain in @xmath273 has vertices in the form of @xmath277 for @xmath278 and integers @xmath279 . let @xmath281 be subsets of @xmath282 $ ] whose fiber over @xmath172 are the sets @xmath283 and @xmath141 , respectively . define the reflection @xmath284 as @xmath285 let @xmath286 be the sets @xmath287 the set @xmath288 is defined similarly as @xmath268 , which is a fiber bundle over @xmath289 such that the fiber above @xmath290 is @xmath291 . now , we consider the maximal domains in @xmath268 for @xmath292 . let @xmath293 be a subinterval of @xmath294 $ ] . let @xmath135 be any one of the six polyhedra in @xmath288 . a maximal domain in @xmath288 is a maximal subset where the first return @xmath295 on @xmath135 is entirely defined and continuous . $ ] at the plane @xmath296 for any subinterval @xmath297 $ ] , if the number of maximal domains in @xmath288 is finite , we can apply the calculation on the vertices of the maximal domains to show the conjugacy of the first return maps . however , the number of maximal domains is not always finite on each arbitrary subintervals of @xmath294 $ ] . the next experimental result provides a classification of subintervals @xmath293 of @xmath298 $ ] such that the number of maximal domains in @xmath288 is finite . let @xmath293 be a subinterval of @xmath299 . if @xmath293 is in one of the following form of continued fraction expansion indexed by @xmath300 , then number of maximal domains in @xmath288 is fixed . furthermore , none of the maximal domains vanishes in the interval @xmath293 . 1 . @xmath301$],@xmath302 2 . @xmath303 $ ] , @xmath304 3 . @xmath305 $ ] , @xmath306 odd , @xmath307 4 . @xmath308$],@xmath309 even , @xmath310 5 . @xmath311 $ ] , @xmath312 even , @xmath313 . for convenience , we introduce some notation in this section . define @xmath314 and the interval @xmath315 as follows : @xmath316 @xmath317 , \quad \bar a_{m , n}=[t_{m , n},t_{m , n-1}].\ ] ] then , we denote @xmath318 to be the union @xmath319 , \quad \bigcup_{n\geq 3 } a_{2,n } = [ \frac{12}{29},\frac{29}{70}],\ ] ] respectively . similarly , denote @xmath320 as @xmath321 , \quad \bigcup_{n\geq 3 } \bar a_{2,n}= [ \frac{2}{5},\frac{5}{12}],\ ] ] respectively . note that @xmath323 \quad \mbox{and } \quad \bar a_{2,3}=r(a_{2,3})=[\frac{7}{17},\frac{5}{12}].\ ] ] @xmath322 is partitioned into 176 maximal domains , each of which is a convex polytope . figure 10 shows cross sections of the union maximal domains in @xmath324 $ ] at the plane @xmath325 . by calculation , the vertices of every maximal domain are in the form of @xmath326 where @xmath327 are end points of the interval @xmath293 and @xmath328 are integers . for each connected component @xmath329 , @xmath295 is a piecewise affine map . for each point @xmath330 , we have @xmath331 where @xmath100 . if we vary the point @xmath332 in a neighborhood of @xmath332 , the integers @xmath333 do not change . since @xmath322 is partitioned into finitely many maximal domains , @xmath295 is a piecewise affine map on @xmath135 . a maximal domain @xmath34 in @xmath334 is a permanent maximal polyhedron if @xmath34 satisfies the following condition : * at least one vertex of @xmath34 has @xmath335-coordinate @xmath336 , * at least one vertex of @xmath34 has @xmath335-coordinate @xmath337 . a maximal domain @xmath34 in @xmath338 is called resident maximal polyhedron if @xmath34 satisfies the following condition : * at least one vertex of @xmath34 has @xmath335-coordinate @xmath336 , * at least one vertex of @xmath34 has @xmath335-coordinate @xmath339 , * all the vertices @xmath340 of @xmath34 has @xmath341 . it s equivalent to say that if a maximal domain @xmath34 in @xmath342 does not vanish between the plane @xmath343 and @xmath344 , then @xmath34 is a permanent polyhederon . note that @xmath345 are two end points of the interval @xmath346 . moreover , if a maximal domain @xmath347 lies between the plane @xmath343 and @xmath348 and the intersection of @xmath34 with each plane is non - empty , then @xmath34 is a resident maximal polyhedron in @xmath349 . these notations help us to classify the maximal domains restricting to the smaller intervals @xmath350 . if a maximal domain @xmath351 is obtained by chopping from a resident maximal domain @xmath352 in @xmath353 , we say @xmath34 is a primary maximal domain in @xmath322 . more precisely , @xmath34 is primary if @xmath354 for some resident maximal domain @xmath352 in @xmath355 . in @xmath322 , there are 176 maximal domains , where 150 are primary . let @xmath356 be the set of resident maximal polyhedra in @xmath349 and @xmath357 be the set of primary maximal domains in @xmath322 . denote @xmath358 to be the set of rest 26 maximal polyhedra in @xmath338 which also produce maximal domains in @xmath322 . these are the polyhedra lying strictly above the plane @xmath343 . we list them in the last section of the paper . before going to the proof , we provide the explicit formula of the similarity @xmath149 which appeared in the renormalization theorem 2.1 . @xmath359 where the scalar @xmath150 . then , we can define the set @xmath360 in theorem 2.1 as @xmath361 and @xmath362 be the fiber bundle over @xmath293 such that the fiber above @xmath363 $ ] is @xmath360 . a maximal domain in @xmath362 is defined in the same way as the maximal in @xmath288 . moreover , a maximal domain @xmath352 in @xmath364 is a permanent maximal polyhedron if @xmath352 satisfies the following properties : * @xmath352 has at least one vertex with @xmath335-coordinate @xmath365 , * @xmath352 has at least one vertex with @xmath335-coordinate @xmath339 . we say @xmath352 a resident maximal polyhedron in @xmath366 , if * @xmath352 has at least one vertex with @xmath335-coordinate @xmath365 * @xmath352 has at least one vertex with @xmath335-coordinate @xmath367 . * the @xmath335-coordinates of all vertices of @xmath352 should satisfy @xmath368 . let @xmath369 be the collection of resident maximal polyhedron in @xmath370 . by direct computation , there are 162 maximal domains in @xmath371 , 136 of which are chopped from resident maximal domains in @xmath372 . let us denote the set of primary maximal domains by @xmath373 . moreover , there are @xmath374 maximal polyhedra in @xmath364 from which the non - primary maximal domains in @xmath371 can be obtained . denote the set of these 26 non - resident maximal polyhedra in @xmath375 by @xmath376 . the goal in this section is to show that for all @xmath378 , @xmath379 and @xmath380 are conjugate by the similarity map @xmath381 . to prove this , we ve attached 1-dimensional parameter space to the planar torus @xmath69 and want to apply the calculation in @xmath382 . for calculation , we always refer to open polyhedra . first , we piece together the similarities @xmath381 on @xmath69 to construct a piecewise affine map @xmath383 on the fiber bundle which is defined as @xmath384 fix a parameter @xmath385 $ ] . let @xmath146 . the first return map @xmath163 satisfies @xmath386 _ step 1 . _ for every non - resident maximal polyhedron @xmath387 , @xmath388 in @xmath356 , we check that @xmath387 satisfies the following properties : 1 . there exists a non - resident maximal domain @xmath389 in @xmath376 such that @xmath390 it follows that there is a one - to - one correspondence between the elements in @xmath358 and the ones in @xmath376 . for computation , it is sufficient to check that the set of vertices of @xmath389 are @xmath391 where @xmath392 are the vertices of @xmath387 . 2 . let @xmath135 be a polyhedra in @xmath393 such that @xmath394 , then @xmath395 . moreover , the maximal domains satisfy the following condition : @xmath396 3 . we denote the polyhedra @xmath397 by @xmath398 if @xmath387 satisfies the inclusion above . we check the fact : @xmath399 4 . @xmath400 _ step 2 . _ next , we consider the points in resident maximal polyhedra . we apply the calculation on the set of resident maximal polyhedra because if the conjugacy is satisfied , then it follows that the renormalization scheme exists for all points in @xmath360 when @xmath378 . since there is no one - to - one correspondence between the resident maximal domains in @xmath356 and the ones in @xmath369 , we can not apply the same calculation as before . however , by computer assistance , we find that each element in @xmath356 is a subpolyhedron of a resident maximal polyhedron in @xmath376 up to a similarity . we apply the similar calculations as in step 1 and check the following properties : 1 . for every resident maximal polyhedron @xmath401 in @xmath402 , there exists a resident maximal polyhedron @xmath403 such that @xmath404 for some @xmath405 . 2 . let @xmath135 be the connected component of @xmath349 such that @xmath394 . denote @xmath406 as the polyhedron @xmath407 @xmath406 satisfies that @xmath408 3 . @xmath409 4 . @xmath410 hence , we ve shown that the map @xmath95 is renormalizable when @xmath411 $ ] . since the size of the data is too large to include in this paper , i provide the code on my website and one can check the data of all the resident maximal polyhedra from my website . the url is : @xmath412 . lemma 4.4 holds for parameter @xmath414 $ ] . we want to apply the same method as used in the previous case . therefore , we need to classify the maximal domains in @xmath415 and @xmath416 first . the primary maximal domains in @xmath415 are the maximal domains chopped from the resident maximal polyhedera defined in section 4.4 . the non - primary maximal domains of @xmath417 are either obtained by chopping from the elements in @xmath358 or they are the newly - appeared maximal domains defined as follows : a maximal domain @xmath34 in @xmath393 is newly - appeared at the parameter @xmath172 if it satisfies the following : * @xmath418 . * the number of vertices in @xmath34 with @xmath335-coordinate being @xmath172 is less than 3 . if a maximal domain @xmath34 is newly - appeared , then @xmath34 lies below the plane @xmath276 and it can only touches the plane @xmath276 at a point or a line segment . the primary and newly - appeared maximal domains of @xmath372 at @xmath172 are defined similarly by replacing @xmath419 with @xmath420 . by computation , @xmath415 is partitioned into 178 maximal domains and 150 of them are primary . then the set @xmath416 has 136 primary and 28 non - primary maximal domains . since we ve shown that the renormalization exists for all points in every resident maximal polyhedron , we are left to check the points in non - primary maximal domains . among the 28 non - primary maximal domains in @xmath416 ( or @xmath415 ) , there are 12 of them are obtained by chopping from non - resident maximal domains in @xmath375 ( or @xmath393 ) , which we have already done the calculation . if @xmath34 is a non - primary maximal domain in @xmath415 but does not belong to the above case , then @xmath34 must be chopped from a newly - appeared polyhedron at the parameter @xmath421 . this is because the maximal z - coordinate of all points in @xmath34 must be @xmath421 . moreover , if @xmath34 has more than 2 vertices with @xmath422 , then @xmath34 is either primary or inherited from a maximal polyhedron appeared in @xmath423 . the same argument works for the case of @xmath416 . the lists of all newly - appeared maximal domains in @xmath415 and @xmath416 are provided in section 6 . there is a one - to - one correspondence between the newly - appeared maximal polyhedra in in @xmath424at @xmath419 and the ones in @xmath364 . we apply the same calculation as in lemma 5.1 . therefore , we show that when the parameter @xmath425 $ ] , it is true that @xmath426 we want to show that @xmath95 is conjugate to @xmath156 when @xmath427 by a piecewise translation @xmath381 . recall that @xmath381 is the map interchanging the upper half and lower half of the torus @xmath69 . therefore , we can piece together the map @xmath381 for @xmath275 to get an affine map @xmath383 in @xmath382 : @xmath428 it is easy to see that the affine map @xmath429 is an involution as well . as discussed in section 4.1 , there is a partition @xmath430 of @xmath272 such that each @xmath387 is a maximal domains in @xmath272 determined by the fiber bundle map @xmath431 @xmath267 there is a partition @xmath432 of @xmath282 $ ] such that each @xmath389 is a maximal subset of @xmath282 $ ] where @xmath274 is entirely defined and continuous . next , we construct a finer partition @xmath433 of @xmath434)$ ] as follows : * if there exists some @xmath435 such that @xmath436 , then the polyhedron @xmath437 is an element in @xmath433 . * if there exists some @xmath438 such that @xmath439 , then the polyhedron @xmath440 is an element in @xmath433 . @xmath433 is partitioned into 26 elements and the bundle map @xmath274 is well - defined on each @xmath441 . then , we check that the following properties hold : 1 . for each @xmath442 , there exists some @xmath443 such that @xmath444 2 . @xmath445 3 . @xmath446 4 . @xmath447 thus , we show that the tetrahedral pet @xmath95 on @xmath69 is renormalizable when @xmath275 . the 26 non - secondary maximal polyhedron of @xmath322 are listed as follows : 10 r. adler , b. kitchens . and c. tresser , _ dynamics of non - ergodic piecewise affine maps of the torus _ , ergodic theory dyn . syst 21 ( 2001 ) , no.4 , 959 - 999 . a. goetz , _ dynamics of piecewise isometries _ , ill . j. math . 44(2000 ) , no . 3 , 465 - 478 ( english ) . yoccoz , _ continued fraction algorithms for interval exchange maps : an introduction _ , frontiers in number theory , physics , and geometry vol . 1 , p. cartier , b. julia , p. moussa , p.vanhove ( ed . ) springer - verlag 4030437 ( 2006 ) .
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we introduce a family of piecewise isometries @xmath0 parametrized by @xmath1 on the surface of a regular tetrahedron , which we call the tetrahedral twists .
this family of maps is similar to the pets constructed by patrick hooper .
we study the dynamics of the tetrahedral twists through the notion of renormalization . by the assistance of computer
, we conjecture that the renormalization scheme exists on the entire interval @xmath2 . in this paper , we show that this system is renormalizable in the subintervals @xmath3 $ ] and @xmath4 .
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many complex systems in various areas of science exhibit a spatio - temporal dynamics that is inhomogeneous and can be effectively described by a superposition of several statistics on different scales , in short a superstatistics @xcite . the superstatistics notion was introduced in @xcite , in the mean time many applications for a variety of complex systems have been pointed out @xcite . essential for this approach is the existence of sufficient time scale separation between two relevant dynamics within the complex system . there is an intensive parameter @xmath1 that fluctuates on a much larger time scale than the typical relaxation time of the local dynamics . in a thermodynamic setting , @xmath1 can be interpreted as a local inverse temperature of the system , but much broader interpretations are possible . the stationary distributions of superstatistical systems , obtained by averaging over all @xmath1 , typically exhibit non - gaussian behavior with fat tails , which can be a power law , or a stretched exponential , or other functional forms as well @xcite . in general , the superstatistical parameter @xmath1 need not to be an inverse temperature but can be an effective parameter in a stochastic differential equation , a volatility in finance , or just a local variance parameter extracted from some experimental time series . many applications have been recently reported , for example in hydrodynamic turbulence @xcite , for defect turbulence @xcite , for cosmic rays @xcite and other scattering processes in high energy physics @xcite , solar flares @xcite , share price fluctuations @xcite , random matrix theory @xcite , random networks @xcite , multiplicative - noise stochastic processes @xcite , wind velocity fluctuations @xcite , hydro - climatic fluctuations @xcite , the statistics of train departure delays @xcite and models of the metastatic cascade in cancerous systems @xcite . on the theoretical side , there have been recent efforts to formulate maximum entropy principles for superstatistical systems @xcite . in this paper we provide an overview over some recent developments in the area of superstatistics . three examples of recent applications are discussed in somewhat more detail : the statistics of lagrangian turbulence , the statistics of train departure delays , and the survival statistics of cancer patients . in all cases the superstatistical model predictions are in very good agreement with real data . we also comment on recent theoretical approaches to develop generalized maximum entropy principles for superstatistical systems . in generalized versions of statistical mechanics one starts from more general entropic measures than the boltzmann - gibbs shannon entropy . a well - known example is the @xmath2-entropy @xcite @xmath3 but other forms are possible as well ( see , e.g. , @xcite for a recent review ) . the @xmath4 are the probabilities of the microstates @xmath5 , and @xmath2 is a real number , the entropic index . the ordinary shannon entropy is contained as the special case @xmath6 : @xmath7 extremizing @xmath8 subject to suitable constraints yields more general canonical ensembles , where the probability to observe a microstate with energy @xmath9 is given by @xmath10 one obtains a kind of power - law boltzmann factor , of the so - called @xmath2-exponential form . the important question is what could be a physical ( non - equilibrium mechanism ) to obtain such distributions . the reason could indeed be a driven nonequilibrium situation with local fluctuations of the environment . this is the situation where the superstatistics concept enters . our starting point is the following well - known formula @xmath11 where @xmath12 is the @xmath0 ( or @xmath13 ) probability distribution . we see that averaged _ ordinary _ boltzmann factors with @xmath0 distributed @xmath1 yield a _ generalized _ boltzmann factor of @xmath2-exponential form . the physical interpretation is that tsallis type of statistical mechanics is relevant for _ nonequilibrium _ systems with temperature fluctuations . this approach was made popular by two prls in 2000/2001 @xcite , which used the @xmath0-distribution for @xmath14 . general @xmath14 were then discussed in @xcite . in @xcite it was suggested to construct a dynamical realization of @xmath2-statistics in terms of e.g. a linear langevin equation @xmath15 with fluctuating parameters @xmath16 . here @xmath17 denotes gaussian white noise . the parameters @xmath18 are supposed to fluctuate on a much larger time scale than the velocity @xmath19 . one can think of a brownian particle that moves through spatial cells with different local @xmath20 in each cell ( a nonequilibrium situation ) . assume the probability distribution of @xmath1 in the various cells is a @xmath0-distribution of degree @xmath21 : @xmath22 then the conditional probability given some fixed @xmath1 in a given cell is gaussian , @xmath23 , the joint probability is @xmath24 and the marginal probability is @xmath25 . integration yields @xmath26 i.e. we obtain power - law boltzmann factors with @xmath27 , @xmath28 , and @xmath29 . here @xmath30 is the average of @xmath1 . the idea of superstatistics is to generalize this example to much broader systems . for example , @xmath1 need not be an inverse temperature but can in principle be any intensive parameter . most importantly , one can generalize to _ general probability densities @xmath14 _ and _ general hamiltonians_. in all cases one obtains a superposition of two different statistics : that of @xmath1 and that of ordinary statistical mechanics . superstatistics hence describes complex nonequilibrium systems with spatio - temporal fluctuations of an intensive parameter on a large scale . the _ effective _ boltzmann factors @xmath31 for such systems are given by @xmath32 some recent theoretical developments of the superstatistics concept include the following : * can prove a superstatistical generalization of fluctuation theorems @xcite * can develop a variational principle for the large - energy asymptotics of general superstatistics @xcite ( depending on @xmath14 , one can get not only power laws for large @xmath33 but e.g. also stretched exponentials ) * can formally define generalized entropies for general superstatistics @xcite * can study microcanonical superstatistics ( related to a mixture of @xmath2-values ) @xcite * can prove a superstatistical version of a central limit theorem leading to @xmath2-statistics @xcite * can relate it to fractional reaction equations @xcite * can consider superstatistical random matrix theory @xcite * can apply superstatistical techniques to networks @xcite * can define superstatistical path integrals @xcite * can do superstatistical time series analysis @xcite ... and some more practical applications : * can apply superstatistical methods to analyze the statistics of 3d hydrodynamic turbulence @xcite * can apply it to atmospheric turbulence ( wind velocity fluctuations @xcite ) * can apply superstatistical methods to finance and economics @xcite * can apply it to blinking quantum dots @xcite * can apply it to cosmic ray statistics @xcite * can apply it to various scattering processes in particle physics @xcite * can apply it to hydroclimatic fluctuations @xcite * can apply it to train delay statistics @xcite * can consider medical applications @xcite while in principle any @xmath14 is possible in the superstatistics approach , in practice one usually observes only a few relevant distributions . these are the @xmath0 , inverse @xmath0 and lognormal distribution . in other words , in typical complex systems with time scale separation one usually observes 3 physically relevant universality classes @xcite * \(a ) @xmath0-superstatistics ( @xmath34 tsallis statistics ) * \(b ) inverse @xmath0-superstatistics * \(c ) lognormal superstatistics what could be a plausible reason for this ? consider , e.g. , case ( a ) . assume there are many microscopic random variables @xmath35 , @xmath36 , contributing to @xmath1 in an additive way . for large @xmath37 , the sum @xmath38 will approach a gaussian random variable @xmath39 due to the ( ordinary ) central limit theorem . there can be @xmath21 gaussian random variables @xmath40 of the same variance due to various relevant degrees of freedom in the system . @xmath1 should be positive , hence the simplest way to get such a positive @xmath1 is to square the gaussian random variables and sum them up . as a result , @xmath41 is @xmath0-distributed with degree @xmath21 , @xmath42 where @xmath43 is the average of @xmath1 . \(b ) the same considerations can be applied if the temperature @xmath44 rather than @xmath1 itself is the sum of several squared gaussian random variables arising out of many microscopic degrees of freedom @xmath35 . the resulting @xmath14 is the inverse @xmath0-distribution : @xmath45 it generates superstatistical distributions @xmath46 that decay as @xmath47 for large @xmath33 . \(c ) @xmath1 may be generated by multiplicative random processes . consider a local cascade random variable @xmath48 , where @xmath37 is the number of cascade steps and the @xmath35 are positive microscopic random variables . by the ( ordinary ) central limit theorem , for large @xmath37 the random variable @xmath49 becomes gaussian for large @xmath37 . hence @xmath39 is log - normally distributed . in general there may be @xmath21 such product contributions to @xmath1 , i.e. , @xmath50 . then @xmath51 is a sum of gaussian random variables , hence it is gaussian as well . thus @xmath1 is log - normally distributed , i.e. , @xmath52 where @xmath53 and @xmath54 are suitable parameters . we will now discuss examples of the three different superstatistical universality classes . our first example is the departure delay statistics on the british rail network . clearly , at the various stations there are sometimes train departure delays of length @xmath55 . the 0th - order model for the waiting time would be a poisson process which predicts that the waiting time distribution until the train finally departs is @xmath56 , where @xmath1 is some parameter . but this does not agree with the actually observed data @xcite . a much better fit is given by a @xmath2-exponential , see fig . 1 . ) .,width=302 ] what may cause this power law that fits the data ? the idea is that there are fluctuations in the parameter @xmath1 as well . these fluctuations describe large - scale temporal or spatial variations of the british rail network environment . examples of causes of these @xmath1-fluctuations : * begin of the holiday season with lots of passengers * problem with the track * bad weather conditions * extreme events such as derailments , industrial action , terror alerts , etc . as a result , the long - term distribution of train delays is then a mixture of exponential distributions where the parameter @xmath1 fluctuates : @xmath57 for a @xmath0-distributed @xmath1 with @xmath21 degrees of freedom one obtains @xmath58 where @xmath59 and @xmath60 . the model discussed in @xcite generates @xmath2-exponential distributions of train delays by a simple mechanism , namely a @xmath0-distributed parameter @xmath1 of the local poisson process . this is an example for @xmath0 superstatistics . our next example is an application in turbulence . consider a single tracer particle advected by a fully developed turbulent flow . for a while it will see regions of strong turbulent activity , then move on to calmer regions , just to continue in yet another region of strong activity , and so on . this is a superstatistical dynamics , and in fact superstatistical models of turbulence have been very successful in recent years @xcite . the typical shape of a trajectory of such a tracer particle is plotted in fig . 2 . this is lagrangian turbulence in contrast to eulerian turbulence , meaning that one is following a single particle in the flow . in particular , one is interested in velocity differences @xmath61 of the particle on a small time scale @xmath62 . for @xmath63 this velocity difference becomes the local acceleration @xmath64 . a superstatistical lagrangian model for 3-dim velocity differences of the tracer particle has been developed in @xcite . one simply looks at a superstatistical langevin equation of the form @xmath65 here @xmath66 and @xmath67 are constants . note that the term proportional to @xmath67 introduces some rotational movement of the particle , mimicking the vortices in the flow . the noise strength @xmath68 and the unit vector @xmath69 evolve stochastically on a large time scale @xmath70 and @xmath71 , respectively , thus obtaining a superstatistical dynamics . @xmath70 is of the same order of magnitude as the integral time scale @xmath72 , whereas @xmath73 is of the same order of magnitude as the kolmogorov time scale @xmath74 . one can show that the reynolds number @xmath75 is basically given by the time scale ratio @xmath76 . the time scale @xmath77 describes the average life time of a region of given vorticity surrounding the test particle . in this superstatistical turbulence model one defines the parameter @xmath1 to be @xmath78 , but it does _ not _ have the meaning of a physical inverse temperature in the flow . rather , one has @xmath79 , where @xmath80 is the kinematic viscosity and @xmath81 is the average energy dissipation , which is known to fluctuate in turbulent flows . in fact , kolmogorov s theory of 1961 suggests a lognormal distribution for @xmath82 , which automatically leads us to lognormal superstatistics : it is reasonable to assume that the probability density of the stochastic process @xmath83 is approximately a lognormal distribution @xmath84 for very small @xmath62 the 1d acceleration component of the particle is given by @xmath85 and one gets out of the model the 1-point distribution @xmath86 this prediction agrees very well with experimentally measured data of the acceleration statistics , which exhibits very pronounced ( non - gaussian ) tails , see fig . 3 for an example . ) ( see @xcite for more details).,width=302 ] it is interesting to see that our 3-dimensional superstatistical model predicts the existence of correlations between the acceleration components . for example , the acceleration @xmath87 in @xmath88 direction is not statistically independent of the acceleration @xmath89 in @xmath90-direction . we may study the ratio @xmath91 of the joint probability @xmath92 to the 1-point probabilities @xmath93 and @xmath94 . for independent acceleration components this ratio would always be given by @xmath95 . however , our 3-dimensional superstatistical model yields prediction @xmath96 this is a very general formula , valid for any superstatistics , for example also tsallis statistics , obtained when @xmath14 is the @xmath0-distribution . the trivial result @xmath95 is obtained only for @xmath97 , i.e. no fluctuations in the parameter @xmath1 at all . 4 shows @xmath91 as predicted by lognormal superstatistics : ) , @xmath14 being the lognormal distribution.,width=302 ] the shape of this is very similar to experimental measurements @xcite . our final example of application of superstatistics is for a completely different area : medicine . we will look at cell migration processes describing the metastatic cascade of cancerous cells in the body @xcite . there are various pathways in which cancerous cells can migrate : via the blood system , the lymphatic system , and so on . the diffusion constants for these various pathways are different . in this way superstatistics enters , describing different diffusion speeds for different pathways ( see fig . 5 ) . but there is another important issue here : when looking at a large ensemble of patients then the spread of cancerous cells can be very different from patient to patient . for some patients the cancer spreads in a very aggressive way , whereas for others it is much slower and less aggressive . so superstatistics also arises from the fact that all patients are different . a superstatistical model of metastasis and cancer survival has been developed in @xcite . details are described in that paper . here we just mention the final result that comes out of the model : one obtains the following prediction for the probability density function of survival time @xmath55 of a patient that is diagnosed with cancer at @xmath98 : @xmath99 or @xmath100 , \label{eq7}\end{aligned}\ ] ] where @xmath101 is the modified bessel function . note that this is inverse @xmath0 superstatistics . the role of the parameter @xmath1 is now played by the parameter @xmath102 , which in a sense describes how aggressively the cancer propagates . the above formula based on inverse @xmath0 superstatistics is in good agreement with real data of the survival statistics of breast cancer patients in the us . the superstatistical formula fits the observed distribution very well , both in a linear and logarithmic plot ( see fig.6 ) . one remark is at order . when looking at the relevant time scales one should keep in mind that the data shown are survival distributions _ conditioned on the fact that death occurs due to cancer_. many patients , in particular if they are diagnosed at an early stage , will live a long happy life and die from something else than cancer . these cases are _ not _ included in the data . , both in a linear and double logarithmic plot . only patients that die from cancer are included in the statistics . the solid line is the superstatistical model prediction @xcite . , title="fig:",width=302 ] , both in a linear and double logarithmic plot . only patients that die from cancer are included in the statistics . the solid line is the superstatistical model prediction @xcite . , title="fig:",width=302 ] we finish this article by briefly mentioning some other recent interesting theoretical developments . one major theoretical concern is that a priori the superstatistical distribution @xmath14 can be anything . but perhaps one should single out the really relevant distributions @xmath14 by a least biased guess , given some constraints on the complex system under consideration . this program has been developed in some recent papers @xcite . there are some ambiguities which constraints should be implemented , and how . a very general formalism is presented in @xcite , which contains previous work @xcite as special cases . the three important universality classes discussed above , namely @xmath0 superstatistics , inverse @xmath0 superstatistics and lognormal superstatistics are contained as special cases in the formalism of @xcite . in principle , once a suitable generalized maximum entropy principle has been formulated for superstatistical systems , one can proceed to a generalized thermodynamic description , get a generalized equation of state , and so on . there is still a lot of scope of future research to develop the most suitable formalism . but the general tendency seems to be to apply maximum entropy principles and least biased guesses to nonequilibrium situations as well . in fact , jaynes @xcite always thought this is possible . another interesting development is what one could call a superstatistical path integral . these are just ordinary path integrals but with an additional integration over a parameter @xmath1 that make the wiener process become something more complicated , due to large - scale fluctuations of its diffusion constant . jizba et al . investigate under which conditions one obtains a markov process again @xcite . it seems some distributions @xmath14 are distinguished as making the superstatistical process simpler than others , preserving markovian - like properties . these types of superstatistical path integral processes have applications in finance , and possibly also in quantum field theory and elementary particle physics . in high energy physics , many of the power laws observed for differential cross sections and energy spectra in high energy scattering processes can also be explained using superstatistical models @xcite . the key point here is to extend the hagedorn theory @xcite to a superstatistical one which properly takes into account temperature fluctuations @xcite . superstatistical techniques have also been recently used to describe the space - time foam in string theory @xcite . superstatistics ( a statistics of a statistics ) provides a physical reason why more general types of boltzmann factors ( e.g. @xmath2-exponentials or other functional forms ) are relevant for _ nonequilibrium _ systems with suitable fluctuations of an intensive parameter . let us summarize : * there is evidence for three major physically relevant universality classes : @xmath0-superstatistics @xmath34 tsallis statistics , inverse @xmath0-superstatistics , and lognormal superstatistics . these arise as universal limit statistics for many different systems . * superstatistical techniques can be successfully applied to a variety of complex systems with time scale separation . * the train delays on the british railway network are an example of @xmath0 superstatistics = tsallis statistics @xcite . * a superstatistical model of _ lagrangian turbulence _ @xcite is in excellent agreement with the experimental data for probability densities , correlations between components , decay of correlations , lagrangian scaling exponents , etc . this is an example of lognormal superstatistics @xcite . * cancer survival statistics is described by inverse @xmath0 superstatistics @xcite . * the long - term aim is to find a good thermodynamic description for general superstatistical systems . a generalized maximum entropy principle may help to achieve this goal . 99 c. beck and e.g.d . cohen , physica a * 322 * , 267 ( 2003 ) c. beck , e.g.d . cohen , and h.l . swinney , phys . e * 72 * , 056133 ( 2005 ) c. beck and e.g.d . cohen , physica a * 344 * , 393 ( 2004 ) h. touchette 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we provide an overview on superstatistical techniques applied to complex systems with time scale separation .
three examples of recent applications are dealt with in somewhat more detail : the statistics of small - scale velocity differences in lagrangian turbulence experiments , train delay statistics on the british rail network , and survival statistics of cancer patients once diagnosed with cancer .
these examples correspond to three different universality classes : lognormal superstatistics , @xmath0-superstatistics and inverse @xmath0 superstatistics .
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the leading models for gamma - ray bursts ( grb ) , bursts of 0.1 - 1 mev photons typically lasting for 0.1 - 100 seconds ( fishman & meegan 1995 ) , involve a relativistic wind emanating from a compact central source . the ultimate energy source is rapid accretion onto a newly formed stellar mass black hole . observations suggest that the prompt @xmath1-ray emission is produced by the dissipation ( perhaps due to internal shocks ) of the kinetic energy of a relativistically expanding wind , i.e. a `` fireball '' . both synchrotron and inverse compton emissions from the shock - accelerated electrons have been proposed as the grb emission mechanism . in this paper , we study in detail the production of neutrinos by protons accelerated along with electrons . we assume that equal energy of the fireball is dissipated in protons and electrons ( or photons ) . this is the case in models where grbs are the sources of the highest energy cosmic rays . the basic idea is that the protons produce pions decaying into neutrinos in interactions with the fireball photons , or with external photons surrounding the newly formed black hole . where previous calculations have estimated the universal diffuse flux of neutrinos produced by all grbs over cosmological time , we estimate the flux from individual grbs observed by the batse ( burst and transient source experiment ) experiment on the compton gamma - ray observatory . the prediction can be directly compared with coincident observations performed with the amanda detector . having these observations in mind , we specialize on neutrino emission coincident in time with the grb . opportunities for neutrino production exist after and , in some models , before the burst of @xmath1-rays , e.g. when the fireball expands through the opaque ejecta of a supernova . the calculations are performed in two models chosen to be representative and rather different versions of a large range of competing models . the first is generic for models where an initial event , such as a merger of compact objects or the instant collapse of a massive star to a black hole , produces the fireball ( waxman & bahcall 1997 ; guetta , spada & waxman 2001a , gsw hereafter ) . we calculate the neutrino production by photomeson interactions of relativistic protons accelerated in the internal shocks and the synchrotron photons that are emitted in these shocks . the neutrinos produced via this mechanism have typical energies of @xmath2 ev , and are emitted in coincidence with the grbs , with their spectrum tracing the grb photon spectrum . for an alternative model , we have chosen the supranova model where a massive star collapses to a neutron star with mass @xmath3 , which loses its rotational energy on a time scale @xmath4 of weeks to years , before collapsing to a black hole , thus triggering the grb . following guetta and granot ( guetta & granot 2002a ) , we calculate the neutrino flux from interactions of the fireball protons with external photons in the rich radiation field created during the spindown of the supra - massive pulsar . production on external photons turns out to be dominant for a wide range of parameters , @xmath5yr for a typical grb and @xmath6yr for x - ray flashes . the neutrinos produced via this mechanism have energies @xmath7ev for typical grb ( x - ray flashes ) and are emitted simultaneously with the prompt @xmath1-ray ( x - ray ) emission . their energy spectrum consists of several power law segments and its overall shape depends on the model parameters , especially @xmath4 . if the mass of the supernova remnant is of the order of @xmath8 , and if the supernova remnant shell is clumpy , then for time separation @xmath9 yr the snr shell has a thomson optical depth larger than unity and obscures the radiation emitted by the grb . therefore , for @xmath10yr , the @xmath11 s would not be accompanied by a detectable grb providing us with an example of neutrino emission not coinciding with a grb display . as previously mentioned , to realistically estimate the neutrino fluxes associated with either of these models , we turn to the grb data collected by batse . the batse records include spectral and temporal information which can be used to estimate neutrino spectra for individual bursts . we will perform these calculations for two version of each of the two models previously described . the two versions correspond to alternative choices of important parameters . the wide variety of grb spectra results , not surprisingly , in a wide range of neutrino spectra and event rates . for approximately 800 bursts in the batse catalog , and for four choices of models , we have calculated the neutrino spectra and the event rates , coincident with grbs , for a generic neutrino telescope . with 800 bursts , the sample should also be representative for data expected from much larger next - generation neutrino observatories . neutrino telescopes can leverage the directional and time information provided by batse to do an essentially background - free search for neutrinos from grbs . individual neutrino events within the batse time and angular window are a meaningful observation . a generic detector with 1km@xmath12 effective telescope area , during one year , should be able to observe 1000 bursts over @xmath13 steradians . using the batse grbs as a template , we predict order 10 events , muons or showers , for both models . the rates in the supranova model depend strongly on @xmath4 . in this model we anticipate @xmath14 events per year assuming @xmath15 yr , but only one event per ten years for @xmath16 yr . we will present detailed tabulated predictions further on . they can be accessed at http://www.arcetri.astro.it/@xmath0dafne / grb/. short duration grbs , characterized by lower average fluences , are less likely to produce observable neutrino fluxes . we find that grbs with lower peak energies , x - ray flash candidates , yield the largest rates in the supranova model . for instance , for @xmath17 yr , we predict one event ( muon or shower ) for every 1000 bursts . if only 100 , or so , x - ray flashes occur per year , as observations suggest , this will be difficult to observe . however , such events may be considerably more common and may contribute significantly to the diffuse high energy neutrino flux . observations of neutrinos from this class of grbs would be strong evidence for a supranova progenitor model . the amanda collaboration has collected neutrino data in coincidence with batse observations ( barouch & hardtke 2001 ) . it operated the detector for 3 years ( 1997 - 1999 ) with an effective area of approximately 5,000m@xmath12 for grbs events . in early 2000 the expanded detector reached an effective area of roughly 50,000m@xmath12 . unfortunately , only @xmath0100 coincident bursts could be observed with the completed detector before batse operations ceased in june 2000 . with effective areas significantly below the canonical square kilometer discussed in this paper , amanda is not large enough to test the grb models considered here . we estimate only 0.08 events in 1997 - 1999 and 0.3 in 2000 . our estimate of 110 events in 1 year for a telescope with 1 kilometer square telescope area is consistent with previous , burst - averaged , determinations of the grb neutrino flux ( alvarez , halzen & hooper 2000 , dermer & atoyan 2003 ) . note that the effective area of icecube for grb will significantly exceed this reference value ( pdd ) . the rate can be understood by a back - of - the - envelope estimate . a typical grb produces a photon fluence on the order of @xmath18 , which we assume to be equal to the energy in protons . if @xmath19 of the proton energy is converted into pions , half to charged pions , and one quarter of the charged pion energy to muon neutrinos in the @xmath20 decay chain , then @xmath21 neutrinos are generated . for a typical neutrino energy of @xmath22 , this yields @xmath23 neutrinos per square kilometer . at this energy the probability that a neutrino converts to a muon within range of the detector is @xmath24 ( gaisser , halzen , stanev 1995 ) . therefore @xmath25 muons are detected in association with a single grb . with over 1000 grbs in one year , we estimate a few muons per year in a kilometer - square detector . fluctuations in fluence and other burst characteristics enhance this estimate significantly ( see figure [ figevn ] , for instance ) , however , absorption of neutrinos in the earth can reduce this number . these effects are included in the calculations of actual event rates throughout this paper . our estimates , while observable with future kilometer - scale observatories , may be conservative . we already mentioned bursts with no counterpart in photons . also , occasional nearby bursts , much like supernova , could exceptionally provide higher event rates than our calculations reflect . we would like to point out the fact that the aim of the paper is not to do more precise calculations than the ones already done in the literature . we just want to generate results that can be compared with experiments that do coincidents observations with satellites . the outline of the paper is somewhat unconventional . the detailed results are collected in a data archive at http://www.arcetri.astro.it/@xmath0dafne/grb . the details of the calculations of the neutrino fluxes in the two models are described in appendices a and b. in appendix c we collected the methods used to evaluate the rates of neutrino - induced muons , taus and showers in a generic detector with a given effective telescope area . the main body of the paper is organized as follows . in 2 , we review grb models and describe the mechanisms for neutrino production in grbs . in 3 and 4 we describe the batse catalog of grbs and we identify subclasses of events : long duration grbs with and without measured redshift , short duration grbs and x - ray flash candidates . in 5 we summarize our simulation of the response of a generic high energy neutrino telescope to the predicted neutrino fluxes . in 6 we analyse some anomalous bursts . results and conclusions are collected in 7 . progenitor models of grbs are divided into two main categories . the first category involves the merger of a binary system of compact objects , such as a double neutron star ( eichler et al . 1989 ) , a neutron star and a black hole ( narayan , pacyski & piran 1992 ) or a black hole and a helium star or a white dwarf ( fryer & woosley 1998 ; fryer , woosley & hartmann 1999 ) . the second category involves the death of a massive star . it includes the failed supernova ( woosley 1993 ) or hypernova ( pacyski 1998 ) models , where a black hole is created promptly , and a large accretion rate from a surrounding accretion disk ( or torus ) feeds a strong relativistic jet in the polar regions . this type of model is known as the collapsar model . an alternative model within this second category is the supranova model ( vietri & stella 1998 ) , where a massive star explodes in a supernova and leaves behind a supra - massive neutron star ( smns ) , of mass @xmath3 . it subsequently loses its rotational energy on a time scale @xmath4 of order weeks to years until it collapses to a black hole . this triggers the grb . long grbs ( with a duration @xmath26 ) are usually attributed to the second category of progenitors , while short grbs are attributed to the first category . we select two models to investigate the opportunities for neutrino production in a grb . the analysis can be easily extended to other models ( see for example razzaque , mszros & waxman 2002 ; dermer & atoyan 2003 ) . the first model is based on the standard fireball phenomenology where electrons and protons are shock accelerated in the fireball . pions and neutrinos are produced by photoproduction interactions when the protons coexist in the fireball with photons . these are produced by synchrotron radiation of accelerated electrons . for a second model we turn to the supranova scenario where the supra - massive pulsar loses its rotational energy through a strong pulsar wind this pulsar wind creates a rich external radiation field before the collapse to the final black hole and the creation of the grb fireball . it is referred to as the pulsar wind bubble ( pwb ) ( knigl & granot 2002 ; inoue , guetta & pacini 2002 ; guetta & granot 2003 ) and provides a target for the photoproduction of neutrinos by fireball protons . let s note in passing that the supranova model has several advantages compared to other collapsar models : ( i ) the jet does not have to penetrate the stellar envelope ( vietri & stella 1998 ) , ( ii ) it can naturally explain the x - ray line features observed in several afterglows ( piro et al . 2000 ; lazzati , et al . 2001 ; vietri et al . 2001 ) , and ( iii ) the large fraction of the internal energy in the magnetic field and in electrons observed in the afterglow emission arise naturally ( knigl & granot 2002 ; guetta & granot 2003 ) . in all of the different scenarios mentioned above , the final stage of the process consists of a newly formed black hole with a large accretion rate from a surrounding torus , and involve a similar energy budget ( @xmath27 ) . observations suggest that prompt @xmath1-ray emission is produced by the dissipation of the kinetic energy within the fireball , due to internal shocks within the flow that arise from variability of the lorentz factor , @xmath28 , on a time scale @xmath29 . the afterglow emission is thought to arise from an external shock that is driven into the ambient medium as it decelerates the ejected matter ( rees & mszros 1994 ; sari & piran 1997 ) . in this so called ` internal - external ' shock model , the duration of the prompt grb is directly related to the time during which the central source is active . the emission mechanism is successfully described by synchrotron radiation from relativistic electrons that radiate in the strong magnetic fields . these are close to equipartition values within the shocked plasma . an additional radiation mechanism that may also play some role is synchrotron self - compton ( ssc ) ( guetta & granot 2002b ) , which is the upscattering of the synchrotron photons by relativistic electrons to higher energy . protons are expected to be accelerated along with the electrons in the region where the wind kinetic energy is converted into internal energy due to a dissipation mechanism like internal shocks . the conditions in the dissipation region allow proton acceleration up to @xmath30ev ( waxman 1995 ; vietri 1995 ) . the energy in @xmath1-rays reflect the fireball energy in accelerated electrons and afterglow observations indicate that accelerated electrons and protons carry similar energy ( freedman & waxman 2000 ) . our basic assumption in calculating neutrino emission from grbs is that equal amounts of energy go into protons and photons . in models where grb protons are the source of the highest energy cosmic rays , this assumption is supported by the approximate equality of the @xmath1-ray fluence of all grbs and the total energy in extragalactic cosmic rays . both internal shocks , responsible for the prompt grb emission , and the external shock , responsible for the afterglow emission , have been proposed as possible sources of the highest energy cosmic rays ( waxman 1995 and vietri 1995 , respectively ) . a comparison between the two mechanisms has been done by vietri , de marco and guetta ( 2003 ) , but it is not easy to conceive , at this point , an observational test capable of distinguishing between the two models . the only hope appears to observe the production of high energy neutrinos which must accompany the _ in situ _ acceleration of particles . occasionally , ultra - high energy protons will produce pions and neutrinos in collisions with photons in photon rich environment provided by the post shock shells or by the pulsar wind bubble . if the protons are accelerated in internal shocks , the neutrinos produced will arrive at earth simultaneously with the photons of the burst proper and will have an energy @xmath31 ( waxman and bachall 1997 ; gsw ; guetta & granot 2002a ) . if accelerated in external shocks , they will arrive at earth simultaneously with the photons of the afterglow and will have a higher energy , @xmath32 ev ( vietri 1998a , 1998b , waxman & bahcall 2000 ) . general phenomenological considerations indicate that gamma - ray bursts are produced by the dissipation of the kinetic energy of a relativistic expanding fireball . internal shocks that are mildly relativistic are believed to dissipate the energy . therefore , the proton energy distribution should be close to that for fermi acceleration in a newtonian ( non - relativistic ) shock , @xmath33 . moreover , the power law index of the electron and proton energy distributions are expected to be the same , and the values inferred for the electron distribution from the observed photon spectrum are @xmath34 with @xmath35 . we shall , therefore , adopt @xmath33 . plasma parameters in the dissipation region allow proton acceleration to energies greater than @xmath36 ev ( waxman 1995 , vietri 1995 ) . we will assume that the fireball is spherically symmetric . note , however , that a jet - like fireball behaves as if it were a conical section of a spherical fireball as long as @xmath37 , where @xmath38 is the jet opening angle and @xmath39 is the wind lorentz factor . therefore , our results apply without modification to a jet - like fireball . for a jet - like wind , the luminosity , @xmath40 , in our equations should be understood as the luminosity of the fireball inferred by assuming spherical symmetry . we have relegated all details of the calculation of neutrino production via photomeson interaction with grb photons and pwb photons to appendices a and b , respectively . throughout the paper , we will refer to the models by the following convention : * model 1 : neutrino flux from the interaction of high - energy protons with grb photons . the fraction of proton energy transfered to pion energy is set to 0.2 as indicated by simulations of gsw ( see apppendix a ) . * model 2 : neutrino flux from the interaction of high - energy protons with grb photons . the fraction of proton energy transfered to pion energy is calculated as described in appendix a ( see appendix a ) . * model 3 : neutrino flux from the interaction of high - energy protons with pwb photons , as in the supranova progenitor model . the time scale between supernova and grb is set to @xmath41 yr ( see appendix b ) . * model 4 : neutrino flux from the interaction of high - energy protons with pwb photons , as in the supranova progenitor model . the time scale between supernova and grb is set to @xmath42 yr ( see appendix b ) . batse , the burst and transient source experiment , was a high energy astrophysics experiment launched on the compton gamma - ray observatory in 1991 . batse , between its launch and the termination of its orbit in 2000 , has observed and recorded data from over 8000 events including gamma - ray bursts , pulsars , terrestrial gamma - ray flashes , soft gamma repeaters and black holes . the data batse recorded from gamma - ray bursts is publically available in the current batse catalog at http://f64.nsstc.nasa.gov / batse / grb / catalog/. for a description see paciesas et al . the catalog includes information on the spectrum , time and location of each triggered burst . each triggered event has been assigned a batse trigger number ( between 105 and 8121 for grbs ) which we use to identify individual bursts . spectral information is recorded in four energy channels , 20 - 50 kev , 50 - 100 kev , 100 - 300 kev and above 300 kev . using these four fluence measurements , we have fitted the spectrum of each grb to a broken power law , treating the break energy , both spectral slopes and the normalization as free parameters ; see eq.([eq : fnu ] ) . we determine the lorentz factor of the relativistic expanding ejecta using the break energy through eq.([eq : epeak ] ) . for bursts with an observed break energy above 300 kev , or below 50 kev , it is difficult to determine both the break energy and the power law slope . for high energy breaks , the impact on this ambiguity is not critical . as explained in appendix a , the lorentz factor is not dependent on the fit because it is fixed by the observed high energy of the event and the requirement that the fireball be optically thin . for very low spectral breaks ( below 50 kev ) , for instance in events we classify as x - ray flash candidates , we acknowledge a significant degree of uncertainty in the lorentz factor calculation and resulting neutrino spectra . in this case , the spectral break is only uncertain to about a factor of 2 or 3 . detailed temporal information is available in the batse catalog , in the form of light curves . the batse time resolution varies between 2.048 seconds and 0.016 seconds . a resolution of 0.064 seconds is available for all bursts during the time following the trigger . in the framework of the internal shock model , we need a variability time , @xmath29 , @xmath43 in order to get the 1mev @xmath44 ( guetta , spada & waxman 2001b , waxman 2001 ) . in fact with a value of the lorentz factor larger than the minimal value needed to be optically thin up to 100 mev ( @xmath45 see eq.[eq : gamma ] ) the variability time , @xmath29 , has to be @xmath46 to get the 1 mev @xmath1-rays . how well the data support the model is still controversial and a detailed analysis on this issue is out of the aim of our paper . since we refer to this model for our analysis we consider a value of @xmath47 ms for long duration grbs in the rest of the paper . for short duration bursts we take @xmath48 ms and 0.050 seconds for x - ray flash candidates . we have divided the list of batse grbs into four different classes : 1 ) long duration bursts ( duration @xmath26 ) with measured redshfit , 2 ) long duration bursts without measured redshift , 3 ) short duration bursts and 4 ) x - ray flashes . major differences in temporal and spectral properties of long duration grbs , short duration grbs and x - ray flashes has lead to some speculation that they may involve different progenitors or mechanisms . by observing the optical afterglow of grbs , it is possible to measure spectral lines and , therefore , the redshift of an individual burst . although to date the x - ray afterglow of on the order of 100 long duration grbs have been observed , only 31 of them have been observed in the optical making a determination of the redshift possible . ( no redshift has been identified for short grbs ) . we first consider 13 of these that have a complete batse record . the relationship between the comoving distance to an object and its redshift is given by : @xmath49 where @xmath50 and @xmath51 are the fractions of the critical density of the universe in dark energy and matter , respectively . @xmath52 is hubble s constant . once the distance to a grb is known , and its fluence ( or flux ) has been measured , the isotropic - equivalent @xmath1-ray luminosity can be calculated . from this and the value of @xmath53 , together with the break energy of the grb photon spectrum , we estimate the bulk lorentz factor using eqs.([eq : gamma ] , [ eq : epeak ] ) . setting the efficiency for pion production @xmath54 for model 1 and using eq.([eq : fpi2 ] ) for model 2 , we estimate the fraction of proton energy transfered into pions . using eqs.([eq : synclos ] , [ eq : synclos2 ] ) the energy scale of synchrotron losses , @xmath55 , is determined . from these informations we can determine the neutrino flux at earth for each of the four models . for a more detailed discussion see appendices a and b. for the majority of the long duration bursts in the batse catalog no redshift is available . in this case , the @xmath1-ray luminosity of long duration bursts , and , therefore , the distance , can be estimated by assuming a relationship between the observed variability and the luminosity of a grb ( lloyd - ronning & ramierz - ruiz 2002 ; zhang & meszaros 2002 ; kobayashi , ryde & macfadyen 2002 ) . note that the variability of a grb is not the same as its variability time , @xmath29 , previously introduced . the variability of burst is a measure of fluctuations in the temporal structure of the burst . it is defined such that pure noise should have a variability of zero , while the most variable bursts have very sudden and distinctive temporal features . we use the following definition of variability ( fenimore & ramirez - ruiz 2000 ) : @xmath56 where @xmath57 , @xmath58 is the background subtracted fluence in a time bin @xmath59 , @xmath60 is the average background in a single time bin , @xmath61 is the maximum fluence and @xmath62 is the average fluence over a time period centered at time bin @xmath59 of length 30% of the @xmath63 time ( duration ) of the burst . the grbs with observed redshifts have been used to empirically derive a relationship between the variability and the luminosity of the burst ( fenimore & ramirez - ruiz 2000 ; reichart et al . 2001 , reichart & lamb 2001 ) : @xmath64 a relationship between luminosity and the time lag between the peaks for light curves in different energy bands , has been observed in the grb redshift data ( norris , marani & bonnell 2000 ) , but appears to be less reliable . therefore , we will only consider the luminosity - variability relationship . together with the fluence ( flux ) of a burst , the distance of a burst can be determined by the luminosity . note that the variability , and therefore luminosity , of a burst depends on the redshift or distance to the burst . therefore , we must do this calculation by iteration . the end result is a value of the luminosity and redshift for each burst . we would like to enphasize the fact that there are a lot of uncertainties in this way to estimate the redshift , however the knoweledge of the redshift is not so important in our analysis since the minimum variability time scales and the fluence are the most important quantities . it is difficult to reliably calculate the variability of bursts with low flux . for this reason , we only consider bursts with a peak flux greater than 1.5 photons/@xmath65 ( over a 0.256 s time scale ) and with at least 30 time bins ( of 0.064 s width ) of 5 sigma or more above the average background . the grbs which do not meet these requirements have low fluence and are therefore likely to yield a low neutrino flux anyway . after these criteria were applied , 566 long duration bursts without measured redshift are left , making it our largest class . even for bursts which meet the above criteria , the luminosity estimated is only accurate to an order of magnitude . this corresponds to uncertainties of a factor of 2 or 3 in the fraction of proton energy transfered into pions and in the synchrotron loss energy . short duration grbs , with no observation of an optical afterglow and , therefore , no measurement of redshift , can not have a relationship between variability and luminosity empirically established . additionally , variability is difficult to measure for short duration bursts . left with no way to measure the @xmath1-ray luminosity of , or distance to , a short duration grbs , we choose to set @xmath66 for each burst . this introduces greater uncertainty than in the long duration grbs case but , given the ambiguities in the burst characteristics , it is the best that can be done at this time . to be consistent , we considered only short bursts with a peak flux greater than 1.5 photons/@xmath67 ( over a 0.256 s time scale ) , as we did with for long duration grbs . after this criteria was applied , 199 short duration bursts remained in this category . temporal structure and variations appear to occur on shorter time scales for short compared to long grbs ( mcbreen et al . we therefore use a time scale of temporal fluctuations of @xmath68 for all short duration bursts , as opposed to the value of @xmath69 used for long duration bursts . it is also interesting to note that short duration bursts generally have a somewhat harder spectrum and higher peak energy than long duration grbs ( paciesas et al . 2001 ) . x - ray flashes are a newly discovered class of fast transient sources . the bepposax experiment s wide field cameras have observed such events at a rate of about four per year ( heise et al . 2001 , kippen et al . 2002 ) , implying a total rate on the order of @xmath70 per year . these events typically have peak energies as low as 2 - 10 kev and durations of 10 - 100 seconds . the bepposax experiment discriminates x - ray flashes from standard grbs by the non - detection of a signal above 40 kev with the bepposax grb monitor . more generally , a large ratio of x - ray to @xmath1-ray fluence is the differentiating characteristic of x - ray flashes from grbs . it has been suggested , however , that x - ray flashes are a low peak energy extension of gamma - ray bursts ( heise et al . 2001 , kippen et al . 2002 ) . the final class considered here consists of batse events which may be x - ray flashes . we identify 15 events in this class with spectra that peak below 50 kev , although it is difficult to determine accurately where the peak occurs because the sensitivity of batse is somewhat poor in this energy range . these are long duration bursts , and typically have a hard spectrum ; several have @xmath71 in eq.([eq : fnu ] ) larger than 1.5 , and no observed flux in batse s fourth energy channel ( above 300 kev ) . again , with no measured redshift , and limited temporal information , we can not deduce the luminosities of these events . we set @xmath66 for each event and calculate its luminosity accordingly . it is thought that the radius of collisions in x - ray flashes is typically larger than in other grbs and , therefore , the time scale of fluctuations will generally be larger ( guetta , spada & waxman 2001b ) . we therefore choose @xmath72 . for some x - ray flashes the peak energy is very low ( 20 - 30 kev ) and the lorentz factor accordingly very high . increasing the time scale of fluctuations has the additional effect of lowering the lorentz factor to a reasonable value in these extreme cases ( see eq.([eq : epeak ] ) ) . for x - ray flashes , with a very large lorentz factor and a long time scale of fluctuations , we expect that a very small fraction of proton energy converted to pions ; see eqs.([eq : fpi1],[eq : fpi2 ] ) . we therefore expect low neutrino fluxes from proton interactions with grb photons . for the models involving proton interactions with photons in a surrounding pulsar wind bubble , however , the rates can be quite high ( guetta & ganot 2002a ) . there is another class of objects that are the non - triggered bursts . they are found in the batse data in off - line analysis ; see e.g. stern and tikhomirova + ( http://www.astro.su.se/groups/head/grb@xmath73archive.html ) . they are not energetic enough to trigger in real time . these can be grbs with low kinetic luminosity and will be very weak neutrino sources for all the four models , therefore we have decided to neglect them in our analysis . we have constructed a database publicly accessible online with a complete list of all of the grb characteristics and associated neutrino event rates for the approximately 800 bursts we have considered in this analysis . the data was populated in a mysql database , interacting through a user - friendly web interface developed with perl . the database is searchable by grb number and/or by grb class , and is capable of searching for only bursts included in the amanda analysis or all bursts in this work . the rates are for a generic neutrino telescope provided the threshold is sufficiently low for observing the neutrino fluxes predicted . how we transform the grb neutrino fluxes into observed event rates is the topic of the next section . representative results are tabulated in tables 1 through 12 . the database also contains the predicted neutrino spectra for each burst ; these can be directly combined with the simulation of a specific detector . the database is accessible at http://www.arcetri.astro.it/@xmath0dafne/grb . large volume neutrino telescopes are required to observe and measure the neutrino flux from grbs . current experiments , such as amanda ( andres et al . 2001 ) at the south pole , or future ( aslanides et al . 1999 ) and next generation experiments , including icecube ( see http://icecube.wisc.edu/ ) with a full cubic kilometer of instrumented detector volume , i.e. with @xmath74 telescope area , are designed to observe high energy cosmic neutrinos with energies expected from grbs . neutrino telescopes detect the cherenkov light radiated by showers ( hadronic and electromagnetic ) , muons , and taus that are produced in the interactions of neutrinos inside or near the detector . muons are of particular interest because , at the energies typical for grb neutrinos , they travel kilometers before losing energy . the dominant signal is , therefore , through - going muons . icecube can measure the energy and direction of any observed muon . the angular resolution is less than @xmath75 while the energy resolution is approximately a factor of three . signal and background muons may , therefore , be differentiated with a simple energy cut . for shower events the energy measurement improves significantly , being better than 20% , but reconstructing their direction is challenging , the angular resolution being of order 10 degrees . for high energies , when tau decay is sufficiently time dilated , taus have a range similar to or larger than muons , and so the dominant tau signal is from through - going taus . these events have a characteristic signature consisting of a `` clean '' minimum - ionizing track despite its long range inside the detector . we will therefore consider all events in which a tau track passes through the detector in the direction and at the time of a grb . we assume that taus and muons are distinguishable at all energies when specializing to rare events in coincidence in time and direction with a grb . we realize that , in general , it may be difficult to distinguish a muon of energy @xmath76 gev , that is expected to lose relatively little energy from catastrophic processes , from a very high energy tau . tau signatures however become dramatic when they decay inside the detector ( lollipop events ) , or when the tau neutrino interacts and the produced tau decays into showers inside the detector ( double bang events ) . to evaluate the prospects for grb neutrino observations , it is essential to determine the rate of muon , shower and tau events . these calculations are each described in appendix c. for a review of high energy neutrino astronomy ; see ( halzen & hooper 2002 ; learned & mannheim 2000 ) . there are a few grbs we have considered in this analysis which are anomalous for a variety of reasons . we briefly mention these in this section . * grb 6707 is a burst with a measured redshift of 0.0085 , yet a relatively low fluence of @xmath77 . together , this implies a luminosity on the order of @xmath78erg / s , well below the normal range considered . our calculation of the lorentz factor , which depends on the luminosity of the grb , yields a value of about 25,000 , much larger than the normally allowed range . results in model 2 are , therefore , not particularly realistic for this particular burst ( the rates are actually very low for this model ) . the other models are only affected by this in the calculation of the synchrotron energy loss scale . * grbs 7648 , 6891 and 1997 each have lorentz factors below 100 . in all three cases , the burst has been found to have a low luminosity , @xmath79erg / s , which contributes to this result . the results of models 2 , 3 or 4 are only affected by this in their low synchrotron energy loss scales , and are , therefore , conservative . results for model 1 should be interpreted with caution for these three grbs . * grbs 1025 and 8086 , both long duration bursts , have been found to have very low variabilities and , therefore , low luminosities . this results in very high lorentz factors in our calculation ( 20,000 and 5,000 , respectively ) . given the uncertainties involved in the variability calculation and the variability - luminosity relationship , we feel that these luminosities and lorentz factors are unlikely to accurately represent these bursts . as already mentioned , our results are summarized in a series of tables . they summarize our fits to the batse data that provide the input spectrum for calculating the neutrino emission . the neutrino event rates for the 4 classes and the 4 models are also tabulated ( see tables 1 through 8) . in tables 9 - 12 , we summarize the event rates for a generic kilometer - scale telescope , such as icecube , for the 4 classes of grbs . for the duration and in the direction of a grb , the background in a neutrino telescope should be negligible . therefore individual events represent a meaningful observation when coincident with a grb . we next evaluate the prospects for such observations . the amanda experiment has operated for approximately four years ( 1997 - 2000 ) that overlap with the batse mission ( barouch & hardtke 2001 ) . data has been collected for several hundred grbs with an effective telescope area of order 5,000 m@xmath12 . amanda - ii , the completed version of the experiment , with approximately 50,000 m@xmath12 effective area for the high energy neutrinos emitted by grbs , was commissioned less than half a year before the end of the batse mission . nevertheless , of order 100 bursts occurred during that period for which coincident observations were made . with effective areas significantly below the reference square kilometer of future neutrino telescopes , amanda is not expected to test the grb models considered here . for long duration grbs , the most common classification , we anticipate on the order of 0.01 events ( muons+showers ) per square kilometer per grb for models 1 and 2 . for amanda - ii , with one twentieth of this area , and only capable of observing northern hemisphere grbs ( @xmath80/yr ) , we predict on the order of .3 events ( muons+showers ) per year of observation . amanda - b10 , with smaller area , should observe one tenth of this rate . it is interesting , however , to note that such experiments are on the threshold of observation at this time . icecube with a square kilometer of effective area , now under construction , will likely cross this threshold . for the high energies considered here , icecube should be able to make observations of 1000 bursts over @xmath13 steradians during one year . where models 1 and 2 are concerned , we expect that an event ( muon or shower ) will be observed from roughly 10 grbs . for models 3 and 4 we predict @xmath14 events per year for models with @xmath17 yr ( model 4 ) , but only around one event per ten years if @xmath81 is somewhat larger , such as 0.4 yr ( model 3 ) . the prospects for observation depends strongly on @xmath81 , as expected . we expect that classes of grbs with lower average fluence , such as short duration grbs and x - ray flash candidates , will be more difficult to observe . x - ray flash candidates , although unlikely to be observable in models 1 and 2 , could possibly be observed in models 3 and 4 . if @xmath17 yr ( model 4 ) , we predict order one event ( muon or shower ) for every 1000 bursts . with only @xmath82 x - ray flashes thought to occur per year , such an observation will be unlikely . if such events are more common , however , they may contribute significantly to the diffuse high energy neutrino flux . observations of neutrinos from low peak energy grbs would be strong evidence for a supranova progenitor model . it is interesting to note that the majority of the neutrino events from grbs come from a relatively small fraction of the grb population ( alvarez , halzen & hooper 2000 ; halzen & hooper 1999 ) . in figure [ figdist ] we show some of the factors which go into this conclusion . the occasional nearby ( low redshift ) , large @xmath83 , high fluence and/or near horizontal burst can dominate the event rate calculation . the distribution of the number of events per burst ( or x - ray flash candidate ) is shown in figure [ figevn ] . in summary , taking advantage of the large body of grb statistics available in the batse catalog , we have attempted to estimate the neutrino fluxes and event rates in neutrino telescopes associated with grbs for a variety of theoretical models . our analysis has yielded several conclusions : * gamma - ray bursts with high fluence , most often long duration bursts , provide the best opportunity for neutrino observations . * for typical gamma - ray bursts , proton interactions with fireball photons provides the largest neutrino signal . we have also shown that the rates are relatively model independent . * for gamma - ray bursts with very low peak energies , possibly associated with x - ray flashes , very little energy is transfered into pions ( and , therefore , neutrinos ) in interactions with fireball photons . interactions with a surrounding pulsar wind bubble , however , can yield interesting neutrino fluxes . this is an illustration that observation of neutrinos is likely to help decipher the progenitor mechanism . * while our calculations indicate that existing neutrino telescopes , such as amanda , are not likely to have the sensitivity to observe gamma - ray burst neutrinos , next generation , kilometer - scale observatories , such as icecube , will be capable of observing on the order of ten bursts each year . we would like to thank jonathan granot , rellen hardtke , robert preece , ricardo vzquez and eli waxman for valuable discussions . this research was supported by the u.s . department of energy under grant de - fg02 - 95er40896 and by the wisconsin alumni research foundation . m . is supported by mcyt ( fpa 2001 - 3837 ) . .characteristics for grbs with redshifts measured from optical afterglow observations . the fluence and zenith angle of each burst are taken from the batse catalog , whereas the luminosity is derived from the redshift and fluence . the break energy is obtained by a fit to the batse data and the lorentz factor is calculated as described in appendix a. note that grb 6707 is an anomalous burst ( see 6 ) . [ table : i ] [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] in this appendix , we describe the production of neutrinos in interactions of protons and photons in the grb fireball . protons predominantly produce the parent pions via the processes @xmath84 and @xmath85 which have very large cross sections of @xmath86 . the charged @xmath87 s subsequently decay producing charged leptons and neutrinos , while neutral @xmath87 s decay into high - energy photons . for the center - of - mass energy of a proton - photon interaction to exceed the threshold energy for producing the @xmath88-resonance , the comoving proton energy must meet the condition : @xmath89 throughout this paper , primed quantities are measured in the comoving frame and unprimed quantities in the observer frame . in the observer s frame , @xmath90 resulting in a neutrino energy @xmath91 where @xmath92 is the plasma expansion ( bulk ) lorentz factor and @xmath93 is the photon energy . @xmath94 is the average fraction of energy transferred from the initial proton to the produced pion . the factor of 1/4 is based on the estimate that the 4 final state leptons in the decay chain @xmath95 equally share the pion energy . these approximations are adequate given the uncertainties in the astrophysics of the problem . for each proton energy , the resulting neutrino spectrum traces the broken power law spectrum of photons which we fit to the batse data using the broken power law parameterization @xmath96 summing over proton energies results in a neutrino spectrum with the same spectra slopes , @xmath97 and @xmath71 , as for the gamma - ray spectra in the batse data , but with a break energy of order 1pev in the observer frame : @xmath98 we here explicitly introduce the dependence on source redshift , z. the highest energy pions may lose some energy via synchrotron emission before decaying , thus reducing the energy of the decay neutrinos . the effect becomes important when the pion lifetime @xmath99s becomes comparable to the synchrotron loss time @xmath100 where @xmath101 is the energy density of the magnetic field in the shocked fluid . @xmath102 , the fraction of the internal energy carried by the magnetic field , is defined by the relation @xmath103 , where @xmath104 is the collision radius . the collision radius @xmath105 is obtained from the consideration that different shells in the shocked fireball have velocities differing by @xmath106 , where @xmath28 is an average value representative of the entire fireball . different shells emitted at times differing by @xmath29 therefore collide with each other after a time @xmath107 , i.e. at a radius @xmath108 . a detailed account of the kinematics can be found in halzen & hooper 2002 . we can now compare the synchrotron loss time with the time over which the pions decay : @xmath109 here @xmath110 is the fraction of internal energy converted to electrons , @xmath111s is the time scale of fluctuations in the grb lightcurve , @xmath112erg / s is the @xmath1-ray luminosity of the grb and @xmath113ev , is the pion energy . in deriving eq.([t_syn ] ) we have assumed that the wind luminosity carried by internal plasma energy , @xmath114 , is related to the observed @xmath1-ray luminosity through @xmath115 . this assumption is justified because the electron synchrotron cooling time is short compared to the wind expansion time and hence electrons lose all their energy radiatively . the radiative losses become important for @xmath116 , which corresponds to @xmath117 , where @xmath118 neutrinos from muon decay have a lifetime 100 times longer than pions , the energy cutoff will therefore be 10 times smaller : @xmath119 above this energy , the slope of the neutrino spectrum steepens by two to ( @xmath120 ) . to normalize the neutrino spectrum to the observed grb luminosity , we must calculate the fraction , @xmath83 , of fireball proton energy lost to pion production . the fraction of energy converted to pions is estimated from the ratio of the size of the shock , @xmath121 , and the mean free path of a proton for photomeson interactions : @xmath122 here , the proton mean free path is given by @xmath123 where @xmath124 is the number density of photons . the photon number density is given by the ratio of the photon energy density and the photon energy in the comoving frame : @xmath125 using these equations , and recalling that @xmath126 , we obtain that @xmath127 and the fraction of proton energy converted to @xmath87 s is @xmath128 this derivation was performed for protons at the break energy . in general , @xmath129 where @xmath130 is given by eq.([eq : epb ] ) . as we can see from eq.([eq : fpi1 ] ) , @xmath83 strongly depends on the bulk lorentz factor @xmath28 . it has been pointed out by halzen & hooper ( 1999 ) and alvarez , halzen & hooper ( 2000 ) that , if the lorentz factor @xmath28 varies significantly between bursts , then the resulting neutrino flux will be dominated by a few bright bursts with @xmath83 close to unity . however , guetta , spada & waxman ( 2001a ) have shown that burst - to - burst variations in the fraction of fireball energy converted to neutrinos are constrained . first , the observational constraints imposed by @xmath1-ray observations , in particular the requirement @xmath131 mev , imply that wind model parameters @xmath132 ) are correlated ( guetta , spada & waxman 2001b ) and that @xmath28 is restricted to values in a range much narrower than @xmath133 . for instance , for values of @xmath1 much smaller than average the fireball becomes very dense with abundant neutrino production . such fireballs will also produce a thermal photon spectrum which is not the case for the events considered here . second , for wind parameters that yield @xmath83 values significantly exceeding 20% , only a small fraction of pion energy is converted to neutrinos because of pion and muon synchrotron losses as can be seen from e q.([eq : synclos ] ) . we will use two methods to determine the value of the bulk lorentz factor , @xmath28 . for bursts with high break energies , @xmath134 kev , @xmath28 can not differ significantly from the minimum value for which the fireball pair production optical depth is @xmath135 near the maximum energy of @xmath1-rays produced , @xmath136 . egret has observed @xmath1-rays with energies in excess of 1 gev for six bursts , although the maximum @xmath1-ray energy should be lower for the majority of grbs . we choose 100 mev as the default value , therefore , @xmath137^{1/6}. \label{eq : gammamin}\ ] ] note that in the end , the value of the lorentz factor depends weakly on luminosity , time structure and maximum @xmath1-ray energy . for bursts with lower break energies eq.([eq : gamma ] ) may not be reliable because the lorentz factor of these grbs may be larger than estimated . guetta spada & waxman ( 2001b ) have argued that the x - ray flashes identified by bepposax could be produced by relativistic winds where the lorentz factor is larger than the minimum value given in eq.([eq : gamma ] ) required to produce a grb with the characteristic photon spectrum . for grbs with low break energy we , instead , relate the lorentz factor to the peak energy of the @xmath1-ray spectrum . the characteristic frequency of synchrotron emission is determined by the minimum electron lorentz factor @xmath138 and by the strength of the magnetic field given above , before eq.([t_syn ] ) . the characteristic energy of synchrotron photons , @xmath139 , at the source redshift is @xmath140 for bursts with @xmath141 kev we will evaluate @xmath28 from the break photon energy given above . at present no theory allows the determination of the values of the equipartition fractions @xmath110 and @xmath102 . eq.([eq : epeak ] ) implies that fractions not far below unity are required to account for the observed @xmath1 ray emission and this is confirmed also by simulations ( guetta spada & waxman 2001b ) . for bursts with very large values of @xmath28 , the peak energy is shifted to values lower than @xmath142 kev . this could be the x - ray bursts detected by bepposax ( guetta spada & waxman 2001b ) . from eq.([eq : fpi1 ] ) , we estimate that the neutrino flux from such events is expected to be small . we have now collected all the information to derive the neutrino from the observed @xmath1-ray fluency @xmath143 : @xmath144 batse detectors measure the grb fluence @xmath143 over two decades of photon energies , @xmath145 mev to @xmath0 2 mev , corresponding to a decade of energy of the radiating electrons . the factor 1/8 takes into account that charged and neutral pions are produced with roughly equal probabilities , and each neutrino carries @xmath146 of the pion energy . using eq.([eq : fpi2 ] ) , @xmath147 where @xmath148 and @xmath149 are given by eq.([eq : enub ] ) , eq.([eq : synclos ] ) and eq.([eq : synclos2 ] ) . this spectrum is shown in fig.([fig2a ] ) . this result depends on a number of somewhat tenuous assumptions . simulations ( gsw ) actually suggest that one can simply fix @xmath150 at the break energy and derive the @xmath11 flux directly from the @xmath1-flux . doing better may require a better understanding of the fireball phenomenology than we have now . first , the variability time may be shorter than what is observed ; in most cases variability is only measured to the smallest time scale that can be detected with adequate statistics . second , the parameters @xmath151 and @xmath152 are uncertain . third , the luminosity - variability relation used to derive the luminosity for bursts with no measured redshift ( see [ zknown ] ) is uncertain , and in addition may have large fluctuations around the prediction . we have therefore decided to do the detailed analysis described above as well as an alternative analysis that assumes @xmath54 , at the break energy , for all bursts and determines the neutrino flux directly from the observed gamma - ray fluence . for this alternative approach , @xmath153 we refer to the models based on eq.([eq : nuflux2 ] ) and eq.([eq : nuflux1 ] ) as models 1 and 2 , respectively . in fig . [ fig2a ] we show the muon neutrino spectrum for our fiducial parameters in models 1 and 2 . in this section , we consider the neutrino photoproduction on external photons in supranova grbs ( guetta & granot 2002a ) . the external radiation field surrounding the final black hole is referred to as the pulsar wind bubble ( pwb ) . a pulsar wind bubble is formed when the relativistic wind ( consisting of relativistic particles and magnetic fields ) that emanates from a pulsar is abruptly decelerated ( typically , to a newtonian velocity ) in a strong relativistic shock , due to interaction with the ambient medium . fractions of the post - shock energy density go to the magnetic field , the electrons and the protons , respectively . the electrons will lose energy through synchrotron emission and inverse - compton ( ic ) scattering . in ( guetta & granot 2003 ) a deep study of the characteristic features of the plerion emission has been carried out . as shown in ( guetta & granot 2003 ) , the electrons are in the fast cooling regime for relevant values of @xmath4 , and therefore most of the emission takes place within a small radial interval just behind the wind termination shock . the mechanism for neutrino production through photomeson interaction with the external photons dominates when the lifetime of the supramassive neutron star , @xmath5yr ( or @xmath6yr for x - ray flashes ) . neutrinos generated in this way have typical energies @xmath7ev ( @xmath154ev for x - ray flashes ) . as in models 1 and 2 , these neutrinos are emitted simultaneously with the prompt @xmath1-ray ( x - ray ) emission . for even shorter lifetimes @xmath10yr , the @xmath11 s would not be accompanied by a detectable grb because the thomson optical depth on the pwb is larger than unity . as before , protons of energy @xmath155 interact mostly with photons that satisfy the @xmath88-resonance condition , @xmath156 , where , in this case , @xmath157 is the pwb photon energy . the minimum photon energy for photomeson interactions corresponds to the maximum proton energy @xmath158ev . for reasonable model parameters ( @xmath159yr ) this energy exceeds self absorption frequency of the pwb spectrum ( guetta & granot 2003 ) . moreover , for @xmath160 yr the electrons are in fast cooling and emit synchrotron radiation . therefore the relevant part of the spectrum consists of two power laws , @xmath161 for @xmath162 and @xmath163 for @xmath164 , where @xmath165hz is the peak frequency of the pwb spectral energy distribution @xmath166 , @xmath124 is the number density of photons and @xmath167 is the power law index of the pwb electrons . the normalization factor of the target photon number density is determined by equating the pulsar wind luminosity in pairs , @xmath168 ( @xmath169 is the fraction of the pulsar wind energy in the @xmath170 component and @xmath171 is the total energy of the pulsar wind ) to the total energy output in photons , which is @xmath172 at @xmath173 , where @xmath174 is the radius of the pulsar wind termination shock , which is a factor , @xmath175 , smaller than the pwb radius , @xmath176 . at @xmath177 , which is relevant for our case , the photons are roughly isotropic and @xmath178 becomes roughly constant , and assumes the value @xmath179 . as we mentioned above , the relevant target photons for photomeson interaction with high energy protons are the synchrotron photons and the fraction of the total photon energy that goes into the synchrotron component is @xmath180 , where @xmath181 and @xmath182 are the fractions of the pwb energy in the electrons and in magnetic field , respectively . this spectrum is consistent with the spectrum of known plerions like the crab , that can be well fit by emission from a power law distribution with a power law index value @xmath183 . in our case , the only difference is that there is a fast cooling synchrotron spectrum , rather than a slow cooling one . there are no observations of the spectrum from very young plerions where there is a fast cooling spectrum ( as they are more rare ) . this radiation will be typically hard to detect ( for @xmath184 yr and @xmath185 ) , but might be detected for closer ( though rarer ) pwbs . the proton energy satisfying the @xmath88-resonance condition with pwb photons of energy @xmath186 , is @xmath187 where @xmath188yr , @xmath189 is the velocity of the snr shell ( in units of @xmath190 ) , @xmath191erg , @xmath192 is the lorentz factor of the pulsar wind , @xmath193 , @xmath194 , @xmath195 , and @xmath196 is the fraction of the wind energy that remains in the pwb . the latter is the fraction that goes into the proton component and is , unlike the electron component , not radiated away . the corresponding neutrino energy is @xmath197 . for @xmath198yr , this energy is similar to those obtained in interactions with grb photons in the previous section . photon emission is only detected in coincidence with these neutrinos if the thompson optical depth is @xmath199 , which is the case for @xmath198yr and a clumpy snr ( guetta & granot 2003 ; inoue , guetta & pacini 2003 ) . in the case of a uniform shell the condition is @xmath200yr , corresponding to @xmath201ev . as before , the internal shocks occur over a distance @xmath202 . thus , the optical depth for photo - pion production by protons of energy @xmath155 , is @xmath203 where @xmath92 , @xmath204s , @xmath205 , @xmath206 , @xmath207 and @xmath71 is the spectral slope of the seed pwb synchrotron photons with @xmath208 ( @xmath209 ) for @xmath210 ( @xmath211 ) . the fraction of the proton energy that is lost to pion production is given by @xmath212 \approx\min\left[1,\tau_{p\gamma}(\varepsilon_p)/5\right]\ .\ ] ] the factor of 5 , as before , takes into account that the proton loses @xmath213 of its energy in a single interaction . we denote @xmath214 for which @xmath215 by @xmath216ev [ i.e. @xmath217 , and obtain @xmath218 as before , the decay of charged pions created in interactions between pwb photons and grb protons , produces high energy neutrinos , @xmath219 , where each neutrino receives @xmath220 of the proton energy . the total energy of the protons accelerated in the internal shocks is expected to be similar to the @xmath1-ray energy produced in the grb ( waxman 1995 ) . this implies a @xmath221 fluence , @xmath222 @xmath223 where @xmath224erg is the isotropic equivalent energy in @xmath1-rays , @xmath225 , while @xmath226 is given in eq.([f_pg_grb ] ) and @xmath227 is the fraction of the original pion energy , @xmath228 , that remains after decay . the pions may lose energy via synchrotron or inverse compton ( ic ) emission . if these energy losses are significant , then the energy of the neutrinos will be reduced as well . following arguments already presented in appendix a , we find that @xmath229 , where @xmath230s is the lifetime of the pion , and @xmath231 is the time for radiative losses due to both synchrotron and ic losses . the time , @xmath232 , is given by eq.([t_sync ] ) and @xmath233 where @xmath234 is the energy density of photons below the klein - nishina limit , @xmath235 . ic losses due to scattering of the grb photons were shown to be unimportant ( waxman & bahcall 1997 ) . we therefore only consider the ic losses from the upscattering of the external pwb photons , and find @xmath236 where @xmath110 and @xmath102 are the equipartition parameters of the grb , and @xmath113ev . the radiative losses become important for @xmath237 , which corresponds to @xmath238 , where @xmath239 the protons may also lose energy via @xmath240 interactions with the grb photons ( waxman & bahcall 1997 ) . however @xmath241 for this process is typically @xmath242 , so that it does not have a large effect on @xmath243 interactions with the pwb photons , on which we focus . because the lifetime of the muons is @xmath244100 times longer than that of the pions , they experience significant radiative losses at an energy of @xmath245 . this causes a reduction of up to a factor of @xmath246 in the total neutrino flux in the range @xmath247 , since only @xmath221 that are produced directly in @xmath248 decay contribute significantly to the neutrino flux . note that since both ratios in eqs . ( [ t_syn ] ) and ( [ t_ic ] ) scale as @xmath249 , we always have @xmath250 , and therefore , the spectrum steepens by a factor of @xmath251 for @xmath252 . this is evident because @xmath253 . in figure [ fig1 ] we show the proton energies that correspond to the neutrino break energies @xmath254 , @xmath255 and @xmath256 , as a function of @xmath4 . from eq . ( [ tau_pg_grb ] ) , ( [ e_syn ] ) and ( [ e_ic ] ) , we conclude that @xmath257 , while @xmath258 and @xmath259 . for a fixed value of @xmath260 , @xmath261 implies @xmath262 and @xmath263ev , while @xmath264 implies @xmath265 . this implies increased neutrino emission reaching higher energies for larger values of @xmath28 . since large @xmath28 implies lower synchrotron frequency for the prompt grb , this may be relevant to x - ray flashes , assuming that they are indeed grbs with relatively large lorentz factors and/or a large variability time , @xmath29 ( guetta , spada & waxman 2001b ) . as can be seen from figure [ fig1 ] , depending on the relevant parameters , there are four different orderings of these break energies : ( i ) @xmath266 , ( ii ) @xmath267 , ( iii ) @xmath268 and ( iv ) @xmath269 . each of these results in a different spectrum consisting of 3 or 4 power laws . analytically these spectra are : @xmath270^{2 } } } = \left\{\begin{array}{lll } ( \varepsilon_{\nu}/ \varepsilon_{\nu\tau})^{s/2 } & \ \ \varepsilon_{\nu}<\varepsilon_{\nu\tau}\\ 1 & \ \ \varepsilon_{\nu\tau}<\varepsilon_{\nu}<\varepsilon_{\nu s}\\ ( \varepsilon_{\nu}/\varepsilon_{\nu s})^{-2 } & \ \ \varepsilon_{\nu}>\varepsilon_{\nu s } \end{array}\right . \ , \ ] ] @xmath271^{2 } } } = \left\{\begin{array}{lll}\sqrt{\varepsilon_{\nu b}\over\varepsilon_{\nu\tau } } \left({\varepsilon_{\nu}\over\varepsilon_{\nu b}}\right)^{s\over 2 } & \ \ \varepsilon_{\nu}<\varepsilon_{\nu b}\\ ( \varepsilon_{\nu}/\varepsilon_{\nu\tau})^{1/2 } & \ \ \varepsilon_{\nu b}<\varepsilon_{\nu}<\varepsilon_{\nu\tau}\\ 1 & \ \ \varepsilon_{\nu\tau}<\varepsilon_{\nu}<\varepsilon_{\nu s}\\ ( \varepsilon_{\nu}/\varepsilon_{\nu s})^{-2 } & \ \ \varepsilon_{\nu}>\varepsilon_{\nu s}\end{array}\right . \ , \ ] ] @xmath272 @xmath273(\varepsilon_{\nu s}/\varepsilon_{\nu b})^{s/2 } \sqrt{\varepsilon_{\nu b}/\varepsilon_{\nu\tau}}}= \left\{\begin{array}{lll } ( \varepsilon_{\nu}/\varepsilon_{\nu s})^{s/2 } & \ \ \varepsilon_{\nu}<\varepsilon_{\nu s}\\ ( \varepsilon_{\nu}/\varepsilon_{\nu s})^{(s-4)/2 } & \ \ \varepsilon_{\nu s}<\varepsilon_{\nu}<\varepsilon_{\nu b}\\ \left({\varepsilon_{\nu b}\over\varepsilon_{\nu s}}\right)^{(s-4)/2 } \left({\varepsilon_{\nu}\over\varepsilon_{\nu b}}\right)^{-3/2 } & \ \ \varepsilon_{\nu b}<\varepsilon_{\nu}<\varepsilon_{\nu\tau}\\ \left({\varepsilon_{\nu b}\over\varepsilon_{\nu s}}\right)^{s-4\over 2 } \left({\varepsilon_{\nu b}\over\varepsilon_{\nu\tau}}\right)^{3\over 2 } \left({\varepsilon_{\nu\tau}\over\varepsilon_{\nu}}\right)^{2 } & \ \ \varepsilon_{\nu}>\varepsilon_{\nu\tau } \end{array}\right . \ .\ ] ] for our analysis we consider two characteristic values of @xmath4 , 0.07 yr and 0.4 yr , and refer to the models as 3 and 4 , respectively . in the case of model 3 , the grb is seen only if the shell is sufficiently clumpy while in model 4 the grb should always be detectable . the muon neutrino spectrum is shown in figure [ fig2 ] for our fiducial parameters and @xmath274yr . the spectrum of the other neutrino flavors is the same . a compilation of the probability that a grb neutrino is actually detected as a muon , a tau or a shower by an underground detector is shown in fig.4 as a function of the neutrino energy . these are required to convert neutrino spectra from grbs to event rates . we present in this appendix the formalism for doing this conversion . the number of shower events in an underground detector from a neutrino flux @xmath275 produced by a single grb with duration @xmath276 is given by @xmath277 where @xmath278 is the zenith angle ( @xmath279 is vertically downward ) . the sum is over neutrino ( and anti - neutrino ) flavors @xmath280 and interactions @xmath281 ( charged current ) and nc ( neutral current ) . @xmath282 is the detector s cross sectional area with respect to the @xmath11 flux , and @xmath283 is the differential neutrino flux that reaches the earth . for @xmath284 , [ rate ] is modified to include the effects of regeneration of neutrinos propagating through the earth , as will be discussed further on . @xmath285 is the probability that a neutrino reaches the detector , i.e. is not absorbed by the earth . it is given by @xmath286 \ , \label{survival}\ ] ] where @xmath287 , and the total neutrino interaction cross section is @xmath288 this is somewhat conservative because it neglects the possibility of a neutrino interacting via a nc interaction and subsequently creating a shower in the detector . @xmath289 is the column density of material a neutrino with zenith angle @xmath278 must traverse to reach the detector . it depends on the depth of the detector and is given by @xmath290 the path length along direction @xmath278 weighted by the earth s density @xmath291 at distance @xmath292 from the earth s center . for the earth s density profile we adopt the piecewise continuous density function @xmath293 of the preliminary earth model ( dziewonski 1989 ) . @xmath294 is the probability that the neutrino interacts in the detector . it is given by @xmath295 \ , \ ] ] where , for showers , @xmath40 is the linear dimension of the detector , and @xmath296 is the mean free path for neutrino interaction of type @xmath297 . for realistic detectors , @xmath298 , and so @xmath299 . to an excellent approximation the event rate scales linearly with detector volume @xmath300 . the inelasticity parameter @xmath301 is the fraction of the initial neutrino energy carried by the hadronic shower ( rather than the primary lepton ) . the limits of integration depend on the type of interaction and on the neutrino flavor . for nc @xmath302 interactions and all @xmath221 and @xmath303 interactions , @xmath304 and @xmath305 , where @xmath306 is the threshold energy for shower detection . for cc @xmath302 interactions , the outgoing electron also showers , therefore @xmath304 and @xmath307 . energetic muons are produced in @xmath221 cc interactions . for a muon to be detected , it must reach the detector with an energy above its threshold @xmath308 . the expression of eq.([rate ] ) then also describes the number of muon events after the replacement @xmath309 \ , \ ] ] where @xmath310 is the range of a muon with initial energy @xmath311 and final energy @xmath308 . we will assume that muons lose energy continuously according to @xmath312 where @xmath313 and @xmath314 ( dutta et al . 2001 ) . the muon range is then @xmath315 \ . \label{murange}\ ] ] in this case , @xmath316 and @xmath317 . the event rate for muons is enhanced by the possibility that muons reach the detector , even if produced in neutrino interactions kilometers from its location . note , however , that this enhancement ( i.e. @xmath310 ) is @xmath278-dependent : for nearly vertical down - going paths , the path length of the muon is limited by the amount of matter above the detector , not by the muon s range . this is taken into account in the simulations . fig.[fig5 ] shows the probability of detecting a muon generated by a muon neutrino as a function of the incidence zenith angle . the figure illustrates the effect of the limited amount of matter above the detector , as well as the neutrino absorption in the earth . absorption is not important at @xmath318 tev , and the muon range at this energy ( for a muon energy threshold of 500 gev as we adopted in the figure ) is not limited by the amount of matter above the detector . as a consequence the probability at 1 tev is weakly dependent on zenith angle . absorption in the earth starts to be important for upgoing neutrinos of energy above @xmath82 tev - 1 pev , and restricts the neutrino observation to the horizontal and downward directions at extremely high energy ( eev range ) as can be seen in the figure . the large muon range at pev and eev energies , much larger than the depth at which the detector is located ( 1.8 km vertical depth ) , reduces considerably the detection probability of downgoing neutrinos . the largest background to a grb signal consists of muons from atmospheric neutrinos . however , for grb observations , the time and angular windows are very small and this background can be easily controlled , as we will illustrate . following ( dermer & atoyan 2003 ) , the number of background events is approximately given by @xmath319 where @xmath320 and @xmath321 are the solid angle and time considered , respectively , and @xmath322 is the probability of a muon neutrino generating a muon in the detector volume . icecube is designed to have angular resolution smaller than 1 degree at the relevant energies . we consider a 1 degree cone for the solid angle calculation . considering a long burst of duration @xmath82 seconds , and using the following approximate atmospheric neutrino spectrum @xmath323 we find that for a natural threshold ( minimum energy ) of @xmath324gev , we expect @xmath325 background events per burst . more practically , an energy threshold in the range of 1 - 10 tev could be imposed which would reduce this background by an addition factor of 50 to 2500 , respectively . for a naive illustration , consider one years of observation , with 1000 bursts , each of duration of 10 seconds and a 1 tev energy threshold imposed . for such a example , less than 0.01 total background events are predicted . taus are produced only by cc @xmath303 interactions . this process differs significantly from the muon case because tau neutrinos are regenerated by the production and subsequent tau decay through @xmath326 ( halzen & saltzberg 1998 ) . as a result , for tau neutrinos , cc and nc interactions in the earth do not deplete the @xmath303 flux , they only reduce the neutrino energy down to a value where , eventually , the earth becomes transparent . we implement this important effect using a dedicated simulation that determines @xmath327 , the average energy of the @xmath303 when reaching the detector . it depends on the initial energy @xmath328 and zenith angle @xmath278 . the tau event rate is then given by @xmath329 @xmath294 depends on the geometry of the neutrino tau induced event . for events consisting on a minimum - ionizing track going through the detector it is given by : @xmath330 \label{eq : pint_tau}\ ] ] where @xmath331 is the range of the produced tau evaluated at the energy of the tau after regeneration . @xmath332 is given by eq.([murange ] ) with @xmath333 ( dutta et al . the last factor takes into account the requirement that the tau track be long enough to be identified in the detector . we require @xmath334gev so that the tau decay length is larger than the 125 m string spacing in icecube . it is not clear with what efficiency through - going tau events can be separated from low energy muons . those tau events that include one shower ( lollipop events ) or two showers ( double bang events ) inside the detector volume will be identifiable . for these cases we use @xmath294 obtained by a dedicated simulation that determines the probabilities for double bang and lollipop geometries shown in fig.([fig4 ] ) . the rate of downgoing lollipop events in a km@xmath335 neutrino telescope is expected to be of the order of the rate of down - going shower events , probably slightly smaller . double bang events will be mostly observed for neutrino energies in a limited range between roughly 10 and 100 pev . finally , those events in which the tau decays into muons that reach the detector are counted as muon events . @xmath294 is given by eq.([eq : pint_tau ] ) where we use as range the sum of the range of the tau evaluated at the regeneration energy and the range of the muon at the energy that carries in the decay of the tau . as with muons , at very high energies taus can travel several kilometers before decaying or suffering significant energy loss . the enhancement to tau event rates from this effect is @xmath278-dependent as discussed above for muons . heise , j. , zand , j. j. , kippen , m. & woods , p. , gamma - ray bursts in the afterglow era , proceedings of the international workshop held in rome , cnr headquarters , 17 - 20 october , 2000 . edited by enrico costa , filippo frontera , and jens hjorth . berlin heidelberg : springer , 2001 , p. 16 . astro - ph/0111246 . paciesas , w. s. , preece , r. d. briggs , m. s. & mallozzi , r. s. , gamma - ray bursts in the afterglow era , proceedings of the international workshop held in rome , cnr headquarters , 17 - 20 october , 2000 . edited by enrico costa , filippo frontera , and jens hjorth . berlin heidelberg : springer , 2001 , p. 13 . astro - ph/0109053 . reichart d. e. & lamb , d. q. , gamma - ray bursts in the afterglow era , proceedings of the international workshop held in rome , cnr headquarters , 17 - 20 october , 2000 . edited by enrico costa , filippo frontera , and jens hjorth . berlin heidelberg : springer , 2001 , p. 233 . astro - ph/0103254 . gamma - ray bursts in the afterglow era , proceedings of the international workshop held in rome , cnr headquarters , 17 - 20 october , 2000 . edited by enrico costa , filippo frontera , and jens hjorth . berlin heidelberg : springer , 2001 , p. 263 . the distribution of ( a ) redshifts , ( b ) lorentz factors , ( c ) fluences and ( d ) zenith angles ( for muon events : dotted line corresponds to model 1 , solid line to model 2 ) for long duration grbs . in ( a ) , the decrease at large redshift is due to sampling bias . the lorentz factors in ( b ) were calculated as described by eqs . [ eq : gamma ] and [ eq : epeak ] . frame ( d ) demonstrates the advantages of long muon range and poor absorption near the horizon for muon track detection.,width=529 ] the distribution of the estimated number of muon events from individual ( a ) long duration grbs ( the dotted line corresponds to model 1 and the solid line to model 2 ) and ( b ) x - ray flash candidates ( dotted line corresponds to model 3 and the solid line to model 4 ) . note that the majority of events result from a relatively small number of grbs.,width=529 ] the muon neutrino spectrum , @xmath336 , for our fiducial parameters in models 1 and 2 ( interactions with grb photons ) . the solid line is for a typical grb with @xmath337 and @xmath47 ms , while the dashed line is for a x - ray flash candidate with @xmath338 ( calculated from eq . [ eq : epeak ] ) and @xmath339 ms.,width=529 ] the muon neutrino spectrum , @xmath336 , for our fiducial parameters in models 3 and 4 ( interactions with pwb photons ) . four choices of @xmath274yr are used , which correspond to the 4 different orderings of the break energies . this figure is taken from guetta & granot 2002a , width=529 ] the probabilities of a neutrino generating various types of events when traveling through the effective area of a neutrino telescope . curves are shown for various choices of zenith angle , which reflects enhancements due to long muon range and the effect of absorption in the earth.,width=529 ] the probability of a muon neutrino generating a detectable muon as a function of zenith angle . the curves show the probabilities for three representative neutrino energies : @xmath341 ev , @xmath342 ev and @xmath343 ev.,width=529 ]
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we estimate the neutrino emission from individual gamma - ray bursts observed by the batse detector on the compton gamma - ray observatory .
neutrinos are produced by photoproduction of pions when protons interact with photons in the region where the kinetic energy of the relativistic fireball is dissipated allowing the acceleration of electrons and protons .
we also consider models where neutrinos are predominantly produced on the radiation surrounding the newly formed black hole . from the observed redshift and photon flux of each individual burst ,
we compute the neutrino flux in a variety of models based on the assumption that equal kinetic energy is dissipated into electrons and protons . where not measured , the redshift is estimated by other methods . unlike previous calculations of the universal diffuse neutrino flux produced by all gamma - ray bursts , the individual fluxes ( compiled at http://www.arcetri.astro.it/@xmath0 dafne / grb/ ) can be directly compared with coincident observations by the amanda telescope at the south pole . because of its large statistics , our predictions are likely to be representative for future observations with larger neutrino telescopes .
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in a typical instance of a combinatorial optimization problem the underlying constraints model a static application frozen in one time step . in many applications however , one needs to solve instances of the combinatorial optimization problem that changes over time . while this is naturally handled by re - solving the optimization problem in each time step separately , changing the solution one holds from one time step to the next often incurs a transition cost . consider , for example , the problem faced by a vendor who needs to get supply of an item from @xmath10 different producers to meet her demand . on any given day , she could get prices from each of the producers and pick the @xmath10 cheapest ones to buy from . as prices change , this set of the @xmath10 cheapest producers may change . however , there is a fixed cost to starting and/or ending a relationship with any new producer . the goal of the vendor is to minimize the sum total of these two costs : an `` acquisition cost '' @xmath11 to be incurred each time she starts a new business relationship with a producer , and a per period cost @xmath12 of buying in period @xmath2 from the each of the @xmath10 producers that she picks in this period , summed over @xmath13 time periods . in this work we consider a generalization of this problem , where the constraint `` pick @xmath10 producers '' may be replaced by a more general combinatorial constraint . it is natural to ask whether simple combinatorial problems for which the one - shot problem is easy to solve , as the example above is , also admit good algorithms for the multistage version . the first problem we study is the _ multistage matroid maintenance _ problem ( ) , where the underlying combinatorial constraint is that of maintaining a base of a given matroid in each period . in the example above , the requirement the vendor buys from @xmath10 different producers could be expressed as optimizing over the @xmath14uniform matroid . in a more interesting case one may want to maintain a spanning tree of a given graph at each step , where the edge costs @xmath12 change over time , and an acquisition cost of @xmath11 has to paid every time a new edge enters the spanning tree . ( a formal definition of the problem appears in section [ sec : formal - defs ] . ) while our emphasis is on the online problem , we will mention results for the offline version as well , where the whole input is given in advance . a first observation we make is that if the matroid in question is allowed to be different in each time period , then the problem is hard to approximate to any non - trivial factor ( see section [ sec : time - varying ] ) even in the offline case . we therefore focus on the case where the same matroid is given at each time period . thus we restrict ourselves to the case when the matroid is the same for all time steps . to set the baseline , we first study the offline version of the problem ( in section [ sec : offline ] ) , where all the input parameters are known in advance . we show an lp - rounding algorithm which approximates the total cost up to a logarithmic factor . this approximation factor is no better than that using a simple greedy algorithm , but it will be useful to see the rounding algorithm , since we will use its extension in the online setting . we also show a matching hardness reduction , proving that the problem is hard to approximate to better than a logarithmic factor ; this hardness holds even for the special case of spanning trees in graphs . we then turn to the online version of the problem , where in each time period , we learn the costs @xmath12 of each element that is available at time @xmath2 , and we need to pick a base @xmath15 of the matroid for this period . we analyze the performance of our online algorithm in the competitive analysis framework : i.e. , we compare the cost of the online algorithm to that of the optimum solution to the offline instance thus generated . in section [ sec : online ] , we give an efficient randomized @xmath16-competitive algorithm for this problem against any oblivious adversary ( here @xmath17 is the universe for the matroid and @xmath6 is the rank of the matroid ) , and show that no polynomial - time online algorithm can do better . we also show that the requirement that the algorithm be randomized is necessary : any deterministic algorithm must incur an overhead of @xmath18 , even for the simplest of matroids . our results above crucially relied on the properties of matriods , and it is natural to ask if we can handle more general set systems , e.g. , @xmath19-systems . in section [ sec : matchings ] , we consider the case where the combinatorial object we need to find each time step is a perfect matching in a graph . somewhat surprisingly , the problem here is significantly harder than the matroid case , even in the offline case . in particular , we show that even when the number of periods is a constant , no polynomial time algorithm can achieve an approximation ratio better than @xmath20 for any constant @xmath21 . we first show that the problem , which is a packing - covering problem , can be reduced to the analogous problem of maintaining a spanning set of a matroid . we call the latter the _ multistage spanning set maintenance _ ( ) problem . while the reduction itself is fairly clean , it is surprisingly powerful and is what enables us to improve on previous works . the problem is a covering problem , so it admits better approximation ratios and allows for a much larger toolbox of techniques at our disposal . we note that this is the only place where we need the matroid to not change over time : our algorithms for work when the matroids change over time , and even when considering matroid intersections . the problem is then further reduced to the case where the holding cost of an element is in @xmath22 , this reduction simplifies the analysis . in the offline case , we present two algorithms . we first observe that a greedy algorithm easily gives an @xmath23-approximation . we then present a simple randomized rounding algorithm for the linear program . this is analyzed using recent results on contention resolution schemes @xcite , and gives an approximation of @xmath24 , which can be improved to @xmath25 when the acquisition costs are uniform . this lp - rounding algorithm will be an important constituent of our algorithm for the online case . for the online case we again use that the problem can be written as a covering problem , even though the natural lp formulation has both covering and packing constraints . phrasing it as a covering problem ( with box constraints ) enables us to use , as a black - box , results on online algorithms for the fractional problem @xcite . this formulation however has exponentially many constraints . we handle that by showing a method of adaptively picking violated constraints such that only a small number of constraints are ever picked . the crucial insight here is that if @xmath26 is such that @xmath27 is not feasible , then @xmath26 is at least @xmath28 away in @xmath29 distance from any feasible solution ; in fact there is a single constraint that is violated to an extent half . this insight allows us to make non - trivial progress ( using a natural potential function ) every time we bring in a constraint , and lets us bound the number of constraints we need to add until constraints are satisfied by @xmath27 . our work is related to several lines of research , and extends some of them . the paging problem is a special case of where the underlying matroid is a uniform one . our online algorithm generalizes the @xmath30-competitive algorithm for weighted caching @xcite , using existing online lp solvers in a black - box fashion . going from uniform to general matroids loses a logarithmic factor ( after rounding ) , we show such a loss is unavoidable unless we use exponential time . the problem is also a special case of classical metrical task systems @xcite ; see @xcite for more recent work . the best approximations for metrical task systems are poly - logarithmic in the size of the metric space . in our case the metric space is specified by the total number of bases of the matroid which is often exponential , so these algorithms only give a trivial approximation . in trying to unify online learning and competitive analysis , buchbinder et al . @xcite consider a problem on matroids very similar to ours . the salient differences are : ( a ) in their model all acquisition costs are the same , and ( b ) they work with fractional bases instead of integral ones . they give an @xmath31-competitive algorithm to solve the fractional online lp with uniform acquisition costs ( among other unrelated results ) . our online lp solving generalizes their result to arbitrary acquisition costs . they leave open the question of getting integer solutions online ( seffi naor , private communication ) , which we present in this work . in a more recent work , buchbinder , chen and naor @xcite use a regularization approach to solving a broader set of fractional problems , but once again can do not get integer solutions in a setting such as ours . shachnai et al . @xcite consider `` reoptimization '' problems : given a starting solution and a new instance , they want to balance the transition cost and the cost on the new instance . this is a two - timestep version of our problem , and the short time horizon raises a very different set of issues ( since the output solution does not need to itself hedge against possible subsequent futures ) . they consider a number of optimization / scheduling problems in their framework . cohen et al . @xcite consider several problems in the framework of the stability - versus - fit tradeoff ; e.g. , that of finding `` stable '' solutions which given the previous solution , like in reoptimization , is the current solution that maximizes the quality minus the transition costs . they show maintaining stable solutions for matroids becomes a repeated two - stage reoptimization problem ; their problem is poly - time solvable , whereas matroid problems in our model become np - hard . the reason is that the solution for two time steps does not necessarily lead to a base from which it is easy to move in subsequent time steps , as our hardness reduction shows . they consider a multistage offline version of their problem ( again maximizing fit minus stability ) which is very similar in spirit and form to our ( minimization ) problem , though the minus sign in the objective function makes it difficult to approximate in cases which are not in poly - time . in dynamic steiner tree maintenance @xcite where the goal is to maintain an approximately optimal steiner tree for a varying instance ( where terminals are added ) while changing few edges at each time step . in dynamic load balancing @xcite one has to maintain a good scheduling solution while moving a small number of jobs around . the work on lazy experts in the online prediction community @xcite also deals with similar concerns . there is also work on `` leasing '' problems @xcite : these are optimization problems where elements can be obtained for an interval of any length , where the cost is concave in the lengths ; the instance changes at each timestep . the main differences are that the solution only needs to be feasible at each timestep ( i.e. , the holding costs are @xmath32 ) , and that any element can be leased for any length @xmath33 of time starting at any timestep for a cost that depends only on @xmath33 , which gives these problems a lot of uniformity . in turn , these leasing problems are related to `` buy - at - bulk '' problems . given reals @xmath34 for elements @xmath35 , we will use @xmath36 for @xmath37 to denote @xmath38 . we denote @xmath39 by @xmath40 $ ] . we assume basic familiarity with matroids : see , e.g. , @xcite for a detailed treatment . given a matroid @xmath41 , a _ base _ is a maximum cardinality independent set , and a _ spanning set _ is a set @xmath42 such that @xmath43 ; equivalently , this set contains a base within it . the _ span _ of a set @xmath44 is @xmath45 . the _ matroid polytope _ @xmath46 is defined as @xmath47 . the _ base polytope _ @xmath48 . we will sometimes use @xmath5 to denote @xmath49 and @xmath6 to denote the rank of the matroid . an instance of the _ multistage matroid maintenance _ ( ) problem consists of a matroid @xmath41 , an _ acquisition cost _ @xmath50 for each @xmath35 , and for every timestep @xmath51 $ ] and element @xmath35 , a _ holding cost _ cost @xmath12 . the goal is to find bases @xmath52}$ ] to minimize @xmath53 where we define @xmath54 . a related problem is the _ multistage spanning set maintenance _ ( ) problem , where we want to maintain a spanning set @xmath55 at each time , and cost of the solution @xmath56}$ ] ( once again with @xmath57 ) is @xmath58 the following lemma shows the equivalence of maintaining bases and spanning sets . this enables us to significantly simplify the problem and avoid the difficulties faced by previous works on this problem . [ lem : pack - cover ] for matroids , the optimal solutions to and have the same costs . clearly , any solution to is also a solution to , since a base is also a spanning set . conversely , consider a solution @xmath59 to . set @xmath60 to any base in @xmath61 . given @xmath62 , start with @xmath63 , and extend it to any base @xmath64 of @xmath15 . this is the only step where we use the matroid properties indeed , since the matroid is the same at each time , the set @xmath63 remains independent at time @xmath2 , and by the matroid property this independent set can be extended to a base . observe that this process just requires us to know the base @xmath65 and the set @xmath15 , and hence can be performed in an online fashion . we claim that the cost of @xmath66 is no more than that of @xmath67 . indeed , @xmath68 , because @xmath69 . moreover , let @xmath70 , we pay @xmath71 for these elements we just added . to charge this , consider any such element @xmath72 , let @xmath73 be the time it was most recently added to the cover i.e . , @xmath74 for all @xmath75 $ ] , but @xmath76 . the solution paid for including @xmath77 at time @xmath78 , and we charge our acquisition of @xmath77 into @xmath64 to this pair @xmath79 . it suffices to now observe that we will not charge to this pair again , since the procedure to create @xmath80 ensures we do not drop @xmath77 from the base until it is dropped from @xmath15 itself the next time we pay an addition cost for element @xmath77 , it would have been dropped and added in @xmath81 as well . hence it suffices to give a good solution to the problem . we observe that the proof above uses the matroid property crucially and would not hold , e.g. , for matchings . it also requires that the _ same _ matroid be given at all time steps . also , as noted above , the reduction is online : the instance is the same , and given an solution it can be transformed online to a solution to . we will find it convenient to think of an instance of as being a matroid @xmath82 , where each element only has an acquisition cost @xmath83 , and it has a lifetime @xmath84 $ ] . there are no holding costs , but the element @xmath77 can be used in spanning sets only for timesteps @xmath85 . or one can equivalently think of holding costs being zero for @xmath86 and @xmath87 otherwise . _ an offline exact reduction . _ the translation is the natural one : given instance @xmath88 of , create elements @xmath89 for each @xmath90 and @xmath91 , with acquisition cost @xmath92 , and interval @xmath93 $ ] . ( the matroid is extended in the natural way , where all the elements @xmath89 associated with @xmath77 are parallel to each other . ) the equivalence of the original definition of and this interval view is easy to verify . _ an online approximate reduction . _ observe that the above reduction created at most @xmath94 copies of each element , and required knowledge of all the costs . if we are willing to lose a constant factor in the approximation , we can perform a reduction to the interval model in an _ online _ fashion as follows . for element @xmath35 , define @xmath95 , and create many parallel copies @xmath96 of this element ( modifying the matroid appropriately ) . now the @xmath97 interval for @xmath77 is @xmath98 $ ] , where @xmath99 is set to @xmath100 in case @xmath101 , else it is set to the _ largest _ time such that the total holding costs @xmath102 for this interval @xmath103 $ ] is at most @xmath11 . this interval @xmath104 is associated with element @xmath105 , which is only available for this interval , at cost @xmath106 . a few salient points about this reduction : the intervals for an original element @xmath77 now partition the entire time horizon @xmath40 $ ] . the number of elements in the modified matroid whose intervals contain any time @xmath2 is now only @xmath107 , the same as the original matroid ; each element of the modified matroid is only available for a single interval . moreover , the reduction can be done online : given the past history and the holding cost for the current time step @xmath2 , we can ascertain whether @xmath2 is the beginning of a new interval ( in which case the previous interval ended at @xmath108 ) and if so , we know the cost of acquiring a copy of @xmath77 for the new interval is @xmath109 . it is easy to check that the optimal cost in this interval model is within a constant factor of the optimal cost in the original acquisition / holding costs model . given the reductions of the previous section , we can focus on the problem . being a covering problem , is conceptually easier to solve : e.g. , we could use algorithms for submodular set cover @xcite with the submodular function being the sum of ranks at each of the timesteps , to get an @xmath23 approximation . in section [ sec : greedy ] , we give a dual - fitting proof of the performance of the greedy algorithm . here we give an lp - rounding algorithm which gives an @xmath110 approximation ; this can be improved to @xmath25 in the common case where all acquisition costs are unit . ( while the approximation guarantee is no better than that from submodular set cover , this lp - rounding algorithm will prove useful in the online case in section [ sec : online ] ) . finally , the hardness results of section [ sec : hardness - offline ] show that we can not hope to do much better than these logarithmic approximations . we now consider an lp - rounding algorithm for the problem ; this will generalize to the online setting , whereas it is unclear how to extend the greedy algorithm to that case . for the lp rounding , we use the standard definition of the problem to write the following lp relaxation . @xmath111 it remains to round the solution to get a feasible solution to ( i.e. , a spanning set @xmath15 for each time ) with expected cost at most @xmath31 times the lp value , since we can use lemma [ lem : pack - cover ] to convert this to a solution for at no extra cost . the following lemma is well - known ( see , e.g. @xcite ) . we give a proof for completeness . [ lem : alon ] for a fractional base @xmath112 , let @xmath113 be the set obtained by picking each element @xmath35 independently with probability @xmath114 . then @xmath115 \geq r(1 - 1/e)$ ] . we use the results of chekuri et al . @xcite ( extending those of chawla et al . @xcite ) on so - called contention resolution schemes . formally , for a matroid @xmath82 , they give a randomized procedure @xmath116 that takes the random set @xmath113 and outputs an independent set @xmath117 in @xmath82 , such that @xmath118 , and for each element @xmath77 in the support of @xmath119 , @xmath120 \geq ( 1 - 1/e)$ ] . ( they call this a @xmath121-balanced cr scheme . ) now , we get @xmath122 & \geq { { \mathbf{e } } } [ { \textsf{rank}}(\pi_z(r(z ) ) ) ] = \sum_{e \in \text{supp}(z ) } \pr [ e \in \pi_z(r(z ) ) ] \\ & = \sum_{e \in \text{supp}(z ) } \pr [ e \in \pi_z(r(z ) ) \mid e \in r(z ) ] \cdot \pr [ e \in r(z ) ] \\ & \geq \sum_{e \in \text{supp}(z ) } ( 1 - 1/e ) \cdot z_e = r(1 - 1/e ) . \end{aligned}\ ] ] the first inequality used the fact that @xmath117 is a subset of @xmath113 , the following equality used that @xmath117 is independent with probability 1 , the second inequality used the property of the cr scheme , and the final equality used the fact that @xmath119 was a fractional base . [ thm : lp - round ] any fractional solution can be randomly rounded to get solution to with cost @xmath24 times the fractional value , where @xmath6 is the rank of the matroid and @xmath13 the number of timesteps . set @xmath123 . for each element @xmath35 , choose a random threshold @xmath124 independently and uniformly from the interval @xmath125 $ ] . for each @xmath126 , define the set @xmath127 ; if @xmath128 does not have full rank , augment its rank using the cheapest elements according to @xmath129 to obtain a full rank set @xmath15 . since @xmath130 = \min\ { l\cdot z_t(e ) , 1\}$ ] , the cost @xmath131 . moreover , @xmath132 exactly when @xmath124 satisfies @xmath133 , which happens with probability at most @xmath134 hence the expected acquisition cost for the elements newly added to @xmath128 is at most @xmath135 . finally , we have to account for any elements added to extend @xmath128 to a full - rank set @xmath15 . [ lem : rand - round ] for any fixed @xmath51 $ ] , the set @xmath128 contains a basis of @xmath82 with probability at least @xmath136 . the set @xmath128 is obtained by threshold rounding of the fractional base @xmath137 as above . instead , consider taking @xmath138 different samples @xmath139 , where each sample is obtained by including each element @xmath35 independently with probability @xmath140 ; let @xmath141 . it is easy to check that @xmath142 \leq \pr [ { \textsf{rank}}(\widehat{s}_t ) = r]$ ] , so it suffices to give a lower bound on the former expression . for this , we use lemma [ lem : alon ] : the sample @xmath143 has expected rank @xmath144 , and using reverse markov , it has rank at least @xmath145 with probability at least @xmath146 . now focusing on the matroid @xmath147 obtained by contracting elements in @xmath148 ( which , say , has rank @xmath149 ) , the same argument says the set @xmath150 has rank @xmath151 with probability at least @xmath152 , etc . proceeding in this way , the probability that the rank of @xmath13 is less than @xmath6 is at most the probability that we see fewer than @xmath153 heads in @xmath154 flips of a coin of bias @xmath152 . by a chernoff bound , this is at most @xmath155 . now if the set @xmath128 does not have full rank , the elements we add have cost at most that of the min - cost base under the cost function @xmath156 , which is at most the optimum value for ( [ eq : lp2 ] ) . ( we use the fact that the lp is exact for a single matroid , and the global lp has cost at least the single timestep cost . ) this happens with probability at most @xmath157 , and hence the total expected cost of augmenting @xmath128 over all @xmath13 timesteps is at most @xmath158 times the lp value . this proves the main theorem . again , this algorithm for works with different matroids at each timestep , and also for intersections of matroids . to see this observe that the only requirements from the algorithm are that there is a separation oracle for the polytope and that the contention resolution scheme works . in the case of @xmath14matroid intersection , if we pay an extra @xmath30 penalty in the approximation ratio we have that the probability a rounded solution does not contain a base is @xmath159 so we can take a union bound over the multiple matroids . 0 we can replace the dependence on @xmath13 by a term that depends only on the variance in the acquisition costs . let us divide the period @xmath160 into `` epochs '' , where an epoch is an interval @xmath161 for @xmath162 such that the total fractional acquisition cost @xmath163 . we can afford to build a brand new tree at the beginning of each epoch and incur an acquisition cost of at most the rank @xmath6 , which we can charge to the lp s fractional acquisition cost in the epoch . by theorem [ thm : lp - round ] , naively applying the rounding algorithm to each epoch independently gives a guarantee of @xmath164 , where @xmath165 is the maximum length of an epoch . now we should be able to use the argument from the online section that says that we can ignore steps where the total movement is smaller than half . thus @xmath165 can be assumed to be @xmath166 . more details to be added once we consistentize notation . in fact , if we define epoch to be a period of acquisition cost @xmath167 , then the at least half means movement cost at least @xmath168 . thus the epoch only has @xmath169 relevant steps in it , so we get log of that . for the special case where all the acquisition costs @xmath11 are all the same , this implies we get rid of the @xmath13 term in the lp rounding , and get an @xmath25-approximation . when the ratio of the maximum to the minimum acquisition cost is small , we can improve the approximation factor above . more specifically , we show that essentially the same randomized rounding algorithm ( with a different choice of @xmath138 ) gives an approximation ratio of @xmath170 . we defer the argument to section [ sec : just - logr ] , as it needs some additional definitions and results that we present in the online section . [ [ an - improvement - avoiding - the - dependence - on - t.-1 ] ] an improvement : avoiding the dependence on @xmath13 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + when the ratio of the maximum to the minimum acquisition cost is small , we can improve the approximation factor above . more specifically , we show that essentially the same randomized rounding algorithm ( with a different choice of @xmath138 ) gives an approximation ratio of @xmath170 . we defer the argument to section [ sec : just - logr ] , as it needs some additional definitions and results that we present in the online section . we defer the hardness proof to appendix [ app : offline ] , which shows that the and problems are np - hard to approximate better than @xmath171 even for graphical matroids . an integrality gap of @xmath172 appears in appendix [ sec : int - gap - matroids ] . [ thm : matr - hard ] the and problems are np - hard to approximate better than @xmath173 even for graphical matroids . we give a reduction from set cover to the problem for graphical matroids . given an instance @xmath174 of set cover , with @xmath175 sets and @xmath176 elements , we construct a graph as follows . there is a special vertex @xmath6 , and @xmath5 set vertices ( with vertices @xmath177 for each set @xmath178 ) . there are @xmath5 edges @xmath179 which all have inclusion weight @xmath180 and per - time cost @xmath181 for all @xmath2 . all other edges will be zero cost short - term edges as given below . in particular , there are @xmath182 timesteps . in timestep @xmath183 $ ] , define subset @xmath184 to be vertices corresponding to sets containing element @xmath185 . we have a set of edges @xmath186 for all @xmath187 , and all edges @xmath188 for @xmath189 . all these edges have zero inclusion weight @xmath11 , and are only alive at time @xmath190 . ( note this creates a graph with parallel edges , but this can be easily fixed by subdividing edges . ) in any solution to this problem , to connect the vertices in @xmath191 to @xmath6 , we must buy some edge @xmath192 for some @xmath193 . this is true for all @xmath190 , hence the root - set edges we buy correspond to a set cover . moreover , one can easily check that if we acquire edges @xmath194 such that the sets @xmath195 form a set cover , then we can always augment using zero cost edges to get a spanning tree . since the only edges we pay for are the @xmath194 edges , we should buy edges corresponding to a min - cardinality set cover , which is hard to approximate better than @xmath196 . finally , that the number of time periods is @xmath182 , and the rank of the matroid is @xmath197 for these hard instances . this gives us the claimed hardness . we now turn to solving in the online setting . in this setting , the acquisition costs @xmath11 are known up - front , but the holding costs @xmath12 for day @xmath2 are not known before day @xmath2 . since the equivalence given in lemma [ lem : pack - cover ] between and holds even in the online setting , we can just work on the problem . we show that the online problem admits an @xmath198-competitive ( oblivious ) randomized algorithm . to do this , we show that one can find an @xmath199-competitive fractional solution to the linear programming relaxation in section [ sec : offline ] , and then we round this lp relaxation online , losing another logarithmic factor . again , we work in the interval model outlined in section [ sec : intervals ] . recall that in this model , for each element @xmath77 there is a unique interval @xmath200 $ ] during which it is alive . the element @xmath77 has an acquisition cost @xmath11 , no holding costs . once an element has been acquired ( which can be done at any time during its interval ) , it can be used at all times in that interval , but not after that . in the online setting , at each time step @xmath2 we are told which intervals have ended ( and which have not ) ; also , which new elements @xmath77 are available starting at time @xmath2 , along with their acquisition costs @xmath11 . of course , we do not know when its interval @xmath201 will end ; this information is known only once the interval ends . we will work with the same lp as in section [ sec : lp - round ] , albeit now we have to solve it online . the variable @xmath202 is the indicator for whether we acquire element @xmath77 . @xmath203 \notag\end{aligned}\ ] ] note that this is not a packing or covering lp , which makes it more annoying to solve online . hence we consider a slight reformulation . let @xmath204 denote the _ spanning set polytope _ defined as the convex hull of the full - rank ( a.k.a . spanning ) sets @xmath205 . since each spanning set contains a base , we can write the constraints of ( [ eq:3 ] ) as : @xmath206 here we define @xmath207 to be the vector derived from @xmath208 by zeroing out the @xmath202 values for @xmath209 . it is known that the polytope @xmath204 can be written as a ( rather large ) set of covering constraints . indeed , @xmath210 , where @xmath211 is the dual matroid for @xmath82 . since the rank function of @xmath212 is given by @xmath213 , it follows that ( [ eq:4 ] ) can be written as @xmath214 thus we get a covering lp with `` box '' constraints over @xmath17 . the constraints can be presented one at a time : in timestep @xmath2 , we present all the covering constraints corresponding to @xmath215 . we remark that the newer machinery of @xcite may be applicable to [ eq : coveringconstraints ] . we next show that a simpler approach suffices will be useful in improving the rounding algorithm . ] . the general results of buchbinder and naor @xcite ( and its extension to row - sparse covering problems by @xcite ) imply a deterministic algorithm for fractionally solving this linear program online , with a competitive ratio of @xmath216 . however , this is not yet a polynomial - time algorithm , the number of constraints for each timestep being exponential . we next give an adaptive algorithm to generate a small yet sufficient set of constraints . [ [ solving - the - lp - online - in - polynomial - time.-1 ] ] solving the lp online in polynomial time . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + given a vector @xmath217^e$ ] , define @xmath218 as follows : @xmath219 clearly , @xmath220 and @xmath221^e$ ] . we next describe the algorithm for generating covering constraints in timestep @xmath2 . recall that @xcite give us an online algorithm @xmath222 for solving a fractional covering lp with box constraints ; we use this as a black - box . ( this lp solver only raises variables , a fact we will use . ) in timestep @xmath2 , we adaptively select a small subset of the covering constraints from ( [ eq : coveringconstraints ] ) , and present it to @xmath222 . moreover , given a fractional solution returned by @xmath222 , we will need to massage it at the end of timestep @xmath2 to get a solution satisfying all the constraints from ( [ eq : coveringconstraints ] ) corresponding to @xmath2 . let @xmath208 be the fractional solution to ( [ eq : coveringconstraints ] ) at the end of timestep @xmath108 . now given information about timestep @xmath2 , in particular the elements in @xmath215 and their acquisition costs , we do the following . given @xmath208 , we construct @xmath218 and check if @xmath223 , as one can separate for @xmath204 . if @xmath223 , then @xmath218 is feasible and we do not need to present any new constraints to @xmath222 , and we return @xmath218 . if not , our separation oracle presents an @xmath42 such that the constraint @xmath224 is violated . we present the constraint corresponding to @xmath42 to @xmath222 to get an updated @xmath208 , and repeat until @xmath218 is feasible for time @xmath2 . ( since @xmath222 only raises variables and we have a covering lp , the solution remains feasible for past timesteps . ) we next argue that we do not need to repeat this loop more than @xmath225 times . [ lem : farfromfeasible ] if for some @xmath208 and the corresponding @xmath218 , the constraint @xmath226 is violated . then @xmath227 let @xmath228 and let @xmath229 . let @xmath230 denote @xmath231 . thus @xmath232 since both @xmath233 and @xmath234 are integers , it follows that @xmath235 . on the other hand , for every @xmath236 , and thus @xmath237 . consequently @xmath238 finally , for any @xmath239 , @xmath240 , so the claim follows . the algorithm @xmath222 updates @xmath208 to satisfy the constraint given to it , and lemma [ lem : farfromfeasible ] implies that each constraint we give to it must increase @xmath241 by at least @xmath28 . the translation to the interval model ensures that the number of elements whose intervals contain @xmath2 is at most latexmath:[$|e_t| \leq time @xmath2 is at most @xmath243 . we summarize the discussion of this section in the following theorem . [ thm : lp - solve ] there is a polynomial - time online algorithm to compute an @xmath244-approximate solution to ( [ eq:3 ] ) . we observe that the solution to this linear program can be trivially transformed to one for the lp in section [ sec : lp - round ] . finally , the randomized rounding algorithm of section [ sec : lp - round ] can be implemented online by selecting a threshold @xmath245 $ ] the beginning of the algorithm , where @xmath246 and selecting element @xmath77 whenever @xmath247 exceeds @xmath248 : here we use the fact that the online algorithm only ever raises @xmath202 values , and this rounding algorithm is monotone . rerandomizing in case of failure gives us an expected cost of @xmath24 times the lp solution , and hence we get an @xmath249-competitive algorithm . the dependence on the time horizon @xmath13 is unsatisfactory in some settings , but we can do better using lemma [ lem : farfromfeasible ] . recall that the @xmath251-factor loss in the rounding follows from the naive union bound over the @xmath13 time steps . we can argue that when @xmath252 is small , we can afford for the rounding to fail occasionally , and charge it to the acquisition cost incurred by the linear program . the details appear in appendix [ sec : just - logr ] . the dependence on the time horizon @xmath13 is unsatisfactory in some settings , but we can do better using lemma [ lem : farfromfeasible ] . recall that the @xmath251-factor loss in the rounding follows from the naive union bound over the @xmath13 time steps . we now argue that when @xmath252 is small , we can afford for the rounding to fail occasionally , and charge it to the acquisition cost incurred by the linear program . let us divide the period @xmath253 $ ] into disjoint `` epochs '' , where an epoch ( except for the last ) is an interval @xmath161 for @xmath162 such that the total fractional acquisition cost @xmath254 . thus an epoch is a minimal interval where the linear program spends acquisition cost @xmath255 $ ] , so that we can afford to build a brand new tree once in each epoch and can charge it to the lp s fractional acquisition cost in the epoch . naively applying theorem [ thm : lp - round ] to each epoch independently gives us a guarantee of @xmath256 , where @xmath165 is the maximum length of an epoch . however , an epoch can be fairly long if the lp solution changes very slowly . we break up each epoch into phases , where each phase is a maximal subsequence such that the lp incurs acquisition cost at most @xmath257 ; clearly the epoch can be divided into at most @xmath258 disjoint phases . for a phase @xmath259 $ ] , let @xmath260}$ ] denote the solution defined as @xmath260}(e ) = \min_{t\in [ t_1,t_2 ] } z_t(e)$ ] . the definition of the phase implies that for any @xmath261 $ ] , the @xmath262 difference @xmath263 } - z_t\|_1 \leq \frac{1}{4}$ ] . now lemma [ lem : farfromfeasible ] implies that @xmath264}$ ] is in @xmath265 , where @xmath266 is defined as in ( [ eq:5 ] ) . suppose that in the randomized rounding algorithm , we pick the threshold @xmath267 $ ] for @xmath268 . let @xmath269}$ ] be the event that the rounding algorithm applied to @xmath260}$ ] gives a spanning set . since @xmath264 } \leq 2z_{[t_1,t_2]}$ ] is in @xmath270 for a phase @xmath259 $ ] , lemma [ lem : rand - round ] implies that the event @xmath269}$ ] occurs with probability @xmath271 . moreover , if @xmath269}$ ] occurs , it is easy to see that the randomized rounding solution is feasible for all @xmath272 $ ] . since there are @xmath273 phases within an epoch , the expected number of times that the randomized rounding fails any time during an epoch is @xmath274 . suppose that we rerandomize all thresholds whenever the randomized rounding fails . each rerandomization will cost us at most @xmath275 in expected acquisition cost . since the expected number of times we do this is less than once per epoch , we can charge this additional cost to the @xmath275 acquisition cost incurred by the lp during the epoch . thus we get an @xmath276-approximation . this argument also works for the online case ; hence for the common case where all the acquisition costs are the same , the loss due to randomized rounding is @xmath25 . we can show that any polynomial - time algorithm can not achieve better than an @xmath277 competitive ratio , via a reduction from online set cover . details appear in appendix [ app : sec : hardness - online ] . in the online set cover problem , one is given an instance @xmath278 of set cover , and in time step @xmath2 , the algorithm is presented an element @xmath279 , and is required to pick a set covering it . the competitive ratio of an algorithm on a sequence @xmath280}$ ] is the ratio of the number of sets picked by the algorithm to the optimum set - cover of the instance @xmath281\},{{\mathcal{f}}})$ ] . korman ( * ? ? ? * theorem 2.3.4 ) shows the following hardness for online set cover : there exists a constant @xmath282 such that if there is a ( possibly randomized ) polynomial time algorithm for online set cover with competitive ratio @xmath283 , then @xmath284 . recall that in the reduction in the proof of theorem [ thm : matr - hard ] , the set of long term edges depends only on @xmath285 . the short term edges alone depend on the elements to be covered . it can then we verified that the same approach gives a reduction from online set cover to online . it follows that the online problem does not admit an algorithm with competitive ratio better than @xmath286 unless @xmath284 . in fact this hardness holds even when the end time of each edge is known as soon as it appears , and the only non - zero costs are @xmath287 . we next consider the _ perfect matching maintenance _ ( ) problem where @xmath17 is the set of edges of a graph @xmath288 , and the at each step , we need to maintain a perfect matchings in @xmath289 . * integrality gap . * somewhat surprisingly , we show that the natural lp relaxation has an @xmath290 integrality gap , even for a constant number of timesteps . the lp and the ( very simple ) example appears in appendix [ sec : match - int - gap ] . the natural lp relaxation is : @xmath291 the polytope @xmath292 is now the perfect matching polytope for @xmath289 . [ lem : int - gap ] there is an @xmath290 integrality gap for the problem . consider the instance in the figure , and the following lp solution for 4 time steps . in @xmath293 , the edges of each of the two cycles has @xmath294 , and the cross - cycle edges have @xmath295 . in @xmath296 , we have @xmath297 and @xmath298 , and otherwise it is the same as @xmath293 . @xmath299 and @xmath300 are the same as @xmath293 . in @xmath301 , we have @xmath302 and @xmath303 , and otherwise it is the same as @xmath293 . for each time @xmath2 , the edges in the support of the solution @xmath304 have zero cost , and other edges have infinite cost . the only cost incurred by the lp is the movement cost , which is @xmath158 . consider the perfect matching found at time @xmath305 , which must consist of matchings on both the cycles . ( moreover , the matching in time 3 must be the same , else we would change @xmath290 edges . ) suppose this matching uses exactly one edge from @xmath306 and @xmath307 . then when we drop the edges @xmath308 and add in @xmath309 , we get a cycle on @xmath310 vertices , but to get a perfect matching on this in time @xmath311 we need to change @xmath290 edges . else the matching uses exactly one edge from @xmath306 and @xmath312 , in which case going from time @xmath313 to time @xmath314 requires @xmath290 changes . * hardness . * moreover , in appendix [ app : sec : match - hard ] we show that the perfect matching maintenance problem is very hard to approximate : for any @xmath315 it is np - hard to distinguish instances with cost @xmath316 from those with cost @xmath317 , where @xmath318 is the number of vertices in the graph . this holds even when the holding costs are in @xmath22 , acquisition costs are @xmath319 for all edges , and the number of time steps is a constant . in this section we prove the following hardness result : for any @xmath315 it is np - hard to distinguish instances with cost @xmath316 from those with cost @xmath317 , where @xmath318 is the number of vertices in the graph . this holds even when the holding costs are in @xmath22 , acquisition costs are @xmath319 for all edges , and the number of time steps is a constant . the proof is via reduction from @xmath313-coloring . we assume we are given an instance of @xmath313-coloring @xmath288 where the maximum degree of @xmath289 is constant . it is known that the @xmath313-coloring problem is still hard for graphs with bounded degree ( * ? ? ? * theorem 2 ) . we construct the following gadget @xmath320 for each vertex @xmath321 . ( a figure is given in figure [ fig : gadget ] . ) there are two cycles of length @xmath322 , where @xmath33 is odd . the first cycle ( say @xmath323 ) has three distinguished vertices @xmath324 at distance @xmath33 from each other . the second ( called @xmath325 ) has similar distinguished vertices @xmath326 at distance @xmath33 from each other . there are three more `` interface '' vertices @xmath327 . vertex @xmath328 is connected to @xmath329 and @xmath330 , similarly for @xmath331 and @xmath332 . there is a special `` switch '' vertex @xmath333 , which is connected to all three of @xmath334 . call these edges the _ switch _ edges . due to the two odd cycles , every perfect matching in @xmath320 has the structure that one of the interface vertices is matched to some vertex in @xmath323 , another to a vertex in @xmath325 and the third to the switch @xmath333 . we think of the subscript of the vertex matched to @xmath333 as the color assigned to the vertex @xmath335 . at every odd time step @xmath126 , the only allowed edges are those within the gadgets @xmath336 : i.e. , all the holding costs for edges within the gadgets is zero , and all edges between gadgets have holding costs @xmath87 . this is called the `` steady state '' . at every even time step @xmath2 , for some matching @xmath337 of the graph , we move into a `` test state '' , which intuitively tests whether the edges of a matching @xmath338 have been properly colored . we do this as follows . for every edge @xmath339 , the switch edges in @xmath340 become unavailable ( have infinite holding costs ) . moreover , now we allow some edges that go between @xmath320 and @xmath341 , namely the edge @xmath342 , and the edges @xmath343 for @xmath344 and @xmath345 . note that any perfect matching on the vertices of @xmath346 which only uses the available edges would have to match @xmath342 , and one interface vertex of @xmath320 must be matched to one interface vertex of @xmath341 . moreover , by the structure of the allowed edges , the colors of these vertices must differ . ( the other two interface vertices in each gadget must still be matched to their odd cycles to get a perfect matching . ) since the graph has bounded degree , we can partition the edges of @xmath289 into a constant number of matchings @xmath347 for some @xmath348 ( using vizing s theorem ) . hence , at time step @xmath349 , we test the edges of the matching @xmath350 . the number of timesteps is @xmath351 , which is a constant . . the test - state edges are on the right . ] suppose the graph @xmath289 was indeed @xmath313-colorable , say @xmath352 is the proper coloring . in the steady states , we choose a perfect matching within each gadget @xmath320 so that @xmath353 is matched . in the test state @xmath354 , if some edge @xmath355 is in the matching @xmath338 , we match @xmath342 and @xmath356 . since the coloring @xmath357 was a proper coloring , these edges are present and this is a valid perfect matching using only the edges allowed in this test state . note that the only changes are that for every test edge @xmath358 , the matching edges @xmath359 and @xmath360 are replaced by @xmath342 and @xmath361 . hence the total acquisition cost incurred at time @xmath354 is @xmath362 , and the same acquisition cost is incurred at time @xmath363 to revert to the steady state . hence the total acquisition cost , summed over all the timesteps , is @xmath364 . suppose @xmath289 is not @xmath313-colorable . we claim that there exists vertex @xmath365 such that the interface vertex not matched to the odd cycles is different in two different timesteps i.e . , there are times @xmath366 such that @xmath367 and @xmath185 ( for @xmath345 ) are the states . then the length of the augmenting path to get from the perfect matching at time @xmath368 to the perfect matching at @xmath369 is at least @xmath33 . now if we set @xmath370 , then we get a total acquisition cost of at least @xmath371 in this case . the size of the graph is @xmath372 , so the gap is between @xmath373 and @xmath374 . this proves the claim . in this paper we studied multistage optimization problems : an optimization problem ( think about finding a minimum - cost spanning tree in a graph ) needs to be solved repeatedly , each day a different set of element costs are presented , and there is a penalty for changing the elements picked as part of the solution . hence one has to hedge between sticking to a suboptimal solution and changing solutions too rapidly . we present online and offline algorithms when the optimization problem is maintaining a base in a matroid . we show that our results are optimal under standard complexity - theoretic assumptions . we also show that the problem of maintaining a perfect matching becomes impossibly hard . our work suggests several directions for future research . it is natural to study other combinatorial optimization problems , both polynomial time solvable ones such shortest path and min - cut , as well np - hard ones such as min - max load balancing and bin - packing in this multistage framework with acquisition costs . moreover , the approximability of the _ bipartite _ matching maintenance , as well as matroid intersection maintenance remains open . our hardness results for the matroid problem hold when edges have @xmath375 acquisition costs . the unweighted version where all acquisition costs are equal may be easier ; we currently know no hardness results , or sub - logarithmic approximations for this useful special case . an extension of /problems is to the case when the set of elements remain the same , but the matroids change over time . again the goal in is to maintain a matroid base at each time . [ thm : diff - matrs - wpb ] the problem with different matroids is np - hard to approximate better than a factor of @xmath376 , even for partition matroids , as long as @xmath377 . the reduction is from 3d - matching ( 3 dm ) . an instance of 3 dm has three sets @xmath378 of equal size @xmath379 , and a set of hyperedges @xmath380 . the goal is to choose a set of disjoint edges @xmath381 such that @xmath382 . first , consider the instance of with three timesteps @xmath383 . the universe elements correspond to the edges . for @xmath384 , create a partition with @xmath10 parts , with edges sharing a vertex in @xmath385 falling in the same part . the matroid @xmath386 is now to choose a set of elements with at most one element in each part . for @xmath387 , the partition now corresponds to edges that share a vertex in @xmath388 , and for @xmath389 , edges that share a vertex in @xmath390 . set the movement weights @xmath391 for all edges . if there exists a feasible solution to 3 dm with @xmath10 edges , choosing the corresponding elements form a solution with total weight @xmath10 . if the largest matching is of size @xmath392 , then we must pay @xmath393 extra over these three timesteps . this gives a @xmath10-vs-@xmath394 gap for three timesteps . to get a result for @xmath13 timesteps , we give the same matroids repeatedly , giving matroids @xmath395 at all times @xmath396 $ ] . in the `` yes '' case we would buy the edges corresponding to the 3d matching and pay nothing more than the initial @xmath10 , whereas in the `` no '' case we would pay @xmath397 every three timesteps . finally , the apx - hardness for 3 dm @xcite gives the claim . the time - varying problem does admit an @xmath24 approximation , as the randomized rounding ( or the greedy algorithm ) shows . however , the equivalence of and does not go through when the matroids change over time . the restriction that the matroids vary over time is essential for the np - hardness , since if the partition matroid is the same for all times , the complexity of the problem drops radically . [ thm : partition ] the problem with partition matroids can be solved in polynomial time . the problem can be solved using min - cost flow . indeed , consider the following reduction . create a node @xmath398 for each element @xmath77 and timestep @xmath2 . let the partition be @xmath399 . then for each @xmath400 $ ] and each @xmath401 , add an arc @xmath402 , with cost @xmath403 . add a cost of @xmath12 per unit flow through vertex @xmath398 . ( we could simulate this using edge - costs if needed . ) finally , add vertices @xmath404 and source @xmath405 . for each @xmath406 , add arcs from @xmath177 to all vertices @xmath407 with costs @xmath408 . all these arcs have infinite capacity . now add unit capacity edges from @xmath405 to each @xmath177 , and infinite capacity edges from all nodes @xmath409 to @xmath2 . since the flow polytope is integral for integral capacities , a flow of @xmath6 units will trace out @xmath6 paths from @xmath405 to @xmath2 , with the elements chosen at each time @xmath2 being independent in the partition matroid , and the cost being exactly the per - time costs and movement costs of the elements . observe that we could even have time - varying movement costs . whereas , for graphical matroids the problem is @xmath410 hard even when the movement costs for each element do not change over time , and even just lie in the set @xmath375 . moreover , the restriction in theorem [ thm : diff - matrs - wpb ] that @xmath411 is also necessary , as the following result shows . [ thm : two ] for the case of two rounds ( i.e. , @xmath412 ) the problem can be solved in polynomial time , even when the two matroids in the two rounds are different . the solution is simple , via matroid intersection . suppose the matroids in the two timesteps are @xmath413 and @xmath414 . create elements @xmath415 which corresponds to picking element @xmath77 and @xmath416 in the two time steps , with cost @xmath417 . lift the matroids @xmath386 and @xmath418 to these tuples in the natural way , and look for a common basis . we note that deterministic online algorithms can not get any non - trivial guarantee for the problem , even in the simple case of a @xmath319-uniform matroid . this is related to the lower bound for deterministic algorithms for paging . formally , we have the 1-uniform matroid on @xmath5 elements , and @xmath419 . all acquisition costs @xmath11 are 1 . in the first period , all holding costs are zero and the online algorithm picks an element , say @xmath420 . since we are in the non - oblivious model , the algorithm knows @xmath420 and can in the second time step , set @xmath421 , while leaving the other ones at zero . now the algorithm is forced to move to another edge , say @xmath422 , allowing the adversary to set @xmath423 and so on . at the end of @xmath419 rounds , the online algorithm is forced to incur a cost of 1 in each round , giving a total cost of @xmath13 . however , there is still an edge whose holding cost was zero throughout , so that the offline opt is 1 . thus against a non - oblivious adversary , any online algorithm must incur a @xmath424 overhead . in this section , we show that if the aspect ratio of the movement costs is not bounded , the linear program has a @xmath426 gap , even when @xmath13 is exponentially larger than @xmath5 . we present an instance where @xmath426 and @xmath427 are about @xmath6 with @xmath428 , and the linear program has a gap of @xmath425 . this shows that the @xmath429 term in our rounding algorithm is unavoidable . the instance is a graphical matroid , on a graph @xmath289 on @xmath430 , and @xmath431 . the edges @xmath432 for @xmath433 $ ] have acquisition cost @xmath434 and holding cost @xmath435 for all @xmath2 . the edges @xmath436 for @xmath437 $ ] have acquisition cost @xmath438 and have holding cost determined as follows : we find a bijection between the set @xmath40 $ ] and the set of partitions @xmath439 of @xmath440 with each of @xmath441 and @xmath442 having size @xmath443 ( by choice of @xmath13 such a bijection exists , and can be found e.g. by arranging the @xmath441 s in lexicographical order . ) . in time step @xmath2 , we set @xmath181 for @xmath444 , and @xmath445 for all @xmath446 . first observe that no feasible integral solution to this instance can pay acquisition cost less than @xmath443 on the @xmath432 edges . suppose that the solution picks edges @xmath447 for some set @xmath448 of size at most @xmath443 . then any time step @xmath2 such that @xmath449 , the solution has picked no edges connecting @xmath450 to @xmath442 , and all edges connecting @xmath441 to @xmath442 have infinite holding cost in this time step . this contradicts the feasibility of the solution . thus any integral solution has cost @xmath290 . finally , we show that on this instance , ( [ eq : lp2 ] ) from section [ sec : lp - round ] , has a feasible solution of cost @xmath158 . we set @xmath451 for all @xmath452 $ ] , and set @xmath453 for @xmath454 . it is easy to check that @xmath455 is in the spanning tree polytope for all time steps @xmath2 . finally , the total acquisition cost is at most @xmath456 for the edges incident on @xmath450 and at most @xmath457 for the other edges , both of which are @xmath158 . the holding costs paid by this solution is zero . thus the lp has a solution of cost @xmath158 the claim follows . the greedy algorithm for is the natural one . we consider the interval view of the problem ( as in section [ sec : intervals ] ) where each element only has acquisition costs @xmath11 , and can be used only in some interval @xmath201 . given a current subset @xmath458 , define @xmath459 . the benefit of adding an element @xmath77 to @xmath385 is @xmath460 and the greedy algorithm repeatedly picks an element @xmath77 maximizing @xmath461 and adds @xmath77 to @xmath385 . this is done until @xmath462 for all @xmath51 $ ] . phrased this way , an @xmath23 bound on the approximation ration follows from wolsey @xcite . we next give an alternate dual fitting proof . we do not know of an instance with uniform acquisition costs where greedy does not give a constant factor approximation . the dual fitting approach may be useful in proving a better approximation bound for this special case . using lagrangian variables @xmath465 for each @xmath77 and @xmath466 , we write a lower bound for @xmath467 by @xmath468 which using the integrality of the matroid polytope can be rewritten as : @xmath469 here , @xmath470 denotes the cost of the minimum weight base at time @xmath2 according to the element weights @xmath471 , where the available elements at time @xmath2 is @xmath472 . the best lower bound is : @xmath473 it is useful to maintain , for each time @xmath2 , a _ minimum weight base _ @xmath64 of the subset @xmath479 according to weights @xmath480 . hence the current dual value equals @xmath481 . we start with @xmath482 and @xmath483 for all @xmath2 , which satisfies the above properties . suppose we now pick @xmath77 maximizing @xmath461 and get new set @xmath484 . we use @xmath485 akin to our definition of @xmath486 . call a timestep @xmath2 `` interesting '' if @xmath487 ; there are @xmath488 interesting timesteps . how do we update the duals ? for @xmath489 , we set @xmath490 . note the element @xmath77 itself satisfies the condition of being in @xmath491 for precisely the interesting timesteps , and hence @xmath492 . for each interesting @xmath86 , define the base @xmath493 ; for all other times set @xmath494 . it is easy to verify that @xmath495 is a base in @xmath496 . but is it a min - weight base ? inductively assume that @xmath64 was a min - weight base of @xmath479 ; if @xmath2 is not interesting there is nothing to prove , so consider an interesting @xmath2 . all the elements in @xmath491 have just been assigned weight @xmath497 , which by the monotonicity properties of the greedy algorithm is at least as large as the weight of any element in @xmath479 . since @xmath77 lies in @xmath491 and is assigned value @xmath498 , it can not be swapped with any other element in @xmath496 to improve the weight of the base , and hence @xmath499 is an min - weight base of @xmath496 . it remains to show that the dual constraints are approximately satisfied . consider any element @xmath500 , and let @xmath501 . the first step where we update @xmath502 for some @xmath503 is when @xmath500 is in the span of @xmath486 for some time @xmath2 . we claim that @xmath504 . indeed , at this time @xmath500 is a potential element to be added to the solution and it would cause a rank increase for @xmath505 time steps . the greedy rule ensures that we must have picked an element @xmath77 with weight - to - coverage ratio at most as high . similarly , the next @xmath2 for which @xmath502 is updated will have @xmath506 , etc . hence we get the sum @xmath507 since each element can only be alive for all @xmath13 timesteps , we get the claimed @xmath23-approximation . note that the greedy algorithm would solve @xmath508 even if we had a different matroid @xmath509 at each time @xmath2 . however , the equivalence of and no longer holds in this setting , which is not surprising given the hardness of theorem [ thm : diff - matrs - wpb ] .
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this paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time .
the challenge then is to continually maintain near - optimal solutions to the underlying optimization problems , without creating too much churn in the solution itself .
we model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions ( one for each time step ) ; while we can change the solution from step to step , we incur an additional cost for every such change .
we first study the multistage matroid maintenance problem , where we need to maintain a base of a matroid in each time step under the changing cost functions and acquisition costs for adding new elements .
the online version of this problem generalizes onine paging , and is a well - structured case of the metrical task systems .
e.g. , given a graph , we need to maintain a spanning tree @xmath0 at each step : we pay @xmath1 for the cost of the tree at time @xmath2 , and also @xmath3 for the number of edges changed at this step .
our main result is a polynomial time @xmath4-approximation to the online multistage matroid maintenance problem , where @xmath5 is the number of elements / edges and @xmath6 is the rank of the matroid .
this improves on results of buchbinder et al .
@xcite who addressed the _ fractional _ version of this problem under uniform acquisition costs , and buchbinder , chen and naor @xcite who studied the fractional version of a more general problem .
we also give an @xmath7 approximation for the offline version of the problem .
these bounds hold when the acquisition costs are non - uniform , in which case both these results are the best possible unless p = np .
we also study the perfect matching version of the problem , where we must maintain a perfect matching at each step under changing cost functions and costs for adding new elements .
surprisingly , the hardness drastically increases : for any constant @xmath8 , there is no @xmath9-approximation to the multistage matching maintenance problem , even in the offline case .
| 18,570 | 537 |
our work is concerned with the kinetics of electrons in a stellar atmosphere , modelled as a parallel - plane slab irradiated on a face . our models of atmospheres start in the deep layers of stars , where the radiative field can be described in the diffusion approximation , and end with the layers of minimal temperature , before the chromospheric raise whose effects are ignored . the free electrons are characterized by their velocity distribution function : the electron distribution function ( edf ) , which is calculated with the other thermodynamical quantities of the atmosphere . our main objective is to understand the mechanism leading to the thermalisation of electrons , where the edf tends toward the maxwell - boltzmann distribution . it is accepted , in stellar atmospheres theory , that the thermalisation of electrons is effective as long as elastic collisions dominate inelastic interactions of electrons with the plasma , a rather well verified hypothesis for electrons having energies greatly below the first excitation energies of atoms and ions composing the atmosphere . this hypothesis is not necessarily correct for faster electrons . our work follows the line drawn by some plasma physicists at the beginning of the 70s ( peyraud 1968 , 1970 ; peyraud 1969 ; oxenius 1970 , 1974 ; shoub 1977 ) . their work demonstrated the important role played by inelastic ( collisional or radiative ) processes in the equilibrium reached by electrons , which can deviate considerably from the maxwellian equilibrium at high energies . we present below a stellar atmosphere model which is not in local thermodynamical equilibrium ( lte ) , confirming the results anticipated in the 70s on the basis of mainly theoretical developments . this problem consists in solving the equations generally used to model a non - lte stellar atmosphere ( equation of radiative transfer , equations of statistical equilibrium , pressure equation , equation of energy , conservation of charge ) , coupled with the kinetic equation of electrons . this non - linear system is difficult to solve numerically because it contains two coupled kinetic equations : one for photons , the other for electrons . therefore we have used the simplest model of non - lte atmosphere : homogeneous ( constant density of heavy particles @xmath0 ) , isotherm ( constant temperature @xmath1 ) , and composed with hydrogen atoms with only two energy levels . the deviation from lte is then due to the escape of photons by the free surface . on the other hand we have included in our model the main collision processes existing in a stellar atmosphere ( elastic collisions , collisional or radiative inelastic interactions ) . the elastic collision term of the kinetic equation of electrons is written in a bgk model with a velocity dependent collision frequency . this model accuratly fits the main properties of the usual landau term ( fokker - planck ) . to solve the equation of radiative transfer , we used the codes of the transfer group in cral ( rutily 1992 ) . in our model , we choosed @xmath2 and @xmath3 , which are typical values in the solar photosphere . the plasma is optically thick at all frequencies ( optical thickness greater than 100 ) , leading to a high geometrical thickness @xmath4 since there is no temperature or heavy particles density gradient . finally the atmosphere is irradiated on its internal boundary layer by a planck radiation of temperature @xmath1 . the figure [ fig1]a is a classical diagram showing the superficial regions where the non - lte effects are important . figure [ fig1]b shows that the edf is not a maxwell - boltzmann distribution in the non - lte region of the atmosphere ( see @xmath5 ) , the deviation from a maxwellian distribution being important very close to the surface ( @xmath6 , corresponding to an optical depth @xmath7 in the ly@xmath8 spectral line ) . in figure [ fig1]c , we drawn the superficial edf at @xmath9 as a function of the electronic velocity . the edf tail of fast electrons is strongly depleted when electron energies are greater than the minimum excitation energy of the hydrogen atom ( @xmath10 , @xmath11 , @xmath12 ) . the edf tail shows successive platforms centered on @xmath13 and @xmath14 . these features were already described by the authors at the origin of this work , referenced at the beginning of this article . the mechanism responsible for this effect is very well explained in oxenius s monograph ( 1986 ) , where the author outlines an interesting _ feedback effect _ tending to amplify the deviation of the edf from a maxwellian distribution . this mechanism starts when elastic and inelastic collision frequencies become comparable at high electronic velocities , which is the case for a weak ionization degree . a ) deviations of the populations from their lte values as a function of the reduced geometrical depth @xmath15 . coefficients @xmath16 , where @xmath17 are the saha densities of the hydrogen atom in energy states @xmath18 , are used to characterize non - lte regions ( @xmath19 ) . b ) deviation of the edf to the maxwellian distribution @xmath20 as a function of the reduced geometrical depth @xmath15 . both curves are drawn for a given velocity @xmath21 , where @xmath22 is the velocity corresponding to the rydberg energy @xmath23 . c ) deviation of the edf to the maxwellian distribution as a function of the electronic velocity @xmath24 , at the surface of the atmosphere @xmath9.,width=472,height=377 ] astrophysical consequences of this work are numerous . in general the deviation of the edf from a maxwellian distribution has a direct effect on all thermodynamic quantities involving the edf , _ e.g. _ collisional transition rates or spectral lines profiles . it has an indirect effect on all other characteristics of the atmosphere , because of the coupling of all equations . transition rates are used to solve the equations of statistical equilibrium , which lead to the populations and ionization degree of the atmosphere . inversion techniques of spectral lines observed by spectroscopy are also sensitive to the edf shape , so that temperatures or densities calculated with these techniques are affected by deviation of the edf from a maxwellian distribution ( shoub 1983 , owocki 1983 , salzmann 1995 ) . finally the non thermodynamical equilibrium of electrons may be at the origin of physical processes which are still not very well understood at present , for example the heating of the sun corona ( scudder 1992,1994 ) . the results presented in this article are based on a very simple atmosphere model , which guarants a numerically stable solution . in non - lte regions close to the surface , the edf shows important deviations from the maxwellian distribution in the fast electrons tail . our model confirm , by means of numerical codes accurate enough to handle this complex problem , most of the physical ideas advanced thirty years ago . our main contribution consists in the construction of a selfconsistent model of a stellar atmosphere with non thermalized electrons . also we have used very accurate radiative transfer codes . it remains to make this model more realistic for comparison with observations . peyraud , n. 1968 , le journal de physique , 29 , 201 ; 29 , 747 ; 29 , 997 . peyraud , n. , peyraud j. 1969 , le journal de physique , 30 , 773 . peyraud , n. 1970 , le journal de physique , 31 , 565 . owocki , s. p. , scudder , j. d. 1983 , astrophys . j. , 270 , 758 . oxenius , j. 1970 , z. naturforsch . , 25 , 101 ; 25 , 1302 . oxenius , j. 1974 , j. quant . transfer , 14 , 731 . oxenius , j. 1986 , kinetic theory of particles and photons , springer - verlag ( berlin ) , vol . 20 . rutily , b. 1992 , solutions exactes de lquation de transfert et applications astrophysiques , thse de doctorat detat , universit claude bernard lyon i. salzmann , d. , lee , y. t. 1995 , j. quant . transfer , 54 , 339 . scudder , j. d. 1992 , astrophys . j. , 398 , 299 ; 398 , 319 . scudder , j. d. 1994 , astrophys . j. , 427 , 446 . shoub , e. c. 1977 , astrophys . , 34 , 259 ; 34 , 277 . shoub , e. c. 1983 , astrophys . j. , 266 , 339 .
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we are interested in electrons kinetics in a stellar atmosphere to validate or invalidate the usually accepted hypothesis of thermalisation of electrons . for this purpose
, we calculate the velocity distribution function of electrons by solving the kinetic equation of these particles together with the equations of radiative transfer and statistical equilibrium .
we note that this distribution can deviate strongly from a maxwell - boltzmann distribution if non - lte effects are important .
some results and astrophysical consequences are examined .
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in the textbooks of quantum mechanics the solution of schrdinger equation and the consequent results are illustrated through simple one - dimensional potentials . for discrete bound states the square well@xcite and double wells@xcite are studied . square well , square barrier and semi - infinite step potentials are used for studying continuous energy ( scattering ) states.@xcite a well with two side barriers is studied for understanding resonances and meta - stable states.@xcite an overlapping well adjacent to a finite barrier is a well known model for discussing discrete complex energy gamow - seigert meta - stable states @xcite in alpha decay . students may wonder as to what happens if a non - overlapping well ( at negative energies ) is adjacent to a finite barrier ( at negative energies ) ( see figs . 1 ) . perhaps for the want of an application this system has gone undiscussed , however , interesting queries do arise for this kind of potentials . one may wonder as to whether the well ( at negative energies ) can change ( increase / decrease ) the transmitivity of the barrier ( at positive energies ) quantitatively and significantly . one may like to know whether there can be qualitative changes in the transmitivity of the barrier @xmath0 due to the presence of the well in some class of cases . in this article we would like to show that a well near a barrier can change the transmitivity of the barrier both quantitatively and qualitatively . in fact a scattering potential well ( vanishing at @xmath4 ) can give rise to a non - overlapping well adjacent to a finite barrier ( nwafb ) as @xmath5 where @xmath6 see figs . however in this case , a change in the depth of the well or its distance from the barrier would also change the height of the barrier . consequently , the effect of the well on the transmission property of the original barrier can not come up explicitly . we , therefore , consider wells of zero - range or finite range . else , if they are scattering wells of infinite range on one side they ought to be joined to the barrier continuously or dis - continuously . in the following we discuss the various possibilities for nwabf . we construct various models of nwafb using three parameters @xmath7 and @xmath8 . here @xmath9 is the depth of the well , @xmath10 is height of the barrier and @xmath8 denotes the separation between the well and the barrier . in these models a change in @xmath8 does not change the depth of the well or the height of the barrier . first let us consider both the well and the barrier of zero range . using the zero range dirac delta potentials we construct a simple solvable model of nwafb as @xmath11 using finite range well , we construct a more general model of nwafb ( see figs . 1(b ) ) @xmath12 where @xmath13 may be chosen as constant ( square or rectangular well ) , @xmath14 ( parabolic well ) , @xmath15 ( triangular well ) , @xmath16 ( gaussian well ) or @xmath17 ( exponential well ) . it may be mentioned that in some cases @xmath10 may not represent the effective barrier height ( @xmath18maximum of @xmath19 ) . for instance in this article we shall be choosing @xmath20 where for @xmath21 we get @xmath22 . using asymptotically converging profiles @xmath23 and @xmath24 , we construct two - parameter @xmath25 models of nwabf wherein a well of infinite range is juxtaposed to a barrier of infinite range continuously as ( see solid curve in figs . 1(c ) ) @xmath26 and discontinuously as ( see dashed curve in figs . 1(c ) ) @xmath27 here the functions @xmath23 may be chosen as rectangular profile or as @xmath28 , @xmath29 , @xmath30 ... , and @xmath24 may be taken as @xmath31 , @xmath32 , @xmath33 , ... . it may be mentioned that the finite range potential like @xmath34 would rather be a nwafb of type ( 3 ) with @xmath35 than of the type ( 4 ) . next we have to solve the schrdinger equation @xmath36 for finding the transmitivity , @xmath2 , of the various potential models discussed above . when the potentials are real and hermitian the time reversal symmetry ensures that the transmitivity and reflectivity are independent of the direction of incidence of particle whether it is from left or right . due to this symmetry , in transmission through nwafb it does not matter whether the incident particle sees the well or the barrier first . the zero range delta potential model of nwafb is exactly solvable . we solve the schrdinger equation ( 6 ) for this potential , @xmath37 given in eq . ( 1 ) using just plane waves : @xmath38 as usual . let the direction of incidence of the particle at the potential be from the left hand , we can write @xmath39 the wavefunction ( 7 ) has to be continuous at @xmath40 and 0 . however , due the point singularity at @xmath41 in delta functions in eq . ( 2 ) , there occurs a mis - match in the first derivative ( see problem no . 20 and 21 in ref.@xcite ) of the wavefunction we get @xmath42-ik[c e^{-ikd } - d e^{ikd } ] = -{2 m \over \hbar^2 } v_w [ c e^{-ikd } + d e^{ikd } ] , \nonumber \\ & & c+d = f,\nonumber \\ & & ik[(c - d)-f]={2 m \over \hbar^2 } v_b f.\end{aligned}\ ] ] by eliminating @xmath43 and @xmath44 from eq . ( 8) , we get @xmath45 these ratios give us the reflectivity @xmath46 and the transmitivity @xmath47 . when @xmath48 the numerator of @xmath49 in eq . ( 9 ) becomes @xmath50 which gives rise reflectivity zeros when @xmath51 these are the positions of transmission resonances with @xmath52 when either of @xmath9 and @xmath10 is zero , from eq . ( 9 ) we get ( see problem no . 21 in @xcite ) @xmath53 this is a particular feature of the delta potential well or barrier that their transmission co - efficients are identical . for all our calculations we choose @xmath54 , so that energies and lengths are in arbitrary units . in figs . 2(a ) , both @xmath2 and @xmath1 are plotted as a function of energy , @xmath55 , when @xmath56 . see the interesting energy - oscillations in solid curve that represent the transmitivity of the total potential @xmath37 : a perturbed barrier . when compared with the transmitivity of the dirac delta barrier ( see the dotted curve ) these energy oscillations in @xmath2 can be seen to be riding around @xmath1 even at large energies ( @xmath57 ) . we find that the smaller values of @xmath9 ( than 1 ) create only small excursions ( ripples ) around the smooth variation of @xmath1 . the depth of the well @xmath9 governs the amplitude of these oscillations . in figs . 2(c ) see that the frequency of these energy - oscillations remain the same but their amplitudes are larger as @xmath9 is increased and made equal to 5@xmath58 . compare figs . 2(a ) with figs . 2(c ) and figs . 2(b ) with figs . 2(d ) to appreciate the effect of the increase in the depth of the well resulting in the increase of amplitude of oscillations . we find that the frequency of these oscillations is governed by the value of @xmath8 . larger the value of @xmath8 , more is the frequency of oscillations . compare figs . 2(a ) with figs . 2(b ) and figs . 2(c ) with figs . 2(d ) to appreciate the effect of the increase in @xmath8 . this simple and exactly solvable model of nwafb suggests that a well near a barrier neither increases nor decreases the transmitivity of the barrier . most interestingly , it does both and hence energy oscillations in @xmath2 . increase in the frequency of these oscillations due to increase in @xmath8 ( perturbation moving away ) is paradoxical . the question arising here is whether energy oscillations in @xmath2 is the essence of nwafb of some type or a particular feature of extremely thin delta potentials making up @xmath37 ( 2 ) . we therefore need to study the other models given eqs . ( 1,3 - 5 ) . as the other models of nwafb are not solvable analytically , in the following we discuss a numerical procedure to find @xmath2 . when the potentials vanish asymptotically one can calculate its transmission co - efficient by solving the schr " odinger equation numerically for scattering solutions . we propose to solve eq . ( 6 ) using runge - kutta method@xcite of step by step integration ( see appendix ) . this method consists of solving two first order , linear , one dimensional coupled differential equations @xmath59 , \quad { dz(x)\over dx}=g[x , y(x),z(x ) ] , \quad y(0)=c_1 , \quad z(0)=c_2.\ ] ] in this setting , we introduce @xmath60 and @xmath61 and split the schrdinger equation in two first order coupled linear differential equations as @xmath62y(x).\end{aligned}\ ] ] the schr " odinger equation which is a second order differential equation will have two linearly independent solutions as @xmath63 and @xmath64 . we start the numerical integration from @xmath65 using the two sets of initial values as ( see problem no . 22 in ref.@xcite and ref.@xcite ) @xmath66 such that the wronskian function @xmath67=\psi_1(x ) \psi_2^\prime(x)- \psi_1^\prime(x ) \psi_2(x)=1 $ ] which is known to be a constant of motion . here the prime denotes first differentiation with respect to @xmath68 . on the right , the rk - integration is carried up to ( say ) @xmath69 for the case of a finite range barrier @xmath70 in @xmath71 ( 3 ) . for infinite range cases like @xmath72 ( 4 ) and @xmath73 ( 5 ) rk - integration is to be carried up to ( say ) @xmath74 such that @xmath75 is very small . similarly , on the other side , the rk - integration is to be carried up to @xmath76 in case of @xmath71 . in case of @xmath72 ( 4 ) and @xmath73 ( 5 ) we integrate up to ( say ) @xmath77 . let us denote the end values @xmath78 as @xmath79 , respectively . the end values @xmath80 are denoted as @xmath81 , respectively . as rk - integration is step by step method wherein the calculated value of the function , @xmath82 , and its slope ( momentum ) @xmath83 at one step serve as initial values for the next step . this suits quantal calculations wherein the wavefunction and its derivative must match everywhere in the domain of the potential . importantly , then it does not matter whether or not the potential is continuous or has a finite jump discontinuity at one or more number of points in the domain of the potential . we finally write the solution of eq . ( 6 ) as @xmath84 in case of @xmath72 ( 4 ) and @xmath73 ( 5 ) , the distances @xmath85 and @xmath86 will be replaced by @xmath87 and @xmath88 , respectively . next by matching @xmath82 and @xmath89 at these points we get @xmath90 solving eqs . ( 15 ) , we get @xmath91 here we have used the constancy of the wronskian @xmath92=w[\phi_1,\phi_2]=1 $ ] . the transmitivity ( transmission probability ) of the total the nwafb is given by @xmath2 as in above equation . this may be denoted fully as @xmath93 where @xmath1 denotes the transmitivity of the ( unperturbed ) barrier and @xmath94 and @xmath8 may be taken to act as perturbation parameters . using the eq . ( 16 ) , we calculate the transmitivity of various analytically intractable models given in section iii . let us discuss the nwafb represented by @xmath71 in eq . 3 presents @xmath2 and @xmath1 when @xmath13 is a rectangular well in @xmath71 ( see dotted well in figs . the form of the barrier is fixed as @xmath95 and its parameter @xmath21 this gives @xmath96 as about 5 units . in figs . 3(a ) , we see only marginal excursions in @xmath2 when the well is shallow , wide and distant . when the well is deeper but juxtaposed to the barrier ( @xmath35 ) the frequency of oscillations decreases ( see figs . when the well is away from the barrier , @xmath2 is more oscillatory compare figs . 3(b ) with figs . when the depth of the well is increased to 10 units ( @xmath97 ) the amplitude of the oscillations increases ( see figs . 3(d ) ) . in nwabf the essence is that the oscillations in @xmath2 are seen riding around @xmath1 . in other words the well induces oscillations in the transmitivity of the adjacent barrier . we would like to remark that a piecewise constant potential mentioned in ref.@xcite ( see eq . ( 22 ) there ) can now be seen as a nwafb of the type ( 3 ) , wherein both the well and the barrier are square ( rectangular ) and @xmath2 is oscillatory ( see fig . 5 there ) . next we study parabolic well in @xmath71 ( 3 ) . in figs . 4(a ) , this time we find that the well - depth has to be comparable to the barrier height of 5 units for changing @xmath2 appreciably when compared to @xmath1 . the effect of increase in the depth of the well can be seen to enhance the amplitude of of oscillations in @xmath2 by comparing figs . 4(a ) with figs . @xmath2 in figs . 4(b ) is less oscillatory as compared to that in figs . 4(c ) because the well and barrier are juxtaposed to each other with @xmath35 . so in this model too the energy oscillations occurring in @xmath2 are due to increase in the width or depth of the well or its distance from the barrier . however , these oscillations are less prominent than those of rectangular potential model seen in figs . the general feature of the nwabf of the type @xmath71 ( 3 ) that the transmitivity is more oscillatory when a thinner barrier is away from the well is well demonstrated when one compares figs . 4(c ) and figs . . the oscillations in the transmitivity of rectangular and parabolic models of nwafb ( 3 ) may be attributed @xcite to their finite range ( finite support ) and also to the distance @xmath8 over which the potential being zero allows the interference of plane waves . further , the prominence of oscillations in @xmath2 of rectangular model lies in the fact that rectangular potential well or barriers are most localized profiles between two points than any other profile of finite support@xcite . 5 , displays the qualitatively similar oscillatory transmitivity when quite thin wells ( @xmath98 ) are used in nwafb of the type given by @xmath71 in eq . the depths of the wells and their distances from the barrier are fixed as @xmath99 and @xmath100 , respectively . these wells taken here are rectangular , parabolic , gaussian , and triangular ( see the line below eq . ( 3 ) ) . from this fig . 5 we conclude that quite thin wells despite being away from the barrier can induce prominent oscillations in @xmath2 provided they are sufficiently deep . if not so deep the amplitude of oscillations will be small . now we study two more modifications of nwafb which are made up of scattering potentials of infinite range . these are @xmath72 ( 4 ) and @xmath73 ( 5 ) . in the case of @xmath72 ( see solid curve in figs . 1(c ) ) when the well and the barrier are juxtaposed continuously at @xmath65 , we find ( see figs . 6(a ) ) that if the well is strong it reduces the transmitivity and then increases it only marginally at energies below the barrier . at energies above the barrier height the changes are inappreciable . in the dis - continuous case ( see dashed curve in figs . 1(c ) ) , we find that the hidden well reduces @xmath2 over all ( below and above the barrier ) energies ( see figs . 6(b ) ) . this is the characteristic feature of the potential being discontinuous at a point ( @xmath65 ) as the well and the barrier are juxtaposed there in a discontinuous way as in the case of a simple potential step@xcite . also the well reduces transmitivity of the barrier in an appreciable way only if it is strong ( e.g. , @xmath101 ) . we have confirmed absence of energy oscillations in these two models by varying @xmath9 and @xmath10 high and low abundantly . moreover , in this regard the exact analytic expression @xcite @xmath2 of the scarf ii potential ( @xmath102 ) readily testifies to a non - oscillatory behaviour of nwafb of the type ( 4 ) as a function of energy @xmath103},\ ] ] with @xmath104 , @xmath105 , @xmath106 , and @xmath107 . however , in the above models @xmath72 ( 4 ) and @xmath73 ( 5 ) if the well and barrier are separated by a distance , @xmath8 , the transmitivity will again acquire oscillations . we would like to emphasize that it is the separation between the well and the barrier that plays a crucial role in causing energy - excursions ( oscillations ) in @xmath2 with respect to @xmath1 . 6(c , d ) demonstrate that in case of single piece nwafb ( 1 ) when @xmath108 and @xmath109 it requires a very deep well @xmath110 to get even small excursions in @xmath2 with respect to @xmath1 . appreciable energy oscillations can be seen in @xmath2 only if the well is much deeper ( @xmath111 ) . this feature is surprising in view of the fact that the nwafb of the types ( eqs . ( 2,3 ) ) in figs . 2 - 5 have displayed good energy oscillations even if @xmath9 is twice of @xmath10 or even less than @xmath10 . in all the results presented in figs . 2 - 6 ( see the dotted curve ) , in nwabf the general trend of @xmath2 is determined by the barrier is irrespective of the strength of the well . broadly , three ( eqs . ( 1 - 3 ) ) types of nwafb ( see figs . 1 ) entailing single well and a single barrier are possible . however , one has choices of the profiles for the well and the barrier in them . apart from the results of various profiles presented here in figs . ( 2 - 6 ) we have also studied several other profiles and explored various parametric regimes in all three types of nwabf to confirm our findings presented here . the transmission through a barrier is the phenomenon of positive energy continuum , we conclude that the well ( at negative energies ) essentially causes energy - excursions ( ripples or oscillations ) in the transmitivity of the barrier . howsoever strong the well is the trend of transmitivity as a function of energy is determined only by the barrier . ordinarily , the finite support(range ) of the well may also be attributed@xcite to cause energy oscillations in the transmitivity . in this regard , the energy - oscillations in the transmitivity of one - piece smooth potential ( 1 ) of infinite range found here are unexpected . however , it has required the well depth to be extremely large ( see figs . the separation between the well and the barrier is _ sufficient _ if not the _ necessary _ condition in giving rise to oscillations in transmitivity . when the well and the barrier are separated away , the potential in the intermediate region is zero . this gives a scope for destructive and constructive interference of plane waves and hence the frequency of energy - oscillations in the transmitivity increases . however , if one views the well as a perturbation to the barrier then the enhanced oscillations in @xmath2 despite the well being distant is paradoxical . the infinite range well and barrier if joined at a point with no separation ( @xmath35 ) between them do not seem to have energy - oscillations in transmitivity until they are separated . the energy - oscillations in transmitivity at energy below the barrier suggests a novelty because usually transmitivity is found@xcite to be oscillatory at energies above the barrier . the transmitivity of various potential systems which converge asymptotiacally @xmath112 to zero or to a constant value and which are either continuous or entail finite jump discontinuities can be found using eq . ( 16 ) presented here . in this article we have presented the first and hopefully an exhaustive study of transmission through non - overlapping well adjacent to a finite barrier . we hope that this investigation will be found pedagogically valuable . [ [ section ] ] the runge - kutta@xcite solution of the coupled first order equations @xmath113 are obtained as @xmath114 and @xmath115 starting with the initial values @xmath116 using the following equations . @xmath117 , \quad z_{n+1}=z_n+{h \over 6}[m_1 + 2m_2 + 2m_3+m_4],~ n \ge 0,~h={d \over n}\nonumber \\ & & k_1=f(x_n , y_n , z_n ) , \quad m_1=g(x_n , y_n , z_n)\nonumber \\ & & k_2=f(x_n+h/2,y_n+h k_1/2,z_n+h k_1/2 ) , \quad m_2=g(x_n+h/2,y_n+h m_1/2,z_n+h m_1/2 ) \nonumber \\ & & k_3=f(x_n+h/2,y_n+h k_2/2,x_n+h k_2/2 ) , \quad m_3=g(x_n+h/2,y_n+h m_2/2,x_n+h m_2/2 ) \nonumber \\ & & k_4=f(x_n+h , y_n+h k_3 , z_n+h k_3 ) , \quad m_4=g(x_n+h , y_n+ h m_3 , z_n+h m_3).\end{aligned}\ ] ] when we solve ( 11 ) for @xmath118 , we get @xmath63 and @xmath119 and we get @xmath64 and @xmath120 when the starting values are @xmath118 . l. i. schiff,_quantum mechanics _ ( mcgraw hill , sydney , 1968 ) ch . 2 and 5 . e. merzbacher , _ quantum mechanics _ ( john wiely and sons , inc . , new - york , 1970 ) ch . 5 and 6 . d. rapp , _ quantum mechanics _ ( holt , rinehart and winston , inc . , new - york , 1970 ) ch . 6 , 7 and 8 . s. flugge , _ practical quantum mechanics _ ( springer - verlag , berlin , 1971 ) ch . a. bhom , m. gadella , g.b . mainland , ` gamow vectors and decaying states ' , am j , phys . * 57 * 1989 1103 - 1108 . john a. jacquez , a first course in computing and numerical methods ( addison wesley publishing company , london , 1070 ) 340 . j. d. chalk , ` a study of barrier penetration in quantum mechanics ' am . j. phys . * 56 * ( 1988 ) 29 - 32 . a. uma maheswari , p. prema , and c. s. shastry , ` resonant states and transmission co - efficient oscillations for potential wells and barriers ' , am . * 78 * , ( 2009 ) 412 - 417 . z. ahmed , ` comment on : ` resonant states and transmission co - efficient oscillations for potential wells and barriers ' by a. uma maheswari , p. prema , and c. s. shastry am . * 78 * , ( 2009 ) 412 - 417 ' am . 79 ( 2011 ) 682 - 683 . m. v. berry , semi - classically weak reflection above analytic and non - analytic potential barriers , " j. phys . a : math . gen . * 15 * , 36933704 ( 1982 ) . + l. v. chebotarev , transmission spectra for one - dimensional potentials in semi - classical approximation , " phys . a * 52 * , 107124 ( 1995 ) . + z. ahmed , reflectionlessness , kurtosis and top - curvature of potential barriers , " j. phys . gen . * 39 * , 73417348 ( 2006 ) . + z. ahmed , c. m. bender , and m. v. berry , reflectionless potentials and pt - symmetry , " j. phys . a : math . gen . * 38 * , l627l630 ( 2005 ) . + ( 3 ) : parabolic well ( dashed line ) , rectangular well ( dotted line ) and very thin rectangular well near a barrier , ( c ) @xmath121 ( 4 ) : a smooth well continuously juxtaposed to a barrier ( solid line ) and @xmath122 ( 5 ) : a smooth well discontinuously juxtaposed to the a barrier . ] , of the delta potential model @xmath123 of nwafb ( 2 ) . the dotted curve represent the transmitivity , @xmath1 , of the barrier only . we have a fixed barrier height @xmath124 and take(a ) : @xmath125 , ( b ) : @xmath126 , ( c ) @xmath127 , ( d ) @xmath128 . ] ( 3 ) . here the barrier @xmath129 is perturbed by a rectangular ( square ) well . the effective height of the barrier @xmath130 is approximately 5 units . we have taken ( a ) : @xmath131 , ( b ) : @xmath132 , ( c ) : @xmath133 , ( d ) : @xmath134 . ] ( 3 ) . here in general the energy - oscillations in @xmath2 are present but these are less prominent than those in figs . 2 . the same barrier(@xmath70 ) is now perturbed by a parabolic well of finite range . we take ( a ) : @xmath135 , ( b ) : @xmath132 , ( c ) : @xmath133 , ( d ) : @xmath134 ] ( solid lines ) and @xmath1 ( dotted curve ) for various nwafb of the type @xmath71 ( 3 ) when the wells are quite thin(@xmath98 ) . we have @xmath137 . these wells are rectangular , parabolic , gaussian , and triangular used in eq . ( 3 ) ( see the text below eq ( 3 ) ) . thin wells away from the barrier give rise to qualitatively similar transmitivity which is oscillatory . this is an essential feature of the nwafb of the type in eqs . ( 2,3 ) . ] for ( a ) : the continuous ( 4 ) and ( b ) : the discontinuous ( 5 ) models ; the dotted line @xmath138 , thin solid line @xmath139 and thick solid line @xmath140 . figs . ( c , d ) represent the transmitivities for the single piece smooth nwafb ( 1 ) . for a fixed distance ( @xmath109 ) between the well and the barrier and @xmath108 figs . ( c ) shows only small excursions in @xmath2 only when the well is very deep ( @xmath141 ) . in figs . ( d ) , significant oscillations in @xmath2 have required even higher value @xmath142 ) . ]
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we point out that a non - overlapping well ( at negative energies ) adjacent to a finite barrier ( at positive energies ) is a simple potential which is generally missed out while discussing the one - dimensional potentials in the textbooks of quantum mechanics .
we show that these systems present interesting situations wherein transmitivity @xmath0 of a finite barrier can be changed both quantitatively and qualitatively by varying the depth or width of the well or by changing the distance between the well and the barrier . using delta ( thin ) well near a delta ( thin ) barrier
we show that the well induces energy oscillations riding over @xmath1 in the transmitivity @xmath2 at both the energies below and above the barrier .
more generally we show that a thick well separated from a thick barrier also gives rise to energy oscillations in @xmath2 .
a well joining a barrier discontinuously ( a finite jump ) reduces @xmath2 ( as compared to @xmath3 over all energies .
when the well and barrier are joined continuously , @xmath2 increases and then decreases at energies below the barrier . at energy above the the barrier the changes are inappreciable . in these two cases
if we separate the well and the barrier by a distance , @xmath2 again acquires oscillations .
paradoxically , it turns out that a distant well induces more energy oscillations in @xmath2 than when it is near the barrier .
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core collapse of massive stars which lead to supernovae ( sne ) of type ib , c and ii are in some cases associated with long duration ( @xmath0-@xmath1 s ) gamma - ray bursts ( grbs ) , as evidenced by observed correlations of grb 980425/sn 1998bw , grb 021211/sn 2002lt , grb 030329/sn 2003dh and grb 0131203/sn 2003lw.@xcite a relativistic jet with bulk lorentz factor @xmath2 , powered by a black hole and an accretion disc which form after the core collapse in the most likely scenario , is believed to lead to the grb event.@xcite observational evidence of only a small fraction of detected sne associated with grbs hints that the frequency of highly relativistic jets in core collapse sne is at best 1 in 1000 , roughly the ratio of grb to sn rates.@xcite however , a significantly larger fraction ( @xmath3 of type ib / c rate@xcite ) of sne ( also called _ hypernovae _ ) may have mildly relativistic jets associated with them.@xcite one or more of the following observations support the jetted sn hypothesis : high expansion velocity ( 30 - 40 @xmath4 km / s ) first observed in sn 1998bw.@xcite radio afterglow not associated with @xmath5-ray emission.@xcite asymmetric explosion supported by polarimetry observations of sn type ib / c.@xcite numerical simulations of core collapse sne , carried out over the last three decades have failed to produce a successful explosion by a prompt shock wave created due to the collapse of its iron core.@xcite the deposition of bulk kinetic energy in a jet form into the stellar envelope may help disrupt and blow it up making the sn possible.@xcite the presence of a jet is also conducive to shock acceleration of particles . in case of a grb , internal shocks of plasma material along the jet accelerate protons and electrons which radiate observed @xmath5-rays.@xcite high energy protons may escape as cosmic rays and/or produce 100 tev neutrinos by interacting with @xmath5-rays _ in situ_.@xcite while the grb jet is making its way out of the collapsing stellar progenitor it is expected to produce 10 tev neutrio precursor burst.@xcite these neutrinos are emitted even in the cases when the jets do not manage to burrow through the stellar envelope and choke inside without producing observable @xmath5-rays . the jets in core collapse sne or hypernovae which is the topic of this review are slow with @xmath6 few and choke inside the stellar envelope.@xcite neutrinos produce from such jets are typically of a hundred gev to tev energy.@xcite as opposed to 10 mev thermal neutrinos produced by the core collapse sn shocks which have been detected from sn 1987a in our own galaxy,@xcite high energy neutrinos from the jets may be detected from a longer distance because of an increasing detection prospect with neutrino energy . kilometer scale ice and water cherenkov detectors such as icecube@xcite and antares@xcite which are currently being built in antarctica and in the mediterranean will have an excellent chance to detect these neutrinos from sne within the nearest 20 mpc . the organization of this brief review is as follows : in sec . [ sec : core - collapse ] a basic core collapse sn picture is outlined and a particular slow jet model in sec . [ sec : jet - model ] . shock acceleration and the maximum energy reachable by protons are discussed in sec . [ sec : proton - acc ] . neutrino flux on earth from a point source and diffuse sources is calculated in sec . [ sec : nu - flux ] and their detection prospects in sec . [ sec : events ] . conclusions are given in sec . [ sec : summary ] . nuclear fusion reactions , similar to the ones which take place in our sun , constantly enrich the interior of a star forming an iron core as the end product . burning up all fusion materials causes hydrodynamic instability due to lack of radiation pressure from inside the star . the immense gravitational pressure of the stellar envelope and/or overlying material causes the core of stars with mass @xmath7 to collapse at this point . the density of the compressed core material reaches a few times the nuclear density and a rising temperature helps iron dissociate into nucleons and alpha particles . infall of stellar material onto the core produces @xmath8 mev neutrinos by the process of electron capture on protons ( @xmath9 ) . the density of neutrons in the core exceeds that of protons in this process , called _ neutronization_. initially the neutrinos are trapped within a radius called _ neutrinosphere _ because of a density @xmath10 g-@xmath11 . for progenitors of mass @xmath12 , the increasing degeneracy pressure of the neutrons leads to a rebound , which sends a shockwave through the core . while traversing through the core , the shockwave heats up material , dissociates more iron atoms and releases trapped @xmath13 from the neutrinosphere . neutrinos carry away @xmath14 erg of energy or roughly @xmath15 of the total gravitational binding energy in this bursting phase which lasts for a few milliseconds . the shockwave , however , does not reach the envelope to drive it away because of heavy energy loss and the star fails to explode into a supernova . the mechanisms envisaged to produce a successful supernova explosion , such as observed in nature , may be divided into two main categories despite many uncertainties such as the mass loss rate of the pre - supernova star and neutrino transport in the core , to name a few . the first is a _ revived shock _ model , for stars initially less massive than @xmath16 , where the core collapses to make a neutron star . in this case , the above - mentioned stalled supernova shock is re - energized by neutrino absorption on nucleons outside the stellar core ( @xmath17 ; @xmath18 ) , re - energizing them . the shock wave then reaches the envelope and expels it away . after the supernova explosion , the stellar core cools down in next 10 s of seconds by emitting @xmath19 erg of energy in neutrinos of all flavors created by lepton pair annihilation ( @xmath20 ) , neutrino pair annihilation ( @xmath21 ) and nucleon bremsstrahlung ( @xmath22 ) . these neutrinos are thermal with a typical energy @xmath8 mev . a neutron star is left over after the stellar core cools down , following ejection of the envelope and outer core . in the second scenario , which is the relevant case for the current review , a star initially more massive than @xmath23 undergoes core collapse . ( a ) if the initial mass is @xmath24 , the core collapses initially to a neutron star , but after fall - back of additional core gas which did not reach ejection velocity , it collapses further to a black hole ( bh ) of mass in excess of the chandrasekhar mass ( the neutron degeneracy pressure not being sufficient to counteract gravity for this mass ) . ( b ) if the initial mass @xmath25 , the core collapses directly to a black hole of mass @xmath26 ( see ref . thermal neutrinos of @xmath27 mev are also produced in the neutronized core , before it falls into the black hole . this neutrino luminosity , of order several solar luminosities , may or may not be able to eject ( via absorption in re - energization ) the outer envelope , which is needed in order for it to appear as supernova detectable by its photon emission . as in the case of stars of mass @xmath28 , numerical simulations have not yet been able to prove whether stars in this mass range eject their envelope ; and , observationally , it is unclear whether any observed supernova can be ascribed to progenitors in this mass range . however , stars more massive than @xmath16 are certainly observed , and from well understood physics , they must core - collapse to a black hole . this is the gist of what happens to stars of mass @xmath29 if the stellar core is rotating slowly . a black hole , and possibly a temporary small accretion disk is formed after core collapse , which may not greatly affect the symmetry of the collapse and/or envelope ejection . the situation is thought to be drastically different for stars in the range @xmath30 whose core is fast - rotating.@xcite in particular , for a core angular momentum in the range ( 3 - 20)@xmath31 @xmath32 s@xmath33 for this model , the black hole and disk accretion can serve as an energy source for powering a long grb , and may be able to eject a stellar envelope with kinetic energies possibly 10 - 100 times higher ( at least in an isotropic - equivalent sense ) than the energy of typical type ib / c or type ii sne . fast core rotation also helps in forming low density channels along the rotation axis , by centrifugal evacuation . part of the material accreting from the disc onto the black hole can then be ejected as narrow jets along the axes , collimated by the gas pressure of the envelope . the jets are powered by neutrino annihilation or magnetohydrodynamic stresses . the pressure and energy deposition from the jet helps eject the star s envelope and thus the sn explosion happens . in the case of a highly relativistic jet which manages to break through the stellar envelope , the grb event takes place outside the star in an optically thin environment . a slow jet , which is modelled in the next section , may never break through , but the ejected material can give rise to an `` orphan '' radio afterglow which is not associated with @xmath5-ray emission . however , non - thermal high energy ( @xmath34 tev ) neutrinos are produced ubiquitously in both cases . the pre - supernova star , after losing its outermost envelope , is typically a wolf - rayet star of radius @xmath35 cm in the case of a type ib or larger in the case of a type ii supernova . the mildly relativistic sn jet may be modelled inside the pre - supernova star ( see fig . [ fig : slowjet ] ) with a bulk lorentz factor @xmath36 and a total jet kinetic energy @xmath37 ergs which is @xmath38 of the total energy released in the sn explosion . because of its relativistic motion , the jet is most likely beamed with an opening half angle @xmath39 which is much wider compared to a grb jet of @xmath40 . assuming a duration @xmath41 s , the isotropic equivalent kinetic luminosity of the jet is @xmath42 . with a jet variability time scale @xmath43 s at the base , internal shocks between plasma materials moving along the jet occur at a radius @xmath44 cm , which is below the stellar surface . [ t ] = 3.in internal shocks convert a fraction @xmath45 of the bulk kinetic energy ( @xmath46 ) into random electron motion analogous to the grb fireball models . in the case of grbs , these relativistic electrons would emit synchrotron photons in an optically thin environment , which are observed as @xmath5-rays on earth . the density of these electrons and baryons ( since they are coupled to the baryons ) in the jet may be estimated as n_e n_p 10 ^ 20.5 cm^-3 , [ particle - density ] in the comoving jet frame . the opacity to thomson scattering by photons _th 10 ^ 6.6 [ thomson - opacity ] in the comoving frame is then very high . the magnetic fields in the jet , built up by turbulent motions in the shock region , have an energy characterized as a fraction @xmath47 of the total jet energy , @xmath48 . the corresponding magnetic field strength in the jet is given by b ( ) ^1/2 10 ^ 9 ( ) ^1/2 g. [ b - field ] electrons and protons are expected to be accelerated to high energies in the internal shocks , via the fermi mechanism . the electrons cool down rapidly by synchrotron radiation in the presence of the magnetic field in eq . ( [ b - field ] ) . however , due to the large thomson optical depth in eq . ( [ thomson - opacity ] ) , these photons thermalize and the corresponding black - body temperature is e _ ( ) ^1/4 4.3 ( ) ^1/4 kev . [ bb - phot - energy ] the volume number density of these thermal photons may be roughly calculated as n _ 10 ^ 24.8 ( ) ^1/4 cm^-3 . [ bb - phot - density ] it may be noted that photons from the shocked stellar plasma do not diffuse into the jet due to the high optical depth @xmath49 in eq . ( [ thomson - opacity ] ) , and the number density in eq . ( [ bb - phot - density ] ) is roughly constant . the shock acceleration time for a proton of energy @xmath50 is proportional to its larmor s radius and may be estimated as t_acc 10 ^ -12 ( ) ( ) ^1/2 s , [ proton - acc - time ] where @xmath51 and the magnetic field in eq . ( [ b - field ] ) was used . the maximum proton energy is limited by requiring this time not to exceed the dynamic time scale for the shock to cross plasma material : @xmath52 , or any other possible proton cooling process time scale which we discuss next . the cooling time scale for protons by synchrotron radiation in the same magnetic field which is responsible for its acceleration is given by t_syn 3.8 ( ) ^-1 ( ) s , [ proton - syn - cool ] with @xmath53 . inverse compton ( ic ) scattering of thermal electron synchrotron photons is another cooling channel for high energy protons . the ic cooling time scale in the thomson and klein - nishina ( kn ) regimes , valid for @xmath50 much less or greater than @xmath54 gev respectively , as t_ic , th = 3.8 ( ) ^-1 ( ) s + t_ic , kn = 10 ^ -10.5 ( ) ( ) ^1/2 s. [ proton - ic - cool ] here we used the thermal photons with peak energy and density given in eqs . ( [ bb - phot - energy ] ) and ( [ bb - phot - density ] ) respectively as targets . because of a high density of thermal photons in the sn jet in eq . ( [ bb - phot - density ] ) , protons may produce @xmath55 pairs by interacting with them , a process known as bethe - heitler ( bh ) . the cross - section for bh interaction : @xmath56 is given by @xmath57 - 106/9 \}$ ] . the logarithmic increase of the cross - section with incident proton energy implies that this is a very efficient cooling mechanism for high energy protons . the @xmath58 pairs are produced at rest in the center of mass ( c.m . ) frame of the collision and acquire an energy @xmath59 each in the comoving frame . here @xmath60 is the lorentz boost factor of the c.m . in the comoving frame . the energy lost by the proton in each bh interaction is thus the energy of the created pairs in the comoving frame , and is given by @xmath61 . the energy loss rate of the proton is proportional to the bh scattering rate as given by @xmath62 , and the corresponding proton cooling time is t_bh = = , [ proton - bh - cool]in the comoving frame . photomeson ( @xmath63 ) and proton - proton ( @xmath64 ) interactions which are responsible for producing high energy neutrinos may also serve as a cooling mechanism for the shock accelerated protons . the @xmath63 at the @xmath65 resonance and the average @xmath64 cross - sections are @xmath66 @xmath32 and @xmath67 @xmath32 respectively . the corresponding optical depths , given by _p & = & 10 ^ 7.8 ( ) ^1/4 and + _pp & = & 10 ^ 5.5 ( ) , [ p - opacities ] are very high . the threshold proton energy for @xmath65 production against the target thermal photons of energy @xmath68 in eq . ( [ bb - phot - energy ] ) is e_p , ^+ & = & 10 ^ 4.8 ( ) ^1/4 gev . [ pg - energy ] adopting the energy loss by a proton @xmath69 and @xmath70 respectively for each @xmath63 and @xmath64 interaction , the hadronic cooling time scales are t_p & = & 10 ^ -7.3 ( ) ^1/4 s + t_pp & = & 10 ^ -5.6 ( ) s , [ proton - had - cool ] using eqs . ( [ bb - phot - density ] ) and ( [ particle - density ] ) . the photomeson cooling time scale @xmath71 above is roughly valid at @xmath72 and one needs to use the thermal photon spectrum to calculate it for different proton energies . the shock acceleration time in eq . ( [ proton - acc - time ] ) for protons in the sn jet ( solid line ) and the different cooling time scales are plotted in fig . [ fig : proton - cool - time ] as functions of the comoving proton energy . the dashed lines are hadronic cooling time scales in eq . ( [ proton - had - cool ] ) cooling with a delta - function approximation . ] and the dot - dashed lines are electromagnetic cooling time scales in eqs . ( [ proton - syn - cool ] ) , ( [ proton - ic - cool ] ) and ( [ proton - bh - cool ] ) . note that the hadronic cooling time ( @xmath73 ) , the bh cooling time ( @xmath74 ) and the synchrotron cooling time ( @xmath75 ) are first longer and then shorter than the maximum proton acceleration time ( @xmath76 ) . the hadronic ( @xmath71 ) and ic scattering ( @xmath77 ) are not efficient cooling mechanisms for protons . the maximum proton energy can be roughly estimated , by equating the @xmath75 to @xmath76 , from eqs . ( [ proton - acc - time ] ) and ( [ proton - syn - cool ] ) as e_p , max = ( ) ^1/2 10 ^ 6.3 ( ) ^1/4 gev , [ max - proton - energy ] since @xmath78 at this energy . [ t ] = 3.75 in shock accelerated protons in the sn jet can produce non - thermal neutrinos by photomeson ( @xmath63 ) interactions with thermal synchrotron photons and/or by proton - proton ( @xmath64 ) interactions with cold protons present in the shock region . as shown in fig . [ fig : proton - cool - time ] , the @xmath63 process is dominant in the energy range @xmath79 - @xmath80 gev and the @xmath64 process is dominant at other energies . in the case of @xmath63 interactions at the @xmath65 resonance , neutrinos are produced from charged pion ( @xmath81 ) decay as @xmath82 . the @xmath64 interactions also produce charged pions ( @xmath83 ) and kaons ( @xmath84 ) and their decay modes are the same as above with @xmath85 and @xmath86 branching ratios respectively . and @xmath87 decay modes are charge conjugate of @xmath81 and @xmath88 decay modes . ] the total pion ( kaon ) multiplicity in each @xmath64 interaction is @xmath89 ( @xmath90 ) in the energy range considered here.@xcite the energy of the shock accelerated protons in the sn jet is expected to be distributed as @xmath91 , following the standard shock acceleration models . charged mesons , produced by @xmath64 and @xmath63 interactions , are expected to follow the proton spectrum with @xmath92 of the proton energy for each pion or kaon . high - energy pions , kaons and muons produced by @xmath63 and @xmath64 interactions do not all decay to neutrinos as electromagnetic ( synchrotron radiation and ic scattering ) and hadronic ( @xmath93 and @xmath94 interactions ) cooling mechanisms reduce their energy . muons are severely suppressed by electromagnetic energy losses and do not contribute much to high - energy neutrino production . suppression factors for pion and kaon decay neutrinos are discussed next . the synchrotron and ic cooling times may be combined into a single electromagnetic cooling rate as @xmath95 . for ic cooling in the thomson regime @xmath96 and in the kn regime @xmath97 . the electromagnetic cooling time scales for mesons may be estimated assuming @xmath98 for simplicity as t_em ( ) s. [ meson - emcool - time ] the hadronic energy losses for mesons is similar to the proton energy losses by @xmath64 interactions in eq . ( [ proton - had - cool ] ) with the same @xmath93 and @xmath94 cross - section of @xmath99 @xmath32 . the corresponding hadronic cooling time scales for mesons are t_had 10 ^ -5.4 ( ) s , [ meson - hadcool - time ] with @xmath100 the electromagnetic ( @xmath101 ) and hadronic ( @xmath102 ) cooling time scales for mesons along with their decay times boosted by the respective lorentz factors , @xmath103 , are plotted in fig . [ fig : meson - cool - time ] . the total cooling time scale @xmath104 is first dominated by the hadronic and then by the electromagnetic cooling channel . the ratio @xmath105 determines the suppression of mesons before they decay to neutrinos . [ t ] = 3.75 in from the condition @xmath106 , one may roughly define a break energy as e_br = [ meson - break - energy ] above and below which the mesons cool by electromagnetic and hadronic interactions respectively . the corresponding suppression factor may be defined from the ratios @xmath107 and @xmath108 as = ; = [ suppression - factor ] another break energy may be defined from the condition @xmath109 below which mesons decay to neutrinos without any suppression . however , this energy : @xmath110 and @xmath111 gev respectively for pions and kaons in the comoving frame is low . the observed energy on earth in both cases is below the detection threshold energy as will be discussed shortly . if the shock accelerated protons could travel unimpeded from the sn jet at a luminosity distance @xmath112 , then their isotropic equivalent fluence on earth would be _ p , ob = . [ proton - flux ] here @xmath113 and @xmath114 are the energy and time related in the observer s and local rest frames for the source location at redshift @xmath115 . of course , defelction by magnetic fields as well as other interactions along the way prevent such direct arrivale of the protons . nearly all protons are expected to convert into @xmath81 ( @xmath83 ) by @xmath63 ( @xmath64 ) interactions in the sn jet because @xmath116 ( @xmath117 ) in eq . ( [ p - opacities ] ) and assuming the charged pion multiplicity is @xmath89 from @xmath64 interactions as a conservative estimate . with a @xmath85 branching ratio for pion decay to neutrinos , one may define a multiplicative factor @xmath118 for protons which will produce neutrinos via pions . a similar factor @xmath119 may be defined for kaons as a product of @xmath84 multiplicity of @xmath90 from @xmath64 interactions and @xmath86 branching ratio for kaon decay to neutrinos . protons produce neutral pions ( kaons ) and charged pions ( kaons ) with equal probability in both the @xmath63 and @xmath64 interactions . interactions only . ] for simplicity one may also assume that a @xmath120 carries 1/4 of the charged pion ( kaon ) energy : @xmath121 , from roughly equipartition of energy between the final decay products of @xmath83 and @xmath84 . from these considerations one may estimate the neutrino ( of one flavor ) fluence on earth as _ , ob & = & ( ) ^- , [ nu - fluence - formula ] per sn burst using eqs . ( [ suppression - factor ] ) and ( [ proton - flux ] ) . here @xmath122 ( @xmath123 ) for @xmath124 greater ( less ) than @xmath125 from eq . ( [ meson - break - energy ] ) . the pre - factors in eq . ( [ suppression - factor ] ) are represented by the parameters @xmath126 and @xmath127 respectively for pions and kaons . for a typical ice cherenkov detector such as icecube , the threshold neutrino detection energy is @xmath128 gev . the neutrino energy range is then @xmath129 - @xmath130 - @xmath131 - @xmath132 gev in the observer s frame on earth . as mentioned earlier , @xmath81 s are produced by @xmath64 or @xmath63 interactions by shock accelerated protons of all energies . in this case @xmath81 decay @xmath120 fluence from a sn at a distance @xmath133 mpc ( @xmath134 cm , @xmath135 ) , e.g. from the virgo cluster , would be _ , , ob^ * & & 10 ^ -5 ( ) ^-4 gev^-1cm^-2 ; + & & 10 ^ 2.5 e_,ob / gev 10 ^ 5.5 , [ nuflux - pi - numbers ] from eq . ( [ nu - fluence - formula ] ) with @xmath136 . the neutrino break energy from pion decay : @xmath137 gev in this case . a similar expression may be derived for neutrino fluence from kaon decays . however , @xmath64 interactions are overwhelmed by @xmath63 interactions in the energy range @xmath79-@xmath80 gev ( fig . [ fig : proton - cool - time ] ) . hence kaon and the corresponding decay neutrino production is expected to be suppressed in the energy range @xmath138-@xmath139 gev . the neutrino break energy from eq . ( [ meson - break - energy ] ) is @xmath140 gev and the fluence is _ , k , ob^ * & & 10 ^ -11.7 ( ) ^-(+2 ) gev^-1cm^-2 ; + & & = [ nuflux - k - numbers ] on earth from kaon decays in a sn jet at a distance @xmath141 mpc . the diffuse neutrino flux is calculated by summing over fluences from all slow - jet endowed sne distributed over cosmological distances in hubble time . the sne rate follows closely the star formation rate ( sfr ) which can be modeled , as a function of redshift per unit comoving volume,@xcite as _ * ( z ) = m _ yr^-1 mpc^-3 . [ sfr ] here @xmath142 km s@xmath33 mpc@xmath33 is the hubble constant . for a friedmann - robertson - walker universe , the comoving volume element is = | cosmic time @xmath143 is ( dt / dz)^-1 = -h_0 ( 1+z ) . [ redshift - time ] for the standard @xmath144cdm cosmology , @xmath145 and @xmath146 . the number of sne per unit star forming mass ( @xmath147 ) depends on the initial mass function and the threshold for stellar mass to produce sn ( @xmath148 m@xmath149 ) . a salpeter model @xmath150 with different power - law indices can generate different values for @xmath147 , e.g. @xmath151 m@xmath152 for @xmath153 and @xmath154 m@xmath152 for @xmath155.@xcite we adopt the model in ref . which corresponds to the local type ii sne rate @xmath156 yr@xmath33 mpc@xmath157 agreeing with data.@xcite the distribution of sne per unit cosmic time @xmath143 and solid angle @xmath158 covered on the sky can be written as = = | | . [ sn - dist ] we assume a fraction @xmath159 of all sne involve jets and a fraction @xmath160 of all such jets are pointing towards us . the observed diffuse sne neutrino flux , using eqs . ( [ nu - fluence - formula ] ) and ( [ sn - dist ] ) , is then _ , ob^diff & = & _ 0^ dz f_,ob ( e_,ob ) + & = & _ 0^ dz | | ( ) ^- + & & ( e _ , th e_,ob e _ , ob , max ) . [ diff - nu - flux ] with @xmath161 specified as before . the diffuse @xmath120 flux from all cosmological slow - jet sne is plotted in fig . [ fig : nuflux ] by numerically integrating eq . ( [ diff - nu - flux ] ) , assuming the maximal fraction @xmath162 . the pion ( kaon ) decay flux is plotted with thick solid ( dashed ) line(s ) . the exponential suppression in the kaon decay @xmath120 flux curves is due to the lack of kaon production by @xmath64 interactions , as @xmath63 interactions , which do not produce kaons , are dominant in the particular proton energy range corresponding to this neutrino energy range , @xmath163-@xmath139 gev . the light dashed curve corresponds to the kaon decay @xmath120 flux if @xmath64 interaction dominates over the whole proton energy range for comparison . the kaon decay neutrino fluxes are first smaller and then larger than the pion decay neutrino flux . this is because of the kaon s heavier mass and shorter decay time compared to pions ( see fig . [ fig : meson - cool - time ] ) . [ t ] = 3.75 in the atmospheric @xmath120 , @xmath164 flux from pion and kaon decays ( conventional flux ) compatible with amanda data@xcite is plotted with the following parametrization@xcite _ , ob^atm = [ atmo - nu - flux ] also shown are the cosmic ray bounds ( wb limits ) on the diffuse neutrino flux.@xcite it is unlikely that neutrino detectors can measure the sne diffuse fluxes plotted in figure [ fig : nuflux ] , as they are below the atmospheric background . however , individual sne in nearby galaxies may be detectable , as discussed in the next section . there are approximately 4000 galaxies known within 20 mpc distance . at the standard rate of 1 snu = @xmath165/yr/@xmath166 blue solar luminosity for average galaxies , the estimated sn rate is @xmath167/yr . the sn rate in the starburst galaxies , such as m82 and ngc253 ( at distances 3.2 and 2.5 mpc in the northern and southern sky , respectively ) is @xmath90/yr , much larger than in the milkyway or in the magellanic clouds.@xcite very strong neutrino signals in future kilometer scale neutrino detectors are expected from these nearby sne , over a negligible atmospheric background , using temporal and positional coincidences with optical detections . the directional sensitivity of the cherenkov detectors is best for muon neutrinos , which create muons by charge current neutrino - nucleon ( @xmath168 ) interactions . muons carry @xmath169 of the incident neutrino energy . it emits cherenkov light as it travels faster than the speed of light in the detection media . photo multiplier tubes ( pmts ) buried in the medium ( ice , e.g. in the case of icecube ) can detect muons by gathering their cherenkov light . the effective detection area of a cherenkov detector depends on the arrival direction of the neutrino and the energy of the muon . in principle it can be larger than the geometrical area ( @xmath170 in the case of icecube e.g. ) as muons can be produced outside the instrumented volume and travel inside . to achieve a good pointing resolution a muon should _ hit _ at least 4 pmts , for reconstructing its track unamiguously , strung on different vertical _ strings_. the pmt spacing is 17 m vertically and 125 m horizontally . with a muon energy loss of @xmath171 gev / m and a pmt efficiency of @xmath172 , the threshold energy for muon detection is @xmath173 gev as a conservative estimate . this corresponds to a neutrino threshold energy @xmath174 gev as used before . the icecube detector is expected to have an angular sensitivity of @xmath175 for muon tracks of energy @xmath176 tev coming from a zenith angle @xmath177 and it gets better at higher energy.@xcite the number of muon events from a nearby jetted sn may be calculated as n _ = a_det _ e_,th^e_,max p ( e _ , ) f_,;k^ * ( e _ ) de _ , [ event - rate ] where the neutrino fluences are given in eqs . ( [ nuflux - pi - numbers ] ) and ( [ nuflux - k - numbers ] ) . the full detection probability @xmath178 depends on the source s angular position ( @xmath179 ) and hence on the earth s shadowing effect , as well as the energy dependent @xmath180 cross section.@xcite the cumulative number of muon events from m82 and ngc253 are plotted in fig . [ fig : nuevents ] per individual sn with its jet pointing towards earth . the @xmath120 flux models are from pion ( solid lines ) and kaon ( dashed lines ) decays in the sn jet as in fig . [ fig : nuflux ] . the two kaon decay models are due to the protons in the sn jet producing kaons in their whole ( thick dashed lines ) and partial ( thin dashed lines ) energy range ( see fig . [ fig : proton - cool - time ] ) . the lower and upper sets of lines at @xmath181 gev correspond to the sn in m82 and ngc253 respectively . the timing uncertainty in the optical detection of sn is @xmath89 day . the corresponding atmospheric background events within @xmath175 angular resolution is 0.07 for both m82 and ngc253 in the same energy range as in fig . [ fig : nuevents ] . [ t ] = 3.75 in neutrino fluxes of all three flavors ( @xmath120 , @xmath182 , @xmath183 ) on earth should be equal because of oscillations over astrophysical distances . however , only @xmath120 s are emitted from the sources under consideration and the total number of neutrino events ( including @xmath182 and @xmath183 ) will remain the same as the @xmath120 events calculated here . the lack of good directional sensitivity for the @xmath182 and @xmath183 events may prevent obtaining their positional coincidence with the sn . timing coincidences of @xmath182 and @xmath183 events with @xmath120 events , however , may still be useful to verify the neutrino oscillations at these energies and test their common origin . at the quoted rate for m82 and ngc253 , a sn from one of these galaxies would be expected within five years . however , only 1/5 of them would be _ visible _ in neutrinos due to the @xmath184 beaming effect of the jet . icecube may detect one such sn in 25 years of its operation . the real situation is not as pessimistic as for the brightest neutrino events expected from m82 and ngc253 . other nearby spiral galaxies ( m31 , m74 , m51 , m101 , etc , and the virgo cluster ) will also have sne . for a hypothetical sn at 20 mpc with its jet pointing towards earth , the number of neutrino events is @xmath185/km@xmath186 from pion and kaon decay fluxes combined . with the suggested rate of @xmath187 sn / yr within 20 mpc , icecube may detect @xmath188 muon events / yr after beaming correction from the jetted sne . while a core collapse sn in a typical galaxy is a rare event , their rate of occurence within 20 mpc could be more than 1/yr , and the physics and astrophysics one can learn from such an explosion is enormous . a buried slow jet from the collapsing core of the supernova progenitor star is an attractive possibility for solving the long standing problem of how to understand the ejection of the envelope in sn explosions , by re - energizing the shock wave through energy deposition by the jet . alternatively it could be that only a fraction of core collapses leads to such jets . these hypothetical mildly relativistic jets in sne may be related to the ultra relativistic jets thought to be responsible for long duration grbs , which are thought to originate from the core collapse of massive progenitor stars , some of which have been positively associated with observed envelope ejection supernova events . while all typical core collapses should produce 10 mev thermal neutrinos , the presence of a jet would allow proton acceleration by shocks , and produce 1 tev non - thermal neutrinos . detection of these high energy neutrinos by upcoming cherenkov detectors would be a smoking gun signal of a sn jet , and would allow one to study the conditions inside a collapsing star which may not be possible otherwise . work supported by an nsf grant ast0307376 and nasa grant nag5 - 13286 . k. nomoto , p. a. mazzali , t. nakamura , k. iwamoto , i. j. danziger , f. patat , in supernovae and gamma - ray bursts : the greatest explosions since the big bang , may , 1999 , eds . m. livio , n. panagia and k. sahu . stsci symposium series , vol . 13 , pg 144 ( 2001 ) .
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it has been hypothesized recently that core collapse supernovae are triggered by mildly relativistic jets following observations of radio properties of these explosions .
association of a jet , similar to a gamma - ray burst jet but only slower , allows shock acceleration of particles to high energy and non - thermal neutrino emission from a supernova .
detection of these high energy neutrinos in upcoming kilometer scale cherenkov detectors may be the only direct way to probe inside these astrophysical phenomena as electromagnetic radiation is thermal and contains little information .
calculation of high energy neutrino signal from a simple and slow jet model buried inside the pre - supernova star is reviewed here .
the detection prospect of these neutrinos in water or ice detector is also discussed in this brief review . jetted core collapse supernovae in nearby galaxies may provide the strongest high energy neutrino signal from point sources .
| 10,790 | 215 |
the shape of the stellar body of a galaxy reflects its formation process . reconstructing the intrinsic , three - dimensional shapes of spiral galaxies from their shapes projected on the sky has a long tradition , and proved to be an exquisitely accurate and precise approach , especially once sample size increased ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? these results provided us with the general notion that the stellar bodies of present - day star - forming galaxies over a wide range in luminosity can be described as thin , nearly oblate ( therefore , disk - like ) systems with an intrinsic short - to - long axis ratio of @xmath10 . such global shapes encompass all galactic components , including bars and bulges . the disk component is generally thinner ( @xmath11 , e.g. , * ? ? ? analogous information about the progenitors of today s galaxies is scarcer . among faint , blue galaxies in deep hubble space telescope imaging , @xcite found a substantial population of elongated ` chain ' galaxies , but several authors argued that chain galaxies are edge - on disk galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , @xcite demonstrated that the ellipticity distribution of a large sample of @xmath12 lyman break galaxies is inconsistent with randomly oriented disk galaxies , lending credence to the interpretation that a class of intrinsically elongated ( or , prolate ) objects in fact exists at high redshift . by modeling ellipticity distributions , @xcite and @xcite concluded that the intrinsic shapes of @xmath13 star - forming galaxies are strongly triaxial . on the other hand , regular rotation is commonly seen amongst @xmath14 samples @xcite , and the evidence for the existence of gaseous disks is ample among massive systems @xcite . one possible explanation for the seeming discrepancy between the geometric and kinematic shape inferences is a dependence of structure on galaxy mass . indeed , for lower - mass galaxies ( @xmath15 ) the evidence for rotation is less convincing ( e.g. , * ? ? ? * ; * ? ? ? * ) and in rare cases rotation is convincingly ruled out ( e.g. , * ? ? ? the prevailing view is that the gas and hence presumably the stars that form from it in those galaxies is supported by random motions rather than ordered rotation . however , the kinematic measurements for low - mass galaxies probe only a small number of spatial resolution elements signs of rotation may be smeared out @xcite and the observed motions may have a non - gravitational origin such as feedback . here we aim to provide the first description of the geometric shape distribution of @xmath4 star - forming galaxies and its dependence on galaxy mass . we examine the projected axis ratio distributions ( @xmath16 ) of large samples of star - forming galaxies out to @xmath17 drawn from the candels @xcite and 3d - hst @xcite surveys . a low - redshift comparison sample is drawn from the sloan digital sky survey ( sdss ) . the methodology developed by @xcite and @xcite will be used to convert @xmath16 into 3-dimensional shape distributions of star - forming galaxies and its evolution from @xmath17 to the present day . we construct volume - limited samples of star - forming galaxies over a large range in stellar mass ( @xmath1 ) and redshift ( @xmath0 ) with @xmath18 measured at an approximately fixed rest - frame wavelength of @xmath19 . @xcite provide wfc3/f125w+f140w+f160w - selected , multi - wavelength catalogs for the candels fields , as well as redshifts , stellar masses and rest - frame colors using the 3d - hst wfc3 grism spectroscopy in addition to the photometry . 36,653 star - forming galaxies with stellar masses @xmath20 and up to redshift @xmath17 are selected based on their rest - frame @xmath21 and @xmath22 colors as described by @xcite , 35,832 of which have @xmath18 measurements . the typical accuracy and precision is better than 10% @xcite . for the @xmath23 galaxies we use the f160w - based values , for the @xmath24 galaxies we use the f125w - based values , such that all @xmath4 galaxies have their shapes measured at a rest - frame wavelength as close as possible to @xmath19 ( and always in the range @xmath25 ) . this avoids the effects due the shape variations with wavelength seen in local galaxies @xcite . below @xmath26 our f125w shape measurements probe longer wavelengths . we compared the f125w - based shapes with hst / acs f814w - based shapes for 1,365 galaxies ( see * ? ? ? * ) . the median f125w - based axis ratio is 0.014 larger than the median f814w - based shape , with a scatter of 0.06 . this is consistent with the measurement errors . we conclude that using f125w axis ratios at @xmath27 does not affect our results . sdss photometry - based stellar masses from @xcite are used to select 36,369 star - forming galaxies with stellar masses @xmath20 and in the ( spectroscopic ) redshift range @xmath28 . the distinction between star - forming and passive galaxies is described by @xcite and is based on the rest - frame @xmath29 and @xmath30 colors , analogous to the use of @xmath21 and @xmath22 colors at higher redshifts . for the sdss sample we use the @xmath18 estimates from fitting the exponential surface brightness model to the @xmath31-band imaging as part of the dr7 photometric pipeline @xcite . these measurements have been verified by @xcite , who showed that systematic offsets and scatter with respect to our galfit -based measurements are negligible . the very pronounced change of the projected shape distribution with redshift ( figure [ hist ] ) immediately reveals that galaxy structure evolves with cosmic time . especially at low stellar masses we see that a larger fraction of galaxies have flat projected shapes than at the present day . this observation underpins the analysis presented in the remainder of the letter . here we provide a brief description of the methodology to infer the intrinsic , 3-dimensional shapes of galaxies , outlined in detail by @xcite . we adopt the ellipsoid as the general geometric form to describe the shapes of galaxies . it has three , generally different , axis lengths ( @xmath32 ) , commonly used to define ellipticity ( @xmath33 ) and triaxiality ( @xmath34 ) . in order to facilitate an intuitive understanding of our results we define three broad geometric types , shown in figure [ class ] : _ disky _ ( @xmath35 ) , _ elongated _ ( @xmath36 ) , and _ spheroidal _ ( @xmath37 ) . the goal is to find a model population of triaxial ellipsoids that , when seen under random viewing angles , has the same @xmath16 as an observed galaxy sample . our model population has gaussian distributions of the ellipticity ( with mean @xmath38 and standard deviation @xmath39 ) and triaxiality ( with mean @xmath40 and standard deviation @xmath41 ) . such a model population has a known @xmath16 which we adjust to include the effect of random uncertainties in the axis ratio measurements these are asymmetric for nearly round objects . then , given that each observed value of @xmath18 corresponds to a known probability , we calculate the total likelihood of the model by multiplying the probabilities of each of the observed values . we search a grid of the four model parameters to find the maximal total likelihood . in figure [ hist ] we show observed axis ratio distributions ( histograms ) , and the probability distributions of the corresponding best - fitting model populations ( smooth lines ) . the models generally match the data very well . even in the worst case ( bottom - right panel ) the model and data distributions are only marginally inconsistent , at the @xmath42 level . a triaxial model population with parameters @xmath43 corresponds to a cloud of points in figure [ class ] and , hence , with certain fractions of the three geometric types . the colored bars in figure [ hist ] represent these fractions for the best - fitting triaxial models . this illustrates the connection between projected shapes and intrinsic shapes : a broad @xmath16 reflects a large fraction of _ disky _ objects , whereas a narrow distribution with a peak at small @xmath18 is indicative of a large fraction of _ elongated _ objects . a narrow distribution with a peak at large @xmath18 would indicate a large fraction of _ spheroidal _ objects . in figure [ res ] we provide the modeling results for the full redshift and mass range probed here : for each stellar mass bin we show the redshift evolution of the four model parameters , including the uncertainties obtained by bootstrapping the samples . finally , in figure [ frac ] we show the full set of results in the form of the color coding defined in figure [ class ] . the small values of @xmath40 and the large values of @xmath38 for present - day star - forming galaxies ( figure [ res ] ) imply that the vast majority are thin and nearly oblate . indeed , according to our classification shown in figure [ hist ] between 80% and 100% are _ disky _ , as is generally known and was demonstrated before on the basis of similar axis - ratio distribution analyses by @xcite and @xcite . importantly , the intrinsic shape distribution of star - forming galaxies does not change over a large range in stellar mass ( @xmath1 ) . toward higher redshifts star - forming galaxies become gradually less disk - like ( figures [ hist ] , [ res ] and [ frac ] ) . this effect is most pronounced for low - mass galaxies . already in the @xmath44 redshift bin in figure [ res ] we see evolution , mostly in the scatter in triaxiality ( @xmath41 ) . that is , there is substantial variety in intrinsic galaxy shape . beyond @xmath26 , galaxies with stellar mass @xmath5 typically do not have a _ disky _ geometry , but are most often _ elongated _ ( figure [ res ] ) . galaxies with mass @xmath6 show similar behavior , but with evolution only apparent at @xmath13 . this geometric evidence for mass - dependent redshift evolution of galaxy structure is corroborated by the analysis of kinematic properties of @xmath45 galaxies by @xcite . _ disky _ objects are the most common type ( @xmath46 ) among galaxies with mass @xmath47 at all redshifts @xmath3 . a population of _ spheroidal _ galaxies is increasingly prominent among massive galaxies at @xmath48 . a visual inspection of such objects reveals that at least a subset are mergers , but an in - depth interpretation of this aspect we defer to another occasion . it is interesting to note that ellipticity hardly depends on mass and redshift ( figure [ res ] ) . that is , despite strong evolution in geometry , the short - to - long axis ratio remains remarkably constant with redshift , and changes little with galaxy mass . a joint analysis of galaxy size and shape is required to explore the possible implications . note that our definition of geometric shape is unrelated to the common distinction between disks and spheroids on the basis of their concentration parameter or srsic index . as a result we distinguish between the observation that most low - mass star - forming galaxies at @xmath8 have exponential surface brightness profiles ( e.g. , * ? ? ? * ) and our inference that these galaxies are not , generally , shaped like disks in a geometric sense . this illustrates that an approximately exponential light profile can correlate with the presence of a disk - like structure but can not be used as a definition of a disk . star formation in the present - day universe mostly takes place in @xmath49 galaxies and in non - starburst galaxies . since essentially all such star - forming galaxies are _ disky _ and star formation in disk galaxies occurs mostly over the full extent of the stellar disk , it follows immediately that essentially all current star formation takes place in disks . the analysis presented in this _ letter _ allows us to generalize this conclusion to include earlier epochs . at least since @xmath8 most star formation is accounted for by @xmath50 galaxies ( e.g. , * ? ? ? figures [ res ] and [ frac ] show that such galaxies have disk - like geometries over the same redshift range . given that 90% of stars in the universe formed over that time span , it follows that the majority of all stars in the universe formed in disk galaxies . combined with the evidence that star formation is spatially extended , and not , for example , concentrated in galaxy centers ( e.g. , * ? ? ? * ; * ? ? ? * ) this implies that the vast majority of stars formed in disks . despite this universal dominance of disks , the elongatedness of many low - mass galaxies at @xmath51 implies that the shape of a galaxy generally differs from that of a disk at early stages in its evolution . according to our results , an elongated , low - mass galaxy at @xmath52 will evolve into a disk at later times , or , reversing the argument , disk galaxies in the present - day universe do not initially start out disks . as can be seen in figure [ res ] , the transition from _ elongated _ to _ disky _ is gradual for the population . this is not necessarily the case for individual galaxies . hydrodynamical simulations indicate that sustained disks form quite suddenly , on a dynamical time scale , after an initial period characterized by rapidly changing dynamical configurations ( e.g. , * ? ? ? this turbulent formation phase may include the subsequent formation and destruction of short - lived disks ( e.g. , * ? ? ? * ) , associated with rapid changes in orientation and resulting in a hot stellar system of rather arbitrary shape . our observation that at @xmath4 the low - mass galaxy population consists of a mix of _ disky _ and _ elongated _ objects in this picture , the latter represent the irregular phase without a sustained disk can be interpreted as some fraction of the galaxies having already transformed into a sustained disk . the probability for this transition is , then , a function of mass which may or may not depend on redshift . given the various estimates of the stellar mass evolution of milky way - mass galaxies as a function of redshift ( e.g. , * ? ? ? * ; * ? ? ? * ) , we suggest that the milky way may have first attained a sustained stellar disk at redshift @xmath53 . our analysis rests on the assumption that stellar light traces the mass distribution of a galaxy . potential spoilers include obscuration by dust , dispersion in age among stars , and large gas fractions . dust has a viewing angle - dependent effect on the measured @xmath18 . massive galaxies at all redshifts are dusty , and a large variety of dust geometries could disturb axis ratio measurements , hiding the disk - like structure of the population when traced by the axis ratio distribution . perhaps this plays a role at @xmath48 where we see an increased fraction of round objects . however , the reverse to create a disk - like axis ratio distribution for a population of dusty non - disks requires unlikely fine tuning . we prefer the more straightforward interpretation that massive , star - forming galaxies truly are disks , at least up to @xmath54 . this is supported by the observed correlation between axis ratio and color ( e.g. , * ? ? ? * ) , also seen in our sample : galaxies with smaller @xmath18 are redder than those with larger @xmath18 , as expected from a population on inclined , dusty disks . dust is also unlikely to affect @xmath16 of low - mass galaxies . at @xmath4 galaxies with stellar masses @xmath15 are generally very blue . for these young , presumably metal - poor galaxies dust is of limited relevance to the shape measurements . this also implies that completeness of our sample is not affected by strong dust obscuration . age variations in the stellar population and large gas fractions both potentially present challenges to our assumption that the rest - frame optical light traces the underlying mass distribution . perhaps the luminous regions are young , bright complexes embedded in disks consisting of cold gas or fainter , older stellar populations . we can not immediately discard this possibility as dynamical masses exceed stellar masses by an average factor of @xmath55 in the stellar mass range @xmath56 galaxies at @xmath4 ( e.g. , * ? ? ? * ; * ? ? ? it is implausible that this difference between stellar mass and dynamical mass is entirely made up of undetected , older stars in a disk - like configuration . the different spatial distributions of the young and old stars would lead to wavelength - dependent shapes , which is not observed . if such a population of older stars is present , it must be spatially coincident with the young population , and not , generally , in a disk . we can not exclude the existence of cold gas disks that are @xmath573@xmath58 more massive than the ( young ) stellar population . hydrodynamical simulations show that low - mass , high - redshift systems can produce elongated stellar bodies embedded in more extended , turbulent gaseous bodies with ordered rotation ( e.g. , * ? ? ? at the moment there is little observational evidence for such extended gaseous disks . for the mass range @xmath59 gas masses in excess of the stellar mass have been inferred based on the star - formation rate and the inverse kennicutt - schmidt relation ( e.g. , * ? ? ? * ) , but this inversion relies on the assumption of a disk - like geometry , weakening the argument . furthermore , even if these cold gas mass estimates are correct it is not clear that the gas should be organized in a disk . generally , gas ionized by star formation and cold gas share global kinematic traits , and in these cases the ionized gas does not generally show rotation . deep alma observations will settle this issue , and for now we will leave this as the main caveat in our analysis . we have analyzed the projected axis ratio distributions , @xmath16 , measured at rest - frame optical wavelenghts , of stellar mass - selected samples of star - forming galaxies in the redshift range @xmath0 drawn from sdss and 3d - hst+candels . the intrinsic , 3-dimensional geometric shape distribution is reconstructed under the assumption that the population consists of triaxial objects view under random viewing angles . in the present - day universe star - forming galaxies of all masses are predominantly oblate and flat , that is , they are disks . massive galaxies ( @xmath2 ) typically have this shape at all redshifts @xmath60 . given the dominance of @xmath61 galaxies in terms of their contribution to the cosmic stellar mass budget and the star formation rate density it follows that , averaged over all cosmic epochs , the majority of all stars formed in disks . lower - mass galaxies have shapes at @xmath4 that differ significantly from those of thin , oblate disks . for galaxies with stellar mass @xmath5 ( @xmath6 ) there exists a mix of roughly equal numbers of elongated and disk galaxies at @xmath7 ( @xmath8 ) . at @xmath4 the @xmath5 galaxies are predominantly elongated . our findings imply that low - mass galaxies at high redshift had not yet formed a regularly rotating , sustained disk . given a range of plausible mass growth rate of milky way - mass galaxies we infer the disk formation phase for such galaxies at @xmath53 .
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we determine the intrinsic , 3-dimensional shape distribution of star - forming galaxies at @xmath0 , as inferred from their observed projected axis ratios . in the present - day universe
star - forming galaxies of all masses @xmath1 are predominantly thin , nearly oblate disks , in line with previous studies .
we now extend this to higher redshifts , and find that among massive galaxies ( @xmath2 ) disks are the most common geometric shape at all @xmath3 .
lower - mass galaxies at @xmath4 possess a broad range of geometric shapes : the fraction of elongated ( prolate ) galaxies increases toward higher redshifts and lower masses .
galaxies with stellar mass @xmath5 ( @xmath6 ) are a mix of roughly equal numbers of elongated and disk galaxies at @xmath7 ( @xmath8 ) .
this suggests that galaxies in this mass range do not yet have disks that are sustained over many orbital periods , implying that galaxies with present - day stellar mass comparable to that of the milky way typically first formed such sustained stellar disks at redshift @xmath9 .
combined with constraints on the evolution of the star formation rate density and the distribution of star formation over galaxies with different masses , our findings imply that , averaged over cosmic time , the majority of stars formed in disks .
| 5,436 | 360 |
dependence logic @xcite is an extension of first - order logic which adds _ dependence atoms _ of the form @xmath0 to it , with the intended interpretation of `` the value of the term @xmath1 is a function of the values of the terms @xmath2 . '' the introduction of such atoms is roughly equivalent to the introduction of non - linear patterns of dependence and independence between variables of branching quantifier logic @xcite or independence friendly logic @xcite : for example , both the branching quantifier logic sentence @xmath3 and the independence friendly logic sentence @xmath4 correspond in dependence logic to @xmath5 in the sense that all of these expressions are equivalent to the skolem formula @xmath6 as this example illustrates , the main peculiarity of dependence logic compared to the others above - mentioned logics lies in the fact that , in dependence logic , the notion of _ dependence and independence between variables _ is explicitly separated from the notion of quantification . this makes it an eminently suitable formalism for the formal analysis of the properties of _ dependence itself _ in a first - order setting , and some recent papers ( @xcite ) explore the effects of replace dependence atoms with other similar primitives such as _ independence atoms _ @xcite , _ multivalued dependence atoms _ @xcite , or _ inclusion _ or _ atoms @xcite . branching quantifier logic , independence friendly logic and dependence logic , as well as their variants , are called _ logics of imperfect information _ : indeed , the truth conditions of their sentences can be obtained by defining , for every model @xmath7 and sentence @xmath8 , an imperfect - information _ semantic game _ @xmath9 between a _ verifier _ ( also called eloise ) and a _ falsifier _ ( also called abelard ) , and then asserting that @xmath8 is true in @xmath7 if and only if the verifier has a winning strategy in @xmath9 . as an alternative of this ( non - compositional ) _ game - theoretic semantics _ , which is an imperfect - information variant of hintikka s game - theoretic semantics for first order logic @xcite , hodges introduced in @xcite _ team semantics _ ( also called _ trump semantics _ ) , a compositional semantics for logics of imperfect information which is equivalent to game - theoretic semantics over sentences and in which formulas are satisfied or not satisfied not by single assignments , but by _ sets _ of assignments ( called _ teams _ ) . in this work , we will be mostly concerned with team semantics and some of its variants . we refer the reader to the relevant literature ( for example to @xcite and @xcite ) for further information regarding these logics : in the rest of this section , we will content ourselves with recalling the definitions and results which will be useful for the rest of this work . let @xmath7 be a first order model and let @xmath10 be a finite set of variables . then an _ assignment _ over @xmath7 with _ domain _ @xmath10 is a function @xmath11 from @xmath10 to the set @xmath12 of all elements of @xmath7 . furthermore , for any assignment @xmath11 over @xmath7 with domain @xmath10 , any element @xmath13 and any variable @xmath14 ( not necessarily in @xmath10 ) , we write @xmath15 $ ] for the assignment with domain @xmath16 such that @xmath17(w ) = \left\{\begin{array}{l l } m & \mbox{if } w = v;\\ s(w ) & \mbox{if } w \in v \backslash \{v\ } \end{array } \right.\ ] ] for all @xmath18 . let @xmath7 be a first - order model and let @xmath10 be a finite set of variables . @xmath19 over @xmath7 with _ domain _ @xmath20 is a set of assignments from @xmath10 to @xmath7 . let @xmath19 be a team over @xmath7 , and let @xmath10 be a finite set of variables . and let @xmath21 be a finite tuple of variables in its domain . then @xmath22 is the relation @xmath23 . furthermore , we write @xmath24 for @xmath25 . as is often the case for dependence logic , we will assume that all our formulas are in negation normal form : let @xmath26 be a first - order signature . then the set of all dependence logic formula with signature @xmath26 is given by @xmath27 where @xmath28 ranges over all relation symbols , @xmath29 ranges over all tuples of terms of the appropriate arities , @xmath30 range over all terms and @xmath14 ranges over the set @xmath31 of all variables . the set @xmath32 of all _ free variables _ of a formula @xmath8 is defined precisely as in first order logic , with the additional condition that all variables occurring in a dependence atom are free with respect to it . [ dl - ts ] let @xmath7 be a first - order model , let @xmath19 be a team over it , and let @xmath8 be a dependence logic formula with the same signature of @xmath7 and with free variables in @xmath33 . then we say that @xmath19 _ satisfies _ @xmath8 in @xmath7 , and we write @xmath34 , if and only if ts - lit : : : @xmath8 is a first - order literal and @xmath35 for all @xmath36 ; ts - dep : : : @xmath8 is a dependence atom @xmath37 and any two assignments @xmath38 which assign the same values to @xmath2 also assign the same value to @xmath1 ; ts-@xmath39 : : : @xmath8 is of the form @xmath40 and there exist two teams @xmath41 and @xmath42 such that @xmath43 , @xmath44 and @xmath45 ; ts-@xmath46 : : : @xmath8 is of the form @xmath47 , @xmath48 and @xmath49 ; ts-@xmath50 : : : @xmath8 is of the form @xmath51 and there exists a function @xmath52 such that @xmath53 } \psi$ ] , where @xmath54 = \{s[f(s)/v ] : s \in x\}\ ] ] ts-@xmath55 : : : @xmath8 is of the form @xmath56 and @xmath57 } \psi$ ] , where @xmath58 = \{s[m / v ] : s \in x , m \in { \texttt{dom}}(m)\}.\ ] ] the disjunction of dependence logic does not behave like the classical disjunction : for example , it is easy to see that @xmath59 is not equivalent to @xmath60 , as the former holds for the team @xmath61 and the latter does not . however , it is possible to define the classical disjunction in terms of the other connectives : [ defin : classic_or ] let @xmath62 and @xmath63 be two dependence logic formulas , and let @xmath64 and @xmath65 be two variables not occurring in them . then we write @xmath66 as a shorthand for @xmath67 [ propo : classic_or ] for all formulas @xmath62 and @xmath63 , all models @xmath7 with at least two elements whose signature contains that of @xmath62 and @xmath63 and all teams @xmath19 whose domain contains the free variables of @xmath62 and @xmath63 @xmath68 the following four proportions are from @xcite : [ propo : emptyteam ] for all models @xmath7 and dependence logic formulas @xmath8 , @xmath69 . if @xmath34 and @xmath70 then @xmath71 . if @xmath34 and @xmath72 then @xmath73 . [ dltosigma ] let @xmath74 be a dependence logic formula with free variables in @xmath21 . then there exists a @xmath75 sentence @xmath76 such that @xmath77 for all suitable models @xmath7 and for all nonempty teams @xmath19 . furthermore , in @xmath76 the symbol @xmath28 occurs only negatively . as proved in @xcite , there is also a converse for the last proposition : [ sigmatodl ] let @xmath76 be a @xmath75 sentence in which @xmath28 occurs only negatively . then there exists a dependence logic formula @xmath74 , where @xmath78 is the arity of @xmath28 , such that @xmath77 for all suitable models @xmath7 and for all nonempty teams @xmath19 whose domain contains @xmath21 . because of this correspondence between dependence logic and existential second order logic , it is easy to see that dependence logic is closed under existential quantification : for all dependence logic formulas @xmath79 over the signature @xmath80 there exists a dependence logic formula @xmath81 over the signature @xmath26 such that @xmath82 for all models @xmath7 with domain @xmath26 and for all teams @xmath19 over the free variables of @xmath8 . therefore , in the rest of this work we will add second - order existential quantifiers to the language of dependence logic , and we will write @xmath81 as a shorthand for the corresponding dependence logic expression . _ game logics _ are logical formalisms for reasoning about games and their properties in a very general setting . whereas the game theoretic semantics approach attempts to use game - theoretic techniques to _ interpret _ logical systems , game logics attempt to put logic to the service of game theory , by providing a high - level language for the study of games . they generally contain two different kinds of expressions : 1 . _ game terms _ , which are descriptions of games in terms of compositions of certain primitive _ atomic games _ , whose interpretation is presumed fixed for any given game model ; 2 . _ formulas _ , which , in general , correspond to assertions about the abilities of players in games . in this subsection , we are going to summarize the definition of a variant of dynamic game logic @xcite . from our formalism . in this , we follow @xcite . ] then , in the next subsection , we will discuss a remarkable connection between first - order logic and dynamic game logic discovered by johan van benthem in @xcite . + one of the fundamental semantic concepts of dynamic game logic is the notion of _ forcing relation : _ let @xmath83 be a nonempty set of _ states_. a _ forcing relation _ over @xmath83 is a set @xmath84 , where @xmath85 is the powerset of @xmath83 . in brief , a forcing relation specifies the abilities of a player in a perfect - information game : @xmath86 if and only if the player has a strategy that guarantees that , whenever the initial position of the game is @xmath11 , the terminal position of the game will be in @xmath19 . a ( two - player ) _ game _ is then defined as a pair of forcing relations satisfying some axioms : let @xmath83 be a nonempty set of states . a _ game _ over @xmath83 is a pair @xmath87 of forcing relations over @xmath83 satisfying the following conditions for all @xmath88 , all @xmath89 and all @xmath90 : monotonicity : : : if @xmath91 and @xmath92 then @xmath93 ; consistency : : : if @xmath94 and @xmath95 then @xmath96 ; non - triviality : : : @xmath97 . determinacy : : : if @xmath98 then @xmath99 , where @xmath100 . , this implies that the other player can force it to belong to the complement of @xmath19 . ] let @xmath83 be a nonempty set of states , let @xmath101 be a nonempty set of _ atomic propositions _ and let @xmath102 be a nonempty set of _ atomic game symbols_. then a _ game model _ over @xmath83 , @xmath101 and @xmath102 is a triple @xmath103 , where @xmath104 is a game over @xmath83 for all @xmath105 and where @xmath10 is a valutation function associating each @xmath106 to a subset @xmath107 . the language of dynamic game logic , as we already mentioned , consists of _ game terms _ , built up from atomic games , and of _ formulas _ , built up from atomic proposition . the connection between these two parts of the language is given by the _ test _ operation @xmath108 , which turns any formula @xmath8 into a test game , and the _ diamond _ operation , which combines a game term @xmath109 and a formula @xmath8 into a new formula @xmath110 which asserts that agent @xmath111 can guarantee that the game @xmath109 will end in a state satisfying @xmath8 . let @xmath101 be a nonempty set of _ atomic propositions _ and let @xmath102 be a nonempty set of _ atomic game formulas_. then the sets of all game terms @xmath109 and formulas @xmath8 are defined as @xmath112 for @xmath113 ranging over @xmath101 , @xmath114 ranging over @xmath102 , and @xmath111 ranging over @xmath115 . we already mentioned the intended interpretations of the test connective @xmath108 and of the diamond connective @xmath110 . the interpretations of the other game connectives should be clear : @xmath116 is obtained by swapping the roles of the players in @xmath109 , @xmath117 is a game in which the existential player @xmath118 chooses whether to play @xmath119 or @xmath120 , and @xmath121 is the _ concatenation _ of the two games corresponding to @xmath119 and @xmath120 respectively . let @xmath122 be a game model over @xmath83 , @xmath102 and @xmath101 . then for all game terms @xmath109 and all formulas @xmath8 of dynamic game logic over @xmath102 and @xmath101 we define a game @xmath123 and a set @xmath124 as follows : dgl - atomic - game : : : for all @xmath105 , @xmath125 ; dgl - test : : : for all formulas @xmath8 , @xmath126 , where + * @xmath127 iff @xmath128 and @xmath36 ; + * @xmath129 iff @xmath130 or @xmath36 + for all @xmath89 and all @xmath19 with @xmath131 ; dgl - concat : : : for all game terms @xmath119 and @xmath120 , @xmath132 , where , for all @xmath88 and for @xmath133 , @xmath134 , + * @xmath135 if and only if there exists a @xmath136 such that @xmath137 and for each @xmath138 there exists a set @xmath139 satisfying @xmath140 such that @xmath141 dgl-@xmath142 : : : for all game terms @xmath119 and @xmath120 , @xmath143 , where + * @xmath127 if and only if @xmath144 or @xmath145 , and * @xmath129 if and only if @xmath146 and @xmath147 + where , as before , @xmath133 and @xmath134;[multiblock footnote omitted ] dgl - dual : : : if @xmath148 then @xmath149 ; dgl-@xmath150 : : : @xmath151 ; dgl - atomic - pr : : : @xmath152 ; dgl-@xmath153 : : : @xmath154 ; dgl-@xmath39 : : : @xmath155 ; dgl-@xmath156 : : : if @xmath148 then for all @xmath8 , @xmath157 if @xmath128 , we say that @xmath8 is _ satisfied _ by @xmath11 in @xmath158 and we write @xmath35 . we will not discuss here the properties of this logic , or the vast amount of variants and extensions of it which have been developed and studied . it is worth pointing out , however , that @xcite introduced a _ concurrent dynamic game logic _ that can be considered one of the main sources of inspiration for the transition logic that we will develop in subsection [ subsect : tdl ] . in this subsection , we will briefly recall a remarkable result from @xcite which establishes a connection between dynamic game logic and first - order logic . in brief , as the following two theorems demonstrate , either of these logics can be seen as a special case of the other , in the sense that models and formulas of the one can be uniformly translated into models of the other in a way which preserves satisfiability and truth : [ theo : repfo1 ] let @xmath122 be any game model , let @xmath8 be any game formula for the same language , and let @xmath89 . then it is possible to uniformly construct a first - order model @xmath159 , a first - order formula @xmath160 and an assignment @xmath161 of @xmath159 such that @xmath162 [ theo : repfo2 ] let @xmath7 be any first order model , let @xmath8 be any first - order formula for the signature of @xmath7 , and let @xmath11 be an assignment of @xmath7 . then it is possible to uniformly construct a game model @xmath163 , a game formula @xmath164 and a state @xmath165 such that @xmath166 we will not discuss here the proofs of these two results . their _ significance _ , however , is something about which is necessary to spend a few words . in brief , what this back - and - forth representation between first order logic and dynamic game logic tells us is that it is possible to understand first order logic as a _ logic for reasoning about determined games _ ! in the next sections , we will attempt to develop a similar result for the case of dependence logic . we will now define a variant of dynamic game logic , which we will call _ transition logic_. it deviates from the basic framework of dynamic game logic in two fundamental ways : 1 . it considers _ one - player _ games against nature , instead of _ two - player games _ as is usual in dynamic game logic ; 2 . it allows for _ uncertainty _ about the initial position of the game . hence , transition logic can be seen as a _ decision - theoretic logic _ , rather than a _ game - theoretic _ one : transition logic formulas , as we will see , correspond to assertions about the abilities of a single agent acting under uncertainty , instead of assertions about the abilities of agents interacting with each other . in principle , it is certainly possible to generalize the approach discussed here to multiple agents acting in situations of imperfect information , and doing so might cause interesting phenomena to surface ; but for the time being , we will content ourselves with developing this formalism and discussing its connection with dependence logic . our first definition is a fairly straightforward generalization of the concept of forcing relation : let @xmath83 be a nonempty set of _ states_. a _ transition system _ over @xmath83 is a nonempty relation @xmath167 satisfying the following requirements : downwards closure : : : if @xmath168 and @xmath169 then @xmath170 ; monotonicity : : : if @xmath168 and @xmath171 then @xmath172 ; non - creation : : : @xmath173 for all @xmath174 ; non - triviality : : : if @xmath175 then @xmath176 . informally speaking , a transition system specifies the abilities of an agent : for all @xmath90 such that @xmath168 , the agent has a strategy which guarantees that the output of the transition will be in @xmath177 whenever the input of the transition is in @xmath19 . the four axioms which we gave capture precisely this intended meaning , as we will see : a _ decision game _ is a triple @xmath178 , where @xmath83 is a nonempty set of _ states _ , @xmath118 is a nonempty set of possible _ decisions _ for our agent and @xmath179 is an _ outcome function _ from @xmath180 to @xmath85 . if @xmath181 , we say that @xmath182 is a _ possible outcome _ of @xmath11 under @xmath183 ; if @xmath184 , we say that @xmath183 _ fails _ on input @xmath11 . let @xmath178 be a decision game , and let @xmath90 . then we say that @xmath102 _ allows _ the transition @xmath185 , and we write @xmath186 , if and only if there exists a @xmath187 such that @xmath188 for all @xmath36 ( that is , if and only if our agent can make a decision which guarantees that the outcome will be in @xmath177 whenever the input is in @xmath19 ) . a set @xmath167 is a transition system if and only if there exists a decision game @xmath178 such that @xmath189 let @xmath167 be any transition system , let us enumerate its elements @xmath190 , and let us consider the game @xmath191 , where @xmath192 suppose that @xmath168 . if @xmath193 , then @xmath194 follows at once by definition . if instead @xmath175 , by * non - triviality * we have that @xmath177 is nonempty too , and furthermore @xmath195 for some @xmath196 . then @xmath197 for all @xmath198 , as required . now suppose that @xmath186 . then there exists a @xmath196 such that @xmath199 for all @xmath36 . if @xmath175 , this implies that @xmath200 and @xmath201 . hence , by * monotonicity * and * downwards closure * , @xmath168 , as required . if instead @xmath193 , then by * non - creation * we have again that @xmath168 . conversely , consider a decision game @xmath178 . then the set of its abilities satisfies our four axioms : downwards closure : : : suppose that @xmath202 and that @xmath169 . by definition , there exists a @xmath187 such that @xmath188 for all @xmath36 . but then the same holds for all @xmath203 , and hence @xmath204 . monotonicity : : : suppose that @xmath202 and that @xmath171 . by definition , there exists a @xmath187 such that @xmath188 for all @xmath36 . but then , for all such @xmath11 , @xmath205 too , and hence @xmath206 . non - creation : : : let @xmath174 and let @xmath187 be any possible decision . then trivially @xmath188 for all @xmath207 , and hence @xmath208 . non - triviality : : : let @xmath209 , and suppose that @xmath186 . then there exists a @xmath183 such that @xmath188 for all @xmath36 , and hence in particular @xmath210 . therefore , @xmath177 is nonempty . what this theorem tells us is that our notion of transition system is the correct one : it captures precisely the abilities of an agent making choices under imperfect information and attempting to guarantee that , if the initial state is in a set @xmath19 , the outcome will be in a set @xmath177 . let @xmath83 be a nonempty set of states . a _ trump _ over @xmath83 is a nonempty , downwards closed family of subsets of @xmath83 . whereas a transition system describes the abilities of an agent to transition from a set of possible initial states to a set of possible terminal states , a trump describes the agent s abilities to reach _ some _ terminal state from a set of possible initial states : let @xmath211 be a transition system and let @xmath212 . then @xmath213 forms a trump . conversely , for any trump @xmath214 over @xmath83 there exists a transition system @xmath211 such that @xmath215 for any nonempty @xmath174 . let @xmath211 be a transition system . then if @xmath168 and @xmath169 , by downwards closure we have at once that @xmath170 . furthermore , @xmath173 for any @xmath177 . hence , @xmath216 is a trump , as required . conversely , let @xmath217 be a trump , and let us enumerate its elements as @xmath218 . then define @xmath211 as @xmath219 it is easy to see that @xmath211 is a transition system ; and by construction , for @xmath220 we have that @xmath221 , where we used the fact that @xmath214 is downwards closed . we can now define the syntax and semantics of transition logic : let @xmath101 be a set of _ atomic propositional symbols _ and let @xmath222 be a set of _ atomic transition symbols_. then a _ transition model _ is a tuple @xmath223 , where @xmath83 is a nonempty set of states , @xmath224 is a transition system over @xmath83 for any @xmath225 , and @xmath10 is a function sending each @xmath106 into a trump of @xmath83 . let @xmath101 be a set of atomic propositions and let @xmath222 be a set of atomic transitions . then the _ transition terms _ and _ formulas _ of our language are defined respectively as @xmath226 where @xmath227 ranges over @xmath222 and @xmath113 ranges over @xmath101 . let @xmath228 be a transition model , let @xmath229 be a transition term , and let @xmath90 . then we say that @xmath229 _ allows _ the transition from @xmath19 to @xmath177 , and we write @xmath230 , if and only if tl - atomic - tr : : : @xmath231 for some @xmath225 and @xmath232 ; tl - test : : : @xmath233 for some transition formula @xmath8 such that @xmath234 in the sense described later in this definition , and @xmath92 ; tl-@xmath235 : : : @xmath236 , and @xmath237 for two @xmath238 and @xmath239 such that @xmath240 and @xmath241 ; tl-@xmath242 : : : @xmath243 , @xmath244 and @xmath245 ; tl - concat : : : @xmath246 and there exists a @xmath247 such that @xmath248 and @xmath249 . analogously , let @xmath8 be a transition formula , and let @xmath250 . then we say that @xmath19 _ satisfies _ @xmath8 , and we write @xmath234 , if and only if tl-@xmath251 : : : @xmath252 ; tl - atomic - pr : : : @xmath253 for some @xmath106 and @xmath254 ; tl-@xmath39 : : : @xmath255 and @xmath256 or @xmath257 ; tl-@xmath46 : : : @xmath258 , @xmath256 and @xmath257 ; tl-@xmath156 : : : @xmath259 and there exists a @xmath177 such that @xmath230 and @xmath260 . for any transition model @xmath261 , transition term @xmath229 and transition formula @xmath8 , the set @xmath262 is a transition system and the set @xmath263 is a trump . by induction . we end this subsection with a few simple observations about this logic . first of all , we did not take the negation as one of the primitive connectives . indeed , transition logic , much like dependence logic , has an intrinsically _ existential _ character : it can be used to reason about which sets of possible states an agent _ may _ reach , but not to reason about which ones such an agent _ must _ reach . there is of course no reason , in principle , why a negation could not be added to the language , just as there is no reason why a negation can not be added to dependence logic , thus obtaining the far more powerful _ team logic _ @xcite : however , this possible extension will not be studied in this work . the connectives of transition logic are , for the most part , very similar to those of dynamic game logic , and their interpretation should pose no difficulties . the exception is the _ tensor operator _ @xmath264 , which substitutes the game union operator @xmath117 and which , while sharing roughly the same informal meaning , behaves in a very different way from the semantic point of view ( for example , it is not in general idempotent ! ) the decision game corresponding to @xmath264 can be described as follows : first the agent chooses an index @xmath265 , then he or she picks a strategy for @xmath266 and plays accordingly . however , the choice of @xmath111 may be a function of the initial state : hence , the agent can guarantee that the output state will be in @xmath177 whenever the input state is in @xmath19 only if he or she can split @xmath19 into two subsets @xmath238 and @xmath239 and guarantee that the state in @xmath177 will be reached from any state in @xmath238 when @xmath267 is played , and from any state in @xmath239 when @xmath268 is played . it is also of course possible to introduce a `` true '' choice operator @xmath269 , with semantical condition tl-@xmath142 : : : @xmath270 iff @xmath244 or @xmath245 ; but we will not explore this possibility any further in this work , nor we will consider any other possible connectives such as , for example , the iteration operator tl-@xmath271 : : : @xmath272 iff there exist @xmath273 and @xmath274 such that @xmath275 , @xmath276 and @xmath277 for all @xmath278 . this subsection contains the central result of this work , that is , the analogues of theorems [ theo : repfo1 ] and [ theo : repfo2 ] for dependence logic and transition logic . + representing dependence logic models and formulas in transition logic is fairly simple : [ defin : dl2tl - mod ] let @xmath7 be a first - order model . then @xmath279 is the transition model @xmath280 such that * @xmath83 is the set of all teams over @xmath7 ; * the set of all atomic transition symbols is @xmath281 , and hence @xmath222 is @xmath282 ; * for any variable @xmath14 , @xmath283 \subseteq y \}$ ] and @xmath284 \subseteq y\}$ ] ; * for any first - order literal or dependence atom @xmath285 , @xmath286 . [ defin : dl2tl - form ] let @xmath8 be a dependence logic formula . then @xmath287 is the transition term defined as follows : 1 . if @xmath8 is a literal or a dependence atom , @xmath288 ; 2 . if @xmath255 , @xmath289 ; 3 . if @xmath258 , @xmath290 ; 4 . if @xmath291 , @xmath292 ; 5 . if @xmath293 , @xmath294 . [ theo : tl - rep1 ] for all first - order models @xmath7 , teams @xmath19 and formulas @xmath8 , the following are equivalent : * @xmath34 ; * @xmath295 ; * @xmath296 ; * @xmath297 . we show , by structural induction on @xmath8 , that the first condition is equivalent to the last one . the equivalences between the last one and the second and third ones are then trivial . 1 . if @xmath8 is a literal or a dependence atom , @xmath298 if and only if @xmath299 , that is , if and only if @xmath34 ; 2 . @xmath300 if and only if @xmath237 for two @xmath301 such that @xmath302 and @xmath303 . by induction hypothesis , this can be the case if and only if @xmath304 and @xmath305 , that is , if and only if @xmath306 . 3 . @xmath307 if and only if @xmath308 and @xmath309 , that is , by induction hypothesis , if and only if @xmath310 . 4 . @xmath311 if and only if there exists a @xmath177 such that @xmath312 $ ] for some @xmath313 and @xmath314 . by induction hypothesis and downwards closure , this can be the case if and only if @xmath53 } \psi$ ] for some @xmath313 , that is , if and only if @xmath315 ; 5 . @xmath316 if and only if @xmath317 for some @xmath318 $ ] , that is , if and only if @xmath57 } \psi$ ] , that is , if and only if @xmath319 . one interesting aspect of this representation result is that dependence logic _ formulas _ correspond to transition logic _ transitions _ , not to transition logic _ formulas_. this can be thought of as one first hint of the fact that dependence logic can be thought of as a logic of transitions : and in the later sections , we will explore this idea more in depth . representing transition models , game terms and formulas in dependence logic is somewhat more complex : let @xmath320 be a transition model . furthermore , for any @xmath225 , let @xmath321 , and , for any @xmath106 , let @xmath322 . then @xmath323 is the first - order model with domain for the _ disjoint union _ of the sets @xmath324 and @xmath325 . ] @xmath326 whose signature contains * for every @xmath225 , a ternary relation @xmath327 whose interpretation is @xmath328 ; * for every @xmath106 , a binary relation @xmath329 whose interpretation is @xmath330 . for any transition formula @xmath8 and variable @xmath331 , the dependence logic formula @xmath332 is defined as 1 . @xmath333 is @xmath251 ; 2 . for all @xmath106 , @xmath334 is @xmath335 ; 3 . @xmath336 is @xmath337 , where @xmath338 is the classical disjunction introduced in definition [ defin : classic_or ] ; 4 . @xmath339 is @xmath340 ; 5 . @xmath341 is @xmath342 , where for any transition term @xmath229 , variable @xmath331 and unary relation symbol @xmath343 , @xmath344 is defined as 1 . for all @xmath225 , @xmath345 is @xmath346 ; 2 . for all formulas @xmath8 , @xmath347 is @xmath348 ; 3 . @xmath349 ; 4 . @xmath350 ; 5 . @xmath351 for a new and unused variable @xmath352 . [ theo : tl - rep2 ] for all transition models @xmath320 , transition terms @xmath229 , transition formulas @xmath8 , variables @xmath331 , sets @xmath353 and teams @xmath19 over @xmath323 with @xmath354 , is a set of states of the transition model . ] @xmath355 and @xmath356 the proof is by structural induction on terms and formulas . let us first consider the cases corresponding to formulas : 1 . for all teams @xmath19 , @xmath357 and @xmath358 , as required suppose that @xmath359 . then there exists a @xmath360 such that @xmath361 } v_p(j , x)$ ] . hence , we have that @xmath362 ; and , by downwards closure , this implies that @xmath363 , and hence that @xmath364 as required . + conversely , suppose that @xmath364 . then @xmath363 , and hence @xmath365 for some @xmath366 . then we have by definition that @xmath361 } v_p(j , x)$ ] , and finally that @xmath367 . 3 . by proposition [ propo : classic_or ] , @xmath368 if and only if @xmath369 or @xmath370 . by induction hypothesis , this is the case if and only if @xmath371 or @xmath372 , that is , if and only if @xmath373 . 4 . @xmath374 if and only if @xmath369 and @xmath375 , that is , by induction hypothesis , if and only if @xmath376 . @xmath377 if and only if there exists a @xmath343 such that @xmath378 and @xmath379 } \lnot py \vee ( \psi)^{dl}_y$ ] . by induction hypothesis , the first condition holds if and only if @xmath380 . as for the second one , it holds if and only if @xmath381 = y_1 \cup y_2 $ ] for two @xmath41 , @xmath42 such that @xmath382 and @xmath383 . but then we must have that @xmath384 and that @xmath385 ; therefore , by downwards closure , @xmath386 and finally @xmath387 . + conversely , suppose that there exists a @xmath343 such that @xmath380 and @xmath388 ; then by induction hypothesis we have that @xmath389 and that @xmath379 } \lnot py \vee ( \psi)^{dl}_x$ ] , and hence @xmath390 . now let us consider the cases corresponding to transition terms : 1 . suppose that @xmath391 . if @xmath193 then @xmath392 , and hence by * non - creation * we have that @xmath393 , as required . + let us assume instead that @xmath175 . then , by hypothesis , there exists a @xmath360 such that * there exists a @xmath313 such that @xmath394[f / y ] } r_t(i , x , y)$ ] ; * @xmath394[t^{dl}/y ] } \lnot r_t(i , x , y ) \vee py$ ] . + from the first condition it follows that for every @xmath395 there exists a @xmath396 such that @xmath397 : therefore , by the definition of @xmath327 , every such @xmath113 must be in @xmath398 . + from the second condition it follows that whenever @xmath397 and @xmath399 , @xmath400 ; and , since @xmath401 , this implies that @xmath402 by the definition of @xmath327 . + hence , by * monotonicity * and * downwards closure * , we have that @xmath403 and that @xmath404 , as required . + conversely , suppose that @xmath405 for some @xmath406 . if @xmath392 then @xmath193 , and hence by proposition [ propo : emptyteam ] we have that @xmath407 , as required . otherwise , by * non - triviality * , @xmath408 let now @xmath409 be any of its elements and let @xmath410 for all @xmath411 $ ] : then @xmath412[f / y ] } r_t(i , x , y)$ ] , as any assignment of this team sends @xmath331 to some element of @xmath398 and @xmath352 to @xmath413 . furthermore , let @xmath414 , and let @xmath396 be such that @xmath415 : then @xmath416 , and hence @xmath412[t^{dl}/y ] } \lnot r_t(i , x , y ) \vee py$ ] . so , in conclusion , @xmath417 , as required . 2 . @xmath418 if and only if @xmath419 and @xmath420 , that is , if and only if @xmath421 . 3 . @xmath422 if and only if @xmath237 for two @xmath423 such that * @xmath237 , and therefore @xmath424 ; * @xmath425 , that is , by induction hypothesis , @xmath426 ; * @xmath427 , that is , by induction hypothesis , @xmath428 ; + hence , if @xmath429 then @xmath430 . + conversely , if @xmath431 for two @xmath324 , @xmath325 such that @xmath432 and @xmath433 , let @xmath434 clearly @xmath237 , and furthermore by induction hypothesis @xmath425 and @xmath427 . hence , @xmath429 , as required . @xmath435 if and only if @xmath436 and @xmath437 , that is , by induction hypothesis , if and only if @xmath438 . 5 . @xmath439 if and only if there exists a @xmath440 such that @xmath441 and there exists a @xmath442 such that @xmath443 . by downwards closure , if this is the case then @xmath444 too , and hence @xmath445 , as required . + conversely , suppose that there exists a @xmath440 such that @xmath441 and @xmath444 . then , by induction hypothesis @xmath446 ; and furthermore , @xmath381 $ ] can be split into @xmath447 : s(y ) \not \in q\}\ ] ] and @xmath448 : s(y ) \in q\}\ ] ] it is trivial to see that @xmath449 ; and furthermore , since @xmath450 and @xmath444 , by induction hypothesis we have that @xmath451 . thus @xmath379 } \forall y ( \lnot qy \vee ( \tau_2)^{dl}_y(p))$ ] and finally @xmath452 , and this concludes the proof . hence , the relationship between transition logic and dependence logic is analogous to the one between dynamic game logic and first - order logic . in the next sections , we will develop variants of dependence logic which are syntactically closer to transition logic , while still being first - order : as we will see , the resulting frameworks are expressively equivalent to dependence logic on the level of satisfiability , but can be used to represent finer - grained phenomena of _ transitions _ between sets of assignments . now that we have established a connection between dependence logic and a variant of dynamic game logic , it is time to explore what this might imply for the further development of logics of imperfect information . if , as theorems [ theo : tl - rep1 ] and [ theo : tl - rep2 ] suggest , dependence logic can be thought of as a logic of imperfect - information decision problems , perhaps it could be possible to develop variants of dependence logic in which expressions can be interpreted directly as transition systems ? in what follows , we will do exactly that , first with _ transition dependence logic _ a variant of dependence logic , expressively equivalent to it , which is also a quantified version of transition logic and then with _ dynamic dependence logic _ , in which _ all _ expressions are interpreted as transitions ! but why would we interested in such variants of dependence logic ? one possible answer , which we will discuss in this subsection , is that transitions between teams are _ already _ a central object of study in the field of dependence logic , albeit in a non - explicit manner : after all , the semantics of dependence logic interprets quantifiers in terms of transformations of teams , and disjunctions in terms of decompositions of teams into subteams . this intuition is central to the study of issues of interdefinability in dependence logic and its variants , like for example the ones discussed in @xcite . as a simple example , let us recall definition [ defin : classic_or ] : @xmath453 where @xmath64 and @xmath65 are new variables . as we said in proposition [ propo : classic_or ] , @xmath454 if and only if @xmath48 or @xmath49 . we will now sketch the proof of this result , and as we will see this proof will hinge on the fact that the above expression can be read as a specification of the following algorithm : 1 . choose an element @xmath455 and extend the team @xmath19 by assigning @xmath456 as the value of @xmath64 for all assignments ; 2 . choose an element @xmath457 and further extend the team by assigning @xmath458 as the value of @xmath65 for all assignments ; 3 . split the resulting team into two subteams @xmath41 and @xmath42 such that 1 . @xmath62 holds in @xmath41 , and the values of @xmath64 and @xmath65 coincide for all assignments in it ; 2 . @xmath63 holds in @xmath42 , and the values of @xmath64 and @xmath65 differ for all assignments in it . since the values of @xmath64 and @xmath65 are chosen to always be respectively @xmath456 and @xmath458 , one of @xmath41 and @xmath42 is empty and the other is of the form @xmath459 $ ] , and since @xmath64 and @xmath65 do not occur in @xmath62 or @xmath63 the above algorithm can succeed ( for some choice of @xmath456 and @xmath458 ) only if @xmath48 or @xmath49 . as another , slightly more complicated example , let us consider the following problem . given four variables @xmath460 , @xmath461 , @xmath462 and @xmath463 , let @xmath464 be an _ exclusion atom _ holding in a team @xmath19 if and only if for all @xmath38 , @xmath465 that is , if and only if the sets of the values taken by @xmath466 and by @xmath467 in @xmath19 are disjoint . by theorem [ sigmatodl ] , we can tell at once that there exists some dependence logic formula @xmath468 such that for all suitable @xmath7 and @xmath19 , @xmath469 if and only if @xmath470 ; but what about the converse ? for example , can we find an expression @xmath471 , in the language of first order logic augmented with these exclusion atoms ( but with no dependence atoms ) , such that for all suitable @xmath7 and @xmath19 @xmath472 if and only if @xmath473 ? as discussed in @xcite in a more general setting , the answer is positive , and one such @xmath471 is @xmath474 , where @xmath475 is some variable other than @xmath331 and @xmath352 . in the second disjunct can be removed , but for simplicity we will keep it . ] why is this the case ? well , let us consider any team @xmath19 with domain containing @xmath331 and @xmath352 , and let us evaluate @xmath476 over it . as shown graphically in figure [ fig : f1 ] , the transitions between teams occurring during the evaluation of the formula correspond to the following algorithm : 1 . first , assign all possible values to the variable @xmath475 for all assignments in @xmath331 , thus obtaining @xmath477 = \{s[m / z ] : s \in x , m \in { { \texttt{dom}}}(m)\}$ ] ; 2 . then , remove from @xmath477 $ ] all assignments @xmath11 for which @xmath478 , keeping only the ones for which @xmath479 ; 3 . then , verify that for any possible fixed value of @xmath331 , the possible values of @xmath352 and @xmath475 are disjoint . this algorithm succeeds only if @xmath352 is a function of @xmath331 . indeed , suppose that instead there are two assignments @xmath38 such that @xmath480 , @xmath481 and @xmath482 for three @xmath483 with @xmath484 . now we have that @xmath485 , s[c / z ] , s'[b / z ] , s'[c / z]\ } \subseteq x[m / z]$ ] : and since @xmath484 , we have that the assignments @xmath486 $ ] and @xmath487 $ ] are not removed from the team in the second step of the proof . but then @xmath486(xz ) = a c = s'[b / z](xy)$ ] , and therefore it is not true that @xmath488 . and , conversely , if in the team @xmath19 the value of @xmath352 is a function of the value of @xmath331 then by splitting @xmath477 $ ] into the two subteams @xmath489 : s \in x , s(y ) = s(z)\}$ ] and @xmath490 : s(y ) \not = s(z)\}$ ] we have that @xmath491 , @xmath492 and @xmath493 ( since for all @xmath494 , @xmath495 ) . on the other hand , one dependence logic expression corresponding to @xmath464 is @xmath496 where @xmath497 , @xmath498 , @xmath64 and @xmath65 are new variable . we encourage the interested reader to verify that this is the case by examining the transitions between teams corresponding to the formula : in brief , the intuition is that first we extend our team by picking all possible pairs of values for @xmath497 and @xmath498 , then for any such pair we flag through our choice of @xmath64 and @xmath65 whether @xmath499 is different from @xmath466 or from @xmath467 . this implies that no such pair is equal to both @xmath466 and @xmath467 , or , in other words , that @xmath466 and @xmath467 have no value in common . more and more complex examples of definability results of this kind can be found in @xcite ; but what we want to emphasize here is that all these examples , like the one we discussed in depth here , have a natural interpretation in terms of algorithms which transform teams and apply simple tests to them , as the above one . hence , we hope that the development of variants of dependence logic in which these transitions are made explicit might prove itself useful for the further study of this interesting class of problems . as stated , we will now define a variant of dependence logic which can also be seen as a quantified variant of transition logic . we will then prove that the resulting transition dependence logic is expressively equivalent to dependence logic , in the sense that any dependence logic formula is equivalent to some transition dependence logic formula and vice versa . let @xmath26 be a first - order signature . then the sets of all _ transition terms _ and of all _ formulas _ of dependence transition logic are given by the rules @xmath500 where @xmath14 ranges over all variables in @xmath31 , @xmath28 ranges over all relation symbols of the signature , @xmath29 ranges over all tuples of terms of the required arities , @xmath501 ranges over @xmath502 and @xmath30 range over the terms of our signature . let @xmath7 be a first - order model , let @xmath229 be a first - order transition term of the same signature , and let @xmath19 and @xmath177 be teams over @xmath7 . then we say that the transition @xmath503 is _ allowed _ by @xmath229 in @xmath7 , and we write @xmath504 , if and only if tdl-@xmath50 : : : @xmath229 is of the form @xmath505 for some @xmath506 and there exists a @xmath313 such that @xmath507\subseteq y$ ] ; tdl-@xmath55 : : : @xmath229 is of the form @xmath508 for some @xmath506 and @xmath509 \subseteq y$ ] ; tdl - test : : : @xmath229 is of the form @xmath108 , @xmath34 in the sense given later in this definition , and @xmath92 ; tdl-@xmath235 : : : @xmath229 is of the form @xmath264 and @xmath237 for some @xmath238 and @xmath239 such that @xmath510 and @xmath511 ; tdl-@xmath242 : : : @xmath229 is of the form @xmath512 , @xmath513 and @xmath514 ; tdl - concat : : : @xmath229 is of the form @xmath515 and there exists a team @xmath136 such that @xmath516 and @xmath517 . similarly , if @xmath8 is a formula and @xmath19 is a team with domain @xmath31 . then we say that @xmath19 _ satisfies _ @xmath8 in @xmath7 , and we write @xmath34 , if and only if tdl - lit : : : @xmath8 is a first - order literal and @xmath35 in the usual first - order sense for all @xmath36 ; tdl - dep : : : @xmath8 is a dependence atom @xmath37 and any two @xmath38 which assign the same values to @xmath2 also assign the same value to @xmath1 ; tdl-@xmath39 : : : @xmath8 is of the form @xmath518 and @xmath519 or @xmath520 ; tdl-@xmath46 : : : @xmath8 is of the form @xmath521 , @xmath519 and @xmath520 ; tdl-@xmath156 : : : @xmath8 is of the form @xmath522 and there exists a @xmath177 such that @xmath504 and @xmath71 . as the next theorem shows , in this semantics formulas and transitions are interpreted in terms of trumps and transition systems : for all transition dependence logic formulas @xmath8 , all models @xmath7 and all teams @xmath19 and @xmath177 , we have that downwards closure : : : if @xmath34 and @xmath70 then @xmath73 ; empty team property : : : @xmath69 . furthermore , for all transition dependence logic transition terms @xmath229 , all models @xmath7 and all teams @xmath19 , @xmath177 and @xmath136 , downwards closure : : : if @xmath504 and @xmath523 then @xmath524 ; monotonicity : : : if @xmath504 and @xmath525 then @xmath526 ; non - creation : : : for all @xmath177 , @xmath527 ; non - triviality : : : if @xmath175 then @xmath528 . the proof is by structural induction over @xmath8 and @xmath229 , and presents no difficulties whatsoever . also , it is not difficult to see , on the basis of the results of the previous section , that this new variant of dependence logic is equivalent to the usual one : for every dependence logic formula @xmath8 there exists a transition dependence logic transition term @xmath529 such that @xmath530 for all first - order models @xmath7 and teams @xmath19 . @xmath529 is defined by structural induction on @xmath8 , as follows : 1 . if @xmath8 is a first - order literal or a dependence atom then @xmath531 ; 2 . if @xmath8 is @xmath518 then @xmath532 ; 3 . if @xmath8 is @xmath521 then @xmath533 ; 4 . if @xmath8 is @xmath51 then @xmath534 ; 5 . if @xmath8 is @xmath56 then @xmath535 it is then trivial to verify , again by induction on @xmath8 , that @xmath34 if and only if @xmath536 , as required . this representation result associates dependence logic _ formulas _ to transition dependence logic _ transition terms_. this fact highlights the dynamical nature of dependence logic operators , which we discussed in the previous subsection : in this framework , quantifiers describe _ transformations _ of teams , the dependence logic connectives are operations over games , and the literals are interpreted as tests . in fact , one might wonder what is the purpose of transition dependence logic formulas : could we do away with them altogether , and develop a variant of transition dependence logic in which _ all _ formulas are transitions ? later , we will explore this idea further ; but first , let us verify that transition dependence logic is no more expressive than dependence logic . for every transition dependence logic formula @xmath8 there exists a dependence logic formula @xmath537 such that @xmath538 for all first - order models @xmath7 and teams @xmath19 . furthermore , for every transition dependence logic transition term @xmath229 and dependence logic formula @xmath211 there is a dependence logic formula @xmath539 such that @xmath540 again for all first - order models @xmath7 and teams @xmath19 . we prove the two claims together , by structural induction over @xmath8 and @xmath229 . first , let us consider the cases corresponding to formulas : 1 . if @xmath8 is a first order literal or a dependence atom , let @xmath537 be @xmath8 itself . as the interpretation of these expressions is the same in dependence logic and in transition dependence logic , there is nothing to prove . 2 . if @xmath8 is of the form @xmath40 , let @xmath537 be @xmath541 . this expression holds in a team if and only if @xmath542 or @xmath543 hold , that is , by induction hypothesis , if and only if @xmath62 or @xmath63 do . if @xmath8 is of the form @xmath47 , let @xmath537 be @xmath544 . then @xmath537 holds if and only if @xmath62 and @xmath63 do , that is , if and only if @xmath8 does . 4 . if @xmath8 is of the form @xmath522 , let @xmath21 be the tuple of all variables occurring in @xmath545 , let @xmath28 be a new @xmath78-ary relation , and let @xmath537 be @xmath546 . indeed , suppose that @xmath547 : then for some relation @xmath28 , there exists a @xmath177 such that @xmath504 and @xmath548 . furthermore , @xmath549 , and therefore for the set @xmath550 we have that @xmath551 . but then , by downwards closure and locality , @xmath552 , and therefore @xmath553 . + conversely , suppose that @xmath554 : then there exists a @xmath177 such that @xmath504 and @xmath71 . now let @xmath28 be @xmath555 : clearly @xmath556 , since @xmath557 , and furthermore @xmath558 , by locality and by the fact that ( by induction hypothesis ) @xmath552 . now let us consider the cases corresponding to transitions : 1 . if @xmath229 is of the form @xmath505 for some variable @xmath14 , let @xmath559 be @xmath560 . indeed , suppose that @xmath561 : then @xmath53 } \theta$ ] for some @xmath313 , and by choosing @xmath562 $ ] we have that @xmath563 and @xmath564 , as required . conversely , suppose that for some @xmath177 , @xmath563 and @xmath564 : then for some @xmath313 , @xmath507 \subseteq y$ ] , and by downwards closure we have that @xmath53 } \theta$ ] . 2 . if @xmath229 is of the form @xmath508 for some variable @xmath14 , let @xmath559 be @xmath565 . indeed , suppose that @xmath566 : then @xmath57 } \theta$ ] , and if we choose @xmath567 $ ] we have at once that @xmath568 and @xmath564 . conversely , if for some @xmath177 @xmath568 and @xmath564 then @xmath509 \subseteq y$ ] and , by downwards closure , @xmath57 } \theta$ ] . 3 . if @xmath229 is of the form @xmath108 , let @xmath559 be @xmath569 . indeed , suppose that @xmath570 : then by induction hypothesis @xmath34 , and , for @xmath571 , we have that @xmath572 . furthermore , @xmath564 , as required . conversely , suppose that for some @xmath177 , @xmath572 and @xmath564 . then @xmath34 , and therefore @xmath547 ; and furthermore @xmath92 , and hence by downwards closure @xmath573 . hence , @xmath574 . 4 . if @xmath229 is of the form @xmath264 and @xmath21 is the tuple of all free variables of @xmath211 then let @xmath559 be @xmath575 , where @xmath28 is a new @xmath576-ary relation symbol . indeed , suppose that @xmath577 : then there exists a relation @xmath28 and two subteams @xmath238 and @xmath239 of @xmath19 such that @xmath237 , @xmath578 and @xmath579 . hence , there are two teams @xmath41 and @xmath42 such that @xmath580 , @xmath581 , @xmath582 and @xmath583 . now , let @xmath177 be @xmath584 : by monotonicity , we have that @xmath510 and @xmath511 , and furthermore @xmath557 too ( that is , for all @xmath585 , @xmath586 is in @xmath28 ) . since @xmath587 , this implies that @xmath564 , by locality and downwards closure . + conversely , suppose that there is a @xmath177 such that @xmath588 and @xmath564 . then let @xmath28 be @xmath589 . now @xmath237 for two @xmath238 and @xmath239 such that @xmath510 and @xmath511 , and by induction hypothesis we have that @xmath590 and @xmath591 . but then @xmath592 ; and furthermore , by locality we have that @xmath587 . hence , @xmath593 , as required . if @xmath229 is of the form @xmath512 and @xmath21 is the tuple of all variables of @xmath211 then let @xmath559 be @xmath594 . indeed , suppose that @xmath577 : then for some relation @xmath28 , by induction hypothesis , there exist teams @xmath41 and @xmath42 such that @xmath595 , @xmath596 , @xmath582 and @xmath583 . now let @xmath177 be @xmath584 : as before , by monotonicity we have that @xmath513 and @xmath597 , and hence @xmath598 . finally , since @xmath587 we have that @xmath599 , as required . + conversely , suppose that there is a @xmath177 such that @xmath598 and @xmath564 . since @xmath598 , @xmath513 and @xmath597 . now let @xmath28 be @xmath555 . by induction hypothesis , @xmath600 and @xmath601 ; and furthermore , since @xmath564 we have that @xmath587 . if @xmath229 is of the form @xmath515 let @xmath559 be @xmath602 . indeed , @xmath603 if and only if there is a @xmath177 such that @xmath604 and @xmath605 , that is , if and only if there are a @xmath177 and a @xmath136 such that @xmath513 , @xmath606 and @xmath607 . however , in a sense , transition dependence logic allows one to consider subtler distinctions than dependence logic does . the formula @xmath608 , for example , could be translated as any of * @xmath609 ; * @xmath610 ; * @xmath611 ; * @xmath612 . the intended interpretations of these formulas are rather different , even though they happen to be satisfied by the same teams : and for this reason , transition dependence logic may be thought of as a proper refinement of dependence logic even though it has exactly the same expressive power . _ dynamic semantics _ is the name given to a family of semantical frameworks which subscribe to the following principle ( @xcite ) : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the meaning of a sentence does not lie in its truth conditions , but rather in the way it changes ( the representation of ) the information of the interpreter . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in various forms , this intuition can be found prefigured in some of the later work of ludwig wittgenstein , as well as in the research of philosophers of language such as austin , grice , searle , strawson and others ( @xcite ) ; but its formal development can be traced back to the work of groenendijk and stokhof about the proper treatment of pronouns in formal linguistics ( @xcite ) . + we refer to @xcite for a comprehensive analysis of the linguistic issues which caused such a development , as well as for a description of the ways in which this framework was adapted in order to model presuppositions , questions / answers and other phenomena ; here we will only present a formulation of _ dynamic predicate semantics _ , the alternative semantics for first - order logic which was developed in the above mentioned paper by groenendijk and stokhof . let @xmath8 be a first - order formula , let @xmath7 be a suitable first - order model and let @xmath11 and @xmath182 be two assignments . then we say that the transition from @xmath11 to @xmath182 is _ allowed _ by @xmath8 in @xmath7 , and we write @xmath613 , if and only if dpl - atom : : : @xmath8 is an atomic formula , @xmath614 and @xmath35 in the usual sense ; dpl-@xmath153 : : : @xmath8 is of the form @xmath615 , @xmath616 and for all assignments @xmath617 , @xmath618 ; dpl-@xmath46 : : : @xmath8 is of the form @xmath47 and there exists an @xmath617 such that @xmath619 and @xmath620 ; dpl-@xmath39 : : : @xmath8 is of the form @xmath40 , @xmath614 and there exists an @xmath617 such that @xmath619 or @xmath621 ; dpl-@xmath622 : : : @xmath8 is of the form @xmath623 , @xmath614 and for all @xmath617 it holds that @xmath624 dpl-@xmath50 : : : @xmath8 is of the form @xmath625 and there exists an element @xmath626 such that @xmath627 \rightarrow s ' } \psi$ ] ; dpl-@xmath55 : : : @xmath8 is of the form @xmath628 , @xmath614 and for all elements @xmath626 there exists an @xmath617 such that @xmath627 \rightarrow h } \psi$ ] . a formula @xmath8 is _ satisfied _ by an assignment @xmath11 if and only if there exists an assignment @xmath182 such that @xmath613 ; in this case , we will write @xmath35 . we will discuss neither the formal properties of this formalism nor its linguistic applications here . all that is relevant for our purposes is that , according to it , formulas are interpreted as _ transitions _ from assignments to assignments , and furthermore that the rule for conjunction allows us to bind occurrences of a variable of the second conjunct to quantifiers occurring in the first one . : by the rules given , it is easy to see that @xmath629 if and only if @xmath630 , that is , if and only if @xmath631 , differently from the case of tarski s semantics . ] the similarity between this semantics and our semantics for transition terms should be evident . hence , it seems natural to ask whether we can adopt , for a suitable variant of dependence logic , the following variant of groenendijk and stokhof s motto : + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the meaning of a formula does not lie in its satisfaction conditions , but rather in the team transitions it allows . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ from this point of view , _ transition terms _ are the fundamental objects of our syntax , and formulas can be removed altogether from the language although , of course , the tests corresponding to literals and dependence formulas should still be available . as in groenendijk and stokhof s logic , satisfaction becomes then a derived concept : in brief , a team @xmath19 can be said to satisfy a term @xmath229 if and only if there exists a @xmath177 such that @xmath229 allows the transition from @xmath19 to @xmath177 , or , in other words , if and only if _ some _ set of non - losing outcomes can be reached from the set @xmath19 of initial positions in the game corresponding to @xmath229 . in the next section , we will make use of these intuitions to develop another , terser version of dependence logic ; and finally , we will discuss some implications of this new version for the further developments and for the possible applications of this interesting logical formalism . we will now develop a formula - free variant of transition dependence logic , along the lines of groenendijk and stockhof s dynamic predicate logic . let @xmath26 be a first - order signature . the set of all formulas of dynamic dependence logic over @xmath26 is given by the rules @xmath632 where , as usual , @xmath28 ranges over all relation symbols of our signature , @xmath29 ranges over all tuples of terms of the required lengths , @xmath501 ranges over @xmath502 , @xmath30 range over all terms , and @xmath14 ranges over @xmath31 . the semantical rules associated to this language are precisely as one would expect : [ ddl - tts ] let @xmath7 be a first - order model , let @xmath229 be a dynamic dependence logic formula over the signature of @xmath7 , and let @xmath19 and @xmath177 be two teams over @xmath7 with domain @xmath31 . then we say that @xmath229 _ allows _ the transition @xmath185 in @xmath7 , and we write @xmath504 , if and only if ddl - lit : : : @xmath229 is a first - order literal , @xmath633 in the usual first - order sense for all @xmath36 , and @xmath92 ; ddl - dep : : : @xmath229 is a dependence atom @xmath37 , @xmath92 , and any two assignments @xmath38 which coincide over @xmath2 also coincide over @xmath1 ; ddl-@xmath50 : : : @xmath229 is of the form @xmath505 for some @xmath506 , and @xmath507 \subseteq y$ ] for some @xmath52 ; ddl-@xmath55 : : : @xmath229 is of the form @xmath508 for some @xmath506 , and @xmath509 \subseteq y$ ] ; ddl-@xmath235 : : : @xmath229 is of the form @xmath264 and @xmath237 for two teams @xmath238 and @xmath239 such that @xmath510 and @xmath511 ; ddl-@xmath242 : : : @xmath229 is of the form @xmath512 , @xmath513 and @xmath597 ; ddl - concat : : : @xmath229 is of the form @xmath515 , and there exists a @xmath136 such that @xmath516 and @xmath517 . a formula @xmath229 is said to be _ satisfied _ by a team @xmath19 in a model @xmath7 if and only if there exists a @xmath177 such that @xmath504 ; and if this is the case , we will write @xmath634 . it is not difficult to see that dynamic dependence logic is equivalent to transition dependence logic ( and , therefore , to dependence logic ) . let @xmath8 be a dependence logic formula . then there exists a dynamic dependence logic formula @xmath635 which is equivalent to it , in the sense that @xmath636 for all suitable teams @xmath19 and models @xmath7 we build @xmath635 by structural induction : 1 . if @xmath8 is a literal or a dependence atom then @xmath637 ; 2 . if @xmath8 is @xmath40 then @xmath638 ; 3 . if @xmath8 is @xmath47 then @xmath639 ; 4 . if @xmath8 is @xmath625 then @xmath640 ; 5 . if @xmath8 is @xmath628 then @xmath641 . let @xmath229 be a dynamic dependence logic formula . then there exists a transition dependence logic transition term @xmath642 such that @xmath643 for all suitable @xmath19 , @xmath177 and @xmath7 , and such that hence @xmath644 build @xmath642 by structural induction : 1 . if @xmath229 is a literal or dependence atom then @xmath645 ; 2 . if @xmath229 is of the form @xmath505 or @xmath508 then @xmath646 ; 3 . if @xmath229 is of the form @xmath264 then @xmath647 ; 4 . if @xmath229 is of the form @xmath512 then @xmath648 ; 5 . if @xmath229 is of the form @xmath515 then @xmath649 . dynamic dependence logic is equivalent to transition dependence logic and to dependence logic follows from the two previous results and from the equivalence between dependence logic and transition dependence logic . in this work , we established a connection between a variant of dynamic game logic and dependence logic , and we used it as the basis for the development of variants of dependence logic in which it is possible to talk directly about transitions from teams to teams . this suggests a new perspective on dependence logic and team semantics , one which allow us to study them as a special kind of _ algebras of nondeterministic transitions between relations_. one of the main problems that is now open is whether it is possible to axiomatize these algebras , in the same sense in which , in @xcite , allen mann offers an axiomatization of the algebra of trumps corresponding to if logic ( or , equivalently , to dependence logic ) . furthermore , we might want to consider different choices of connectives , like for example ones related to the theory of database transactions . the investigation of the relationships between the resulting formalisms is a natural continuation of the currently ongoing work on the study of the relationship between various extensions of dependence logic , and promises of being of great utility for the further development of this fascinating line of research . the author wishes to thank johan van benthem and jouko vnnen for a number of useful suggestions and insights . furthermore , he wishes to thank the reviewers for a number of highly useful suggestions and comments . hintikka , j. and g. sandu : 1989 , ` informational independence as a semantic phenomenon ' . in : j. fenstad , i. frolov , and r. hilpinen ( eds . ) : _ logic , methodology and philosophy of science_. elsevier , pp . 571589 . kontinen , j. and v. nurmi : 2009 , ` team logic and second - order logic ' . in : h. ono , m. kanazawa , and r. de queiroz ( eds . ) : _ logic , language , information and computation _ , vol . 5514 of _ lecture notes in computer science_. springer berlin / heidelberg , pp . 230241 . parikh , r. : 1985 , ` the logic of games and its applications ' . in : _ selected papers of the international conference on `` foundations of computation theory '' on topics in the theory of computation_. new york , ny , usa , pp . 111139 . vnnen , j. : 2007b , ` team logic ' . in : j. van benthem , d. gabbay , and b. lwe ( eds . ) : _ interactive logic . selected papers from the 7th augustus de morgan workshop_. msterdam university press , pp .
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we examine the relationship between dependence logic and game logics .
a variant of dynamic game logic , called _ transition logic _ , is developed , and we show that its relationship with dependence logic is comparable to the one between first - order logic and dynamic game logic discussed by van benthem .
this suggests a new perspective on the interpretation of dependence logic formulas , in terms of assertions about _ reachability _ in games of imperfect information against nature .
we then capitalize on this intuition by developing expressively equivalent variants of dependence logic in which this interpretation is taken to the foreground .
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if one combines today s standard model of particle physics ( sm ) and that of cosmology , one finds inevitably that particles and their antiparticles annihilate at a very early moment in the evolution of the universe , leaving just radiation behind . the absence of a sizable matter - antimatter asymmetry at this epoch would imply that the universe as we know it could never form . the question about the origin of the observed asymmetry therefore represents a major challenge for modern physics . in the sm baryon and lepton number are ( accidental ) global symmetries . if baryon number was also conserved in the early universe a dynamical emergence of the asymmetry would have been impossible . in grand - unified extensions ( guts ) of the sm baryon number ( and also lepton number ) is explicitly broken . according to past reasoning , this could provide a solution to the apparent discrepancy . in the class of ` gut - baryogenesis ' scenarios the matter - antimatter imbalance is generated by asymmetric decays of new super - heavy bosons . anomalous electroweak processes @xcite ( sphalerons ) which violate baryon and lepton number but conserve their difference essentially eliminated the prospects for gut - baryogenesis @xcite . at the same time , it inspired the now widely appreciated scenarios of ` electroweak baryogenesis ' @xcite and ` baryogenesis via leptogenesis ' @xcite . according to the latter scenario , the asymmetry is initially generated in the leptonic sector by the decay of heavy majorana neutrinos at an energy scale far above the electroweak scale . subsequently it is converted into the observed baryon asymmetry by sphalerons . the mass scale of the heavy majorana neutrinos required for leptogenesis @xcite fits together very well with the mass - differences inferred from observations of solar- , atmospheric- and reactor - neutrino oscillations . we focus here on the conventional , but most popular , high - energy ( type - i ) seesaw extension : @xmath0 where @xmath1 are the heavy majorana fields , @xmath2 are the lepton doublets , @xmath3 is the conjugate of the higgs doublet , and @xmath4 are the corresponding yukawa couplings . the majorana mass term violates lepton number and the yukawa couplings can violate _ therefore the model fulfills essential requirements for baryogenesis @xcite . they can also be realized for more complicated sm extensions and a wide range of values for couplings and neutrino masses @xcite . in general the right - handed neutrinos do not necessarily get into thermal equilibrium and _ cp_-violating oscillations between them can contribute to the asymmetry . this effect of leptogenesis through neutrino oscillations @xcite is crucial for neutrino - minimal extensions of the sm ( @xmath5msm ) @xcite and poses interesting questions for non - equilibrium quantum field theory @xcite . in the considered scenario of thermal leptogenesis the heavy majorana neutrinos experience only a moderate deviation from thermal equilibrium at the time when the bulk of the asymmetry is produced . also , for a hierarchical mass spectrum , effects related to oscillations are negligible . the amount of the generated asymmetry is determined by the out of equilibrium evolution of the heavy majorana neutrinos . therefore , statistical equations for the abundance of the neutrinos and the generated asymmetry are needed . the conventional approach here follows the lines developed for gut - baryogenesis @xcite . cp_-violating amplitudes for the decay and scattering processes involving the heavy majorana neutrinos are computed in terms of feynman graphs at lowest loop order . they are used to build generalized boltzmann collision terms for these processes . each of them contributes to the evolution of the distributions of majorana neutrinos and leptons or , upon momentum integration , their entire abundances . however this approach is plagued by the so - called double - counting problem which manifests itself in the generation of a non - vanishing asymmetry even in thermal equilibrium . this technical issue is expression of the fact that the ` naive ' generalization of the collision terms is quantitatively inexact , and inconsistent in the presence of _ cp_-violation . after a real intermediate state ( or ris ) subtraction procedure and a number of approximations , it can be made consistent with fundamental requirements . nevertheless this pragmatic solution remains unsatisfactory . the requirement of unitarity guarantees a consistent approximation for the amplitudes , realized by the ris subtraction , if the statistical system is in thermal equilibrium . however , the deviation from equilibrium is a fundamental requirement for leptogenesis and it is not obvious how the equations have to be generalized for a system out of equilibrium . furthermore , the _ cp_-violation arises from one - loop contributions due to the exchange of virtual quanta . as such they seem to be beyond a boltzmann approximation . but the relevant imaginary part is due to intermediate states in which at least some of the particles are on - shell . these can also be absorbed or emitted by the medium and it is not obvious how such contributions enter the amplitudes . it is , however , clear that the influence of medium effects on the one - loop contributions enters directly the _ cp_-violating parameter and therefore the source for the lepton asymmetry . their size can be of the same order as that of the vacuum contributions . those questions can be addressed within a first - principle approach based on non - equilibrium quantum field theory ( neqft ) . several aspects of leptogenesis have already been investigated within this approach @xcite . the influence of medium effects on the generation of the asymmetry has been studied e.g. in @xcite , and an analysis with special emphasis on off - shell effects was performed in @xcite . the role of flavor effects as well as the range of applicability of the conventional approach to the analysis of flavored leptogenesis has been investigated in @xcite . the resonant enhancement of the lepton asymmetry has been addressed within a first - principle approach in @xcite . in addition , steps towards a consistent inclusion of gauge interactions have been taken @xcite . in this work we use the 2pi - formalism of neqft to derive boltzmann - like quantum kinetic equations for the lepton asymmetry . in particular , we show how two - body scattering processes that violate lepton number by two units and contribute to the washout of the asymmetry emerge within the 2pi - formalism . this approach treats quantum field theory and the out of equilibrium evolution on an equal footing and allows to overcome the conceptional difficulties inherent in the conventional approach . it allows us to obtain quantum - generalized boltzmann equations which include medium effects and which are free of the double - counting problem . in other words , the structure of the obtained quantum kinetic equations automatically ensures that the asymmetry vanishes in thermal equilibrium and no need for ris subtraction arises . the resulting equation for the lepton asymmetry @xmath6 is given by @xmath7 together with the ` effective amplitudes ' @xmath8 this is the main result of this paper . in eq . we introduced @xmath9 { { \ , , } } \end{aligned}\ ] ] with @xmath10 for fermions ( bosons ) . note that @xmath11 vanishes in equilibrium due to detailed balance . this ensures that the asymmetry vanishes in thermal equilibrium as mentioned before . the effective amplitudes contain medium effects ignored in the corresponding canonical expressions . we find that , in the amplitudes of the scattering processes medium effects are sub - dominant and can be neglected . the total decay amplitude of the majorana neutrino is barely affected as well . however , at high temperatures the available phase space shrinks when taking gauge interactions in the form of effective thermal masses of higgs and leptons into account . this leads to a suppression of the decay and scattering rates . since the _ cp_-violation appears as loop effect it is more sensitive to influences of the surrounding medium . even though there is a partial cancellation of the fermionic and bosonic contributions , the _ cp_-violating parameter is enhanced by medium effects . however , the thermal masses reduce the enhancement and turn it into suppression at high temperatures . we review the conventional approach to leptogenesis based on ris subtraction in section[conventionalapproach ] . in section [ risquantstat ] we demonstrate explicitly that _ in thermal equilibrium _ the success of this procedure is guaranteed by the requirement of unitarity . in section[rateequations ] we review the derivation of rate equations for total abundances and discuss in how far quantum statistical and medium corrections can be incorporated in the reaction densities . in section[neqftapproach ] we review the application of the 2pi approach of neqft to leptogenesis . equation and explicit expressions for the effective in - medium decay and scattering amplitudes are derived within this framework in section[majorana ] . we compare the results obtained within the 2pi - formalism to those of a conventional analysis with manual ris subtraction . in section[higgsdecay ] we derive rate equations and the _ cp_-violating amplitudes for higgs decay within the framework of neqft . finally , we summarize the results and present our conclusions in sec.[summary ] . the amount of produced asymmetry depends on the details of the non - equilibrium evolution of the majorana neutrinos as well as on the strength of _ cp_-violation . the latter is usually quantified by _ cp_-violating parameters@xcite : @xmath12 where @xmath13 and @xmath14 are the vacuum decay rates to a particle or anti - particle pair respectively.for a hierarchical mass spectrum @xmath15 can be computed perturbatively as the interference tree - level , one - loop self - energy and one - loop vertex contributions to the decay of the heavy majorana neutrino . ] of the tree - level , one - loop vertex @xcite and one - loop self - energy @xcite amplitudes in fig.[treevertexself ] . the contribution of the loop diagrams can be accounted for by effective yukawa couplings @xcite : [ effective couplings ] @xmath16 where the loop function @xmath17 is defined as @xmath18{{\ , .}}\nonumber\end{aligned}\ ] ] the first term in eq . is related to the self - energy and the second term to the vertex contribution . the decay widths are proportional to the absolute values of the effective couplings , @xmath19 and @xmath20 respectively , where we have summed over flavors of the leptons and @xmath21 indices ( hence the factor @xmath22 ) in the final state . since the phase space for the decay into particles and antiparticles is the same , one gets for the _ cp_-violating parameter : @xmath23 let us note in passing that the divergence of the loop function for @xmath24 is not physical and can be removed by a resummation of the self - energy contribution @xcite . here we work in a regime where the mass splittings @xmath25 are large enough to render effects related to the enhancement of the self - energy contribution irrelevant ( non - resonant leptogenesis ) . we do not require a strictly hierarchical mass - spectrum , however . to describe the statistical evolution of the lepton asymmetry one usually employs generalized boltzmann equations for the one - particle distribution functions of the different species @xcite . taking into account decay and inverse decay processes one writes for the distribution function of the leptons ( for a single flavor ) : @xmath26{{\ , , } } \label{eqn : collision terms decay}\end{aligned}\ ] ] where @xmath27 $ ] is the invariant phase space element , @xmath28 denotes spin degrees of freedom of @xmath29 , and @xmath30 is the covariant derivative . the corresponding equation for antileptons may be obtained by interchanging @xmath31 and @xmath32 . _ cpt_-invariance implies that @xmath33 and @xmath34 . furthermore , in _ thermal equilibrium _ detailed balance requires that @xmath35 . subtracting the two relations we find for the contribution of the ( inverse ) decay terms : @xmath36{{\ , .}}\end{aligned}\ ] ] if the decay amplitudes in square brackets differ , the right - hand side of eq . represents the ( non - zero ) _ cp_-violating source term for the asymmetry generation . the total asymmetry is given by the sum over all flavors and @xmath21 components : @xmath37 . neglecting the quantum - statistical terms , @xmath38 , and integrating eq . over the lepton phase space we obtain for its time derivative : @xmath39 where @xmath40 is the modified bessel function of the second kind , @xmath41 is the total tree - level decay width of @xmath29 , and the factor @xmath42 emerges from the sum over the majorana spin degrees of freedom in , see appendix[kinematics ] for more details . this implies that the source - term for the lepton asymmetry differs from zero even in equilibrium . on the other hand , combined with time translational invariance of an equilibrium state , _ cpt_-invariance requires the asymmetry to vanish in thermal equilibrium . thus , we arrive at an apparent contradiction . the generation of an asymmetry in equilibrium within the @xmath43-matrix formalism is a manifestation of the so - called double - counting problem . in vacuum an inverse decay immediately followed by a decay is equivalent to a scattering process where the intermediate particle is on the mass shell ( real intermediate state or ris ) . two - body scattering process @xmath44 . both graphs contribute with all @xmath29 as intermediate states . note that we read ( b ) as t - channel contribution . ] thus , the same contribution is taken into account twice : once by the amplitude for ( inverse ) decay processes , and once by that for the @xmath45 scattering processes , see fig.[lhbarlbarh].(a ) . let us convince ourselves that this is indeed the case . including scattering processes we have for the distribution function of the leptons : @xmath46{{\ , , } } \end{aligned}\ ] ] where the dots denote the contribution of the ( inverse ) decay processes , the sum is over flavors and @xmath21 components of the antileptons and we have introduced @xmath47 to shorten the notation . in the unflavored regime , to which we restrict our analysis , the distribution functions of leptons of all flavors are equal . if the majorana neutrinos are close to equilibrium the difference between the distribution functions of the two spin degrees of freedom can be neglected as well . therefore , in the expression for the total asymmetry @xmath48 the summation over spin , flavor and @xmath21 components reduces to summation of the corresponding decay and scattering amplitudes . we will denote these sums over internal degrees of freedom by @xmath49 and call them effective amplitudes in the following . for the effective amplitude of @xmath50 scattering one obtains @xcite @xmath51\nonumber{{\ , , } } \end{aligned}\ ] ] where @xmath52 are the momenta of initial and final leptons respectively , and @xmath53 and @xmath54 are the usual mandelstam variables . the amplitude @xmath55 is obtained by interchanging @xmath56 . note that the loop corrections to the yukawas vanish for negative momentum transfer , i.e. in the @xmath54-channel . for this reason the above scattering amplitude contains combinations of the yukawa couplings and their one - loop corrected counterparts . the propagators @xmath57 are given by @xmath58 where @xmath59 is the heaviside step function . the ris contribution appears for the flavor diagonal ( @xmath60 ) terms in the product of the @xmath53-channel amplitudes since only in this case the @xmath61 terms vanish simultaneously in both @xmath57 and @xmath62 . in other words : @xmath63 and a similar result for @xmath64 . using the definitions of the effective couplings and the expression for the _ cp_-violating parameter we find that @xmath65 . furthermore , for a small decay width , we can approximate the breit - wigner propagator by a delta - function using , @xmath66 ^ 2}= 2\pi\delta(\omega){{\ , , } } \end{aligned}\ ] ] where @xmath67 and @xmath68 in the considered case . the ris contribution to the scattering amplitude then takes the form @xmath69 where @xmath70 is the decay amplitude squared summed over all internal degrees of freedom ( and a similar expression for anti - particles ) . just as one would expect , it is proportional to the product of the corresponding inverse decay and decay amplitudes . the additional momentum dependence ( momenta of the leptons ) arises because the initial and final states contain fermions . close to thermal equilibrium @xmath71 and @xmath72 . neglecting the quantum - statistical terms we can write the ris contribution to the source - term as : @xmath73{{\ , .}}\end{aligned}\ ] ] taking into account that with maxwell - boltzmann distributions @xmath74 in the presence of the dirac - delta and performing the phase space integration using eq . we obtain a result _ identical _ to eq .. to correct the double - counting _ in equilibrium _ we may therefore subtract the ris contribution from the scattering amplitude : @xmath75 and similarly for the conjugate process @xmath76 . at first sight it might seem that the ris subtracted scattering amplitudes @xmath77 do not contribute to the generation of the lepton asymmetry in equilibrium , @xmath78{{\ , , } } \end{aligned}\ ] ] but also can not compensate the asymmetry generated in equilibrium by the decay processes , see eq .. however , upon phase space integration the difference of the unsubtracted scattering amplitudes vanishes at leading order in @xmath4 . the remaining difference of the ris - amplitudes precisely compensates the contribution of the ( inverse ) decay processes . the ris subtracted scattering amplitude can be conveniently rewritten in terms of a ` ris subtracted propagator ' @xmath79 . motivated by eq . we define its diagonal components such , that they vanish upon integration over @xmath53 in the vicinity of the mass pole : @xmath80 ^ 2}{{\ , .}}\end{aligned}\ ] ] since the second of the expressions approaches the delta - function faster than the first it is common to write eq . in the form @xmath81 for @xmath82 there is no need to perform the ris subtraction and therefore @xmath83 . in the following we will also need the sum of the ris subtracted tree - level scattering amplitudes . it does not contribute to the generation of the asymmetry but plays a role for its washout . it is defined as @xmath84\nonumber\\ & = 4({{r}}_1 { { r}}_2 ) \textstyle\sum_{ij } m_i m_j\,\re ( h^\dagger h)^2_{ij } \nonumber\\ & \times \left[\,2{\cal p}_{ij}(s)+2p^{*}_i(t)p_j(t)\right.\nonumber\\ & + \left.p^{*}_i(s)p_j(t)+p^*_i(t)p_j(s)\,\right]{{\ , .}}\end{aligned}\ ] ] since it contains only the real part of @xmath85 this process is _ cp_-conserving . two - body scattering process @xmath86 . ] a further important washout process is @xmath87 scattering which receives the @xmath54- and @xmath88-channel contributions , see fig.[llbarhbarh ] . by analogy with eq . it is convenient to introduce @xmath89 \nonumber\\ & = 2({{r}}_1 { { r}}_2)\textstyle\sum_{ij } m_i m_j\,\re(h^\dagger h)^2_{ij } \nonumber\\ & \times \left[2 p^{*}_i(t)p^{*}_j(t)+2 p^{*}_i(u)p^{*}_j(u)\right.\nonumber\\ & + \left . p^{*}_i(u ) p_j(t)+p^{*}_i(t ) p_j(u)\,\right]{{\ , .}}\end{aligned}\ ] ] since the intermediate majorana neutrino can not go on - shell in the @xmath54- and @xmath88-channel , there is no need to use the ris subtracted propagator in eq .. above we have briefly reviewed the canonical approach to the computation of the lepton asymmetry , which is based on generalized boltzmann equations . boltzmann equations , according to conventional reasoning , describe scattering processes of particles which propagate freely over timescales large compared to the duration of individual interactions . this picture seems to be consistent with the use of @xmath43-matrix elements which are intended to describe transitions between asymptotically free initial and final states . however , in leptogenesis the crucial processes ( _ cp_-violating decays ) involve unstable particles which spoils this picture . in vacuum the amplitudes for such processes can be computed in terms of their feynman graphs . however the ` naive ' way of generalizing the boltzmann equation by multiplying the obtained amplitudes by the one - particle distributions of the initial states and integrating over phase space leads to inconsistent equations . the origin of this problem is that the obtained collision terms for particle decay and inverse decay in eq . miscount the rate of particle generation . in a short time - interval a finite number of unstable majorana neutrinos - formed by inverse decay of particles and antiparticles - decays immediately back to either particles or antiparticles . these contributions to particle generation are not included in eq . where the amplitudes are defined in terms of feynman graphs . for leptogenesis , in the presence of _ cp_-violation , it leads to inconsistent equations and must be corrected . since the missing contribution can be constructed as the rate of a two - body scattering process with on - shell intermediate state this issue can be addressed by the ris - subtracting procedure presented above . it modifies the amplitudes for two - body scattering in order to cure the problem which appears due to the collision terms for particle decay . however the corresponding correction appears at order @xmath90 , which is usually neglected . ] it is well known that unitarity has important consequences for baryogenesis and leptogenesis @xcite as it implies restrictions for the _ cp_-violating amplitudes . the issue of ris subtraction is as well tightly related to unitarity as has been mentioned in e.g. @xcite . as noted in section[conventionalapproach ] , the use of ` naive ' boltzmann equations of the kind for unstable particles leads to problems such as the spurious asymmetry generation in the presence of _ cp_-violation in the decay of the heavy neutrinos . in this section we show explicitly that the success of the ris subtraction _ in thermal equilibrium _ is guaranteed by the unitarity of the @xmath43-matrix and how it can be generalized to include quantum - statistical terms . the approach to ris subtraction differs slightly from the one discussed in the previous section . to illustrate it we work in thermal equilibrium , @xmath91 and @xmath92 , where @xmath93 subtracting from the boltzmann equation the corresponding equation for antiparticles , summing over internal degrees of freedom of the leptons and integrating with @xmath94 $ ] we obtain in thermal equilibrium : contribution which describes the dilution due to the expansion of the universe . ] @xmath95 { ( 1-{{f}_{{{\ell}}}^{eq}})}{(1+{{f}_{{\phi}}^{eq}})}{{f}_{{n}_i}^{eq}}\nonumber\\ & + 2 \int { { { \,d}\pi_{{{\ell}}{\phi}{{\ell}}{\phi}}^{{{r}}_1 k_1 p_2 k_2 } } } ( 2\pi)^4\ , \delta({{r}}_1 + k_1 - p_2 - k_2 ) \nonumber\\ & \times \bigl [ { \xi^{'}_{\bar{{{\ell}}}\bar{{\phi}}\rightarrow{{\ell}}{\phi } } } - { \xi^{'}_{{{\ell}}{\phi}\rightarrow\bar{{{\ell}}}\bar{{\phi}}}}\bigr ] { ( 1-{{f}_{{{\ell}}}^{eq}})}{(1+{{f}_{{\phi}}^{eq}})}{{f}_{{{\ell}}}^{eq}}{{f}_{{\phi}}^{eq}}\nonumber{{\ , .}}\end{aligned}\ ] ] we can exploit the unitarity of the @xmath43-matrix and _ cpt_-symmetry to obtain a requirement for a consistent approximation of the decay and scattering amplitudes . to this end we multiply eq . , which follows from the generalized optical theorem at order @xmath96 , by @xmath97 and integrate over @xmath98 . assuming maxwell - boltzmann equilibrium distributions we may use @xmath99 in the presence of the energy conserving dirac - delta on the right - hand - side : @xmath100{{f}_{{n}_i}^{eq}}\nonumber\\ & = -\int { { { \,d}\pi_{{{\ell}}{\phi}{{\ell}}{\phi}}^{{{r}}_1 k_1 p_2 k_2 } } } ( 2\pi)^4{\delta({{r}}_1 + k_1 - p_2 - k_2 ) } \nonumber\\ & \hspace{15mm}\times\big [ { \xi^{'}_{\bar{{{\ell}}}\bar{{\phi}}\rightarrow{{\ell}}{\phi } } } - { \xi^{'}_{{{\ell}}{\phi}\rightarrow\bar{{{\ell}}}\bar{{\phi}}}}\big]{{f}_{{{\ell}}}^{eq}}{{f}_{{\phi}}^{eq}}{{\ , .}}\end{aligned}\ ] ] we see that imposing this as a condition for the scattering amplitudes will correctly yield @xmath101 if we neglect the quantum - statistical terms in . equation represents the zeroth - order term in an expansion about equilibrium . using eq . we can therefore obtain consistent equations at this order without the need to specify the detailed form of @xmath102 and @xmath103 . at higher order ( for washout contributions ) we also need to know the sum @xmath104 , see section[rateequations ] . we know from section[conventionalapproach ] that relation can be satisfied by subtracting ris contributions from the tree - level two - body scattering amplitudes and taking the zero width limit : [ eqn : matrix element ris subtracted ] @xmath105 note that , strictly speaking , the ris terms in eq . include @xmath106 factors . however , upon the phase space integration in eq . the two expressions give identical results and are therefore equal in an average sense . it is obvious from comparison of eqs . and that the above definition of the ris subtracted scattering amplitudes is not sufficient to guarantee zero asymmetry in equilibrium if quantum - statistical terms are included . however this can be achieved if we replace the vacuum decay width in eq . by the thermal one @xcite : @xmath107 ( 1-{{f}_{{{\ell}}}^{eq}}+{{f}_{{\phi}}^{eq}}){{\ , .}}\end{aligned}\ ] ] using the identity @xmath108 and the fact that the ( inverse ) decay amplitudes are related by _ cpt_-symmetry we can rewrite the ris - contribution to the second term of eq . in the form : @xmath109\nonumber\\ & \times \int { { \,d}\pi^{{{\ell}}}_{{{r}}_2 } } { { \,d}\pi^{{\phi}}_{{{k}}_2}}(2\pi)^4\delta(q-{{r}}_2-{{k}}_2)\nonumber\\ & \hspace{5 mm } \times ( 1-{{f}_{{{\ell}}}^{eq}})(1+{{f}_{{\phi}}^{eq}})\bigl [ { \xi^{}_{{n}_i\rightarrow{{\ell}}{\phi } } } - { \xi^{}_{{n}_i\rightarrow\bar{{{\ell}}}\bar{{\phi}}}}\bigr]{{\ , .}}\end{aligned}\ ] ] the integration over @xmath53 is trivial . the @xmath110 term ensures that after integration over @xmath111 the intermediate majorana neutrino is on - shell , @xmath112 . using @xmath113 together with the definition we can rewrite the second term of eq . as @xmath114 which cancels the factors coming from ris subtraction . the resulting expression reads @xmath115 { ( 1-{{f}_{{{\ell}}}^{eq}})}{(1+{{f}_{{\phi}}^{eq}})}{{f}_{{n}_i}^{eq}}{{\ , , } } \end{aligned}\ ] ] and cancels the first term on the right - hand side of eq .. since @xmath116 at @xmath117 the new ris subtracted source - term for the asymmetry vanishes in equilibrium . the thermal width @xmath118 defined in eq . would also be obtained if one computes it using thermal cutting rules instead of the optical theorem ( which applies in vacuum ) , see appendix[generalizedoptth ] . we have seen that the unitarity of the @xmath43-matrix can be employed to generalize the concept of ris subtraction to rate equations which include quantum - statistical factors . as we shall see in section[higgsdecay ] , the majorana neutrino decay is at high temperature replaced by higgs decay if the higgs acquires a large effective thermal mass . in this case thermal cutting rules enforce relations between the amplitudes which can be used to obtain consistent equations , analogous to the optical theorem , see appendix[generalizedoptth ] . note again that in eq . we had to assume that the majorana neutrinos are in exact thermal equilibrium . for leptogenesis this is an inconsistent assumption since the deviation of their distribution from equilibrium realizes the third sakharov condition and drives the generation of the asymmetry . not surprisingly , the neqft approach leads to a ( slightly ) different result for the kinetic equations . however the differences between the two approaches enter only at an order beyond the usual approximation as we will discuss in the next section . in this section we review the derivation of rate equations , discuss in how far quantum - statistical and medium corrections can be incorporated , and compare the structure obtained when starting from the neqft result with the conventional form . solving a system of boltzmann - like equations in general requires the use of numerical codes capable of treating large systems of stiff differential equations for the different momentum modes a cumbersome task if one wants to study a wide range of model parameters . in the context of baryogenesis , a commonly employed simplification is to approximate the boltzmann equations by the corresponding network of ` rate equations ' for number densities @xmath119 or abundances @xmath120 , where @xmath53 is the comoving entropy density . the resulting equations correspond to the hydrodynamical limit of the boltzmann kinetic equations , in the comoving frame of _ homogeneous _ frw space - time . to obtain evolution equations for @xmath121 in the conventional approach , i.e. from eq . , we therefore integrate the corresponding boltzmann equations over @xmath122 $ ] to obtain , on the left - hand sides : @xmath123 where we have introduced the dimensionless inverse temperature @xmath124 and the hubble rate @xmath125 . in the homogeneous and isotropic universe the derivative of the quantity @xmath126 can be related to the divergence of the lepton - current @xmath127 a quantity which is particularly easy to access in the first - principles computation by @xmath128 on the right - hand sides we get sums of integrated collision terms representing the effect of the different interactions . we separate contributions attributed to decays and scattering : @xmath129 the decay contributions @xmath130 to @xmath131 are very similar to the decay contributions @xmath132 to @xmath133 and we can treat them in the same way . reordering the contributions to @xmath134 we find @xmath135{{\ , , } } \nonumber\end{aligned}\ ] ] where the upper ( lower ) signs and arrows correspond to the rate equations for @xmath136 ( @xmath29 ) abundance and we defined @xmath137\nonumber{{\ , , } } \end{aligned}\ ] ] which corresponds to eq . , as well as @xmath138{{\ , .}}\end{aligned}\ ] ] we used _ cpt_-symmetry of the amplitudes in the derivation of eqs . and . later we will see that the second term in eq . appears also in the first - principle approach , compare eq . , while the terms in eqs . and are absent . this motivates the separation into ` regular ' and ` extra ' terms performed in eq .. for the contributions attributed to scattering we get : @xmath139{{\ , , } } \nonumber\end{aligned}\ ] ] with @xmath140{{\ , , } } \nonumber\end{aligned}\ ] ] corresponding to eq .. again , eq . does not appear in the first - principle approach . since in equilibrium the regular terms in each of eqs . and vanish by detailed balance we retain eq . in the sum of decay and scattering contributions . the latter vanishes as well in equilibrium if we adopt e.g. eq . with thermal width for the ris subtracted amplitudes @xmath141 . out of equilibrium the last terms constitute a structural difference compared to the results obtained from first - principles . this difference carries over to the rate equations . we will therefore analyze these contributions separately . the computational advantage of rate equations over full boltzmann equations is maximized by a number of common approximations . in particular , assuming that all species are close to equilibrium and that the majorana neutrino distribution function @xmath142 is proportional to its equilibrium distribution for all values of the momentum @xmath143 . the temperature for all kinetic equilibrium distributions is set to a common value @xmath144 while finite deviations of the chemical potential with small @xmath145 are permitted . these approximations result for @xmath146 in a closed network of rate equations for the abundances of the form ( compare with @xcite ) : [ rateequations1 ] @xmath147 where we have introduced [ amplsqandepsdef ] @xmath148 the factor @xmath149 ( we neglect the thermal lepton masses here ) relates the chemical potential of the leptons to their number density , @xmath150 and the coefficient @xmath151 takes into account that in the sm the chemical potentials of leptons and higgs are related by @xmath152 with @xmath153 through equilibrium gauge , yukawa and sphaleron interactions @xcite . hence , the evolution of the abundances close to equilibrium is roughly governed by a few average quantities called _ reaction densities _ which describe decay and scattering processes . we will refer to @xmath154 , @xmath155 , @xmath156 as _ cp_-violating decay reaction density , decay reaction density and washout reaction density respectively . for comparison with standard results we want to maintain the form of eqs . and repeat their derivation from eqs . and to obtain expressions for the reaction densities which take the quantum statistical factors of the boltzmann equation into account . this is important in the present context because the thermal corrections to the _ cp_-violating parameter , to be derived later , are of a similar kind . to this end we use that the sm gauge and yukawa interactions keep the higgs and leptons very close to kinetic equilibrium : @xmath157 with a common temperature @xmath158 and chemical potentials @xmath159 , @xmath160 . we shall also use @xmath161 for the equilibrium distribution functions with zero chemical potential defined in eq .. since a chemical potential with positive sign will appear for either the higgs or its antiparticle , we need to include at least the thermal mass of the higgs to be consistent . in the dense plasma gauge- , yukawa- and higgs self - interactions induce a large thermal higgs mass of about @xmath162 . with @xmath163 , the higgs can not acquire a condensate component . it is then safe to use a bose - einstein equilibrium distribution function to describe the distribution of the higgs particles . using that @xmath164 and hence , for a general decay collision term @xmath165 in the presence of the energy conserving dirac - delta , , @xmath166 and @xmath167 can be any species for which the above conditions apply . here we identify @xmath168 , @xmath169 , @xmath170 . ] @xmath171 , we may write : @xmath172\nonumber\\ & \times{(1-\xi^{n}{f^{}_{n}})}\frac{{{f}_{n}^{eq}}}{{(1-\xi^{n}{{f}_{n}^{eq}})}}{{\ , .}}\end{aligned}\ ] ] we can now expand the exponential in square brackets in the small quantity @xmath173 . if this quantity is tiny at all times the integral will not change much if we neglect quadratic and higher order terms . as well . ] for the zeroth - order ( first ) term in square brackets we use the linear expansion @xmath174 $ ] of the prefactor . the linear order ( second ) term in square brackets will appear preceded by just the zeroth - order factor @xmath175 with @xmath176 to write the results in a compact form we introduce decay reaction densities with quantum - statistical factors included : @xmath177 and @xmath178 where @xmath179 is the total majorana decay amplitude . similarly we define the scattering reaction densities as @xmath180 since @xmath181 refers here to a cp - symmetric ( tree - level ) amplitude squared we have @xmath182 if @xmath183 is symmetric as well . with help of eq . we may separate the contributions to @xmath184 into terms proportional to @xmath185 , terms proportional to @xmath186 , or just proportional to @xmath187 ( see appendix[rateequationsapp ] for details ) : [ eqn : quantum corrected boltzmann equation li result ] @xmath188 in addition we get with eq . for the extra term in eq . : @xmath189 equations describe the generation of a net asymmetry due to out of equilibrium decays of heavy majorana neutrinos . once @xmath187 has a non - zero value , there will be a slight difference in the decay rates to particles and antiparticles respectively which is not due to _ cp_-violation in the decay amplitude , but due to the presence of slightly different occupation numbers of leptons and higgs in the final states of the decays . at linear order this combined effect of blocking and stimulated emission is accounted for by eq .. depending on the ` typical ' sign of @xmath190 it can add to or diminish an existing asymmetry . finally , eq . describes washout due to inverse decays . in section[neqftapproach ] we will see that the functional dependence on @xmath191 in the integrated collision terms is the same as that encountered in the _ cp_-violating parameter @xmath15 itself . considering the last two terms in eq . we find for the scattering contributions : @xmath192 { { \ , , } } \end{aligned}\ ] ] where we defined the ` ris subtracted reaction density ' @xmath193 . if we adopt the amplitudes defined in eq . in the framework of ris subtraction , it is given by @xmath194 \gamma^{{{\ell}}{\phi}}_{\bar{{{\ell}}}\bar{{\phi}}}\bigg > { { \ , .}}\nonumber\end{aligned}\ ] ] note that the contribution proportional to @xmath90 is of higher order in @xmath195 . furthermore , we get for the extra term : @xmath196 here we used @xmath197 we have written eq . schematically in order to show how it compares to other washout terms . note that the extra terms indicate that there will be a slight difference between the equations obtained in the 2pi approach and those obtained with ris subtraction at finite temperature . comparing eqs . and we see that the first term in the former equation will cancel the latter contribution in thermal equilibrium ( @xmath198 ) if the decay contributions are summed up . the second term in eq . is due to quantum statistics . since it is proportional to @xmath199 it can be large only if @xmath15 is large ( as in the case of resonant leptogenesis ) . anticipating our knowledge about the structure obtained within neqft , we will ignore the extra term in what follows . at the time being , everything is still exact with respect to deviations of @xmath200 from equilibrium . this distribution is necessarily distorted due to the fact that it is subject to conflicting equilibrium conditions corresponding to the decay into particles and antiparticles , by the effects of the expansion and , possibly , due to non - equilibrium initial conditions . in order to obtain the full momentum - dependent distribution function we would have to solve the corresponding full kinetic equations however @xcite . to proceed we shall as usual assume that the deviation of the majorana neutrinos from equilibrium is small . this allows us to neglect the @xmath201 contribution and to replace @xmath202 in eq .. the extra terms cancel at this level of approximation up to the quantum - statistical term . in order to bring the remaining source - term eq . into the conventional form , we need to assume that the non - equilibrium distribution of the majorana neutrino is proportional to its equilibrium value ( with momentum independent prefactor ) @xmath203 with this approximation we can write @xmath204 the total contribution to the evolution equations for the lepton asymmetry is then given by @xmath205 i.e. we obtain eq .. we see that , at this level of approximation , there are no contributions due to extra terms apart from those which cancel due to the ris - subtraction . quantitative differences can arise if the deviation of the majorana neutrinos from equilibrium is large or @xmath15 is of order @xmath206 . for the evolution of the majorana neutrino we obtain with appendix[rateequationsapp ] , similar to eq . , the decay contributions [ eqn : quantum corrected boltzmann equation ni result ] @xmath207 and for the extra term in eq . : @xmath208 neglecting again @xmath201 and @xmath209 contributions we obtain @xmath210 i.e. eq .. if higher order contributions are taken into account , we get a difference between the conventional equations and those derived in the 2pi - formalism . ignoring the contribution and the second term in eq . , which are due to quantum statistics , we obtain a contribution @xmath211 to @xmath133 . here the upper sign applies if the extra terms are included and the lower sign if not . this can therefore result in the inclusion of this term with wrong sign even if quantum statistics are neglected , compare e.g. @xcite . the reaction densities for decay , @xmath154 , @xmath212 , @xmath156 , and scattering , @xmath193 , @xmath213 , represent the hydrodynamical coefficients which govern the evolution of the number densities ( abundances ) . we will compute them numerically once the additional medium dependence of the amplitudes ( in particular the _ cp_-violating parameters ) has been derived . in addition , it is useful to define a thermally averaged _ cp_-violating parameter as @xmath214 which equals @xmath15 if it is momentum independent , such as in the zero temperature case , but will differ once thermal effects are included . this quantity is meaningful for the comparison with conventional results because it takes into account that the deviation of the majorana neutrino abundance from equilibrium , which appears in the source - term for the lepton abundance , is influenced by the ( _ cp_-conserving ) decay reaction density in the denominator . inserting conventional vacuum amplitudes in eqs . with eqs . and and dropping quantum - statistical factors one obtains the conventional results for the reaction densities . for the readers convenience we quote them here . for the decay reaction density we obtain @xmath215 and @xmath216 , see appendix[kinematics ] . for the two - body scattering the reaction density is given by @xmath217 where @xmath218 is so - called reduced cross section : @xmath219 for the @xmath87 process it reads @xmath220 where we have replaced @xmath53 by @xmath221 and introduced dimensionless quantities @xmath222 and @xmath223 . the case @xmath60 is included in this expression in the limiting sense @xmath224 . note that eq . only contains the real part of @xmath225 . the contribution of the imaginary part vanishes because @xmath226 is antisymmetric with respect to @xmath227 whereas the sum in the curly brackets is symmetric under this transformation . the integration of eq . yields for the reduced ` ris subtracted cross section ' of the @xmath228 process : @xmath229 \left[(x - a_j)^2+(a_jc_j)^2\right ] } \right.\nonumber\\ & + 2\frac{x+a_i}{a_j - a_i}\ln\left(\frac{x+a_i}{a_i}\right)+ 2\frac{x+a_j}{a_i - a_j}\ln\left(\frac{x+a_j}{a_j}\right)\nonumber\\ & + \frac{x - a_i}{(x - a_i)^2+(a_ic_i)^2}\left[x-(x+a_j)\ln\left(\frac{x+a_j}{a_j } \right)\right]\nonumber\\ & + \left.\frac{x - a_j}{(x - a_j)^2+(a_jc_j)^2}\left[x-(x+a_i)\ln\left(\frac{x+a_i}{a_i } \right)\right]\right\}\nonumber{{\ , .}}\end{aligned}\ ] ] the reduced ` cross section ' is _ _ negative _ _ is not a physical cross section but denotes the contribution to the reaction density arising from the difference of the full and the ris term . we stress that all physical rates are manifestly positive , e.g. the washout term , to which @xmath230 yields a sub - leading correction that is relatively suppressed by yukawa couplings . see also @xcite . ] in the vicinity of the mass shells , @xmath231 . this is due to the @xmath232 term in the numerator of the ris subtracted propagator . note that because we have not approximated this term by the dirac - delta the structure of eq . is slightly different from the one usually used in the literature @xcite . in this section we briefly review the description of leptogenesis within non - equilibrium quantum field theory @xcite . this framework has been shown recently to be suitable for the derivation of quantum dynamic equations for the lepton asymmetry within a first - principle approach , and to incorporate medium , off - shell , coherence and possibly further quantum effects in a self - consistent way @xcite . we continue these efforts by deriving consistent quantum corrected boltzmann equations that describe the generation and washout of the lepton asymmetry and include the ( inverse ) decay as well as scattering processes mediated by majorana neutrinos . the lepton asymmetry is given by the @xmath233-component of the expectation value of the lepton - current operator : @xmath234 it can be expressed in terms of the leptonic two - point function . we define the two - point functions for the higgs , lepton and majorana fields with time arguments attached to the closed time path ( ctp ) shown in fig.[fig : ctp ] by [ twopointfunctions ] @xmath235 where the sub- and superscripts refer to @xmath21 and flavor indices and @xmath236 denotes time - ordering with respect to the ctp . closed time path . ] we will frequently use matrix notation for the flavor indices , where e.g. @xmath237 denotes the flavor - matrix @xmath238 , etc . using the definition we find for the lepton - current : @xmath239{{\ , .}}\ ] ] two - point functions @xmath240 ( where @xmath241 stands for @xmath242 , @xmath243 or @xmath244 ) defined on the ctp can be decomposed into a _ spectral function _ @xmath245 and _ statistical propagator _ @xmath246 : @xmath247 the signum function is either @xmath248 or @xmath249 depending on whether @xmath250 or @xmath251 occur ` later ' on the contour @xmath252 . @xmath246 and @xmath245 encode information on the state and the spectrum of the system , respectively . for example , for the leptons they are given by @xmath253_- \rangle { { \ , , } } \\ { s^{\alpha\beta}_{ab\,\rho}}(x , y ) & = i \langle \left [ { { \ell}}_{\alpha}^a(x ) , \bar { { \ell}}_{\beta}^b(y ) \right]_+ \rangle { { \ , , } } \end{aligned}\ ] ] where @xmath254_\pm$ ] denote ( anti-)commutators . statistical and spectral functions of majorana neutrino and higgs can be expressed similarly , with @xmath255 and @xmath256 exchanged for bosons . although there are only two independent two - point functions for each species , it is convenient to introduce additional combinations of them , namely the _ wightman _ functions @xmath257 as well as _ retarded _ and _ advanced _ functions , [ retadvdef ] @xmath258 from the above definitions one can see that the difference of the retarded and advanced propagators gives the spectral one , whereas the sum yields the _ hermitian _ propagator @xmath259 : [ hermdef ] @xmath260 finally , we will also need the _ cp _ conjugated propagators on the ctp : [ cpconjugates ] @xmath261 here @xmath262 , @xmath263 and @xmath264 are the charge conjugation and parity matrices , respectively , and the transposition refers to spinor indices . _ cp _ conjugated statistical and spectral functions immediately follow from the above definition by inserting the decomposition . the time - evolution of the two - point functions is described self - consistently by the kadanoff - baym ( kb ) equations . these equations can be obtained from a variational principle using the so - called 2pi effective action @xcite . the resulting equations of motion have the form of schwinger - dyson equations for the non - equilibrium propagators formulated on the ctp : @xmath265 here @xmath266 is the inverse of the _ full _ lepton propagator in coordinate space , and @xmath267 is the inverse of the _ free _ lepton propagator , @xmath268 the information about the interaction processes is encoded in the self - energies @xmath269 . they can be obtained by cutting one line of the 2pi contributions to the effective actions . the two- and three - loop contributions are presented in fig.[fig:2pi contributions ] . two- and three - loop contributions to the 2pi effective action and the corresponding contributions to the lepton self - energy . note that the propagator lines used here denote full resummed propagators in contrast to those employed in the previous feynman graphs . the contributions ( a ) and ( c ) to the 2pi effective action are known as ` setting - sun ' and ` mercedes ' diagrams respectively.,title="fig : " ] + two- and three - loop contributions to the 2pi effective action and the corresponding contributions to the lepton self - energy . note that the propagator lines used here denote full resummed propagators in contrast to those employed in the previous feynman graphs . the contributions ( a ) and ( c ) to the 2pi effective action are known as ` setting - sun ' and ` mercedes ' diagrams respectively.,title="fig : " ] the kb equations can be obtained by convoluting the schwinger - dyson equation with the full propagator , which yields : @xmath270 here @xmath271 . after decomposing the resulting equation into statistical and spectral components , one obtains : [ kblepton ] @xmath272 the equations for majorana and higgs propagators have a similar structure , with the klein - gordon instead of the dirac operator for the latter . the schwinger - dyson equations and the corresponding kadanoff - baym equations are formally very similar to the schwinger - dyson equation in vacuum . however , out of equilibrium the propagators depend not only on the relative coordinate @xmath273 , but also on the central coordinate @xmath274 , which makes their solution much more involved . in contrast to the schwinger - dyson equation in vacuum , the kb equations determine the spectral properties of the system including medium corrections , as well as the non - equilibrium dynamics of the statistical propagator self - consistently . since the latter represents the quantum field theoretical generalization of the classical particle distribution functions , kb equations can be seen as the quantum field theoretical generalizations of boltzmann equations . as pointed out above , an equation of motion for the lepton asymmetry can be derived by considering the divergence of the lepton - current @xmath275 . using the kb equations one obtains and using the covariant derivative @xmath30 . as has been demonstrated in @xcite , this is the case for scalar fields . a manifestly covariant generalization of center and relative coordinates @xmath183 and @xmath53 to curved space - time can be found in @xcite . ] : @xmath276 \nonumber\\ & = g_w\ , i\int_0^{x^0 } { \mathscr{d}}^4z \ , { { \rm tr}}\bigl [ { \sigma^{\alpha\beta}_{\rho}}(x , z ) { s^{\beta\alpha}_{f}}(z , x ) \nonumber\\ & - { \sigma^{\alpha\beta}_{f}}(x , z ) { s^{\beta\alpha}_{\rho}}(z , x ) - { s^{\alpha\beta}_{\rho}}(x , z ) { \sigma^{\beta\alpha}_{f}}(z , x ) \nonumber\\ & + { s^{\alpha\beta}_{f}}(x , z ) { \sigma^{\beta\alpha}_{\rho}}(z , x ) \bigr ] { { \ , .}}\end{aligned}\ ] ] here summation over repeated indices is implicitly assumed . the two equations above represent the quantum generalization of the boltzmann equation for the lepton asymmetry . thus , they may be considered as the master equations for a quantum field theoretical treatment of leptogenesis @xcite . the dependence of the two - point functions on the relative coordinate @xmath53 is characterized by the hard scales like the majorana neutrino mass @xmath277 or the temperature @xmath144 of the surrounding plasma . in contrast to that , the variation with the central coordinate @xmath183 is given by the macroscopic time - evolution of the system , e.g. the hubble rate @xmath278 or the majorana decay rate @xmath279 . therefore , it is possible to perform an expansion in slow relative to fast time - scales , i.e. in powers of e.g. @xmath280 or @xmath281 . technically , this can be realized by a so - called gradient or derivative expansion with respect to @xmath183 , and a fourier transformation with respect to @xmath53 , known as wigner transformation , see appendix[selfenergy ] for more details . then , to leading order in the gradients , the evolution equation for the lepton asymmetry becomes markovian , and after some straightforward algebra , can be written as @xmath282\nonumber\\ & - \bigl[{\bar{\sigma}^{\beta\alpha}_{<}}(t,{{r } } ) { \bar{s}^{\alpha\beta}_{>}}(t,{{r } } ) - { \bar{\sigma}^{\beta\alpha}_{>}}(t,{{r } } ) { \bar{s}^{\alpha\beta}_{<}}(t,{{r } } ) \bigr]\bigr\}{{\ , .}}\nonumber\end{aligned}\ ] ] note that it is possible to investigate higher orders in the derivative expansion systematically @xcite . in eq . all two - point functions are evaluated in wigner space , where @xmath283 is the _ physical _ momentum @xcite that corresponds to @xmath53 . for a spatially homogeneous system ( like frw ) the two - point functions depend only on the time coordinate @xmath284 , and on the momentum @xmath283 , because of spatial translational invariance . strictly speaking , this is true only in the rest frame of the medium ( comoving frame ) . in a general frame the two - point functions depend on @xmath285 , where @xmath286 is the four - velocity of the medium . the latter satisfies the normalization condition @xmath287 , and is given by @xmath288 in the medium rest frame . in order to allow for a physical interpretation of eq we have written it such that the integration is over positive frequencies only , and expressed the lepton propagator and self - energy in terms of the wigner transformed wightman functions eq .. in thermal equilibrium , the wightman functions depend only on the momentum @xmath283 and satisfy the kubo - martin - schwinger ( kms ) relation @xmath289 for fermions / bosons , respectively . when inserting the kms relations for propagators and self - energies into eq . , one immediately finds that the divergence of the lepton - current vanishes in thermal equilibrium as it should ( see also @xcite ) . in other words , the quantum equation for the lepton asymmetry is in accordance with the third sakharov condition . we emphasize that it is _ not _ necessary to apply ris subtraction to obtain this result within the ctp approach @xcite . the four terms on the right - hand side of eq . may be interpreted as gain and loss terms of leptons and anti - leptons respectively @xcite . in particular , one may define generalized lepton distribution functions @xmath290 via the so - called kadanoff - baym ansatz @xmath291 thus the contribution on the right - hand side of eq . that contains @xmath292 corresponds to the lepton loss term , while the contribution proportional to @xmath293 represents the lepton gain term . analogous definitions relate the _ cp _ conjugate propagators with the anti - lepton distribution . note that the kms relations ensure that in equilibrium @xmath294 approaches the fermi - dirac distribution @xmath295 . the flavor off - diagonal components encode coherent flavor correlations @xcite . in the unflavored regime considered here @xmath296 and @xmath297 . in the quasiparticle ( qp ) approximation , the spectral function is given by @xmath298 where we assume that leptons obey conventional dispersion relation and @xmath299 is the effective thermal mass . these assumptions might be modified in the presence of a medium @xcite . due to the presence of the dirac - delta - function in eq . the integration over @xmath300 in eq . is trivial and leaves only the integration over spatial momenta of on - shell leptons . therefore the right - hand side of eq . can be interpreted as a difference of two ( integrated ) boltzmann - like equations one for the particles and one for the antiparticles @xcite . according to the physical interpretation of eq . in terms of gain and loss terms , the wightman components of the lepton self - energy and of its _ cp _ conjugate are the analogs of the collision integrals . since we limit our analysis to the unflavored regime , it is convenient to perform the summation over the flavor indices : @xmath301 . then the one - loop contribution , see fig.[fig:2pi contributions ] ( b ) , takes the form : @xmath302 where @xmath303 . the explicit expression for the two - loop contribution is rather lengthy and it is convenient to split it into three distinct terms : @xmath304 the first term on the right - hand side reads @xmath305\nonumber { { \ , , } } \end{aligned}\ ] ] where we have introduced two functions containing loop corrections : @xmath306{{\ , , } } \end{aligned}\ ] ] and @xmath307 to shorten the notation . comparing eqs . and we see that they have a very similar structure . first , the integration is over momenta of the higgs and majorana neutrino and the delta - function contains the same combination of the momenta . second , both self - energies include one wightman propagator of the higgs field and one wightman propagator of the majorana field . upon the use of the kadanoff - baym ansatz the wightman propagators can be interpreted as cut - propagators which describe on - shell particles created from or absorbed by the plasma @xcite . on the other hand , the retarded and advanced propagators can be associated with the off - shell intermediate states . we therefore conclude that eqs . and describe ( inverse ) decays of the heavy neutrino into a lepton - higgs pair . the second term on the right - hand side of eq . contains two wightman propagators of the higgs field and one wightman propagator of the lepton field . the majorana propagator appears only in the intermediate state : @xmath308{{\ , .}}\end{aligned}\ ] ] we therefore conclude that this term describes lepton number violating scattering processes mediated by the heavy neutrino . finally the last term in eq . contains two wightman propagators of the majorana field and one of the lepton field , whereas the higgs field is in the intermediate state : @xmath309{{\ , .}}\end{aligned}\ ] ] therefore it can be identified with the higgs mediated scattering processes . these conserve lepton number and do not contribute to generation of the lepton asymmetry . the _ cp _ conjugate of the wigner transforms can be obtained using eq .. in practice this amounts to replacing the propagators by their _ conjugate and the couplings by their complex conjugate in the above expressions . for instance for the _ cp _ conjugate of the one - loop self - energy we find : @xmath310 expression for the _ cp _ conjugate of the two - loop lepton self - energy can be obtained in a similar way . for the higgs propagators in the above self - energies we can also use the kadanoff - baym ansatz , @xmath311 and the simple quasiparticle approximation for the spectral function , @xmath312 where @xmath313 is the effective thermal mass . effects of the finite thermal higgs mass will be studied in section[higgsdecay ] . in this section we will analyze the lepton number and _ cp_-violating ( inverse ) decay of the majorana neutrino as well as the two - body scattering processes mediated by the heavy neutrino . in particular , we will derive expressions for the in - medium _ cp_-violating parameters , decay widths and scattering amplitudes . we will also explicitly demonstrate that the obtained equation for the lepton asymmetry is free of the double - counting problem . in the previous section we have used the kadanoff - baym ansatz and quasiparticle approximation for the higgs and lepton fields . let us now assume that similar approximations also hold for majorana neutrinos . that is , we assume that in eqs . and the spectral function @xmath314 is diagonal in flavor space and can be approximated by @xmath315 and that it is related to the wightmann components via the kadanoff - baym ansatz : @xmath316 substituting eqs . and in eq . and making the above approximations we find after some algebra that the lepton - current can be represented in the form : @xmath317{{\ , , } } \end{aligned}\ ] ] where @xmath318 have been introduced in eq . and we have defined : [ effectivedecayamplitudestree ] @xmath319{{\ , , } } \\ \label{effamplsebarlbarh } { \xi^{t}_{\bar { { \ell}}\bar { \phi}\leftrightarrow { n}_i}}&\equiv g_w ( h^\dagger h)_{ii } { { \rm tr}}[(\slashed{{{q}}}+m_i)p_l\slashed{{{r}}}\,]{{\ , .}}\end{aligned}\ ] ] the superscript ` t ' stands for ` tree - level ' . the expression strongly resembles the boltzmann equation . therefore the functions @xmath320 and @xmath321 can be interpreted as effective in - medium amplitudes squared , summed over internal degrees of freedom , for the decays into leptons and antileptons respectively . tree - level contribution . ] the two effective amplitudes can be replaced by the total decay amplitude and the _ cp_-violating parameter . we find that within the used approximations the resulting decay amplitude coincides with the outcome of the vacuum calculation , @xmath322 , and that @xmath323 . in the presence of a nonzero lepton asymmetry @xmath324 and @xmath325 . therefore @xmath326 and this leads to a washout of the asymmetry . despite the fact that eq . correctly describes the ( leading - order ) washout processes , it fails to describe processes which generate lepton asymmetry : in the considered approximation @xmath323 because the _ cp_-violating effects , which are required to produce the asymmetry , are of fourth order in the yukawa couplings of the majorana neutrino . in eq . we have taken into account only terms quadratic in the coupling . in other words , this approximation corresponds to the tree - level approximation in the canonical approach . terms of higher order in the couplings emerge from three- and higher - loop contributions to the lepton self - energy , see eqs.- , as well as from expansion of the full majorana propagators entering the self - energies . in order to define an effective cp - violating parameter and decay width that incorporate medium corrections we have to identify the quasiparticle excitations in the system . to perform this analysis we follow the discussion of the self - energy contribution within a toy - model as presented in @xcite . as has been demonstrated there , it is important to take the matrix structure of the majorana propagator in flavor space into account . our starting point is the schwinger - dyson equation for the majorana two - point function : @xmath327 let us split the self - energy into diagonal and off - diagonal components in flavor space and introduce a diagonal propagator @xmath328 defined by the equation : @xmath329 where @xmath330 is the free propagator and @xmath331 denotes the diagonal components of the self - energy . the poles of the diagonal propagator define the quasiparticle excitations . it can be shown that the dynamics of these is described by a boltzmann - like quantum kinetic equation . inserting this decomposition into the schwinger - dyson equation we find , using matrix notation : @xmath332 where @xmath333 denotes the off - diagonal components of the self - energy and @xmath237 the full neutrino propagator including flavor - diagonal and flavor off - diagonal contributions . multiplying eq . by @xmath237 from the left , by @xmath334 from the right and integrating over the contour @xmath335 we obtain a formal solution for the full non - equilibrium propagator : @xmath336 after decomposing the propagators and self - energies into the spectral and statistical components , we can rewrite eq . in the form : @xmath337{{\ , .}}\end{aligned}\ ] ] here we are using the retarded and advanced propagators defined by eq . , so that the integration can be extended to the whole @xmath338-plane . using their definitions and eq . , we can also derive formal solutions for the retarded and advanced propagators : @xmath339 next we wigner transform eqs . and and perform the leading order gradient expansion as has been outlined in section[neqftapproach ] . combining both results , we find for the full statistical and spectral propagators and the corresponding causal two - point functions of the system in to equilibrium : [ fullpropagator ] @xmath340 where all propagators and self - energies are evaluated at the same point @xmath341 in configuration space . we can express the full statistical and spectral propagators in terms of the diagonal ones and the off - diagonal self - energies , @xmath342 { \hat{\theta}^{}_{a}}{{\ , , } } \end{aligned}\ ] ] where @xmath343 and @xmath344 are defined by @xmath345 and @xmath346 respectively , with @xmath347 being the @xmath348 unit matrix in the dirac and flavor space of the @xmath349 generations . solution eq . reduces the dynamics of the full statistical and spectral propagators to the dynamics of two quasiparticle excitations . their masses , decay widths and _ cp_-violating parameters are determined by the medium and the abundances are described by the corresponding one - particle distribution functions . strictly speaking , the solution is valid only in thermal equilibrium . however , we assume that it also holds for small deviations from equilibrium . to consistently analyze processes of the fourth order in the coupling one has to use so - called extended quasiparticle approximation ( eqp ) for the statistical propagator and spectral function @xcite . the eqp approximation represents the diagonal propagator as a sum of two terms : @xmath350 the first describes decay processes , whereas the second can be associated with scattering processes . inserting eq . into eq we get a solution for the resummed majorana propagator consistent up to the fourth order in the couplings : @xmath351{\hat{\theta}^{}_{a}}{{\ , .}}\end{aligned}\ ] ] the first term in the above formula describes majorana decay , see section[cpviolationindecay ] , whereas the remaining three terms describe the two - body scattering processes mediated by the majorana neutrino . these are discussed in section[majoranamediatedscattering ] . using definition of the retarded and advanced two - point functions , eq . , and the schwinger - dyson equation for the diagonal propagators , eq . , we find that the causal propagators in eq . are given by @xmath352 splitting the retarded and advanced self - energies into the vector and scalar components we can write the solution of eq . in the form : @xmath353 where we have omitted flavor indices to shorten the notation and introduced @xmath354 from eq . we can extract the spectral and hermitian propagators . to leading order in the yukawas they read [ shrhoapprox ] @xmath355 the on - shell condition is defined by @xmath356 . expanding @xmath357 to linear order in the yukawas we find : @xmath358 where @xmath359 is the medium - induced component of the hermitian self - energy in the on - shell renormalization scheme . in vacuum the on - shell condition is fulfilled for @xmath360 , i.e. @xmath361 is the physical vacuum mass . at non - zero temperatures the mass receives medium - induced corrections . to linear order in the yukawas the effective mass is given by @xmath362 . for a hierarchical mass spectrum , which we consider here , the contributions of the hermitian self - energy are always negligible and we will use @xmath363 and @xmath364 in the following . from eq . we can also deduce the effective width . to leading order in the yukawas it is given by @xmath365 . the minus sign in this definition ensures that the effective decay width is positive . one - loop contribution to the majorana self - energy is derived in appendix[selfenergy ] . in a _ cp_-symmetric medium it is given by @xmath366 l_{\rho}{{\ , .}}\end{aligned}\ ] ] therefore we can write the effective decay width in the form @xmath367 , where @xmath368 is the total vacuum decay width . for positive @xmath369 and @xmath370 the loop integral @xmath371 takes the form : @xmath372{{\ , .}}\end{aligned}\ ] ] for massless final states @xmath373 . therefore the definition of the effective decay width inferred in section[risquantstat ] from the requirement of successful ris subtraction is consistent with that implied by eq .. for the eqp wightman propagators we can use the kadanoff - baym ansatz . as can be inferred from eq . , the corresponding spectral function reads @xmath374 substituting eq . into eq . we obtain @xmath375 where we have again omitted the flavor indices . the second and the third terms in eq . vanish on the mass shell and can be neglected . commuting @xmath376 and @xmath377 in the first term and again neglecting contributions which are tiny on the mass shell we finally obtain for the eqp spectral function : @xmath378 ^ 2 } { { \ , .}}\end{aligned}\ ] ] note that structures of eqs . and are very similar . furthermore , as follows from eq . , in the limit of vanishing decay width both of them approach the delta - function . however , for a small but _ finite _ decay width the eqp spectral function is a better approximation to the delta - function than eq .. therefore , we can approximate it by the usual expression , @xmath379 and at the same time keep finite - width terms in the diagonal propagators . to go beyond the tree - level approximation and take into account _ cp_-violating effects we need to consider contributions to the lepton self - energy that are of the fourth order in the yukawa couplings . one of them comes from expansion of the majorana propagator in the one - loop self - energy . substituting the decay term of eq . into eq . we can write it in the form : @xmath380 substituting eq . and its _ cp _ conjugate into eq . we find that the resulting contribution to the divergence of the lepton - current has precisely the form . however , the corresponding effective amplitudes are no longer equal : [ effectivedecayamplitudesse ] @xmath381{{\ , , } } \\ \label{effamplsecpbarlbarh } { \xi^{t}_{\bar{{\ell}}\bar{\phi}\leftrightarrow { n}_i } } & + { \xi^{s}_{\bar{{\ell}}\bar{\phi}\leftrightarrow { n}_i}}\equiv g_w { \textstyle\sum}_{mn}(h^\dagger h)^*_{mn}\nonumber\\ \times&{{\rm tr}}[{\bar{\theta}^{ni}_{r}}(t,{{q}})(\slashed{{{q}}}+m_i ) { \bar{\theta}^{im}_{a}}(t,{{q}})p_l\slashed{{{r}}}p_r\,]{{\ , .}}\end{aligned}\ ] ] the matrices @xmath343 and @xmath344 are evaluated on the mass shell of the @xmath382th majorana neutrino . the bar denotes _ cp_-conjugation and the trace is over dirac indices . interference of tree - level and one - loop self - energy corrections . ] as compared to tree - level result it additionally contains interference of the tree - level and one - loop self - energy contributions to the majorana decay amplitude , see fig.[tpluss ] . for a hierarchical mass spectrum we can use the approximation @xmath383 and a similar approximation for @xmath344 . using furthermore eqs . and we find for the _ cp_-violating parameter : @xmath384 where @xmath283 and @xmath385 are on - shell momenta of the outgoing lepton and decaying majorana neutrino respectively . in vacuum @xmath386 and the _ cp_-violating parameter takes the form : @xmath387 the ` regulator ' in the denominator of eq . differs from the result @xmath388 found in @xcite by the ratio of the masses . for a hierarchical neutrino mass spectrum the ` regulator ' term is sub - dominant and this difference is numerically small . note also that although eq . does not diverge in the limit of vanishing mass difference the approximations made in the course of its derivation are not applicable for a quasidegenerate mass spectrum @xcite . for a consistent treatment of resonant enhancement within neqft we refer to @xcite . the two - loop lepton self - energy is of the fourth order in the couplings to begin with . therefore , for a hierarchical mass spectrum one can safely neglect the off - diagonal components of the majorana propagators and replace @xmath244 by the eqp one : @xmath389 { { \ , .}}\end{aligned}\ ] ] substituting eq . and its _ cp _ conjugate into eq . we again find that the resulting contribution to the divergence of the lepton - current has the form . the corresponding effective amplitudes read [ majoranavertexamplitudes ] @xmath390\\ & -g_w(h^{\dagger}h)_{ji}^{2}\,m_i\ , { { \rm tr}}\bigr [ c v_{jj}(q , k)p_l\slashed p p_r\bigr]{{\ , , } } \nonumber\\ \label{effamplvertbarlbarh } { \xi^{v}_{n_i \leftrightarrow \bar { { \ell}}\bar { \phi } } } \equiv & -g_w(h^{\dagger}h)_{ij}^2\ m_i\ , { { \rm tr}}\bigr [ c v_{jj}(q , k)p_l\slashed p p_r\bigr]\\ & -g_w(h^{\dagger}h)_{ji}^{2}\,m_i\ , { { \rm tr}}\bigl[\varlambda_{jj}(q , k ) c p_l\slashed pp_r\bigr]\nonumber{{\ , .}}\hphantom{aa}\end{aligned}\ ] ] they describe interference of the tree - level and one - loop vertex contributions to the majorana decay amplitude , see fig.[interferencev ] . interference of tree - level and one - loop vertex corrections . ] to account for the contribution of the vertex correction to the decay width and the _ cp_-violating parameter we have to substitute the sum of @xmath391 and @xmath392 and a similar sum for the antiparticles into eq .. the vertex contribution to the decay amplitude is of fourth order in the coupling and is negligible compared to the tree - level term . since we assume the medium to be almost _ cp_-symmetric we can use , at leading order , _ cp_-symmetric two - point functions in the loop integrals @xmath393 and @xmath394 . then , at leading order in the yukawa couplings , we find for the vertex contribution to the _ cp_-violating parameter : @xmath395\nonumber{{\ , .}}\end{aligned}\ ] ] the quasiparticle approximation and the kb - ansatz enforce two of the intermediate lines of the vertex loop to be on - shell whereas the remaining line described by the hermitian part of the retarded and advanced propagators remains off - shell . the three lines in square brackets in eq . therefore correspond to different cuts through two of the three internal lines of the loop diagram fig.[treevertexself].(c ) . note also that only for one of the three internal lines the corresponding distribution function enters the result . the first possible cut described by the first line in square brackets corresponds to cutting the propagators of higgs and lepton . one can interpret this cut as decay of the majorana neutrino into a lepton - higgs pair which is followed by a subsequent @xmath54-channel scattering mediated by a virtual majorana neutrino . introducing @xmath396 m_i^2 { \boldsymbol{\mathcal{s}}^{ii}_{h}}({{k}}-{{r}}_2 ) { { \ , , } } \end{aligned}\ ] ] we can rewrite the first term in eq . in a form which strongly resembles the form of the self - energy _ cp_-violating parameter : @xmath397 in vacuum @xmath398 can be computed explicitly and we recover the well - known result @xcite : @xmath399{{\ , .}}\end{aligned}\ ] ] adding up eqs . and we obtain the canonical expression for the vacuum _ cp_-violating parameter , eq .. if the intermediate majorana neutrino is much heavier than the decaying one then @xmath400 and therefore @xmath401 . in this case we can also neglect the ` regulator ' term in the denominator of eq .. in this approximation the two _ cp_-violating parameters have the same structure and their sum can be written in the form : @xmath402 note that the combination of the distribution functions that enters the self - energy and vertex _ cp_-violating parameters , see eqs . and , is the same as that of @xmath403 encountered in the derivation of the rate equations , see eq .. this result is in agreement with the findings of @xcite using neqft and of @xcite based on imaginary - time thermal qft . note that older results featured a different dependence on the distribution functions , with an additional term quadratic in the one - particle distribution functions which is absent in eq . as well as in eq . : @xmath404 in @xcite it was demonstrated that the result obtained using thermal field theory can be reconciled with the result of neqft calculation once causal green s functions are used in the former . the two other cuts in eq . are proportional to @xmath405 and to @xmath406 respectively . they vanish in the zero temperature limit and are usually boltzmann - suppressed at finite temperatures , but can be relevant in specific cases @xcite . the quantities that enter the rate equations are the decay , washout and _ cp_-violating decay reaction densities . in the canonical approximation , i.e. when the quantum - statistical effects and effective masses of the higgs and leptons are neglected , they are given by eq .. if the thermal masses are neglected but the quantum - statistical effects are taken into account , there is an enhancement of the decay and washout reaction densities at high temperature , see fig.[gammadratio ] . decay and _ cp_-violating reaction densities with thermal lepton and higgs masses , @xmath407 , and with zero masses , @xmath408 , for the two majorana neutrinos @xmath409 and @xmath410 . the values are normalized to the corresponding reaction density in the conventional approximation @xmath411 . the thermal enhancement due to quantum - statistical factors is overcompensated by the phase space suppression due to thermal masses at high temperatures . note that we show only the self - energy contribution to the _ cp_-violating reaction densities . ] however , the inclusion of the thermal masses turns this enhancement into a suppression at high temperatures . it is explained by the decrease of the decay phase space . at intermediate temperatures the thermal masses become small relative to the majorana mass and we observe a minor enhancement . for the _ cp_-violating reaction density we observe a very similar behavior . given that for a hierarchical mass spectrum most of the asymmetry is typically generated by the lightest majorana neutrino at @xmath412 , where @xmath40 is the washout parameter ( see appendix[numericalparameters ] ) , we expect the medium effects to induce a moderate enhancement of the total generated asymmetry . two - body scattering processes mediated by majorana neutrinos violate lepton number by two units and play an important role in the washout of the generated asymmetry . in this section we derive the effective scattering amplitudes using neqft . this is an important part of our results . the last three terms in eq . contain the wigner - transformed one - loop majorana self - energy : @xmath413 { { \ , , } } \hphantom{a}\end{aligned}\ ] ] see appendix[majoranaselfen ] for more details . combining them with eq . we find that their contribution to the divergence of the lepton - current contains two wightman propagators of leptons and two of the higgs field . as we have argued above , these correspond to initial and final states in the kinetic equations . therefore , we conclude that these terms describe scattering processes depicted in fig.[lhbarlbarh ] and fig.[llbarhbarh ] . as higgs and leptons are maintained close to equilibrium we can safely use the kadanoff - baym ansatz for their propagators in the majorana self - energy . inserting eq . into the scattering terms of eq . we can then split the majorana propagator into a lepton number conserving and lepton number violating part : [ propmajapsp ] @xmath414 { { \ , , } } \\ \label{propmajapsp < } { \mathscr{s}^{ij}_{<}}({{q}})&=- g_w\int { { \,d}\pi^4_{{{k } } } } { { \,d}\pi^4_{{{r}}}}(2\pi)^4 \delta(q - p - k){\delta^{}_{\rho}}({{k } } ) { \mathbf{s}^{}_{\rho}}({{r } } ) \nonumber \\ & \times \bigl[f_{{\ell}}^{{r}}f_{\phi}^{{k}}{\mathscr{s}^{ij}_{lc}}({{q}},{{r } } ) + f_{\bar { { \ell}}}^{{r}}f_{\bar { \phi}}^{{k}}{\mathscr{s}^{ij}_{lv}}({{q}},{{r } } ) \bigr ] { { \ , , } } \end{aligned}\ ] ] where we have defined : [ lvlc ] @xmath415{{\ , , } } \nonumber \\ \label{lv } & { \mathscr{s}^{ij}_{lv}}=(h^\dagger h)_{ji } \bigl [ ( 1-\delta^{ij}){\mathcal{s}^{ii}_{r}}(q)p_r\slashed{{{r } } } p_l { \mathcal{s}^{jj}_{a}}({{q } } ) \\ & + \frac{\delta^{ij}}{2 } \big({\mathcal{s}^{ii}_{r } } ( q)p_r\slashed{{{r } } } p_l { \mathcal{s}^{jj}_{r}}({{q } } ) + { \mathcal{s}^{ii}_{a}}(q)p_r \slashed{{{r } } } p_l { \mathcal{s}^{jj}_{a}}({{q}})\big ) \bigr]{{\ , .}}\nonumber \end{aligned}\ ] ] here we neglected higher order terms coming from the matrices @xmath416 and @xmath417 . the first terms in eqs . and corresponds to the second term in eq . , whereas the remaining terms correspond to the last two terms in eq .. substituting the majorana propagators into the lepton - current together with the one - loop lepton self - energy we finally obtain @xmath418{{\ , .}}\end{aligned}\ ] ] note that the zeroth component of the momenta in the above equation can have both signs . the effective amplitudes of the lepton number conserving and lepton number violating processes read [ traceap ] @xmath419{{\ , , } } \hphantom{aaaa}\\ \label{traceaplc } a^{lc}({{q}},{{r}}_1,{{r}}_2)\equiv & ( h^\dagger h)_{ji}\ , { { \rm tr}}\bigl[{\mathscr{s}^{ij}_{lc}}(q,{{r}}_2 ) p_l \slashed{{{r}}}_1p_r\bigr]{{\ , .}}\hphantom{aaaa}\end{aligned}\ ] ] the functions @xmath420 and @xmath421 are symmetric under the exchange of the momenta @xmath422 and @xmath423 . this implies that the contribution of @xmath424 in eq . the terms of @xmath425 diagonal in flavor space correspond to the ris - propagator . substituting eq . into eq . and taking the trace we find that it contains only the scalar components of the retarded and advanced propagators and is proportional to : @xmath426 ^ 2}\nonumber\\ & \approx 2m^2\frac{(q^2-m^2)^2-(m{{\boldsymbol \gamma}^{}_{}})^2}{\left[(q^2-m^2)^2 + ( m{{\boldsymbol \gamma}^{}_{}})^2\right]^2}{{\ , .}}\end{aligned}\ ] ] equation differs from the canonical result only in that the vacuum masses and decay widths are replaced by thermal ones . for a hierarchical mass spectrum this difference can be safely neglected . introducing an analogue of the ris subtracted propagator , @xmath427 we can rewrite the lepton number violating effective amplitude in a compact form : @xmath428 next we perform the trivial integrations over the frequencies using the dirac - deltas in the quasiparticle spectral functions and . each dirac - delta can be decomposed into two terms , one with positive and one with negative frequency . therefore , the integration over the four frequencies gives rise to @xmath429 terms , but only six of them satisfy energy conservation ensured by the remaining delta - function . in a homogeneous and isotropic medium the one - particle distribution functions satisfy @xmath430 and the diagonal majorana propagators have the properties @xmath431 upon substitution of the resulting self - energy into eq . and the use of eqs . and the remaining six contributions in the lepton - current can be conveniently written as @xmath432{{\ , , } } \end{aligned}\ ] ] where we have defined the effective scattering amplitudes : [ m1 ] @xmath433 the momenta of the majorana neutrinos are related to the momenta of the initial and final states by @xmath434 and @xmath435 . from eq . we see that the obtained amplitudes contain only @xmath436 and @xmath437 interference terms . indeed , in the products of the majorana propagators in eq . both of them depend on the same momentum . the missing cross terms emerge from the two - loop ( vertex ) contribution to the lepton self - energy . as we have mentioned in section[neqftapproach ] , within the discussed assumptions and approximations the second and third terms on the right - hand side of eq . describe scattering processes . since they are of the fourth order in the yukawas , we can replace the full majorana propagators by the diagonal propagators . the third term , eq . , corresponds to lepton number conserving processes and does not need to be discussed further . the second one , eq . , is given by @xmath438{{\ , .}}\end{aligned}\ ] ] we substitute eq . into the equation for the lepton - current and perform the steps preceding eq .. using furthermore relations we find @xmath439{{\ , , } } \end{aligned}\ ] ] where we have introduced [ m2 ] @xmath440 and @xmath441 . the combinations of the momenta appearing in the products of the majorana propagators clearly indicate that the above amplitudes correspond to the interference terms of the @xmath53- , @xmath54- , and @xmath88-channel contributions . combining eqs . and we obtain for the effective amplitude of @xmath442 scattering : @xmath443{{\ , , } } \end{aligned}\ ] ] whereas the effective amplitude of @xmath444 scattering reads @xmath445{{\ , .}}\end{aligned}\ ] ] to compare the obtained expressions with the vacuum results of the canonical computation eqs . and , we evaluate the retarded and advanced propagators at zero temperature . in vacuum @xmath446 . therefore for positive @xmath447 it follows from eq . that @xmath448^{-1}{{\ , .}}\end{aligned}\ ] ] in the vicinity of the mass shell of the respective majorana neutrino @xmath112 and we find @xmath449 for @xmath54- and @xmath88-channels the imaginary parts of eqs . and vanish , so that in the vacuum limit @xmath450 and @xmath451 is symmetric with respect to @xmath227 . the squares of the @xmath54- and @xmath88-channel propagators in eq . give identical contributions to the reduced cross section of @xmath87 process . therefore , upon substitution of eq . into eq . and the use of the @xmath227 symmetry , we recover for the reduced cross section the canonical result . relations imply that the off - diagonal components of @xmath452 and @xmath451 coincide . for the @xmath53-channel the diagonal components of @xmath453 are given in the vacuum limit by @xmath454 ^ 2}{{\ , .}}\end{aligned}\ ] ] thus , in the vicinity of the mass shell , @xmath455 , the canonical expression for the ris - subtracted propagator , eq . , coincides with the expression obtained from first principles . this justifies results of the earlier calculations . for the reduced cross section of @xmath228 scattering we recover eq .. at finite temperatures @xmath456 is not zero even for @xmath54- and @xmath88-channels . in other words , the medium effects induce additional contributions to the effective decay amplitudes . however , these contributions are proportional to the coefficients @xmath457 . numerical analysis shows that for the two chosen sets of parameters , see appendix[numericalparameters ] , the additional correction typically do not have any sizable impact on the reaction densities . a quantity relevant for the numerical analysis is the ratio @xmath458 . the dependence of this ratio on the dimensionless inverse temperature is presented in fig.[blzmnvsfdbe1 ] and fig.[blzmnvsfdbe2 ] . if the approximate expression is used , then the reaction density of @xmath228 scattering becomes negative for @xmath459 for the first set of the parameters whereas for the second set of the parameters it turns negative for @xmath460 . a qualitatively similar behavior has also been observed in @xcite . washout reaction densities due to @xmath87 and @xmath228 scattering processes for benchmark point 1 in the approximation of massless leptons and higgs . shown are the reaction densities computed using the boltzmann approximation ( [ reactdenscanon ] ) ( thin lines ) , and taking into account the quantum - statistical effects , ( [ reactdensexact ] ) ( thick lines ) . the ( ris subtracted ) reaction densities @xmath193 may be negative as they are not the physical rates for the @xmath461 scattering process . ] this rather counter - intuitive result can be traced back to the behavior of the ris part of the effective amplitude which is negative in the vicinity of the mass shell . its sign is not fixed by physical requirements , since it constitutes a sub - leading contribution to the washout rate . washout reaction densities due to @xmath87 and @xmath228 scattering processes for benchmark point 2 . compare fig.[blzmnvsfdbe1 ] . ] as can be inferred from fig.[blzmnvsfdbe1 ] , for the first set of model parameters the quantum - statistical corrections render the ris subtracted reaction density of @xmath228 scattering positive in the whole range of temperatures . however , this is merely a numerical coincidence . for the second set of parameters , see fig.[blzmnvsfdbe2 ] , the reaction density of @xmath228 scattering remains negative for @xmath462 . it is nevertheless important to note that even in the region where the reaction density is negative the quantum - statistical corrections shift it upwards as compared to the result of the canonical computation . as far as @xmath463 scattering is concerned , the quantum - statistical corrections enhance the reduced reaction density at high temperatures by about 50% . as the temperature decreases , the reaction density computed using eq . approaches the one computed using eq . as one would expect . a similar behavior is also observed for the reaction density of @xmath228 scattering . in figs.[blzmnvsfdbe1 ] and [ blzmnvsfdbe2 ] we neglected the thermal masses of initial and final states . ratio of the reaction density computed using thermal masses with ( thick lines ) and without ( thin lines ) the quantum - statistical terms to the canonical one for the two sets of parameters . ] to estimate the size of the mass corrections , in fig.[scatteringwithmasses ] we plot ratio of the reaction density of the @xmath87 scattering computed using thermal masses of the higgs and leptons with and without the quantum - statistical terms to the canonical one for the two sets of parameters . if the quantum - statistical effects are neglected , the thermal masses lead to a @xmath464 suppression of the reaction densities . on the other hand , an enhancement induced by the quantum - statistical terms to a large extent compensates the mass - induced suppression . as a result , the deviation from the canonical reaction density does not exceed @xmath465 in the whole range of temperatures . to compare the relative importance of the ( inverse ) decay and scattering processes in fig.[washoutrdbm1 ] and fig.[washoutrdbm2 ] we also present the uniformly normalized reaction densities . washout reaction densities due to @xmath87 and @xmath228 scattering processes for benchmark point 1 in the approximation of massless leptons and higgs . for comparison the washout reaction densities for @xmath409 and @xmath410 ( inverse ) decays are shown as well . note that the normalization differs from the one used in fig.[blzmnvsfdbe1 ] and fig.[blzmnvsfdbe2 ] for the present choice of parameters contributions by @xmath461 scatterings are strongly suppressed by the smallness of the couplings . ] in both cases we observe a qualitatively similar picture : for the chosen sets of parameters reaction densities of the scattering processes are strongly suppressed by the smallness of the yukawa couplings as compared to those of the decay processes . washout reaction densities due to @xmath87 and @xmath228 scattering processes as well as ( inverse ) decays for benchmark point 2 . compare fig.[washoutrdbm1 ] . ] in the rate equation for the lepton asymmetry , see eq . , the total washout rate is given by a sum of reaction densities for decays and scattering . whereas @xmath156 and @xmath466 are positive , the ris subtracted reaction density @xmath467 can be negative at some temperatures . if the total washout rate would turn negative , it would lead to a spurious self - enhanced generation of the asymmetry . in figs.[washoutrdbm1 ] and [ washoutrdbm2 ] we show the sum of the reactions densities . for both parameter sets it is positive . this sum should always be positive . if the quantum - statistical terms are neglected this can be demonstrated explicitly . in the absence of the quantum - statistical corrections , the results of this section revert to the ones discussed in section[conventionalapproach ] . using the expression for the ris - propagator , eq . , we can rewrite the ris subtracted scattering amplitude as a difference of the unsubtracted one and the ris term . since the latter is proportional to @xmath468 the integration in eqs . and is trivial and we obtain after some algebra : @xmath469 the total washout rate is then given by @xmath470 and is positive as a sum of two positive functions . in other words , even though @xmath467 can be negative at some temperatures , the total washout rate is always positive . the form of the unsubtracted scattering reaction density that can be inferred from eq . is another manifestation of the double - counting . if we had not subtracted the ris contribution , we would have counted contributions of the inverse decay processes twice and ended up with an incorrect prediction for the generated asymmetry . in the preceding section we have approximately taken the gauge interactions into account in the form of effective masses of the higgs and leptons . the thermal masses are of order of @xmath471 and large enough to influence the values of the reaction densities quantitatively . in particular , the majorana neutrino decay can become kinematically forbidden when the sum of the masses of lepton and higgs exceeds the heavy majorana neutrino mass . for even higher temperatures ( with @xmath472 ) the higgs decay channel into a lepton - majorana pair becomes kinematically allowed instead and can contribute to the asymmetry since it violates _ cp_. for simplicity we do not take modified dispersion relations into account here , see @xcite , but use the simple picture of temperature dependent thermal masses as an estimate : @xmath473 where we use the temperature dependent values of the @xmath21 , @xmath474 , top yukawa and higgs self - couplings @xmath475 , @xmath476 , @xmath477 and @xmath478 assuming a higgs mass of @xmath479 @xcite . we also ignore that the thermal mass of leptons might be better approximated by the ` asymptotic thermal mass ' @xmath480 in a kinematic regime in which their momentum is such that @xmath481 @xcite . we do also not take into account , in our quantitative analysis , the thermal corrections to the majorana neutrino masses as they are negligible compared to their vacuum masses . as they can influence the resonance in this case . ] tree level contribution and one - loop corrections to the ( anti-)higgs decay amplitude . the additional arrows illustrate the direction of momentum flow . ] the _ cp_-violating decay of the higgs arises from the interference of the tree - level , self - energy and vertex graphs depicted in fig.[higgsdecayplot ] . although in this case the decaying particle the higgs doublet is very close to thermal equilibrium due to the yukawa and gauge interactions of the standard model , the majorana neutrino in the final state may deviate from equilibrium , so that the third sakharov condition is fulfilled . using the expression for the divergence of the lepton - current , eq . , we can extract the corresponding _ cp_-violating parameter . to calculate the self - energy contribution it is convenient to rewrite the one - loop lepton self - energy in the form : @xmath482 where the transposition is only in dirac space and we have used one of the properties of majorana propagator : @xmath483 in its _ cp _ conjugate the yukawa couplings are replaced by their complex conjugates and the propagators by the _ cp _ conjugate ones . similar to the case of the majorana decay , substituting eq . into eq . we can define effective higgs decay amplitudes : [ higgsdecayamplitides ] @xmath484{{\ , , } } \hphantom{aa}\\ \label{higgsdecayamplitidesbarln } { \xi^{}_{{\phi}\leftrightarrow \bar { { \ell}}{n}_i}}&\equiv g_w\sum_{mn } ( h^\dagger h)^*_{mn}\nonumber\\ & \times { { \rm tr}}\left[{\bar{\theta}^{mi}_{r}}({{q } } ) ( \slashed{{{q}}}+m_i){\bar{\theta}^{in}_{a}}({{q } } ) p_r\slashed{{{r}}}p_l\right]{{\ , .}}\end{aligned}\ ] ] the overall factor @xmath485 in eqs . comes from summation over the doublet components of the ( decaying ) higgs particle . to leading order in the couplings : [ higgsamplitudes ] @xmath486{{\ , , } } \\ \label{antihiggsamplitude } { \xi^{}_{{\phi}\leftrightarrow \bar{{\ell}}{n}_i } } & \approx 2g_w\bigl[(h^\dagger h)_{ii}({{r}}\cdot { { q}})\nonumber\\ & - { \textstyle\frac{g_w}{16\pi } } \im ( h^\dagger h)_{ij}^2 m_i m_j { \boldsymbol{\mathcal{s}}^{jj}_{h } } ( { { r}}\ , l_\rho)\bigr]{{\ , , } } \end{aligned}\ ] ] which , up to the relative sign in square brackets , coincides with the amplitudes . the corresponding _ cp_-violating parameter reads @xmath487 where @xmath283 and @xmath385 are the momenta of the on - shell final lepton and majorana neutrino with positive zeroth components respectively . the direction of momentum flow is as defined in fig.[higgsdecayplot ] . although @xmath371 in eqs . and is one and the same function , because of the different kinematic regimes the explicit result in terms of the distribution functions differs for the higgs decay : @xmath488{{\ , , } } \end{aligned}\ ] ] see appendix[majoranaselfen ] . note that our result is different from the one presented in @xcite . instead of the @xmath489 dependence , it is proportional to a sum , @xmath490 , of the two distribution functions . this dependence can also be obtained in the framework of real time thermal field theory using causal @xmath349-point functions , compare @xcite . the derivation within the kadanoff - baym formalism gives certainty concerning the sign of the contribution by higgs decay . cp_-violating parameter has an opposite sign relative to that for majorana neutrino decay . however , it is canceled by the relative sign in eq .. to calculate the vertex contribution we use eq . and represent the two - loop self - energy in the form @xmath491 \nonumber{{\ , .}}\end{aligned}\ ] ] its _ cp _ conjugate again differs by the conjugation of the yukawas and propagators . substituting eq . and its _ cp _ conjugate into eq . we obtain for the corresponding effective amplitudes : [ higgsvertexamplitudes ] @xmath492\nonumber\\ & \hspace{3mm}+g_w(h^{\dagger}h)_{ji}^{2}\,m_i\ , { { \rm tr}}\bigr[c v_{jj}(-q ,- k)p_l\slashed{{{r } } } p_r\bigr]{{\ , , } } \\ \label{effamplbarhbarln } & { \xi^{v}_{{\phi}\leftrightarrow \bar { { \ell}}n_i } } \equiv g_w(h^{\dagger}h)_{ij}^2\ m_i { { \rm tr}}\bigr[c v_{jj}(-q ,- k)p_l\slashed{{{r } } } p_r\bigr]\nonumber\\ & \hspace{3mm}+g_w(h^{\dagger}h)_{ji}^{2}\ , m_i { { \rm tr}}\bigl[\varlambda_{jj}(-q ,- k ) c p_l\slashed{{{r } } } p_r\bigr]{{\ , .}}\end{aligned}\ ] ] just like for the self - energy contribution we observe that the overall sign of the vertex contribution to the higgs decay amplitude is opposite to that in the majorana decay , compare eqs . and . the corresponding _ cp_-violating parameter reads @xmath493\nonumber{{\ , .}}\end{aligned}\ ] ] the last three lines of eq . correspond to the three possible cuts of the vertex graph . similarly to the majorana decay only two of the intermediate states can be on - shell and for only one of them the corresponding distribution function enters the result . the value of the vertex _ cp_-violating parameter depends on the temperature as well as on masses of the majorana neutrinos . for definiteness , let us assume a strongly hierarchical mass spectrum , @xmath494 . in this case contribution of the last two lines in eq . is strongly suppressed . integrating out the delta - functions we find for the contribution of the first cut : @xmath495 where the loop function @xmath496 is now defined as : @xmath497 m_i^2 { \boldsymbol{\mathcal{s}}^{ii}_{h}}({{k}}-{{r}}_2 ) { { \ , .}}\end{aligned}\ ] ] note that the vertex _ cp_-violating parameter has an opposite sign relative to that for majorana neutrino decay . similarly to the self - energy contribution we observe that the @xmath498 combination is replaced in by @xmath499 . for a milder mass hierarchy the two other cuts can become important . their contributions are proportional to @xmath500 and @xmath501 respectively . the first - principle computation gives for the higgs decay contribution to the evolution of the lepton current an expression similar to eq . : @xmath502{{\ , .}}\end{aligned}\ ] ] we do not discuss ` extra ' terms here which would arise from ` naive ' boltzmann equations . to write this as a rate equation we need to repeat the steps in section[rateequations ] which lead to eq .. for a general process @xmath503 ( where we allow for deviations from equilibrium in @xmath504 ) we have eq .. therefore we obtain , for higgs decay , the following contributions to the rate equations for lepton number and majorana neutrino abundance ( see appendix[rateequationsapp ] ) : @xmath505 we introduced the decay and washout reaction densities , @xmath506 and @xmath507 , for higgs decay : @xmath508 where now @xmath509 . in complete analogy to eq . the total amplitude and the _ cp_-violating parameter for ( anti-)higgs decay are defined as [ amplsqandepsdefhiggs ] @xmath510 similarly to the majorana neutrino decays we also define an averaged _ cp_-violating parameter as @xmath511 by comparing eqs . and we observe that , ignoring @xmath512 contributions , the difference to the majorana neutrino decay contributions amounts to the replacements @xmath513 and @xmath514 . we therefore obtain eq . with an opposite sign for the _ cp_-violating source term . this sign cancels the relative sign of the _ cp_-violating parameter such that majorana neutrino decay and higgs decay contribute effectively with same sign . similarly , for the contribution to the majorana neutrino rate equation : @xmath515 to compute the thermally averaged _ cp_-violating parameter we take into account the temperature dependent evolution of the lepton and higgs masses @xcite . the averaged _ cp_-violating parameter in the higgs decay and in the majorana decay as functions of the inverse temperature are presented in fig.[epsilongammahiggsbm1 ] and fig.[epsilongammahiggsbm2 ] . averaged self - energy _ cp_-violating parameters for majorana neutrino and higgs decay for benchmark point 1 as a function of the inverse temperature . thin lines represent the value in the zero temperature limit . with conventional dispersion relations , the decay @xmath516 is active at @xmath517 but replaced at high temperatures ( @xmath518 ) by @xmath519 . the decay @xmath520 is active at @xmath521 but replaced at high temperatures ( @xmath522 ) by @xmath523 . ] at very high temperatures the magnitude of @xmath524 can be much larger for higgs decay . averaged self - energy _ cp_-violating parameters for majorana neutrino and higgs decay for benchmark point 2 . see caption of fig.[epsilongammahiggsbm1 ] . ] as the temperature decreases the _ cp_-violating parameter for the higgs decay approaches zero . this is explained by the shrinking of the available phase space in the loop integrals and . in this paper we have studied leptogenesis in the type - i seesaw extension of the standard model using the 2pi - formalism of non - equilibrium quantum field theory . the asymmetry generation can , in the case of thermal leptogenesis , be approximately described by rate equations . usually these statistical equations are treated as a ` black - box ' in the sense that their form is assumed given and model specific amplitudes are inserted by hand . indeed , this approach is supported by the observation that it describes the free decay in the zero temperature limit correctly and inherent inconsistencies ( namely the ` double - counting problem ' ) can be resolved in exact equilibrium . however , out of equilibrium it is not obvious whether the subtraction of real intermediate states works to all orders and how amplitudes computed in thermal field theory enter kinetic equations . these issues can be completely avoided in a systematic treatment within non - equilibrium quantum field theory . only in recent years fundamental questions related to the non - equilibrium statistical description received more attention . the progress here is mainly based on the 2pi - formalism which is known to yield consistent quantum kinetic equations without double - counting . these equations can be reduced to a system of boltzmann - like kinetic equations for quasiparticles which can easily be compared to the conventional results . in the course of the derivation necessary approximations and the related physical assumptions have to be specified explicitly . therefore , this approach enables a deeper insight into the dynamics of the asymmetry generation . in the conventional analysis the minimal set of interactions is obtained at order @xmath117 of the perturbative expansion . we have complemented existing analyses based on the 2pi - formalism by the computation of further processes which appear at this order . starting from a system of kadanoff - baym and ( equivalent ) schwinger - dyson equations for leptons and heavy majorana neutrinos we have derived boltzmann - like quantum - kinetic equations for the lepton asymmetry . they include ( inverse ) decays of the heavy majorana neutrinos as well as two - body scattering processes mediated by the heavy neutrinos : @xmath525 because all terms in this equation are proportional to @xmath318 , a combination of the distribution functions which vanishes in equilibrium , the obtained equations are free of the double - counting problem and no need for the real intermediate state subtraction arises . together with the systematic derivation of the effective decay and scattering amplitudes @xmath49 this is the main result of the present work . the individual amplitudes arise as combinations of different 2pi contributions . the vertex contribution to the _ cp_-violating decay amplitude is obtained as cut of the 2pi ` mercedes ' diagram . the same graph yields also the @xmath526 contributions to @xmath228 and the @xmath527 contribution to @xmath87 process . to extract the self - energy contribution , the off - diagonal elements of the majorana neutrino propagator have to be taken into account . in addition , an extended quasiparticle approximation needs to be employed in order to obtain the @xmath436 and @xmath437 contributions to @xmath228 as well as @xmath528 and @xmath437 contributions to @xmath87 scattering from the ` setting - sun ' diagram . in the zero temperature limit the effective amplitudes reduce to the canonical ones . in particular , the form of the resulting amplitudes for @xmath228 scattering coincides with the ris subtracted amplitudes encountered in existing calculations . at finite temperatures the effective amplitudes receive thermal corrections . medium corrections to the majorana decay amplitudes into leptons and antileptons are @xmath117 . they are small compared to the tree - level vacuum contribution and are therefore negligible for the total decay width . on the other hand , they play an important role for the _ cp_-violating source - terms , which are proportional to the difference of the two amplitudes . we find that medium corrections to the _ cp_-violating parameter are linear in the particle number densities . although there is a partial cancellation of the bosonic and fermionic contribution , the _ cp_-violating parameter is enhanced . in the effective scattering amplitudes the medium corrections affect only the ` regulator ' term in the denominator of the breit - wigner propagators . due to the smallness of the majorana decay width , which is constrained by the light neutrino masses , numerically these corrections are very small in the case of non - degenerate majorana neutrinos . taking sm interactions into account in the form of thermal lepton and higgs masses results in a suppression of the phase space for the majorana neutrino decay and the enhancement of the _ cp_-violating parameters is overcompensated . at even higher temperatures , when the effective higgs mass exceeds the majorana masses , the _ cp_-violating decay of the higgs into a lepton - majorana pair can become kinematically allowed instead . at these temperatures the averaged _ cp_-violating parameters for higgs decay exceeds that obtained for majorana decay in vacuum by orders of magnitude . the signs of the corresponding _ cp_-violating parameters are opposite but their contribution to the lepton asymmetry has the same sign ( at least in the limit of hierarchical majorana masses ) . these results are in qualitative agreement with earlier studies based on thermal field theory and may ultimately be compared to a full treatment of _ cp_-violating decays out of equilibrium , including gauge interactions . we have also derived the corresponding rate equations for abundances of the participating species . they are obtained as expansion in small deviations from equilibrium ( @xmath145 and @xmath529 ) and represent the hydrodynamical approximation of the boltzmann kinetic equations . as compared to the standard ( zero temperature ) result they are improved in that the obtained coefficients include medium corrections to the quasiparticle properties and take into account quantum - statistical effects . we compare with the result obtained if the amplitudes are computed in thermal quantum field theory and the ris subtraction is performed manually . we find that there are differences at higher order in the expansion parameters . the coefficients reaction densities reflect the interplay between the medium enhancement of the effective amplitudes and the phase space suppression induced by the thermal masses of higgs and leptons . at very low temperatures the reaction densities approach their canonical limit . since for a hierarchical mass spectrum most of the asymmetry is typically generated by the lightest majorana neutrino at temperatures of the order and smaller than its mass , we expect a moderate enhancement of the total generated asymmetry is possible for a typical average to strong washout scenario . for a detailed phenomenological analysis it is necessary to include further phenomena such as flavour effects and @xmath530 scattering processes which contribute to the washout at @xmath531 . additional quantum effects beyond the present analysis are relevant for non - standard scenarios in which the majorana neutrinos have degenerate masses or if they are not as close to thermal equilibrium . this work has been supported by the german science foundation ( dfg ) under grant ka-3274/1 - 1 `` systematic analysis of baryogenesis in non - equilibrium quantum field theory '' and the collaborative research center 676 `` particles , strings and the early universe '' . t.f . acknowledges support by the imprs - ptfs . we thank s. blanchet and m. shaposhnikov for useful discussions . the reaction densities contain distribution functions of the initial and final states , which depend on the corresponding energies . therefore to compute the reaction densities we need to analyze the kinematics of the decay and scattering processes . to compute the _ cp_-violating reaction density and the decay reaction density we need to evaluate the integral : @xmath532{{\ , .}}\end{aligned}\ ] ] for the washout reaction densities or reaction densities for higgs decay we have similar expressions . the integration over @xmath533 can be performed trivially and yields @xmath534 . using latexmath:[$|\vec k|=(\vec q\,^2 + \vec p\,^2 - @xmath59 we remove the remaining dirac - delta and obtain @xmath536 the integration limits are given by @xmath537{{\ , , } } \end{aligned}\ ] ] where @xmath538 , @xmath539 and @xmath540 is the usual kinematical function . for an on - shell heavy neutrino @xmath360 . if @xmath541 then @xmath542 and the above expression simplifies to @xmath543 . on the other hand , if @xmath544 then @xmath545 and therefore @xmath546 . since the integration limits coincide in this case , the integral vanishes . combining it with the integration over @xmath547 and using the isotropy of the medium we find @xmath548 { { \ , .}}\end{aligned}\ ] ] if quantum - statistical effects are neglected then both the _ cp_-violating parameter and the total tree - level decay amplitude are momentum independent and the integration can be performed analytically . we get : @xmath549 where @xmath42 is the number of the majorana spin degrees of freedom . similar results are obtained for higgs decay and washout reaction densities . for @xmath461 scattering processes the reaction density is defined by @xmath550 to reduce it to a form suitable for the numerical analysis we insert an identity : @xmath551 into eq .. the resulting expression can be interpreted as a product of the inverse decay and decay amplitudes integrated over the ` mass ' and energy of the intermediate state : @xmath552 for the third line we will use eq .. for the last line it is more convenient to us a different representation . integrating out the delta - function we obtain for the last line in eq . : @xmath553 note that not all angles are kinematically allowed , see eq . below . as a product of lorentz - invariant quantities the integral is also lorentz - invariant . we can therefore boost to the center - of - mass frame where @xmath554 and @xmath555 . by energy - momentum conservation @xmath556 , where @xmath557 now . the angle integration can be partially reduced to integration over the mandelstam variable @xmath558 using the relation @xmath559 . note that the azimuthal angle @xmath560 is lorentz - invariant by itself . therefore , boosting back to the rest frame of the medium we can write the left - hand side of eq . in the form @xmath561 integrating furthermore over @xmath562 and using the fact the the integrand is independent of the orientation of @xmath563 , we finally obtain @xmath564 where @xmath565 . just like for particle decay , the integration limits @xmath566 are given by eq . but with @xmath567 and @xmath568 replaced by @xmath569 and @xmath570 respectively and the three - momentum given by @xmath571 . the range of integration over @xmath54 is given by @xmath572{{\ , .}}\end{aligned}\ ] ] in particular , for massless initial and finial states it reduces to @xmath573 and @xmath574 . if the quantum - statistical effects are neglected then @xmath575 . the integration over @xmath576 can be easily performed in this case and , combined with the @xmath577 prefactor , gives @xmath578 . in the same approximation the last line of does not depend on the distribution functions and gives so - called ` reduced cross section ' : @xmath579 using eq . and integrating over @xmath580 we recover the usual expression for the scattering reaction density : @xmath581 to take the quantum - statistical effects into account we need to express energies of initial and final states in terms of the integration variables . by energy conservation @xmath582 . therefore @xmath576 and @xmath583 as well as the related momenta @xmath584 and @xmath585 are completely fixed by the second and third integration variables . next we consider the final states . by energy conservation @xmath586 . it remains to express @xmath587 in terms of the integration variables . let us choose the coordinates such that @xmath588 points along the @xmath589-axis and @xmath590 lies in the @xmath591-plane . then the momentum transfer @xmath592 also lies in the same plane . its components are given by @xmath593 where : @xmath594 the components of @xmath595 can be written in the form @xmath596 . then the angle between the vectors @xmath595 and @xmath592 is given by @xmath597 using energy - momentum conservation we can express @xmath587 in terms of this angle and the integration variables : @xmath598{{\ , .}}\end{aligned}\ ] ] note that for @xmath599 the difference under the square root in eq . can become negative for some angles . this means that such scattering angles are forbidden kinematically and should not be integrated over . since eq . implicitly depends on @xmath600 it is convenient to use this angle as an integration variable instead of @xmath54 . the integration measure in eq . is then modified according to @xmath601 . to calculate the scattering amplitudes we need the three mandelstam variables . @xmath53 is an integration variable . @xmath54 is given by @xmath602 the remaining one , @xmath88 , can be inferred from the mandelstam relation @xmath603 . the generalized optical theorem is a consequence of the unitarity of the @xmath43-matrix and can be seen as a consistency condition for the amplitudes to ensure conservation of probability . it can also be seen as a consequence of the cutkosky cutting rules @xcite for the computation of the discontinuities of feynman diagrams . as such it can be applied to unstable particles at any given order of perturbation theory . we may write it as @xmath604=\nonumber\\ & = \sum_{i}\left ( \prod_{i_l } \int { { { \,d}\pi^{}_{i_l}}}\right)(2\pi)^4{\delta(\sum_j k_j -\sum_{i_l } q_{i_l})}\nonumber\\ & \times{{\mathcal{m}}_{{a}\rightarrow{i}}}(\{k_i\},\{q_{i_l}\}){{\mathcal{m}}_{{b}\rightarrow{i}}}^*(\{p_i\},\{q_{i_l}\}){{\ , .}}\label{eqn : generalized optical theorem}\end{aligned}\ ] ] the amplitudes @xmath605 include all contributing diagrams ( at a given order of perturbation theory ) and the sum on the right - hand side is over all possible real intermediate states @xmath382 which contribute to @xmath605 . the generalized optical theorem can be exploited to see explicitly why the ris subtraction works . scattering at @xmath606 . the cuts through the internal lepton and higgs lines yield the @xmath436 , @xmath526 and @xmath437-contributions to @xmath76 . the cuts through single internal majorana lines yield the interference terms which contribute to the _ cp_-violating parameter . [ fig : gneral diagram bb - bb ] ] to this end , we apply it to the forward scattering processes @xmath607 and @xmath608 , see fig.[fig : gneral diagram bb - bb ] , above the energy thresholds @xmath609 ( the contribution by @xmath410 real intermediate states etc . can be addressed analogously ) and @xmath610 . we include all possible graphs up to @xmath606 . furthermore , we sum eq . over all internal degrees of freedom of initial and final states and absorb these in the ` effective amplitudes ' defined in section[conventionalapproach ] . we get for the process involving particles @xmath611 the amplitudes squared on the right - hand side contain the relevant graphs including the vertex and self - energy contributions , see fig.[treevertexself ] , whose interference terms lead to _ cp_-violation in the particle decay . in the same way one finds for @xmath612 : @xmath613 as a consequence of _ cpt _ we have @xmath614 . therefore the difference of the left - hand sides of eqs . and as well as that of the third terms on the right - hand - sides vanish . subtracting the right - hand sides we obtain @xmath615=\nonumber\\ & -\int { { { \,d}\pi^{{n}_1}_{{q}}}}(2\pi)^4{\delta({k}+{p}-{q } ) } \big [ { \xi^{}_{{{\ell}}{\phi}\rightarrow{n}_1 } } - { \xi^{}_{\bar{{{\ell}}}\bar{{\phi}}\rightarrow{n}_1}}\big ] , \label{eqn : generalized optical theorem applied at order four difference}\end{aligned}\ ] ] as a requirement for a consistent approximation of the amplitudes compatible with unitarity and _ cpt_. it is obvious that this can not be satisfied if the scattering amplitudes are in tree - level approximation meanwhile the decay amplitudes violate _ cp_. we show in section[risquantstat ] how eq . can be satisfied by replacing the two - body scattering amplitudes @xmath616 . the solution for @xmath617 amounts to subtracting the real intermediate state contributions from @xmath8 . in order to obtain an equivalent result for the higgs decay at high temperature ( i.e. for @xmath618 ) we need to consider different processes since the amplitude @xmath619 can not be obtained as cut of the graphs in fig.[fig : gneral diagram bb - bb ] . one could try to draw and to cut graphs for @xmath620 scattering , but the obtained cuts are lepton number conserving and would drop out in the difference @xmath621 . instead the relevant contributions appear as ` thermal cuts ' of the @xmath54-channel contributions to @xmath622 depicted in fig.[fig : diagram bb - bb contribution ] . scattering at @xmath606 . due to medium effects the particles in the loop can be on - shell.[fig : diagram bb - bb contribution ] ] since they exist only at finite density we need to use finite temperature ` circling rules ' , which can be seen as a generalization of eq . , to compute the imaginary part of causal @xmath349-point functions in the real time formalism @xcite : @xmath623 where ` not all ' means that not all @xmath624 should be circled at the same time . it was shown in @xcite that causal @xmath349-point functions are the ones relevant for the computation of _ cp_-violating parameters . the vertex @xmath625 with largest or smallest time is always circled . we can use this equation together with the circling rules given in @xcite to compute the imaginary parts of the graphs in fig.[fig : diagram bb - bb contribution ] and the _ cpt_-conjugated process @xmath626 . taking the difference of both we obtain , similar to eq . : @xmath627=\nonumber\\ & -\int { { { \,d}\pi^{{\phi}}_{{q}}}}(2\pi)^4{\delta({k}+{p}-{q})}\big [ { \xi^{}_{{{\ell}}{n}_1\rightarrow\bar{{\phi } } } } - { \xi^{}_{\bar{{{\ell}}{n}_1}\rightarrow{\phi } } } \big]{{\ , .}}\end{aligned}\ ] ] we can also use eq . to compute the thermal widths which cutoff the @xmath53- and @xmath54-channel resonances by cutting the self - energy graphs as shown in fig.[rate through thermal cutting ] . the thermal width can be obtained using ( causal ) finite temperature cutting rules . ] furthermore , the thermal _ cp_-violating parameters can be obtained using thermal cutting rules , see e.g. fig.[fig : selfenergy loop corrections imaginary part causal products ] and @xmath628 can not be on - shell simultaneously for @xmath629 . contributions by graphs ( c - e ) are suppressed if the cut is through an internal majorana neutrino line.[fig : selfenergy loop corrections imaginary part causal products],title="fig : " ] + and @xmath628 can not be on - shell simultaneously for @xmath629 . contributions by graphs ( c - e ) are suppressed if the cut is through an internal majorana neutrino line.[fig : selfenergy loop corrections imaginary part causal products],title="fig : " ] and @xmath628 can not be on - shell simultaneously for @xmath629 . contributions by graphs ( c - e ) are suppressed if the cut is through an internal majorana neutrino line.[fig : selfenergy loop corrections imaginary part causal products],title="fig : " ] and @xcite . altogether , the concept of ris subtraction can be generalized to include quantum - statistical effects using thermal quantum field theory in the real time formalism and a complete set of reaction densities can be computed . note however that inconsistencies are inherent out of equilibrium and arise e.g. at higher order in the expansion performed in section[rateequations ] . similar computations where performed in @xcite in the imaginary time formalism of thermal quantum field theory . in this appendix we present some detailed intermediate steps in the derivation of rate equations with quantum - statistical terms . to obtain these we need to assume that the system is close to thermal equilibrium . under certain conditions it is then possible to reduce boltzmann - like equations to a set of rate equations for the systems evolution . assuming that the detailed conditions given in section[rateequations ] are fulfilled and using eq . , we get for the integrated boltzmann equations with quantum - statistical terms : @xmath630 \\ = \int { { { \,d}\pi_{{n}_i{{\ell}}{\phi}}^{{{q}}{{r}}{{k } } } } } ( 2\pi)^4\,{\delta({{q}}-{{r}}-{{k } } ) } \big\ { \pm & { \xi^{}_{{n}_i\rightleftarrows{{\ell}}{\phi } } } ( 1-{f^{}_{{{\ell}}}})(1+{f^{}_{{\phi}}})\big [ \frac{{f^{}_{{n}_i } } -{{f}_{{n}_i}^{eq}}}{{(1-{f^{}_{{n}_i}})}{{f}_{{n}_i}^{eq } } } - ( e^{+\frac{\mu_{{{\ell}}}+\mu_{{\phi}}}{t}}-1)\big]\nonumber\\ - & { \xi^{}_{{n}_i\rightleftarrows\bar{{\ell}}\bar{\phi } } } ( 1-{f^{}_{\bar{{{\ell}}}}})(1+{f^{}_{\bar{{\phi } } } } ) \big [ \frac{{f^{}_{{n}_i } } -{{f}_{{n}_i}^{eq}}}{{(1-{f^{}_{{n}_i}})}{{f}_{{n}_i}^{eq } } } - ( e^{-\frac{\mu_{{{\ell}}}+\mu_{{\phi}}}{t}}-1)\big]\big\ } { ( 1-{f^{}_{{n}_i } } ) } \frac{{{f}_{{n}_i}^{eq}}}{{(1-{{f}_{{n}_i}^{eq}})}}{{\ , .}}\nonumber \end{aligned}\ ] ] this expression is still exact with respect to deviations from equilibrium and , taking into account results before eq . , it has an obvious expansion in @xmath145 . we can see that there will be contributions proportional to @xmath631 , to @xmath145 and proportional to @xmath632 at linear order . the coefficients of these terms were introduced in section[rateequations ] ( eqs . and ) and dubbed reaction densities . we find for the @xmath633 expansion of each of the two terms in eq . , i.e. for the general collision term of @xmath634 in eq . : @xmath635 where we defined the decay reaction densities as @xmath636 with @xmath637 . for a general process @xmath503 ( where we allow again for deviations from equilibrium in @xmath504 ) we find in the same way : @xmath638 where @xmath639 with , now , @xmath640 . note that these reaction densities will tend to zero in the zero temperature limit because of their dependence on the distribution functions . using the general result we obtain the contributions to eq . , and therefore to eq . , proportional to @xmath529 , @xmath641 contributions proportional to @xmath186 , @xmath642 or just proportional to @xmath187 : @xmath643 this leads immediately to the result in eq .. similarly , we get using eq . for the contributions by ( anti-)higgs decay to the rate equations : @xmath644 in addition we get for the extra terms in eqs . and : @xmath645 this means that for @xmath646 only the first term and for @xmath133 the last two terms contribute . the terms proportional to the difference of @xmath647 and @xmath648 lead to contributions proportional to the _ cp_-violating parameter . to reduce the two - body scattering terms in eq . we can use the same methods to find for the general expression , @xmath649{{\ , , } } \nonumber\end{aligned}\ ] ] the @xmath633 expansion : @xmath650\nonumber\\ = & \frac{\mu_c + \mu_d \pm ( \mu_a + \mu_b)}{t}\langle\gamma^{ab}_{ij}\rangle + ( 1\pm 1)\langle\gamma^{ab}_{ij}\rangle\nonumber\\ - & ( 1\pm 1)\langle ( \xi_a\frac{\mu_a}{t}{{f}_{a}^{eq}}+\xi_b\frac{\mu_b}{t}{{f}_{b}^{eq}}-\xi_i\frac{\mu_i}{t}{{f}_{i}^{eq}}-\xi_j\frac{\mu_j}{t}{{f}_{j}^{eq}})\gamma^{ab}_{ij}\rangle{{\ , , } } \nonumber\end{aligned}\ ] ] where the two - body scattering reaction density was introduced in eq .. the last expression results after first order expansion in @xmath145 . in this appendix we derive one - loop contribution to the self - energy of the majorana field as well as one and two - loop contributions to the self - energy of leptons . the 2pi effective action is defined as a functional of the one- and two - point functions consisting of an infinite sum of all 2pi vacuum diagrams @xcite . in practice , its expansion can be characterized in terms of the number of loops appearing in each diagram : @xmath651=\sum_{n } i\gamma_{2{\ensuremath{\,\mathrm{pi}}}}^{(n)}[{s^{}_{}},{\mathscr{s}^{}_{}},{\delta^{}_{}}]{{\ , .}}\end{aligned}\ ] ] the two lowest order contributions , @xmath652 and @xmath653 , relevant for leptogenesis are shown in fig.[fig:2pi contributions ] . their contributions to the 2pi action read [ 2picontrib ] @xmath654{{\ , , } } \hphantom{aa}\\ \label{g3 } i\gamma_{2{\ensuremath{\,\mathrm{pi}}}}^{(3)}= & \frac{1}{2}\int_{\cal c } d^4u\ d^4w\ d^4\eta\ d^4\xi \ { { \rm tr}}\big[h p_r { \mathscr{s}^{}_{}}(u , w)\nonumber\\ & \times cp_r h^t { s^{t}_{}}(\eta , w)\epsilon { \delta^{*}_{}}(w,\xi)\epsilon h^ * p_lc \nonumber\\ & \times { \mathscr{s}^{}_{}}(\eta,\xi)p_l h^{\dagger } { s^{}_{}}(\xi , u)\ \epsilon { \delta^{*}_{}}(u,\eta ) \epsilon \big]{{\ , .}}\end{aligned}\ ] ] in eq . the trace is taken over flavor , dirac and @xmath21 indices whereas the transposition only acts in flavor and dirac space . by functional differentiation of the 2pi effective action with respect to the two - point function we obtain the corresponding self - energy which enters the schwinger - dyson equation : @xmath655}{\delta { s^{\ , t}_{\beta \alpha}}(y , x)}{{\ , .}}\end{aligned}\ ] ] here , flavor indices are shown explicitly whereas the @xmath21 and dirac structure is embodied implicitly in matrix notation . the resulting self - energy is given by a combination of the majorana , lepton and higgs propagators . since we do not consider the flavor effects and the early universe was in an @xmath21-symmetric state : @xmath656 since @xmath657 the lepton self - energy also becomes diagonal : @xmath658 . furthermore , in the unflavored approximation it is convenient to sum over lepton flavors : @xmath659 . then the one- and two - loop order contributions to the lepton self - energy , see fig.[fig:2pi contributions ] , take the form : @xmath660 eventually , it is the wightman components that we are interested in since they enter the gain- and loss terms on the right - hand side of eq .. therefore , we insert the usual decomposition of the propagators @xmath661 into the spectral and statistical parts , eq . , into the self - energies and . a formal decomposition of the self - energy in analogy to allows us to identify its spectral and statistical part and define the wightman components in coordinate space as @xmath662 . for the one - loop self - energy they read @xmath663 in the case of the two - loop contribution , eq . , the computation becomes slightly elaborate . the complication is due to the appearance of 32 different terms after inserting the decomposition for each of the five propagators into eq . as well as due to the two remaining integrations over the internal space - time arguments @xmath664 and @xmath665 . the decomposition makes the path - ordering explicit and allows us to convert the integration along the ctp into an integration along the positive branch . the 32 terms contain different combinations of the @xmath666-functions . these can be rewritten by using relations given in appendix c of @xcite . after some simple but lengthy algebra we obtain for the wightman components : @xmath667{{\ , .}}\end{aligned}\ ] ] specific approximations will allow us to interpret both expressions , eq . and , as describing decay , inverse decay and scattering processes of quasiparticles in the medium . the expressions for the one- and two - loop self - energies given by eqs . and depend explicitly on two coordinates in four dimensional space - time . however , the self - energies which govern the gain- and loss term on the right - hand side of eq . are expressed in terms of phase space coordinates . let us therefore exchange the pair of space - time arguments @xmath668 for an equivalent set of center and relative coordinates , @xmath669 and @xmath670 . in contrast to thermal equilibrium , the out of equilibrium propagators depend not only on the relative coordinate @xmath53 but also on the center coordinate @xmath183 . performing a so - called wigner transformation @xcite , i.e. a fourier transformation with respect to the relative coordinate @xmath53 , we can trade the latter for a momentum space variable : [ wt ] @xmath671 where we have used @xmath672 and @xmath673 , @xmath674 according to the various combinations appearing in eqs . and . note that the factor @xmath675 in the definition is conventional and makes the wigner transform of the spectral propagator a hermitian matrix . definitions of the wigner transforms of the advanced and retarded propagators coincide with that for the statistical propagator . the wigner transform of eq . is obtained straightforwardly : @xmath676 note that motivated by the homogeneity and isotropy of the early universe we only indicate time - dependence of the propagators and self - energy , @xmath677 . to obtain the wigner transform of eq . we will use an additional approximation : each of the wigner transforms of the propagators we replace by @xmath678 . this means that we neglect the variations of @xmath679 from the center coordinate @xmath680 at which the self - energy is evaluated . technically , it corresponds to a gradient expansion to lowest order and therefore disregards all memory effects . this can be compared to the ` stozahlansatz ' within the usual approach to the boltzmann equation . it is convenient to represent the resulting expression as a sum of three terms : @xmath681 the first term on the right - hand side of eq . corresponds to the wigner transform of lines one to three and four to six in eq . : @xmath682\nonumber { { \ , .}}\end{aligned}\ ] ] where we have introduced two functions containing loop corrections : @xmath683{{\ , , } } \end{aligned}\ ] ] and @xmath684 . as we will see , describes _ cp_-violating decay of the heavy majorana neutrino . the second term on the right - hand side of eq . is given by the wigner transform of the seventh line of eq . and describes lepton number violating scattering processes : @xmath685{{\ , .}}\end{aligned}\ ] ] finally the last term in eq . corresponds to the last line of eq . , @xmath686{{\ , , } } \end{aligned}\ ] ] and can be identified with lepton number conserving processes which do not contribute to generation of the lepton asymmetry . differentiating eq . with respect to the two - point function of the majorana neutrino and using definitions of the _ cp_-conjugate two - point functions we obtain for the majorana self - energy @xmath687{{\ , , } } \nonumber\end{aligned}\ ] ] where we have assumed the @xmath21 symmetry of the medium and neglected flavor effects . the factor @xmath22 in eq . comes from the summation over the @xmath21 indices . conjugate self - energy differs from eq . only in the propagators replaced by their _ conjugate counterparts and the couplings replaced by their complex conjugates . from eq . we can read off the wightman components of the self - energy : @xmath688{{\ , .}}\end{aligned}\ ] ] its _ cp_-conjugate can be obtained by complex conjugating the couplings and replacing the two - point functions by their _ cp_-conjugates . to calculate amplitudes of the scattering processes we will need its wigner transform : @xmath689 { { \ , .}}\hphantom{aaaa}\end{aligned}\ ] ] from eq . we can deduce the wigner transform of the corresponding spectral self - energy : @xmath690{{\ , , } } \end{aligned}\ ] ] where we have introduced @xmath691{{\ , , } } \\ { \pi^{}_{\rho}}(t,{{q } } ) & \equiv 16\pi{\textstyle\int}{{\,d}\pi^4_{{{k } } } } { { \,d}\pi^4_{{{r } } } } ( 2\pi)^4{\delta({{q}}-{{k}}-{{r}})}\nonumber\\ & \times p\bigl[{\bar{\delta}^{}_{f}}(t,{{k}}){\bar{s}^{}_{\rho}}(t,{{r } } ) + { \bar{\delta}^{}_{\rho}}(t,{{k } } ) { \bar{s}^{}_{f}}(t,{{r}})\bigr]p{{\ , , } } \end{aligned}\ ] ] and @xmath692 . in the quasiparticle approximation the wigner transforms of the two - point functions of leptons and the higgs are given by eqs. and eqs. respectively . in a _ cp_-symmetric medium , which the early universe was to a very good approximation , @xmath693 and @xmath694 . the homogeneity and isotropy of the early universe furthermore imply , that there is no dependence on the momentum direction and the spatial central coordinate so that @xmath695 and @xmath696 . just like for scalars , in a _ cp_-symmetric medium the fermion two - point functions are related by @xmath697 and @xmath698 . as for the @xmath699 transformation , the terms in the lepton propagators which carry spinor structure are not invariant under it : @xmath700 since @xmath701 , eq . implies that in a homogeneous , isotropic and _ cp_-symmetric medium @xmath702 and @xmath703 are left and right projections of the same ` vector ' integral @xmath704 : @xmath705 l_{\rho}{{\ , .}}\end{aligned}\ ] ] explicit form of @xmath371 depends on the kinematic regime and is presented below . to evaluate the decay amplitudes as well as amplitudes of the @xmath53-channel scattering processes we need to evaluate it for positive @xmath360 and @xmath706 . if @xmath707 is also positive then @xmath371 takes the form : @xmath708{{\ , .}}\end{aligned}\ ] ] to obtain eq . we have used the quasiparticle approximation for the two - point functions and integrated over the zeroth components of the momenta . integrating out the energy - momentum conserving delta - function we obtain for its lorentz components : [ sigmavtherm ] @xmath709{{\ , , } } \end{aligned}\ ] ] where @xmath710 and @xmath711 . the integral functions @xmath712 are defined by @xmath713 where , in complete analogy with eq . , the integration limits are given by @xmath714{{\ , .}}\end{aligned}\ ] ] for positive @xmath369 and negative @xmath715 the components of @xmath371 are related to the ones above by @xmath716 and @xmath717 respectively . to evaluate amplitudes of the @xmath54- and @xmath88-channel processes we also need to calculate @xmath371 for negative square of the momentum transfer . in this case momentum - energy conservation ensures that @xmath718 and @xmath719 can not be positive or negative simultaneously . if they have different signs then , assuming homogeneity and isotropy of the medium and using relations , we find @xmath720\nonumber\\ + & { \delta({{q}}-{{k}}+{{r}})}[f_{\phi}(t,{{k } } ) + f_{\bar{{\ell}}}(t,{{r}})]\bigr\}{{\ , .}}\end{aligned}\ ] ] eq . implies that for negative square of the momentum transfer @xmath371 vanishes in vacuum . although it is in principle possible to retain the thermal masses of leptons and the higgs in the calculation , the resulting expressions are quite lengthy in this case . neglecting the thermal masses we obtain for the lorentz components of @xmath371 in this regime : [ lrhonegmom ] @xmath721{{\ , , } } \end{aligned}\ ] ] where the integral functions are given by @xmath722 note that in this regime @xmath723 and therefore the lower integration limit is positive . to compute the scattering amplitude we need to calculate the product @xmath724 for the @xmath53-channel we find : @xmath725 whereas the corresponding expression for the @xmath54- and @xmath88-channels reads @xmath726 at low temperatures eq . is exponentially small and vanishes in the vacuum limit . to analyze the higgs decay we need to evaluate the spectral loop integral in a region of the phase space where the effective higgs mass exceeds the sum of the majorana and lepton masses . using properties of the distribution functions under the @xmath727 transformation we find after some algebra from eq . : @xmath728.\end{aligned}\ ] ] just like in eq . , the integration is over the ( on - shell ) momenta of the higgs and lepton and the majorana momentum serves as a constraint . note that in this case we are interested only in the on - shell majorana momenta and therefore @xmath360 . after integrating out the delta - function we obtain a result similar to eq . : @xmath729{{\ , , } } \end{aligned}\ ] ] where the integral function is defined as : @xmath730 the integration limits are given by an expression similar to eq . but with @xmath731 replaced by @xmath732 . when the effective higgs mass approaches the kinematic limit , @xmath733 the upper integration limit approaches the lower one , and the integral vanishes to perform the quantitative analysis we need to specify the yukawa couplings . for simplicity we focus on the case of a very heavy third majorana neutrino @xmath734 ( msm ) . in this limit the yukawa couplings can be expressed in terms of the observed active neutrino masses and mixing angles and only one complex additional free parameter @xmath735 . where @xmath740 gev is the higgs vev and we have assumed normal hierarchy . in this case the physical neutrino masses are given by @xmath741 for illustration we choose the benchmark points @xmath742 and @xmath743 denoted by bm1 and bm2 in the plots . as masses of the right - handed neutrinos we choose @xmath744 gev and @xmath745 respectively . this choice of parameters is such that effects related to the resonant enhancement will be unimportant but contributions from both heavy majorana neutrinos can be relevant . note however that there are lower bounds on the washout parameters @xmath746 ( with ` equilibrium neutrino mass ' @xmath747 ) , see figs.[m1tildevsreomegaimomega ] and [ m2tildevsreomegaimomega ] . the freeze - out of the asymmetry will therefore typically occur late ( i.e. @xmath752 ) which renders medium effects in general small . however for the qualitative issues discussed in this paper our preference is to specify a consistent set of parameters for which we can discuss the generation of the lepton asymmetry in terms of two heavy majorana neutrinos . this is of course not a general restriction for the employed techniques .
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in this work we study thermal leptogenesis using non - equilibrium quantum field theory . starting from fundamental equations for correlators of the quantum fields we describe the steps necessary to obtain quantum kinetic equations for quasiparticles .
these can easily be compared to conventional results and overcome conceptional problems inherent in the canonical approach . beyond _
cp_-violating decays we include also those scattering processes which are tightly related to the decays in a consistent approximation of fourth order in the yukawa couplings .
it is demonstrated explicitly how the s - matrix elements for the scattering processes in the conventional approach are related to two- and three - loop contributions to the effective action .
we derive effective decay and scattering amplitudes taking medium corrections and thermal masses into account . in this context
we also investigate _
cp_-violating higgs decay within the same formalism . from the kinetic equations we derive rate equations for the lepton asymmetry improved in that they include quantum - statistical effects and medium corrections to the quasiparticle properties .
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most biopolymers , such as rnas @xcite , proteins @xcite and genomic dna @xcite , are found in folded configurations . folding involves the formation of one or more intramolecular interactions , termed contacts . proper folding of these molecules is often necessary for their function . intensive efforts have been made to measure the geometric and topological properties of protein and rna folds , and to find generic relations between those properties and molecular function , dynamics and evolution @xcite . likewise , topological properties of synthetic molecules have been subject to intense research , and their significance for polymer chemistry @xcite and physics @xcite has been widely recognized . topology is a mathematical term , which is used to describe the properties of objects that remain unchanged under continuous deformation @xcite . different approaches have been discussed in the literature to describe the topology of branched @xcite or knotted polymers @xcite . however , many important biopolymers , such as proteins and nucleic acids , are unknotted linear chains . the circuit topology approach has recently been introduced to characterize the folded configuration of linear polymers . circuit topology of a linear chain elucidates generically the arrangement of intra - chain contacts of a folded - chain configuration @xcite ( see fig . [ fig1 ] ) . the arrangement of the contacts has been shown to be a determinant of the folding rates and unfolding pathways of biomolecules , @xcite and has important implications for bimolecular evolution and molecular engineering @xcite . topology characterization and sorting of polymers has been the subject of intense research in recent years ; bulk purification of theta - shaped and three - armed star polymers is performed using chromatography @xcite ; linear and circular dna are separated in nano - grooves embedded in a nano - slit @xcite ; and star - branched polymers with different number of arms are shown to travel with different speeds through a nano - channel @xcite . in the context of characterization , linear and circular dna molecules are probed by confining them in a nano - channel and using fluorescence microscopy @xcite . we know little about how to sort folded linear polymers based on topology . this is in contrast to size sorting of folded linear polymers which has been studied extensively in the literature @xcite . nano - pore technology represents a versatile tool for single - molecule studies and biosensing . a typical setting involves a voltage difference across the nano - pore in an ionic solution containing the desired molecule . the ion current through the nano - pore decreases as the molecule enters the pore . the level of current reduction and its duration reveals information about the molecule @xcite . prior to the current project , different properties of nucleic acids and proteins have been studied using nano - pore technology , for example : dna sequencing @xcite , unzipping of nucleic acids @xcite , protein detection @xcite , unfolding of proteins @xcite , and interactions between nucleic acids and proteins @xcite . in our study , we used simple models of polymer chains and molecular dynamic simulations to determine how the circuit topology of a chain influences its passage through a nano - pore . we investigated whether nano - pores can be used for topology - based sorting and characterization of folded chains . two scenarios were considered : ( 1 ) passage through pores large enough to permit the chain to pass through without breaking its contacts , and ( 2 ) passage of chains through small nano - pores , during which contacts were ripped apart . in the first scenario , nano - pore technology enabled purification of chains with certain topologies and allowed us to read the topology of a folded molecule as it passed through the pore . in the second scenario , we used the nano - pore to read the circuit topology of a single fold . we also asked if translocation time and chain topology are correlated . this technology has been subject to intense research for simple - structured polynucleotides @xcite ; however , the current study is the first to use nano - pores to systematically measure contact arrangements of folded molecules @xcite ( fig . [ fig1 ] ) . the polymer is modeled by beads connected by fene bonds @xmath0 $ ] . @xmath1 and @xmath2 are the strength and the maximum extension of the bonds , respectively . the fene potential is used to eliminate unrealistic extension of the bonds due to the pulling event . the short - range repulsive interaction between monomers is taken into account by the shifted - truncated lennard - jones potential @xmath3 $ ] at @xmath4 . @xmath5 is the energy scale of the simulations . @xmath6 is the monomer size and the length scale of the simulations . all simulations were performed by espresso @xcite as detailed below . initially , the first monomer is fixed inside the nano - pore . after the whole polymer is equilibrated , the first monomer is unfixed and force , @xmath7 , is applied to pull it through the nano - pore . for pore diameters smaller than two monomers , passage of the polymer inevitably leads to breakage of the contacts . in this case , the bond between the contact sites is replaced with a simple lennard - jones potential @xmath8 $ ] after equilibration . the depth of the attraction well , @xmath9 , is a measure of the strength of the bond between the contact sites . number of passed monomers and position of the first monomer versus time are studied in simulations . these quantities are averaged over different realizations . for longer passages , the averages are again window - averaged over intervals equal to 10 time units . window - averaging is used to reduce the data points and the noise in the plots . to minimize the effect of determinants other than topology , we take equal spacing , @xmath10 , between the contact sites ( connected monomers ) and two tails on the sides equal to the spacings . the total length of the polymer is @xmath11 . if the monomers are numbered consecutively from one end , then , the position of the contact sites along the chain would be @xmath12 , @xmath13 , @xmath14 and @xmath15 . the spacing is taken equal to 12 monomers , unless otherwise stated . some chains become knotted when the bonds are formed in the chain or when the chain is pulled suddenly with a strong force . passage times of these knotted chains are extremely long . the data related to these unusual passages is removed before averaging . consider translocation of 2-contact chains through a nano - pore with an internal diameter equal to @xmath16 . first we assume contacts are permanent and unbreakable . two different strengths for the pulling force , @xmath17 and @xmath18 , are examined . 50 realizations are performed for each of the three topologies ( fig . [ fig1](a ) ) and the two forces . the average number of monomers passed through the nano - pore versus time is shown in figs . [ fig2](a ) and [ fig2](b ) . shoulders in the curves correspond to pauses during the passage of the polymer when the contacts encounter the nano - pore . we first examine the passage dynamics under stronger force , @xmath19 ( fig . [ fig2](a ) ) . one shoulder is observed during the passages of the cross and the parallel topologies , while the series topology is markedly different with two clear shoulders during its passage . the average number of passed monomers at the shoulders coincides with the position of the contact sites ( shown with horizontal lines in the plot ) . this confirms interpretation of the shoulders as the pauses related to the passage of the contacts . the average number of monomers inside the nano - pore versus time is also significantly different for the series topology ( inset of fig . [ fig2](a ) ) . two distinct peaks are observed for the series topology , while only one peak is seen for the cross and the parallel topologies . the peaks in the inset plot occur simultaneously with the shoulders in the main plot . force has a dramatic effect on the passage dynamics of chains . [ fig2](b ) plots the average number of monomers passed through the nano - pore versus time under @xmath17 . here , the maximum passage time for the chain with parallel topology is larger relative to other topologies , while the one with the series topology had the largest maximum passage time under the stronger force . by reducing the force , the passage time gets much longer and the entropic effects become dominant . as a result , the shoulders in the number of passed monomers are not as clear as before . one shoulder is observed for all topologies at the position of the first contact site . two other shoulders are observed during the passage of the chain with parallel contact arrangemenet . the second shoulder corresponds to the time when the large loop of the chain is midway inside the nano - pore . furthermore , the third shoulder is due to the second contact and the pause caused by the entropy of the second small loop in the parallel topology ( shown schematically in the inset ) . additionally , two peaks are seen in the time profile of the average number of monomers inside the nano - pore . these peaks appear simultaneously with the described shoulders in the time profile of the number of passed monomers ( inset of fig . [ fig2](b ) ) . we note that the first and the third shoulders are also seen in the average position of the first monomer versus time ( fig . s2 ) . and ( b ) @xmath17 . the shoulders correspond to pauses in the passage process and can be used to read the chain topology . the series topology shows two shoulders under the strong force ( a ) and the parallel topology shows three shoulders under the weak force ( b ) . insets : average number of monomers inside the nano - pore versus time . the peaks in the inset plots occur at the same time with the shoulders in the main plots . schematics shows the parallel topology and the arrow points to the smaller loop . entropy of the smaller loop causes the last pause in the passage of the parallel topology under the weak force . ] the results show that it is possible to distinguish the series and parallel from other topologies using nano - pores with strong and weak forces , respectively . the number of passed monomers and the number of monomers inside the nano - pore can be readily measured in experiments . the former can be measured by pulling the end of the polymer using optical tweezers @xcite , while the latter is readable by measuring the ion current through the nano - pore . the ion current has been shown to take discrete values with the number of monomers in the nano - pore @xcite . finally , the maximum passage time in each pulling force is also different for the three topologies and can be used alternatively for identifying the topology of a 2-contact unbreakable chain . to generalize the obtained results to molecules of various sizes ( chain lengths ) , we investigate the passage of a chain with two unbreakable contacts and double spacing between the contact sites under weak and strong pulling forces . position of the shoulders and the peaks are in agreement with the above descriptions ( fig . also , it is seen that the maximum passage time is longer for the series topology under the strong force and for the parallel topology under the weak force . ( see esi section 1 ) next , we consider folded molecules with more than two intramolecular contacts . nano - pores of various sizes are needed to pass these complex unbreakable topologies under usual pulling forces . this gives the opportunity to use the nano - pore for purifying topologies or for enrichment of a certain topology from a mixture of different topologies . to test this idea , we examine the passage of 3-contact chains through a pore with an internal diameter equal to @xmath16 under the pulling force @xmath19 . there are 15 topologically different configurations for a 3-contact chain . among these , three configurations have two parallel relations in their topologies , shown in fig 1(b ) . two of them ( among all 15 configurations ) do not pass the nano - pore in usual time intervals . this is in agreement with the expectation that chains with a higher number of parallel topologies tend to interlock more , and do not pass through smaller pores . the three chains shown in fig . [ fig1](b ) behave similarly when they enter the nano - pore from either end . this means that the chain direction is not important in purifying the topologies using a nano - pore . . for the smaller nano - pores , 150 random configurations are averaged . however , up to 700 random configurations are tested for the nano - pores with @xmath20 . all the chains can pass through the nano - pore when the nano - pore diameter is larger than or equal to @xmath21 . ] we then extended the simulations to 5-contact chains as a representation of real chains with increasing complexities . there are ( 2 * 5 - 1)!!=1500 topologically different configurations for a 5-contact chain , so we chose configurations at random and passed them through the nano - pore under the pulling force @xmath19 . the chains are examined to see whether they pass through the nano - pore in a reasonable time . a first - order measure of the circuit topology of a chain is the number of contact pairs that are in series , cross , or parallel arrangements . 3 shows the average numbers of the three topological arrangements versus the internal diameter of the nano - pore , @xmath22 , for chains that pass and do not pass through the nano - pore . the average number of series topology is higher in the passed chains compared to the chains that do not pass . the average number of parallel topology is smaller in the passed chains , for pore diameters smaller than four times the monomer size . average number of cross topology is smaller for passed chains . in a realistic setting , when a mixture of randomly connected chains are allowed to pass through a nano - pore ( smaller than @xmath23 ) , we predict that the flow through would be enriched in series topology . however , the fraction of the mixture that fails to pass through the pore would contain chains with high number of parallel and cross topologies . this can be justified by the fact that in parallel and cross topologies , the contact sites are relatively far from each other along the chain . thus , chains with a high number of contacts with parallel and cross arrangements are bulkier and have more interlocking configurations . in contrast , in series topology , the contacts are local and do not connect distant points along the chain . as a result , the chains with a high number of contacts with series arrangements are more extended in configuration and can pass through the nano - pore more easily . as it is evident from our study , the excluded volume interaction is the main deriving force behind separation in a narrow nano - pore ; this interaction has not been considered in previous theoretical works @xcite . finally , simulations with a four times stronger pulling force @xmath24 shows that no purification is possible under higher forces even with the smallest nano - pore , @xmath25 , reflecting the importance of tuning the applied force to its optimal values . there are two parameters that determine the time needed for the passage of a chain coupled to bond breakup ; the bond strength and the pulling force . we first studied pulling of 2-contact chains through a nano - pore with internal diameter equal to @xmath26 , under a force comparable to the bond strength ( see esi section 2 for a theoretical description ) . for very weak bonds , with a bond strength equal to @xmath27 , the contacts break before reaching the pore . this is due to the tension propagated along the chain from the pulled end @xcite . for medium to strong bonds between @xmath28 , it is possible to see shoulders in the time profile of the position of the first monomer , using suitable pulling forces ( figs . 4(a ) and 4(b ) ) . for shoulders to become prominent , a large pulling force is required to dominate the entropic fluctuations . however , it should not be too strong to completely eliminate the effect of topology . as the leading end of the chain is stretched completely with large pulling forces , the shoulders can be used to find position of the contact sites along the chain ( horizontal lines in figs . 4(a ) and 4(b ) ) . and @xmath29 . the pulling force should be chosen carefully : large enough to minimize the effect of entropy but not too large to eliminate the effect of topology . the shoulders are due to the pauses at the contact sites . position of the shoulders can give information about the position of the connected monomers along the chain ( horizontal lines ) . insets : position of the contact sites can not be tracked in smaller pulling forces . ] for large forces , there is no difference between the passage times of the three topologies . the difference increases by decreasing the pulling force . for very weak forces , however , the entropic effects become significant and hide the effect of topology on the translocation time . moreover , the simulation time becomes very large and simulations ( experiments ) are not cost - effective . the results indicate that moderately weak forces can be used to discriminate the three topologies . for this purpose , we calculate the average passage times of the three topologies using a suitable pulling force ( considering the bond strength ) . then , the maximum and the minimum average passage time is found among the three topologies . figs . [ fig5](a ) and [ fig5](b ) show the topologies that have the minimum and the maximum of the average passage times , respectively . we note that changing the dataset used for averaging does not alter the order of the average passage times of the three topologies for all bond strengths and the corresponding suitable forces ( fig . s7 and table s1 ) . therefore , the average translocation time can be regarded as a tool for reading the chain topology . ( see esi section 3 ) to generalize our findings , we investigate translocation of 5-contact chains to find a correlation between topology of the chain and its average passage time . in three sets of simulations , one of the cross , parallel , and series relations is taken to be dominant in the topology of the chain , meaning that the majority of the contact pairs have the dominant arrangement . we call such states as pure states . more specifically , 8 out of 10 total binary relations are taken the same ; however , the numbers of the other two relations are not determined . the average passage time in each set is calculated over 150 randomly chosen chains that fulfill the mentioned conditions ; @xmath30 , @xmath31 or @xmath32 . for each bond energy and pulling force , we also calculate the average passage time for a fourth set which contains 150 completely random chains . again , we calculate the average passage times for the three pure states . then , the maximum and the minimum passage times are found between the three sets . pure states that have the maximum and the minimum of the average passage times are shown in figs . [ fig5](c ) and [ fig5](d ) . extremely large passage times , which occur due to chain knotting , are removed from the data prior to averaging . the order of the average passage time among pure states does not depend on the data set used , while the data set contains enough data points ( table s2 ) . this shows that the dominant topology in pure states can be recognized by using the passage time through a nano - pore . ( see esi section 4 ) in summary , we studied translocation of folded polymers through nano - pores using molecular dynamics simulations . we found settings that are required for a nano - pore setup to be able to read and sort molecules based on their molecular topology . we showed that nano - pores can be used to efficiently enrich certain topologies from mixtures of random 5-contact chains and that this purification is not sensitive to chain orientation in the nano - pore . we also showed that nano - pores can be used to determine the chain topology for 2-contact chains when the intact folded chains pass through the pore . when the chain unfolds upon passing through the nano - pore , we showed that the nano - pore enables determining the position of the contacts along a 2-contact chain in large pulling forces . in this condition , by using moderate forces , we could discriminate between pure states ( i.e. , states for which the majority of contacts were arranged identically ) by using the average passage time . the authors thank mahdieh mikani for technical help and fatemeh ramazani for careful reading of the paper . vakhrushev a. v. ; gorbunov a. a. ; tezuka y. ; tsuchitani a. ; oike h. liquid chromatography of theta - shaped and three - armed star poly ( tetrahydrofuran)s : theory and experimental evidence of topological separation . chem . _ * 2008 * , 80 , 8153 - 8162 . mikkelsen m. b. ; reisner w. ; flyvbjerg h. ; kristensen a. pressure - driven dna in nanogroove arrays : complex dynamics leads to length - and topology - dependent separation . _ nano lett . _ * 2011 * , 11 , 1598 - 1602 . dorfman k. d. ; king s. b. ; olson d. w. ; thomas j. d. ; tree d. r. beyond gel electrophoresis : microfluidic separations , fluorescence burst analysis , and dna stretching . rev . _ * 2012 * , 113 , 2584 - 2667 . oukhaled g. ; mathe j. ; biance a. l. ; bacri l. ; betton j. m. ; lairez d. ; pelta j. ; auvray l. unfolding of proteins and long transient conformations detected by single nano - pore recording . _ lett . _ * 2007 * , 98 , 158101 . langecker m. ; ivankin a. ; carson s. ; kinney s. r. ; simmel f. c. ; wanunu m. nano - pores suggest a negligible influence of cpg methylation on nucleosome packaging and stability . _ nano lett . _ * 2014 * , 15 , 783 - 790 .
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here we report on the translocation of folded polymers through nano - pores using molecular dynamic simulations .
two cases are studied ; one in which a folded molecule unfolds upon passage and one in which the folding remains intact as the molecule passes through the nano - pore .
the topology of a folded polymer chain is defined as the arrangement of the intramolecular contacts , known as circuit topology . in the case where intramolecular contacts remain intact ,
we show that the dynamics of passage through a nano - pore varies for molecules with differing topologies : a phenomenon that can be exploited to enrich certain topologies in mixtures .
we find that the nano - pore allows reading of topology for short chains .
moreover , when the passage is coupled with unfolding , the nano - pore enables discrimination between pure states , i.e. , states for which the majority of contacts are arranged identically . in this case , as we show here , it is also possible to read the positions of the contact sites along a chain .
our results demonstrate the applicability of nano - pore technology to characterize and sort molecules based on their topology .
| 5,672 | 283 |
the identification of the physical mechanisms responsible for the dissipation of turbulence in the solar wind , and for the resulting heating of the solar wind plasma , remains an important and unsolved problem of heliospheric physics . an important clue to this problem is the observed non - zero fluctuating magnetic helicity signature at scales corresponding to the dissipation range of solar wind turbulence . @xcite first proposed the `` fluctuating '' magnetic helicity as a diagnostic of solar wind turbulence , defining the `` reduced fluctuating '' magnetic helicity spectrum derivable from observational data ( see [ sec : red ] below ) . a subsequent study , corresponding to scales within the inertial range , found values that fluctuated randomly in sign , and suggested an interpretation that `` a substantial degree of helicity or circular polarization exists throughout the wavenumber spectrum , but the sense of polarization or handedness alternates randomly '' @xcite . based on a study of the fluctuating magnetic helicity of solutions to the linear vlasov - maxwell dispersion relation , @xcite suggested instead that , at inertial range scales , all eigenmodes have a very small _ intrinsic _ normalized fluctuating magnetic helicity , eliminating the need to invoke an ensemble of waves with both left- and right - handed helicity to explain the observations . subsequent higher time resolution measurements , corresponding to scales in the dissipation range , exhibited a non - zero net reduced fluctuating magnetic helicity signature , with the sign apparently correlated with the direction of the magnetic sector @xcite . assuming dominantly anti - sunward propagating waves , the study concluded that these fluctuations had right - handed helicity . the proposed interpretation was that left - hand polarized alfvn / ion cyclotron waves were preferentially damped by cyclotron resonance with the ions , leaving undamped right - hand polarized fast / whistler waves as the dominant wave mode in the dissipation range , producing the measured net reduced fluctuating magnetic helicity . we refer to this as the _ cyclotron damping interpretation_. a subsequent analysis of more solar wind intervals confirmed these findings for the dissipation range @xcite . @xcite argued that a comparison of the normalized cross - helicity in the inertial range ( as a proxy for the dominant wave propagation direction in the dissipation range ) to the measured normalized reduced fluctuating magnetic helicity provides evidence for the importance of ion cyclotron damping , which would selectively remove the left - hand polarized waves from the turbulence ; using a simple rate balance calculation , they concluded that the ratio of damping by cyclotron resonant to non - cyclotron resonant dissipation mechanisms was of order unity . a recent study performing the same analysis on a much larger data set concurred with this conclusion @xcite . in this letter , we demonstrate that a dissipation range comprised of kinetic alfvn waves produces a reduced fluctuating magnetic helicity signature consistent with observations . a dissipation range of this nature results from an anisotropic cascade to high perpendicular wavenumber with @xmath0 ; such a cascade is consistent with existing theories for low - frequency plasma turbulence @xcite , numerical simulations @xcite , and observations in the solar wind @xcite . our results imply that no conclusions can be drawn about the importance of ion cyclotron damping in the solar wind based on the observed magnetic helicity signature alone . the magnetic helicity is defined as the integral over the plasma volume @xmath1 , where @xmath2 is the vector potential which defines the magnetic field via @xmath3 . this integral is an invariant of ideal magnetohydrodynamics ( mhd ) in the absence of a mean magnetic field @xcite . @xcite chose to set aside the complications associated with the presence of a mean magnetic field , defining the _ fluctuating magnetic helicity _ by @xmath4 , where the fluctuating quantities denoted by @xmath5 do not include contributions from the mean field . modeling the turbulent magnetic field by @xmath6 in a periodic cube of plasma with volume @xmath7 , we obtain @xmath8 , where the _ fluctuating magnetic helicity density _ for each wave vector @xmath9 is defined by @xmath10 . here @xmath11 and @xmath12 are reality conditions and @xmath13 is the complex conjugate of the fourier coefficient . specifying the coulomb gauge @xmath14 , we obtain @xmath15 where the components @xmath16 arise from the eigenfunctions of the linear wave mode . it is easily shown that this result is invariant to rotation of the wave vector @xmath9 , along with its corresponding linear eigenfunction , about the direction of the mean magnetic field . the _ normalized fluctuating magnetic helicity density _ is defined by @xmath17 where @xmath18 . this normalized measure has values within the range @xmath19 $ ] , where negative values denote left - handed helicity and positive values denote right - handed helicity . we numerically calculate @xmath20 over the @xmath21@xmath22 plane for the eigenmodes of the linear vlasov - maxwell dispersion relation @xcite for a proton and electron plasma with an isotropic maxwellian equilibrium distribution function for each species and no drift velocities ( see * ? ? ? * for a description of the code ) . the dispersion relation depends on five parameters @xmath23 , for ion larmor radius @xmath24 , ion plasma beta @xmath25 , ion to electron temperature ratio @xmath26 , and ion thermal velocity to the speed of light @xmath27 . we specify plasma parameters characteristic of the solar wind at 1 au : @xmath28 , @xmath29 , and @xmath30 . figure [ fig : mhel ] is a contour plot of @xmath20 obtained by solving for the alfvn wave root over the @xmath21@xmath22 plane , then using the complex eigenfunctions to determine @xmath20 . the mhd regime corresponds to the lower left corner of the plot , @xmath31 and @xmath32 ; here , the alfvn wave with @xmath33 is linearly polarized with @xmath34 . as one moves up vertically on the plot to the regime @xmath35 , the solution becomes left - handed with values of @xmath36 . in this regime of nearly parallel wave vectors , the solution represents alfvn waves in the limit @xmath37 and ion cyclotron waves in the limit @xmath38 . the linear wave mode becomes strongly damped via the ion cyclotron resonance at a value of @xmath38 @xcite . this is precisely the behavior supporting the cyclotron damping interpretation of the measured magnetic helicity in the solar wind . but the alfvn wave solution does not always produce left - handed magnetic helicity . if one moves instead from the mhd regime horizontally to the right , the solution becomes right - handed with @xmath39 as @xmath40 , a behavior previously found by @xcite . in this regime of nearly perpendicular wave vectors with @xmath41 , the solution represents alfvn waves in the limit @xmath32 and kinetic alfvn waves in the limit @xmath42 . thus , if the dissipation range is comprised of kinetic alfvn waves , as suggested by theories for critically balanced , low - frequency plasma turbulence @xcite , one would expect to observe a positive normalized fluctuating magnetic helicity signature in that regime . unfortunately , due to the limitations of single - point satellite measurements , equations ( [ eq : hm ] ) and ( [ eq : sigm1 ] ) can not be used directly to calculate the fluctuating magnetic helicity from observations ; approximations must be introduced to define a related measurable quantity . in this section , we calculate the reduced fluctuating magnetic helicity density , as defined by @xcite and used by subsequent authors , for the magnetic field defined by equation ( [ eq : b ] ) , but without assuming the taylor hypothesis . the two - point , two - time magnetic field correlation function is @xmath43 where the angle brackets specify an ensemble average , defined here by @xmath44 . we find @xmath45 where the reality conditions ensure that this quantity is real . we choose to sample this correlation function at a moving probe with position given by @xmath46 ; this corresponds to satellite measurements of the solar wind , where the probe is stationary and the solar wind is streaming past the probe at velocity @xmath47 . thus , we may determine the reduced magnetic field correlation function , @xmath48 , obtaining the form @xmath49 the reduced frequency spectrum , defined by @xmath50 , is then given by @xmath51 . \label{eq : sr}\ ] ] this demonstrates that the frequency @xmath52 of the fluctuations sampled by the moving probe is the doppler shifted frequency @xmath53 . note that adopting the taylor hypothesis @xcite , as often done in studies of solar wind turbulence , corresponds to dropping @xmath54 in equation ( [ eq : sr ] ) . the _ reduced fluctuating magnetic helicity density _ is defined by @xmath55/k_1 . \label{eq : mhr}\ ] ] where the effective wavenumber is calculated from the measured frequency using @xmath56 , assuming the taylor hypothesis is satisfied @xcite , and we have chosen an orthonormal basis with direction 1 along the direction of sampling @xmath57 and directions 2 and 3 in the plane perpendicular to @xmath58 . the _ normalized reduced fluctuating magnetic helicity density _ is given by @xmath59 , where @xmath60 is the trace power . the relation between the reduced fluctuating magnetic helicity density @xmath61 and the fluctuating magnetic helicity density @xmath62 can be seen by writing the spectrum in terms of the doppler - shifted frequency @xmath52 instead of @xmath63 , @xmath64/(\omega'/v)$ ] . using equation ( [ eq : sr ] ) and @xmath65=i(ab^ * - a^*b)$ ] , the reduced fluctuating magnetic helicity density can be written as @xmath66}{\omega'/v } \right ) \nonumber \\ & \times&\delta[\omega'- ( { \mathbf{k}}\cdot { \mathbf{v}}+\omega ) ] \label{eq : hmrom}\end{aligned}\ ] ] equation ( [ eq : hmrom ] ) , the experimentally accessible quantity , is in terms of the magnetic field measurements in a frame defined by the solar wind velocity @xmath47 . to write this in terms of the theoretically calculable @xmath62 ( eq . [ eq : hm ] ) , we express the magnetic field components @xmath67 and @xmath68 in the @xmath69 coordinate system . to do so , define the probe velocity in spherical coordinates about the direction of the mean magnetic field : @xmath70 . the orthonormal basis specified with respect to @xmath58 can be written as @xmath71 finally , we exploit the fact that the solutions of the vlasov - maxwell dispersion relation depend only on the perpendicular and parallel components of the wave vector @xmath21 and @xmath22 with respect to the mean magnetic field , and not on the angle about the field ; thus the eigenfunction for a wave vector @xmath72 can be rotated by an angle @xmath73 about the mean magnetic field to yield the solution for any wave vector @xmath74 . using the above , the reduced fluctuating magnetic helicity density @xmath75 in equation ( [ eq : hmrom ] ) becomes @xmath76 , \label{eq : hmr}\end{aligned}\ ] ] where we have specified the azimuthal angle of the probe velocity @xmath77 without loss of generality . it is clear from this equation that all possible wave vectors @xmath78 that give the same doppler shifted frequency @xmath52 will contribute to the sum for the reduced fluctuating magnetic helicity density at the frequency @xmath52 . predicting the values of @xmath79 for solar wind turbulence based on equation ( [ eq : hmr ] ) requires understanding three issues : the scaling of the magnetic fluctuation spectrum with wavenumber , the imbalance of alfvn wave energy fluxes in opposite directions along the mean magnetic field , and the variation of the angle @xmath80 between the solar wind velocity @xmath47 and the mean magnetic field . the 1-d magnetic energy spectrum in the solar wind typically scales as @xmath81 in the inertial range and @xmath82 in the dissipation range , where @xmath83 @xcite and the effective wavenumber is @xmath84 . it is clear from equation ( [ eq : hmr ] ) that , when the plasma frame frequency @xmath54 is negligible , the doppler - shifted observed frequency always results in an effective wavenumber @xmath85 , with equality occurring only when the velocity @xmath47 is aligned with the wave vector @xmath9 . we assume that , for homogeneous turbulence at the dissipation range scales , turbulent energy at fixed @xmath21 and @xmath22 is uniformly spread over wave vectors with all possible angles @xmath73 about the mean magnetic field . because the fluctuation amplitude deceases for larger effective wavenumbers , the contribution to @xmath79 is maximum at angle @xmath86 ; for angles @xmath73 yielding a doppler shift to lower effective wavenumbers @xmath87 , the higher amplitude fluctuations at those lower wavenumbers will contribute more strongly to @xmath79 . an accurate calculation of the magnetic helicity signature based on equation ( [ eq : hmr ] ) must take into account the scaling of the magnetic energy spectrum . to compare to @xmath88 derived from observations ( for example , see figure 1 of @xcite ) , we construct the normalized quantity @xmath89 } { \sum_{{\mathbf{k } } } [ |{\mathbf{b}}({\mathbf{k}})|^2/k ] \delta[\omega'- ( { \mathbf{k}}'\cdot { \mathbf{v}}+\omega)]}. \label{eq : numsigm}\ ] ] in evaluating equation ( [ eq : numsigm ] ) , we assume a model 1-d energy spectrum logarithmic gridpoints over @xmath90 $ ] , the model weights @xmath91 as a function of @xmath92 using @xmath93/[1+(k \rho_i)^{2}]\}^2 $ ] . ] that scales as @xmath94 for @xmath95 and @xmath96 for @xmath97 , consistent with theories for critically balanced turbulence @xcite and solar wind observations @xcite . in figure [ fig : mhelr ] , we plot @xmath98 vs. effective wavenumber @xmath84 for a turbulent spectrum filling the mhd alfvn and kinetic alfvn wave regimes ( @xmath99 and @xmath100 ) for @xmath28 , @xmath29 , @xmath30 , @xmath101 , and @xmath102 . the contributions to @xmath98 for all angles @xmath73 of each wave vector are collected in 120 logarithmically spaced bins in doppler - shifted frequency . the results are rather insensitive to the scaling of the 1-d magnetic energy spectrum over the range from @xmath103 to @xmath104 . the solid line in figure [ fig : mhelr ] corresponds to the model spectrum assumed above , while the dashed line corresponds to a @xmath103 energy spectrum . figure [ fig : mhelr ] demonstrates that turbulence consisting of alfvn and kinetic alfvn waves produces a positive ( right - handed ) magnetic helicity signature in the dissipation range at @xmath105 . the analysis presented in figure [ fig : mhelr ] considers only waves with @xmath106 , so all of the waves in the summation in equation ( [ eq : hmr ] ) are traveling in the same direction . if there were an equal alfvn wave energy flux in the opposite direction a case of balanced energy fluxes , or zero cross helicity the net @xmath98 would be zero due to the odd symmetry of @xmath62 in @xmath22 . it is often observed , at scales corresponding to the inertial range , that the energy flux in the anti - sunward direction dominates , leading to a large normalized cross helicity @xcite . if this imbalance of energy fluxes persists to the smaller scales associated with the dissipation range , a non - zero value of @xmath98 is expected . however , theories of imbalanced mhd turbulence ( * ? ? ? * and references therein ) predict that the turbulence is `` pinned '' to equal values of the oppositely directed energy fluxes at the dissipation scale . this implies that , at sufficiently high wavenumber @xmath63 , the value of @xmath98 should asymptote to zero . thus , @xmath98 in figure [ fig : mhelr ] would likely drop to zero more rapidly than shown , leaving a smaller positive net @xmath98 around @xmath107 , consistent with observations @xcite . we defer a detailed calculation of the effects of imbalance to a future paper . the angle @xmath80 between @xmath108 and @xmath47 is likely to vary during a measurement ; this angle does not typically sample its full range @xmath109 , but has some distribution about the parker spiral value . calculations of @xmath98 over @xmath110 yield results that are qualitatively similar to figure [ fig : mhelr ] , so this averaging will not significantly change our results . taken together , we have demonstrated that a solar wind dissipation range comprised of kinetic alfvn waves produces a magnetic helicity signature consistent with observations , as presented in figure [ fig : mhelr ] . the underlying assumption of the cyclotron damping interpretation of magnetic helicity measurements , an interpretation that dominates the solar wind literature @xcite , is the slab model , @xmath111 and @xmath112 , i.e. , purely parallel wave vectors . as shown in figure [ fig : mhel ] , only in the limit @xmath113 does the alfvn wave root generate a left - handed helicity @xmath36 as @xmath114 ; in the same limit , the fast / whistler root generates a right - handed helicity @xmath39 in a quantitatively similar manner ( see figure 9 of @xcite ) . strong ion cyclotron damping of the alfvn / ion cyclotron waves as @xmath115 @xcite would leave a remaining spectrum of right - handed fast / whistler waves , as proposed by cyclotron damping interpretation . however , only if the majority of the turbulent fluctuations have @xmath116 is the slab limit applicable , and only if significant energy resides in slab - like fluctuations are the conclusions drawn about the importance of cyclotron damping valid . there is , on the other hand , strong theoretical and empirical support for the hypothesis that the majority of the energy in solar wind turbulence has @xmath0 ( see @xcite and references therein ) . in this case , there is a transition to kinetic alfvn wave fluctuations at the scale of the ion larmor radius . this letter demonstrates that a dissipation range comprised of kinetic alfvn waves produces a reduced fluctuating magnetic helicity signature consistent with observations . g. g. h. thanks ben chandran for useful discussions . g. g. h. was supported by the doe center for multiscale plasma dynamics , fusion science center cooperative agreement er54785 . e. q. and g. g. h. were supported in part by the david and lucille packard foundation . e. q. was also supported in part by nsf - doe grant phy-0812811 and nsf atm-0752503 .
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measurements of small - scale turbulent fluctuations in the solar wind find a non - zero right - handed magnetic helicity .
this has been interpreted as evidence for ion cyclotron damping .
however , theoretical and empirical evidence suggests that the majority of the energy in solar wind turbulence resides in low frequency anisotropic kinetic alfvn wave fluctuations that are not subject to ion cyclotron damping .
we demonstrate that a dissipation range comprised of kinetic alfvn waves also produces a net right - handed fluctuating magnetic helicity signature consistent with observations .
thus , the observed magnetic helicity signature does not necessarily imply that ion cyclotron damping is energetically important in the solar wind .
| 5,466 | 180 |
block copolymers , constructed by linking together chemically distinct subchains or blocks , spontaneously assemble into exquisitely ordered soft materials@xcite . the self - assembled order structures , spanning length scales from a few nanometers to several micrometers , offer a diverse and expanding range of practical applications in , for example , optical materials , microelectronic materials , drug delivery , advanced plastics , and nanotemplates@xcite . the development of nanotechnology using block copolymers requires a good understanding of the phase behavior of the block copolymers . the self - assembling mechanism of block copolymers sensitively depends on block types , the number of blocks , block - block interactions , architecture , and topology of the block polymers . the equilibrium ordered patterns can be formed due to the delicate balance between these competing factors . in abc triblock copolymers , the number of controlled parameters is at least five , including three interaction parameters @xmath2 , @xmath3 , @xmath4 , and two independent block compositions @xmath5 and @xmath6 . @xmath7 is the degree of polymerization and @xmath8 is the flory - huggins interaction parameter characterizing the interaction between two chemically different blocks @xmath9 and @xmath10 . compared with linear copolymers , the star - shaped copolymers have complicated phase behavior or physical properties induced by different molecular architecture . in the ordered phases of abc star triblock copolymers , the most distinct feature is the arrangement of the junction points . if the chain lengths of three blocks are comparable , junction points are aligned on a one - dimensional ( 1d ) straight line , then cylindrical morphologies can be formed naturally . their cross sections tend to show two - dimensional ( 2d ) patterns since polymer / polymer interfaces can be flat surfaces due to the repulsion forces between `` unconnected '' branches . these factors lead to the formation of 2d polygonal tiling patterns , or archimedean tiling . the tiling patterns can be encoded by a set of integers @xmath11 $ ] , indicating that a @xmath12-gon , an @xmath13-gon , and an @xmath14-gon , etc . , meet consecutively at each vertex . if some asymmetry , to the contrary , is introduced to the compositions , the junction points are mostly aligned on the curved trails . consequently three - dimensional ( 3d ) structures can be formed . furthermore , when the interactions are strong enough , the abc star triblock copolymers can self - assemble into hierarchical structures@xcite . owing to the rich phase behavior , a number of experiments have been carried out on the phase and phase transition of abc star triblock copolymers in past decades@xcite . in particular , matsushita and co - workers have conducted systematic studies on the morphologies formation of ( polyisoprene - polystyrene - poly(2-vinylpyridine ) ) ( isp ) star triblock copolymers . several ordered tiling patterns , such as [ 6.6.6 ] , [ 8.8.4 ] , [ 12.6.4 ] , [ 3.3.4.3.4 ] , and even a dodecagonal symmetric quasicrystalline tiling@xcite , have been observed in the isp star triblock copolymers@xcite . meanwhile , several hierarchical structures including cylinders - in - lamella , lamellae - in - cylinder , lamellae - in - sphere , and hierarchical double gyroid structures are also discovered with the asymmetries of composition@xcite . recently , nunns et al . used polyisoprene ( i ) , polystyrene ( s ) and poly-(ferrocenylethylmethylsilane ) ( f ) to synthesize isf star triblock copolymers@xcite . three 2d patterns , including [ 8.8.4 ] , [ 12.6.4 ] , and lamellae with alternating cylinders , were observed . both of the experimental systems satisfy asymmetrical interaction between different blocks , i.e. , @xmath15 . in particular , in isp star triblock copolymers , the interaction strengths are known to follow the order @xmath16@xcite , while in isf star - shaped triblock copolymers , @xmath17@xcite . besides experimental works , theoretical studies provide a good understanding of phase behavior of abc star triblock copolymers . bohbot - raviv and wang@xcite used a coarse - grained free energy functional to numerically investigate some morphologies of abc star triblock copolymers . in 2002 , gemma and co - workers@xcite carried out monte carlo ( mc ) simulations on abc star triblock copolymers with equal interactions between the three components . the phase behavior of abc star triblock copolymers with composition ratio of , @xmath18 , was investigated in detail in strong segregation region . five kinds of 2d cylindrical phases , three kinds of lamellar - type phases and two kinds of continuous matrix phases were obtained from the mc simulations . huang and co - workers@xcite studied the effects of composition and interaction parameter on the phase behavior of abc star copolymers with equal interactions among the three components using dissipative particle dynamics ( dpd ) simulations . several efforts have also been made to study the phase behavior of abc star triblock copolymers using the scft@xcite . in 2004 , tang et al.@xcite started to use a 2d scft simulation to study the phase behavior of abc star triblock copolymers . based on the scft , zhang et al.@xcite and li et al.@xcite examined the weak and intermediate segregation cases of mainly 2d structures with equal interactions . zhang et al.@xcite also chose @xmath19 , @xmath20 to model the isp star triblock copolymer system of the type @xmath21 . accordingly , a 1d phase diagram was obtained as a function of @xmath22 from @xmath23 to @xmath24 . the stability of the different lamellar morphologies formed from abc star triblock copolymers has been examined by xu et al.@xcite . these scft studies mainly focus on the symmetrically interacting systems and the phase behavior of 2d structures . despite these previous experimental and theoretical studies , a comprehensive understanding of abc star triblock copolymers is still lacking , especially for the asymmetric interaction systems . in this work , we will explore the phase behavior of abc star triblock copolymers , including 2d and 3d structures , with asymmetric interaction parameters between chemically different blocks . theoretical approaches to investigating the phase behavior of block copolymers , often involve minimizing an appropriate free energy functional of the system , and comparing the free energies of different candidate structures . therefore a systematic examination of the emergence and stability of ordered phases requires the availability of suitable free energy functionals and accurate methods to compute the free energy of ordered phases . owing to a large number of studies , it has been well proven that the scft provides a powerful theoretical framework for the study of the phase behavior of block copolymers@xcite . in particular , the scft can be used to determine the relative stability of different phases because it provides an accurate estimate of the free energy . the essence of the scft is that the free energy of the system can be written as a functional of the spatially varying polymer densities and a set of conjugate fields . minimizing the free energy functional with respect to the densities and conjugate fields leads to a set of equations , encoded as scft equations . the scft equations are a set of highly nonlinear equations with multi - solutions which correspond to the different ordered phases of block copolymers . the equations also have a strong nonlocality that emerges from the connection of propagators and densities , conjugate fields . solving the scft equations requires iterative techniques . owing to the nature of iterative methods , the solutions sensitively depend on the initial configuration at the start of the iteration . a series of efficient strategies of screening initial conditions are developed based on the fact that all periodic structures belong to one of @xmath25 space groups@xcite . in each iterative step , efficient numerical schemes are required to solve the propagator equations . in the past years , two complementary methods , including spectral methods@xcite and real - space methods@xcite to solve the scft equations have been developed . in recent years , an efficient pseudospectral method has been introduced to solve propagator equations@xcite . this algorithm takes advantage of the best features of real- and fourier - space with the computational effort scale of @xmath26 based on the fast fourier transformation ( fft ) . @xmath7 is the number of spectral modes or discrete points in real - space . to obtain the solutions of the scft equations corresponding to the saddle points of the scft free energy functional , the iterative methods are required to make the iteration convergent . the quasi - newton methods were employed in the fully spectral approach to the scft by matsen and schick@xcite . a simple mixing method by a linear combination of two consecutive fields was introduced by drolet and fredrickson@xcite for the scft simulations . recently the anderson mixing method@xcite has been proven by itself to greatly reduce the number of scft iterations . from the perspective of nonlinear optimization , ceniceros and fredrickson@xcite devised a class of efficient semi - implicit schemes for solving the scft equations using the asymptotic expansion technology . later , jiang et al.@xcite have extended these algorithms to the scft calculations for multicomponent polymer systems . a generic strategy of theoretical studying phase behavior of complex block copolymer systems includes two steps@xcite . the first step involves an efficient strategy to produce a library of possible candidate structures . in the second step , the candidate structures are used as initial conditions in the more accurate methods to compute free energies , which are used to construct phase diagrams of the systems . in this work , we apply this strategy to examine the phase behavior of abc star triblock copolymers using the scft . specifically , the strategies developed in our previous work@xcite are used as a screening technique to obtain candidate structures as many as possible . these strategies include ( 1 ) knowledge from previous experiments and theories ; ( 2 ) knowledge from related systems , for example , diblock copolymers ; ( 3 ) combination and interpolation of known structures ; and ( 4 ) random initial configurations . using these candidate structures as initial conditions , a fourth - order pseudospectral method combined with anderson mixing method is employed to study the stability of ordered phases . to model the asymmetric interacting experimental systems of isp and isf , the interaction parameters of @xmath27 are used in our study . it should be emphasized that , to broaden the scope of the research , the 2d and 3d ordered phases are included in our calculations . the resulting free energies of the different ordered phases are used to construct phase diagrams . there are two main steps in studying the phase behavior of block copolymers . the first step is to obtain possible candidate structures as many as possible . several strategies of exploring ordered phases have been proposed in our previous works@xcite . beyond the random initial values , these approaches include ( 1 ) knowledge from experiments and theories , such as small - angle x - ray scattering images , space group theory for periodic structures ; ( 2 ) knowledge from related systems ; ( 3 ) combination and interpolation of known structures . using these diverse strategies of initialization , a large number of ordered phases can be generated as the solutions of the scft equations . in addition , the possible candidate structures from previous theoretical and experimental studies are considered in our calculations . the second step is to identify the stability of these phases by comparing their free energies using efficient numerical methods . in this work , we combine an improved pseudosepectral method with anderson mixing algorithm to solve the scft equations for periodic block - copolymer morphologies . a fourth - order accurate adams - bashford scheme@xcite is used to discretize the mdes . the initial values required to apply this formula are obtained using a special extrapolation method@xcite , based on the second - order operator - splitting scheme@xcite . a modified integral formula for closed interval is chosen to solve the integrated equations - that can guarantee fourth - order precision in @xmath46-direction whether the number of discretization points is even or odd@xcite . to ensure the accuracy , we require that these substeps of contour length @xmath46 are smaller than @xmath53 in the fourth - order accurate scheme . the fft is used to translate the data between real- and fourier - space in the pseudospectral method . to obtain the equilibrium morphologies of scft equations , iterative methods shall be required to update the conjugate fields . for this step we choose the anderson mixing algorithm , firstly proposed by anderson@xcite , then introduced into polymer theoretical calculations by schmid and mller@xcite , thompson et al.@xcite owing to the local convergence of this method , we use the simple mixing method alone at the start of the algorithm to obtain better initial values for the fields , followed by anderson mixing approach alone to accelerate the convergent procedure to the prescribed accuracy in the fields . the anderson mixing method requires the fields of previous @xmath28 steps when update new fields . using the previous @xmath28 step fields , the anderson mixing method produces an @xmath28-order linear equations system from a least square problem . then the new fields will be obtained by a combination of the fields of previous @xmath28 steps , with the solution of linear system being the weight factors . for relatively simpler systems , such as diblock copolymers , the anderson mixing algorithm can significantly reduce the required number of iterations with few histories@xcite . for more complex situation , a larger @xmath28 shall be taken to update fields to accelerate the convergent procedure . we find that assembling the @xmath28-order linear system spends more computation time than solving this linear system . when @xmath28 becomes too large , it will slow the iteration . here we overcome this problem by rearranging the elements of the @xmath54 matrix . the technical details can be found in the appendix section . by using our approach , the anderson mixing method can robustly converge without slowing the scft iterations . in practice , we use the available histories of @xmath55 steps . here we only consider the periodic structures , therefore , periodic boundary conditions are imposed on each direction . in our calculations , all spatial functions are all expanded in terms of plane waves . for 2d morphologies , a square box is simulated . @xmath56 plane - wave basis functions are used to discretize the 2d box . for 3d structures , we use a cubic unit cell in most of our calculations . the number of plane - wave basis functions is @xmath57 . the size of computation box plays an important role in determining the stability of ordered phases . for a given phase , its free energy is minimized with respect to the box sizes by the steepest descent approach coupled with solving the scft equations@xcite . based on the discussion above , we summarize the iteration procedure by sketching the numerical recipes : step 1 : : starting from the given initial conditions and computational box , the fourth - order pseudosepectral method combined with anderson mixing method is applied to obtain the ordered phases when the field s change is smaller than the prescribed error @xmath58 . step 2 : : optimizing the size of unit cell by minimizing the free energy with the steepest descent method . step 3 : : goto * step 1 * until the free energy change is smaller than a given error @xmath59 . to ensure enough accuracy , in our implementation , each calculation is terminated until the field s change ( defined in appendix ) at each iteration is reduced to @xmath60 ( corresponding to a free energy change of about @xmath61 ) , and @xmath62 . for the cases of @xmath63 , @xmath64 in the current work , the relative values of free energy among different phases in determining the phase boundary are from @xmath65 to @xmath66 . therefore our numerical resolution is adequately accurate for constructing phase diagrams . the previously theoretical studies mainly focus on the equally interacting abc star triblock copolymer systems , i.e. , @xmath67@xcite . however , under experimental circumstances as mentioned above , many abc star triblock copolymers with asymmetric interactions have been synthesized to observe their phase behaviors , such as isp@xcite and isf@xcite star triblock copolymers . in particular , the interaction parameters of the above systems satisfy the relationship @xmath68 . in order to make a meaningful comparison of the theoretical study and experimental investigation , the interaction parameters are chosen in the calculations so that they are appropriate for these experimental systems . although accurate values of the flory - huggins parameters are not available in the literatures , qualitative interaction strengths are known to follow the order @xmath69@xcite , @xmath70@xcite . in what follows we choose asymmetric interaction parameters , @xmath63 , @xmath64 , and equal statistical segment lengths . this is a rough approximation to isp and isf copolymers , with the important difference that , in the idealized system , we neglect the small differences between @xmath2 and @xmath4 . as mentioned above , there are two main steps in studying the phase behavior of block copolymers . the first step is to obtain as many possible candidate structures as possible . the second step is to identify the stability of these patterns by comparing their free energies and construct the phase diagrams . in order to further analyse the stability of candidate patterns , it is helpful to split the free energy into two parts : internal ( @xmath71 ) and entropic ( @xmath72 ) . the internal and entropic contributions to the free energy can be expressed as@xcite @xmath73 where @xmath37 are the block labels . using different initialization procedures , a large number of candidate structures have been obtained in our previous work@xcite . here we only present the stable phases in the case of @xmath63 , @xmath64 , as shown in fig.[fig : phases ] . among these candidate phases , several polygon tiling patterns , or the cylindrical structures , with translation invariant along the third direction are found as 2d phases . these polygon patterns include [ 6.6.6 ] , [ 8.8.4 ] , [ 12.6.4 ] , [ 3.3.4.3.4 ] , [ 8.6.6;8.6.4 ] , [ 10.6.6;10.6.4;8.6.6;8.6.4 ] . the first pattern is designated as [ 6.6.6 ] because each vertex in this tiling is surrounded by three hexagonal polygons . similarly the second and third patterns are named [ 8.8.4 ] and [ 12.6.4 ] , respectively . the fourth pattern looks more complex . there are two types of vertices : one is surrounded by 10-gon , 8-gon and 4-gon , whereas the other is formed by a decagon , a hexagon and a tetragon . from our naming convention , it should be named [ 10.8.4;10.6.4 ] . however , with the help of triangles and squares of hypothetical tiling , as fig.[fig : phases ] shows the superimposed tiling on a schematic drawing , it is noted all the meeting vertices are surrounded by three regular triangles and two squares , which is one of the archimedean tilings . in order to compare with experimental results , we encode it as [ 3.3.4.3.4]@xcite . the fifth pattern also possesses two kinds of vertices : one is surrounded by 8-gon , 6-gon , and 4-gon , whereas the other is formed by an 8-gon and two 6-gons . consequently , this pattern is designated as [ 8.6.4;8.6.6 ] . similarly , the fifth pattern has four kinds of vertices , therefore , it is named [ 10.6.6;10.6.4;8.6.6;8.6.4 ] . besides these polygonal phases , additional three 2d structures , i.e. , three color lamellae ( lam ) , the core - shell cylinders ( hc ) , and hierarchical cylinders - in - lamella phases ( l+c ) are also obtained in our simulations . at the same time , a series of 3d structures are obtained . from the morphology of patterns , these 3d patterns can be classified into core - shell phases and hierarchical structures . the former includes core - shell spheres in body - centered - cubic lattice ( bcc ) and core - shell double - gyroid phases ( dg ) . the hierarchical patterns consist of two kinds of hierarchical cylinders packed hexagonally ( hhc ) , two kinds of cylinders - in - lamella phases , with the cylinders being packed hexagonally ( hpl ) and tetragonally ( tpl ) , and hierarchical gyroid phases ( hdg ) . note that due to the equal interaction parameters of @xmath74 , these structures have their mirror phases along the phase path of isopleth @xmath75 . ( @xmath77 ) . note that in the @xmath78 ^ 1 $ ] phase , the minority c blocks form the 4-coordinated domains , and blocks a and b alternatively form 8-coordinated microdomains . while in the @xmath78 ^ 2 $ ] morphology , the a and c blocks form the 8-coordinated polygons , and b blocks form the domains with 4-coordinations . in one periodicity , the @xmath79 phase has the cacb lamellar sequence , whereas the @xmath80 structure has the babc layers . ] in a series of experiments@xcite , two of the three arms are kept to be of equal length and the arm - length ratio is expressed as @xmath81 . motivated by these experiments , we start with the calculation of the phase stability along this phase path . here we assume that a and b arms have equal length and the c arm holds the arm - length ratio of @xmath82 . the free energy differences of candidate phases from the value of the homogeneous phase as a function of the volume fraction of @xmath83 are given in fig.[fig:11x](a ) . the phase stability regions as a function of @xmath22 are presented in fig.[fig:11x](b ) . in fig.[fig:11x](a ) , the free energies of some metastable phases along this path are not shown , such as that of [ 10.6.6;10.6.4;8.6.6;8.6.4 ] in the region of @xmath84 , where it has higher free energy than [ 6.6.6 ] , or @xmath78 ^ 2 $ ] , or [ 3.3.4.3.4 ] . with the increase of @xmath22 , the phase sequence is @xmath79 @xmath85 @xmath78 ^ 1 $ ] @xmath85 [ 8.6.6;8.6.4 ] @xmath85 [ 6.6.6 ] @xmath85 @xmath78 ^ 2 $ ] @xmath85 [ 3.3.4.3.4 ] @xmath85 [ 12.6.4 ] @xmath85 tpl @xmath85 @xmath80 . the corresponding stable regions are @xmath86 ( @xmath79 ) , @xmath87 ( @xmath78 ^ 1 $ ] ) , @xmath88 ( [ 8.6.6;8.6.4 ] ) , @xmath89 ( [ 6.6.6 ] ) , @xmath90 ( @xmath78 ^ 2 $ ] ) , @xmath91 ( 3.3.4.3.4 ) , @xmath92 ( [ 12.6.4 ] ) , @xmath93 ( tpl ) , and @xmath94 ( @xmath80 ) , respectively . when @xmath83 is small , in one period , the lamellar structure of @xmath95 of cacb type includes two thick a and b layers , and two thin c layers . the symbols of @xmath78 ^ 1 $ ] and @xmath78 ^ 2 $ ] are used to distinguish the 4-coordinated polygons segregated by different block . in the @xmath78 ^ 1 $ ] phase , the minority c - arms form the 4-coordinated domains , and blocks a and b alternatively form 8-coordinated polygons . while the unit cell of @xmath78 ^ 2 $ ] contains one 4-coordinated b - domain , 8-coordinated and 4-coordinated microdomains formed by blocks a and c , respectively . from the phase path , in tiling patterns , the coordination number of c - domains is proportional to the volume fraction of @xmath83 . with the increment of @xmath83 from @xmath96 to @xmath97 , the coordinations of c - domains change from @xmath98 , to @xmath99 , to @xmath100 , then to @xmath101 ( in the [ 3.3.4.3.4 ] phase ) , finally to @xmath102 . further increasing @xmath83 , the asymmetry of a ( b ) arm and c arm results in arrangement of junctions on a curve line . then a 3d structure of tpl appears when @xmath103 . and ( b ) entropic energy of @xmath104 of various structures as a function of @xmath83 on the phase path of @xmath77 . the morphologies of @xmath78 ^ 1 $ ] , @xmath78 ^ 2 $ ] , @xmath79 and @xmath80 are explained in fig.[fig:11x ] . ] when @xmath105 , the lamellar phase @xmath106 is stable in which the arrangement manner is babc type in one period , including one thick c layer and three thin bab layers . we can analyze the stability of these ordered phases after splitting the scft energy functional into the internal and entropic parts ( see eq . ) . fig.[fig : energysplit11x ] gives the internal energy @xmath107 , subtracted by that of the homogeneous phase@xcite , together with the entropic energy of @xmath108 , as a function of @xmath83 . from fig.[fig : energysplit11x](a ) , we can find that @xmath79 ( cacb layers ) has very high internal energy at small @xmath83 which is induced by the penetrations of a and b arms through c domains@xcite . at the same time , the lamellae of @xmath79 is favorable from the aspect of entropic energy , because the a and b blocks can get the largest entropy in the lamellar structure when they have equal large lengths compared with the c blocks . the combination of the two contributions makes the @xmath79 be the stable phase when @xmath109 . when increasing @xmath83 , the internal energy becomes dominant and the stable phase transfers from @xmath79 to the cylindrical structures where the arm penetrations in dissimilar phase regions are diminished . when the stable phase transfers from @xmath78 ^ 2 $ ] to [ 3.3.4.3.4 ] ( also termed as [ 10.8.4 ; 10.6.4 ] ) , the entropic energy plays a dominant role on the stability . it is attributed to the increment of c arm which tends to form large c - domains . the enlarged c - domains have an opportunity to meet much more a - domains which will increase the internal energy due to the largest interaction parameter @xmath3 . on the other hand , as the arm c increases , the a and b arms become shorter . the a and b arms can freely stretch in their rich domains which can greatly reduce the entropic energy . the explanation is also available to the appearance of [ 12.6.4 ] , tpl as stable phases . further increasing @xmath83 to @xmath110 , the lamellar phase of @xmath80 ( babc layers ) is stable again . the reason is similar to that of the stability of @xmath79 phase when @xmath83 is smaller than @xmath96 . along this phase path , matsushita and co - workers@xcite synthesized a set of @xmath111 copolymers and observed a number of ordered structures . four isp star triblock copolymers with volume ratios of 1.0:1.0:0.7 , 1.0:1.0:1.2 , 1.0:1.0:1.3 , and 1.0:1.0:1.9 were investigated . the cylindrical structures of @xmath112 , @xmath113 , @xmath114 , and @xmath115 are @xmath116 $ ] , @xmath78 $ ] , @xmath117 $ ] , and @xmath118 $ ] , respectively . from our simulations , the resulting phase behavior is in good agreement with these experimental measurements when @xmath119 . at higher asymmetries , takano et al.@xcite observed an l+s phase in @xmath120 system , and an l+c phase in @xmath121 and @xmath122 copolymers . however , in our calculations , the lam phase dominates the regions . this might be attributed to the relatively weak interaction parameters of @xmath2 or @xmath4 in our simulations so that a further segregation between components a and b ( or b and c ) can not occur . in 2007 , hayashida et al.@xcite have discovered a cylinders - in - lamella phase in which the stacking manner of the cylinders seems to be random in the experiments of isp star triblock copolymer melts . in our calculations , the cylinders - in - lamella , tpl , is found to be stable when @xmath123 , in which the cylinders are stacking tetragonally . the discrepancy is attributed to the thermodynamic fluctuations which may affect the arrangement of cylinders under experimental circumstances . while within the mean - field level theory , the fluctuations have been neglected . along the similar phase path , nunns et al.@xcite synthesized a set of isf star triblock copolymers and observed the polygonal tilings [ 8.8.4 ] and [ 12.6.4 ] . our resulting phase behavior is generally consistent with the isf experiments . a deviation between our theoretical results and experiments is that the [ 12.6.4 ] was observed in @xmath124 system . the discrepancy can be attributed to the different interactions , and different monomer sizes . our computational results also agree with the previous theoretical calculations@xcite . among these works , zhang et al.@xcite considered 2d tiling patterns and chose the asymmetric flory - huggins interaction parameters of @xmath125 , @xmath126 to model the system of abc star triblock copolymers . a 1d phase diagram of the star triblock copolymers @xmath76 was obtained . with the increase of @xmath22 , the phase transition changes from @xmath78 ^ 1 $ ] to @xmath78 ^ 2 $ ] , then to [ 8.6.6;8.6.4 ] and finally to [ 12.6.4 ] . the corresponding stable regions are @xmath127 , @xmath128 , @xmath129 and @xmath130 , respectively . the phase behaviors of @xmath78 ^ 1 $ ] , @xmath78 ^ 2 $ ] , and [ 12.6.4 ] are qualitatively consistent with our results . there are some discrepancies between their results and our computer simulations . in their phase diagram , the archimedean tiling of [ 3.3.4.3.4 ] is not included in their simulations , the order pattern [ 6.6.6 ] is metastable , and the stable area of [ 8.6.6;8.6.4 ] is different from ours . the discrepancies can be attributed to two aspects . the first one is that more candidate structures are involved in our simulations . the second one is the difference of the flory - huggins interaction parameters . on the above phase path , many candidate structures , such as 2d polygonal phases [ 10.6.6;10.6.4 ; 8.6.6;8.6.4 ] and 3d hierarchical structures of hhc , hdg , do not appear . to obtain the stability regions of the 2d and 3d phases which have been observed in experiments , we turn to another phase path of isopleth @xmath75 , i.e. , by fixed equal length of arms a and c and the arm - length ratio of @xmath132 with an increment of volume fraction @xmath6 of @xmath53 . the free energy difference from the value of the homogeneous phase as a function of @xmath6 varying from @xmath133 to @xmath134 is plotted in fig.[fig:1x1](a ) . the phase stability regions as a function of @xmath22 are presented in fig.[fig:1x1](b ) . for abc star triblock copolymers with symmetric a and c arms . ( b ) phase stability regions as a function of the arm - length ratio of @xmath135 ( @xmath75 ) of abc star triblock copolymers with asymmetric interactions , @xmath63 , @xmath64 as a function of the arm - length ratio of @xmath135 . ] along this phase path , the phase sequence with increasing @xmath6 is lam @xmath85 [ 8.8.4 ] @xmath85 [ 6.6.6 ] @xmath85 [ 10.6.6;10.6.4;8.6.6;8.6.4 ] @xmath85 [ 12.6.4 ] @xmath85 l+c @xmath85 hdg @xmath85 hhc . the stable region of lam phase is @xmath136 . at the center part of the phase path , the chain lengths of three arms are close to one another , junction points are aligned on a straight line , and hence 2d tiling patterns can be formed . the stability regions of 2d cylindrical phases are @xmath137 ( [ 8.8.4 ] ) , @xmath138 ( [ 6.6.6 ] ) , @xmath139 ( [ 10.6.6;10.6.4;8.6.6;8.6.4 ] ) , @xmath140 ( [ 12.6.4 ] ) , respectively . from the phase path , we can also find that , in polygonal patterns , the coordinations of b - domains are proportional to the volume fraction of @xmath6 . with the increase of @xmath6 from @xmath141 to @xmath142 , the coordination number of b - domains goes through a gradual change from @xmath98 ( @xmath143 ^ 1}$ ] ) , @xmath99 ( [ 6.6.6 ] ) , @xmath100 or @xmath101 ( [ 10.6.6;10.6.4;8.6.6;8.6.4 ] tiling ) , to @xmath102 ( [ 12.6.4 ] ) . further increasing @xmath6 , the a and c arms become shorter . also , owing to the largest interaction parameter @xmath3 , a further segregation between two minority blocks a and c occurs within the large - length - scale phase , and the system can form some hierarchical morphologies . the interesting hierarchical patterns include 2d l+c phase , and 3d patterns of hdg , hhc , and their stability regions are @xmath144 ( l+c ) , @xmath145 ( hdg ) , @xmath146 ( hhc ) , respectively . and ( b ) entropic energy of @xmath104 of various structures as a function of @xmath6 on the phase path of @xmath75 . ] after splitting the scft energy functional into internal energy and entropic energy as expressed in eqn . ( see fig.[fig : energysplit1x1 ] ) , the stability of 3d hierarchical structures hhc and hdg along the phase path can be understood more readily . when the ratio of @xmath135 ( @xmath75 ) is large enough , the asymmetric abc star triblock copolymers will exhibit similar phase behavior of asymmetric diblock copolymer . the longest b arm plays an equal role as the long block in an asymmetric diblock copolymer , whereas the short a arm , together with c arm , are just like the short block in the diblock copolymer . in asymmetric diblock copolymers , the systems tend to curve the interface towards the minority domain . it requires the minority block to stretch , and the cost is more than compensated for by relaxation of the longer blocks , which increases the internal energies@xcite . besides the asymmetry of blocks , the segregation power between arms a and c , @xmath3 , is sufficiently large to separate the a- , and c - domains , leading to the formation of hierarchical structures . in particular , for the asymmetric abc star copolymers , the blocks a and c have shorter chains than the short block in ab diblock when the curve interface can be formed . the a and c arms can freely relax in their packing frustration domains which greatly reduces the entropic energy of the system . although the internal energies of 3d hierarchical structures are higher than that of l+c structure , their entropic energies are much lower than that of the 2d cylinders - in - lamella phase . as a consequence , the combination of the two competing energies makes 3d hierarchical structures stable rather than l+c when @xmath147 . at higher asymmetries , block copolymers prefer to form structures with a higher interfacial curvature@xcite . therefore hhc phase with larger spontaneous curvature than that of hdg phase , has lower free energy when @xmath148 . the phase transition sequence for systematically varying volume fractions can be obtained by repeating the free energy comparison among the candidate structures . the results of the phase transition sequences can be summarized in terms of phase diagrams . for the case of asymmetric interaction parameters of @xmath149 , the triangular phase diagram is mirror symmetric with the axes of @xmath75 . therefore one - half of the whole triangular phase diagram should be calculated . the triangular phase diagram obtained by our scft simulations is presented in fig.[fig : phasediagram ] . sixteen structures , including 2d , 3d ordered phases and disordered phase ( d ) , are predicted to be stable in the phase diagram . besides the [ 6.6.6 ] , [ 8.8.4 ] , [ 8.6.6 ; 8.6.4 ] , [ 3.3.4.3.4 ] , [ 10.6.6;10.6.4;8.6.6;8.6.4 ] , [ 12.6.4 ] , l+c , tpl , hhc , hdg and lam phases discussed above , four more ordered structures , hpl , core - shell structures of hc , dg , and bcc , and disordered phase are included in the triangular phase diagram . the regions of stability of the different phases are obtained by comparing the free energy of these candidate structures . the phase boundaries are determined by calculating the cross over point of the free energies of the two neighbouring phases . the most significant feature of the triangular phase diagram is the rich phase behavior with a large number of stable ordered phases . it should be noted that the ordered structures emerging in diblock copolymers , i.e. , no core - shell cylinders , spheres , gyroid , two - color lamella , are not included in our calculations . that is , near the boundary of the triangular phase diagram , simulations are not carried out in our study . , @xmath64 . the ordered structures emerging in diblock copolymers , i.e. , no core - shell cylinders , spheres , gyroid , two - color lamella , are not included in our calculations . that is , near the boundary of the triangular phase diagram , simulations are not carried out in our study . ] when the chain lengths of three arms are close to each other , junction points are aligned on a straight line , and hence the system tends to form polygonal tilings to get smaller internal energy . as a consequence , the central region of the phase map is dominated by the 2d tiling patterns . the coordination number of each domain is proportional to the composition of corresponding block . for example , as discussed above , along the phase path of @xmath150 = @xmath81 , the coordination number of c - domains changes from @xmath98 , to @xmath99 , to @xmath100 , then to @xmath101 , and finally to @xmath102 with increasing of @xmath22 from @xmath151 to @xmath152 . besides the cylindrical structures observed by experiments , two tilings of [ 8.6.6 ; 8.6.4 ] and [ 10.6.6;10.6.4;8.6.6;8.6.4 ] are predicted as the stable phases in our theoretical calculations . the stability has been analyzed in the above context . it is very different from the phase behavior of abc linear triblock copolymer melt in which the lamellar phase occupies the large central region@xcite . the reason is attributed to the topology of the star copolymer chain for which the junction points tend to be aligned on a straight line when the chain lengths of three arms are comparable . as the compositions of star copolymer become asymmetric , more 2d and 3d phases , including lam , hc , dg , bcc , hpl , tpl and cylinders - in - lamella of l+c , appear and surround these 2d tilings in the triangular phase diagram . among these phases , lam phase has the following phase transitions . near the ab edge , cacb layers can be formed whereas abac layers formed near the bc edge , and babc layers formed near the ac edge . consider the structural evolution that starts from the central 2d tilings region , toward the a- and c - rich corners of the triangular phase diagram . the phase transition sequence is the cylinders - in - lamella structures ( including hpl and tpl ) , l+c , lam , dg , hc , bcc and disordered phase . near the a - rich corner , in the cylinders - in - lamellar structure , the c arms form internal cylinders in the b perforated lamellar , then blocks b and c form lamellar together with a - layers . the packed manners of the cylinders determine the morphologies of hpl and tpl , as shown in fig.[fig : phases ] . the 2d hierarchical pattern , l+c , where arms b and c form alternating cylinders in the lamellar - based phase , has a stable region between cylinders - in - lamellar structures and lam phase . the prediction of stability region of l+c phase is generally consistent with the recent experiment in which nunns et al . observed the phase in @xmath153 star copolymer system@xcite . because of the smallness of @xmath154 , a further segregation between the minority components b and c ( a and b ) will not occur , which implies that the 3d hierarchical structures will not be global stable in this region . instead , the lam phase dominates this area . owing to @xmath155 , there is similar phase behavior near the c - rich corner when exchanging the position of arms a and c. the bcc , hc , and dg phases form continuous areas across the a- and c - rich corners of the triangular phase diagram , where a or c block is the largest arm . the continuous areas formed by these phases in the triangular phase diagram reflect a continuous evolution in the compositions of the core and shell blocks . for example , consider the evolution of structure along a path within the bcc phase in the a - rich corner , starting from the ab edge , where the structure contains b spheres in an a matrix , to the ac edge . as the c arm increases its length , a spherical `` shell '' of c grows in the middle of each b - sphere , while the composition of the surrounding shell of b shrinks , until a structure of c spheres in an a matrix is obtained at the ac edge . an analogous change in the volume fractions of the `` shell '' and `` core '' components occurs in the hc and dg phases in both the a- and c - rich corners . the phase behavior in the b - rich corner is more complicated . owing to the largest value of @xmath156 , the minority components a and c tend to a further segregation near the b - rich corner , which leads to the formation of the hierarchical structures . from the cylindrical regions to b - rich corner , the phase sequence of hierarchical morphologies is l+c @xmath85 hdg @xmath85 hhc . near the @xmath75 line , the stable regions of hc and dg are separated by hierarchical structures of hhc , hdg , l+c and three archimedean tilings of [ 12.6.4 ] , [ 3.3.4.3.4 ] , [ 8.8.4 ] . the stable region of bcc phase in the b - rich corner is continuous above that of hhc structure . consider the structural evolution along a path that starts from the ab edge , where the length of arms satisfies the relationship @xmath157 , toward the @xmath158 isopleth . along this path , for example , in the bcc phase , a arms segregate into spherical domains , surrounded by c - rich pockets shell within the b matrix . the core - shell hc and dg phases have similar phase behavior with a - core , c - shell and b - matrix . similarly , structures , such as dg , hc , and bcc , evaluating along a path that starts from bc edge toward the @xmath75 isopleth , are of the c - core , a - shell and b - matrix patterns . in addition , the disordered phase emerges in the b - rich corner of the triangular phase diagram . in general , the theoretical phase behavior is in agreement with the experimental observations . however , there are some discrepancies between our theoretical computationus and experimental observations . some structures observed by experiments in isp star triblock copolymers , such as l+s phase@xcite , zinc - blende type structure@xcite are not obtained in our simulations . the stable regions of some structures obtained by experiments are not exactly located in the predicted phase regions of our phase diagram . our theoretical phase diagram predicted [ 8.6.6 ; 8.6.4 ] and [ 10.6.6;10.6.4;8.6.6;8.6.4 ] as stable phases which have been not observed in the experiments . there are three main possible reasons for these discrepancies . the first one is that in the experiments many star triblock copolymers are obtained by blending two kinds of copolymers , or adding additional homopolymers . for example , the zinc - blende type structure was observed in the @xmath159 system which was realized by blending the s homopolymer to @xmath160 . the homopolymer s can definitely affect the phase stabilities . the second one is that the predicted phases in the theoretical calculation , such as the [ 8.6.6;8.6.4 ] and [ 10.6.6;10.6.4;8.6.6;8.6.4 ] , may be metastable in the recent experimental systems . the third one is that there are many differences between theoretical systems and experimental conditions , such as the different interactions , different monomer sizes . in our calculations , we neglect the differences between @xmath2 and @xmath4 . the difference is small , however , it might influence the self - assembling behavior . at the same time , the values of interaction may be different from the experimental systems . in this work , we have investigated the phase behavior of abc star triblock copolymers with asymmetric interaction parameters using the scft . based on the previous work of screening initialization strategies@xcite , we can obtain a large number of ordered phases in studying the complex polymer systems . then we used a fourth - order pseudospectral method , combined with the anderson mixing algorithm to calculate the free energy of the observed phases . motivated by previously experimental studies , the flory - huggins interaction parameters of @xmath161 , @xmath63 are used to model the experimental systems of isp and isf star copolymers in bulk . to extend the scope of theoretical study , a large number of 2d and 3d ordered structures have been involved in our calculations . in order to shed light on the phase behavior of the abc star triblock copolymers , we first determined the phase stability along the phase path @xmath162 . our results agree with those of isp star triblock copolymers for cylindrical phases well . then we calculated the phase regions along the phase path @xmath163 . on this phase path , besides the cylindrical structures , we emphatically analyzed the stability of hierarchical structures of l+c , hdg , hhc . the phase stability has been analyzed by splitting the scft energy functional into internal and entropic parts in detail . finally we constructed a very complicated triangular phase diagram with these candidate structures . owing to the case of interactions @xmath164 , the phase diagram has only one mirror symmetric axis @xmath75 . fifteen ordered structures and the inhomogeneous phase constitute the triangular phase diagram . in general , the phase regions predicted by our scft calculations are consistent with previous theoretical studies of either scft calculations or mc , dpd simulations which mainly focus on the equal interaction systems . however , in our calculations , the interaction parameters are more closely related to experimental systems , such as isp and isf star triblock copolymers , i.e. , the asymmetric interaction parameters @xmath27 . it has been found that the asymmetry of the interaction parameters plays a profound role in the complex phase formation . furthermore , our predicted phase diagram involves more phases , especially 3d structures , and presents more comprehensive phase behavior . our calculations and analysis can be helpful to understand the self - assembling mechanism of complex structures . the resulting phase behavior extends the theoretical study to the asymmetrically interacting abc star triblock copolymers . and the presented phase diagram will be a useful guide for further study of abc star triblock copolymers . 10 bates , f. s. ; hillmyer , m. a. ; lodge , t. p. ; bates , c. m. ; delaney , k. t. ; fredrickson , g. h. multiblock polymers : panacea or pandora s box ? , 336 , 434 - 440 . hamley , i. w. , wiley ; new york , 2004 . park , c. ; yoon , j. ; thomas , e. l. enabling nanotechnology with self assembled block copolymer patterns . , 44 , 6725 - 6760 . meng , f. ; zhong , z. ; feijen , j. stimuli - responsive polymersomes for programmed drug delivery . , 10 , 197 - 209 . ruiz , r. ; kang , h. ; detcheverry , f. a. ; dobisz , e. ; kercher , d. s. ; albrecht , t. r. ; de pablo , j. j. ; nealey , p. f. density multiplication and improved lithography by directed block copolymer assembly . , 321 , 936 - 939 . matsushita , y. ; hayashida , k. ; takano , a. jewelry box of morphologies with mesoscopic length scales abc star - 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separated structures from abc star - shaped terpolymers investigated by electron computerized tomography . , 37 , 9941 - 9946 . takano , a ; kawashima , w. ; noro a. ; isono , y. ; tanaka , n. ; dotera , t. ; matsushita , y. a mesoscopic archimedean tiling having a new complexity in an abc star polymer . , 43 , 2427 - 2432 . hayashida , k. ; kawashima , w. ; takano , a. ; shinohara , y. ; amemiya , y. ; nozue , y. ; matsushita , y. archimedean tiling patterns of abc star - shaped terpolymers studied by microbeam small - angle x - ray scattering . , 39 , 4869 - 4872 . matsushita , y. creation of hierarchically ordered nanophase structures in block polymers having various competing interactions . , 40 , 771 - 776 . hayashida , k. ; saito , n. ; arai , s. ; takano , a. ; tanaka , n. ; matsushita , y. hierarchical morphologies formed by abc star - shaped terpolymers . , 40 , 3695 - 3699 . hckstdt , h. ; gpfert , a. ; abetz , v. synthesis and morphology of abc heteroarm star terpolymers of polystyrene , polybutadiene and poly(2-vinylpyridine ) . , 201 , 296 - 307 . nunns , a. ; ross , c. a. ; manners , i. synthesis and bulk self - assembly of abc star terpolymers with a polyferrocenylsilane metalloblock . , 46 , 2628 - 2635 . park , j. ; jang , s. ; kim , j. k. morphology and microphase separation of star copolymers . , 53 , 1 - 21 . hayashida , k. ; dotera , t. ; takano , a. ; matsushita , y. polymeric quasicrystal : mesoscopic quasicrystalline tiling in abc star polymers . , 98 , 195502 . bohbot - raviv , y. ; wang , z .- g . discovering new ordered phases of block copolymers . , 85 , 3428 - 3431 . gemma , t. ; hatano , a. ; dotera , t. monte carlo simulations of the morphology of abc star polymers using the diagonal bond method . , 35 , 3225 - 3237 . huang , c. i. ; fang , h. k. ; lin , c. h. morphological transition behavior of abc star copolymers by varying the interaction parameters . , 77 , 031804 . tang , p. ; qiu , f. ; zhang , h. ; yang , y. morphology and phase diagram of complex block copolymers : abc star triblock copolymers . , 108 , 8434 - 8428 . zhang , g. ; qiu , f. ; zhang , h. ; yang , y. ; shi , a .- c . scft study of tiling patterns in abc star terpolymers . , 43 , 2981 - 2989 . li , w. ; xu , y. ; zhang , g. ; qiu , f. ; yang , y. ; shi , a .- c . real - space self - consistent mean - field theory study of abc star triblock copolymers . , 133 , 064904 . xu , y. ; li , w. ; qiu , f. ; zhang , h. ; yang , y. ; shi , a .- c . stability of hierarchical lamellar morphologies formed in abc star triblock copolymers . , 48 , 1101 - 1109 . xu , w. ; jiang , k. ; zhang , p. ; shi , a .- c . a strategy to explore stable and metastable ordered phases of block copolymers . , 117 , 5296 - 5305 . matsen , m. w. the standard gaussian model for block copolymer melts . , 14 , r21-r47 . fredrickson , g. h. ; oxford university press : new york , 2006 . jiang , k. ; huang , y. ; zhang , p. spectral method for exploring patterns of diblock copolymers . , 229 , 7796 - 7805 . jiang , k. ; wang , c. ; huang , y. ; zhang , p. discovery of new metastable patterns in diblock copolymers . , 14 , 443 - 460 . matsen , m. w. ; schick , m. stable and unstable phases of a diblock copolymer melt . , 72 , 2660 - 2663 . guo , z. ; zhang , g. ; qiu , f. ; zhang , h. ; yang , y. ; shi , a .- c . discovering ordered phases of block copolymers : new results from a generic fourier - space approach . , 101 , 28301 . drolet , f. ; fredrickson , g. h. combinatorial screening of complex block copolymer assembly with self - consistent field theory . , 83 , 4317 - 4320 . rasmussen , k. . ; kalosakas , g. improved numerical algorithm for exploring block copolymer mesophases . , 40 , 1777 - 1783 . cochran , e. w. ; garcia - cervera , c. j. ; fredrickson , g. h. stability of the gyroid phase in diblock copolymers at strong segregation . , 39 , 2449 - 2451 . ranjan , a. ; qin , j. ; morse , d. c. linear response and stability of ordered phases of block copolymer melts . , 41 , 942 - 954 . ceniceros , h. d. ; fredrickson , g. h. numerical solution of polymer self - consistent field theory . , 2 , 452 - 474 . jiang , k. ; xu , w. ; zhang p. analytic structure of the scft energy functional of multicomponent block copolymers . , 17 , 1360 - 1387 . schmid , f. ; mller m. quantitative comparison of self - consistent field theories for polymers near interfaces with monte carlo simulations . , 28 , 8639 - 8645 . thompson r. b. ; rasmussen k. . ; lookman t. improved convergence in block copolymer self - consistent field theory by anderson mixing . , 120 , 31 - 34 . see the formula ( 4.1.14 ) on p.160 in _ numerical recipes : the art of scientific computing _ , 3rd edition , press , w. h. ; teukolsky , s. a. ; vetterling , w. t. ; flannery , b. p. , ed . ; cambridge university press ; new york , 2007 . anderson , d. g. iterative procedures for nonlinear integral equations . , 12 , 547 - 560 . matsen , m. w. fast and accurate scft calculations for periodic block - copolymer morphologies using the spectral method with anderson mixing . , 30 , 361 - 369 . v. abetz , block copolymers , ternary triblocks , in : _ encyclopedia of polymer science and technology _ , 3rd edition , kroschwitz , j. i. ed . ; john wiley & sons , inc , vol . 1 , 2003 . takano a. ; kawashima w. ; wada s. ; hayashida k. ; sato s. ; kawahara s. ; isono y. ; makihara m. ; tanaka n. ; kawaguchi d. ; matsushita y. composition dependence of nanophase - separated structures formed by star - shaped terpolymers of the @xmath165 type . , 45 , 2277 - 2283 . matsen , m. w. ; bates , f. s. origins of complex self - assembly in block copolymers . , 29 , 7641 - 7644 . tyler , c. a. ; qin , j. ; bates , f. s. ; morse , d. c. scft study of nonfrustrated abc triblock copolymer melts . , 40 , 4654 - 4668 the @xmath12 iteration in the anderson mixing method begins with the evaluation of new fields from eqns.- @xmath166 in the above expressions , @xmath167 , @xmath168 , are the old fields , @xmath169 , @xmath170 , @xmath171 . next we evaluate the deviation , @xmath172 where @xmath173 , @xmath174 , @xmath175 @xmath176 . from the deviation we can specify an error tolerance through the inner product @xmath177 where @xmath178 and @xmath179 are arbitrary functions . the error tolerance is defined by @xmath180^{1/2 } \label{}\end{aligned}\ ] ] as a measure of the numerical inaccuracy in the field eqns.- . the simple mixing method is performed until a certain tolerance is reached where a morphology has begun to develop . from our experience , @xmath181 , is sufficient in most cases . we then switch to the anderson mixing procedure by the previous @xmath28 steps to update fields . we assemble the symmetric matrix in this way @xmath182 for @xmath183 , and vector @xmath184 from these , we calculate the coefficients @xmath185 and combine the previous histories as @xmath186 finally , the new fields are obtained from @xmath187 where @xmath188@xcite . in our implementation , the used previous steps are usually much less than the number of basis functions or grid points . therefore assembling the @xmath28-order linear system spends more computation time than solving this system . to save computational amount , we decompose eqn . into @xmath189 in @xmath12 iteration , only the terms related to @xmath190 in the right term of eqn . are required to calculate , but the last terms @xmath191 , @xmath192 , should be not computed repeatedly which will save the main computational cost in assembling matrix @xmath71 . in practical calculations , we store the following inner product matrix @xmath193 then we can assemble the matrix @xmath71 and vector @xmath29 according to the expressions of eqns . and using the elements of matrix @xmath194 . note that the inner product matrix is symmetric , therefore only a row ( or a column , equivalently ) of @xmath194 is required to update in each iteration .
|
the phase behavior of asymmetrically interacting abc star triblock copolymer melts is investigated by the self - consistent field theory ( scft ) .
motivated by the experimental systems , in this study , we focus on the systems in which the flory - huggins interaction parameters satisfy @xmath0 . using various initialization strategies ,
a large number of periodic structures have been obtained in our calculations . a fourth - order pseudospectral algorithm combined with anderson mixing method
is used to compute the free energy of candidate structures carefully .
the stability has been detailedly analyzed by splitting the free energy into internal and entropic parts .
a complete and complex triangular phase diagram is presented for a model with @xmath1 in which fifteen ordered phases , including two- , and three - dimensional structures , have been predicted to be stable from the scft calculations . generally speaking , with the asymmetrical interactions , the hierarchical structures tend to be formed near the b - rich corner of the triangular phase diagram .
this work broadens the previous theoretical results from equal interaction systems to unequal interaction systems .
the predicted phase behavior is in good agreement with experimental observations and previous theoretical results .
| 17,634 | 286 |
the possibility of altering and controlling the spin - state of a single magnetic ion or of a small magnetic cluster with an external probe represents a unique opportunity towards the understanding and the exploitation of the magnetic interaction at the most microscopic level . possible areas of application for such ability may include spin - based quantum logic , where one necessitates to prepare , manipulate and read spin - qubits . it is then crucial to develop tools capable of addressing the single spin - limit . low - temperature scanning tunneling microscopy provides one of such tools . in general the method exploits a scanning tunneling microscope ( stm ) operated in spectroscopical mode , by which the inelastic electron tunneling spectroscopy ( iets ) at the spin - excitations of a given system is measured @xcite . this scheme is known as spin - flip iets ( sf - iets ) . the same stm can also be used to position and manipulate the magnetic atoms on a non - magnetic substrate @xcite , so that stm appears both as a fabrication and subtle characterization tool . transition metal magnetic atoms on insulating surfaces , in particular mn@xcite , co@xcite and fe@xcite , have been the focus of intensive research in the last few years . these have all partially filled @xmath0-shells , which are highly localized and responsible for the magnetic moment , and extended @xmath1-like electrons , which are responsible for the electron conduction . in general @xmath1 and @xmath0 electrons interact via exchange coupling so that the magnetic structure is coupled to the conducting electrons . the magnetic atoms are usually deposited on carefully prepared cun - decorated cu surfaces , where the typical electronic coupling is weak enough that the magnetism is preserved , but it is sufficiently strong to break the atomic central symmetry so that magnetic anisotropy develops . stm experiments are then conducted and the fingerprint of a magnetic excitation is a step in the differential conductance , @xmath2 , as a function of bias , @xmath3 ( @xmath4 is the stm current ) . these appear at the critical voltage necessary to open a new inelastic transport channel , i.e. at voltages corresponding to the given magnetic excitation energy . several methods aimed at modeling sf - iets have been recently developed . early theoretical work has focused on second order perturbation theory to describe the experimental conductance spectra of equilibrium spins by either using a master equation approach @xcite or a non - equilibrium green s function one @xcite . more recently this scheme has been extended to third order , which allows us to describe additional features in the @xmath5 line - shape that can not be accounted for at the second order level @xcite . these works have been very successful in describing conductance profiles , which appear symmetric with the external bias polarity , i.e. that they can not distinguish whether the current flows from the sample to the tip or in the opposite direction . however , recent experiments have shown that regardless on whether a non - spin - polarized @xcite or a spin - polarized @xcite tip is used the iets profiles exhibit an intrinsic asymmetry with respect to the applied voltage , i.e. @xmath6 . in the case of a spin - polarized stm tip , where the tip density of states is spin split between majority ( spin up ) and minority ( spin down ) carriers , the asymmetry has been theoretically well explained @xcite . it has been shown that spin selection rules enforce a suppression of the inelastic scattering , which depends on the direction of the electrons flow . this results in a asymmetric conductance profile , where the magnitude of the asymmetry depends directly on the spin - polarization of the tip . it is also well understood that by driving spins out of equilibrium ( e.g. by decreasing the tip - sample distance ) the conductance line - shape changes @xcite . in this case we must assume that the tunneling electrons influence the spin - state of the atom as the time between inelastic events is small compared to the spin relaxation time . a tunneling electron can then encounter the local spin in an excited state far from the ground state . the non - equilibrium population of the various accessible spin - states then becomes bias - dependent and , for spin - polarized tips , this enhances the asymmetry of the @xmath5 line - shape . also in the case of a non - spin - polarized tip a bias asymmetry has been revealed experimentally @xcite . in particular this appears to be quite prominent for both single mn atoms and mn mono - atomic chains . this feature has been previously ascribed to a shift in the magnetic atom on - site energy , i.e. to an effect arising from the details of the density of states of the atom producing scattering . such a density of state effect produces a non - trivial slope in the conductance as a function of bias @xcite . the on - site energy shift however does not account for the asymmetry seen in the inelastic step heights , which also depends on bias . here we provide an alternative theoretical description , which allows us to better fit the experimentally found conductance line - shape . in our previous works @xcite we have combined the non - equilibrium green s function ( negf ) @xcite formalism with a perturbative expansion of the electron - spin interaction in order to describe sf - iets spectra in a manner , which is fully amenable to an implementation within density functional theory ( dft)@xcite . the scheme essentially consists in constructing an electron - spin interaction self - energy , which describes the inelastic tunneling events . the interacting self - energy was previously expanded first up to second order @xcite and then to the third order @xcite , with this latter describing the logarithmic decays of @xmath5 at each conductance step . although both non - equilibrium effects and spin - polarized iets have been well described up to the second order by the master equation approach , in this work we extend our formalism to include the extra line - shape features that can be ascribed to the third order self - energy . we also propose that the real part of the interacting self - energy , which has been well studied in the case of electron - phonon interactions @xcite , is a necessary addition to the negf formalism in order to account for asymmetric features in non - spin polarized systems . the layout of the paper is as follows . in the next section we extend our negf formalism to account for spin polarized leads . in the same section we also derive a second - order electron - spin self - energy , which includes a second order expansion of the spin propagator . this provides the means to study non - equilibrium effects that result from high current densities . furthermore , in the case of non - spin polarized tips we derive an expression for the real part of the scattering self - energy up to second order . all of the above is combined with the third order contribution to the electron propagator . then we move to the results . first we study the non - equilibrium effects arising from an increase in current density for the case of a mn dimer . then we recreate the spin - polarized experiments of loth et al . on single mn and fe atoms @xcite . finally we test how the inclusion of the real part of the scattering self - energy modifies the sf - iets for a non - polarized tip probing mn monomers and trimers . we consider here the same single - orbital tight - binding model used in our previous works @xcite to describe a magnetic system ( s ) coupled to two non - interacting electrodes representing respectively the stm tip ( tip ) and the substrate ( sub ) . the scattering region containing the magnetic nanostructure consists of a one - dimensional chain of @xmath7 magnetic atoms . each of the @xmath8-th atoms carries a quantum mechanical spin @xmath9 and it is characterized by an on - site energy @xmath10 . we assume that the tip and the substrate can only couple to one atom at a time in the scattering region thus to broaden the electronic level @xmath10 through the interaction with the electrode by @xmath11 . the hamiltonian in the scattering region is then described by @xmath12 , where @xmath13 is the tight - binding electronic part , @xmath14 is the spin part and @xmath15 describes the electron - spin interaction . the various terms can be written explicitly as @xmath16\big\}\:,\\ \label{eq:3 } & { h}_\mathrm{e - sp}=j_\mathrm{sd}\sum_{\lambda\:\alpha,\alpha'}(c_{\lambda\alpha}^{\dagger}[\boldsymbol{\sigma}]_{{\alpha}{\alpha'}}c_{\lambda\alpha'})\cdot\mathbf{s}_\lambda\ : \\ \nonumber & + j_\mathrm{0}\sum_{\lambda\:\alpha}c_{\lambda\alpha}^{\dagger}c_{\lambda\alpha}\:,\end{aligned}\ ] ] where the electron ladder operator @xmath17 ( @xmath18 ) creates ( annihilates ) an electron at site @xmath8 with spin @xmath19 ( @xmath20 ) and on - site energy @xmath10 . we model the spin - spin interaction between the localized @xmath21 spins by a nearest neighbour heisenberg hamiltonian with coupling strength @xmath22 . furthermore we include interaction with an external magnetic field @xmath23 ( @xmath24 is the bohr magneton and @xmath25 is the gyromagnetic ratio ) and both uni - axial and transverse anisotropy of magnitude @xmath26 and @xmath27 respectively @xcite . the electron - spin interaction hamiltonian is constructed within the @xmath1-@xmath0 model @xcite where the transport electron , @xmath1 , are locally exchanged coupled to quantum spins , @xmath21 ( @xmath0 indicates that the local moments originating from the atomic @xmath0 shell ) . in equation ( [ eq:3 ] ) the interaction strength is @xmath28 and @xmath29 is a vector of pauli matrices . the second term in eq . ( [ eq:3 ] ) represents the potential scattering elastic contribution to the @xmath1-@xmath0 interaction given by the exchange parameter @xmath30 ( note that this enters as a shift of the on - site potential of a given atom ) . the ratio , @xmath31 , is typically in the range 1 - 2 @xcite . this term was not included in our previous works @xcite as it only becomes important for spin - polarized electrodes . in order to construct an electron - spin interacting self - energy we must first consider the keldysh @xcite contour - ordered single - body green s functions ( propagators ) for both the electronic ( @xmath32 ) and the spin ( @xmath26 ) sub - systems in the electron - spin many - body ground state @xmath33 @xmath34_{\sigma\sigma'}=-i{\langle}|t_c\{c_{\sigma}(\tau)c^{\dagger}_{\sigma'}(\tau')\}|{\rangle}\:,\ ] ] @xmath35_{nm}=-i{\langle}|t_c\{d_{n}(\tau)d^{\dagger}_{m}(\tau')\}|{\rangle}\:.\end{aligned}\ ] ] these propagators describe a non - equilibrium system at zero - temperature . here @xmath36 is a quasi - particle creation operator defined by the relation @xmath37 where @xmath38 and @xmath39 are eigenstates of @xmath40 of energy @xmath41 . the matrix elements @xmath42 determine the transition rates from the initial state @xmath43 to the final state @xmath44 . the quasi - particle operators are assumed fermionic in nature@xcite . therefore the equilibrium population of a given state @xmath44 is given by @xmath45 where @xmath27 is the energy , @xmath46 the temperature and @xmath47 the boltzmann constant . in the non - interacting case ( @xmath48 ) the electronic system is not in equilibrium as the interaction with the electrodes establishes a steady state current . in contrast the spin system is in thermal equilibrium at the temperature @xmath46 . in this case the energy resolved lesser and greater green s functions take the form @xmath49_{\sigma\sigma'}=\frac{[\sigma^{\lessgtr}_{\mathrm{tip - s}}(e)]_{\sigma\sigma'}+[\sigma^{\lessgtr}_{\mathrm{sub - s}}(e)]_{\sigma\sigma'}}{(e-\varepsilon_0)^2+\gamma^2}\:,\\ \label{eq:72 } & [ d_0^{\lessgtr}(e)]_{mn}=\frac{[\pi^{\lessgtr}_{0}(e)]_{mn}+[\pi_0^{\lessgtr}(e)]_{mn}}{(e-\varepsilon_m)^2+(k_\mathrm{b}t)^2}\:.\end{aligned}\ ] ] in the electronic non - interacting green s function [ see eq . ( [ eq:71 ] ) ] the coupling of the sample to the tip and the substrate causes a broadening of the bare on - site level @xmath10 of magnitude @xmath50_{\sigma\sigma'}$ ] , where @xmath51 . the non - interacting self - energies take the form @xmath52_{\sigma\sigma'}=[1-f_{\eta}(e , v)][\gamma_{{\eta}-s}]_{\sigma\sigma'}$ ] and @xmath53_{\sigma\sigma'}=f_{\eta}(e , v)[\gamma_{{\eta}-s}]_{\sigma\sigma'}$ ] , where @xmath54 is the fermi function in each of the @xmath55-th leads at a bias @xmath3 . in contrast the local spin green s function of equation ( [ eq:72 ] ) describes a system , which is adiabatically coupled to a heat - bath of temperature @xmath46 . this provides a very weak broadening of the single spin states @xmath56 of magnitude @xmath57 . such a heat bath keeps the spin - system in equilibrium and in the non - interacting case the population then resides mostly in the ground state [ see eq . ( [ eq:7 ] ) ] . for a ground state population of the spin system @xmath58 we have @xmath59_{mn}=\delta_{mn}(1-p^0_m)k_\mathrm{b}t$ ] and @xmath60_{mn}=\delta_{mn}p^0_mk_\mathrm{b}t$ ] . in order to evaluate the effects that the interaction has on the electronic motion we must calculate the electron - spin self - energy . here we take a perturbative approach and formally expand equation ( [ eq:4 ] ) up to the @xmath43-th order in the interaction hamiltonian , @xmath61 , as @xmath62_{\sigma\sigma'}=\sum_n\frac{(-i)^{n+1}}{n!}\int\limits_c{d}\tau_1\dots\int\limits_c{d}\tau_n\ \times \nonumber \\ & \frac{{\langle}0|t_c\{{h}_\mathrm{e - sp}(\tau_1)\dots{h}_\mathrm{e - sp}(\tau_n)c_{\sigma}(\tau)c_{\sigma'}^{\dagger}(\tau')\}|0{\rangle}}{u(-\infty,-\infty)},\end{aligned}\ ] ] where @xmath63 is the time - evolution unitary operator and the time - averages are performed over the known non - interacting @xmath64 ground state @xmath65 . the time integration over @xmath66 is ordered on the contour @xmath67 going from @xmath68 to @xmath69 and then returning from @xmath69 to @xmath68 , since the ground state of the non - equilibrium system can only be defined at @xmath68 @xcite . in the following we consider the tip to have a spin polarization @xmath55 ( @xmath70 ) . this is defined as the spin asymmetry in the electronic coupling between the tip and the sample . an such the spin resolved electronic broadening is given by @xmath71_{\uparrow\uparrow}=\frac{(1+\eta)}{2}\gamma_\mathrm{tip - s}$ ] and @xmath71_{\downarrow\downarrow}=\frac{(1-\eta)}{2}\gamma_\mathrm{tip - s}$ ] , where @xmath72 is the non - spin - polarized broadening . the substrate is assumed to remain non - magnetic . as a result @xmath73_{\uparrow\uparrow}\neq[g^{\lessgtr}_0(e)]_{\downarrow\downarrow}$ ] so that we now must retain the spin indexes when constructing the self - energy . the self - energy for the majority ( @xmath74 ) and minority ( @xmath75 ) spins writes respectively as @xmath76_{\uparrow\uparrow}^{(2)}=-j^2_\mathrm{sd}\sum_{mn}[g^{\lessgtr}_0(e\pm\omega_{mn})]_{\uparrow\uparrow } \times\nonumber \\ & \big(\delta_{nm}\chi p_ns_{mn}^z+ p_n(1-p_m)|s_{mn}^z|^2\big)\nonumber \\ & -j^2_\mathrm{sd}\sum_{mn}[g^{\lessgtr}_0(e\pm\omega_{mn})]_{\downarrow\downarrow}p_n(1-p_m)|s^+_{mn}|^2\end{aligned}\ ] ] and @xmath77_{\downarrow\downarrow}^{(2)}=-j^2_\mathrm{sd}\sum_{mn}[g^{\lessgtr}_0(e\pm\omega_{mn})]_{\downarrow\downarrow } \times\nonumber \\ & \big(-\delta_{nm}\chi p_ns_{mn}^z+p_n(1-p_m)|s_{mn}^z|^2\big)\nonumber \\ & -j^2_\mathrm{sd}\sum_{mn}[g^{\lessgtr}_0(e\pm\omega_{mn})]_{\uparrow\uparrow}p_n(1-p_m)|s^-_{mn}|^2\:.\end{aligned}\ ] ] the lesser ( greater ) self - energy describe an incoming ( outgoing ) electron that excites ( relaxes ) the spin system by @xmath78 , with a probability that depends on the occupation of the spin levels @xmath79 and @xmath80 and on the spin selection rules @xmath81 ( note @xmath82 and @xmath83 ) . the first term in both the equations ( [ eq:10a ] ) and ( [ eq:10b ] ) , proportional to @xmath84 , corresponds to the magnetoresistive elastic term of the @xmath1-@xmath0 hamiltonian of equation ( [ eq:3 ] ) . the remaining contributions are inelastic in nature and depend on the spin orientation of the electron transferred from the tip . when is magnetic and the tunneling current tip carries a finite spin - polarization the spin system can be dragged out of equilibrium , in particular if the current density is intense . this means that the equilibrium conditions employed previously @xcite , namely @xmath85 and @xmath86 is no longer valid . as a consequence we must now derive also an expression for the propagator and thus for the self - energy associated to the local spins . the derivation , up to second order in the electron - spin interaction is described in details in the appendix . in particular the total spin - self - energy also includes a zeroth - order contribution , which accounts for the non - interacting ( @xmath48 ) case . this is approximated by @xmath87_{kk}=p_k^0k_\mathrm{b}t$ ] and @xmath88_{kk}=(1-p_k^0)k_\mathrm{b}t$ ] where @xmath89 is the ground state population . therefore , in absence of inelastic scattering , the spin system will remain in thermal equilibrium with the heat bath and only the ground state will be occupied . by combining the zeroth and second order contributions to self - energy we can write down a master equation , describing the non - equilibrium spin - population , in terms of the total self energy @xmath90@xcite @xmath91_{nm}[d_0^<(e)]_{mn } \nonumber \\ & -\pi^{<}(e)]_{nm}[d_0^>(e)]_{mn}\big\}\:.\end{aligned}\ ] ] after some rearrangement this can be written in more compact form as @xmath92\nonumber \\ & + ( p_n^0-p_n)k_\mathrm{b}t\:,\end{aligned}\ ] ] where the bias dependent transition rate from an initial state @xmath93 to a final state @xmath94 is calculated after evaluating the integral described in equation ( [ eq:14 ] ) of the appendix . this finally writes @xmath95_{\uparrow\uparrow}[\gamma_{\eta'}]_{\uparrow\uparrow}-[\gamma_{\eta}]_{\downarrow\downarrow}[\gamma_{\eta'}]_{\downarrow\downarrow}\big ) + \nonumber \\ & + |s^z_{nl}|^2[\gamma_{\eta}]_{\uparrow\uparrow}[\gamma_{\eta'}]_{\uparrow\uparrow}+|s^z_{nl}|^2[\gamma_{\eta}]_{\downarrow\downarrow}[\gamma_{\eta'}]_{\downarrow\downarrow } + \nonumber \\ & + |s^+_{nl}|^2[\gamma_{\eta}]_{\downarrow\downarrow}[\gamma_{\eta'}]_{\uparrow\uparrow}+|s^-_{nl}|^2[\gamma_{\eta}]_{\uparrow\uparrow}[\gamma_{\eta'}]_{\downarrow\downarrow}\big\}\:,\end{aligned}\ ] ] where @xmath96 and @xmath97 is the chemical potential in lead @xmath51 . note that @xmath98 is such that for @xmath99 the resulting transition rates @xmath100 are bias independent and do not contribute to the current . however , they do contribute to the spin relaxation time i.e. to the time taken by the localized spin system to relax back to its equilibrium state . such a relaxation time is reduced if the coupling between the sample and the leads is increased . furthermore , the smaller is the inelastic energy transition @xmath78 , the longer the spins will remain in the excited state before relaxing back to equilibrium . finally , we note that the above expression is based on the assumption that the on - site energy is large enough for the density of states of the spin system to remain constant in the small energy window of interest . therefore @xmath101 . returning to equation ( [ eq:17 ] ) note that we are only interested in the steady state non - equilibrium population of the spin states at a given bias . therefore we can set @xmath102 and reduce eq . ( [ eq:17 ] ) to system of linear equations , which can be solved self - consistently . for an initial guess of the populations ( @xmath103 ) we can iterate eq . ( [ eq:17 ] ) , which define @xmath80 , with equation ( [ eq:14 ] ) of the appendix , which define the self - energy @xmath104 , until self - consistency is reached . we can then combine the resulting non - equilibrium population with the second order electronic self - energy calculated in eq . ( [ eq:4 ] ) to obtain the current . in order to provide an explanation to the inherent asymmetry that has been observed in most of the stm experiments on magnetic atoms using a non - magnetic tip we return to the expression for the full retarded self - energy . this is defined by the hilbert transform ( note we will only consider the non - spin polarized case for simplicity ) @xmath105 by using the expressions derived in section [ spse ] for the 2@xmath106 order lesser and greater self - energies we can find an analytic expression for the real contribution to the retarded self energy @xmath107=2\rho j^2_\mathrm{sd}\sum_{i , m , n}|s^i_{mn}|^2p_n(1-p_m)\times \\ \nonumber & \frac{1}{\gamma}\big\{2\pi\varepsilon_0+\sum_{\eta}\gamma_{\mathrm{\eta - s}}\:\mathrm{ln}\big[\frac{(e+\omega_{mn}-\mu_{\eta})^2+(k_bt)^2}{(e-\omega_{mn}-\mu_{\eta})^2+(k_bt)^2}\big]\big\}\:.\end{aligned}\ ] ] such a final expression is heavily dependent on the on - site energy @xmath10 but is also an odd function of the energy and the bias via its logarithmic dependence on the spin level eigenvalues with opposite polarity for @xmath108 and @xmath109 . we will show in the results section that this is at the origin of the conductance asymmetry found in experiments . we can finally unveil the effects that a non - equilibrium spin - population bares on the conductance profile of a magnetic nanostructure by taking the derivative of the current with respect to the bias voltage @xmath3 . the current , @xmath110 , flowing at the electrode @xmath111 can be written as @xmath112-[\sigma_{\eta}^{<}(e)g^{>}(e)]\}\:,\end{aligned}\ ] ] where @xmath113 are the full many - body lesser / greater electronic green s functions . these are finally defined as @xmath114=\frac{[\sigma^{\lessgtr}_{\mathrm{tip - s}}(e)]+[\sigma^{\lessgtr}_{\mathrm{sub - s}}(e)]+[\sigma^{\lessgtr}_{\mathrm{int}}(e)]}{(e-\varepsilon_0-\mathrm{re}[\sigma_{\mathrm{int}}(e)])^2+(\gamma-\mathrm{im}[\sigma_{\mathrm{int}}(e)])^2},\end{aligned}\ ] ] and @xmath115=[\sigma^{\lessgtr}_{\mathrm{int}}(e)]^{(2)}+[\sigma^{\lessgtr}_{\mathrm{int}}(e)]^{(3)}$ ] where the third order self - energies are calculated following ref . here for simplicity we take the expression for the third order contribution to @xmath116 obtained by neglecting any explicit spin - polarization . such approximation is justified by the fact that the effects due to spin - polarization are small at the third order and that in doing so we avoid a rather cumbersome formulation . we do however consider the combination of non - spin polarized 3rd order effects with 2nd order spin - polarized self energies to highlight subtle differences in the spectra . we start our analysis by first looking at the effects originating from driving the spin system out of equilibrium with an electronic current . this attempts at explaining the experiments reported in ref . [ ] , in which a stm tip ( non - magnetic ) is positioned above a mn dimer deposited onto a cun substrate . the conductance spectra are measured for different tip to sample distances . varying the stm tip height is equivalent to changing both the current density and the electronic coupling between the tip and the sample . non - equilibrium effects then appear as variations of the conductance profiles as a function of the stm tip height . many of the parameters needed by our model can be extrapolated from a similar experiment carried on over mn linear atomic chains deposited on cun @xcite . the five unpaired electrons in the mn 3@xmath0 shell suggest a @xmath117 ground state , as confirmed both by experiments and theory @xcite . the spin - spin exchange interaction between two mn atoms is antiferromagnetic and has an estimated value of @xmath118 mev . as a result the ground state of the dimer is a singlet ( total spin @xmath119 ) . the first excited state is a triplet with total spin @xmath120 and the energy splitting between the ground state and such first excited state is exactly @xmath22 . the next excited level is the quintuplet with total spin @xmath121 and it is separated from the first excited state by @xmath122 . this pattern continues throughout the spin manifold ( see figure [ fig1 ] ) . the axial and transverse anisotropies are found to be @xmath123 mev and @xmath124 mev respectively . these cause the lifting of the spin multiplets degeneracy . the temperature is set at @xmath125 k. the value of @xmath28 is estimated from density functional theory ( dft ) to be of the order of 500 mev @xcite , while @xmath126 is also found from dft to be approximately 100 mev . in contrast @xmath72 remains an adjustable parameter with the chosen values ranging from 0.125 mev to 200 mev . finally , in order to ensure a nearly constant density of states around the fermi energy ( @xmath127 ) we set the on - site energy of the atom under the stm tip to be @xmath128 ev . figure [ fig2 ] shows the conductance spectra obtained by simply taking the numerical derivative of the current [ eq . ( [ eq:23 ] ) ] with respect to the bias . we consider three different tip to sample distance , corresponding respectively to weak ( @xmath129 mev ) , intermediate ( @xmath130 mev ) and strong ( @xmath131 mev ) electronic coupling . the evolution of the conductance lineshape as a function of @xmath72 is a direct consequence of the spin system being driven out of equilibrium . for @xmath129 mev the stm tip is far enough from the sample to ensure that the spin system is always in its ground state between two subsequent electron tunneling events . therefore the only transition detected in the @xmath5 profile is that between the @xmath119 ground state and the first excited state with @xmath120 . this has an excitation energy equal to @xmath22 and it does manifest itself as a conductance step at a voltage @xmath132 , with @xmath133 being the electron charge . as the tip is brought closer to the sample ( @xmath130 mev ) the first excited triplet level ( @xmath120 ) starts to populate . now an incoming electron with sufficiently large energy ( @xmath122 ) can induce a second transition from from the first to the second excited state . note that the @xmath121 state is not accessible with a single electron tunneling process from the ground state and it can be reached only if the spin system does not have enough time between tunneling events to relax back to the ground state . for this case the transition appears as a reduction of the conductance at the critical voltage @xmath134 . the same spectroscopical feature is further enhanced at an even larger current density ( @xmath135 200 mev ) , when a third conductance step appears at @xmath136 . this is associated to a transition from the @xmath121 to the @xmath137 spin state and it becomes possible only if the occupation of the @xmath121 level is not zero , i.e. if the system is driven to this highly excited state . these results are in almost perfect quantitative agreement with the experimental data ( see fig . 2 of ref . [ ] ) . the evolution of the population of the various spin states ( up to @xmath121 ) as a function of bias is presented in figure [ fig3 ] . this is calculated in the case of strong tip to sample electronic coupling @xmath135 200 mev . in the figure one can note the strong spin - pumping from the ground state into both the first and the second excited state . the excitation to the @xmath138 excited state occurs at approximately 18 mev but is too weak to be observed on this scale . we now move on to consider the situation where the tip is magnetic , i.e. when the injected current is spin - polarized . again we use as guide the experimental work of loth _ et al._@xcite . the stm tip is now spin - polarized by placing an additional mn atom at its apex , while also applying a strong magnetic field perpendicular to the substrate of 3 t. in this case the spectrum is collected from a single mn or fe ion on the surface ( not from a dimer ) . in the case of mn , the atom exhibits a weak anisotropy on cun ( see previous section ) and the strong magnetic field effectively produces a zeeman split of the six levels of the @xmath117 mn spin manifold . the direction of the magnetic field in these experiments is chosen so that the ground state of the mn spin corresponds to the magnetic quantum number @xmath139 . since the same magnetic field is applied to the mn atom on the tip s apex , the tip and atom are both spin - polarized and collinear . figure [ fig4 ] shows the calculated spectra for the system described above . in particular we consider magnetic field strength of 3 t and either weak ( @xmath140 mev ) , intermediate ( @xmath141 mev ) or strong ( @xmath142 mev ) tip to sample couplings . the on - site energy is fixed at @xmath143ev and the value of @xmath144mev is infered from the work of lucignano _ _ et al.__@xcite . the tip spin - polarization constant and the inelastic ratio that best fit the experimental data are respectively @xmath145 and @xmath146 . in the weak coupling regime ( when the local spin remains always close to equilibrium ) the local spin resides almost entirely in its @xmath139 ground state . due to the spin - exchange selection rules and to the collinearity of the tip and the sample , only the minority carriers can excite the local spin out of the ground state . for a tip spin - polarization of @xmath145 , there are more minority electrons coming from the tip than those coming from the substrate . as a result , the intensity of the inelastic interaction will change depending on the direction of the current . this creates an asymmetry in the conductance spectrum with respect to the applied bias . the additional lineshape features appearing in the weak coupling case ( the conductance decay following a conductance step ) are due to the third order kondo - like self - energy , which produces a logarithmic decay at the conductance steps . this result is in good agreement with experiments ( see fig . 4 of ref . ) . when the spin of the mn ion is driven further out of equilibrium , in particular in the strong coupling case , the bias asymmetry becomes more pronounced . such spin - pumping phenomenon can be appreciated by looking at figure [ fig5](a ) , where we show the populations of the six spin states of the mn atom as a function of bias for strong tip to sample coupling ( @xmath142 mev ) . from the figure one can see that as the bias increases the @xmath139 ground state gets depleted in favour of populating the other five excited states . in particular already at @xmath147 mv the population of the @xmath148 level is larger than that of the ground state . [ fig5 ] in th figure we also plot the average magnetization , which is defined as @xmath149 [ see panel ( b ) ] . intriguingly we find that for negative biases the spin is effectively flipped from @xmath139 to @xmath150 over 25mev range . such spin flipping results in a large dip in the conductance for negative biases as the tip is no longer collinear to the sample . in fig [ fig6 ] we present the calculated spectra for the fe atom in the spin polarized case . we choose parameters in this case that conform with the experimental data of loth _ the fe atom is assumed to carry a quantum mechanical spin of @xmath121 and it also exhibits a transverse easy axis anisotopy of @xmath151mev and an axial anisotropy of @xmath152mev . we again assume a large value of the onsite energy , @xmath153ev , and we examine the spectra in the strong coupling case of @xmath142mev with a tip polarisation of @xmath154 and magnetic field strength 3 t as used in experiments . in fig 6(a ) we present the conductance spectra for the two cases when the magnetic field is parallel and perpendicular to the easy axis of the atom ( the @xmath155axis in this model ) . as previously , we present this for both second and third order calculations . firstly , we notice that the spin polarized tip affects the spectra only in the case of parallel magnetic field where a clear bias assymetry is produced . no significant assymetry is found in the perpendicular case . this conforms with the experimental findings and is due to the fact that electron spins in the tip are no longer colinear with localised spin of the fe atom . as found in previous works @xcite the inclusion of third order effects is vital in reproducing the corresct logarithmic decay at each of the conductance steps , which is particularly noticable for the perpendicular magnetic field . more significantly , experimental spectra for the parallel case exhibit a zero bias conductance dip which is absent in the 2nd order spectra but appears strongly when third order effects are included . this can also be seen the calculation of the second derivative of the current in fig 6(b ) where a clear zero bias anomaly is evident in the third order case . we finally move to discuss the inherent asymmetry measured in the conductance profile , which is usually observed even if the tip is not spin - polarized @xcite . we model this lineshape feature by including the real part of the full interacting electron - spin self - energy in the description [ see equation ( [ eq:22 ] ) ] . the structure of this contribution to the self - energy shows an explicit dependence on the on - site energy , @xmath10 , and also a logarithmic peak of width @xmath57 at the onset of an inelastic transition ( @xmath156 ) . the asymmetry arises from the difference in polarity of the logarithmic peak for @xmath157 . the self - energy is thus an odd function of both energy and bias . this results in the conductance profile having a decrease of the step heights for @xmath158 and a increase of them for @xmath159 . we test this approach by considering the case of a non - spin - polarized tip and a single mn atom . we use the same anisotropy parameters as for the mn dimer but , for the sale of simplicity , we keep the spin always its equilibrium state and choose @xmath140mev . figure [ fig7 ] shows the resulting conductance spectra for three different choices of the on - site energy @xmath160 . it is cear that the closer @xmath160 is to the fermi energy ( 0 ev ) , the greater is the bias asymmetry , while as @xmath160 is increased , the conductance profile becomes more symmetric . in this respect , the formalism outlined here is in agreement with the fano lineshape argument @xcite where the degree of asymmetry for electrons tunneling through a single impurity is given by a ratio of the real to the imaginary contributions to the interacting green s function@xcite . as an additional test we consider the case of a mn trimer , whose spectrum was shown first by hijibehedin _ _ et al.__@xcite to exhibit a large bias asymmetry when measured with a non - magnetic tip . we model this system by choosing an antiferromagnetic nearest neighbour exchange coupling @xmath1612.3 mev . furthermore , in order to accurately describe the position of the principle conductance steps in the conductance profile , we also include a ferromagnetic second - nearest - neighbour interaction between the local spins of magnitude @xmath162 - 1.0 mev @xcite . we again choose to keep the spin system in equilibrium and therefore consider weak coupling between the stm tip and the second atom in the trimer chain ( @xmath140 mev ) . the best fit to the experimental data is found with @xmath163ev . figure [ fig8 ] shows the model fit to the experimental data ( from ref . [ ] ) ) . whereas previous calculations did not predict any conductance asymmetry @xcite it is clear from the figure that the inclusion of the real part of the self - energy in the description produces a significant conductance asymmetry this is most prominent at the principle step height @xmath164 for each bias polarity . although the step height for the negative bias is not as small as that found experimentally , the qualitative trends are similar . in particular we notice the logarithmic conductance increase ( reduction ) that occurs before ( after ) the onset of the step at @xmath165mev , which also originates from the third order contribution to the self - energy . in this work , based on a perturbative approach of the s - d model , we have shown that the entire lineshape description can be re - conciliated with experiments by considering an expansion of the self - energy to the third order , which also includes its real part . as such we have shown that the conductance asymmetry can be described also if the electronic orbitals forming the sample s spin are not explicitly taken into account as also suggested by delgado and fernandez - rossier @xcite . in conclusion we have studied the lineshape details of the conductance profile of mn atoms deposited on cun and probed with a stm tip either or not carrying spin - polarization . in particular we have looked closely at the asymmetry of the conductance with the bias polarity . firstly , we have extended our perturbative approach to spin - scattering to the spin - polarized case and considered an expansion of the complex part of the electronic propagator up to the third order . this allows us to reproduce the logarithmic decay of the conductance subsequent a conductance step , which is observed in experiments but could not be explained by a second order theory . when the current density is increased and the tip is spin - polarized the conductance profile starts to develop a significant asymmetry with respect to the bias polarity . these are indicative of the spin system being driven out of equilibrium . we have then derived a second order expansion of the spin - propagator capable of evaluating the non - equilibrium population of the various spin energy levels . this was put favorably to the test against a series of experiments probing a single mn and fe ions with a spin - polarized stm tip in an intense magnetic field . furthermore the same formalism was capable of describing excitations occurring away from the ground state for a mn dimer probed by a non - magnetic tip . also in this case the agreement with experiments is very satisfactory . finally , in an attempt to describe the bias asymmetry in the case of non - spin - polarized stm tips we have derived an analytic expression for the real part of the electron - spin interacting self - energy . this contains logarithmic peaks at the excitation energies that are odd with respect to energy and voltage . such parity results in an asymmetry in the conductance profiles . such a scheme was tested for the case of a mn monomer and a mn trimer and compares reasonably well with experiments . this work is sponsored by the irish research council for science , engineering & technology ( ircset ) . nb and ss thank science foundation of ireland ( grant no . 08/era / i1759 ) and crann for financial support . computational resources have been provided by the trinity centre for high performance computing ( tchpc ) . we wish to thank cyrus hirjibehedin for making the experimental data shown in figures [ fig7 ] available to us . here we wish to derive a method for calculating the steady - state non - equilibrium distribution of the spin energy levels populations , @xmath166 , due to the coupling with the electrodes . in order to do so we expand equation ( [ eq:5 ] ) up to the @xmath43-th order in the interaction hamiltonian @xmath167_{nm}=\sum_n\frac{(-i)^{n+1}}{n!}\int\limits_c{d}\tau_1\dots\int\limits_c{d}\tau_n\ \times \nonumber \\ & \frac{{\langle}0| t_c\{{h}_\mathrm{e - sp}(\tau_1)\dots{h}_\mathrm{e - sp}(\tau_n)d_{n}(\tau)d_{m}^{\dagger}(\tau')\}|0{\rangle}}{u(-\infty,-\infty)}\:,\end{aligned}\ ] ] where @xmath63 is the time - evolution unitary operator and the time - averages are over the known non - interacting @xmath64 ground state @xmath65 . as in equation ( [ eq:8 ] ) the time integration over @xmath66 is ordered on the contour @xmath67 going from @xmath68 to @xmath69 and then returning from @xmath69 to @xmath68 @xcite . by inserting the expression for @xmath61 from equation ( [ eq:3 ] ) into the equation above and by expanding up to the second order we obtain [ note for the ease of the description we omit the elastic contribution of @xmath168 , which is then included in the final expression in equation ( [ eq:12 ] ) ] @xmath169^{(2)}_{nm}=\frac{(-i)^{3}}{2!}j^2_\mathrm{sd}\sum_{\alpha,\alpha',\beta,\beta'}\int\limits_c{d}\tau_1\int\limits_c{d}\tau_2\ \times \nonumber \\ & { \langle}0|t_c\{c_{\alpha}^{\dagger}(\tau_1)c_{\alpha'}(\tau_1)c_{\beta}^{\dagger}(\tau_2)c_{\beta'}(\tau_2)d_{n}(\tau)d_{m}^{\dagger}(\tau')\}|0{\rangle } \nonumber \\ & \times \sum_{i , j}{\langle}0|t_c\{{s^{i}}(\tau_1){s^{j}}(\tau_2)\}|0{\rangle}[{\sigma}^i]_{{\alpha}{\alpha'}}[{\sigma}^j]_{{\beta}{\beta'}}\:,\end{aligned}\ ] ] where the indices @xmath8 and @xmath170 run over the cartesian coordinates @xmath171 , @xmath172 and @xmath173 for the given spin coupled to the tip ( the tip make electronic contact with one spin only ) . we now substitute into equation ( [ eq:9 ] ) the operator breakdown of the spin from equation ( [ eq:6 ] ) @xmath174^{(2)}_{nm}=\frac{(-i)^{3}}{2!}j^2_\mathrm{sd}\sum_{k , k',l , l'}\int\limits_c{d}\tau_1\int\limits_c{d}\tau_2 \nonumber \\ & \times { \langle}0|t_c\{d_{n}(\tau)d_{k}^{\dagger}(\tau_1)d_{k'}(\tau_1)d_{l}^{\dagger}(\tau_2)d_{l'}(\tau_2)d_{m}^{\dagger}(\tau')\}|0{\rangle } \nonumber \\ & \times \sum_{\alpha,\alpha',\beta,\beta'}{\langle}0|t_c\{c_{\alpha'}(\tau_1)c_{\beta}^{\dagger}(\tau_2)c_{\beta'}(\tau_2)c_{\alpha}^{\dagger}(\tau_1)\}|0{\rangle } \nonumber \\ & \times \sum_{i , j}s_{kk'}^is_{ll'}^j[{\sigma}^i]_{{\alpha}{\alpha'}}[{\sigma}^j]_{{\beta}{\beta'}}\:.\end{aligned}\ ] ] the time - ordered contractions of the two brackets in equation ( [ eq:10 ] ) can be re - written in terms of their respective non - interacting green s functions , @xmath175 and @xmath176 as follows @xmath177^{(2)}_{nm}=-j^2_\mathrm{sd}\sum_{k , k',l , l'}\int\limits_c{d}\tau_1\int\limits_c{d}\tau_2 \nonumber \\ & \times \delta_{nk}\delta_{lk'}\delta_{ml'}[d_0(\tau,\tau_1)]_{nn}[d_0(\tau_1,\tau_2)]_{ll}[d_0(\tau_2,\tau')]_{mm } \nonumber \\ & \times \sum_{\alpha,\alpha',\beta,\beta'}\delta_{\alpha'\beta}\delta_{\alpha\beta'}[g_0(\tau_1,\tau_2)]_{\beta\beta}[g_0(\tau_2,\tau_1)]_{\alpha\alpha } \nonumber \\ & \times \sum_{i , j}s_{kk'}^is_{ll'}^j[{\sigma}^i]_{{\alpha}{\alpha'}}[{\sigma}^j]_{{\beta}{\beta'}},\end{aligned}\ ] ] where the extra factor of 2 emerges from the fact that a second contraction of the time - ordered bracket merely exchanges @xmath178 and @xmath179 . then , by using dyson s equation @xcite , one can write the second order contribution to the interacting spin self - energy ( @xmath180 ) . this reads @xmath181^{(2)}_{nm}=-2j^2_\mathrm{sd}\sum_{\alpha,\beta}[g_0(\tau_1,\tau_2)]_{\beta\beta}[g_0(\tau_2,\tau_1)]_{\alpha\alpha } \nonumber \\ & \times \sum_{l}[d_0(\tau_1,\tau_2)]_{l , l}\sum_{i , j}s^i_{nl}s^j_{lm}[{\sigma}^i]_{{\alpha}{\beta}}[{\sigma}^j]_{{\beta}{\alpha}}\:,\end{aligned}\ ] ] where we have evoked the assumption that the electrons are spin degenerate thus omitting the spin index on @xmath182 and including a factor of 2 . we now calculate the real - time quantities , such as the lesser ( greater ) self - energies , by using langreth s theorem for the time ordering over the defined contour @xcite . after including the elastic contribution we obtain @xmath183^{(2)}_{nm}=-2j^2_\mathrm{sd}\sum_{\alpha,\beta}[g^{\lessgtr}_0(t_1,t_2)]_{\beta\beta}[g^{\gtrless}_0(t_2,t_1)]_{\alpha\alpha } \nonumber \\ & \times \sum_{l}[d^{\lessgtr}_0(t_1,t_2)]_{ll}\nonumber \\ & \times\sum_{i , j}\big(s^i_{nl}s^j_{lm}[{\sigma}^i]_{{\alpha}{\beta}}[{\sigma}^j]_{{\beta}{\alpha}}+\delta_{ij}\delta_{\alpha\beta}\chi s^i_{nm}[{\sigma}^i]_{{\alpha}{\beta}}\big)\:.\end{aligned}\ ] ] on computing the fourier transform we note the two different expressions for the lesser and greater green s functions are @xmath184^{(2)}_{nm}=-\frac{j^2_\mathrm{sd}}{\pi}\sum_lp^{\lessgtr}_l\times \nonumber \\ & \sum_{\alpha,\beta}\int\limits_{-\infty}^{+\infty}{d}{\omega}[g_0^{<}(\omega)]_{\beta\beta}[g_0^{>}(\omega\pm(e-\varepsilon_l))]_{\alpha\alpha}\times\nonumber \\ & \sum_{i , j}\big(s^i_{nl}s^j_{lm}[{\sigma}^i]_{{\alpha}{\beta}}[{\sigma}^j]_{{\beta}{\alpha}}+ \delta_{ij}\delta_{\alpha\beta}\chi s^i_{nm}[{\sigma}^i]_{{\alpha}{\beta}}\big),\end{aligned}\ ] ] where we have defined @xmath185 and @xmath186 and @xmath187 $ ] . by assuming that the spin system is in thermal contact with a heat bath kept at the temperature @xmath46 , the energy levels @xmath188 should be broadened by the factor @xmath189 . this can be neglected for the ease of the calculation since in general @xmath190 . however we do not disregard the broadening in the electronic green s function due to contact to tip and substrate as this is pivotal to the calculation of the non - equilibrium spin populations . j. j. parks , a. r. champagne , t. a. costi , w. w. shum , a. n. pasupathy , e. neuscamman , s. flores - torres , p. s. cornaglia , a. a. aligia , c. a. balseiro , g. k .- l . chan , h. d. abrua and d. c. ralph1 _ et al . _ , science * 328 * , 1370 ( 2010 ) .
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the conductance profiles of magnetic transition metal atoms , such as fe , co and mn , deposited on surfaces and probed by a scanning tunneling microscope ( stm ) , provide detailed information on the magnetic excitations of such nano - magnets .
in general the profiles are symmetric with respect to the applied bias .
however a set of recent experiments has shown evidence for inherent asymmetries when either a normal or a spin - polarized stm tip is used . in order to explain such asymmetries here we expand our previously developed perturbative approach to electron - spin scattering to the spin - polarized case and to the inclusion of out of equilibrium spin populations . in the case of a magnetic stm tip
we demonstrate that the asymmetries are driven by the non - equilibrium occupation of the various atomic spin - levels , an effect that reminds closely that electron spin - transfer .
in contrast when the tip is not spin - polarized such non - equilibrium population can not be build up . in this circumstance
we propose that the asymmetry simply originates from the transition metal ion density of state , which is included here as a non - vanishing real component to the spin - scattering self - energy .
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recall a polynomial @xmath0 is _ postcritically finite _ if the forward orbits of its critical points are finite . then the filled julia set of @xmath1 contains a forward invariant , finite topological tree , called the _ hubbard tree _ @xcite . the _ core entropy _ of @xmath1 is the topological entropy of the restriction of @xmath1 to its hubbard tree . we shall restrict ourselves to quadratic polynomials . given @xmath2 , the external ray at angle @xmath3 determines a postcritically finite parameter @xmath4 in the mandelbrot set @xcite . we define @xmath5 to be the core entropy of @xmath6 . the main goal of this paper is to prove the following result : [ t : main ] the core entropy function @xmath7 extends to a continuous function from @xmath8 to @xmath9 . the theorem answers a question of w. thurston , who first introduced and explored the core entropy of polynomials . as thurston showed , the core entropy function can be defined purely combinatorially , but it displays a rich fractal structure ( figure [ f : core ] ) , which reflects the underlying geometry of the mandelbrot set . [ f : core ] the concept of core entropy generalizes to complex polynomials the entropy theory of real quadratic maps , whose monotonicity and continuity go back to milnor and thurston @xcite : indeed , the invariant real segment is replaced by the invariant tree , which captures all the essential dynamics . on the topological side , hubbard trees have been introduced to classify postcritically finite maps @xcite , @xcite , and their entropy provides a new tool to study the parameter space of polynomials : for instance , the restriction of @xmath5 to non - real veins of the mandelbrot set is also monotone @xcite , which implies that the lamination for the mandelbrot set can be reconstructed by looking at level sets of @xmath10 . note by comparison that the entropy of @xmath11 on its julia set is constant , independent of @xmath3 , hence it does not give information on the parameter . furthermore , the value @xmath5 equals , up to a constant factor , the hausdorff dimension of the set of biaccessible angles for @xmath11 ( @xcite , @xcite ) . for dendritic julia sets , the core entropy also equals the asymptotic stretch factor of @xmath11 as a rational map @xcite . a rational external angle @xmath3 determines a postcritically finite map @xmath11 , which has a finite hubbard tree @xmath12 . the simplest way to compute its entropy is to compute the markov transition matrix for the map @xmath11 acting on @xmath12 , and take ( the logarithm of ) its leading eigenvalue : however , this requires to know the topology of @xmath12 , which changes wildly after small perturbations of @xmath3 . in this paper , we shall by - pass this issue by leveraging an algorithm , devised by thurston , which considers instead a larger matrix , whose entries are _ pairs _ of postcritical angles ( see section [ s : algo ] ) ; no knowledge of the topology of the hubbard tree is required . in order to prove @xmath10 is continuous , we develop an infinite version of such algorithm , which is defined for _ any _ angle @xmath13 . in particular , instead of taking the leading eigenvalue of a transition matrix , we shall encode the possible transitions in a directed graph , which will now have countably many vertices . by taking the _ spectral determinant _ of such graph ( see section [ s : spectral ] for a definition ) , for each angle @xmath13 , we construct a power series @xmath14 which converges in the unit disk and such that its smallest zero " @xmath15 is related to the core entropy by @xmath16 we then produce an algorithm to compute each coefficient of the taylor expansion of @xmath14 , and show that essentially these coefficients vary continuously with @xmath3 : the result follows by rouch s theorem . as a corollary of our method , we shall prove that the entropy function is actually hlder continuous at angles @xmath3 such that @xmath17 ( using renormalization , it can be proven that @xmath10 is not hlder continuous where @xmath18 ) . on the way to our proof , we shall develop a few general combinatorial tools to deal with growth rates of countable graphs , which may be of independent interest . we define a countable graph @xmath19 with bounded ( outgoing ) degree to have _ bounded cycles _ if it has finitely many closed paths of any given length . in this case , we define the _ growth rate _ of @xmath19 as the growth rate with @xmath20 of the number of closed paths of length @xmath20 ( see definition [ d : growth ] ) ; as it turns out ( theorem [ t : rootofp ] ) , the inverse of the growth rate equals the smallest zero of the following function @xmath21 , constructed by counting the multi - cycles in the graph : @xmath22 where @xmath23 is the number of components of the multi - cycle @xmath24 , and @xmath25 is its length . ( for the definition of multi - cycle , see section [ s : graphs ] ; note that our graphs have finite outgoing degree but possibly infinite ingoing degree , hence the adjacency operator has infinite @xmath26-norm and the usual spectral theory ( see e.g. @xcite ) does not apply ) . we then define a general combinatorial object , called _ labeled wedge _ , which consists of pairs of integers which can be labeled either as being _ separated _ ( representing two elements of the postcritical set which lie on opposite sides of the critical point ) or _ non - separated _ ; to such object we associate an infinite graph , and prove that it has bounded cycles , hence one can apply the theory developed in the first part ( see theorem [ t : zero ] ) . finally , we shall apply these combinatorial techniques to the core entropy ( section [ s : core ] ) ; indeed , we associate to any external angle @xmath3 a labeled wedge @xmath27 , hence an infinite graph @xmath28 , and verify that : 1 . the growth rate of @xmath28 varies continuously as a function of @xmath3 ( theorem [ t : newmain ] ) ; 2 . the growth rate of @xmath28 coincides with the core entropy for rational angles ( theorem [ t : coincide ] ) . let us remark that , as a consequence of monotonicity along veins and theorem [ t : main ] , the continuous extension we define also coincides with the core entropy for parameters which are not necessarily postcritically finite , but for which the julia set is locally connected and the hubbard tree is topologically finite . the core entropy of polynomials has been introduced by w. thurston around 2011 ( even though there are earlier related results , e.g. @xcite , @xcite , @xcite ) but most of the theory is yet unpublished ( with the exception of section 6 in @xcite ) . several people are now collecting his writings and correspondence into a foundational paper @xcite . in particular , the validity of thurston s algorithm has been proven by tan l. , gao y. , and w. jung ( see @xcite , @xcite ) . continuity of the core entropy along principal veins in the mandelbrot set is proven by the author in @xcite , and along all veins by w. jung @xcite . note that the previous methods used for veins do not easily generalize , since the topology of the tree is constant along veins but not globally . after @xcite , alternative proofs of monotonicity and continuity of entropy for real maps are given in @xcite , @xcite ( the present proof independently yields continuity ) . biaccessible external angles and their dimension have been discussed in @xcite , @xcite , @xcite , @xcite , @xcite . the method we use to count closed paths in the graph bears many similarities with the theory of _ dynamical zeta functions _ @xcite , and several forms of the spectral determinant are used in thermodynamic formalism ( see e.g. @xcite , @xcite ) . moreover , the spectral determinant @xmath21 we use is an infinite version of the _ clique polynomial _ used in @xcite to study finite directed graphs with small entropy . the motivation for our combinatorial construction is thurston s algorithm to compute the core entropy for rational angles , which we shall now describe . let @xmath2 a rational number @xmath29 . then the external ray at angle @xmath3 in the mandelbrot set lands at a misiurewicz parameter , or at the root of a hyperbolic component : let @xmath1 denote the corresponding postcritically finite quadratic map . we shall call @xmath30 the critical point of @xmath1 , and for each @xmath31 , @xmath32 the @xmath33 iterate of the critical point . recall that the _ hubbard tree _ of @xmath1 is the union of the regulated arcs @xmath34 $ ] for all @xmath35 ( for more details , see @xcite ) . in the postcritically finite case , it is a finite topological tree , and it is forward - invariant under @xmath1 ( see figure [ f : tree ] ) . it is possible to compute the core entropy of @xmath1 by writing the markov transition matrix for the action of @xmath1 on the tree , and take the logarithm of its leading eigenvalue . however , given the external angle @xmath3 it is quite complicated to figure out the topology of the tree : the following algorithm by - passes this issue by looking at pairs of external angles . in order to explain the algorithm in more detail , let us remark that a rational angle @xmath3 is eventually periodic under the doubling map @xmath36 ; that is , there exist integers @xmath37 and @xmath38 such that the elements of the set @xmath39 are all distinct modulo @xmath40 , and @xmath41 . the elements of @xmath42 will be called _ postcritical angles _ ; the number @xmath43 is called the _ period _ of @xmath3 , and @xmath44 is the _ pre - period_. if @xmath45 , we shall call @xmath3 _ purely periodic_. denote @xmath46 the partition of the circle @xmath47 in the two intervals @xmath48 moreover , for each @xmath49 let us denote @xmath50 , which we see as a point in @xmath51 . let us now construct a matrix @xmath52 which will be used to compute the core entropy . denote @xmath53 the set of unordered pairs of ( distinct ) elements of @xmath42 : this is a finite set , any element of which will be denoted @xmath54 . we now define a linear map @xmath55 by defining it on its basis vectors in the following way : * if @xmath56 and @xmath57 belong to the same element of the partition @xmath46 , or at least one of them lies on the boundary , we shall say that the pair @xmath54 is _ non - separated _ , and define @xmath58 * if @xmath56 and @xmath57 belong to the interiors of two different elements of @xmath46 , then we say that @xmath54 is _ separated _ , and define @xmath59 ( in order to make sure the formulas are defined in all cases , we shall set @xmath60 whenever @xmath61 ) . by abuse of notation , we shall also denote @xmath52 the matrix representing the linear map @xmath52 in the standard basis . as suggested by thurston in his correspondence , the leading eigenvalue of @xmath52 gives the core entropy : [ t : algo ] let @xmath3 a rational angle.then the core entropy @xmath5 of the quadratic polynomial of external angle @xmath3 is related to the largest real eigenvalue @xmath62 of the matrix @xmath52 by the formula @xmath63 the explanation of this algorithm is the following . let @xmath64 be a complete graph whose vertices are labeled by elements of @xmath42 , and which we shall consider as a topological space . ( more concretely , we can take the unit disk and draw segments between all possible pairs of postcritical angles on the unit circle : the union of all such segments is a model for @xmath64 ( see figure [ f : chords_algo ] ) ) . let us denote by @xmath65 the hubbard tree of @xmath1 . we can define a continuous map @xmath66 which sends the edge with vertices @xmath56 , @xmath57 homeomorphically to the regulated arc @xmath34 $ ] in @xmath65 ( except in the case @xmath67 , where we map all the edge to a single point ) . finally , we can lift the dynamics @xmath68 to a map @xmath69 such that @xmath70 . in order to do so , note that : * if the pair of angles @xmath54 is non - separated , then @xmath1 maps the arc @xmath34 $ ] homeomorphically onto @xmath71 $ ] ; hence to define @xmath72 we have to lift @xmath1 so that it maps the edge @xmath73 $ ] homeomorphically onto @xmath74 $ ] ; * if instead @xmath54 is separated , then the critical point @xmath30 lies on the regulated arc @xmath34 $ ] in the hubbard tree ; thus , @xmath1 maps the arc @xmath75 = [ c_i , c_0 ] \cup [ c_0 , c_j]$ ] onto @xmath76 \cup [ c_1 , c_{j+1}]$ ] ; so we define @xmath72 by lifting @xmath1 and so that it maps @xmath73 $ ] continuously onto the union @xmath77 \cup [ x_1 , x_{j+1}]$ ] . the map @xmath72 is a markov map of a topological graph , hence its entropy is the logarithm of the leading eigenvalue of its transition matrix , which is by construction the matrix @xmath52 . in case @xmath3 is purely periodic , one can prove that the map @xmath78 is surjective and finite - to - one , and semiconjugates the dynamics @xmath72 on @xmath64 to the dynamics @xmath1 on @xmath65 , hence @xmath79 more care is needed in the pre - periodic case , since @xmath78 can collapse arcs to points ( for details , see @xcite , @xcite , or @xcite ) . in the following , by _ graph _ we mean a directed graph @xmath19 , i.e. a set @xmath80 of vertices ( which will be finite or countable ) and a set @xmath81 of edges , such that each edge @xmath82 has a well - defined _ source _ @xmath83 and a _ target _ @xmath84 ( thus , we allow edges between a vertex and itself , and multiple edges between two vertices ) . given a vertex @xmath85 , the set @xmath86 of its _ outgoing edges _ is the set of edges with source @xmath85 . the _ outgoing degree _ of @xmath85 is the cardinality of @xmath86 ; a graph is _ locally finite _ if the outgoing degree of all its vertices is finite , and has _ bounded outgoing degree _ if there is a uniform upper bound @xmath87 on the outgoing degree of all its vertices . note that we do _ not _ require that the ingoing degree is finite , and indeed in our application we will encounter graphs with vertices having countably many ingoing edges . we denote as @xmath88 the set of vertices of @xmath19 , and as @xmath89 its set of edges . moreover , we denote as @xmath90 the number of edges from vertex @xmath85 to vertex @xmath91 . a _ path _ in the graph based at a vertex @xmath85 is a sequence @xmath92 of edges such that @xmath93 , and @xmath94 for @xmath95 . the _ length _ of the path is the number @xmath20 of edges , and the set of vertices @xmath96 visited by the path is called its _ vertex - suppoert _ , or just _ support _ for simplicity . similarly , a _ closed path _ based at @xmath85 is a path @xmath92 such that @xmath97 . note that in this definition a closed path can intersect itself , i.e. two of the sources of the @xmath98 can be the same ; moreover , closed paths with different starting vertices will be considered to be different . on the other hand , a _ simple cycle _ is a closed path which does not self - intersect , modulo cyclical equivalence : that is , a simple cycle is a closed path @xmath92 such that @xmath99 for @xmath100 , and two such paths are considered the same simple cycle if the edges are cyclically permuted , i.e. @xmath92 and @xmath101 designate the same simple cycle . finally , a _ multi - cycle _ is the union of finitely many simple cycles with pairwise disjoint ( vertex-)supports . the length of a multi - cycle is the sum of the lengths of its components . given a countable graph with bounded outgoing degree , we define the _ adjacency operator _ @xmath102 on the space of summable sequences indexed by the vertex set @xmath80 . in fact , for each vertex @xmath103 we can consider the sequence @xmath104 which is @xmath40 at position @xmath49 and @xmath105 otherwise , and define for each @xmath106 the @xmath107 component of the vector @xmath108 to be @xmath109 equal to the number of edges from @xmath49 to @xmath110 . since the graph has bounded outgoing degree , the above definition can actually be extended to all @xmath111 , and the operator norm of @xmath112 induced by the @xmath113-norm is bounded above by the outgoing degree @xmath114 of the graph . note moreover that for each pair @xmath115 of vertices and each @xmath20 , the coefficient @xmath116 equals the number of paths of length @xmath20 from @xmath49 to @xmath110 . we say a countable graph @xmath19 has _ bounded cycles _ if it has bounded outgoing degree and for each positive integer @xmath20 , @xmath19 has at most finitely many simple cycles of length @xmath20 . note that , if @xmath19 has bounded cycles , then for each @xmath20 it has also a finite number of closed paths of length @xmath20 , since the support of any closed path of length @xmath20 is contained in the union of the supports of simple cycles with length @xmath117 . thus , for such graphs we shall denote @xmath118 the number of closed paths of length @xmath20 . note that in this case the trace @xmath119 is also well - defined for each @xmath20 , and equal to @xmath120 . [ d : growth ] if @xmath19 is a graph with bounded cycles , we define the _ growth rate _ @xmath121 as the exponential growth rate of the number of its closed paths : that is , @xmath122{c(\gamma , n)}.\ ] ] if @xmath19 is a finite graph with adjacency matrix @xmath112 , then is well - defined its characteristic polynomial @xmath123 , whose roots are the eigenvalues of @xmath112 . in the following we shall work with the related polynomial @xmath124 which we call the _ spectral determinant _ of @xmath19 . we shall now extend the theory to countable graphs with bounded cycles . it is known that the spectral determinant @xmath21 of a finite graph @xmath19 is related to its multi - cycles by the following formula ( see e.g. @xcite ) : @xmath125 where @xmath25 denotes the length of the multi - cycle @xmath24 , while @xmath23 is the number of connected components of @xmath24 . if @xmath19 is now a ( directed ) graph with countably many vertices and bounded cycles , then the number of multi - cycles of any given length is finite , hence the formula above is still defined as a formal power series . note that we include also the empty cycle , which has zero components and zero length , hence @xmath21 begins with the constant term @xmath40 . now , let @xmath126 denote the number of multi - cycles of length @xmath20 in @xmath19 , and let us define @xmath127{k(\gamma , n)}\ ] ] its growth rate . then the main result of this section is the following : [ t : rootofp ] suppose we have @xmath128 ; then the formula defines a holomorphic function @xmath129 in the unit disk @xmath130 , and moreover the function @xmath129 is non - zero in the disk @xmath131 ; if @xmath132 , we also have @xmath133 . the proof uses in a crucial way the following combinatorial statement . [ l:1/pt ] let @xmath19 be a countable graph with bounded cycles , and @xmath112 its adjacency operator . then we have the equalities of formal power series @xmath134 where @xmath21 is the spectral determinant . note that , since @xmath21 is a power series with constant term @xmath40 , then @xmath135 is indeed a well - defined power series . the second equality is just obtained by expanding the exponential function in power series . to prove the first equality , let us first suppose @xmath19 is a finite graph with @xmath20 vertices , and let @xmath136 be the eigenvalues of its adjacency matrix ( counted with algebraic multiplicity ) . then @xmath137 hence @xmath138 hence the claim follows since @xmath139 . now , let @xmath19 be infinite with bounded cycles , and let @xmath140 . note that both sides of the equation depend , modulo @xmath141 , only on multi - cycles of length @xmath142 , which by the bounded cycle condition are supported on a finite subgraph @xmath143 . thus by applying the previous proof to @xmath143 we obtain equality modulo @xmath141 , and since this holds for any @xmath144 the claim is proven . by the root test , the radius of convergence of @xmath145 is @xmath146 . thus , since the exponential function has infinite radius of convergence , the radius of convergence of @xmath147 is at least @xmath146 , and since @xmath148 where @xmath149 has positive coefficients , then the radius of convergence of @xmath147 is exactly equal to @xmath146 . hence , by lemma [ l:1/pt ] , the radius of convergence of @xmath135 around @xmath150 is also @xmath146 . on the other hand , since @xmath128 , then by the root test the power series @xmath21 converges inside the unit disk , and defines a holomorphic function ; thus , @xmath135 is meromorphic for @xmath151 , and holomorphic for @xmath152 , hence @xmath153 if @xmath152 . moreover , if @xmath154 , then the radius of convergence of @xmath135 equals the smallest modulus of one of its poles , hence @xmath146 is the smallest modulus of a zero of @xmath21 . finally , since @xmath135 has all its taylor coefficients real and nonnegative , then the smallest modulus of its poles must also be a pole , so @xmath133 . [ l : perroneig ] if @xmath19 is a finite graph , then its growth rate equals the largest real eigenvalue of its adjacency matrix . note that , by the perron - frobenius theorem , since the adjacency matrix is non - negative , it has at least one real eigenvalue whose modulus is at least as large as the modulus of any other eigenvalue . moreover , if @xmath62 is the largest real eigenvalue , then @xmath155 is the smallest root of the spectral determinant @xmath21 , hence the claim follows from theorem [ t : rootofp ] . let @xmath156 two ( locally finite ) graphs . a _ graph map _ from @xmath157 to @xmath158 is a map @xmath159 on the vertex sets and a map on edges @xmath160 which is compatible , in the sense that if the edge @xmath82 connects @xmath85 to @xmath91 in @xmath157 , then the edge @xmath161 connects @xmath162 to @xmath163 in @xmath158 . we shall usually denote such a map as @xmath164 . weak cover _ of graphs is a graph map @xmath164 such that : * the map @xmath159 between the vertex sets is surjective ; * the induced map @xmath165 between outgoing edges is a bijection for each @xmath166 . note that the map between outgoing edges is defined because @xmath167 is a graph map , and @xmath167 also induces a map from paths in @xmath157 to paths in @xmath158 . as a consequence of the definition of weak cover , you have the following _ unique path lifting _ property : let @xmath164 a weak cover of graphs . then given @xmath166 and @xmath168 , for every path @xmath24 in @xmath158 based at @xmath91 there is a unique path @xmath169 in @xmath157 based at @xmath85 such that @xmath170 . let @xmath166 and @xmath168 , and let @xmath171 be a path in @xmath158 based at @xmath91 . since the map @xmath172 is a bijection , there exists exactly one edge @xmath173 such that @xmath174 . by compatibility , the target @xmath175 of @xmath176 projects to the target of @xmath177 , hence we can apply the same reasoning and lift the second edge @xmath178 uniquely starting from @xmath175 , and so on . an immediate consequence of the property is the following [ l : easyineq ] each graph map @xmath164 induces a map @xmath179 for each @xmath180 and each @xmath181 . moreveor , if @xmath167 is a weak cover , then @xmath182 is injective . a general way to construct weak covers of graphs is the following . suppose we have an equivalence relation @xmath183 on the vertex set @xmath80 of a locally finite graph , and denote @xmath184 the set of equivalence classes of vertices . such an equivalence relation is called _ edge - compatible _ if whenever @xmath185 , for any vertex @xmath91 the total number of edges from @xmath186 to the members of the equivalence class of @xmath91 equals the total number of edges from @xmath187 to the members of the equivalence class of @xmath91 . when we have such an equivalence relation , we can define a quotient graph @xmath188 with vertex set @xmath184 . namely , we denote for each @xmath189 the respective equivalence classes as @xmath190 $ ] and @xmath191 $ ] , and define the number of edges from @xmath190 $ ] to @xmath191 $ ] in the quotient graph to be @xmath192 \to [ w ] ) : = \sum_{u \in [ w ] } \#(v \to u).\ ] ] by definition of edge - compatibility , the above sum does not depend on the representative @xmath85 chosen inside the class @xmath190 $ ] . moreover , it is easy to see that the quotient map @xmath193 is a weak cover of graphs . let us now relate the growth of a graph to the growth of its weak covers . [ l : growthcover ] let @xmath164 a weak cover of graphs with bounded cycles , and @xmath194 a finite set of vertices of @xmath157 . 1 . suppose that every closed path in @xmath157 passes through @xmath194 . then for each @xmath20 we have the estimate @xmath195 which implies @xmath196 2 . suppose that @xmath197 is a set of closed paths in @xmath158 such that each @xmath198 crosses at least one vertex @xmath91 with the property that : the set @xmath199 is non - empty , and any lift of @xmath24 from an element of @xmath200 ends in @xmath200 . then there exists @xmath201 , which depends on @xmath194 , such that for each @xmath20 we have @xmath202 \(1 ) let @xmath24 be a closed path in @xmath157 of length @xmath20 , based at @xmath85 . let now @xmath203 be the first vertex of @xmath24 which belongs to @xmath194 ( by hypothesis , there is one ) . if we call @xmath204 the cyclical permutation of @xmath24 based at @xmath203 , the projection of @xmath204 to @xmath158 now yields a closed path @xmath205 in @xmath158 which is based at @xmath206 . now note that for each such pair @xmath207 , there are at most @xmath208 possible choices for @xmath209 ; then , given @xmath203 there is a unique lift of @xmath205 from @xmath203 , and @xmath20 possible choices for the vertex @xmath85 on that lift , giving the estimate . \(2 ) recall first that in every directed graph we can define an equivalence relation by saying @xmath210 if there is a path from @xmath85 to @xmath91 and a path from @xmath91 to @xmath85 . the equivalence classes are known as _ strongly connected components _ , or @xmath211 for short . moreover , we can form a graph @xmath212 whose vertices are the strongly connected components and there is an edge from the s.c.c . @xmath213 to @xmath214 if in the original graph there is a path from an element of @xmath213 to an element of @xmath214 . note that by construction this graph has no cycles . moreover , let us consider a pair @xmath215 of distinct elements of @xmath194 ; then either there is no path from @xmath186 to @xmath187 , or we can pick some path from @xmath186 to @xmath187 , which will be denoted @xmath216 . then the set @xmath217 of such paths is finite , and let @xmath72 be the maximum length of an element of @xmath218 . let now @xmath24 be a closed path in @xmath158 based at @xmath219 . by hypothesis , @xmath24 passes through some @xmath91 such that @xmath220 , and each lift of @xmath24 from @xmath200 ends in @xmath200 . let us now consider the set @xmath221 of all s.c.c . of @xmath157 which intersect @xmath200 : since the graph @xmath212 constructed above has no cycles , there is a component @xmath222 which has the property that there is no path from @xmath223 to some other component of @xmath200 . thus , if we pick @xmath224 and lift @xmath24 from @xmath85 to a path @xmath169 in @xmath157 , the endpoint @xmath175 of @xmath169 must lie inside @xmath200 by hypothesis . thus , by the property of @xmath223 , the vertex @xmath175 lies in @xmath223 , hence there is a path @xmath225 from @xmath175 to @xmath85 in the previously chosen set @xmath218 , hence if we take @xmath226 , this is a closed path in @xmath157 of length between @xmath20 and @xmath227 . thus we have a map @xmath228 given by @xmath229 . now , given @xmath85 we can recover @xmath230 , and given @xmath205 we can recover @xmath24 , since @xmath24 is the path given by the first @xmath20 edges of @xmath231 starting at @xmath91 . finally , we have at most @xmath20 choices for the starting point @xmath219 on @xmath24 , hence the fibers of the above map have cardinality at most @xmath20 , proving the claim . an immediate corollary of the lemma is the following , when @xmath157 and @xmath158 are both finite . [ l : sameentro ] let @xmath156 be finite graphs , and @xmath164 a weak cover . then the growth rate of @xmath157 equals the growth rate of @xmath158 . apply the lemma with @xmath232 , and @xmath197 equal to the set of all closed paths in @xmath158 . let us consider the set @xmath233 of pairs of disjoint positive integers . the set @xmath234 will be sometimes called the _ wedge _ and , given an element @xmath235 , the coordinate @xmath49 will be called the _ height _ of @xmath85 , while the coordinate @xmath110 will be called the _ width _ of @xmath85 . the terminology becomes more clear by looking at figure [ f : wedge ] . @xmath236 we call a _ labeled wedge _ an assignment @xmath237 of a label @xmath238 ( which stands for _ non - separated _ ) or @xmath194 ( which stands for _ separated _ ) to each element of @xmath234 . now , to each labeled wedge @xmath239 we assign a graph @xmath240 in the following way . the vertex set of @xmath240 is the wedge @xmath234 , while the edges of @xmath240 are labeled by the set @xmath241 and determined according to the following rule . * if @xmath242 is non - separated , then @xmath243 has as its ( unique ) successor the vertex @xmath244 ; we say that the edge @xmath245 is an _ upward edge _ and we label it with @xmath246 ; * if @xmath247 is separated , then @xmath247 has two successors : * * first , we add the edge @xmath248 , which we call _ forward edge _ and label it with @xmath249 . * * second , we add the edge @xmath250 , which we call _ backward edge _ and label it with @xmath251 . in order to explain the names , note that following an upward or forward edge increases the width by @xmath40 , while following a backward edge ( weakly ) decreases it . moreover , following an upward edge increases the height by @xmath40 , while the targets of both backward and forward edges have height @xmath40 . @xmath252\txt{(2,4 ) } & ( 2,5 ) & \cdots \\ * + [ f]\txt{(1,2 ) } & ( 1,3 ) & * + [ f]\txt{(1,4 ) } & ( 1,5 ) & \cdots \\ } \ ] ] @xmath253^u & ( 3,5 ) & \cdots \\ & ( 2,3 ) \ar[ur]^u & * + [ f]\txt{(2,4 ) } \ar[dr]^f \ar@/_/[dl]_b & ( 2,5 ) & \cdots \\ * + [ f]\txt{(1,2 ) } \ar@(u , l)_b \ar[r]^f & ( 1,3 ) \ar[ur]_u & * + [ f]\txt{(1,4 ) } \ar[r]^f \ar@/^/[ll]^b & ( 1,5 ) & \cdots \\ } \ ] ] [ p : properties ] let @xmath240 the graph associated to the labeled wedge @xmath239 . then the following are true : 1 . each vertex of a closed path of length @xmath20 has height at most @xmath20 ; 2 . the support of each closed path of length @xmath20 intersects the set @xmath254 ; 3 . each vertex of a closed path of length @xmath20 has width at most @xmath255 ; 4 . for each @xmath256 , there exists at most one separated vertex in the @xmath257 diagonal @xmath258 which is contained in the support of at least one closed path ; 5 . there are at most @xmath259 multi - cycles of length @xmath20 . note that ( 1 ) , ( 2 ) , ( 3 ) are sharp , as seen by the simple cycle @xmath260^u & 25 \ar@/^/[r]^u & * + [ f]\txt{36 } \ar@/^/[ll]^b } \ ] ] \(1 ) let us first note that every closed path contains at least one backward edge , since the upward and forward edges always increase the height . moreover , the endpoint of a backward edge has always height @xmath40 , and each edge increases the height by at most @xmath40 , hence the height of a vertex along the closed path is at most @xmath20 . \(2 ) since the target of each backward edge is @xmath261 , where @xmath49 is the height of the source of the edge , which is at most @xmath20 by the previous point , then the target of each backward edge along the closed path belongs to the set @xmath262 . \(3 ) by the previous point , there is at least a vertex along the closed path with width at most @xmath263 . since every move increases the width by at most @xmath40 , then the largest possible width of a vertex along the path is @xmath264 . \(4 ) let @xmath265 be the separated vertex in @xmath266 with smallest height , if there is one . we claim that no vertex @xmath267 of @xmath266 with @xmath268 belongs to any closed path . note by looking at the rules that , if a vertex @xmath269 of height @xmath270 is the target of some edge , then it must be the target of an upward edge , more precisely an edge from the vertex @xmath271 immediately to the lower left of @xmath85 , which then must be non - separated . thus , since @xmath272 is separated , the vertex @xmath273 does not belong to any closed path ; the claim then follows by induction on @xmath274 , since , by the same reasoning , if @xmath275 belongs to some closed path , then also @xmath276 must belong to the same path . \(5 ) let @xmath277 the backward edges along a multi - cycle @xmath24 of length @xmath20 , and denote @xmath278 the source of @xmath98 , and @xmath279 the height of @xmath278 . note that the set @xmath280 determines @xmath24 , as you can start from @xmath186 , follow the backward edge , and then follow upward or forward edges until you either close the loop or encounter another @xmath278 , and then continue this way until you walk along all of @xmath24 . moreover , we know that for each @xmath278 there are at most @xmath255 possible choices , as each @xmath278 is separated and by ( 4 ) there is at most one for each diagonal @xmath266 , and by ( 3 ) it must lie on some @xmath266 with @xmath281 . we now claim that @xmath282 which is then sufficient to complete the proof . let us now prove the claim . by definition of multi - cycle , then the targets of the @xmath98 must be all distinct , and by the rule these targets are precisely @xmath283 with @xmath284 , hence all @xmath279 must be distinct . moreover , let us note that each @xmath98 must be preceded along the multi - cycle by a sequence @xmath285 of upward edges of length @xmath286 , and all such sequences for distinct @xmath49 must be disjoint . hence we have that all @xmath279 are distinct and their total sum is at most the length of the multi - cycle , i.e. @xmath20 . thus we have @xmath287 which proves the claim . [ t : zero ] let @xmath239 be a labeled wedge . then its associated graph @xmath19 has bounded cycles , and its spectral determinant @xmath21 defines a holomorphic function in the unit disk . moreover , the growth rate @xmath288 of the graph @xmath19 equals the inverse of the smallest real positive root of @xmath129 , in the following sense : @xmath289 for @xmath131 and , if @xmath132 , then @xmath133 . by construction , the outgoing degree of any vertex of @xmath19 is at most @xmath290 . moreover , by proposition [ p : properties ] ( 5 ) the graph has bounded cycles , and the growth rate of the number @xmath126 of multi - cycles is @xmath291 , since @xmath292{(2n)^{\sqrt{2n } } } = \limsup_{n } e^{\frac{\sqrt{2 } \log(2n)}{\sqrt{n } } } = 1.\ ] ] the claim then follows by theorem [ t : rootofp ] . we shall sometimes denote as @xmath293 the growth rate of the graph associated to the labeled wedge @xmath239 . we say that a sequence @xmath294 of labeled wedges converges to @xmath239 if for each finite set of vertices @xmath295 there exists @xmath238 such for each @xmath296 the labels of the elements of @xmath194 for @xmath297 and @xmath239 are the same . [ l : contwedge ] if a sequence of labeled wedges @xmath294 converges to @xmath239 , then the growth rate of @xmath297 converges to the growth rate of @xmath239 . let @xmath298 and @xmath21 denote respectively the spectral determinants of @xmath297 and @xmath239 . then for each @xmath256 , the coefficient of @xmath299 in @xmath298 converges to the coefficient of @xmath299 in @xmath21 , because by proposition [ p : properties ] ( 1 ) and ( 3 ) the support of any multi - cycle of length @xmath300 is contained in the finite subgraph @xmath301 . thus , since the modulus of the coefficient of @xmath299 is uniformly bounded above by @xmath302 , then @xmath303 uniformly on compact subsets of the unit disk . thus , by rouch s theorem , the smallest real positive zero of @xmath298 converges to the smallest real positive zero of @xmath21 , hence by theorem [ t : zero ] we have @xmath304 . given integers @xmath37 and @xmath38 we define the equivalence relation @xmath305 on @xmath306 by saying that @xmath307 if : * either @xmath308 and @xmath309 ; * or @xmath310 and @xmath311 . note that if @xmath45 the equivalence relation @xmath312 is simply the congruence modulo @xmath43 . a set of representatives for the equivalence classes of @xmath312 is the set @xmath313 . the equivalence relation induces an equivalence relation on the set @xmath314 of ordered pairs of integers by saying that @xmath315 if @xmath316 and @xmath317 . moreover , it also induces an equivalence relation on the set of _ unordered pairs _ of integers by saying that the unordered pair @xmath318 is equivalent to @xmath319 if either @xmath320 or @xmath321 . a labeled wedge is _ periodic _ of period @xmath43 and pre - period @xmath44 if the following two conditions hold : * any two pairs @xmath247 and @xmath322 such that @xmath323 have the same label ; * if @xmath307 , then the pair @xmath247 is non - separated . if @xmath324 , the labeled wedge will be called _ purely periodic_. a pair @xmath247 with @xmath325 will be called _ diagonal _ ; hence the second point in the definition can be rephrased as every diagonal pair is non - separated " . @xmath326 & \dots \\ & & ( 3,4 ) \ar[ur ] & ( 3 , 5 ) \ar[ur ] & \dots\\ & * + [ f]\txt{(2,3 ) } \ar[d ] \ar[dr ] & * + [ f]\txt{(2,4 ) } \ar[dl ] \ar[dr ] & * + [ f]\txt{(2 , 5 ) } \ar[dll ] \ar[dr ] & \dots \\ ( 1,2 ) \ar[ur ] & * + [ f]\txt{(1,3 ) } \ar[r ] \ar[l ] & * + [ f]\txt{(1 , 4 ) } \ar[r ] & * + [ f]\txt{(1 , 5 ) } \ar[r ] & \dots}\ ] ] @xmath327\txt{(2,3 ) } \ar@<0.5ex>[d ] \ar@<-0.5ex>[d ] \\ ( 1,2 ) \ar[ur ] & * + [ f]\txt{(1,3 ) } \ar[l ] \ar@(d , r ) } \ ] ] given a periodic wedge @xmath239 of period @xmath43 and pre - period @xmath44 , with associated ( infinite ) graph @xmath19 , we shall now construct a finite graph @xmath328 which captures the essential features of the infinite graph @xmath19 , in particular its growth rate . the set of vertices of @xmath328 is the set of @xmath312-equivalence classes of non - diagonal , unordered pairs of integers . a set of representatives of @xmath329 is the set @xmath330 the edges of @xmath328 are induced by the edges of @xmath19 , that is are determined by the following rules : if the unordered pair @xmath318 is non - separated , then @xmath318 has one outgoing edge , namely @xmath331 ; while if @xmath318 is separated , then @xmath318 has the two outgoing edges @xmath332 , and @xmath333 . the main result of this section is the following . [ p : finiteinfinite ] let @xmath239 be a periodic labeled edge , with associated ( infinite ) graph @xmath19 . then the growth rate of @xmath19 equals the growth rate of its finite model @xmath328 . in order to prove the proposition , we shall also introduce an intermediate finite graph , which we call the _ finite 2-cover _ of @xmath328 , and denote @xmath334 . the set of vertices of @xmath334 is the set of @xmath305-equivalence classes of non - diagonal , ordered pairs of integers , and the edges are induced by the edges of @xmath19 in the usual way . the reason to introduce the intermediate graph @xmath334 is that @xmath328 does not inherit the labeling of edges from @xmath19 , as backward and forward edges in @xmath19 may map to the same edge in @xmath328 , while @xmath334 naturally inherits the labels . let @xmath19 be the graph associated to a periodic labeled wedge . 1 . first , let us observe that no diagonal vertex is contained in the support of any closed path of @xmath19 : in fact , every diagonal vertex is non - separated , and its outgoing edge leads to another diagonal vertex with larger height , hence the path can never close up . thus , we can construct the subgraph @xmath335 by taking as vertices all pairs which are non - diagonal , and as edges all the edges of @xmath19 which do not have either as a source or target a diagonal pair . by what has been just said , the growth rate of @xmath19 and @xmath335 is the same , @xmath336 2 . since the maps @xmath337 are given by quotienting with respect to equivalence relations , they are both weak covers of graphs . thus , since both @xmath334 and @xmath328 are finite , by lemma [ l : sameentro ] , the growth rates of @xmath334 and @xmath328 are the same . we are now left with proving that the growth rate of @xmath335 is the same as the growth rate of @xmath334 . since the cardinality of the fiber of the projection @xmath338 is infinite , the statement is not immediate . note that the finite @xmath290-cover @xmath334 is a graph with labeled edges : indeed , if @xmath320 , then @xmath339 , and so on , thus the labeling of @xmath334 inherited from @xmath19 is well - defined , and the graph map @xmath338 preserves the labels . 4 . let us call _ backtracking _ a path in @xmath19 or @xmath334 such that at least one of its edges is labeled by @xmath251 (= backward ) , and _ non - backtracking _ otherwise . now let us note the following : 1 . every closed path in @xmath19 is backtracking ; in fact , following any edge which is upward or forward increases the height , thus a path in @xmath19 made entirely of @xmath246 and @xmath249 edges can not close up . 2 . every closed path in @xmath19 passes through the finite set @xmath340 in order to prove this , we first prove that any element in the support of any closed path in @xmath19 has height at most @xmath341 . in order to do so , let us fix a diagonal @xmath342 . by periodicity , either there is a separated pair @xmath247 in @xmath266 of height @xmath343 , or all elements of @xmath266 are non - separated ; in the latter case , no element of @xmath266 is part of any closed path , since any path based at an element of @xmath266 is non - backtracking . in the first case , let @xmath344 be the separated pair with smallest height in @xmath266 ; then , only the elements with height less than @xmath345 can be part of any closed path , and @xmath346 , so the first claim is proven . as a consequence , the target of any @xmath251-labeled edge which belongs to some closed path is of type @xmath347 , where @xmath348 is the height of the source , hence the target belongs to @xmath194 . by the same reasoning , if @xmath24 is any path in @xmath19 based at a vertex of height @xmath40 , then the target of any @xmath251-labeled edge along @xmath24 lies in @xmath194 . 3 . every backtracking closed path in @xmath334 has at least one @xmath251-labeled edge , hence the target of such @xmath251-labeled edge lies in the set @xmath349 , and every lift to @xmath19 starting from an element of @xmath194 must end in the target of a @xmath251-labeled edge in @xmath19 , hence must end in @xmath194 . 5 . finally , let us note that for each @xmath20 , the number of non - backtracking paths of length @xmath20 in @xmath334 is at most the cardinality of @xmath350 ; indeed , from each vertex there is at most one edge labeled @xmath246 or @xmath249 , thus for each vertex of @xmath334 there is at most one non - backtracking path of length @xmath20 based at it . let us now put together the previous statements . indeed , by ( 4)(a)-(b ) every closed path in @xmath19 passes through @xmath194 , hence we can apply lemma [ l : growthcover ] ( 1 ) and get that @xmath351 to prove the other inequality , let us note that by ( 4)(b)-(c ) we know that lemma [ l : growthcover ] ( 2 ) applies with @xmath197 the set of backtracking closed paths in @xmath334 . moreover , by point ( 5 ) above we have that the number of non - backtracking paths is bounded independently of @xmath20 : thus we can write @xmath352 @xmath353 from which follows @xmath354 as required . when dealing with purely periodic external angles , we shall also need the following lemma . [ l : twoperiodics ] let @xmath355 and @xmath356 be two labeled wedges which are purely periodic of period @xmath43 . suppose moreover that for every pair @xmath247 with @xmath357 the label of @xmath243 in @xmath355 equals the label in @xmath356 . then the finite models @xmath358 and @xmath359 are isomorphic graphs . as a consequence , the growth rates of @xmath355 and @xmath356 are equal . let @xmath360 an equivalence class of unordered pairs . if neither @xmath49 nor @xmath110 are divisible by @xmath43 , then the label of @xmath318 is the same in @xmath355 and @xmath356 , hence the outgoing edges from @xmath318 are the same in @xmath358 and @xmath359 . suppose on the other hand that @xmath361 ( hence , @xmath362 because the pair is non - diagonal ) . then , if the pair @xmath363 is non - separated , then its only outgoing edge goes to @xmath364 in @xmath365 . on the other hand , if @xmath363 is separated , then its two possible outgoing edges are @xmath366 and @xmath367 . however , the pair @xmath366 is diagonal , hence no vertex in the graph has such label . thus , independently of whether @xmath363 is separated , it has exactly one outgoing edge with target @xmath367 , proving the claim . we shall now apply the theory of labeled wedges to the core entropy . as we have seen in section [ s : algo ] , thurston s algorithm allows one to compute the core entropy for periodic angles @xmath3 ; in order to interpolate between periodic angles of different periods , we shall now define for _ any _ angle @xmath13 a labeled wedge @xmath27 , and thus an infinite graph @xmath368 as described in the previous sections . recall that for each @xmath31 we denote @xmath50 , which we see as a point in @xmath51 . for each pair @xmath247 which belongs to @xmath234 , we label @xmath247 as _ non - separated _ if @xmath369 and @xmath370 belong to the same element of the partition @xmath46 , or at least one of them lies on the boundary of the partition . if instead @xmath369 and @xmath370 belong to the interiors of two different intervals of @xmath46 , then we label the pair @xmath247 as _ separated_. the labeled wedge just constructed will be denoted @xmath27 , and its associated graph @xmath28 . the main quantity we will work with is the following : let @xmath3 have period @xmath43 and pre - period @xmath44 . by definition of the equivalence relation @xmath312 , we have @xmath325 if and only if @xmath373 , which proves the first condition in the definition of periodic wedge . moreover , if @xmath247 is a diagonal pair , then @xmath374 , hence the pair @xmath247 is non - separated , verifying the second condition . using the results of the previous section , we are now ready to prove that the logarithm of the growth rate of @xmath28 coincides with the core entropy for rational angles , proving the first part of theorem [ t : newmain ] . let @xmath3 a rational angle , @xmath28 the infinite graph associated to the labeled wedge @xmath27 , and @xmath376 the finite model of @xmath28 as described in section [ s : finitemodel ] by unraveling the definitions , the matrix @xmath52 constructed in section [ s : algo ] is exactly the adjacency matrix of the finite model @xmath376 . by proposition [ p : finiteinfinite ] , the growth rate of @xmath28 coincides with the growth rate of its finite model @xmath376 . moreover , by lemma [ l : perroneig ] , the growth rate of @xmath376 coincides with the largest real eigenvalue of its adjacency matrix , that is the largest real eigenvalue of @xmath52 . thus , by theorem [ t : algo ] its logarithm is the core entropy @xmath5 . as @xmath379 , the labeled wedge @xmath380 stabilizes . in fact , for each @xmath49 consider the position of @xmath50 with respect to the partition @xmath46 . if @xmath381 , then , for all @xmath382 in a neighborhood of @xmath3 , @xmath383 lies on the same side of the partition . otherwise , note that for any @xmath384 and @xmath385 , the functions @xmath386 and @xmath387 are both continuous and orientation preserving but have different derivative : thus , for all @xmath388 close enough to @xmath3 , the point @xmath383 lies on one side of the partition , and for @xmath389 close enough to @xmath3 lies on the other side . thus , the limits @xmath390_^+ : = _ ^+ _ , _ ^- : = _ ^- _ @xmath390 exist , and are both equal to @xmath380 if @xmath3 is not purely periodic , because then no @xmath369 with @xmath31 lies on the boundary of the partition . the claim then follows by lemma [ l : contwedge ] . this proves continuity of @xmath371 at all angles which are not purely periodic . we shall now deal with the purely periodic case , where for the moment we only know that the left - hand side and right - hand side limits of @xmath371 exist . [ l : diffpairs ] let @xmath3 be purely periodic of period @xmath43 . then @xmath27 , @xmath391 and @xmath392 are purely periodic of period @xmath43 , and differ only in the labelings of pairs @xmath247 with either @xmath361 or @xmath393 . let @xmath3 be purely periodic of period @xmath43 . note that @xmath369 lies on the boundary of the partition @xmath46 if and only if @xmath361 . thus , for all the pairs @xmath247 for which neither component is divisible by @xmath43 , the label of @xmath247 is continuous across @xmath3 , proving the second statement . now we shall show that if for all @xmath388 and close enough to @xmath3 one has @xmath394 on one side of the partition , then also @xmath395 is on the same side for each @xmath256 . this will prove that @xmath396 is purely periodic of period @xmath43 . the proof for @xmath397 is symmetric , and @xmath27 is purely periodic of period @xmath43 by lemma [ l : periodic ] . in order to prove the remaining claim , let us denote @xmath398 , @xmath399 ; since @xmath3 is purely periodic , there exists @xmath400 such that @xmath401 for all @xmath256 . now , note that for each @xmath256 the derivatives satisfy the inequality @xmath402 , thus for each @xmath256 there exists @xmath403 such that for each @xmath404 one has @xmath405 , hence the points @xmath406 all belong to the same side of the partition independently of @xmath300 , as required . [ p : bothsides ] if @xmath3 is purely periodic of period @xmath43 , then the ( infinite ) graphs @xmath28 , @xmath407 and @xmath408 , associated respectively to @xmath27 , @xmath391 and @xmath392 have the same growth rate , i.e. we have the equality @xmath409 by lemma [ l : diffpairs ] and lemma [ l : twoperiodics ] , the graphs @xmath28 , @xmath407 and @xmath408 have the same finite form , hence by proposition [ p : finiteinfinite ] they have the same growth rate , proving the claim . let @xmath2 , and denote by @xmath413 the smallest real zero of @xmath14 . the proof is a simple quantitative version of rouch s theorem . for simplicity , let us assume that @xmath3 is pre - periodic , and let @xmath414 be the minimum distance , in @xmath51 , between ( distinct ) elements of the set @xmath415 . given another angle @xmath416 , denote @xmath417 the smallest real zero of @xmath418 , and choose @xmath20 such that @xmath419 then , by looking at how @xmath369 moves with respect to the partition @xmath46 , we realize that , for @xmath420 , the pair @xmath247 is separated for @xmath3 if and only if it is separated for @xmath416 . by construction , the @xmath421 coefficient of @xmath14 depends only on multi - cycles of total length @xmath20 , and by proposition [ p : properties ] ( 3 ) all such multi - cycles live on the subgraph @xmath422 , hence the first @xmath20 coefficients of @xmath14 and @xmath418 coincide . thus we can write @xmath423 for some @xmath424 and @xmath425 , where we used that @xmath426 by proposition [ p : properties ] ( 5 ) . now , using @xmath427 we have @xmath428 where @xmath256 is the order of zero of @xmath46 at @xmath429 and @xmath430 is a holomorphic function , with @xmath431 . by combining , and , we get @xmath432 with @xmath433 , which implies the claim as @xmath434 . in the periodic case the proof the same , except one should argue separately for @xmath435 and @xmath436 , and use proposition [ p : bothsides ] . a. douady , _ topological entropy of unimodal maps : monotonicity for quadratic polynomials _ , in _ real and complex dynamical systems ( hillerd , 1993 ) _ , nato adv . sci . * 464 * , 6587 .
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the core entropy of polynomials , recently introduced by w. thurston , is a dynamical invariant which can be defined purely in combinatorial terms , and provides a useful tool to study parameter spaces of polynomials .
the theory of core entropy extends to complex polynomials the entropy theory for real unimodal maps : the real segment is replaced by an invariant tree , known as hubbard tree , which lives inside the filled julia set .
we prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle , answering a question of thurston .
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solving the motion of a system of @xmath0 particles interacting through their mutual gravitational forces has long been one of the important themes in physics @xcite . though an exact solution is known in the @xmath1 case in newtonian theory , in the context of the general theory of relativity the motion of the @xmath0 bodies can not be solved exactly due to dissipation of energy in the form of gravitational radiation , even when @xmath1 . hence analysis of a two body system in general relativity ( e.g. binary pulsars ) necessarily involves resorting to approximation methods such as a post - newtonian expansion @xcite . however in the past decade lower dimensional versions of general relativity ( both in ( 1 + 1 ) and ( 2 + 1 ) dimensions ) have been extensively investigated from both classical and quantum perspectives . here the reduced dimensionality yields an absence of gravitational radiation . although this desirable physical feature is lost in such models ( at least in the vacuum ) , most ( if not all ) of the remaining conceptual features of relativistic gravity are retained . hence their mathematical simplificity offers the hope of obtaining a deep understanding of the nature of gravitation in a wide variety of physical situations . it is with this motivation that we consider the @xmath0-body problem in lower dimensional gravity . specifically , we consider the gravitational @xmath0-body problem in two spacetime dimensions . such lineal theories of gravity have found widespread use in other problems in physics . the simplest such theory ( sometimes referred to as jackiw - teitelboim ( jt ) theory @xcite ) sets the ricci scalar equal to a constant , with other matter fields evolving in this constant - curvature two - dimensional spacetime . another such theory ( sometimes referred to as @xmath2 theory ) sets the ricci scalar equal to the trace of the stress - energy of the prescribed matter fields and sources in this manner , matter governs the evolution of spacetime curvature which reciprocally governs the evolution of matter @xcite . this theory has a consistent newtonian limit @xcite ( a problematic issue for a generic @xmath3-dimensional gravity theory @xcite ) , and reduces to jt theory if the stress - energy is that of a cosmological constant . the @xmath0-body problem , then , can be formulated in relativistic gravity by taking the matter action to be that of @xmath0 point - particles minimally coupled to gravity . in previous work we developed the canonical formalism for this action in @xmath2 lineal gravity @xcite and derived the exact hamiltonian for @xmath1 as a solution to a transcendental equation which is valid to infinite order in the gravitational coupling constant @xcite . in the slow motion , weak field limit this hamiltonian coincides with that of newtonian gravity in @xmath3 dimensions , and in the limit where all bodies are massless , spacetime is flat . more recently we have extended this case to include a cosmological constant @xmath4 , so that in the limit where all bodies are massless , spacetime has constant curvature ( ie the jt theory is obtained ) , and when @xmath5 vanishes the situation described in the previous paragraph is recovered @xcite . for @xmath1 , we derived an exact solution for the hamiltonian as a function of the proper separation and the centre - of - inertia momentum of the bodies . in the equal mass case an exact solution to the equations of motion for the proper separation of the two point masses as a function of their mutual proper time was also obtained . the trajectories showed characteristic structures depending on the values of a cosmological constant @xmath5 . the purpose of this paper is to more fully describe these results and to expand upon them . specifically , we generalize our previous formalism with @xmath6 @xcite to a system of @xmath0 particles in ( 1 + 1 ) dimensional gravity with cosmological constant . when @xmath1 we obtain exact solutions for the motion of two bodies of unequal ( and equal ) mass . since the einstein action is a topological invariant in ( 1 + 1 ) dimensions , we must incorporate a scalar ( dilaton ) field into the action @xcite . by a canonical reduction of the action , the hamiltonian is defined as a spatial integral of the second derivative of the dilaton field , which is a function of the canonical variables of the particles ( coordinates and momenta ) and is determined from the constraint equations . for a system of two particles an equation which determines the hamiltonian in terms of the remaining degrees of freedom of the system is derived from the matching conditions of the solution to the constraint equations . we refer to this equation as the determining equation ; it is a transcendental equation which allows one to determine the hamiltonian in terms of the centre of inertia momentum and proper separation of the bodies . the canonical equations of motion are derived from the hamiltonian . for the equal mass case they can be transformed so that the separation and momentum are given by differential equations in terms of the proper time . in this latter form they can be solved exactly in terms of hyperbolic and/or trigonometric functions . corresponding to the values of the magnitudes ( and signs ) of the energy and other parameters ( e.g. gravitational coupling constant , masses , cosmological constant ) several different types of motion are expected in the 2 body case . broadly speaking , the two particles could remain either bounded or unbounded , or else balanced between these two conditions . we shall analyze these various states of motion , and discuss the transitions which occur between them . we find several surprising situations , including the onset of mutual repulsion for a range of values of negative @xmath7 and the masses , and the diverging separation of the two bodies at finite proper time for a range of values of positive @xmath7 . we shall also consider the unequal mass case . in this situation the proper time is no longer the same for the two particles , and so a description of the motion requires a more careful analysis . we find that we are able to obtain phase space trajectories from the determining equation . we also can obtain explicit solutions for the proper separation in terms of a transformed time coordinate which reduces to the mutual proper time in the case of equal mass . in sec.ii we describe the canonical reduction of the theory and define the hamiltonian for the @xmath0-body system . in sec.iii we solve the constraint equations for the two - body case and get the determining equation of the hamiltonian , from which the canonical equations of motion are explicitly derived . we investigate the motion for @xmath8 in sec.iv , by using the exact solutions to the canonical equations . the motion of equal masses for @xmath9 are analyzed in sec.v where a general discussion on the structure of the determining equation , the plots of phase space trajectories , the analysis of the explicit solutions in terms of the proper time are developed . we treat the unequal mass case in sec.vi . sec.vii contains concluding remarks and directions for further work . the solution of the metric tensor , a test particle approximation in the small mass limit of one of the particles and the causal relationships between particles in unbounded motion are given in appendices . the action integral for the gravitational field coupled to @xmath0 point masses is @xmath10{}\left.+\sum_{a}\int d\tau_{a}\left\ { -m_{a}\left(-g_{\mu\nu}(x)\frac{dz^{\mu}_{a}}{d\tau_{a } } \frac{dz^{\nu}_{a}}{% d\tau_{a}}\right)^{1/2}\right\ } \delta^{2}(x - z_{a}(\tau_{a } ) ) \right]\;,\end{aligned}\ ] ] where @xmath11 is the dilaton field , @xmath7 is the cosmological constant , @xmath12 and @xmath13 are the metric and its determinant , @xmath14 is the ricci scalar , and @xmath15 is the proper time of @xmath16-th particle , respectively , with @xmath17 . the symbol @xmath18 denotes the covariant derivative associated with @xmath19 . the field equations derived from the action ( [ act1 ] ) are @xmath20{}r - g^{\mu \nu } \nabla _ { \mu } \nabla _ { \nu } \psi = 0\ ; , \label{eq - r } \\ & & \frac{1}{2}\nabla _ { \mu } \psi \nabla _ { \nu } \psi -\frac{1}{4}g_{\mu \nu } \nabla ^{\lambda } \psi \nabla _ { \lambda } \psi + g_{\mu \nu } \nabla ^{\lambda } \nabla _ { \lambda } \psi -\nabla _ { \mu } \nabla _ { \nu } \psi = \kappa t_{\mu \nu } + \frac{1}{2}g_{\mu \nu } \lambda \ ; , \label{eq - psi } \\ & & \frac{d}{d\tau _ { a}}\left\ { g_{\mu \nu } ( z_{a})\frac{dz_{a}^{\nu } } { d\tau _ { a}}\right\ } -\frac{1}{2}g_{\nu \lambda , \mu } ( z_{a})\frac{dz_{a}^{\nu } } { % d\tau _ { a}}\frac{dz_{a}^{\lambda } } { d\tau _ { a}}=0\ ; , \label{eq - z}\end{aligned}\ ] ] where the stress - energy due to the point masses is @xmath21 eq.([eq - psi ] ) guarantees the conservation of @xmath22 . inserting the trace of eq.([eq - psi ] ) into eq.([eq - r ] ) yields @xmath23 eqs . ( [ eq - z ] ) and ( [ rt ] ) form a closed sytem of equations for the matter - gravity system . the evolution of the dilaton then follows from inserting the solutions to these equations into([eq - r ] ) , and then solving for its motion , the traceless part of ( [ eq - psi ] ) being identities once the other equations are satisfied . alternatively , one can solve the independent parts of equations ( [ eq - r ] ) , ( [ eq - psi ] ) , and ( [ eq - z ] ) for the metric , dilaton and matter degrees of freedom , which is the approach we shall take in this paper . if the masses of all particles are taken to be zero then the field equations reduce to those of constant curvature lineal gravity , or jt theory @xcite . consider next the transformation of the action ( [ act1 ] ) to canonical form . we decompose the scalar curvature in terms of the extrinsic curvature @xmath24 via @xmath25 where @xmath26 , and the metric is @xmath27 so that @xmath28 and @xmath29 , and then rewrite the particle action in first - order form . after some manipulation we find that the action ( [ act1 ] ) may be rewritten in the form @xmath30 where @xmath31 and @xmath32 are conjugate momenta to @xmath33 and @xmath34 , respectively , and @xmath35 with the symbols @xmath36 and @xmath37 denoting @xmath38 and @xmath39 , respectively . the action ( [ act2 ] ) leads to the system of equations @xmath40{}\left . -\sum_{a}\frac{p^{2}_{a}}{2\gamma^{2}\sqrt{\frac{% p^{2}_{a}}{\gamma } + m^{2}_{a}}}\;\delta(x - z_{a}(t))\right\ } \nonumber \\ & + & n_{1}\left\{-\frac{1}{\gamma^{2}}\pi\psi^{\prime } + \frac{\pi^{\prime}}{% \gamma } + \sum_{a}\frac{p_{a}}{\gamma^{2}}\;\delta(x - z_{a}(t))\right\ } + n^{\prime}_{0}\frac{1}{2\kappa\sqrt{\gamma}\gamma}\psi^{\prime } + n^{\prime}_{1}\frac{\pi}{\gamma}=0\;,\end{aligned}\ ] ] @xmath41{}+n_{1}\frac{p_{a}}{\gamma^{2}}\frac{\partial\gamma}{% \partial z_{a}}=0\ ; , \label{e - p } \\ & & \dot{z_{a}}-n_{0}\frac{\frac{p_{a}}{\gamma}}{\sqrt{\frac{p^{2}_{a}}{\gamma% } + m^{2}_{a } } } + \frac{n_{1}}{\gamma}=0 \;. \label{e - z}\end{aligned}\ ] ] in the equations ( [ e - p ] ) and ( [ e - z ] ) , all metric components ( @xmath42 , @xmath43 , @xmath44 ) are evaluated at the point @xmath45 and @xmath46 this system of equations can be shown to be equivalent to the set of equations ( [ eq - r ] ) , ( [ eq - psi ] ) and ( [ eq - z ] ) . since @xmath42 and @xmath43 are lagrange multipliers , equations ( [ e - r0 ] ) and ( [ e - r1 ] ) are constraints ; specifically they are the energy and momentum constraints of the @xmath3 dimensional gravitational system we consider . we may solve them for @xmath47 and @xmath48 in terms of the dynamical and gauge ( _ i.e. _ co - ordinate ) degrees of freedom , since these are the only linear terms in these constraints . we identify these coordinate degrees of freedom by writing the generator arising from the variation of the action at the boundaries in terms of @xmath47 and @xmath48 , and then finding which quantities serve to fix the frame of the physical space - time coordinates in a manner similar to the @xmath49-dimensional case . carrying out the same procedure as in the @xmath50 case @xcite we find that we can consistently choose the coordinate conditions @xmath51 { } \mbox{and } \makebox[2em ] { } \pi=0 \;.\ ] ] eliminating the constraints , the action reduces to @xmath52 and the reduced hamiltonian for the @xmath0-body system is @xmath53 where @xmath11 is a function of @xmath54 and @xmath55 , determined by solving the constraints which are under the coordinate conditions ( [ cc ] ) @xmath56 @xmath57 the expression for the hamiltonian ( [ ham1 ] ) is analogous to the reduced hamiltonian in @xmath49 dimensional general relativity . in @xmath3 dimensions it is determined by the dilaton field at spatial infinity . the consistency of this canonical reduction may be demonstrated in a manner analogous to that employed in the @xmath50 case : namely the canonical equations of motion derived from the reduced hamiltonian ( [ ham1 ] ) are identical with the equations ( [ e - p ] ) and ( [ e - z ] ) . the methodology at this point is then as follows . first we must solve ( [ psi ] ) and ( [ pi ] ) for @xmath11 and @xmath58 in terms of the @xmath59 and the @xmath60 , consistently matching solutions across the boundaries of the particles . then we compute from ( [ ham1 ] ) the hamiltonian in terms of the independent momenta and coordinates of the particles . this expression is sufficient to obtain the phase - space trajectories for a given set of initial conditions . finally we solve equations ( [ e - pi ] [ e - z ] ) to obtain a complete solution for the @xmath0 body system . throughout the remainder of this paper we shall consider only @xmath1 , i.e. 2-body dynamics . the standard approach for investigating the dynamics of particles is to derive an explicit expression of the hamiltonian , from which the equations of motion and the solution of trajectories are obtained . in this section we explain how to derive the hamiltonian from the solution to the constraint equations ( [ psi ] ) and ( [ pi ] ) and get the explicit hamiltonian for two particles in a spacetime with a cosmological constant . we first express the equations ( [ psi ] ) and ( [ pi ] ) as @xmath61 @xmath62 where we set @xmath63 . rewriting ( [ psi1 ] ) as @xmath64 and using ( [ chi1 ] ) , we can rewrite ( [ ham1 ] ) as @xmath65 which can also be obtained by inserting ( [ psi1 ] ) and ( [ chi1 ] ) into ( [ ham1 ] ) and iterating by partial integration ( assuming convergence ) . we shall refer to this formula later when we consider boundary conditions . defining @xmath66 by @xmath67 the constraints ( [ psi1 ] ) and ( [ chi1 ] ) for a two - particle system become @xmath68 @xmath69 the general solution to ( [ chi - eq ] ) is @xmath70 the factor @xmath71 ( @xmath72 ) has been introduced in the constants @xmath73 and @xmath74 so that the t - inversion ( time reversal ) properties of @xmath75 are explicitly manifest . by definition , @xmath71 changes sign under time reversal and so , therefore , does @xmath75 . consider first the case @xmath76 , for which we may divide space into three regions : @xmath77 ( ( + ) region ) , @xmath78 ( ( 0 ) region ) and @xmath79 ( ( - ) region ) . in each region @xmath80 is constant : @xmath81{}\mbox{(+ ) region } , \\ -\epsilon x+\frac{1}{4}(p_{1}-p_{2 } ) & \makebox[3em]{}\mbox{(0 ) region } , \\ -\epsilon x+\frac{1}{4}(p_{1}+p_{2 } ) & \makebox[3em]{}\mbox{(- ) region}\;. \end{array } \right.\ ] ] general solutions to the homogeneous equation @xmath82 in each region are @xmath83 where @xmath84 for these solutions to be the actual solutions to eq.([phi - eq ] ) with delta function source terms , they must satisfy the following matching conditions at @xmath85 : @xmath86 the conditions ( [ match1 ] ) and ( [ match3 ] ) lead to @xmath87 @xmath88 yielding @xmath89 { } + \frac{\kappa\sqrt{p^{2}_{1}+m^{2}_{1}}-2k_{0}+2k_{+}}{4k_{+ } } e^{-\frac{1}{2}(k_{0}+k_{+})z_{1}}b_{0}\ ; , \label{a+ } \\ b_{+}&=&-\frac{\kappa\sqrt{p^{2}_{1}+m^{2}_{1}}+2k_{0}-2k_{+}}{4k_{+ } } e^{\frac{1}{2}(k_{0}+k_{+})z_{1}}a_{0 } \nonumber \\ & & \makebox[10em ] { } -\frac{\kappa\sqrt{p^{2}_{1}+m^{2}_{1}}-2k_{0}-2k_{+}}{4k_{+ } } e^{-\frac{1}{2}(k_{0}-k_{+})z_{1}}b_{0 } \;. \label{b+}\end{aligned}\ ] ] similarly from ( [ match2 ] ) and ( [ match4 ] ) we have @xmath90 @xmath91 and then @xmath92 { } -\frac{\kappa\sqrt{p^{2}_{2}+m^{2}_{2}}+2k_{0}-2k_{-}}{4k_{- } } e^{-\frac{1}{2}(k_{0}+k_{-})z_{2}}b_{0}\ ; , \label{a- } \\ b_{-}&=&\frac{\kappa\sqrt{p^{2}_{2}+m^{2}_{2}}-2k_{0}+2k_{-}}{4k_{- } } e^{\frac{1}{2}(k_{0}+k_{-})z_{2}}a_{0 } \nonumber \\ & & \makebox[10em ] { } + \frac{\kappa\sqrt{p^{2}_{2}+m^{2}_{2}}+2k_{0}+2k_{-}}{4k_{- } } e^{-\frac{1}{2}(k_{0}-k_{-})z_{2}}b_{0}\;. \label{b-}\end{aligned}\ ] ] since the magnitudes of both @xmath66 and @xmath75 increase with increasing @xmath93 , it is necessary to impose a boundary condition which guarantees that the surface terms which arise in transforming the action vanish and simultaneously preserves the finiteness of the hamiltonian . a consideration of ( [ ham2 ] ) implies that we may choose the boundary condition @xmath94 with @xmath95 being constants to be determined . this boundary condition means @xmath96x + \epsilon c_{\chi}+\frac{1}{4}(p_{1}z_{1}+p_{2}z_{2})\right\}^{2 } \nonumber \\ & & \makebox[1em]{}+2\lambda x^{2}=c_{+}x \ ; , \label{cond2 } \\ & & \left\{2k_{-}x-4\mbox{log}|b_{-}|\right\}^{2 } -4\kappa^{2}\left\{-\left[\epsilon x-\frac{1}{4}(p_{1}+p_{2})\right]x + \epsilon c_{\chi}-\frac{1}{4}(p_{1}z_{1}+p_{2}z_{2})\right\}^{2 } \nonumber \\ & & \makebox[1em]{}+2\lambda x^{2}=c_{-}x \;. \label{cond3}\end{aligned}\ ] ] it may seem that instead of ( [ cond1 ] ) we could have made the alternate choices @xmath97 or @xmath98 . however the definitions ( [ k+-0 ] ) imply that @xmath99 are positive quantities , which in turn leads to a negative hamiltonian for the choice @xmath100 . for the choices @xmath101 and @xmath98 the hamiltonian identically vanishes . the terms quadratic in @xmath102 from ( [ cond2 ] ) and ( [ cond3 ] ) merely recapitulate the definitions of @xmath99 . equating terms linear in @xmath102 yield the relations @xmath103 \left[\epsilon c_{\chi}+\frac{1}{4}(p_{1}z_{1}+p_{2}z_{2})\right]&=&c_{+}\ ; , \label{cond4 } \\ -16k_{-}\mbox{log}|b_{-}|+8\kappa^{2 } \left[\epsilon x-\frac{1}{4}(p_{1}+p_{2})\right ] \left[\epsilon c_{\chi}-\frac{1}{4}(p_{1}z_{1}+p_{2}z_{2})\right]&= & c_{- } \;. \label{cond5}\end{aligned}\ ] ] equating the constant terms of ( [ cond2 ] ) and ( [ cond3 ] ) leads to @xmath104 ^{2}=0\ ; , \label{cond6 } \\ & & 16\left(\mbox{log}|b_{-}|\right)^{2 } -4\kappa^{2}\left[\epsilon c_{\chi}-\frac{1}{4}(p_{1}z_{1}+p_{2}z_{2})\right ] ^{2}=0 \;. \label{cond7}\end{aligned}\ ] ] we choose the solutions @xmath105\ ; , \nonumber \\ \\ \mbox{log}|b_{-}|&=&\frac{\kappa}{2}\left[c_{\chi } -\frac{\epsilon}{4}(p_{1}z_{1}+p_{2}z_{2})\right]\;. \nonumber\end{aligned}\ ] ] before proceeding , we add a remark to ( [ a+b-1 ] ) . in solving ( [ cond6 ] ) and ( [ cond7 ] ) , there are actually four combinations @xmath106 of sign choices for @xmath107 and @xmath108 . however the choices @xmath109 and @xmath110 do not lead to any relations among the coefficients and gives us an unphysical hamiltonian . the choices @xmath111 and @xmath112 lead to identical physical results once the signs of the momenta @xmath113 and the coefficient @xmath74 are reversed . the condition ( [ cond1 ] ) leads to @xmath114 and @xmath115 from ( [ a / b-1 ] ) and ( [ a / b-2 ] ) we obtain @xmath116{}=\left(\kappa\sqrt{p^{2}_{1}+m^{2}_{1}}+2k_{0}-2k_{+}\right ) \left(\kappa\sqrt{p^{2}_{2}+m^{2}_{2}}+2k_{0}-2k_{-}\right ) e^{k_{0}(z_{1}-z_{2})}\ ; , \nonumber \\\end{aligned}\ ] ] which we shall refer to as the determining equation for @xmath73 . the @xmath11 fields in @xmath117 regions are @xmath118 and the hamiltonian is @xmath119^{\infty}_{-\infty } \nonumber \\ & = & \frac{2(k_{+}+k_{-})}{\kappa}\;.\end{aligned}\ ] ] once the solution for @xmath73 is obtained from ( [ x ] ) , the hamiltonian is explicitly determined from ( [ ham1 ] ) in terms of the degrees of freedom of the system ( i.e. the coordinates and momenta of the particles ) . > from ( [ a+ ] ) , ( [ b- ] ) , ( [ a / b-1 ] ) and ( [ a / b-2 ] ) , @xmath120 and @xmath121 are expressed in terms of @xmath122 as @xmath123 and from ( [ a+b-1 ] ) and ( [ a+b-2 ] ) the coefficients @xmath122 and @xmath74 ( and hence @xmath120 , @xmath121 and @xmath124 ) are also determined @xmath125 the parameters @xmath95 are determined from ( [ cond4 ] ) and ( [ cond5 ] ) . from ( [ a / b-1 ] ) , ( [ a / b-2 ] ) and ( [ a+b-2 ] ) it is evident that an overall common sign of @xmath126 and @xmath124 has no physical meaning , and so we can choose all these coefficients to be positive . the previous expressions are somewhat cumbersome . we can express them more compactly by making use of the following notation : @xmath127\ ; , \nonumber \\ y_{0}&\equiv & \kappa\left[x-\frac{\epsilon}{4}(p_{1}-p_{2})\right]\ ; , \nonumber \\ y_{-}&\equiv & \kappa\left[x-\frac{\epsilon}{4}(p_{1}+p_{2})\right]\;\;. \nonumber\end{aligned}\ ] ] the coefficients @xmath126 and @xmath124 can then be rewritten as @xmath128 and the solution for @xmath66 is then @xmath129 repeating the analysis for @xmath130 yields a similar solution with @xmath131 . hence the full solution is obtained from the preceding expressions by replacing @xmath113 and @xmath132 by @xmath133 and @xmath134 , respectively . the determining equation ( [ x ] ) of the hamiltonian is then expressed as @xmath135 or @xmath136 [ \kappa\sqrt{p_{2}^{2}+m^{2}_{2}}-2k_{-}]\right ) \mbox{tanh}\left(\frac{1}{2}k_{0}|z_{1}-z_{2}|\right ) \nonumber \\ & & \makebox[5em]{}=-2k_{0 } \left([\kappa\sqrt{p_{1}^{2}+m^{2}_{1}}-2k_{+ } ] + [ \kappa\sqrt{p_{2}^{2}+m^{2}_{2}}-2k_{-}]\right)\;,\end{aligned}\ ] ] where the momentum @xmath113 is replaced by @xmath137 . for the expression ( [ ham2 ] ) to have a definite meaning as the hamiltonian , @xmath99 should be real . this imposes the restriction @xmath138 . however @xmath139 need not be real . if @xmath7 takes a sufficiently large positive value @xmath139 will be imaginary and the above analysis must be repeated . in the ( 0 ) region the soluton to the @xmath66 equation ( [ phi - eq ] ) becomes @xmath140 where @xmath141 under the same matching conditions ( [ match1]-[match4 ] ) and the boundary condition ( [ bound ] ) we get , instead of ( [ h2 ] ) , a new determining equation for the hamiltonian @xmath142 [ \kappa\sqrt{p_{2}^{2}+m^{2}_{2}}-2k_{-}]\right ) \mbox{tan}\left(\frac{1}{2}\tilde{k}_{0}|z_{1}-z_{2}|\right ) \nonumber \\ & & \makebox[5em]{}=2\tilde{k}_{0 } \left([\kappa\sqrt{p_{1}^{2}+m^{2}_{1}}-2k_{+ } ] + [ \kappa\sqrt{p_{2}^{2}+m^{2}_{2}}-2k_{-}]\right)\;,\end{aligned}\ ] ] which is just the equation derived from ( [ h2 ] ) by formally replacing @xmath139 with @xmath143 . similarly , the solution for @xmath66 for imaginary @xmath139 is also identical with that obtained from ( [ phi - sol2 ] ) by the same replacement . hence equation ( [ h2 ] ) is valid for all values of @xmath139 , and may be regarded as a transcendental equation which determines @xmath144 as a function of the independent coordinates and momenta of the system . we have previously shown that in the case of zero cosmological constant the solution for @xmath144 can be expressed in terms of the lambert @xmath145 function . in this more general case with @xmath146 the solution for @xmath144 from ( [ h2 ] ) can not expressed in terms of known functions . rather we must regard @xmath144 as being implicitly determined in terms of the coordinates and momenta via ( [ h2 ] ) . finally , the components of the metric may be computed from the equations ( [ e - pi ] ) , ( [ e - gamma ] ) , ( [ e - pi ] ) and ( [ e - psi ] ) under the coordinate conditions ( [ cc ] ) . the derivation and the explicit solutions of the metric are given in appendix a. the canonical equations for the 2-body system can be derived by differentiating the determining equation ( [ h1 ] ) with respect to the variables @xmath147 and @xmath113 . for the variables @xmath148 and @xmath149 this yields @xmath150 @xmath151 where @xmath152 similarly , for particle 2 the equations are @xmath153 it is straightforward to show that these canonical equations guarantee the conservation of the hamiltonian ( i.e. @xmath154 ) and the total momentum @xmath155 ( i.e. @xmath156 ) . alternatively , the equations of motion ( [ e - p ] ) and ( [ e - z ] ) derived from the action ( [ act2 ] ) become @xmath157 under the coordinate conditions ( [ cc ] ) . insertion of the solutions of the metric components given in the appendix a into ( [ pa ] ) and ( [ za ] ) reproduces the canonical equations of motion ( [ p1 ] ) , ( [ z1 ] ) , ( [ p2 ] ) and ( [ z2 ] ) when the partial derivatives at @xmath158 are defined by @xmath159 thus consistency between the geodesic equations and the canonical equations of motion is explicitly verified , while the formal proof of the consistency in the case of @xmath50 @xcite can be easily generalized to the @xmath160 case . in a previous paper @xcite we showed that in the @xmath50 case the determining equation ( [ h1 ] ) can be solved explicitly and the hamiltonian for the equal mass is expressed in the center of inertia ( c.i . ) system @xmath161 as @xmath162}{\kappa |r|}\;,\ ] ] where @xmath163 is the lambert @xmath145 function defined via @xmath164 the @xmath163 has two real branches @xmath165 and @xmath166 for real @xmath102 @xcite . the hamiltonian ( [ ham-0 ] ) is exact to infinite order in the gravitational constant and for arbitrary values of @xmath167 and @xmath168 . we can view the whole structure of the theory from the weak field to the strong field limits . by setting @xmath169 we can draw a phase space trajectory in @xmath170 space . this phase space trajectory should be , as a matter of course , obtainable directly from the solution @xmath171 to the canonical equations by eliminating the time variable @xmath172 . indeed , this can be verified by numerically solving the equations @xmath173 which are obtained in the case of @xmath50 from ( [ p1 ] ) , ( [ z1 ] ) , ( [ p2 ] ) and ( [ z2 ] ) . however for certain values of the parameters , superficial singularities appear in @xmath174 and @xmath175 due to the zero points of the denominator @xmath176 . these singularities correspond to @xmath177 representing the transit point between two branches @xmath165 and @xmath166 . in a spacetime description the singularities are coordinate singularities and are a consequence of @xmath172 being a coordinate time . we can deal with this problem by describing the trajectories of the particles in terms of some invariant parameter . the natural candidate is the proper time @xmath178 of each particle . from the metric components given in appendix a and the canonical equations ( [ za ] ) , the proper time is @xmath179 for the equal mass case it is common for both particles @xmath180 and the canonical equations ( [ p - eq0 ] ) and ( [ r - eq0 ] ) become @xmath181\sqrt{p^2 + m^2}}{h - 2\epsilon\tilde{p } } - 1\right\}\mbox{sgn}(r ) \;. \label{r - eqtau0}\end{aligned}\ ] ] remarkably , the equations ( [ p - eqtau0 ] ) and ( [ r - eqtau0 ] ) have an exact solution . we can obtain it in the following way . first we solve eq.([p - eqtau0 ] ) for @xmath182 . from this @xmath183 can be extracted by directly solving ( [ r - eqtau0 ] ) after substituting the solution for @xmath168 or by solving ( [ h1 ] ) for @xmath184 . this yields an exact expression for the proper separation @xmath184 of the two bodies as a function of their mutual proper time . note that since @xmath185 , at a fixed time @xmath186 ( and hence a fixed @xmath187 ) , the separation @xmath188 is the proper distance between the two particles . in the @xmath189 region the solution is @xmath190\ ; , \label{r - exact00}\end{aligned}\ ] ] with @xmath191\;,\ ] ] where @xmath192 is the initial momentum at @xmath193 . in the @xmath194 region the solution is @xmath195\ ; , \label{r - exact01}\end{aligned}\ ] ] with @xmath196\;.\ ] ] in fig.1 and 2 we show the typical plots of @xmath183 and phase space trajectories for various values of @xmath197 with @xmath198 . ( in all plots in this paper we choose @xmath199 . ) in fig.1 it is seen that as @xmath197 increases , not only the amplitude and the period of the bounded motion become large , but also the shape of @xmath183 deforms from the parabolic ( newtonian ) shape . this deformation is also present in the phase space trajectories shown in fig.2 . at higher energy the trajectories become more and more @xmath200 shaped . this is due to the fact that the trajectory smoothly moves over the two branches @xmath165 and @xmath166 @xcite . we see in fig.3 the striking distinction between the separations of the particles in the non - relativistic and relativistic cases once the value of the energy becomes large relative to the mass ( here @xmath201 and @xmath198 ) . the maximal separation of the particles is much smaller than in its newtonian counterpart ( the dashed curve ) , and is achieved far more quickly . after maximal separation , the particles move toward each other at a slower velocity until they are very close together . at this point ( less than 10% of their maximal separation ) , they accelerate toward the same point , after which the motion repeats with the particles interchanged . fig.1 _ the exact @xmath184 vs @xmath202 curves in the case of @xmath50 for @xmath198 and four different values of @xmath197 . _ fig.2 _ phase space trajectories corresponding to the @xmath183 curves in fig.1 . _ fig.3 _ the exact @xmath184 vs @xmath202 curve for @xmath201 with @xmath198 and the newtonian curve for the same @xmath197 . _ in this section we consider a system of two particles with equal mass for the @xmath160 case . in the c.i . system , depending upon the sign of @xmath203 the determining equations ( [ h2 ] ) and ( [ h2new ] ) become @xmath204 and @xmath205 respectively , where @xmath206 equation ( [ deteq-1 ] ) may be further divided into two types : @xmath207 or @xmath208 when @xmath50 the tanh - type b equation is excluded because @xmath209 exceeds 1 . when a cosmological constant is introduced , this equation has solutions in some restricted range of the parameters . likewise , eq.([deteq-2 ] ) may also be divided into @xmath210 or @xmath211 for all of these four types of determining equations the canonical equations of motion are identical : @xmath212 where @xmath213 for a given value of @xmath7 the equations ( [ deteq-1 ] ) or ( [ deteq-2 ] ) describe the surface in @xmath214 space of all allowed phase - space trajectories , from which the @xmath170 trajectory is obtained by setting @xmath169 . fig.4 shows phase - space plots for a small @xmath215 and three different values of @xmath7 under identical initial conditions . first we note that the trajectories of the relativistic motion ( solid curve ) are are slightly distorted compared to the newtonian motion ( dashed curve ) . second , a trajectory with @xmath216 ( dash - dot curve ) is expanded , reflecting the repulsive effect of the cosmological constant , while a trajectory of @xmath217 ( dotted curve ) shrinks , due to the additional attractive effect . the relativistic plots in fig.4 correspond to the choice of @xmath218 . plots for @xmath219 ( the time - reversed solutions ) are obtained by reflection in the @xmath220 axis . the phase space plots for a large @xmath221 are shown in fig.5 . the trajectories for @xmath222 become extremely @xmath200-shaped , while for @xmath217 the trajectory is still a distorted oval due to the attractive effect of @xmath7 . the effects of the cosmological constant ( @xmath216 repulsive ; @xmath217 attractive ) are precisely analyzed in terms of the exact @xmath183 plots in the next subsection . fig.4 _ non - relativistic ( newtonian ) and relativistic trajectories for @xmath223 and three different values of @xmath7 . the undistorted oval ( which is really a pair of parabolas intersecting at the @xmath224 axis ) is the newtonian trajectory . _ fig.5 _ relativistic trajectories for @xmath225 and three different values of @xmath7 . _ the phase - space trajectories discussed in the previous subsection can be obtained from the solution to the canonical equations ( [ can - p ] ) and ( [ can - z ] ) . for the equal mass case there is a common proper time for both particles @xmath226 via which the canonical equations ( [ can - p ] ) and ( [ can - z ] ) may be expressed in the form @xmath227 first we solve eq.([p - tau ] ) for @xmath168 and then ( [ tanha])-([tanb ] ) for @xmath184 . in the @xmath228 region eq.([p - tau ] ) leads to @xmath229 provided the condition @xmath230 is satified . hence for @xmath231 the motion is allowed as long as @xmath144 satisfies @xmath232 we perfom the integration of the lhs of ( [ p - int ] ) for three separate cases which depend on the value of @xmath7 relative to @xmath167 and @xmath144 . the solution @xmath182 is @xmath233 with @xmath234 } { \frac{m^2}{h}\sqrt{-\gamma_m } -\sigma \tan\left[\frac{\epsilon\kappa m}{8}\sqrt{-\gamma_m}(\tau-\tau_{0})\right ] } } & \qquad \gamma_{m } < 0 \ ; , \end{array } \right.\ ] ] where @xmath235 with @xmath236 being the initial momentum at @xmath237 . similarly the solution in @xmath238 region is @xmath239 with @xmath240 } { \frac{m^2}{h}\sqrt{-\gamma_m } -\bar{\sigma } \tan\left[\frac{\epsilon\kappa m}{8}\sqrt{-\gamma_m}(\tau-\tau_{0})\right ] } } & \qquad \gamma_{m } < 0 \ ; , \end{array } \right.\ ] ] where @xmath241 the solution for @xmath183 for each of the determining equations ( [ tanha])- ( [ tanb ] ) is obtained as follows tanh - type a : @xmath242 & \qquad r>0\ ; , \\ & \\ \frac{-16}{\sqrt{\left(\sqrt{\kappa^2 h^2 + 8\lambda } -m\kappa({\bar f}(\tau)-\frac{1}{{\bar f}(\tau)})\right)^{2 } -8\lambda } } \ ; \mbox{tanh}^{-1}\left[\frac{\kappa\left(h- m\left|{\bar f}(\tau)+\frac{1 } { { \bar f}(\tau)}\right|\right ) } { \sqrt{\left(\sqrt{\kappa^2 h^2 + 8\lambda } -m\kappa({\bar f}(\tau)-\frac{1}{{\bar f}(\tau)})\right)^{2 } -8\lambda } } \right ] & \qquad r<0 \ ; , \end{array } \right.\ ] ] tanh - type b : @xmath243 & \qquad r>0\ ; , \\ & \\ \frac{-16}{\sqrt{\left(\sqrt{\kappa^2 h^2 + 8\lambda } -m\kappa({\bar f}(\tau)-\frac{1}{{\bar f}(\tau)})\right)^{2 } -8\lambda } } \ ; \mbox{tanh}^{-1}\left[\frac{\sqrt{\left(\sqrt{\kappa^2 h^2 + 8\lambda } -m\kappa({\bar f}(\tau)-\frac{1}{{\bar f}(\tau)})\right)^{2 } -8\lambda } } { \kappa\left(h - m\left|{\bar f}(\tau)+\frac{1}{{\bar f}(\tau)}\right|\right ) } \right ] & \qquad r<0 \ ; , \end{array } \right.\ ] ] tan - type a : @xmath244+n\pi\right ) & \quad r>0\ ; , \\ & \\ \frac{-16}{\sqrt{8\lambda - \left(\sqrt{\kappa^2 h^2 + 8\lambda } -m\kappa({\bar f}(\tau)-\frac{1}{{\bar f}(\tau)})\right)^{2 } } } \ ; \left(\mbox{tan}^{-1}\left[\frac{\kappa\left(m \left|{\bar f}(\tau ) + \frac{1}{{\bar f}(\tau)}\right|-h\right ) } { \sqrt{8\lambda - \left(\sqrt{\kappa^2 h^2 + 8\lambda } -m\kappa({\bar f}(\tau)-\frac{1}{{\bar f}(\tau)})\right)^{2 } } } \right]+n\pi\right ) & \quad r<0 \ ; , \end{array } \right.\ ] ] tan - type b : @xmath245+n\pi\right ) & \quad r>0\ ; , \\ & \\ \frac{-16}{\sqrt{8\lambda - \left(\sqrt{\kappa^2 h^2 + 8\lambda } -m\kappa({\bar f}(\tau)-\frac{1}{{\bar f}(\tau)})\right)^{2 } } } \ ; \left(\mbox{tan}^{-1}\left[\frac{\sqrt{8\lambda - \left(\sqrt{\kappa^2 h^2 + 8\lambda } -m\kappa({\bar f}(\tau)-\frac{1}{{\bar f}(\tau)})\right)^{2 } } } { \kappa\left(h - m \left|{\bar f}(\tau)+\frac{1}{{\bar f}(\tau)}\right|\right ) } \right]+n\pi\right ) & \quad r<0 \;\;. \end{array } \right.\ ] ] the exact @xmath183 solutions corresponding to the phase space trajectories in fig.4 are given by tanh - type a solution ( [ sol - tanha ] ) and are plotted in fig.6 . the motions are bounded and periodic . comparison of three curves in fig.6 indicates that a negative cosmological constant @xmath246 acts effectively as an attractive force : for the same value of @xmath197 , the particles do not achieve as wide a proper separation , and the frequency of oscilliation is more rapid . as well , a positive @xmath247 acts as a repulsive force : the frequency of oscillation decreases and the particles achieve a wider proper separation . fig.6 _ the exact @xmath184 vs @xmath202 curves corresponding to the phase space trajectories in fig.4 . _ the period @xmath248 for the bounded motion is obtained from tanh - type a solution with the condition @xmath224 and @xmath249 : @xmath250 in figure 7 we plot @xmath183 for fixed @xmath251 and @xmath252 for several different values of @xmath167 . though the attractive effect of a negative @xmath7 is common in all cases , a special ( and rather surprising ) situation arises . as the motion becomes more relativistic ( i.e. @xmath167 gets smaller ) we find that a second maximum develops in the curve ( see @xmath253 curve ) . the description of the motion is as follows . the two particles start at @xmath224 depart in opposite directions , reaching a maximum separation . they then go back toward one another for a certain period of proper time . however at some point they each reverse direction , reaching a second maximal separation . they then reverse direction again , finally returning to their starting point where the motion then repeats itself . as the mass becomes very small , the second maximum prevails . this peculiar behavior @xcite is due to a subtle interplay between the gravitational attraction , cosmological constant and relativistic motion of the particles . to our knowledge it has never been previously observed . the changes of the peaks are clearly grasped in the phas@xmath254e space trajectories in fig.8 . ( in fig.7 the first maximum of @xmath255 curve could not be drawn due to plotting precision . ) fig.7 _ a sequence of equal mass curves for @xmath251 and @xmath252 . + note the presence of the second maximum for @xmath253 . _ fig.8 _ change of peaks for @xmath251 and @xmath252 as @xmath167 gets smaller . _ as the negative value of @xmath7 approaches its lower bound of @xmath256 , the form of the phase space trajectories transforms from an @xmath200-shaped curve to a double peaked one and then to a diamond shape . figure 9 illustrates these characteristics for the case of @xmath198 and @xmath257 , in which a double peak structure appears for @xmath258 . for each trajectory in fig.9 the corresponding @xmath183 plot is shown in fig.10 . fig.9 _ phase space trajectories for @xmath259 and different values of negative @xmath7 . _ fig.10 _ the @xmath184 vs @xmath202 curves corresponding the phase space trajectories in fig.9 . _ the double - peak structure shown in figs.7 - 10 is a consequence of having a momentum - dependent potential . we can gain some insight into this behaviour by computing a perturbative hamiltonian . the structure of the determining equations ( [ deteq-1 ] ) and ( [ deteq-2 ] ) suggests that we can carry out a 2-parameter expansion of the hamiltonian in terms of @xmath260 and @xmath261 . to the third order the result is @xmath262 it is straightforward to show that the terms in @xmath263 have the form @xmath264 to arbitrary order in @xmath261 , where the @xmath265 are positive integers and @xmath266 . one of the characteristics common to all such terms is that they vanish as @xmath267 , since in this case the determining equation of the hamiltonian becomes @xmath268 which is @xmath7-independent . another important characteristic is that they have a single maximum at @xmath269 if @xmath270 in the case of @xmath231 . consider the situation depicted in figs.7 - 10 , that of two particles initially at the origin , each having initial momentum @xmath192 , with @xmath271 . in this case @xmath272 , @xmath273 , and the particles initially move apart as though they were free . from ( [ p - exact1 ] ) the momentum is a monotonically decreasing function of time . for @xmath274 sufficiently small , the particles will execute a motion which is a perturbation of that described in section [ exactl0 ] since the momentum never becomes large enough to cross the maximum in @xmath275 . however if @xmath274 is sufficiently large , the terms in @xmath275 will grow as @xmath168 decreases , and the @xmath5-dependent part of the potential will continue to increase . the terms in @xmath276 will decrease as @xmath168 decreases , even though @xmath184 is increasing . eventually a maximum in @xmath184 is reached , after which both @xmath184 and @xmath168 are decreasing . in the generic case @xmath184 will continue to decrease toward zero . however a second extremum will appear if the @xmath275 terms get too small too rapidly before @xmath220 , which can happen for @xmath5 within a certain range . since @xmath277 is a constant of the motion , the only way to preserve the constancy of @xmath144 will be for @xmath184 to increase again . essentially the particles are repelled due to their kinetic energy within this range of @xmath5 . of course @xmath184 can not increase too much , because @xmath168 continues to get small eventually @xmath184 must reach a 2nd maximum , and then turns around again until @xmath220 , after which the motion continues to @xmath224 whereupon the particles interchange roles . we can see this effect in a simple non - relativistic model with @xmath278 where in the latter equation @xmath168 and @xmath184 have been rescaled in units of @xmath167 . the potential has been chosen so that @xmath279 in terms of ( [ general ] ) above . since @xmath280 is a constant of the motion , we can write @xmath281 which has extrema at @xmath282 . for @xmath283 , the extreme values of @xmath284 are not real , and so there will be no double peak structure in the phase - space trajectory . however for @xmath285 two extrema appear , and the particle gets a bounce . this motion is viewed qualitatively in a potential diagram shown in fig.11 . as the momentum decreases from a initial value @xmath192 to zero , the potential curve ( which is linear in @xmath284 ) changes from @xmath286 to @xmath287 . accordingly the particle moves from points @xmath288 to @xmath289 in numerical order . the equations of motion for the hamiltonian ( [ nonrelmod ] ) are easily solved , yielding @xmath290 where @xmath163 is the lambert - w function ( [ lambertw ] ) . the function @xmath175 is monotonically decreasing . provided @xmath291 the particle gets a bounce at @xmath292 before moving out to infinity , as illustrated in figs . 12 and 13 for the case of * @xmath293 . * in these figures the numbers on the curves denote the corresponding numbers in the potential diagram of fig.11 . fig.11 _ a schematic view of the motion with a large @xmath236 in a potential diagram . _ fig.12 _ the @xmath175 and @xmath174 curves in the non - relativistic model for @xmath293 . + a bounce occurs at @xmath294 . _ fig.13 _ the phase space trajectory in the non - relativistic model for @xmath293 . _ the expansion ( [ happrox ] ) has similar features , except that there is an additional gravitational and cosmological attraction which prevents the separation from diverging . to order @xmath7 the potential is a sum of two terms of the form in ( [ general ] ) , with @xmath295 in the first term and @xmath296 in the second term . the latter provides the bounce effect described above , but is always overwhelmed by the first term for small @xmath284 . however the @xmath297 term has @xmath298 and is a pure bounce term . hence there exists a range of @xmath7 which can provide a bounce . fig.14 shows the phase space trajectories in @xmath228 region for @xmath299 and two different masses ( @xmath300 ) , in which the dotted curves represent the motions to the first order of @xmath7 in the hamiltonian ( [ happrox ] ) and the solid curves do the motions corresponding to the second order of @xmath7 . we see that to order @xmath7 , as the motion becomes relativistic ( @xmath167 gets smaller ) the trajectory simply expands and changes from @xmath200-shape to a diamond shape . when the @xmath297 term is included , the double peak structure ( a solid curve with @xmath301 ) appears . fig.14 _ phase space trajectories for the perturbative hamiltonian + for @xmath299 and @xmath302 . _ this perturbative analysis indicates that the bounce effect is a result of the negative cosmological constant inducing a momentum dependent potential with positive coefficients . for @xmath303 , odd powers of @xmath261 are strongly repulsive , and suppress the attractive effects of even powers of @xmath261 , eliminating the double peak structure . more generally , a positive cosmological constant acts effectively as a repulsive force . figure 15 shows @xmath183 plots for fixed @xmath304 and several different values of @xmath167 . the motion becomes unbounded between @xmath305 and @xmath306 . the @xmath183 plots in fig.16 are for fixed @xmath307 and different values of @xmath7 , showing also the transition from bounded to unbounded motion . this transition occurs at @xmath308 and the critical value of @xmath7 is given by @xmath309 . as @xmath310 the particles rapidly separate , remaining nearly stationary for an increasingly large period of proper time before coming together again . at @xmath311 this separation time becomes infinite , and for @xmath312 , the separation diverges at finite @xmath202 . fig.15 _ a sequence of curves of equal mass for @xmath313 . _ fig.16 _ a sequence of curves near @xmath314 for @xmath315 . _ though all the above solutions are derived from tanh - type equations ( [ tanha ] ) and ( [ tanhb ] ) , for a positive cosmological constant there exist also a countably infinite set of unbounded motions specified by tan - type a , b equations ( [ tana ] ) and ( [ tanb ] ) . then for @xmath316 , both bounded and unbounded motions are realized for a fixed value of @xmath144 , as shown in fig.17 . in the unbounded motion two particles simply approach one another at some minimal value of @xmath284 and then reverse direction toward infinity . in the trajectories the dotted curves come from tan - type a solution ( [ sol - tana ] ) and the dashed curves do from tan - type b solution ( [ sol - tanb ] ) . as @xmath7 approaches @xmath317 , two bulges of the solid curve ( tanh - type a ) and the dotted curve ( tan - type a : n=0 ) come close and contact . when @xmath7 exceeds @xmath317 two curves switch to the unbounded trajectories as shown in the solid curves in fig.18 . the particles cross one another before receding toward infinity . the upper solid curve represents the motion in which @xmath168 approaches the asymptotic values @xmath318 as @xmath319 . as noted previously , one peculiar feature of this motion is that the two particles diverges to infinite separation at finite proper time . the time @xmath320 for @xmath321 is @xmath322}\right)\;\;.\ ] ] the lower solid curve represents the motion in reversed direction . for @xmath312 only unbounded motions are realized . fig.17 _ phase space trajectories of the bouned and the unbounded motions + for @xmath323 and @xmath324 . _ fig.18 _ phase space trajectories of the unbounded motions + for @xmath325 and @xmath324 . _ we discuss in appendix c the causal relationship between the two particles in the unbounded case . for the unequal masses the proper time ( [ tau-0 ] ) of each particle is @xmath326 where @xmath327 and @xmath328 . in this situation , choosing the time coordinate to be the proper time of one particle introduces an asymmetry into the description of the motion . instead we seek a time variable which is symmetric with respect to @xmath329 and reduces to the proper time ( [ tau ] ) when @xmath330 . from ( [ tau-2 ] ) we choose @xmath331 in terms of this variable the canonical equations are expressed as @xmath332{}\times\left\{\frac{\epsilon j}{8k_{0 } } + \frac{k_{1}}{m_{1}}\left(\frac{p}{\sqrt{p^2+m_{1}^{2 } } } -\epsilon\frac{y}{k}\right ) + \frac{k_{2}}{m_{2}}\left(\frac{p}{\sqrt{p^2+m_{2}^{2 } } } -\epsilon\frac{y}{k}\right)\right\}\;. \label{unequal - r}\end{aligned}\ ] ] note that @xmath184 still describes the proper distance between the particles at any fixed instant . unlike the equal mass case , the integration @xmath333 can not be performed within the framework of elementary calculus . hence we solve ( [ unequal - p ] ) numerically . in the case of a negative cosmological constant the @xmath183 plots in fig.19 show the trajectories for various mass ratios @xmath334 in the fixed @xmath335 and @xmath336 . compared with the equal mass case @xmath337 , as the mass ratio gets larger , the gravitational attraction is stronger and the proper distance between two particles as well as the period become shorter . when the mass ratio gets a small value than unity , the gravity becomes weak . however , for quite a small mass ratio a strong attractive effect of the cosmological constant prevails and the period changes to become shorter . at the same time the double peak structure ( the second maximum ) appears and finally the first maximum fades out . these characteristics are very clear in the unequal mass case . fig.19 _ @xmath183 plots for the different values of the mass ratio @xmath334 + for @xmath338 and @xmath339 . _ for a positive cosmological constant the situation is simple . as shown in fig.20 , as the mass ratio becomes small , the particles separate with an increasingly larger period of bounded motion . this is due to a repulsive effect of the cosmological constant and beyond the critical value the motion bocomes unbounded . fig.20 _ @xmath183 plots for the different values of the mass ratio @xmath334 + for @xmath340 and @xmath339 . _ in general relativity the relationship between the motion of a set of @xmath0 bodies and the structure of space - time is non - linear and quite complicated , even for @xmath1 . expanding upon the solution presented in @xcite , we have obtained an exact solution to the 2-body problem in @xmath3 dimensions with a cosmological constant . to our knowledge this is the first non - perturbative relativistic curved - spacetime treatment of this problem , providing new avenues for investigation of one - dimensional self - gravitating systems . we recapitulate the main results of our paper : ( 1 ) we formulated the canonical formalism for a system of @xmath0 bodies in a lineal theory of gravity with a cosmological constant @xmath5 . the system is described by a conservative hamiltonian . the effect of @xmath5 is incorporated into the potential . ( 2 ) for @xmath1 the determining equation of the hamiltonian is a transcendental equation derived from the matching conditions and appropriate boundary conditions at infinity . from these the canonical equations of motion may be derived . the metric components are also completely determined . ( 3 ) for the equal mass case we obtained explicitly the exact solutions to the canonical equations in terms of the mutual proper time of the particles . using the solutions we analyzed the motion in both @xmath341 plots and phase - space trajectories . ( 4 ) as expected , a positive cosmological constant yields a repulsive effect on the motion relative to their mutual gravitational attraction . for @xmath342 both bounded and unbounded motions are realized , while for @xmath343 only the unbounded motions are allowed . as @xmath344 the particles separate to an infinite proper distance in infinite proper time . for @xmath345 this infinite separation occurs in finite proper time.(5 ) a negative cosmological constant has an additional attractive effect , and the motion of the particles is bounded . however for a certain range of the parameters , a repulsive effect sets in , resulting the double - peaked structures of figs.7 - 10 . this effect is due to a subtle interplay between the momentum - dependent @xmath5 potential and the gravitational attraction . ( 6 ) in the unequal mass case the same basic features also occur ; indeed the double peak behavior shows up more clearly than in the equal mass case . although eq . ( [ unequal - p ] ) can not be integrated in terms of elementary functions , it is straightforward to numerically integrate . an exact solution in the small mass limit of the particle 1 was also obtained . several interesting features of the motion remain to be explored . the divergent separation of the bodies at finite proper time needs to be better understood . another issue concerns the condition ( [ h - bound ] ) which means that for a given value of @xmath346 the motion is allowed for the total energy larger than @xmath347 . what is the physical meaning of this condition ? it seems to suggest that as the attractive effect of @xmath246 exceeds a critical value the two particle system is no longer stable and transforms into some other system ( probably making a black hole ) . to formulate the canonical formalism to treat this problem is our next subject . under the coordinate conditions ( [ cc ] ) the field equations ( [ e - pi ] ) , ( [ e - gamma ] ) , ( [ e - pi ] ) and ( [ e - psi ] ) become @xmath348 the solution to ( [ eq - n0 ] ) is @xmath349 @xmath286 being an integration constant . eq.([eq - n1 ] ) is @xmath350 the solution in each region is @xmath351 + 2ay_{0}a_{0}b_{0}x+d_{0}\right\ } & \qquad \mbox{(+ ) region}\ ; , & \\ n_{1(-)}=-\epsilon\left\{a\;\frac{y_{-}}{k_{-}}\phi_{-}^{2}-d_{-}\right\ } & \qquad \mbox{(+ ) region}\ ; , & \end{array } \right.\ ] ] where @xmath352 and @xmath353 are integration constants . the matching conditions @xmath354 and @xmath355 lead to @xmath356 \right . \nonumber \\ & & \left.-2\left[\left(\frac{y_{0}}{k_{0}}-\frac{y_{+}}{k_{+}}\right ) \frac{1}{{\cal m}_{1}}-\left(\frac{y_{0}}{k_{0}}-\frac{y_{-}}{k_{-}}\right ) \frac{1}{{\cal m}_{2}}\right]k_{1}k_{2 } -\frac{y_{0}}{k_{0}}k_{1}k_{2}(z_{1}+z_{2})\right\}\;.\end{aligned}\ ] ] in deriving these relations the expressions ( [ ab+-0 ] ) for @xmath357 and the determining equation ( [ h1 ] ) were used . as for the equation ( [ eq - pi1 ] ) , first take the @xmath358 function at @xmath359 : @xmath360 where @xmath361 and the canonical equation were inserted . then the integration constant @xmath362 should be @xmath363 and similary @xmath364 now the metric tensor is completely determined : @xmath365^{2}\ ; , \nonumber \\ n_{0(-)}(x)&=&\frac{8}{j}\left(\frac{y_{+}}{k_{+}}+\frac{y_{-}}{k_{-}}\right ) \frac{k_{0}k_{2}}{{\cal m}_{2}}\;e^{-k_{-}(x - z_{2})}\ ; , \nonumber \\ \\ n_{1(+)}&=&\epsilon\frac{y_{+}}{k_{+ } } \left\{\frac{8}{j}\left(\frac{y_{+}}{k_{+}}+\frac{y_{-}}{k_{-}}\right ) \frac{k_{0}k_{1}}{{\cal m}_{1}}\;e^{k_{+}(x - z_{1})}-1\right\}\ ; , \nonumber \\ n_{1(0)}&=&\epsilon\left\{\frac{y_{0}}{2jk_{0}^{2 } } \left(\frac{y_{+}}{k_{+}}+\frac{y_{-}}{k_{-}}\right ) \left[k_{2}m_{2}\;e^{k_{0}(x - z_{2})}-k_{1}m_{1}\;e^{-k_{0}(x - z_{1 } ) } \right.\right . \nonumber \\ & & \makebox[5em]{}\left.\left.+2k_{0}(k_{1}k_{2}m_{1}m_{2})^{1/2 } \;e^{\frac{1}{2}k_{0}(z_{1}-z_{2})}\;x\right]+d_{0}\right\}\ ; , \nonumber \\ n_{1(-)}&=&-\epsilon\frac{y_{-}}{k_{- } } \left\{\frac{8}{j}\left(\frac{y_{+}}{k_{+}}+\frac{y_{-}}{k_{-}}\right ) \frac{k_{0}k_{2}}{{\cal m}_{2}}\;e^{-k_{-}(x - z_{2})}-1\right\}\;. \nonumber\end{aligned}\ ] ] with this solution and the canonical equations , the field equation ( [ eq - pi1 ] ) can be proved to hold in a whole @xmath102 space . as we showed in the previous paper , to satisfy ( [ eq - psi1 ] ) the dilaton field @xmath11 needs an extra function @xmath366 , which has no effect on the dynamics of particles . after lengthy calculation eq.([eq - psi1 ] ) leads to @xmath367 thus @xmath366 is uniquely determined . for a single static source @xmath368 the solution to the field equations ( [ e - pi ] ) -([e - psi ] ) under the coordinate conditions ( [ cc ] ) is @xmath369 \ ; , \nonumber \\ & & \\ \pi & = & -\frac{\epsilon m}{4}\sqrt{1+\frac{8\lambda } { \kappa ^{2}m^{2}}}\;,\qquad \psi = -\frac{\kappa m}{2}\;|x|+\frac{\kappa \epsilon m}{2}\sqrt{1+\frac{8\lambda } { \kappa ^{2}m^{2}}}\;\;t\;. \nonumber\end{aligned}\ ] ] \{}from ( [ e - p ] ) and ( [ e - z ] ) the canonical equations for a test particle ( mass @xmath370 ) under the gravity of a static source are @xmath371 the hamiltonian leading to these equations is @xmath372 \;.\end{aligned}\ ] ] this hamiltonian is also derived from the determining eq . ( [ h1 ] ) by setting @xmath373 and retaining only the linear terms of @xmath374 and @xmath375 . in terms of the proper time of the test particle @xmath376 the canonical equations ( [ test - can - p ] ) and ( [ test - can - z ] ) are expressed as @xmath377 eq.([test - p2 ] ) can be integrated and in @xmath378 region the solution @xmath182 is @xmath379 with @xmath380 } { \frac{\sqrt{p_{0}^{2}+\mu^2}-\epsilon p_{0}}{\mu } + \frac{\sqrt{1-\gamma_{m}}}{1+\sqrt{\gamma_m } } \;\mbox{tan}\left[\epsilon\sqrt{-\frac{\lambda}{8 } } ( \tau-\tau_{0})\right ] } & \qquad \lambda < 0 \ ; , \end{array } \right.\ ] ] where @xmath381 with @xmath236 being the initial momentum at @xmath237 . in @xmath382 region the solution is @xmath383 with @xmath384 } { \frac{\sqrt{p_{0}^{2}+\mu^2}+\epsilon p_{0}}{\mu } + \frac{\sqrt{1-\gamma_{m}}}{1+\sqrt{\gamma_m } } \;\mbox{tan}\left[\epsilon\sqrt{-\frac{\lambda}{8 } } ( \tau-\tau_{0})\right ] } & \qquad \lambda < 0 \ ; , \end{array } \right.\ ] ] where @xmath385 when @xmath386 and @xmath387 at @xmath237 , the total energy is @xmath388 . the solution for @xmath389 is obtained from ( [ test - ham ] ) and @xmath182 as @xmath390 for the test particle solution the critical value of @xmath7 is @xmath391 . fig.21 and 22 show typical trajectories of the test particle @xmath392 for @xmath393 and @xmath394 and @xmath395 . the characteristics of these plots are common to those of the unequal mass case . fig.21 _ phase space trajectories of a test particle for different values of @xmath7 . _ fig.22 _ the @xmath184 plots corresponding to the trajectories in fig.21 . _ we can explicitly verify that the particles lose causal contact with one another for @xmath396 . consider the unbounded motion of tan - type @xmath397 with @xmath398 and @xmath399 . the path @xmath400 of light emitted from particle 2 at time @xmath248 is governed by @xmath187 , which reads @xmath401 and so the equation of the light signal directed to particle 1 is @xmath402 the light emitted in the opposite direction is described by @xmath403 numerically solving ( [ eq1light ] ) and ( [ eq2light ] ) yields the solutions shown in figs.23 and 24 , where the trajectories of light signals emitted from particle 2 at various times @xmath248 are plotted . for small @xmath404 , the particles are in causal contact ( a dotted curve in ( + ) direction in fig.23 ) , but for @xmath405 the signal just barely catches up with particle 1 , which is almost in light - like motion ( a dashed curve in ( + ) direction in fig.23 ) . for @xmath406 the world line @xmath400 in the ( + ) direction is parallel to @xmath407 at large @xmath172 and in the ( - ) direction it goes nearly on the same trajectory with the particle 2 . for large @xmath248 ( @xmath408 ) the particles are out of causal contact with each other ( fig.24 ) : a light ray sent from particle 2 toward particle 1 receives a strong repulsive effect and ultimately reverses direction , following behind particle 2 . in fig.24 the trajectories of the light signal emitted to ( - ) direction can not be discriminated from those of the particle 2 . a flat - space model of these effects can be constructed as follows . consider the following expression for the 2-velocity @xmath413 where @xmath414 is some function and @xmath415 is the flat metric . we have @xmath416 , @xmath417 and so @xmath418 the general expression for the acceleration of a particle with 2-velocity ( [ eq2v ] ) is @xmath419 where @xmath420 . we have @xmath421 and @xmath422 and @xmath423 for the magnitude of the acceleration . in general we have the following possibilites : \2 ) the function @xmath427 as @xmath426 . in this case the particle becomes lightlike , but it takes an infinite proper time ( and coordinate time ) for this to happen . the standard example is @xmath428 , the constant acceleration example . \3 ) the function @xmath427 as @xmath429 , where @xmath430 is finite . in this case the particle becomes lightlike in a finite amount of proper time , but an infinite amount of coordinate time . an example would be @xmath431 . the acceleration is not constant , but increases as a function of proper time , diverging at @xmath237 . this last situation is realized by our exact solutions ( [ sol - tana][sol - tanb ] ) with @xmath432 .
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we develop the canonical formalism for a system of @xmath0 bodies in lineal gravity and obtain exact solutions to the equations of motion for @xmath1 .
the determining equation of the hamiltonian is derived in the form of a transcendental equation , which leads to the exact hamiltonian to infinite order of the gravitational coupling constant . in the equal mass case
explicit expressions of the trajectories of the particles are given as the functions of the proper time , which show characteristic features of the motion depending on the strength of gravity ( mass ) and the magnitude and sign of the cosmological constant .
as expected , we find that a positive cosmological constant has a repulsive effect on the motion , while a negative one has an attractive effect
. however , some surprising features emerge that are absent for vanishing cosmological constant . for a certain range of the negative cosmological constant
the motion shows a double maximum behavior as a combined result of an induced momentum - dependent cosmological potential and the gravitational attraction between the particles . for a positive cosmological constant ,
not only bounded motions but also unbounded ones are realized .
the change of the metric along the movement of the particles is also exactly derived .
* exact solutions of relativistic two - body motion in lineal gravity * + r.b .
mann and d. robbins + dept . of physics ,
university of waterloo waterloo , ont n2l 3g1 , canada + and + t. ohta + department of physics , miyagi university of education , aoba - aramaki , sendai 980 , japan + pacs numbers : 13.15.-f , 14.60.gh , 04.80.+z + +
| 22,772 | 447 |
a central open question in classical fluid dynamics is whether the incompressible three - dimensional euler equations with smooth initial conditions develop a singularity after a finite time . a key result was established in the late eighties by beale , kato and majda ( bkm ) . the bkm theorem @xcite states that blowup ( if it takes place ) requires the time - integral of the supremum of the vorticity to become infinite ( see the review by bardos and titi @xcite ) . many studies have been performed using the bkm result to monitor the growth of the vorticity supremum in numerical simulations in order to conclude yes or no regarding the question of whether a finite - time singularity might develop . the answer is somewhat mixed , see _ e.g. _ references @xcite and the recent review by gibbon @xcite . other conditional theoretical results , going beyond the bkm theorem , were obtained in a pioneering paper by constantin , fefferman and majda @xcite . they showed that the evolution of the direction of vorticity posed geometric constraints on potentially singular solutions for the 3d euler equation @xcite . this point of view was further developed by deng , hou and yu in references @xcite and @xcite . an alternative way to extract insights on the singularity problem from numerical simulations is the so - called analyticity strip method @xcite . in this method the time is considered as a real variable and the space - coordinates are considered as complex variables . the so - called `` width of the analyticity strip '' @xmath5 is defined as the imaginary part of the complex - space singularity of the velocity field nearest to the real space . the idea is to monitor @xmath1 as a function of time @xmath6 . this method uses the rigorous result @xcite that a real - space singularity of the euler equations occurring at time @xmath7 must be preceded by a non - zero @xmath1 that vanishes at @xmath7 . using spectral methods @xcite , @xmath1 is obtained directly from the high - wavenumber exponential fall off of the spatial fourier transform of the solution @xcite . this method effectively provides a `` distance to the singularity '' given by @xmath1 @xcite , which can not be obtained from the general bkm theorem . note that the bkm theorem is more robust than the analyticity - strip method in the sense that it applies to velocity fields that do not need to be analytic . however , in the present paper we will concentrate on initial conditions that are analytic . in this case , there is a well - known result that states : _ in three dimensions with periodic boundary conditions and analytic initial conditions , analyticity is preserved as long as the velocity is continuously differentiable _ ( @xmath8 ) _ in the real domain _ @xcite . the bkm theorem allows for a strengthening of this result : analyticity is actually preserved as long as the vorticity is finite @xcite . the analyticity - strip method has been applied to probe the euler singularity problem using a standard periodic ( and analytical ) initial data : the so - called taylor - green ( tg ) vortex @xcite . we now give a short review of what is already known about the tg dynamics . numerical simulations of the tg flow were performed with resolution increasing over the years , as more computing power became available . it was found that except for very short times and for as long as @xmath1 can be reliably measured , it displays almost perfect exponential decrease . simulations performed in @xmath9 on a grid of @xmath10 points obtained @xmath11 ( for @xmath6 up to @xmath12 ) @xcite . this behavior was confirmed in @xmath13 at resolution @xmath14 @xcite . more than @xmath15 years after the first study , simulations performed on a grid of @xmath16 points yielded @xmath17 ( for @xmath6 up to @xmath18 ) @xcite . if these results could be safely extrapolated to later times then the taylor - green vortex would never develop a real singularity @xcite . the present paper has two main goals . one is to report on and analyze new simulations of the tg vortex that are performed at resolution @xmath0 . these new simulations show , for the first time , a well - resolved change of regime , leading to a faster decay of @xmath1 happening at a time where preliminary @xmath3 visualizations show the collision of vortex sheets . that was reported in mhd for the so - called imtg initial data at resolution @xmath16 in reference @xcite . ] the second goal of this paper is to answer the following question , motivated by the new behavior of the tg vortex : how fast does the analyticity - strip width have to decrease to zero in order to sustain a finite - time singularity , consistent with the bkm theorem ? to the best of our knowledge , this question has not been formulated previously . to answer this question we introduce a new bound of the supremum norm of vorticity in terms of the energy spectrum . we then use this bound to combine the bkm theorem with the analyticity - strip method . this new bound is sharper than usual bounds . we show that a finite - time blowup exists only if the analyticity - strip width goes to zero sufficiently fast at the singularity time . if a power - law behavior is assumed for @xmath1 then its exponent must be greater than some critical value . in other words , we provide a powerful test that can potentially rule out the existence of a finite - time singularity in a given numerical solution of euler equations . we apply this test to the data from the latest @xmath0 taylor - green numerical simulation in order to see if the change of behavior in @xmath1 can be consistent with a singularity . the paper is organized as follows : section [ sec : theo ] is devoted to the basic definitions , symmetries and numerical method related to the inviscid taylor - green vortex . in sec . [ sec : numerics_classical ] , the new high - resolution taylor - green results are presented and are analyzed classically in terms of analyticity - strip method and bkm . in sec . [ sec : as_bkm ] , the analyticity - strip method and bkm theorem are bridged together . the section starts with heuristic arguments that are next formalized in a mathematical framework of definitions , hypotheses and theorems . in sec . [ sec : newanal ] , our new theoretical results are used to analyze the behavior of the decrement . section [ sec : conclusion ] is our conclusion . the generalization to non tg - symmetric periodic flows of the results presented in sec . [ sec : as_bkm ] are described in an appendix . let us consider the 3d incompressible euler equations for the velocity field @xmath19 defined for @xmath20 and in a time interval @xmath21 : @xmath22 the taylor - green ( tg ) flow @xcite is defined by the @xmath23-periodic initial data @xmath24 , where @xmath25 the periodicity of @xmath26 allows us to define the ( standard ) fourier representation @xmath27 the kinetic energy spectrum @xmath28 is defined as the sum over spherical shells @xmath29 and the total energy @xmath30 is independent of time because @xmath26 satisfies the 3d euler equations ( [ eq : euler ] ) . a number of the symmetries of @xmath31 are compatible with the equation of motions . they are , first , rotational symmetries of angle @xmath32 around the axis @xmath33 and @xmath33 ; and of angle @xmath34 around the axis @xmath35 . a second set of symmetries corresponds to planes of mirror symmetry : @xmath36 , @xmath37 and @xmath38 . on the symmetry planes , the velocity @xmath31 and the vorticity @xmath39 are ( respectively ) parallel and perpendicular to these planes that form the sides of the so - called impermeable box which confines the flow . it is demonstrated in reference @xcite that these symmetries imply that the fourier expansions coefficients of the velocity field in eq . @xmath40 vanishes unless @xmath41 are either all even or all odd integers . this fact can be used in a standard way @xcite to reduce memory storage and speed up computations . the euler equations are solved numerically using standard @xcite pseudo - spectral methods with resolution @xmath42 . time marching is done with a second - order runge - kutta finite - difference scheme . the solutions are dealiased by suppressing , at each time step , the modes for which at least one wave - vector component exceeds two - thirds of the maximum wave - number @xmath43 ( thus a @xmath0 run is truncated at @xmath44 ) . the simulations reported in this paper were performed using a special purpose symmetric parallel code developed from that described in @xcite . the workload for a timestep is ( roughly ) twice that of a general periodic code running at a quarter of the resolution . specifically , at a given computational cost , the ratio of the largest to the smallest scale available to a computation with enforced taylor - green symmetries is enhanced by a factor of @xmath45 in linear resolution . this leads to a factor of @xmath46 savings in total computational time and memory usage . the code is based on fftw and a hybrid mpi - openmp scheme derived from that described in @xcite . the runs were performed on the idris bluegene / p machine . at resolution @xmath0 we used @xmath47 mpi processes , each process spawning @xmath45 openmp threads . ( see eq . ) at @xmath48 and b ) maximum of vorticity @xmath49 . results from runs performed at different resolutions are displayed together : @xmath50 ( brown triangles ) , @xmath51 ( blue squares ) , @xmath16 ( green diamonds ) and @xmath0 ( red circles).,height=377 ] runs were performed at resolutions @xmath50 , @xmath51 , @xmath16 and @xmath52 . the behavior of the energy spectra and the spatial maximum of the norm of the vorticity @xmath53 are presented in fig . [ fig : energy_spectra_maxvort ] . visualization of tg vorticity @xmath54 at resolution @xmath0 : a ) full impermeable box @xmath55 , @xmath56 and @xmath57 at @xmath58 . zooms over the subbox marked near @xmath59 , @xmath60 are displayed in b ) at @xmath61 , in c ) at @xmath58 and in d ) at @xmath62.,height=359 ] it is apparent in fig . [ fig : energy_spectra_maxvort](a ) that resolution - dependent even - odd oscillations are present , at certain times , on the tg energy spectrum . note that this behavior is produced when the tail of the spectrum rises above the round - off error @xmath63 . this phenomenon can be explained in terms of a _ resonance _ @xcite , along the lines developed in reference @xcite . in practice we will deal with this problem by averaging the spectrum over shells of width @xmath64 . apart from this it can be seen that spectra computed using different resolutions are in good agreement for all times . in contrast , it is visible in fig . [ fig : energy_spectra_maxvort](b ) that the maximum of vorticity @xmath49 computed at different resolutions are in agreement only up to some resolution - dependent time ( see the inset ) . the fact that @xmath49 at a given time @xmath65 decreases if one truncates the higher wavenumbers of the velocity field ( see fig . [ fig : energy_spectra_maxvort](b ) ) strongly suggests that @xmath49 has significant contributions coming from high - wavenumbers modes . this forms the basis of the heuristic argument presented below in sec . [ subsec : heur ] . figure [ fig : vort_3d_viz ] shows @xmath3 visualizations ( using the vapor software ) of the high vorticity regions in the impermeable box , corresponding to the @xmath0 run at late times . a thin vortex sheet is apparent in fig.[fig : vort_3d_viz](a ) on the vertical faces @xmath66 , @xmath32 and @xmath67 , @xmath32 of the impermeable box . the emergence of this thin vortex sheet is well understood by simple dynamical arguments about the flow on the faces of the impermeable box that were first given in reference @xcite . we now briefly review these arguments . the initial vortex on the bottom face is first forced by centrifugal action to spiral outwards toward the edges and then up the side faces . a corresponding outflow on the top face and downflow from the top edges onto the side faces leads to a convergence of fluid near the horizontal centreline of each side face , from where it is forced back into the centre of the box and subsequently back to the top and bottom faces . the vorticity on the side faces is efficiently produced in the zone of convergence , and builds up rapidly into a vortex sheet ( see figs . 1 and 2 of reference @xcite and fig . 8 of reference @xcite ) . while these considerations explain the presence of the thin vortex sheet in fig.[fig : vort_3d_viz](a ) , the dynamics presented in fig.[fig : vort_3d_viz](b - d ) also involves the collision of vortex sheets happening near the edge @xmath59 , close to @xmath60 . note that , as stated above in sec . [ subsec : symm ] , the vortex lines are perpendicular to the faces of the impermeable box . thus , because the collision takes place near an edge , the corresponding vortex lines must be highly curved , with strong variations of the direction of vorticity . the geometric constraints on potential singularities posed by the evolution of the direction of vorticity developed in references @xcite could be applied to the situation described in fig . [ fig : vort_3d_viz ] . however , such an analysis goes beyond the bkm theorem and involves extensive post - processing of very large datasets . this task is thus left for further work and we concentrate here on simple bkm diagnostics for the vorticity supremum and analyticity strip analysis of energy spectra . the analyticity - strip method @xcite is based on the fact that when the velocity field is analytic in space the energy spectrum satisfies @xmath68 in the asymptotic ` ultraviolet region ' @xmath69 with a proportionality factor that may contain an algebraic decay in @xmath70 a multiplicative function of time and , depending on the complexity of the physical flow , even an oscillatory ( in @xmath71 ) modulation @xcite . ( red markers ) ; times and fit intervals are indicated in the legend . ] the basic idea is thus to assume that @xmath28 can be well approximated by a function of the form @xmath72 in some wave numbers interval between @xmath73 and @xmath74 ( the maximum wavenumber permitted by the numerical resolution @xmath42 ) . the common procedure to determine @xmath75 is to perform a least - square fit at each time @xmath6 on the logarithm of the energy spectrum @xmath28 , using the functional form @xmath76 the error on the fit interval @xmath77 , @xmath78 is minimized by solving the equations @xmath79 , @xmath80 and @xmath81 . note that these equations are linear in the parameters @xmath82 , @xmath83 and @xmath84 the transient oscillations of the energy spectrum observed at the highest wavenumbers ( see above fig . [ fig : energy_spectra_maxvort](a ) are eliminated by averaging the tg spectrum on shells of width @xmath64 before performing the fit @xcite . we present in fig . [ fig : fit_comp ] , examples of tg energy spectra fitted in such a way on the intervals @xmath85 , where @xmath86 denotes the beginning of round off noise . it is apparent that the fits are globally of a good quality . the time evolution of the fit parameters @xmath87 , @xmath88 and @xmath89 computed at different resolutions are displayed in fig . [ fig : fit_evolution ] . the measure of the fit parameters is reliable as long as @xmath1 remains larger than a few mesh sizes , a condition required for the smallest scales to be accurately resolved and spectral convergence ensured . thus the dimensionless quantity @xmath90 is a measure of spectral convergence . it is conventional @xcite to define a ` reliability time ' @xmath91 by the condition @xmath92 and to say that the numerical simulation is reliable for times @xmath93 . this reliability time can be extended only by increasing the spatial resolution available for the simulation , so the more computer power is available the larger is the reliability time . the resolution - dependent reliability condition is marked by the horizontal lines in fig . [ fig : fit_evolution](c ) . the exponential law @xmath94 that was previously reported at resolution @xmath16 in reference @xcite is also indicated in fig . [ fig : fit_evolution](c ) by a dashed black line . it is thus apparent that our lower - resolution results well reproduce the previous computations that were discussed above in sec . [ sec : intro ] ( see text preceding references @xcite ) . in table [ tab : table_rel ] , the reliability time obtained from the fit parameter @xmath88 of fig . [ fig : fit_evolution ] is compared with the reliability time stemming from the exponential behavior . .reliability time deduced from the exponential behavior compared with the reliability time obtained from the fit parameter @xmath88 of fig . [ fig : fit_evolution ] . [ cols="^,^,^",options="header " , ] the results for exponent and predicted singular time of table [ tab : table_dels ] have to be read carefully . because of the local @xmath95-point method used to derive them from the data in table [ tab : table_int ] , they use the values of @xmath88 at @xmath96 , the last one being marginally reliable ( see sec.[sec : numefits ] ) . in fact , they amount to linear @xmath97-point extrapolation of the data in fig . [ fig : delta1 ] ( see the inset ) : @xmath7 is the intersection of the straight line extrapolation with the time axis and @xmath98 is the inverse of the slope . one can guess that there is room for a power - law type of behavior , with exponent @xmath99 if we consider the data at @xmath100 and @xmath101 if we include the data at @xmath102 . we now use corollary 11 ( see sec . [ sec : as_bkm ] ) to test if these estimates of power - law are consistent with the hypothesis of finite - time singularity . there , the product @xmath103 must be greater than or equal to one if finite - time singularity is to be expected . with the conservative estimate @xmath104 obtained by inspection of fig . [ fig : fit_evolution](b ) ( or equivalently using the values of @xmath89 in table [ tab : table_int ] ) , we obtain that @xmath105 for the data at @xmath106 and @xmath58 , but @xmath107 for the data at @xmath102 . these results are insensitive to the fit interval , see table [ tab : table_dels ] . therefore , if the latest data is considered , corollary 11 can not be used to negate the validity of the hypothesis of finite - time singularity . however , there is no sign that the data values of @xmath98 and @xmath7 in table [ tab : table_dels ] are settling down into constants , corresponding to a simple power - law behavior . another piece of analysis consists of comparing the singular time predicted from the data for the decrement @xmath1 with the singular time predicted from the direct data for the vorticity supremum norm . they seem both to be close to @xmath108 ( compare table [ tab : table_dels ] to table [ tab : table_omegasup ] ) . in this context , we should perhaps mention feynman s rule : `` never trust the data point furthest to the right '' , a comment attributed to richard feynman , saying basically that he would never trust the last points on an experimental graph , because if the people taking data could have gone beyond that , they would have . higher - resolution simulations are clearly needed to investigate whether the new regime is genuinely a power law and not simply a crossover to a faster exponential decay . our conclusion for this section is thus similar to that of sec . [ subsec : fit_methods_omegas ] : although our late - time reliable data for @xmath1 shows @xmath109 and is therefore not inconsistent with our corollary 11 , clear power - law behavior of @xmath1 is not achieved . in summary , we presented simulations of the taylor - green vortex with resolutions up to @xmath0 . we used the analyticity strip method to analyze the energy spectrum . we found that , around @xmath110 , a ( well - resolved up to @xmath111 ) change of regime is taking place , leading to a faster decay of the width of the analyticity strip @xmath1 . in the same time - interval , preliminary @xmath3 visualizations displayed a collision of vortex sheets . applying the bkm criterium to the growth of the maximum of the vorticity on the time - interval @xmath2 we found that the occurrence of a singularity around @xmath112 was not ruled out but that higher - resolution simulations were needed to confirm a clear power - law behavior for @xmath113 . we introduced a new sharp bound for the supremum norm of the vorticity in terms of the energy spectrum . this bound allowed us to combine the bkm theorem with the analyticity - strip method and to show that a finite - time blowup can exist only if @xmath1 vanishes sufficiently fast . applying this new test to our highest - resolution numerical simulation we found that the behavior of @xmath1 is not inconsistent with a singularity . however , due to the rather short time interval on which @xmath1 is both well - resolved and behaving as a power - law , higher - resolution studies are needed to investigate whether the new regime is genuinely a power law and not simply a crossover to a faster exponential decay . let us finally remark that our formal assumptions of section [ subsec : main_results ] are motivated and to some extent justified by the fact that , in systems that are known to lead to finite - time singularity , the analogous of the working hypothesis ( [ eq : fit_bound ] ) is verified . for the analogy to apply , a version of the bkm theorem must be available . this is the case of the @xmath73-d inviscid burgers equation for a real scalar field @xmath114 defined on the torus : @xmath115 , \,\forall \ , t \in [ 0,t_*),\ ] ] which admits a bkm - type of theorem @xcite , with singularity time @xmath7 defined by @xmath116 . in the 1-d case , the analogous of our bound is @xmath117 using the simple trigonometric initial data @xmath118 , the energy spectrum can be expressed in terms of bessel functions that admit simple asymptotic expansions . it is straightforward to show ( see @xcite for details ) that , for @xmath119 , one has the large-@xmath71 asymptotic expansion @xmath120 with @xmath121 while , at @xmath122 , @xmath123 in fact , the @xmath124 power law appears already before @xmath7 ( see the remark following eq . ( 3 - 10 ) of reference @xcite ) . it is easy to check that the analytical solution admits , for all @xmath71 and for all @xmath6 sufficiently close to @xmath7 , a working hypothesis ( [ eq : fit_bound ] ) of the form @xmath125 with analytically - obtainable functions @xmath126 and @xmath127 with @xmath128 . the analogous of corollary [ cor : beta ] gives the inequality @xmath129 which is saturated by the analytically - obtained exponents @xmath130 , @xmath128 . we acknowledge useful scientific discussions with annick pouquet , uriel frisch and giorgio krstulovic who also helped us with the visualizations of fig . [ fig : vort_3d_viz ] . the computations were carried out at idris ( cnrs ) . support for this work was provided by ucd seed funding projects sf304 and sf564 , and ircset ulysses project `` singularities in three - dimensional euler equations : simulations and geometry '' . here we provide the generalization to non tg - symmetric periodic flows of the results presented in section [ subsec : main_results ] . definition [ defn : spectrum ] and the working hypothesis ( hypothesis [ hypo : working ] ) are modified slightly in the general case . accordingly , the new bounds leading to lemma [ lem : main ] and theorem [ thm : main ] need to be modified slightly to accommodate the general case . the crucial derived relations between @xmath131 and @xmath132 in lemma [ lem : strong ] and corollaries [ cor : finite - time - sing ] and [ cor : beta ] will apply directly to the general periodic case and will not be discussed . the main technical difference is that the new bounds presented in section [ subsec : main_results ] apply for a flow with tg symmetries ( see section [ subsec : symm ] ) which imply that only modes with even - even - even and odd - odd - odd wavenumber components are populated . the general periodic case does not follow this restriction , which slightly modifies the bounds . we will assume , to simplify matters , that the so - called zero - mode of the velocity field is identically zero : @xmath133 + notice that all remaining wave numbers are populated . this means that all sums involving the scalar @xmath71 in equations ( [ eq : ineq_1 ] ) and ( [ eq : ineq_2_tyg ] ) will start effectively from @xmath134 also , because modes with mixed even - odd wavenumber components are allowed , the definitions of @xmath135 in lemma 2 and constant @xmath136 in equation ( [ eq : ineq_2_tyg ] ) must be replaced by more appropriate quantities . therefore , the corresponding general periodic versions of lemma [ lem : main ] ( equation ( [ eq : ineq_1 ] ) ) and practical bound ( equation ( [ eq : ineq_2_tyg ] ) ) are : + * lemma [ lem : main] ( general periodic version of lemma [ lem : main ] ) . * _ let @xmath137 be a velocity field with energy spectrum defined by equation ( [ eq : spectrum ] ) and let @xmath138 be its vorticity , defined on the periodicity domain @xmath139 ^ 3.$ ] then the following inequality is verified for all times @xmath21 when the sum in the rhs is defined , and independently of any evolution equation that @xmath140 might satisfy : _ @xmath141 _ where @xmath142 is the number of lattice points in a spherical shell of width 1 and radius @xmath143 . _ + * practical bound , general case . * @xmath144 where @xmath145 . we can easily check that the bounds for taylor - green , equations ( [ eq : ineq_1 ] ) and ( [ eq : ineq_2_tyg ] ) , are sharper ( by a factor close to 2 ) to their respective general bounds , equations ( [ eq : ineq_1_gp ] ) and ( [ eq : ineq_2_gp ] ) . finally , theorem [ thm : main ] is replaced by + * theorem [ thm : main]. * _ let a solution of the 3d euler equations satisfy the working hypothesis ( [ eq : fit_bound ] ) with @xmath146 included . then the maximal regularity time @xmath7 of the solution must satisfy _ @xmath147
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numerical simulations of the incompressible euler equations are performed using the taylor - green vortex initial conditions and resolutions up to @xmath0 .
the results are analyzed in terms of the classical analyticity strip method and beale , kato and majda ( bkm ) theorem .
a well - resolved acceleration of the time - decay of the width of the analyticity strip @xmath1 is observed at the highest resolution for @xmath2 while preliminary @xmath3 visualizations show the collision of vortex sheets .
the bkm criterium on the power - law growth of supremum of the vorticity , applied on the same time - interval , is not inconsistent with the occurrence of a singularity around @xmath4 .
these new findings lead us to investigate how fast the analyticity strip width needs to decrease to zero in order to sustain a finite - time singularity consistent with the bkm theorem .
a new simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the bkm theorem with the analyticity - strip method .
it is shown that a finite - time blowup can exist only if @xmath1 vanishes sufficiently fast at the singularity time .
in particular , if a power law is assumed for @xmath1 then its exponent must be greater than some critical value , thus providing a new test that is applied to our @xmath0 taylor - green numerical simulation .
our main conclusion is that the numerical results are not inconsistent with a singularity but that higher - resolution studies are needed to extend the time - interval on which a well - resolved power - law behavior of @xmath1 takes place , and check whether the new regime is genuine and not simply a crossover to a faster exponential decay .
| 7,805 | 463 |
most elementary treatments of reflecting surfaces restrict their attention to the spherical case . in this standard case , and assuming the paraxial approximation ( all angles are small and all rays are close to the optical axis ) , the resulting equation relating the _ axial _ object and image positions and the radius of curvature of the reflecting spherical surface is @xmath0 where all parameters are one dimensional coordinates which locate the image ( @xmath1 ) , object ( @xmath2 ) , and center of curvature ( @xmath3 ) with respect to the vertex ( the intersection of the surface with the optical axis ) @xcite . a convention is typically assumed in which light rays travel from left to right in all figures . the origin of the one dimensional coordinate system employed coincides with the vertex , and locations to the right ( left ) of the vertex are positive ( negative ) . [ ptbh ] the paraxial approximation is equivalent to a first order approximation in the height ( @xmath4 ) of the incidence point ( on the surface ) of a reflecting ray . to higher order , it is found that @xmath5 consequently , spherical mirrors are aberrant at higher order since the image location is not independent of the height , @xmath4 . this paper represents a more general treatment of a mirror than is typically found in the literature . the reflecting surface is assumed to be a conicoid , the surface of revolution generated by a conic . equation ( [ gauss ] ) is then derived as the special case of a spherical surface and to first order in @xmath4 . special cases are analyzed as a function of asphericity , or departure from the spherical , of the reflecting surface . the parabolic surface is shown to be uniquely special in that @xmath6 to all orders for objects at infinity ( @xmath7 ) . [ ptbh ] in fig . [ fig02 ] , a conicoid reflecting surface is depicted with equation @xmath8 where @xmath3 is the radius of curvature of the surface at the vertex , and @xmath9 is the shape factor and is related to the standard eccentricity ( see appendix i or , for example , @xcite ) . for a sphere , @xmath10 , whereas for a paraboloid @xmath11 . note that the @xmath12 coordinate system is set on its side so that @xmath13 coincides with the negative direction on the optical axis ( o.a . ) as defined in fig . [ fig01 ] of the introduction . consequently , the radius of curvature , @xmath3 , at the origin for any concave conicoid ( _ i.e. _ , opening to the left ) will be considered negative . in fig . [ fig02 ] , a representative case is depicted with @xmath14 , the location of the object , and @xmath15 , the location of the image . the figure displays an incident ray , @xmath16 , emanating from the object at @xmath17 and a reflected ray , @xmath18 , passing through the image at @xmath19 . from the figure , the line @xmath16 has equation in the @xmath12-plane @xmath20 similarly , the line @xmath18 has equation @xmath21 consequently , @xmath22 where @xmath23 is the point of reflection , @xmath24 , on the surface . from the figure , it follows that @xmath25 where @xmath26 and @xmath27 therefore @xmath28 substituting for the tangents from above yields @xmath29 -\left ( \frac{1}{uv}\right ) \frac{2y_{0}\left [ 1 + 2\left ( 1-\sigma\right ) ay_{0}\right ] } { \left ( 1 - 2\sigma ay_{0}\right ) ^{2}}=-\frac{4a}{1 - 2\sigma ay_{0 } } \label{eq09a}\]]@xmath30 -\left ( \frac{1}{uv}\right ) & 2y_{0}\left [ 1 + 2\left ( 1-\sigma\right ) ay_{0}\right ] \hspace{0.45in}\nonumber\\ & \hspace{0.6in}=-4a\left ( 1 - 2\sigma ay_{0}\right ) . \label{eq09b}\ ] ] now let @xmath31 be the height of the incidence point @xmath24 for a particular ray from the source object at @xmath17 , then in the paraxial approximation ( @xmath32 ) , @xmath33 equation ( [ eq09b ] ) can then be rewritten to fourth order as @xmath34 h^{2}\nonumber\\ & \hspace{0.5in}+\left [ 4\sigma a^{4}\left ( \frac{1}{v}+\frac{1}{u}\right ) + 2\left ( 3\sigma+2\right ) a^{3}\left ( \frac{1}{uv}\right ) -24\sigma ^{2}a^{5}\right ] h^{4}. \label{eq11}\ ] ] note that there is aberration in imaging a finite axial point since there is no confluence in the rays from @xmath17 . also note that there is no fixed shape factor @xmath35 that eliminates aberration to second order and higher . to first order , all conicoids obey the same relation @xmath36 which coincides , of course , with the gaussian ( first order approximation ) equation for a spherical mirror with focal length @xmath37 . from eq . ( [ eq09b ] ) it follows that for objects at infinity ( @xmath38 ) and a parabolic shape ( @xmath11 ) , the image forms at @xmath39 regardless of the height of the incidence ray , therefore , there is no aberration for such imaging . it is desirable to know to what extend the results of the previous section are pathological to conicoids . with this in mind consider the most general axi - symmetric surface of revolution ( about the y - axis ) as a reflector @xmath40 equation ( [ eq08b ] ) is easily generalized to @xmath41 where @xmath42 . in general , for a given axial object location , the image location ( or intersection point of the reflected ray with the optical axis ) is a function of the object location and the reflection point @xmath43 a reflecting surface is free of aberration if @xmath44 equation ( [ eq14 ] ) can be implicitly differentiated to yield @xmath45 } { y^{\prime}}\right\ } _ { 2}. \label{eq17}\ ] ] the aberration - free surface must satisfy @xmath46 . however , it is evident from eq . ( [ eq17 ] ) that this can not be obtained trivially . for the special case in which the object is at infinity though , the aberration - free surface must only satisfy @xmath47 , and this leads to a defining equation for the surface @xmath48 this is a linear differential equation whose general solution can most easily be found by the reduction in order method to give the general solution @xmath49 . this further reduces to the particular solution of eq . ( [ eq32 ] ) , found by another method , after the two needed boundary conditions are invoked . most elementary treatments of mirrors lack a discussion of the first order equation relating object and image locations in the case of arbitrary mirror shape . the default reflecting surface is always the spherical one . in fact , a simple analysis yields that all axi - symmetric , conic , reflecting surfaces of revolution ( conicoids ) in the first order , paraxial approximation satisfy the same ( gaussian ) equation @xmath50 where @xmath3 is the radius of curvature of the surface at its vertex . aberrations enter at second order and can not be eliminated for finite object locations by any fixed shape . however , for objects at infinity , or specifically , for incoming light parallel to the optical axis , there is a unique reflecting shape that is free of aberration the parabolic one . starting with the general form of a conic section in cartesian coordinates , @xmath51 assume @xmath52-reflection symmetry , so that the equation reduces to @xmath53 next the curve is shifted so the vertex coincides with the origin , @xmath54 with @xmath55 . if the form is further constrained so that the curve lies in @xmath56 half - plane , then the positive root is required , and this yields @xmath57 or in terms of new parameters @xmath58 where @xmath59 . the signed curvature of this curve at the origin is @xmath60 given the optics conventions adopted here as described in the introduction and depicted in figures 1 and 2 , the radius , @xmath3 , of the osculating circle at the origin for a concave conicoid is considered negative . the radius of curvature is therefore related to the parameter @xmath61 @xmath62 and @xmath63 from eq . ( [ eq22 ] ) it follows that @xmath11 corresponds to a parabola . by putting eq . ( [ eq22 ] ) into canonical form @xmath64 it becomes clear that @xmath10 corresponds to a circle with radius @xmath65 . the equation describes a hyperbola when @xmath66 . for @xmath67 , the equation describes an oblate ellipse ( with respect to the y - axis ) , and it describes a prolate ellipse for @xmath68 . in fact , from eq . ( [ eq26 ] ) , the shape factor , @xmath35 , can be related to the standard eccentricity @xmath69 an alternate solution ( to that of section iii ) is presented for the exact conicoid shape in the limit that the object distance approaches infinity ( @xmath70 ) . applying the law of reflection ( based on fermat s principle of stationary optical path ) to a parallel ( to the optical axis ) ray ( from a distant object ) incident on an unknown conicoid surface , results in the optical path displayed in fig . [ fig03 ] . applying eq . ( [ eq04b ] ) to the present special case , it follows that @xmath71 if the notation is changed and the @xmath72 variable is shifted for convenience , @xmath73 , then eq . ( [ eq29 ] ) can be reduced to either a homogeneous nonlinear ordinary differential equation ( ode ) of the form @xmath74 or to a nonlinear clairaut ode @xcite of the form @xmath75 recall that clairaut solutions are of the form @xmath76 and have envelopes that are also exact singularity solutions . solving eq . ( [ eq30 ] ) or ( [ eq31 ] ) yields the final form for the unknown conicoid ( and shifting back @xmath77 ) @xmath78 which is the equation for the ( meridional ) cross section of a paraboloid with focus at @xmath79 . it is also of note that eq . ( [ eq30 ] ) with @xmath80 can be used to model various and sundry airplane , ship , and predator / prey pursuit problems @xcite . foreman , `` the conic sections revisited , '' am . * 59 * , 1002 - 1005 ( 1991 ) ; d.m . watson , _ astronomy 203/403 : astronomical instruments and techniques on - line lecture , _ university of rochester ( 1999 ) , * http://www.pas.rochester.edu/dmw/ast203/lectures.htm . *
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the first order equation relating object and image location for a mirror of arbitrary conic - sectional shape is derived .
it is also shown that the parabolic reflecting surface is the only one free of aberration and only in the limiting case of distant sources .
| 3,213 | 61 |
in spite of being iso - structural and iso - valent to the cubic perovskite @xmath8k superconductor @xmath5 @xcite , @xmath4 remains in the normal metal state down to @xmath9k @xcite . the specific heat measurements indicate that the absence of superconductivity in @xmath4 may be due to a substantial decrease in the density of states at the fermi energy @xmath10 resulting from its relatively low unit cell volume in comparison with @xmath5 @xcite . however , electronic structure calculations show that the decrease in @xmath10 is not sizable enough to make @xmath4 non - superconducting @xcite . for both @xmath5 @xcite and @xmath4 @xcite the density of states spectra display similar characteristics , particularly in the distribution of electronic states near the fermi energy @xmath11 . the electronic states at @xmath11 are dominated by @xmath12 @xmath13 states with a little admixture of @xmath2 @xmath14 states . there exists a strong van hove singularity - like feature just below @xmath11 , which is primarily derived from the @xmath12 @xmath13 bands . to account for the lack of superconductivity in @xmath4 , the density - functional based calculations emphasize that the material subjected to the specific heat measurements may be non - stoichiometric in the @xmath2 sub - lattice @xcite . this would then make it similar to the @xmath15 phase of @xmath5 , which has a low unit cell volume and remains non- superconducting @xcite . it has been shown earlier that exact @xmath2 content in @xmath1 depends on the nature of synthesis and other experimental conditions @xcite . according to johannes and pickett @xcite , the arguments that favor non - stoichiometry are the following : ( i ) total energy minimization en - route to equilibrium lattice constant within the local - density approximation ( lda ) finds an overestimated value for @xmath4 in comparison with the experimental values . in general , overestimation is not so common in lda . meanwhile , when one uses similar technique for @xmath5 , the calculations find a slightly underestimated value which is consistent within the limitations of the density - functional theory @xcite . ( ii ) the authors also find @xmath10 in @xmath5 estimated as @xmath16 states / ry atom , while for @xmath4 , under similar approximations , it was found to be @xmath17 states / ry atom . note that it has been shown both experimentally as well as from first - principles calculations that a decrease in the lattice constant or a decrease in the @xmath2 occupancy would lead to a decrease in @xmath10 @xcite . ( iii ) a decrease in the unit cell dimensions can induce phonon hardening . this is well supported by the experiments which find the debye temperature approximately 1.6 times higher for @xmath4 in comparison to @xmath5@xcite . earlier synthesis of @xmath0 @xcite finds the lattice constant to be @xmath18 a.u . , for which the occupancy in the @xmath2 sub - lattice was just @xmath19% . the authors have employed similar preparation technique for @xmath5 @xcite and have found that the @xmath2 occupancy ranges between @xmath20-@xmath21 which is consistent with the recent reports @xcite . lattice constant for @xmath4 , as high as @xmath22 a.u . has also been reported elsewhere @xcite , which then becomes consistent with the recent total energy minimized value using density - functional based methods . hence , it seems that @xmath4 which was subjected to specific heat experiments @xcite may indeed suffer from non - stoichiometry . to understand and compare the effects of @xmath2 stoichiometry on the structural and electronic properties of @xmath1 and @xmath0 , we carry out a detail study using the korringa - kohn - rostoker ( kkr ) green s function method @xcite formulated in the atomic sphere approximation ( asa ) @xcite . for disorder , we employ the coherent - potential approximation ( cpa ) @xcite . characterization of @xmath1 and @xmath0 with @xmath23 mainly involves the changes in the equation of state parameters viz . , the equilibrium lattice constant , bulk modulus and its pressure derivative . the electronic structure is studied with the help of total and sub - lattice resolved density of states . the propensity of magnetism in these materials is studied with the help of fixed - spin moment method @xcite in conjunction with the landau theory of phase transition @xcite . the hopfield parameter @xmath24 which generally maps the local `` chemical '' property of an atom in a crystal is also calculated as suggested by skriver and mertig @xcite , and its variation as a function of lattice constant has also been studied . in general , we find that both @xmath5 and @xmath4 display very similar electronic structure . evidences point that the non - superconducting nature of @xmath4 may be related to the crystal structure characteristics , namely phonon spectra . the ground state properties of @xmath1 and @xmath0 are calculated using the kkr - asa - cpa method of alloy theory . for improving alloy energetics , the asa is corrected by the use of both the muffin - tin correction for the madelung energy @xcite and the multi - pole moment correction to the madelung potential and energy @xcite . these corrections have brought significant improvement in the accuracy of the total energy by taking into account the non - spherical part of polarization effects @xcite . the partial waves in the kkr - asa calculations are expanded up to @xmath25 inside atomic spheres , although the multi - pole moments of the electron density have been determined up to @xmath26 which is used for the multi - pole moment correction to the madelung energy . in general , the exchange - correlation effects are taken into consideration via the local - density approximation with perdew and wang parametrization @xcite , although a comparison in the equation of state parameters has been made in this work with the generalized gradient approximation ( gga ) @xcite . the core states have been recalculated after each iteration . the calculations are partially scalar - relativistic in the sense that although the wave functions are non - relativistic , first order perturbation corrections to the energy eigenvalues due to the darwin and the mass - velocity terms are included . the atomic sphere radii of @xmath6 ( @xmath7 ) , @xmath2 and @xmath12 were kept as @xmath27 , @xmath28 , and @xmath29 of the wigner- seitz radius , respectively . the vacancies in the @xmath2 sub - lattice are modeled with the help of empty spheres , and their radius is kept same as that of @xmath2 itself . the overlap volume resulting from the blow up of the atomic spheres was less than @xmath30% , which is legitimate within the accuracy of the approximation @xcite . the electron - phonon coupling parameter @xmath31 can be expressed as @xmath24/@xmath32 , where @xmath24 is the hopfield parameter , expressed as the product of @xmath10 and the mean square electron - ion matrix element @xmath33 , with @xmath34 and @xmath35 being the ionic mass and average phonon frequency @xcite . however , one may note that the above decomposition of the problem into electronic and phonon contributions is only approximate since in principle @xmath35 is also determined by the electronic states . it follows that the hopfield parameter is the most simple basic quantity which one may obtain from first - principles as suggested by gaspari and gyorffy @xcite . the latter assumes a rigid muffin - tin approximation ( rmta ) in which the potential enclosed by a sphere rigidly moves with the ion and the change in the crystal potential , caused by the displacement , is given by the potential gradient . within the rmta the spherically averaged part of the hopfield parameter may be calculated as , @xmath36 where @xmath10 is the total density of state per spin at the fermi energy , and @xmath37 the @xmath38 partial density of state calculated at the fermi energy @xmath11 , on the site considered . the term @xmath39 is the electron - phonon matrix element given as @xcite , @xmath40 which are obtained from the gradient of the potential and the radial solutions @xmath41 and @xmath42 of the schrodinger equation evaluated at @xmath11 . the special form of the eqs.[eq-123d ] and [ eq-123 m ] stems from the asa in which the radial wave functions are normalised to unity in the atomic sphere of radius @xmath43 , i.e. , @xmath44=@xmath45 . in asa , @xmath46 is expressed in terms of logarithmic derivatives @xmath47=@xmath48/@xmath41 evaluated at the sphere boundary . skriver and mertig derive the expression for @xmath39 as @xmath49\left[d_{l+1}+l+2)\right]+\left[e_{f}-v(s)\right]s^{2}\right\ } \end{array}\label{eq-234r}\ ] ] where @xmath50 is the one - electron potential and @xmath51 the sphere boundary amplitude of the @xmath52 partial wave evaluated at @xmath11 . numerical estimate to the magnetic energy are carried out using the fixed - spin - moment method @xcite . in the fixed - spin - moment method the total energy is obtained for a given magnetization @xmath34 , i.e. , by fixing the numbers of electrons with up and down spins . in this case , the fermi energies in the up and down spin bands are not equal to each other because the equilibrium condition would not be satisfied for arbitrary @xmath34 . at the equilibrium @xmath34 two fermi energies will coincide with each other . the total magnetic energy becomes minimum or maximum at this value of @xmath34 . note that the two approaches , i.e. , the self - consistent , floating - spin - moment method as well as the fixed - spin moment - method are equivalent in the sense that for a given lattice constant the magnetic moment calculated by the standard floating - spin moment approach is the same as the magnetic moment for which the fixed - spin moment total energy has its minimum @xcite . in practice , the floating - spin moment approach sometimes runs into some convergence problem . from experience , to avoid such predicaments in convergence , one may carefully monitor the mixing of the initial and final charges during the iterations and increase the number of @xmath53 points . thus , for a better resolution to determine the change in the total energy with respect to the input magnetization , the @xmath54- mesh had @xmath55 @xmath53 points in the irreducible wedge of the cubic brillouin zone . by the fixed - spin - moment method the difference @xmath56 ( = @xmath57 ) for given values of @xmath34 is calculated . the calculated @xmath56 is fitted to the phenomenological landau equation of phase transition which is given as @xmath58 for @xmath59 , where the sign of the coefficient @xmath60 for @xmath61 determines the nature of the magnetic ground state , i.e. , @xmath62>0 refers to a paramagnetic ground state while @xmath63 refers to a ferromagnetic phase . we have applied the approach described above to the study of carbon vacancy in @xmath5 @xcite and @xmath13 transition - metal-@xmath5 alloys @xcite . both x - ray and neutron diffraction techniques unambiguously report @xmath5 and @xmath4 as cubic perovskites with their lattice constants determined as @xmath64 and @xmath65 a.u . , respectively . assuming an underlying rigid cubic lattice , with @xmath6(@xmath7 ) at cube corners , @xmath12 at the faces and @xmath2 at the octahedral interstitial site , the total energy minimization were carried out to determine the equation of state parameters . the total energies calculated , self - consistently , for six lattice constants close to equilibrium were fed as input to a third - order birch - murnaghan equation of state @xcite . note that the birch - murnaghan equation is derived from the theory of finite strain , by considering an elastic isotropic medium under isothermal compression , with the assumption that the pressure - volume relation remains linear . hence , in the optimization procedure we have restricted the choice close to the equilibrium . .[tab - zncni3mg - eos]comparison of the equation of state parameters of cubic perovskite @xmath4 with that of @xmath66 using the kkr - asa method as described in the text . [ cols="^,^,^,^,^ " , ] the reported values of @xmath10 for @xmath5 are at variance with the existing reports @xcite . it appears that the value is sensitive to the basic approximations made in each type of the electronic structure method , and also to the parameters like that of the choice of wigner - seitz radii , choice of the exchange - correlation potential and others . however , under similar approximations , it is clear that for @xmath4 the @xmath10 reduces by @xmath67% in comparison with @xmath5 . this is consistent with the earlier first - principles fp - lapw calculations @xcite . the reduction in @xmath10 may be largely due to the smaller lattice constant of @xmath4 , in comparison with @xmath5 . the change in the density of states , as well as in the @xmath10 as a function of lattice constant is shown in fig.[pressure - dos ] . approximating the variation of @xmath10 to be linear with respect to the lattice constant , we find @xmath68/@xmath69 to be @xmath70 and @xmath71 st / ry atom / a.u for @xmath5 and @xmath4 , respectively . to understand the changes in the electronic structure upon the introduction of @xmath2 vacancies , we in figs.[cxtotdos ] , [ c2pdos ] , [ ni3ddos ] and [ c2pni3ddos ] show the changes in the total and sub - lattice resolved @xmath2 @xmath14 and @xmath12 @xmath13 partial densities of states of @xmath0 and @xmath1 alloys calculated at their equilibrium lattice constants . it follows from the figures that the change in the distribution of states is more or less insignificant near the fermi energy , but states lower in energy undergo substantial changes . upon creation of vacancies , a few of the @xmath2 2@xmath72 - @xmath12 @xmath13 bonds break , and result in charge redistribution . note that the @xmath73 octahedral is a covalently built complex to which the cations at the cube corners ( @xmath7 and @xmath6 ) are thought to have donated their outermost valence electrons . the crystal geometry suggests six @xmath12 atoms as the first nearest neighbors to @xmath2 and eight @xmath6/@xmath7 atoms as its second nearest neighbors . for @xmath12 the second nearest coordination shell carries four @xmath6/@xmath7 atoms . the charge redistribution arising due to the breaking of the @xmath72-@xmath74 bonds would be proportional to the electro - positivity of the cation- @xmath6/@xmath7 . since @xmath6 is more electro - positive than @xmath7 , charge redistribution to the @xmath6/@xmath7 sub - lattices , as a function of vacancies would be more significant in @xmath5 when compared to @xmath4 . this is consistent with the fact that a larger fraction of the charge would be transferred back to the @xmath6 sub - lattice , in @xmath5 in comparison with that of the @xmath12 sub - lattice . the change in the @xmath10 as a function of lattice constant in @xmath75 alloys is shown in fig.[znxmg - latdos ] . one may find that @xmath10 decreases for all values of @xmath76 , with respect to lattice constant . however , @xmath10 as a function of @xmath76 , at the equilibrium lattice constant , was found to deviate a little , as is evident from fig.[znxmg - latdos ] . this clearly suggests that the electronic structure properties are mainly governed by the @xmath73 octahedra . the atoms occupying the cube corners i.e. , @xmath6 and @xmath7 , however , play a non - trivial role in determining the structural properties . the hopfield parameter @xmath24 has been regarded as a local `` chemical '' property of an atom in a crystal . it has been emphasized earlier that the most significant single parameter in understanding the @xmath77 of a conventional superconductor is the hopfield parameter @xcite . for strong - coupling systems , the variation in @xmath24 is more important than the variation of @xmath35 in changing @xmath77 . softening @xmath35 often does enhance @xmath77 , but a significant change in the magnitude of @xmath77 depends largely on a significant change in the @xmath24 value rather than a small change in the corresponding @xmath35 . as a matter of fact , we look for the changes in the @xmath24 from the three sub - lattices of these perovskites as a function of lattice constant as well as @xmath78 in @xmath1 and @xmath0 alloys . note that for @xmath5 , it has been reported that the superconducting transition temperature @xmath77 increases upon application of external pressure @xcite . besides , experiments remain controversial on the strength of the electron - phonon interaction in @xmath5 @xcite . it has been suggested that @xmath5 may be a strongly - coupled superconductor , however , the magnitude of @xmath77 being marginally reduced due to the paramagnon interactions @xcite . in fig.[hopeta ] we show the changes in the @xmath24 of @xmath5 and @xmath4 as a function of lattice constant . it is clear from fig.[hopeta ] that the @xmath79 and @xmath80 linearly increase as a function of decreasing volume in either alloys . if the change in the average phonon frequency remains small , then either of these alloys could enhance the transition temperature with respect to external pressure . for @xmath5 this view is consistent with the previous experimental results . similar characteristic feature holds for the vacancy - rich disordered alloys , the variation of which is shown in fig.[mgcxeta00 ] and [ zncxeta00 ] to have an understanding in the variation of @xmath79 , @xmath81 and @xmath80 where @xmath81 can be considered as the local chemical property of the electrons in the empty sphere , we show in fig.[cxeta ] the change in these parameters as a function of @xmath78 in both @xmath1 and @xmath0 alloys . one may find that the variation of @xmath24 remains similar for both the alloys as a function of decreasing @xmath2 content . total energies from both the self - consistent , spin polarized and spin unpolarized calculations remain degenerate for @xmath5 and @xmath4 alloys at their equilibrium lattice constants . this unambiguously shows that the materials are non - magnetic in nature . however , having suggested that @xmath5 is on the verge of a ferromagnetic instability @xcite , and also that incipient magnetism in the form of spin - fluctuations reside in the material , we attempt to compare the magnetic properties of @xmath5 and @xmath4 alloys using the fixed - spin moment approach of alloy theory @xcite . numerical calculations of magnetic energy @xmath56 for @xmath5 and @xmath4 are carried out at over a range of lattice constants . the calculated results of @xmath56 in the fixed - spin - moment method are shown in fig.[fsm - de ] . the calculated @xmath56 curves are fit to the form of a power series of @xmath82 up to @xmath59 , for the polynomial as mentioned above . the variations of the coefficients , @xmath62 in units of @xmath83 , @xmath84 in @xmath85 , and @xmath86 in @xmath87 as a function of lattice constant are shown in fig.[fsm - coeff ] . the propensity of magnetism can be inferred from the sign of the coefficient which is quadratic in @xmath34 , i.e. , @xmath62 . the coefficient @xmath62 is the measure of the curvature and is positive definite when the total energy minimum is at @xmath88 , thus referring to a paramagnetic ground state . in general , when @xmath62 becomes negative , it infers that there would exist a minimum in the @xmath89 curve at a value other than @xmath88 referring to a ferromagnetic phase at that value of @xmath34 . the higher - order coefficients @xmath84 and @xmath86 however are significant and they control the variation of @xmath90 with respect to @xmath34 . for example , for larger values of @xmath34 , @xmath84 and successively @xmath86 would dominate , and if @xmath84(@xmath86 ) tends to be negative it would show a dip in the @xmath89 variation pointing towards a magnetic transition at a higher value of @xmath34 . this , in the first - principles characterization of the magnetic properties of a material would refer to a possibility of a metastable phase at relatively large values of external magnetic fields . however , it has to be noted that calculations for large values of @xmath34 can result in ambiguous results . hence , it is suggested to carry out calculations for smaller values of @xmath34 and use the above mentioned polynomial function up to the minimum order , where the curve fits with sufficient accuracy . [ fsm - coeff ] shows that for smaller values of lattice constant , the alloys show an enhanced paramagnetic character . one may also note that the variation in @xmath84 and @xmath86 coefficients are oppositely complimented and hence in the renormalized approach to include corrections due to spin - fluctuations , as suggested by yamada and terao @xcite , they would cancel out in proportion preserving the trend in the variation of @xmath62 . thus , it becomes likely that the incipient magnetic properties associated with @xmath5 and @xmath4 would decrease as a function of decreasing lattice constant . first - principles syudy of the electronic properties of @xmath5 and @xmath4 , and also their non - stoichiometric alloys are carried out using the density - functional - based kkr - asa method . we find that the lattice constant for @xmath4 is overestimated , while for @xmath5 it is underestimated . this suggests that the material @xmath4 subjected to experiments may be non - stoichiometric . as a function of decreasing @xmath2 content in @xmath1 and @xmath0 alloys , one finds an opposite trend in the variation of pressure derivative of the bulk modulus , which is proportional to the averaged phonon frequency . with electronic structure remaining essentially the same for @xmath1 and @xmath0 , the results hint that non - stoichiometry may have opposite effects . note that for @xmath91 , @xmath1 alloys are feebly superconducting , while according to the conjecture that has been made @xmath0 is not . it can thus be inferred that the associated phonon modes in @xmath4 and its disordered alloys may be characteristically different when compared to the @xmath5 counterparts . a comparison of the phonon spectra of these alloys thus become quite necessary to understand the absence of superconductivity in @xmath4 , although it is iso - structural and iso - valent with @xmath5 . 1 t. he , k.a . regan , m.a . hayward , a.p . ramirez , y. wang , p. khalifah , t. he , j.s . slusky , n. rogado , k. inumaru , m.k . haas , h.w . zandbergen , n.p . ong , and r.j . cava , nature , 411 , 54 ( 2001 ) m.- s. park , j. giim , s. h. park , h. c. ri , s. i. lee , and e. j. choi , supercond . 17 , 274 ( 2004 ) . m. d. johannes and w. e. pickett , phys . rev . b 70 , 060507 ( 2004 ) d. j. singh and i. i. mazin , phys . rev . b 64 , 140507 ( 2001 ) h. rosner , r. weht , m. d. johannes , w. e. pickett , and e. tosatti , * * * * phys . 88 , 027001 ( 2002 ) s. b. dugdale and t. jarlborg , * * * * phys . rev . b 64 , 100508 ( 2001 ) szajek a , j. phys . : matter 13 , l595 ( 2001 ) a. ren , g. c. che , s. l. jia , h. chen , y. m. ni , g. d. liu and z. x. zhao , * * * * physica c 371 , 1 ( 2002 ) j. q. li , l. j. wu , l. li , and y. zhu , phys . rev . b 65 , 052506 ( 2002 ) j. h. kim , j. s. ahn , j. kim , m .- s . park , s. i. lee , e. j. choi , and s .- j . oh , phys . b 66 , 172507 ( 2002 ) g. kinoda , m. nishiyama , y. zhao , m. murakami , n. koshizuka and t. hasegawa , japan . j. appl 40 l1365 , ( 2001 ) l. shan , k. xia , z. y. liu , h. h. wen , z. a. ren , g. c. che and z. x. zhao , * * * * phys . rev . b 68 024523 ( 2003 ) p. j. t. joseph and p. p. singh , phys . b 72 , 064519 ( 2005 ) p. j. t. joseph and p. p. singh , phys . rev . b 72 , 214206 ( 2005 ) h. h. stadelmaier and l. j. huetter , z. metallk 50 , 199 ( 1959 ) h. h. stadelmaier and l. j. huetter , acta metall . , 7 , 415 ( 1959 ) h. h. stadelmaier and f. h. hammad , metall 15 , 124 ( 1961 ) t.g . amos , q. huang , j.w . lynn , t. he , r.j . cava , solid state commun 121 , 73 ( 2002 ) h. h. stadelmaier and t. s. yun , z. metallk 52 , 477 ( 1961 ) j. b. goodenough and j. m. longo , landolt- bornstein series , vol iiia j. korringa , physica 13 , 392 ( 1947 ) w. kohn and n. rostoker , phys . rev 94 , 1111 ( 1954 ) i. turek , v. drchal , j. kudrnovsky , m. sob and p. weinberger , electronic structure of disordered alloys , surfaces andinterfaces , kluwer academic publishers , 1997 . p. soven phys . 156 , 809 ( 1967 ) k schwarz and p mohn , j. phys . 14 , l129 ( 1984 ) statistical physics , l. d. landau and e. m. lifshitz , pergamom press london ( 1962 ) h. l. skriver and i. mertig , phys b 32 , 4431 ( 1985 ) n. e. christensen and s. satpathy , phys . 55 , 600 ( 1985 ) . a. v. ruban and h. l. skriver phys . b 66 , 024201 ( 2002 ) a. v. ruban , s. i. simak , p. a. korzhavyi , and h. l. skriver phys . b 66 , 024202 ( 2002 ) a. v. ruban and h. l. skriver computational materials science , 15 , 119 ( 1999 ) j. p. perdew and y. wang , phys . rev . b 45 , 13244 ( 1992 ) . j. p. perdew , k. burke , and y. wang , phys . rev . b 54 , 16533 ( 1996 ) hans . l. skriver , the lmto method , muffin tin orbitals and electronic structure , springer- verlag ( 1984 ) superconductivity in @xmath92 and @xmath93 band metals , edited by d. h. douglass , plenum press newyork ( 1976 ) g. d. gaspari and b. l. gyorffy , phys . 28 , 801 ( 1972 ) r. hayn and v. drchal phys . rev . b 58 , * * * * 4341 ( 1998 ) f. birch , j. geophys . 57 , 227 ( 1952 ) f. d. murnaghan , wiley , new york , 140 pp . 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first - principles , density - functional - based electronic structure calculations are employed to study the changes in the electronic properties of @xmath0 and @xmath1 using the korringa - kohn - rostoker coherent - potential approximation method in the atomic sphere approximation ( kkr - asa cpa ) . as a function of decreasing @xmath2 @xmath3%
, we find a steady decrease in the lattice constant and bulk modulus in either alloys .
however , the pressure derivative of the bulk modulus displays an opposite trend .
following the debye model , which relates the pressure derivative of the bulk modulus with the average phonon frequency of the crystal , it can thus be argued that @xmath4 and its disordered alloys posses a different phonon spectra in comparison to its @xmath5 counterparts .
this is further justified by the marked similarity we find in the electronic structure properties such as the variation in the density of states and the hopfield parameters calculated for these alloys .
the effects on the equation of state parameters and the density of states at the fermi energy , for partial replacement of @xmath6 by @xmath7 are also discussed .
| 9,892 | 302 |
owing to typical astronomical timescales , a galaxy s spiral arms are often considered as a fixed pattern . so , too , for the numerous , tightly wound spiral waves detected in saturn s rings . in fact , both systems are dynamically active , with waves traveling away from resonant sites . this is manifest only in saturn s case , where a pair of moons janus and epimetheus occupy nearly identical orbits that are interchanged every 4 years , causing the resonance locations in the rings to skip back and forth by tens of km . since spiral density waves are initiated in saturn s rings at locations where ring particle orbits are in a lindblad resonance with a perturbing moon , the starting points of waves jump as well , allowing wave trains to interfere in complex ways @xcite . high - resolution images of the rings were obtained by the cassini spacecraft s imaging science subsystem ( iss ) on 2004 july 1 and on 2005 may 20/21 . the calibration and image processing of these data , resulting in a series of brightness scans with orbital radius , along with a catalog of important satellite gravitational resonances falling within the rings , are presented by ( * ? ? ? * hereafter ) . further analysis of these data , employing techniques derived from the wavelet transform , is presented by ( * ? ? ? * hereafter ) . examination of the cassini images to date shows that density waves raised by the co - orbitals are both unusual and variable in their morphology . in this paper , we describe a model that accounts for much of the observed structure . we proceed to use this model to predict the future morphology of selected waves at the times and locations of planned cassini observations , which we expect will test our predictions . , width=302 ] the orbits of janus and epimetheus about saturn constitute a form of the three - body problem of celestial mechanics that is unique in the solar system ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? when viewed in a rotating frame of reference , whose angular velocity equals the mass - weighted average of the mean motions of the two moons , epimetheus executes a modified `` horseshoe orbit '' encompassing janus slowly drifting @xmath0 , @xmath1 , and @xmath2 lagrange points ( fig . [ janepiorb ] ) . however , since epimetheus mass is not negligible compared to janus the mass ratio is 0.278 @xcite janus executes its own libration about the average orbit . because the orbits of the two moons are so similar ( @xmath3 km ) , they are commonly known as the `` co - orbital satellites '' ( or , more briefly , the `` co - orbitals '' ) . every 4.00 years , they execute their mutual closest approach and effectively `` trade '' orbits , the inner moon moving outward and vice versa . the most recent reversal event occurred on 2006 january 21 , at which time janus became the inner satellite and epimetheus the outer . spiral density waves are raised in the rings at locations where ring - particle orbits are in a lindblad resonance with a perturbing saturnian moon . the density perturbations propagate outward from the resonance location . as reviewed in ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , five parameters characterize the idealized functional form of the perturbation : 1 ) the background surface density @xmath4 , which fixes the wavelength dispersion , 2 ) the resonance location @xmath5 , specifying a linear translation of the wave , 3 ) the wave s initial phase @xmath6 , 4 ) the damping parameter @xmath7 , indicating a characteristic distance over which the wave propagates before damping away , and 5 ) the wave s amplitude @xmath8 , which is proportional to the perturbing moon s mass . considering only co - planar motions , the pattern speed of the resonant perturbation is described by positive integers @xmath9 and @xmath10 + 1 ; the first giving the number of spiral arms , and the second the order of the resonance ( first - order being strongest ) . a ( @xmath10 + 1)th - order lindblad resonance is generally labeled as @xmath11:@xmath12 . at a given ring longitude @xmath13 , the initial phase of a particular density wave is @xmath14 where @xmath15 and @xmath16 are the perturbing moon s mean longitude and longitude of periapse , respectively . although previous authors have usually analyzed spiral density waves in saturn s rings as static phenomena , in fact they propagate with a finite group velocity @xcite @xmath17 where @xmath18 is the radial ( epicyclic ) frequency of ring particle orbits , and @xmath19 is newton s gravitational constant . this is the speed at which information ( e.g. , effects of any change in the resonant forcing ) propagates . in the a ring , @xmath20 is on the order of 10 - 20 km / yr . since spiral density waves commonly propagate over many tens of km , and the forcing from the co - orbital satellites changes every 4 yr , discontinuities resulting from reversal events should be observable in density waves raised by the co - orbitals . in the radial scans of ring brightness taken from cassini iss images ( ) , spiral density waves due to the co - orbitals can be discerned at first - order , second - order , and third - order resonances . first - order waves ( e.g. , 2:1 , 4:3 , 5:4 , 6:5 ) were seen by voyager , and are some of the strongest waves in the ring system . however , non - linear effects , which occur when the density perturbations are comparable to the background surface density , greatly complicate their analysis @xcite . not only does the wavelength dispersion deviate significantly from linear theory , but simple superposition of multiple wave segments is not valid . second - order waves ( e.g. , 7:5 , 9:7 , 11:9 , 13:11 ) were first clearly resolved in the present cassini data set ( fig . [ jewaveimages ] ) . since the density perturbations comprising these weaker waves are much smaller than the background surface density , they remain well - described by linear theory , and overlapping wave segments can be simply superposed . accordingly , second - order waves are best - suited for comparison with our simple model , and we will focus exclusively on them . additionally , several third - order co - orbital waves can be discerned through wavelet analysis of the radial scans ( ) ; however , these are too weak for much detailed structure to be resolved . our model is based on a simple assumption : when the co - orbital satellites are in a given configuration ( we ll call the janus - inward configuration `` je '' and the janus - outward configuration `` ej '' ) , spiral density waves propagate outward from the current lindblad resonance locations at the group velocity @xmath20 . when a reversal occurs ( e.g. , the satellites changing configurations from je to ej ) , the je resonance locations become inactive ; however , the waves previously generated there continue to propagate outwards , resulting in a `` headless '' wave . meanwhile , new `` tailless '' spiral density waves begin to propagate outward at a speed @xmath20 from the ej lindblad resonance locations . , and from ej to je at @xmath21 yr ) . parameters are based on the 9:7 density wave , but here @xmath22 for all wave segments . in the right - hand column are wave segments created by janus in its inner ( purple ) and outer ( red ) configurations , and epimetheus in its inner ( blue ) and outer ( green ) configurations . only amplitudes larger than 1% are shown . dotted lines show the resonance locations , with the same color coding . the left - hand column shows the sum of the wave segments , which is the model prediction . [ jemodeldemo],width=604 ] at any given time , the locations of `` headless '' and `` tailless '' wave segments can be easily calculated from @xmath20 and the elapsed interval since a reversal event . we simply superpose these wave segments atop one another to arrive at a predicted wave morphology ( fig . [ jemodeldemo ] ) . this time - domain approach to the problem is an alternative to the frequency - domain treatment of @xcite . we set the relative amplitude to unity for janus , and to the moons mass ratio of 0.278 for epimetheus . the relevant resonance locations are easily derived from the moons orbital parameters @xcite . the absolute navigation of iss images is not simple ( see ) ; nonetheless , all that is necessary for wave morphology is to know the separations among the resonance locations . the phase of each wave segment ( eq . [ dwphase ] ) is determined by what the orbital parameters of the perturbing moon _ would have been _ had no reversal taken place . this is calculated using high - precision numerically - integrated orbits @xcite . we make a linear fit to @xmath15 and to @xmath16 over a single inter - reversal time period , then extrapolate forward to the observation time . at the end of this process , the only remaining free parameters in our model are the background surface density @xmath4 and the damping parameter @xmath7 , which respectively control the wavelength dispersion and the propagation distance . we manually adjust these parameters to find the optimum agreement of feature locations between model and data . for locations at which a wave segment begins or ends , an unrealistically sharp cutoff can occur . we soften such discontinuities by using a half - gaussian , with a characteristic width of 1 km , to bring the perturbation back to zero . , and damping parameter @xmath7 are given in each figure . radial traces were taken from the following images : a ) n1467345385 , b ) n1495327885 , c ) n1467345916 , d ) n1495326975 , e ) n1495326431 . image resolutions are 250 m / pixel ( a , c ) and 1.4 km / pixel ( b , d , e ) . model resolution has been degraded to correspond to image resolution . [ jemodelresults],width=604 ] our model results are shown , along with corresponding cassini image traces , in fig . [ jemodelresults ] . we find good qualitative agreement between the two , especially in the `` upstream '' ( left - hand ) regions . there is also good agreement between model values of @xmath4 and @xmath7 and values for nearby waves in the a ring ( 9:7 , 11:9 ) and cassini division ( 7:5 ) ( ) . we assume that image n1467345916 ( fig . [ jemodelresults]c ) is in the reverse - contrast regime ( see ) . the most glaring failure of the model occurs in regions where the predicted perturbation is zero ( figs . [ jemodelresults]c and [ jemodelresults]d ) . in such regions , the data instead show an oscillatory mode for which we can not account . it is possible that such oscillations could be raised in the several months during which the resonance locations are migrating from one configuration to the other , which we neglect in treating the reversals as instantaneous events . another potentially interesting explanation for such oscillations is that a leading spiral density wave may be propagating back towards the resonance location @xcite . this mode is allowed by the mathematical formalism , but has never been observed and has been considered impossible to excite ; however , it is conceivable that a `` headless '' wave could send such a mode into an otherwise undisturbed region . , width=604 ] predictions of our model for the times and locations of cassini iss observations designed to image the co - orbital waves @xcite are shown in fig . [ jemodelpredict ] . owing to the spacecraft s low orbital inclination so far this year , the main rings have not been observed at high resolution since the last reversal event ( from ej to je ) on 2006 january 21 , so the wave patterns in the new configuration have yet to be observed . we predict that the innermost wavecycles ( due to epimetheus in the ej configuration ) will become headless and shift outwards , while a blank gap ( possibly containing oscillations as in figs . [ jemodelresults]c and [ jemodelresults]d ) will appear between the new inner janus perturbation and the headless perturbation from the outer resonance location . the next reversal event , expected on 2010 january 21 , will be observable only if the cassini mission is extended beyond the current plan . saturn s rings are a nearby and accessible natural laboratory @xcite , in which we can observe phenomena of broader interest , such as how a disk responds to variable forcing . our model is a first step towards a new understanding of the observed complex and time - variable waveforms . we will revisit this topic in more detail , after the accuracy of our predictions ( at least for data that will be obtained this year ) becomes apparent . , c. c. , west , r. a. , squyres , s. , mcewen , a. , thomas , p. , murray , c. d. , delgenio , a. , ingersoll , a. p. , johnson , t. v. , neukum , g. , veverka , j. , dones , l. , brahic , a. , burns , j. a. , haemmerle , v. , knowles , b. , dawson , d. , roatsch , t. , beurle , k. , & owen , w. 2004 , space sci . rev . , 115 , 363
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we describe a model that accounts for the complex morphology of spiral density waves raised in saturn s rings by the co - orbital satellites , janus and epimetheus .
our model may be corroborated by future cassini observations of these time - variable wave patterns .
| 3,851 | 72 |
( anti-)strangeness enhancement was first observed at cern - sps energies by comparing anti - hyperons , multi - strange baryons , and kaons to @xmath10-data . it was considered a signature for quark gluon plasma ( qgp ) because , using binary strangeness production and exchange reactions , chemical equilibrium could not be reached within a standard hadron gas phase , i.e. , the chemical equilibration time was on the order of @xmath11 whereas the lifetime of a fireball in the hadronic stages is only @xmath12 @xcite . it was then proposed that there exists a strong hint for qgp at sps because strange quarks can be produced more abundantly by gluon fusion , which would account for strangeness enhancement following hadronization and rescattering of strange quarks . later , however , multi - mesonic reactions were used to explain secondary production of @xmath13 and anti - hyperons @xcite . at sps they give a chemical equilibration time @xmath14 using an annihilation cross section of @xmath15 and a baryon density of @xmath16 , which is typical for evolving strongly interacting matter at sps before chemical freeze - out . therefore , the time scale is short enough to account for chemical equilibration within a cooling hadronic fireball at sps . a problem arises when the same multi - mesonic reactions were employed in the hadron gas phase at rhic temperatures where experiments again show that the particle abundances reach chemical equilibration close to the phase transition @xcite . at rhic at @xmath17 mev , where @xmath18 and @xmath19 , the equilibrium rate for ( anti-)baryon production is @xmath20 . moreover , @xmath20 was also obtained in ref . @xcite using a fluctuation - dissipation theorem . from hadron cascades a significant deviation was found from the chemically saturated strange ( anti-)baryons yields in the @xmath21 most central au - au collisions @xcite . these discrepancies suggest that hadrons are born " into equilibrium , i.e. , the system is already in a chemically frozen out state at the end of the phase transition @xcite . in order to circumvent such long time scales it was suggested that near @xmath22 there exists an extra large particle density overpopulated with pions and kaons , which drive the baryons / anti - baryons into equilibrium @xcite . but it is not clear how this overpopulation should appear , and how the subsequent population of ( anti-)baryons would follow . moreover , the overpopulated ( anti-)baryons do not later disappear @xcite . therefore , it was conjectured that hagedorn resonances ( heavy resonances near @xmath23 with an exponential mass spectrum ) could account for the extra ( anti-)baryons @xcite . hadrons can develop according to @xmath24 where @xmath25 can be substituted with @xmath0 , @xmath1 , @xmath2 , or @xmath3 . ( [ eqn : decay ] ) provides an efficient method for producing of @xmath25 pairs because of the large decay widths of the hagedorn states . in eq . ( [ eqn : decay ] ) , @xmath26 is the number of pions for the decay @xmath27 and @xmath28 is the number of pions that a hagedorn state will decay into when a @xmath25 is present . since hagedorn resonances are highly unstable , the phase space for multi - particle decays drastically increases when the mass increases . therefore , the resonances catalyze rapid equilibration of @xmath25 near @xmath23 and die out moderately below @xmath29 @xcite . unlike in pure glue @xmath30 gauge theory where the polyakov loop is the order parameter for the deconfinement transition ( which is weakly first - order ) , the rapid crossover seen on lattice calculations involving dynamical fermions indicates that there is not a well defined order parameter that can distinguish the confined phase from the deconfined phase . because of this it is natural to look for a hadronic mechanism for quick chemical equilibration near the phase transition . one such possibility could be the inclusion of hagedorn states . recently , hagedorn states have been shown to contribute to the physical description of a hadron gas close to @xmath29 . the inclusion of hagedorn states leads to a low @xmath31 in the hadron gas phase @xcite , which nears the string theory bound @xmath32 @xcite . calculations of the trace anomaly including hagedorn states also fits recent lattice results well and correctly describe the minimum of the speed of sound squared , @xmath33 near the phase transition found on the lattice @xcite . estimates for the bulk viscosity including hagedorn states in the hadron gas phase indicate that the bulk viscosity , @xmath34 , increases near @xmath29 , which agrees with the general analysis done in @xcite . furthermore , it has been shown @xcite that hagedorn states provide a better fit within a thermal model to the hadron yield particle ratios . additionally , hagedorn states provide a mechanism to relate @xmath29 and @xmath35 , which then leads to the suggestion that a lower critical temperature could possibly be preferred , according to the thermal fits @xcite . previously , in ref . @xcite we presented analytical results , which we will derive in detail here . moreover , we saw that both the baryons and kaons equilibrated quickly within an expanding fireball . the initial saturation of pions , hagedorn states , baryons , and kaons played no significant role in the ratios such as @xmath7 and @xmath36 . here we consider the effects of various initial conditions on the chemical freeze - out temperature and we find that while they play a small role on the total particle number , they still reproduce fast chemical equilibration times . additionally , we assume lattice values of the critical temperatures ( @xmath4 mev @xcite and @xmath5 mev @xcite ) and find that chemical equilibrium abundances are still reached close to the temperature given by thermal fits ( @xmath37 mev ) . this paper is structured in the following manner . in section [ sec : model ] we discuss the details of our statistical model that calculates the chemical equilibrium values of the hagedorn states and other hadrons . furthermore in this section , fits are shown to thermodynamical properties calculated in lattice qcd , which are used to determine the mass spectrum of the hagedorn states and the rate equations are discussed in detail . in section [ tau ] we are able to extract the chemical equilibration time of an @xmath25 pair when the pions and hagedorn states are held constant . in section [ ar ] we derive an analytical result of the rate equations when we consider only the decay @xmath38 . we then discuss the case of an expanding fireball and the results for the various @xmath25 pair production in section [ expansion ] . the production of @xmath39 particles will also be considered in section [ omega ] . we summarized and discussed our results in section [ conclusions ] . in appendix [ app ] we present some analytical and numerical results for the various equilibration stages in the hadron and hagedorn states gas mixture . hagedorn resonances have an exponentially growing mass spectrum @xcite . their large masses open up the phase space for multi - particle decays . recent analysis involving hagedorn states is given in @xcite . moreover , thoughts on observing hagedorn states in experiments are given in @xcite and their usage as a thermostat in @xcite . hagedorn states can also explain the phase transition _ above _ the critical temperature and , depending on the intrinsic parameters , the order of the phase transition @xcite . for the following discussion , the overall density of hagedorn states in our extended hagedorn gas model are straightforwardly described by , @xmath40^{\frac{5}{4}}}e^{\frac{m}{t_{h}}}dm.\ ] ] where @xmath41 gev and @xmath42 gev . we note that in this work we consider only mesonic hagedorn states with no net strangeness . the exponential in eq . ( [ eqn : fitrho ] ) arises from hagedorn s original idea that there is an exponentially growing mass spectrum . thus , as @xmath43 is approached , hagedorn states become increasingly more relevant and heavier resonances appear " . the factor in front of the exponential has various forms @xcite . while the choice in this factor can vary , it was found in @xcite that the present form gives lower values of @xmath43 , which more closely match the predicted lattice critical temperature @xcite . returning to eq . ( [ eqn : fitrho ] ) , its parameters ( a , m , and @xmath43 ) are dependent on the critical temperature . we assume that @xmath44 , and then we consider the two different different lattice results for @xmath29 : @xmath5 mev @xcite , which uses an almost physical pion mass , and @xmath4 mev @xcite . furthermore , we need to take into account the repulsive interactions and , therefore , we use the following volume corrections ( as also seen in @xcite ) : @xmath45 which ensure that the our model is thermodynamically consistent . note that @xmath46 is a free parameter that is based upon the idea of the mit bag constant . in order to find the maximum hagedorn state mass @xmath47 and the degeneracy " a , we fit our model to the thermodynamic properties of the lattice . in the rbc - bielefeld collaboration the thermodynamical properties are derived from the quantity @xmath48 , the so called interaction measure , which is what we fit in order to obtain the parameters for the hagedorn states . thus , we obtain @xmath49 mev , @xmath50 , @xmath51 gev , and @xmath52 . the fit for the trace anomaly @xmath53 is shown in fig . [ fig : et4 ] . we also show the fit for the entropy density in fig . [ fig : st4 ] . both fits are within the error of lattice and mimic the behavior of the lattice results . as discussed in @xcite , a hadron resonance gas model with hagedorn states uniquely fits the lattice data whereas a hadron resonance gas without hagedorn states completely misses the behavior . mev . ] mev . ] bmw calculates the thermodynamical properties separately and , therefore , we fit only the energy density as shown in fig . [ fig : et4_173 ] . from that we obtain @xmath54 mev , @xmath55 , @xmath51 gev , and @xmath56 . we also show a comparison to the entropy density in fig . [ fig : st4_173 ] our results with the inclusion of hagedorn states are able to match lattice data near the critical temperature but do not match as well at lower temperatures in fig . [ fig : et4 ] and fig . [ fig : st4 ] . mev . ] mev . ] our idea is that these very massive hagedorn states exist , as pictured in fig . [ fig : bag ] , and are so large that they decay almost immediately into multiple pions and @xmath25 pairs . while it can be argued that hagedorn states are more likely to decay into a pair of particles : a lighter hagedorn state and another particle , these reactions are so quick that we can consider the end results , which would be multiple particles ( mostly pions ) . that being said , it would be possible to put hagedorn states into a transport approach such as urqmd @xcite using binary reactions with possible cross - sections as described in @xcite . we leave this as a challenge for the future . moreover , we need to consider the back reactions of multiple particles combining to form a hagedorn state in order to preserve detailed balance . rate equations provide us with a perfect tool for this because there is a loss and gain term that describe both the forward and back reactions . moreover , the state of chemical equilibrium is a fixed point of the rate equations . pair . ] the rate equations for the hagedorn resonances @xmath57 , pions @xmath58 , and the @xmath25 pair @xmath59 , respectively , are given by @xmath60\nonumber\\ & + & \gamma_{i , x\bar{x}}\left [ n_{i}^{eq } \left(\frac{n_{\pi}}{n_{\pi}^{eq}}\right)^{\langle n_{i , x}\rangle } \left(\frac{n_{x\bar{x}}}{n_{x\bar{x}}^{eq}}\right)^2 -n_{i}\right]\nonumber\\ \dot{n}_{\pi } & = & \sum_{i } \gamma_{i,\pi } \left[n_{i}\langle n_{i}\rangle - n_{i}^{eq}\sum_{n } b_{i , n}n\left(\frac{n_{\pi}}{n_{\pi}^{eq}}\right)^{n } \right]\nonumber\\ & + & \sum_{i } \gamma_{i , x\bar{x } } \langle n_{i , x}\rangle\left[n_{i}- n_{i}^{eq } \left(\frac{n_{\pi}}{n_{\pi}^{eq}}\right)^{\langle n_{i , x}\rangle } \left(\frac{n_{x\bar{x}}}{n_{x\bar{x}}^{eq}}\right)^2\right ] \nonumber\\ \dot{n}_{x\bar{x}}&=&\sum_{i}\gamma_{i , x\bar{x}}\left [ n_{i}- n_{i}^{eq}\left(\frac{n_{\pi}}{n_{\pi}^{eq}}\right)^{\langle n_{i , x}\rangle } \left(\frac{n_{x\bar{x}}}{n_{x\bar{x}}^{eq}}\right)^2\right].\end{aligned}\ ] ] the decay widths for the @xmath61 resonance are @xmath62 and @xmath63 , the branching ratio is @xmath64 ( see below ) , and the average number of pions that each resonance will decay into is @xmath65 . the equilibrium values @xmath66 are both temperature and chemical potential dependent . however , here we set @xmath67 . ( [ eqn : setpihsbb ] ) can also be rewritten in terms of fugacities ( @xmath68 , @xmath69 , and @xmath70 ) , which are found by dividing each total number by its respective equilibrium value , for example , @xmath71 ( as seen for the baryon anti - baryon pairs in @xcite ) . additionally , a discrete spectrum of hagedorn states is considered , which is separated into mass bins of 100 mev . each bin is described by its own rate equation . the branching ratios , @xmath64 , are the probability that the @xmath61 hagedorn state will decay into @xmath26 pions . since we are dealing with probabilities , then @xmath72 must always hold . in order to include a distribution for our branching ratios we assume that they follow a gaussian distribution for the reaction @xmath38 @xmath73 which has its peak centered at @xmath65 and the width of the distribution is @xmath74 . assuming a statistical , micro - canonical branching for the decay of hagedorn states , we can take a linear fit to the average number of pions in fig . 1 in ref . @xcite ( multiplying @xmath75 by three to include all pions ) to find @xmath76 such that @xmath77 is the average pion number that each hagedorn state decays into . within the microcanonical model a hagedorn state is defined by its mass and corresponding volume where the volume is taken as @xmath78 . the mean energy density of a hagedorn state is @xmath79 ( taken as @xmath80 ) . further discussions regarding this can be found in @xcite . the width of the distribution is @xmath81 . both our choice in @xmath82 and @xmath83 roughly match the canonical description in @xcite . furthermore , we have the condition that each hagedorn resonance must decay into at least 2 pions . because of the nature of a gaussian distribution there is a non - zero probability that a hagedorn state can decay into less than 2 pions . therefore , we calculate the percentage of the distribution that falls below 2 pions and redistribute that over @xmath84 so that @xmath72 . this in turn leads to a new @xmath65 and @xmath85 , which we find by calculating @xmath86 and @xmath87 . thus , we after normalize for the cutoff @xmath84 , we have @xmath88 and @xmath89 . for the average number of pions when a @xmath25 pair is present , we again refer to the micro - canonical model in @xcite . we use @xmath76 but then readjust it to the average pion number according to fig . 2 in ref . @xcite for when a baryon anti - baryon pair is present ( there the distribution is for a resonance of mass @xmath90 gev ) . thus , @xmath91 where @xmath92 is in gev . in this paper we do not consider a distribution but rather only the average number of pions when a @xmath25 pair is present . we assume that @xmath93 for when a kaon anti - kaon pair , @xmath2 , or @xmath3 pair is present . ideally , @xmath94 , @xmath95 , and @xmath96 should be derived separately and will be done in a future paper using a canonical model @xcite . we used a linear fit for the total decay width similar to that used in ref . the total decay width @xmath97 ( @xmath98 and @xmath92 in terms of gev ) , which ranges from @xmath99 mev , is a linear fit extrapolated from the data in ref . however , in eq . ( [ eqn : setpihsbb ] ) the total decay width is separated into two parts : one for the reactions @xmath38 , @xmath62 , and one for the reaction in eq . ( [ eqn : decay ] ) , @xmath63 , whereby @xmath100 . then relative decay width @xmath63 is the average number of @xmath25 in the system @xmath101 multiplied by the total decay width @xmath102 . essentially , a fraction of the decay of the @xmath61 hagedorn state goes into @xmath25 ( set by the number of @xmath25 the @xmath61 hagedorn state on average decays into ) and the remainder goes into pions . s and @xmath103 s . the @xmath103 s are calculated within our canonical ensemble and the @xmath104 s are calculated both in our canonical ensemble and a micro - canonical ensemble . ] we find @xmath105 by linearly fitting the proton in fig . 2 in ref . @xcite so that @xmath106 where @xmath92 is in gev and @xmath107 . thus , @xmath108 is between @xmath109 and @xmath110 mev . clearly , @xmath62 is then @xmath111 . analogously for the kaons , the decay width is @xmath112 where @xmath113 where @xmath92 is in gev , which is also taken from fig . 2 in ref . we find that @xmath114 to @xmath115 @xcite . thus , @xmath116 is between @xmath117 and @xmath118 mev . for @xmath104 we use a canonical model assuming that the baryon number @xmath119 , the strangeness @xmath120 , and the electrical charge @xmath121 in order to calculate the average lambda number . the results of this are shown in fig . [ fig : lamom ] . we find that our @xmath122 is lower than that from the micro - canonical ensemble in @xcite , which is also shown in fig . [ fig : lamom ] . this corresponds to a decay width of @xmath123 mev . furthermore , the average number of @xmath103 s is also shown in fig . [ fig : lamom ] from our canonical model again assuming that the baryon number @xmath119 , the strangeness @xmath120 , and the electrical charge @xmath121 . in fig . [ fig : lamom ] we multiple @xmath124 in order to better view the results . the resulting decay width is @xmath125 mev . the equilibrium values are found using a statistical model @xcite , which includes 104 particles from the the pdg @xcite ( only light and strange particles ) . as in @xcite , we also consider the effects of feeding ( the contributions of higher lying resonances such as the @xmath126 or @xmath127 resonances on the number of `` pions '' in our system , i.e. , @xmath128 includes all " the pions from resonances from the pdg @xcite ) . feeding is also considered for the protons , kaons , and lambdas . additionally , throughout this paper our initial conditions are the various fugacities at @xmath129 ( at the point of the phase transition into the hadron gas phase ) @xmath130 which are chosen by holding the contribution to the total entropy from the hagedorn states and pions constant i.e. @xmath131 and the corresponding initial condition configurations we choose later can be seen in tab . [ tab : ic ] . @xmath132 is the entropy density at the initial temperature , i.e. , the critical temperature multiplied by our choice in @xmath133 . because the hadron resonance is dominated by pions we can assume that @xmath133 represents the initial fraction of pions in equilibrium . @xmath134 represents the entropy contribution from the hagedorn states at @xmath29 multiplied by the initial fraction of hagedorn states in equilibrium . we hold @xmath133 constantly and then find the appropriate @xmath135 . the volume expansion , @xmath136 is discussed in detail following section entitled expanding fireball. as a starting point of our analysis , we first estimate the chemical equilibration time of the @xmath25 by looking at the fugacity of the @xmath25 rate equation , i.e. , eq . ( [ eqn : setpihsbb ] ) can be rewritten in terms of @xmath137 as shown for @xmath138 in eq . ( 3 ) in @xcite , when both the pions and hagedorn states are held constant . and @xmath139 where @xmath54 mev ( top ) and @xmath49 mev ( top ) . the gray band is the range of chemical equilibrium times for the hagedorn states ( see tab . [ tab : tau ] ) . , title="fig : " ] + and @xmath139 where @xmath54 mev ( top ) and @xmath49 mev ( top ) . the gray band is the range of chemical equilibrium times for the hagedorn states ( see tab . [ tab : tau ] ) . , title="fig : " ] the @xmath25 rate equation then becomes @xmath140 which we can integrate @xmath141\ ] ] where @xmath142 @xmath143 , and @xmath144 . substituting in @xmath145 and @xmath139 when the pions and hagedorn states are in chemical equilibrium , we rederive eq . ( 7 ) in ref . @xcite @xmath146 which is shown in fig . [ fig : taubb ] . from eq . ( [ eqn : ateqbb ] ) we see that the time scale has an indirect dependence on the decay width . since the decay width has a linear dependence on the mass , the time scale decreases when more hagedorn states are included . however , @xmath147 also decreases with increasing mass so above a certain point very many hagedorn states need to be included in order to see an effect in the time scale . furthermore , the chemical equilibrium values have a dependence on the temperature , which makes the time scale shortest for the highest temperatures . + + + + in fig . [ fig : hspireineq176 ] and fig . [ fig : hspireineq196 ] we hold the hagedorn states and pions and let the @xmath25 pairs reach chemical equilibrium . that means that in eq . ( [ eqn : setpihsbb ] ) we set @xmath148 and @xmath149 in the @xmath150 equation . fig . [ fig : hspireineq176 ] shows the results for @xmath0 , @xmath1 , and @xmath2 , respectively , for @xmath54 mev and fig . [ fig : hspireineq196 ] shows the same for @xmath49 mev . in all cases the temperature is held constant while the rate equations are solved over time . at @xmath151 all @xmath25 reach chemical equilibrium almost immediately ( on the order of @xmath152 ) . as t is decreased the chemical equilibrium time obviously increases , which is clear from fig . [ fig : taubb ] . even as the temperature is lowered we still see quick chemical equilibrium times . for the @xmath0 and @xmath2 pairs at @xmath153 the chemical equilibrium time is still about @xmath154 . the @xmath1 pairs do have a slower chemical equilibrium time due to their larger chemical equilibrium abundances , which is directly related to the chemical equilibration time through eq . ( [ eqn : setpihsbb ] ) . this again represents the main idea , which is the importance of potential hagedorn states in understanding fast chemical equilibration of hadrons close and below @xmath29 . the hagedorn states increase dramatically in number close to the critical temperature and , thus , by its subsequent decay and re - population they will quickly produce the various hadronic particles . the equilibration of @xmath25 pairs then shown in fig . [ fig : hspireineq176 ] and fig . [ fig : hspireineq196 ] where the analytical result in eq . ( [ eqn : lambba1 ] ) matches the numerical result exactly . from fig . [ fig : hspireineq176 ] and fig . [ fig : hspireineq196 ] it can be seen that all @xmath25 pairs equilibrate quickly close to the critical temperature @xmath155 . clearly , though , as the temperature decreases the chemical equilibration time lengthens . however , at @xmath49 mev chemical equilibrium is still reached quickly , @xmath155 . while the chemical equilibration time derived in the previous section is a good estimate , it can only be strictly applied when the pions and hagedorn states are assumed to stay in chemical equilibrium at a constant temperature ( fig . [ fig : hspireineq176 ] and fig . [ fig : hspireineq196 ] ) . otherwise , non - linear effects that appear when the pions and hagedorn states are allowed to equilibrate appear . to understand the dynamics in more detail , we consider the simplified case when the hagedorn resonances decay only into pions @xmath38 , which gives @xmath156 \nonumber\\ \dot{n}_{\pi}&=&\sum _ { i } \gamma_{i } \left[n_{i}\langle n_i\rangle- n_{i}^{eq } \sum _ { n=2 } b_{i , n}n\left(\frac{n_{\pi } } { n_{\pi}^{eq}}\right)^{n } \right].\end{aligned}\ ] ] assuming that the pions and the hagedorn states described in eq . ( [ eqn : setpihs ] ) are then allowed to equilibrate near @xmath22 in a static system , we are able to derive analytical solutions , the derivation of which is shown in detail in appendix [ app ] . for the analytical solutions we divide the chemical equilibration into three stages , the chemical equilibration times of which are shown in tab . [ tab : tau ] . the first stage ( described by @xmath157 in tab . [ tab : tau ] ) of the evolution is dominated by the chemical equilibration of the pions when the pions are still far away from their chemical equilibrium values . after the pions are close to chemical equilibrium , new dynamics take over , which are described by @xmath158 in tab . [ tab : tau ] and fig . [ fig : taupi ] . .chemical equilibration times from analytical estimates where qe is quasi - equilibrium at @xmath159 of each respective @xmath43 . [ cols="^,^,^,^ " , ] + + and @xmath160 . along the top axis of each graph the corresponding time is shown in @xmath161.,title="fig : " ] + and @xmath160 . along the top axis of each graph the corresponding time is shown in @xmath161.,title="fig : " ] more interestingly , we consider the case when the pions , hagedorn states , and @xmath25 all are allowed to chemical equilibrate . we then vary the initial conditions and observe their effects . the results for @xmath0 pairs are shown in fig . [ fig : pp176 ] and fig . [ fig : pp196 ] . in fig . [ fig : pp176 ] and fig . [ fig : pp196 ] we show the evolution of both the @xmath0 pairs and the pions for the reaction @xmath162 . note that in all the following figures the effective numbers are shown so that the contribution of the hagedorn states is included . one can see that the chemical equilibration time does depend slightly on our choice of @xmath163 , i.e. , a larger @xmath163 means a quicker chemical equilibration time . for instance , if the hagedorn states were overpopulated coming out of the qgp phase than chemical equilibrium times would be slightly shorter . however , even when the hagedorn resonances start underpopulated the @xmath0 pairs are able to reach chemical equilibrium immediately . additionally , when the @xmath0 pairs start at about half their chemical equilibrium values , it only helps the @xmath0 pairs to reach equilibrium at a slightly higher temperature ( on the order of a couple of mev ) . additionally , we see a greater dependence on @xmath163 for @xmath54 mev than for @xmath54 mev . throughout the evolution we see from the pions that they remain roughly in chemical equilibrium . thus , our initial analytical approximation appears reasonable . in fig . [ fig : pppi ] the ratio of protons s to @xmath164 s is shown . we also compare our results to that of experimental data . we see that for @xmath54 mev that our results enter the band of experimental data before @xmath17 mev and remain there throughout the entire expansion regardless of the initial conditions . however , for @xmath54 mev the results are slightly different . in this case , the ratios match the experimental data early on at around @xmath165 mev . however , they become briefly overpopulated around @xmath166 mev but then quickly return to the experimental values , except for the case when we have the initial conditions such that the pions are overpopulated . this could imply that there are a few too many hagedorn states and a fit for the hagedorn states with a lower @xmath167 ( degeneracy of the hagedorn states ) may produce better results . + + and @xmath168 . along the top axis of each graph the corresponding time is shown in @xmath161.,title="fig : " ] + and @xmath168 . along the top axis of each graph the corresponding time is shown in @xmath161.,title="fig : " ] as with the protons , the total number of kaons are also slightly dependent on our chosen initial conditions , more specifically , our choice in @xmath163 . in fig . [ fig : kk176 ] and fig . [ fig : kk196 ] the temperature of the evolving system after the phase transition at which chemical equilibrium among standard hadrons is basically reached and maintained is between @xmath166 for @xmath54 mev and they have also already reached chemical equilibrium by @xmath17 for @xmath49 mev , below which the hagedorn states basically die out . the one exception is when the hagedorn states begin underpopulated i.e. that @xmath169 . in this case , the kaon pairs take longer to reach chemical equilibrium . however , when we look at @xmath7 in fig . [ fig : kkpi ] , lower @xmath163 actually fits the data better . moreover , the pions again remain roughly at chemical equilibrium throughout the expansion as seen in fig . [ fig : kk176 ] and fig . [ fig : kk196 ] . while the pion graphs look roughly similar in figs . [ fig : pp176]-[fig : kk196 ] , they are not . the difference is how the pions are affected in the presence of a @xmath0 pair compared to a decay that includes a kaon anti - kaon pair . in fig . [ fig : kkpi ] the ratio of kaons to pions is shown for @xmath54 mev and for @xmath49 mev . for @xmath54 mev our results are roughly at the upper edge of the experimental values . however , for @xmath49 mev our results are slightly higher than the experimental values . although , the results at @xmath170 mev are almost exactly those of the uppermost experimental data point . + + s with various initial conditions . note that for star @xmath171 and @xmath172 . along the top axis of each graph the corresponding time is shown in @xmath161.,title="fig : " ] + s with various initial conditions . note that for star @xmath171 and @xmath172 . along the top axis of each graph the corresponding time is shown in @xmath161.,title="fig : " ] we can also observe the affects of the expansion on the @xmath2 pairs as seen in fig . [ fig : ll176 ] and fig . [ fig : ll196 ] . we see that both reach the experimental values almost immediately ( @xmath173 for @xmath54 mev and around @xmath165 for @xmath49 mev ) . the one exception is again for an underpopulation of hagedorn states , which reaches chemical equilibrium at @xmath174 for @xmath54 mev and already by @xmath17 for @xmath49 mev ) . the ratio of @xmath8 s is shown in fig . [ fig : llpi ] . in both cases the @xmath8 s match the experimental values extremely well . for @xmath54 mev our results reach the equilibrium values at @xmath175 mev and for @xmath49 mev the experimental values are reached already by @xmath175 mev . . the points show the ratios at @xmath170 mev for the various initial conditions ( circles are for @xmath54 mev and diamonds are for @xmath49 mev ) . the experimental results for star and phenix are shown by the gray error bars . ] a summary graph of all our results is shown in fig . [ fig : summary ] . the gray error bars cover the range of error for the experimental data points from both star and phenix . the points show the range in values for the various initial conditions at @xmath170 mev . we see in our graph that our freezeout results match the experimental data well . what the graphs in figs . [ fig : pp176]-[fig : llpi ] show us is that a dynamical scenario is able to explain chemical equilibration values that appear in thermal fits by @xmath176 mev . in general , @xmath54 mev and @xmath49 give chemical freeze - out values in the range between @xmath166 mev . these results agree well with the chemical freeze - out temperature found in @xcite . moreover , the initial conditions have little effect on the ratios and give a range in the chemical equilibrium temperature of about @xmath177 mev , which implies that information from the qgp regarding multiplicities is washed out due to the rapid dynamics of hagedorn states . lower @xmath135 does slow the chemical equilibrium time slightly . however , as seen in fig . [ fig : summary ] they still fit well within the experimental values . furthermore , in @xcite we showed the the initial condition play pretty much no roll whatsoever in the ratios of @xmath178 and @xmath179 . thus , strengthening our argument that the dynamics are washed out following the qgp . while the variance in the chemical equilibration time arising from the initial conditions may seem contradictory to the @xmath178 and @xmath179 ratios in @xcite , it can be explained with the pion populations . in figs . [ fig : pp176]-[fig : kk196 ] quicker chemical equilibration times and , thus , larger total baryon / kaon numbers translated into a larger number of pions in the system . thus , the @xmath178 and @xmath179 ratios do not depend on the initial conditions . + + s with various initial conditions . note that for star @xmath180 and @xmath181 . along the top axis of each graph the corresponding time is shown in @xmath161.,title="fig : " ] + s with various initial conditions . note that for star @xmath180 and @xmath181 . along the top axis of each graph the corresponding time is shown in @xmath161.,title="fig : " ] we can also use our model to investigate the possibility of @xmath103 s . in @xcite , they discussed the possibility of @xmath103 s being produced from the following decay channels : @xmath182 the first decay channel of a mesonic non - strange hagedorn state we can implement straightforwardly with our model by employing the canonical branching ratio via fig . [ fig : lamom ] . the results are shown in fig . [ fig : oo176 ] for @xmath54 mev , in fig . [ fig : oo196 ] for @xmath49 mev , and the @xmath183 ratio is shown in fig . [ fig : oopi ] . we are able to find the average number of @xmath103 s from @xcite as seen in fig . [ fig : lamom ] . we see that , using only the first reaction , we are still impressively able to adequately populate the @xmath3 pairs so that they roughly match the experimental data . on the other hand , from fig . [ fig : taubb ] we see that for the @xmath39 particle the equilibration time are short only very close to @xmath29 . the scenario is thus more delicate . if one would take eg one half , or one fourth , respectively , of the decay width of that of eq . [ hsdecaywidth ] , the total production of @xmath39 is not sufficient up to 25 % , or up to 50% , respectively , to meet the experimental yield ( the other ratios are not significantly affected by such a change of the decay width ) . in a future work , it would be interesting to observe the other decay channels as given in eq . [ omdecay ] and advertised in @xcite . the second reaction includes a mesonic , three times strange hagedorn state whereas the third decay channel includes a baryonic , strange hagedorn state . both states are much more likely to directly decay into a @xmath39 . these are , admittedly , exotic states , but should also occur in the spirit of hagedorn states . in order to observe these decay channels a method , e.g. a microscopic quark model , must be found to find the appropriate hagedorn spectrum for strange mesonic / baryonic hagedorn states . in this paper we found that hadronic matter , at rhic or sps energies , can reach chemical equilibrium within a dynamical scenario using hagedorn states close to the critical temperature . these states were able to produce quick chemical equilibration times in ( anti-)proton , ( anti-)kaons , and ( anti-)lambdas close to the critical temperature due to their strong increase in their abundancy . the existence of such a mixture of standard hadrons and hagedorn states just below the phase transition can explain dynamically the chemical equilibration of the hadronic species at around temperatures of 160 mev to 170 mev as seen within the thermal models . from our analytical results we found that the chemical equilibration time depends on the temperature , decay widths , and branching ratios , but not the initial conditions . while this changes slightly when an expanding fireball is considered , the initial condition still only play a small role and only minimally affect the ` freeze - out ' temperature at which chemical equilibrium is reached . this demonstrates that regardless of the population of hadrons coming out of the qgp phase , the initial conditions are washed out and everything can reach abundances which correspond to those of chemical equilibrium by the chemical freezeout temperatures found in @xcite . moreover , from our previous paper @xcite we showed that particle ratios ( @xmath178 and @xmath179 ) are not affected by the initial conditions and here we showed that @xmath6 , @xmath7 , @xmath8 and also @xmath9 match the experimental values regardless of the initial conditions . specially , fig . [ fig : summary ] demonstrates this nicely and summarizes our findings : regardless of the initial conditions , our dynamical scenario can match experimental data . we do find , however , that @xmath49 fits within the experimental data box for @xmath7 whereas @xmath54 is slightly above . this appears to reconfirm the findings in @xcite . our results imply that both lattice temperature can ensure that the hadrons reach their chemical equilibrium values by @xmath166 mev . although the ratios for @xmath54 do fit the data somewhat better , both math the experimental values reasonably well . this implies that independent of the critical temperature the hadrons are able to reach chemical freeze - out . we see sufficiently short time scales for the chemical equilibrium of hadrons . the protons , kaons , and lambdas reach chemical equilibrium on the order of @xmath184 . moreover , hagedorn states states provide a very efficient way for incorporating multi - hadronic interactions ( with parton rearrangements ) . in an upcoming paper we will use a canonical model to derive all the branching ratios included in our calculations . we can then look at reactions that include a mixture of strange and non - strange baryons ( for instance , @xmath185 ) and multi - strange baryons . however , considering that our initial results produce quick chemical equilibration times for the baryons , kaons , and lambdas , it is reasonable to believe that this will occur for mixed reactions and multi - strange baryons as well . in addition , the machinery of standard hadronic reactions , i.e. binary scattering processes and resonance production processes , help also to equilibrate the various hadronic degrees of freedom . still , our work indicates that the population and repopulation of potential hagedorn states close to phase boundary can be the key source for a dynamical understanding of generating and chemically equilibrating the standard and measured hadrons . jnh would like to thank j. noronha , b. cole , and m. gyulassy for productive discussions . this work was supported by the helmholtz international center for fair within the framework of the loewe program ( landes - offensive zur entwicklung wissenschaftlich - konomischer exzellenz ) launched by the state of hesse . thanks the members of the institut fr theoretische physik of johann wolfgang goethe universitt for their hospitality during the final stages of this work . the work of i.a.s . was supported in part by the start - up funds from the arizona state university . if our initial conditions are such that both the pions and hagedorn states begin far out of chemical equilibrium , we can find an analytical solution by subdividing the analysis into three distinct stages . initially , during stage 1 the pions are underpopulated such that we can say that they approximately begin at @xmath186 ( we can also start the pions above zero and the approximation works well ) . because the pions reach chemical equilibrium much quicker than the hagedorn states due to all the hagedorn states decaying quickly into pions , then we can make the approximation that the hagedorn states are held at their initial value of @xmath163 . one can see this from the difference in the time scales from tab . [ tab : tau ] where @xmath187 and @xmath188 . since @xmath189 we let @xmath190 , then substituting this into eq . ( [ eqn : setpihs ] ) we obtain which is the fugacity of the pions in stage 1 and gives @xmath192 . again using the approximation @xmath189 and substituting eq . ( [ eqn : npia2 ] ) into the hagedorn state rate equation in eq . ( [ eqn : setpihs ] ) , with the solution @xmath193,\nonumber\\ \lambda_{i}&= & \left[1-\langle n_i\rangle\left(\frac{-t}{\tau_{i}}\right)^{-\langle n_i\rangle } e^{- \left(\frac{t}{\tau_{i}}\right ) } \int_{0}^{-\frac{t}{\tau_{i}}}x^{\langle n_i\rangle-1}e^{-x}dx \right]\nonumber\\ & \cdot & \left(\frac{t}{\tau_{\pi}^{0}}\right)^{\langle n_i\rangle}+\beta_{i}e^{-\left(\frac{t}{\tau_{i}}\right ) } .\end{aligned}\ ] ] substituting @xmath194 into the integral in eq . ( [ eqn : s1nipre ] ) , expanding the exponential inside the integral so @xmath195 , and integrating over @xmath196 , provides us with the fugacity of the hagedorn states in stage 1 @xmath197\nonumber\\ & + & \beta_{i}e^{- \left(\frac{t}{\tau_{i}}\right)}\;.\end{aligned}\ ] ] therefore , eq . ( [ eqn : npia2 ] ) and eq . ( [ eqn : hsstage1 ] ) describe the behaviour of the pions and hagedorn states during the initial stage of the evolution towards chemical equilibrium . they are then compared to the numerical results in fig . [ fig : pifree ] . as the pions near equilibrium our approximation of @xmath198 no longer holds and we switch to stage 2 where we assume @xmath199 at time @xmath200 . here @xmath200 is a time when the pions are almost in chemical equilibrium , which is normally taken when the pions reach about @xmath201 of their chemical equilibrium value . returning to the pion equation in eq . ( [ eqn : setpihs ] ) , we can substitute in @xmath202 and use the approximation @xmath203 @xmath204 additionally , we substituted in @xmath135 for @xmath205 as an approximation since the hagedorn states do not change significantly in stage 1 ( the majority of the evolution is done by the pions ) . recall that @xmath206 and it is a constant . in its present form , eq . ( [ eqn : hsineqsub ] ) can be integrated . we also define @xmath207 where @xmath208 is close to 1 ( @xmath208 is the measurement of how close the pions are to their equilibrium value when we switch from stage 1 to stage 2 ) . then , after integration @xmath209 where @xmath210 and @xmath211 . analogously to stage 1 , we substitute the pion equation , i.e. , eq . ( [ eqn : npi ] ) into the hagedorn resonance equation in eq . ( [ eqn : setpihs ] ) and integrate @xmath212\end{aligned}\ ] ] where @xmath213 and @xmath214 . thus , our equations for the evolution of the pions and hagedorn states are eq . ( [ eqn : npi ] ) and eq . ( [ eqn : nallep1 ] ) , respectively . as with stage 1 , the evolution equation for the hagedorn states is dictated by that of the pions . at @xmath215 mev for @xmath54 mev when @xmath216 and @xmath217 ( top ) and the numerical results for the same initial conditions including @xmath1 pairs with @xmath218.,title="fig : " ] + at @xmath215 mev for @xmath54 mev when @xmath216 and @xmath217 ( top ) and the numerical results for the same initial conditions including @xmath1 pairs with @xmath218.,title="fig : " ] stage 3 i.e. quasi - equilibrium begins once the pions and at least one species of hagedorn resonances ( @xmath219 is the shortest chemical equilibration time ) has surpassed its equilibration time ( @xmath220 and @xmath221 , respectively ) . to understand quasi - equilibrium we must use the effective pion number @xmath222 because we need a variable that can observe the effects of both the pions and resonances . the effective pion number essentially includes the number of effective pions that each hagedorn state could decay into . thus , we start by taking the derivative of eq . ( [ eqn : effpions ] ) in terms of its fugacity @xmath223\nonumber\\ & = & \frac{\sum_{i}\gamma_{i}n_{i}^{eq}}{\tilde{n}_{\pi}^{eq}}\left[\langle n_i\rangle\sum_{n}b_{i , n}\lambda_{\pi}^n-\sum_{n}b_{i , n}n\lambda_{\pi}^n\right]\;.\end{aligned}\ ] ] once again we make the substitution @xmath202 so that where @xmath225 in the gaussian distribution of our branching ratios . to relate @xmath226 and @xmath227 we return to eq . ( [ eqn : effpions ] ) and separate @xmath68 into a sum over the resonances in quasi - equilibrium and one over the freely " equilibrating resonances @xmath228.\ ] ] since the pions reach quasi - equilibrium first , i.e. , @xmath229 near @xmath22 , we set the @xmath164 rate equation in eq . ( [ eqn : setpihs ] ) equal to zero , which gives @xmath230 , so @xmath231 eq . ( [ eqn : eptil ] ) then has the form @xmath232 where @xmath233 we can then solve for @xmath226 in eq . ( [ eqn : e2etil ] ) and substitute @xmath226 into eq . ( [ eqn : needep ] ) , which in turn can be integrated . this leads us to the solution @xmath234 where @xmath235 stands for the latest resonance to reach chemical equilibrium at that point in time and @xmath236 is the quasi - equilibrium time . clearly , once all the hagedorn states have reached chemical equilibrium than @xmath235 symbolizes the resonance of @xmath237 gev , since it is the slowest hagedorn state to equilibrate . the sums over free " is the sum over the hagedorn states that have not yet surpassed their respective chemical equilibrium time , @xmath238 . once @xmath239 is reached those sums equal zero . therefore , after @xmath239 all that remains is @xmath240 where @xmath241 is shown in tab . [ tab : tau ] . finally , we rewrite eq . ( [ eqn : remaine ] ) in terms of the pion evolution equation @xmath242 where @xmath243 . because the resonance equation depends on the population of the pions we substitute eq . ( [ eqn : piinqe ] ) into the hagedorn resonance rate equation in eq . ( [ eqn : setpihs ] ) , assuming the pions are near equilibrium ( i.e. , we use the approximation @xmath244 and @xmath203 ) @xmath245 where @xmath246 . thus , for stage 3 the population equations for the pions and the hagedorn states are eq . ( [ eqn : piinqe ] ) and eq . ( [ eqn : nimostinqe ] ) so long as @xmath247 . fig . [ fig : pifree ] reveals a remarkable close fit with our numerical results for @xmath215 mev i.e. @xmath248 . thus , the quasi - chemical equilibrium time , @xmath249 , depends only on @xmath102 , @xmath82 , @xmath250 , and @xmath66 , which is temperature dependent , but not on our initial conditions . as mentioned in the text , though , @xmath249 includes many non - linear affects that only occur close to the chemical equilibrium . thus , the more appropriate time scale is @xmath157 in order to describe the dynamics . we also see from fig . [ fig : pifree ] that when @xmath1 pairs are included that the pions and hagedorn resonances equilibrate in roughly the same amount of time , which implies that our analytical solution can still be approximately applied when @xmath1 pairs are present . 99 p. koch , b. muller and j. rafelski , phys . * 142 * , 167 ( 1986 ) . r. rapp and e. v. shuryak , phys . lett . * 86 * ( 2001 ) 2980 . c. greiner , aip conf . proc . * 644 * , 337 ( 2003 ) ; heavy ion phys . * 14 * , 149 ( 2001 ) ; c. greiner and s. leupold , j. phys . g * 27 * , l95 ( 2001 ) . braun - munzinger _ b * 344 * ( 1995 ) 43 ; phys . b * 365 * ( 1996 ) 1 ; eur . j. c * 2 * , 351 ( 1998 ) p. braun - munzinger , i. heppe and j. stachel , phys . b * 465 * ( 1999 ) 15 . j. i. kapusta and i. shovkovy , phys . c * 68 * ( 2003 ) 014901 ; j. i. kapusta , j. phys . g * 30 * ( 2004 ) s351 . p. huovinen and j. i. kapusta , phys . c * 69 * ( 2004 ) 014902 . r. stock , phys . b * 456 * ( 1999 ) 277 ; arxiv : nucl - th/0703050 . u. heinz and g. kestin , arxiv : nucl - th/0612105 . braun - munzinger , j. stachel and c. wetterich , phys . b * 596 * ( 2004 ) 61 . c. greiner _ et al . _ j. phys . g * 31 * , s725 ( 2005 ) . j. noronha - hostler , c. greiner and i. a. shovkovy , phys . lett . * 100 * , 252301 ( 2008 ) . j. noronha - hostler , j. noronha , h. ahmad , i. shovkovy and c. greiner , arxiv:0907.4963 [ nucl - th ] , to appear in nucl . a ; j. noronha - hostler , c. greiner and i. shovkovy , eur . j. st * 155 * , 61 ( 2008 ) ; arxiv : nucl - th/0703079 . j. noronha - hostler , j. noronha and c. greiner , arxiv:0811.1571 [ nucl - th ] , to appear in phys . rev . p. kovtun , d. t. son and a. o. starinets , phys . lett . * 94 * , 111601 ( 2005 ) . d. kharzeev and k. tuchin , arxiv:0705.4280 [ hep - ph ] . j. noronha - hostler , h. ahmad , j. noronha and c. greiner , arxiv:0906.3960 [ nucl - th ] . y. aoki , z. fodor , s. d. katz and k. k. szabo , jhep * 0601 * , 089 ( 2006 ) ; phys . b * 643 * , 46 ( 2006 ) m. cheng _ et al . _ , phys . d * 77 * , 014511 ( 2008 ) . a. bazavov _ et al . _ , arxiv:0903.4379 [ hep - lat ] . r. hagedorn , nuovo cim . suppl . * 6 * 311 ( 1968 ) ; nuovo cim . suppl . * 3 * , 147 ( 1965 ) . w. broniowski , w. florkowski and l. y. glozman , phys . d * 70 * , 117503 ( 2004 ) [ arxiv : hep - ph/0407290 ] . k. a. bugaev , v. k. petrov and g. m. zinovjev , arxiv:0801.4869 [ hep - ph ] . l. g. moretto , l. phair , k. a. bugaev and j. b. elliott , pos c * pod2006 * ( 2006 ) 037 ; l. g. moretto , k. a. bugaev , j. b. elliott and l. phair , arxiv : nucl - th/0601010 ; arxiv : hep - ph/0511180 . i. zakout , c. greiner and j. schaffner - bielich , nucl . a * 781 * , 150 ( 2007 ) ; i. zakout and c. greiner , phys . rev . c * 78 * , 034916 ( 2008 ) ; l. ferroni and v. koch , arxiv:0812.1044 [ nucl - th ] . j. i. kapusta and k. a. olive , nucl . a * 408 * , 478 ( 1983 ) . d. h. rischke , m. i. gorenstein , h. stoecker and w. greiner , z. phys . c * 51 * , 485 ( 1991 ) . s. a. bass _ et al . . part . nucl . phys . * 41 * , 255 ( 1998 ) [ prog . part . phys . * 41 * , 225 ( 1998 ) ] [ arxiv : nucl - th/9803035 ] . m. bleicher _ et al . _ , j. phys . g * 25 * , 1859 ( 1999 ) [ arxiv : hep - ph/9909407 ] . s. pal and p. danielewicz , phys . b * 627 * , 55 ( 2005 ) [ arxiv : nucl - th/0505049 ] . f. m. liu , k. werner and j. aichelin , phys . c * 68 * ( 2003 ) 024905 ; f. m. liu , et . al . , j. phys . g * 30 * ( 2004 ) s589 ; phys . c * 69 * ( 2004 ) 054002 . f. becattini and l. ferroni , eur . j. c * 38 * , 225 ( 2004 ) m. beitel , j. noronha - hostler , and c. greiner , diplomthesis , to appear . i. senda , phys . b * 263 * , 270 ( 1991 ) ; f. lizzi and i. senda , nucl . b * 359 * , 441 ( 1991 ) ; f. lizzi and i. senda , phys . lett . b * 244 * , 27 ( 1990 ) . s. eidelman _ et al . b * 592 * ( 2004 ) 1 . c. spieles , h. stoecker and c. greiner , eur . j. c * 2 * , 351 ( 1998 ) ; c. greiner , d. h. rischke , h. stoecker and p. koch , phys . rev . d * 38 * , 2797 ( 1988 ) ; c. greiner and h. stoecker , phys . rev . d * 44 * , 3517 ( 1991 ) . i. g. bearden _ et al . _ [ brahms collaboration ] , phys . * 94 * ( 2005 ) 162301 . c. greiner , c. gong and b. muller , phys . b * 316 * , 226 ( 1993 ) . f. becattini and l. ferroni , eur . j. c * 38 * , 225 ( 2004 ) [ arxiv : hep - ph/0407117 ] .
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quick chemical equilibration times of hadrons ( specifically , @xmath0 , @xmath1 , @xmath2 , and @xmath3 pairs ) within a hadron gas are explained dynamically using hagedorn states , which drive particles into equilibrium close to the critical temperature . within this scheme ,
we use master equations and derive various analytical estimates for the chemical equilibration times .
we compare our model to recent lattice results and find that for both @xmath4 mev and @xmath5 mev , the hadrons can reach chemical equilibrium almost immediately , well before the chemical freeze - out temperatures found in thermal fits for a hadron gas without hagedorn states .
furthermore the ratios @xmath6 , @xmath7 , @xmath8 , and @xmath9 match experimental values well in our dynamical scenario .
| 17,481 | 221 |
in the case of small antiferromagnetic correlation lengths @xmath48 , corresponding to the overdoped regime , the rpa should be a reasonable approximation for the spin susceptibility . the dashed line in fig . 1 shows the imaginary part of @xmath49 for @xmath50 using the parameters tb2 in table i of ref.@xcite and a chemical potential corresponding to the doping @xmath51 . the energy unit is 1 ev in the following and lengths are measured in units of the lattice constant a of the square lattice . the superconducting order parameter is @xmath52 with @xmath53 , @xmath3 equal to 0.135 and @xmath54 . the dashed line in fig . 1 illustrates that most of the spectral weight resides in the bound state at the energy 0.038 and that only a small part of it has been left in the continuum at higher energies . away from @xmath55 the dashed curve in fig . 1 does not change dramatically as long as the bound state lies still in the gapped region . entering the particle - hole continuum by going further away from @xmath55 destroys the bound state and most of the spectral weight shifts to high energies of the order of @xmath2 . @xmath56 without ( dashed line ) and with ( solid line ) memory function @xmath30 for a doping @xmath57 and @xmath58 . the dashed curve corresponds to the rpa . inset : real part @xmath59 and imaginary part @xmath43 of @xmath60 . , width=302 ] the parameters used in fig . 1 yield @xmath61 . for such a small @xmath48 practically all momenta in the sum over momenta in eq.([m ] ) contribute substantially which means that @xmath43 is mainly determined by contributions away from the small region around @xmath55 so that the bound state and its low - energy scale is rather irrelevant for @xmath30 . this is confirmed by an explicit calculation of @xmath30 using rpa results for the various quantities in eq.([m ] ) . the result is shown in the inset of fig . 1 for t=0 . @xmath43 ( solid line ) is structureless except at small energies where it vanishes rapidly due to the smallness of @xmath46 in this region and the cutoff @xmath62 for the integration over @xmath63 in eq.([d ] ) due to the bosonic factors . taking @xmath30 into account in eq.([phi1 ] ) yields the solid line in fig . 1 which differs only marginally from the dashed line this shows that at short correlation lengths the rpa result for @xmath64 is essentially correct and that the correction @xmath30 to @xmath36 is rather small . the underlying physical picture is that the momentary local axis of preferred spin directions fluctuates very rapidly due to the random forces induced by @xmath23 . the spectrum of these forces is given by @xmath65 and characterized by the large energy scale @xmath2 in agreement with the inset of fig . for large @xmath48 the spectral function @xmath66 is strongly peaked at @xmath67 . this means that the integration over @xmath68 in eq.([m ] ) is restricted to momenta near @xmath69 or near @xmath1 . since we are interested in external momenta @xmath70 the momentum of one of the two spectral functions in eq.([d ] ) is small . due to spin conservation this spectral function describes spin diffusion and is mainly restricted to small values of @xmath63 . as a result one may neglect the small frequency transfer in the second spectral function in eq.([d ] ) . taking also the real part of @xmath30 into account we obtain from eqs.([m ] ) and ( [ d ] ) , @xmath71 with @xmath72 and the equal - time correlation function @xmath73 in deriving eq.([m ] ) we used the fact that the two memory functions in eq.([phi1 ] ) depend for our parameters only slowly on momentum around the wave vector @xmath74 so that the combination @xmath75 on the right - hand side of eq.([m ] ) may be evaluated at @xmath1 . the sum over @xmath76 in eq.([omega ] ) runs over half of the brillouin zone centered around @xmath1 . the evaluation of the above expressions using the rpa encounters a problem : @xmath48 , calculated in the rpa , is in the optimal and moderately underdoped region around one or smaller and increases substantially only near the transition to the antiferromagnetic state in disagreement with the experiment . for instance , we have for @xmath77 @xmath78 , ref.@xcite @xmath79 using quite different parameter values , whereas the experimental values for @xmath48 are larger by about a factor 5 or more@xcite . since this large discrepancy would affect severely the momentum sum in eq.([omega ] ) we prefer to use a realistic @xmath27 as input in calculating @xmath30 and write @xmath80 for @xmath81 considering @xmath48 as a parameter to be determined from experiment . it is instructive to study the frequency dependence of the denominator of eq.([phi1 ] ) . in order to describe a slightly underdoped case we choose the same parameters as in fig . 1 , a chemical potential corresponding to @xmath85 , @xmath86 , and the cutoff @xmath87 for the sum over @xmath88 in eq.([omega ] ) . the solid and dotted line in fig . 2 show @xmath89 for h=0.5 and 0.38 , respectively , writing @xmath90 . this quantity is practically independent of momentum , increases monotonically with @xmath62 and is zero at the rpa resonance energy @xmath91 . the dashed and dash - dotted lines in fig . 2 show @xmath92 for the same momenta . these curves resemble the real part of an oscillator located at @xmath93 with an oscillator strength being very small at @xmath94 and strongly increasing with @xmath95 . the poles of eq.([phi1 ] ) are given by the common points of the two curves denoted by squares and circles . since the common point at @xmath96 ( not shown in fig . 2 ) has vanishing pole strength there are two branches of collective spin excitations . for vanishing damping @xmath97 their dispersion is shown in fig . 3 by solid lines . they approximately touch each other at @xmath98 and disperse up- and downwards with increasing @xmath95 . for not too large @xmath95 both branches lie below the continuum in agreement with experiment@xcite . performing the calculation in the normal state at @xmath99 the solid and dotted lines in fig . 2 lie everywhere above zero but the solid and dashed and also the dotted and dash - dotted lines have still one common point at larger frequencies . in this case only the upper but not the lower branch exists in agreement with experiment@xcite . at very low dopings @xmath100 due to the constraint and the pole condition @xmath101 yields in the presence of long - range order the correct spin wave dispersion@xcite . several prerequisites are necessary to obtain the above hourglass dispersion for spin excitations . there must exist two different kinds of spin excitations to account for the two branches . the first one are rpa spin fluctuations where all induced spin moments have the same direction . they may be characterized by the fact that the internal fields induced by the heisenberg interaction conserves frequency , momentum and spin direction which is a direct consequence of the one - mode behavior of @xmath22 in eq.([l1 ] ) . the second one are local rotations of spins under the influence of @xmath23 in eq.([l2 ] ) . in this case a spin in z direction acquires in its time evolution also a component in x direction due to the presence of a spin fluctuation in y direction . the pure form of the two kind of spin excitations are obtained for @xmath102 and @xmath103 , respectively , and are realized approximately at large and small dopings . in the hourglass regime @xmath30 and @xmath36 are of similar magnitude . another prerequisite for hourglass behavior is that @xmath48 is substantially larger than 1 . only then is the momentum integration in eq.([m ] ) restricted to the resonance and the spin diffusion modes yielding oscillator - like behavior of @xmath30 near @xmath93 . the up- and downwards dispersion and their approximate degeneracy at @xmath1 is mainly determined by @xmath104 , which according to eq.([omega ] ) is roughly proportional to @xmath105 . the first factor tends to zero at low temperatures for @xmath106 and saturates at large @xmath107 . as a result @xmath104 is very small at @xmath98 causing the approximate touching of the two branches at @xmath108 and @xmath109 . with increasing @xmath107 @xmath104 increases strongly leading to a downward dispersion of the lower branch even if @xmath93 was practically dispersionless as in our case . according to fig . 3 the upper branch increases at large @xmath95 roughly as @xmath110 , i.e. , with an effective spin wave velocity which is reduced by about a factor 2 - 3 compared to spin wave theory similar as in experiment@xcite . using the same parameters as in figs . 2 and 3 fig . 4 shows @xmath64 as a function of @xmath4 with the frequency as a parameter . as suggested by fig . 3 @xmath64 exhibits a hourglass dispersion with intensities which are largest near @xmath93 and decay rather fast and slow towards lower and higher frequencies , respectively . fig . 5 shows @xmath64 as a function of @xmath62 for a fixed @xmath4 as a parameter . in agreement with fig . 4 the strong peak at @xmath111 splits into two peaks with decreasing @xmath4 which disperse up- and downwards in frequency . the curve for @xmath111 calculated for the small damping @xmath112 has the shape of a lorentzian . at smaller dampings this peak splits into a double peak due to the small gap between upper and lower branch shown in fig . 3 . for @xmath102 only the lower , weaker peak is obtained . in conclusion , we have shown that the memory function of the spin susceptibility contains in general two distinct contributions due to rpa - like and due to rotational spin fluctuations . the first one dominates at large , the second one at small dopings . at intermediate dopings both are of similar magnitude which leads to one upwards and one downwards dispersing branch of excitations . at low temperatures the two branches are approximately degenerate at @xmath1 which explains , at least qualitatively , the observed hourglass dispersion at intermediate dopings .
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a theory for the dispersion of collective magnetic excitations in superconducting cuprates is presented with the aim to cover both high and low doping regimes . besides of spin fluctuations describable in the random phase approximation ( rpa ) we allow for local spin rotations within a mode - coupling theory . at low temperatures and moderately large correlation lengths
we obtain two branches of excitations which disperse up- and downwards exhibiting the hourglass behavior observed experimentally at intermediate dopings . at large and small dopings our theory essentially reduces to the rpa and spin wave theory , respectively .
the low - temperature magnetic response of many high - t@xmath0 superconductors is characterized by a resonant mode inside the superconducting gap around the antiferromagnetic wave vector @xmath1 .
this collective mode manifests itself as a single peak at @xmath1 which splits into two peaks dispersing up- and downwards in frequency away from @xmath1 .
this unusual dispersion resembles the shape of a hourglass@xcite .
theories to explain this phenomena use either a more local@xcite or an itinerant@xcite description of the magnetism .
the second approach considers particle - hole excitations with spin flips which interact within the random phase approximation ( rpa ) forming a dispersing bound state in the superconducting gap .
this approach yields only one branch of excitations below the stoner continuum whereas it has been established recently that the lower branch , the center of the hourglass as well as part of the upper branch lie below this continuum in the gapped region@xcite .
a more theoretical argument for the incompleteness of a rpa description comes from the fact that different spin directions do not mix as a function of time in this approximation which excludes local rotations of spins known from spin wave theory .
below we will present a theory which contains both spin wave theory and rpa as special cases . at intermediate dopings
we will show that both rpa and spin wave like spin fluctuations are important and produce the two branches of the hourglass dispersion .
we consider the @xmath2-@xmath3 model@xcite with the hamiltonian @xmath4 , @xmath5 @xmath6 are creation and annihilation operators , respectively , for electrons with momentum @xmath7 and spin projection @xmath8 excluding any double occupancies of sites .
@xmath9 are spin operators in momentum space , @xmath10 and @xmath11 are the bare electron dispersion and the fourier transform of the heisenberg coupling , respectively .
a useful approximation for @xmath4 , used in the following , is obtained by taking the large @xmath12 limit of the @xmath2-@xmath3 model , where @xmath10 describes a renormalized dispersion of quasi - particles and the fermionic operators can be treated as usual creation and annihilation operators . in the following we are interested in the time evolution of the spin operator @xmath13 , @xmath14 where @xmath8 denotes the vector of the three pauli matrices .
it obeys the equation of motion @xmath15 with @xmath16 @xmath17 @xmath18 is equal to @xmath19 , where @xmath20 denotes the thermodynamic expectation value , and @xmath21 stands for the vector product .
since we are only interested in the spin response we have dropped terms on the right - hand side of eq.([l2 ] ) which involve fluctuations in the density .
we also dropped an overall prefactor denoting the number of primitive cells .
the unperturbed liouville operator @xmath22 describes the time evolution of the system in the rpa . from its explicit expression in eq.([l1 ] ) follows that it does not mix different cartesian components of the spin operators .
in contrast to that the time evolution described by @xmath23 involves product states of spin operators , mixes different spin components and thus can describe rotations of spins due to fluctuating fields .
the spin susceptibility @xmath24 can conveniently be calculated from the associated kubo relaxation function @xmath25 where @xmath26 is a complex frequency and @xmath27 is equal to @xmath28 . due to the rotational invariance in spin space
we may assume that @xmath29 , @xmath30 etc .
always refer to the z - direction . using the mori formalism @xmath31
can be written as@xcite @xmath32 the first memory kernel @xmath33 describes the time evolution of spin operators by @xmath22 .
according to eq.([l1 ] ) the direction of the spin operators is conserved and they remain always linear in the operators @xmath34 . eliminating the @xmath34 operators in the equation of motion in favor of the orginal @xmath35 operators yields an explicit expression for @xmath36 which may be expressed in terms of the rpa spin susceptibility @xmath37 , @xmath38 with @xmath39 @xmath40 is the free susceptibility .
the second memory contribution @xmath41 is due to the time evolution of single spin operators @xmath35 into product states of spin operators described by @xmath23 . using the mode - coupling assumption and performing the analytic continuation @xmath42
, we obtain for the imaginary part of @xmath30 , denotd by @xmath43 , @xmath44 @xmath45 @xmath46 is the spectral function of the spin propagator and @xmath47 the bose function .
| 2,930 | 1,423 |
consider supercritical bond - percolation on @xmath0 , @xmath1 , and the simple random walk on the ( unique ) infinite cluster . in @xcite sidoravicius and sznitman asked the following question : is it true that for a.e . configuration in which the origin belongs to the infinite cluster , the random walk started at the origin exits the infinite symmetric slab @xmath2 through the `` top '' side with probability tending to @xmath3 as @xmath4 ? sidoravicius and sznitman managed to answer their question affirmatively in dimensions @xmath5 but dimensions @xmath6 remained open . in this paper we extend the desired conclusion to all @xmath1 . as in @xcite , we will do so by proving a quenched invariance principle for the paths of the walk . random walk on percolation clusters is only one of many instances of `` statistical mechanics in random media '' that have been recently considered by physicists and mathematicians . other pertinent examples include , e.g. , various diluted spin systems , random copolymers @xcite , spin glasses @xcite , random - graph models @xcite , etc . from this general perspective , the present problem is interesting for at least two reasons : first , a good handle on simple random walk on a given graph is often a prerequisite for the understanding of more complicated processes , e.g. , self - avoiding walk or loop - erased random walk . second , information about the scaling properties of simple random walk on percolation cluster can , in principle , reveal some new important facts about the structure of the infinite cluster and/or its harmonic properties . let us begin developing the mathematical layout of the problem . let @xmath0 be the @xmath7-dimensional hypercubic lattice and let @xmath8 be the set of nearest neighbor edges . we will use @xmath9 to denote a generic edge , @xmath10 to denote the edge between @xmath11 and @xmath12 , and @xmath13 to denote the edges from the origin to its nearest neighbors . let @xmath14 be the space of all percolation configurations @xmath15 . here @xmath16 indicates that the edge @xmath9 is occupied and @xmath17 implies that it is vacant . let @xmath18 be the borel @xmath19-algebra on @xmath20defined using the product topology and let @xmath21 be an i.i.d . measure such that @xmath22 for all @xmath23 . if @xmath24 denotes the event that the site @xmath11 belongs to an infinite self - avoiding path using only occupied bonds in @xmath25 , we write @xmath26 for the set @xmath27 by burton - keane s uniqueness theorem @xcite , the infinite cluster is unique and so @xmath28 is connected with @xmath21-probability one . for each @xmath29 , let @xmath30 be the `` shift by @xmath11 '' defined by @xmath31 . note that @xmath21 is @xmath32-invariant for all @xmath29 . let @xmath33 denote the percolation threshold on @xmath0 defined as the infimum of all @xmath34 s for which @xmath35 . let @xmath36 and , for @xmath37 , define the measure @xmath38 by @xmath39 we will use @xmath40 to denote expectation with respect to @xmath38 . for each configuration @xmath41 , let @xmath42 be the simple random walk on @xmath43 started at the origin . explicitly , @xmath42 is a markov chain with state space @xmath0 , whose distribution @xmath44 is defined by the transition probabilities @xmath45 and @xmath46 with the initial condition @xmath47 thus , at each unit of time , the walk picks a neighbor at random and if the corresponding edge is occupied , the walk moves to this neighbor . if the edge is vacant , the move is suppressed . our main result is that for @xmath38-almost every @xmath41 , the linear interpolation of @xmath48 , properly scaled , converges weakly to brownian motion . for every @xmath49 , let @xmath50,{\mathscr{w } } _ t)$ ] be the space of continuous functions @xmath51\to{\mathbb r}$ ] equipped with the @xmath19-algebra @xmath52 of borel sets relative to the supremum topology . the precise statement is now as follows : [ thm : mainthm ] let @xmath1 , @xmath53 and let @xmath41 . let @xmath42 be the random walk with law @xmath44 and let @xmath54 then for all @xmath49 and for @xmath38-almost every @xmath25 , the law of @xmath55 on @xmath50,{\mathscr{w } } _ t)$ ] converges weakly to the law of an isotropic brownian motion @xmath56 whose diffusion constant , @xmath57 , depends only on the percolation parameter @xmath34 and the dimension @xmath7 . the markov chain @xmath42 represents only one of two natural ways to define a simple random walk on the supercritical percolation cluster . another possibility is that , at each unit of time , the walk moves to a site chosen uniformly at random from the _ accessible _ neighbors , i.e. , the walk takes no pauses . in order to define this process , let @xmath58 be the sequence of stopping times that mark the moments when the walk @xmath42 made a move . explicitly , @xmath59 and @xmath60 using these stopping times which are @xmath44-almost surely finite for all @xmath41we define a new markov chain @xmath61 by @xmath62 it is easy to see that @xmath61 has the desired distribution . indeed , the walk starts at the origin and its transition probabilities are given by @xmath63 a simple modification of the arguments leading to theorem [ thm : mainthm ] allows us to establish a functional central limit theorem for this random walk as well : [ thm:2ndmainthm ] let @xmath1 , @xmath53 and let @xmath41 . let @xmath64 be the random walk defined from @xmath42 as described in and let @xmath65 be the linear interpolation of @xmath66 defined by with @xmath67 replaced by @xmath68 . then for all @xmath49 and for @xmath38-almost every @xmath25 , the law of @xmath69 on @xmath50,{\mathscr{w } } _ t)$ ] converges weakly to the law of an isotropic brownian motion @xmath56 whose diffusion constant , @xmath70 , depends only on the percolation parameter @xmath34 and the dimension @xmath7 . de gennes @xcite , who introduced the problem of random walk on percolation cluster to the physics community , thinks of the walk as the motion of `` an ant in a labyrinth . '' from this perspective , the `` lazy '' walk @xmath48 corresponds to a `` blind '' ant , while the `` agile '' walk @xmath71 represents a `` myopic '' ant . while the character of the scaling limit of the two `` ants '' is the same , there seems to be some distinction in the rate the scaling limit is approached , cf @xcite and references therein . as we will see in the proof , the diffusion constants @xmath72 and @xmath73 are related via @xmath74 , where @xmath75 is the expected degree of the origin normalized by @xmath76 , cf . there is actually yet another way how to `` put '' simple random walk on @xmath28 , and that is to use continuous time . here the corresponding result follows by combining the clt for the `` lazy '' walk with an appropriate renewal theorem for exponential waiting times . the subject of random walk in random environment has a long history ; we refer to , e.g. , @xcite for recent overviews of ( certain parts of ) this field . on general grounds , each random - media problem comes in two distinct flavors : _ quenched _ , corresponding to the situations discussed above where the walk is distributed according to an @xmath25-dependent measure @xmath44 , and _ annealed _ , in which the path distribution of the walk is taken from the averaged measure @xmath77 . under suitable ergodicity assumptions , the annealed problem typically corresponds to the quenched problem averaged over the starting point . yet the distinction is clear : in the annealed setting the slab - exit problem from sect . [ sec1.1 ] is trivial by the symmetries of the averaged measure , while its answer is _ a priori _ very environment - sensitive in the quenched measure . an annealed version of our theorems was proved in the 1980s by de masi _ et al _ @xcite , based on earlier results of kozlov @xcite , kipnis and varadhan @xcite and others in the context of random walk in a field of random conductances . ( the results of @xcite were primarily two - dimensional but , with the help of @xcite , they apply to all @xmath1 ; cf @xcite . ) a number of proofs of quenched invariance principles have appeared in recent years for the cases where an annealed principle was already known . the most relevant paper is that of sidoravicius and sznitman @xcite which established theorem [ thm:2ndmainthm ] for random walk among random conductances in all @xmath78 and , using a very different method , also for random walk on percolation in @xmath5 . ( thus our main theorem is new only in @xmath6 . ) the @xmath5 proof is based on the fact that two independent random walk paths will intersect only very little something hard to generalize to @xmath6 . as this paper shows , the argument for random conductances is somewhat more flexible . another paper of relevance is that of rassoul - agha and sepplinen @xcite where a quenched invariance principle was established for _ directed _ random walks in ( space - time ) random environments . the directed setting offers the possibility to use independence more efficiently every time step the walk enters a new environment but the price to pay for this is the lack of reversibility . the directed nature of the environment also permits consideration of distributions with a drift for which a clt is not even expected to generally hold in the undirected setting ; see @xcite for an example of `` pathologies '' that may arise . finally , there have been been a number of results dealing with harmonic properties of the simple random walk on percolation clusters . grimmett , kesten and zhang @xcite proved via `` electrostatic techniques '' that this random walk is transient in @xmath79 ; extensions concerning the existence of various `` energy flows '' appeared in @xcite . a great amount of effort has been spent on deriving estimates on the heat - kernel i.e . , the probability that the walk is at a particular site after @xmath80 steps . the first such bounds were obtained by heicklen and hoffman @xcite . later mathieu and remy @xcite realized that the right way to approach heat - kernel estimates was through harmonic function theory of the infinite cluster and thus significantly improved the results of @xcite . finally , barlow @xcite obtained , using again harmonic function theory , gaussian upper and lower bounds for the heat kernel . we refer to @xcite for further references concerning this area of research . _ note _ : at the time a preprint version of this paper was first circulated , we learned that mathieu and piatnitski had announced a proof of the same result ( albeit in continuous - time setting ) . their proof , which has in the meantime been posted @xcite , is close in spirit to that of theorem 1.1 of @xcite ; the main tools are poincar inequalities , heat - kernel estimates and homogenization theory . let us outline the main steps of our proof of theorems [ thm : mainthm ] and [ thm:2ndmainthm ] . the principal idea which permeates in various disguises throughout the work of papanicolau and varadhan @xcite , kozlov @xcite , kipnis and varadhan @xcite , de masi _ et al _ @xcite , sidoravicius and sznitman @xcite and others is to consider an embedding of @xmath43 into the euclidean space that makes the corresponding simple random walk a martingale . formally , this is achieved by finding an @xmath81-valued discrete harmonic function on @xmath28 with a linear growth at infinity . the distance between the natural position of a site @xmath82 and its counterpart in this _ harmonic embedding _ is expressed in terms of the so - called _ corrector _ @xmath83 which is a principal object of study in this paper . see fig . [ fig1 ] for an illustration . it is clear that the corrector can be defined in any finite volume by solving an appropriate discrete dirichlet problem ( this is how fig . 1 was drawn ) ; the difficult part is to define the corrector in infinite volume while maintaining the natural ( distributional ) invariance with respect to shifts of the underlying lattice . actually , there is an alternative , probabilistic definition of the corrector , @xmath84 however , the only proof we presently have for the existence of such a limit is by following , rather closely , the constructions from sect . [ sec2.3 ] . once we have the corrector under control , the proof splits into two parts : ( 1 ) proving that the martingale i.e . , the walk on the deformed graph converges to brownian motion and ( 2 ) proving that the deformation of the path caused by the change of embedding is negligible . the latter part ( which is the principal contribution of this work ) amounts to a sublinear bound on the corrector @xmath83 as a function of @xmath11 . here , somewhat unexpectedly , our level of control is considerably better in @xmath85 than in @xmath79 . in particular , our proof in @xmath85 avoids using any of the recent sophisticated discrete - harmonic analyses but , to handle all @xmath1 uniformly , we need to invoke the main result of barlow @xcite . the proof is actually carried out along these lines only for the setting in theorem [ thm : mainthm ] ; theorem [ thm:2ndmainthm ] follows by noting that the time scales of both walks are comparable . here is a summary of the rest of this paper : in sect . [ sec : defcor ] we introduce the aforementioned corrector and prove some of its basic properties . [ sec : erg ] collects the needed facts about ergodic properties of the markov chain `` on environments . '' both sections are based on previously known material ; proofs have been included to make the paper self - contained . the novel parts of the proof sublinear bounds on the corrector appear in sects . [ sec : sec4]-[sec : sublin ] . the actual proofs of our main theorems are carried out in sect . [ sec : invprnp ] . the appendix ( sects . [ appa ] and [ appb ] ) contains the proof of an upper bound for the transition probabilities of our random walk , further discussion and some conjectures . in this section we will define and study the aforementioned corrector which is the basic instrument of our proofs . the main idea is to consider the markov chain `` on environments '' ( sect . [ sec2.1 ] ) . the relevant properties of the corrector are listed in theorem [ thm2.1 ] ( sect . [ sec2.2 ] ) ; the proofs are based on spectral calculus ( see sect . [ sec2.3 ] ) . as is well known , cf kipnis and varadhan @xcite , the markov chain @xmath42 in ( [ e:1.3][e:1.4 ] ) induces a markov chain on @xmath86 , which can be interpreted as the trajectory of `` environments viewed from the perspective of the walk . '' the transition probabilities of this chain are given by the kernel @xmath87 $ ] , @xmath88 our basic observations about the induced markov chain are as follows : [ lemma1.1 ] for every bounded measurable @xmath89 and every @xmath13 with @xmath90 , @xmath91 where @xmath92 is the bond that is opposite to @xmath13 . as a consequence , @xmath38 is reversible and , in particular , stationary for markov kernel @xmath93 . proof first we will prove . neglecting the normalization by @xmath94 , we need that @xmath95 this will follow from @xmath96 and the fact that , on @xmath97 we have @xmath98 . indeed , these observations imply @xmath99 and then follows by the shift invariance of @xmath21 . from we deduce that for any bounded , measurable @xmath100 , @xmath101 where @xmath102 is the function @xmath103 indeed , splitting the last sum into two terms , the second part reproduces exactly on both sides of . for the first part we apply and note that averaging over @xmath13 allows us to neglect the negative sign in front of @xmath13 on the right - hand side . but is the _ definition _ of reversibility and , setting @xmath104 and noting that @xmath105 , we also get the stationarity of @xmath38 . lemma [ lemma1.1 ] underlines our main reason to work primarily with the `` lazy '' walk . for the `` agile '' walk , to get a stationary law on environments , one has to weigh @xmath38 by the degree of the origin a factor that would drag through the entire derivation . next we will adapt the construction of kipnis and varadhan @xcite to the present situation . let @xmath106 be the space of all borel - measurable , square integrable functions on @xmath20 . abusing the notation slightly , we will use `` @xmath107 '' both for @xmath108-valued functions as well as @xmath81-valued functions . we equip @xmath107 with the inner product @xmath109with `` @xmath110 '' interpreted as the dot product of @xmath111 and @xmath112 when these functions are vector - valued . let @xmath93 be the operator defined by . note that , when applied to a vector - valued function , @xmath93 acts like a scalar , i.e. , independently on each component . from we know @xmath113 and so @xmath93 is symmetric . an explicit calculation gives us @xmath114 and so @xmath115 . in particular , @xmath93 is self - adjoint and @xmath116 $ ] . let @xmath117 be the local drift at the origin , i.e. , @xmath118 ( we will only be interested in @xmath119 for @xmath41 , but that is of no consequence here . ) clearly , since @xmath120 is bounded , we have @xmath121 . for each @xmath122 , let @xmath123 be the solution of @xmath124 since @xmath125 is a non - negative operator , @xmath126 is well - defined and @xmath127 for all @xmath122 . the following theorem is the core of the whole theory : [ thm2.1 ] there is a function @xmath128 such that for every @xmath29 , @xmath129 moreover , the following properties hold : 1 . ( shift invariance ) for @xmath38-almost every @xmath41 , @xmath130 holds for all @xmath131 . ( harmonicity ) for @xmath38-almost every @xmath41 , the function @xmath132 is harmonic with respect to the transition probabilities ( [ e:1.3][e:1.3b ] ) . ( square integrability ) there exists a constant @xmath133 such that @xmath134{\operatorname{\sf 1}}_{\{x\in{\mathscr{c}}_\infty\}}{\operatorname{\sf 1}}_{\{\omega_{e}=1\}}\circ\tau_x\bigr\vert_2<c\ ] ] is true for all @xmath29 and all @xmath13 with @xmath90 . the rest of this section is spent on proving theorem [ thm2.1 ] . the proof is based on spectral calculus and it closely follows the corresponding arguments from @xcite . alternative constructions invoke projection arguments , cf @xcite . let @xmath135 denote the spectral measure of @xmath136 associated with function @xmath120 , i.e. , for every bounded , continuous @xmath137\to{\mathbb r}$ ] , we have @xmath138 ( since @xmath93 acts as a scalar , @xmath135 is the sum of the `` usual '' spectral measures for the cartesian components of @xmath120 . ) in the integral we used that , since @xmath139 $ ] , the measure @xmath135 is supported entirely in @xmath140 $ ] . the first observation , made already by kipnis and varadhan , is stated as follows : [ lemma2.2 ] @xmath141 proof with some caution concerning the infinite cluster , the proof is a combination of arguments right before theorem 1.3 of @xcite and those in the proof of theorem 4.1 of @xcite . let @xmath142 be a bounded real - valued function and note that , by lemma [ lemma1.1 ] and the symmetry of the sums , @xmath143 hence , for every @xmath144 we get @xmath145 the first term on the right - hand side equals a constant times @xmath146 , while lemma [ lemma1.1 ] allows us to rewrite the second term into @xmath147 we thus get that there exists a constant @xmath148 such that for all bounded @xmath142 , @xmath149 applying for @xmath111 of the form @xmath150 , summing @xmath151 over coordinate vectors in @xmath81 and invoking , we find that for every bounded continuous @xmath152\to{\mathbb r}$ ] and @xmath153 , @xmath154 substituting @xmath155 for @xmath156 and noting that @xmath157 , we get @xmath158 and so @xmath159 the monotone convergence theorem now implies @xmath160 proving the desired claim . using spectral calculus we will now prove : [ lemma2.3 ] let @xmath126 be as defined in . then @xmath161 moreover , for @xmath13 with @xmath90 let @xmath162 . then for all @xmath29 and all @xmath13 with @xmath90 , @xmath163 proof the main ideas are again taken more or less directly from the proof of theorem 1.3 in @xcite ; some caution is necessary regarding the containment in the infinite cluster in the proof of . by the definition of @xmath126 , @xmath164 the integrand is dominated by @xmath165 and tends to zero as @xmath166 for every @xmath167 in the support of @xmath135 . then follows by the dominated convergence theorem . the second part of the claim is proved similarly : first we get rid of the @xmath11-dependence by noting that , due to the fact that @xmath168 enforces @xmath82 , the translation invariance of @xmath21 implies @xmath169 next we square the right - hand side and average over all @xmath13 . using that @xmath170 also enforces @xmath171 and applying , we thus get @xmath172 now we calculate @xmath173 the integrand is again bounded by @xmath165 , for all @xmath174 , and it tends to zero as @xmath175 . the claim follows by the dominated convergence theorem . now we are ready to prove theorem [ thm2.1 ] : proof of theorem [ thm2.1 ] let @xmath176 be as in lemma [ lemma2.3 ] . using we know that @xmath176 converges in @xmath107 as @xmath166 . we denote @xmath177 . since @xmath176 is a gradient field on @xmath28 , we have @xmath178 and , more generally , @xmath179 whenever @xmath180 is a closed loop on @xmath28 . thus , we may define @xmath181 where @xmath182 is a nearest - neighbor path on @xmath43 connecting @xmath183 to @xmath184 . by the above `` loop '' conditions , the definition is independent of this path for almost every @xmath185 . the shift invariance now follows from this definition and @xmath186 . in light of shift invariance , to prove the harmonicity of @xmath187 it suffices to show that , almost surely , @xmath188{\operatorname{\sf 1}}_{\{\omega_e=1\}}=v.\ ] ] since @xmath189 , the left hand side is the @xmath166 limit of @xmath190{\operatorname{\sf 1}}_{\{\omega_e=1\}}= ( 1-q)\psi_{\epsilon}.\ ] ] the definition of @xmath126 tells us that @xmath191 . from here we get by recalling that @xmath192 tends to zero in @xmath107 . to prove the square integrability in part ( 3 ) we note that , by the construction of the corrector , @xmath193{\operatorname{\sf 1}}_{\{x\in{\mathscr{c}}_\infty\}}{\operatorname{\sf 1}}_{\{\omega_{e}=1\}}\circ\tau_x = g_{x , x+e}.\ ] ] but @xmath194 is the @xmath107-limit of @xmath107-functions @xmath176 whose @xmath107-norm is bounded by that of @xmath195 . hence follows with @xmath196 . here we will establish some basic claims whose common feature is the use of ergodic theory . modulo some care for the containment in the infinite cluster , all of these results are quite standard and their proofs ( cf sect . [ sec3.2 ] ) may be skipped on a first reading . readers interested only in the principal conclusions of this section should focus their attention on theorems [ thm3.1 ] and [ thm3.2 ] . our first result concerns the convergence of ergodic averages for the markov chain on environments . the claim that will suffice for our later needs is as follows : [ thm3.1 ] let @xmath197 . then for @xmath38-almost all @xmath198 , @xmath199 similarly , if @xmath200 is measurable with @xmath201 , then @xmath202 for @xmath38-almost all @xmath25 and @xmath44-almost all trajectories of @xmath203 . the next principal result of this section will be the ergodicity of the `` induced shift '' on @xmath86 . to define this concept , let @xmath13 be a vector with @xmath90 and , for every @xmath41 , let @xmath204 by birkhoff s ergodic theorem we know that @xmath205 has positive density in @xmath206 and so @xmath207 almost surely . therefore we can define the map @xmath208 by @xmath209 we call @xmath210 the _ induced shift_. then we claim : [ thm3.2 ] for every @xmath13 with @xmath90 , the induced shift @xmath208 is @xmath38-preserving and ergodic with respect to @xmath38 . both theorems will follow once we establish of ergodicity of the markov chain on environments ( see proposition [ prop3.5 ] ) . for finite - state ( irreducible ) markov chains the proof of ergodicity is a standard textbook material ( cf @xcite ) , but our state space is somewhat large and so alternative arguments are necessary . since we could not find appropriate versions of all needed claims in the literature , we include complete proofs . we begin by theorem [ thm3.2 ] which will follow from a more general statement , lemma [ claim : genergd ] , below . let @xmath211 be a probability space , and let @xmath212 be invertible , measure preserving and ergodic with respect to @xmath213 . let @xmath214 be of positive measure , and define @xmath215 by @xmath216 the poincar recurrence theorem ( cf ( * ? ? ? 2.3 ) ) tells us that @xmath217 almost surely . therefore we can define , up to a set of measure zero , the map @xmath218 by @xmath219 then we have : [ claim : genergd ] @xmath220 is measure preserving and ergodic with respect to @xmath221 . it is also almost surely invertible with respect to the same measure . proof ( 1 ) @xmath220 is measure preserving : for @xmath222 , let @xmath223 . then the @xmath224 s are disjoint and @xmath225 . first we show that @xmath226 to do this , we use the fact that @xmath227 is invertible . indeed , if @xmath228 for @xmath229 , then @xmath230 for some @xmath231 with @xmath232 and @xmath233 . but the fact that @xmath227 is invertible implies that @xmath234 , which means @xmath235 , a contradiction . to see that @xmath220 is measure preserving , we note that the restriction of @xmath220 to @xmath224 is @xmath236 , which is measure preserving . hence , @xmath220 is measure preserving on @xmath224 and , by , on the disjoint union @xmath237 as well . ( 2 ) @xmath220 is almost surely invertible : @xmath238 is a one - point set by the fact that @xmath227 is itself invertible . ( 3 ) @xmath220 is ergodic : let @xmath239 be such that @xmath240 and @xmath241 . assume that @xmath242 is @xmath220-invariant . then @xmath243 for all @xmath244 and all @xmath245 . this means that for every @xmath244 and every @xmath246 such that @xmath247 , we have @xmath248 . if follows that @xmath249 is ( almost - surely ) @xmath227-invariant and @xmath250 , a contradiction with the ergodicity of @xmath227 . proof of theorem [ thm3.2 ] we know that the shift @xmath251 is invertible , measure preserving and ergodic with respect to @xmath21 . by lemma [ claim : genergd ] the induced shift @xmath208 is @xmath38-preserving , almost - surely invertible and ergodic with respect to @xmath38 . in the present circumstances , theorem [ thm3.2 ] has one important consequence : [ lemma3.3 ] let @xmath252 be a subset of @xmath86 such that for almost all @xmath253 , @xmath254 then @xmath242 is a zero - one event under @xmath38 . proof the markov property and imply that @xmath255 for all @xmath245 and @xmath38-almost every @xmath253 . we claim that @xmath256 for @xmath38-almost every @xmath253 . indeed , let @xmath253 be such that @xmath257 for all @xmath245 , @xmath44-almost surely . let @xmath258 be as in and note that we have @xmath259 . by the uniqueness of the infinite cluster , there is a path of finite length connecting @xmath260 and @xmath261 . if @xmath262 is the length of this path , we have @xmath263 . this means that @xmath264 , i.e. , @xmath242 is almost surely @xmath210-invariant . by the ergodicity of the induced shift , @xmath242 is a zero - one event . our next goal will be to prove that the markov chain on environments is ergodic . let @xmath265 and define @xmath266 to be the product @xmath19-algebra on @xmath267 ; @xmath268 . the space @xmath267 is a space of two - sided sequences @xmath269the trajectories of the markov chain on environments . ( note that the index on @xmath25 is an index in the sequence which is unrelated to the value of the configuration at a point . ) let @xmath213 be the measure on @xmath270 such that for any @xmath271 , @xmath272 where @xmath93 is the markov kernel defined in . ( since @xmath38 is preserved by @xmath93 , these finite - dimensional measures are consistent and @xmath213 exists and is unique by kolmogorov s theorem . ) clearly , @xmath273 has the same law in @xmath274 as @xmath275 has in @xmath213 . let @xmath212 be the shift defined by @xmath276 . then @xmath227 is measure preserving . [ prop3.5 ] @xmath227 is ergodic with respect to @xmath213 . proof let @xmath277 denote expectation with respect to @xmath213 . pick @xmath278 that is measurable and @xmath227-invariant . we need to show that @xmath279 let @xmath280 be defined as @xmath281 . first we claim that @xmath282 almost surely . indeed , since @xmath283 is @xmath227-invariant , there exist @xmath284 and @xmath285 such that @xmath283 and @xmath286 differ only by null sets from one another . ( this follows by approximation of @xmath283 by finite - dimensional events and using the @xmath227-invariance of @xmath283 . ) now conditional on @xmath287 , the event @xmath288 is independent of @xmath289 and so lvy s martingale convergence theorem gives us @xmath290 & = e_\mu({\operatorname{\sf 1}}_{a_-}|\omega_0,\omega_{-1},\ldots,\omega_{-n})\,\underset{n\to\infty}\longrightarrow\ , { \operatorname{\sf 1}}_{a_-}={\operatorname{\sf 1}}_a , \end{aligned}\ ] ] with all equalities valid @xmath213-almost surely . next let @xmath291 be defined by @xmath292 . clearly , @xmath242 is @xmath18-measurable and , since the @xmath287-marginal of @xmath213 is @xmath38 , @xmath293 hence , to prove , we need to show that @xmath294 but @xmath283 is @xmath227-invariant and so , up to sets of measure zero , if @xmath295 then @xmath296 . this means that @xmath242 satisfies condition of lemma [ lemma3.3 ] and so holds . now we can finally prove theorem [ thm3.1 ] : proof of theorem [ thm3.1 ] recall that @xmath273 has the same law in @xmath274 as @xmath275 has in @xmath213 . hence , if @xmath297 then @xmath298 the latter limit exists by birkhoff s ergodic theorem and ( by proposition [ prop3.5 ] ) equals @xmath299 almost surely . the second part is proved analogously . equipped with the tools from the previous two sections , we can start addressing the main problem of our proof : the sublinearity of the corrector . here we will prove the corresponding claim along the coordinate directions in @xmath0 . fix @xmath13 with @xmath90 and let @xmath258 be as defined in . define a sequence @xmath300 inductively by @xmath301 and @xmath302 . the numbers @xmath303 , which are well - defined and finite on a set of full @xmath38-measure , represent the successive `` arrivals '' of @xmath28 to the positive part of the coordinate axis in direction @xmath13 . let @xmath304 be the corrector defined in theorem [ thm2.1 ] . the main goal of this section is to prove the following theorem : [ thm4.1 ] for @xmath38-almost all @xmath41 , @xmath305 the proof is based on the following facts about the moments of @xmath306 : [ lemma : next_corrector ] abbreviate @xmath307 . then 1 . 2 . @xmath309 . the proof of this proposition will in turn be based on a bound on the tails of the length of the shortest path connecting the origin to @xmath310 . we begin by showing that @xmath311 has exponential tails : [ lemma - tails ] for each @xmath37 there exists a constant @xmath312 such that for all @xmath13 with @xmath90 , @xmath313 proof the proof uses a different argument in @xmath85 and @xmath79 . in @xmath79 , we will use the fact that the slab - percolation threshold coincides with @xmath314 , as was proved by grimmett and marstrand @xcite . indeed , given @xmath37 , let @xmath315 be so large that @xmath316 contains an infinite cluster almost surely . by the uniqueness of the percolation cluster in @xmath0 , this slab - cluster is almost surely a subset of @xmath28 . our bound in is derived as follows : let @xmath317 be the event that at least one of the sites in @xmath318 is contained in the infinite connected component in @xmath316 . then @xmath319 . since the events @xmath320 , @xmath321 , are independent , letting @xmath322 we have @xmath323 from here follows by choosing @xmath151 appropriately . in dimension @xmath85 , we will instead use a duality argument . let @xmath324 be the box @xmath325 . on @xmath326 , none of the boundary sites @xmath327 are in @xmath28 . so either at least one of these sites is in a finite component of size larger than @xmath80 or there exists a dual crossing of @xmath324 in the direction of @xmath13 . by the exponential decay of truncated connectivities ( theorem 8.18 of grimmett @xcite ) and dual connectivities ( theorem 6.75 of grimmett @xcite ) , the probability of each of these events decays exponentially with @xmath80 . our next lemma provides the requisite tail bound for the length of the shortest path between the origin and @xmath310 : [ claim : pathlen ] let @xmath328 be the length of the shortest occupied path from @xmath260 to @xmath310 . then there exist a constant @xmath133 and @xmath329 such that for every @xmath245 , @xmath330 proof let @xmath331 be the length of the shortest path from @xmath260 to @xmath11 in configuration @xmath25 . pick @xmath122 such that @xmath332 is an integer . then @xmath333 in light of lemma [ lemma - tails ] , the claim will follow once we show that the probability of all events in the giant union on the right - hand side is bounded by @xmath334 with some @xmath335 ( independently of @xmath336 ) . we will use the following large - deviation result from theorem 1.1 of antal and pisztora @xcite : there exist constants @xmath337 such that @xmath338 once @xmath339 is sufficiently large . unfortunately , we can not use this bound in directly , because @xmath340 can be arbitrarily close to @xmath260 ( in @xmath341 distance on @xmath0 ) . to circumvent this problem , let @xmath342 be the site @xmath343 such that @xmath344 and let @xmath345 . then , on @xmath346 , either @xmath347 or at least one site `` between '' @xmath348 and @xmath349 is connected to either @xmath260 or @xmath340 by a path longer than @xmath350 . since on @xmath351 we must have @xmath352 for at least one @xmath353 , we have @xmath354 now all events in the first giant union have the same probability , which is exponentially small by lemma [ lemma - tails ] . as to the second union , by we know that @xmath355 whenever @xmath356 is so small that @xmath357 , and a similar bound holds for @xmath358 as well ( except that here we need @xmath359 ) . the various unions then contribute a linear factor in @xmath80 , which is absorbed into the exponential once @xmath80 is sufficiently large . it is possible that a proper merge of the arguments in the previous two proofs might yield the same result without relying on antal and pisztora s bound . ( indeed , the main other `` external '' ingredient of our proofs is grimmett and marstrand s paper @xcite which lies at the core of @xcite as well . ) however , we find the argument using conceptually cleaner and so we are content with the present , even though not necessarily optimal , proof . next we state a trivial , but interesting technical lemma : [ claim : caucschwa ] let @xmath360 and @xmath361 . suppose that @xmath362 are random variables such that @xmath363 and let @xmath364 be a random variable taking values in positive integers such that @xmath365 for some @xmath366 satisfying @xmath367 then @xmath368 . explicitly , @xmath369}},\ ] ] where @xmath370 is a finite constant depending only on @xmath34 , @xmath371 and @xmath366 . proof let us define @xmath372 by @xmath373 . from the hlder inequality and the uniform bound on @xmath374 we get @xmath375 under the assumption that @xmath364 has @xmath366 moments , we get @xmath376 by invoking the hlder inequality one more time . the first term on the right - hand side is finite whenever @xmath366 obeys the bound . proof of proposition [ lemma : next_corrector ] let @xmath83 be the corrector . by theorem [ thm2.1 ] , on the set @xmath377 , @xmath378 is an @xmath107-limit of functions @xmath379 , as @xmath166 . to prove that @xmath380 , recall the notation @xmath381 from lemma [ lemma2.3 ] and let as in lemma [ claim : pathlen]@xmath328 be the length of the shortest path from @xmath260 to @xmath310 . then @xmath382 but theorem [ thm2.1 ] ensures that @xmath383 for all @xmath11 and @xmath13 and all @xmath122 , while the number of terms in the sum does not exceed @xmath384 . by lemma [ claim : pathlen ] , @xmath364 has all moments and so , by lemma [ claim : caucschwa ] , @xmath385 for all @xmath386 . in particular , @xmath380 . in order to prove part ( 2 ) , we first note that a uniform bound on @xmath387-norm of @xmath388 for some @xmath389 implies that the family @xmath390 is uniformly integrable . since @xmath391 in probability , @xmath391 in @xmath392 and it thus suffices to prove @xmath393 this is implied by theorem [ thm3.2 ] and the fact @xmath394 with @xmath126 absolutely integrable . proof of theorem [ thm4.1 ] let @xmath395 , and let @xmath210 be the induced shift in the direction of @xmath13 . then we can write @xmath396 by proposition [ lemma : next_corrector ] we have @xmath397 and @xmath398 . since theorem [ thm3.2 ] ensures that @xmath210 is @xmath38-preserving and ergodic , the claim follows from birkhoff s ergodic theorem . here we will prove the principal technical estimates of this work . the level of control is different in @xmath85 and @xmath79 , so we treat these cases separately . ( notwithstanding , the @xmath79 proof applies in @xmath85 as well . ) we begin with an estimate of the corrector in large boxes in @xmath399 : [ thm5.1 ] let @xmath85 and let @xmath304 be the corrector defined in theorem [ thm2.1 ] . then for @xmath38-almost every @xmath41 , @xmath400 the proof will be based on the following concept : [ def5.6 ] given @xmath401 and @xmath122 , we say that a site @xmath29 is @xmath402-_good _ ( or just _ good _ ) in configuration @xmath198 if @xmath403 and @xmath404 holds for every @xmath405 of the form @xmath406 , where @xmath407 and @xmath13 is a unit coordinate vector . we will use @xmath408 to denote the set of @xmath402-good sites in configuration @xmath25 . on the basis of theorem [ thm4.1 ] it is clear that for each @xmath122 there exists a @xmath409 such that the @xmath410 . our first goal is to estimate the size of the largest interval free of good points in blocks @xmath411 $ ] on the coordinate axes : [ lemma5.3d ] let @xmath13 be one of the principal lattice vectors in @xmath399 and , given @xmath122 , let @xmath412 be so large that @xmath410 . for all @xmath245 and @xmath198 , let @xmath413 be the ordered set of all integers from @xmath411 $ ] such that @xmath414 . let @xmath415 ( if no such @xmath416 exists , we define @xmath417 . ) then @xmath418 proof since @xmath21 is @xmath419 invariant and @xmath419 is ergodic , we have @xmath420 @xmath21-almost surely . a similar statement applies to the limit @xmath421 . but if @xmath422 does not tend to zero , at least one of these limits would not exist . proof of theorem [ thm5.1 ] fix @xmath423 and let @xmath424 be such that @xmath425 for all @xmath426 ( we are using that @xmath427 increases with @xmath412 ) . let @xmath428 be the set of configurations such that the conclusion of lemma [ lemma5.3d ] applies for both @xmath11 and @xmath12-axes , and that shift - invariance holds for all @xmath429 in the infinite cluster . we will show that for every @xmath430 the limsup in is less than @xmath431 almost surely . let @xmath432 and @xmath433 denote the coordinate vectors in @xmath399 . fix @xmath430 and adjust @xmath426 so that @xmath434 . ( this is possible by the definition of @xmath435 . ) then we define @xmath436 to be the increasing two - sided sequence of all integers such that @xmath437 exhausts all @xmath402-good points on the @xmath432-axis , i.e. , @xmath438 if @xmath439 be the maximal gap between consecutive @xmath440 s that lie in @xmath411 $ ] , cf , we define @xmath441 be the least integer such that @xmath442 for all @xmath443 . similarly we identify a two - sided increasing sequence @xmath444 of integers exhausting the sites such that @xmath445 and let @xmath446 be the quantity corresponding to @xmath441 in this case . let @xmath447 . we claim that for all @xmath448 , @xmath449 of _ good lines _ @xmath450 and @xmath451 see fig . [ fig3 ] . as a first step we will use the harmonicity of @xmath187 to deal with @xmath452 . indeed , any such @xmath11 is enclosed between two horizontal and two vertical grid lines and every path on @xmath28 connecting @xmath11 to `` infinity '' necessarily intersects one of these lines at a point which is also in @xmath28 . applying the maximum ( and minimum ) principle for harmonic functions we get @xmath453 apart and , in particular , they all intersect the block @xmath454\times[-2n,2n]$ ] . to estimate the maximum on the grid , we pick , say , a horizontal grid line with @xmath12-coordinate @xmath455 and note that , by , for every @xmath82 on this line , @xmath456 by and the fact that @xmath457 we have @xmath458 whenever @xmath11 is such that @xmath459 . applying the same argument to the vertical line through the origin , and @xmath11 replaced by @xmath460 , we get @xmath461 for every @xmath462 with @xmath463 . combining this with , the estimate and the whole claim are finally proved . interestingly , a variant of the above strategy for controlling the corrector in @xmath85 has independently been developed by chris hoffman @xcite to control the geodesics in the first - passage percolation on @xmath399 . in @xmath79 we have the following weaker version of theorem [ thm5.1 ] : [ thm5.4 ] let @xmath79 . then for all @xmath122 and @xmath38-almost all @xmath25 , @xmath464 here we fix the dimension @xmath7 and run an induction over @xmath465-dimensional sections of the @xmath7-dimensional box @xmath466 . specifically , for each @xmath467 , let @xmath468 be the @xmath465-dimensional box @xmath469 the induction eventually gives for @xmath470 thus proving the theorem . since it is not advantageous to assume that @xmath471 , we will carry out the proof for _ differences _ of the form @xmath472 with @xmath473 . for each @xmath198 , we thus consider the ( upper ) density @xmath474 note that the infimum is taken only over sites in one - dimensional box @xmath475 . our goal is to show by induction that @xmath476 almost surely for all @xmath467 . the induction step is encapsulated into the following lemma : [ lemma5.5 ] let @xmath477 . if @xmath476 , @xmath21-almost surely , then also @xmath478 , @xmath21-almost surely . before we start the formal proof , let us discuss its main idea : suppose that @xmath476 for some @xmath479 , @xmath21-almost surely . pick @xmath122 . then for @xmath21-almost every @xmath25 and all sufficiently large @xmath80 , there exists a set of sites @xmath480 such that @xmath481 and @xmath482 moreover , @xmath80 sufficiently large , @xmath483 could be picked so that @xmath484 and , assuming @xmath485 , the non-@xmath402-good sites could be pitched out with little loss of density to achieve even @xmath486 ( all these claims are direct consequences of the pointwise ergodic theorem and the fact that @xmath487 converges to the density of @xmath28 as @xmath488 . ) as a result of this construction we have @xmath489 for any @xmath490 and any @xmath491 of the form @xmath492 . thus , if @xmath493 are of the latter form , @xmath494 and @xmath495see fig . [ fig4 ] for an illustration then implies @xmath496 , implying a bound of the type but one - dimension higher . unfortunately , the above is not sufficient to prove for all but a vanishing fraction of all sites in @xmath497 . the reason is that the @xmath371 s and @xmath366 s for which holds need to be of the form @xmath492 for some @xmath498 . but @xmath28 will occupy only about @xmath499 fraction of all sites in @xmath468 , and so this argument does not permit us to control more than fraction about @xmath500 of @xmath501 . to fix this problem , we will have to work with a `` stack '' of translates of @xmath468 at the same time . ( these correspond to the stack of horizontal lines on the left of of fig . [ fig4 ] . ) explicitly , consider the collection of @xmath465-boxes @xmath502 here @xmath503 is a deterministic number chosen so that , for a given @xmath504 , the set @xmath505 is so large that @xmath506 once @xmath80 is sufficiently large . these choices ensure that @xmath507-fraction of @xmath468 is now `` covered '' which by repeating the above argument gives us control over @xmath508 for nearly the same fraction of all sites @xmath509 . proof of lemma [ lemma5.5 ] let @xmath479 and suppose that @xmath476 , @xmath21-almost surely . fix @xmath510 with @xmath511 and let @xmath503 be as defined above . choose @xmath122 so that @xmath512 for a fixed but large @xmath412 , and @xmath21-almost every @xmath25 and @xmath80 exceeding an @xmath25-dependent quantity , for each @xmath513 , we can find @xmath514 satisfying the properties ( [ 5.19][5.21])with @xmath468 replaced by @xmath515 . given @xmath516 , let @xmath517 be the set of sites in @xmath501 whose projection onto the linear subspace @xmath518 belongs to the corresponding projection of @xmath519 . note that the @xmath520 could be chosen so that @xmath521 . by their construction , the projections of the @xmath520 s , @xmath513 , onto @xmath522 `` fail to cover '' at most @xmath523 sites in @xmath524 , and so at most @xmath525 sites in @xmath468 are not of the form @xmath526 for some @xmath527 . it follows that @xmath528 i.e. , @xmath517 contains all except at most @xmath529-fraction of all sites in @xmath497 that we care about . next we note that if @xmath412 is sufficiently large , then for every @xmath530 , the set @xmath522 contains at least @xmath531-fraction of sites @xmath11 such that @xmath532 since we assumed , once @xmath533 , for each pair @xmath534 with @xmath530 such @xmath535 and @xmath536 can be found so that @xmath537 and @xmath538 . but the @xmath520 s were picked to make true and so via these pairs of sites we now show that @xmath539 for every @xmath540 ; see again ( the left part of ) fig . [ fig4 ] . from and we now conclude that for all @xmath541 , @xmath542 provided that @xmath543 . if @xmath544 denotes the right - hand side of before taking @xmath166 , the bounds and and @xmath521 yield @xmath545 for @xmath21-almost every @xmath25 . but the left - hand side of this inequality increases as @xmath166 while the right - hand side decreases . thus , taking @xmath166 and @xmath546 proves that @xmath547 holds @xmath21-almost surely . proof of theorem [ thm5.4 ] the proof is an easy consequence of lemma [ lemma5.5 ] . first , by theorem [ thm4.1 ] we know that @xmath548 for @xmath38-almost every @xmath25 . invoking appropriate shifts , the same conclusion applies @xmath21-almost surely . using induction on dimension , lemma [ lemma5.5 ] then tells us that @xmath549 for @xmath38-almost every @xmath25 . let @xmath41 . by theorem [ thm4.1 ] , for each @xmath122 there is @xmath550 with @xmath551 such that for all @xmath448 , we have @xmath552 for all @xmath553 . using this to estimate away the infimum in , the fact that @xmath554 now immediately implies for all @xmath122 . here we will finally prove our main theorems . first , in sect . [ sec6.1 ] , we will show the convergence of the `` lazy '' walk on the deformed graph to brownian motion and then , in sect . [ sec6.2 ] , we use our previous results on corrector growth to extend this to the walk on the original graph . this separation will allow us to treat the parts of the proof common for @xmath85 and @xmath79 in a unified way . theorem [ thm:2ndmainthm ] , which concerns the `` agile '' walk , is proved in sect . [ sec6.3 ] . we begin with a simple observation that will drive all underlying derivations : [ lemma6.2 ] fix @xmath41 and let @xmath555 be the corrector . given a path of random walk @xmath42 with law @xmath44 , let @xmath556 then @xmath557 is an @xmath107-martingale for the filtration @xmath558 . moreover , conditional on @xmath559 , the increments @xmath560 have the same law as @xmath561 . proof since @xmath562 is bounded , @xmath563 is bounded and so @xmath564 is square integrable with respect to @xmath44 . since @xmath187 is harmonic with respect to the transition probabilities of the random walk @xmath48 with law @xmath44 , we have @xmath565 @xmath44-almost surely . since @xmath564 is @xmath566-measurable , @xmath567 is a martingale . the stated relation between the laws of @xmath560 and @xmath561 is implied by the shift - invariance and the fact that @xmath567 is a simple random walk on the deformed infinite component . next we will establish the convergence of the above martingale to brownian motion . the precise statement is as follows : [ thm6.1 ] let @xmath1 , @xmath37 and @xmath41 . let @xmath42 be the random walk with law @xmath44 and let @xmath557 be as defined in . let @xmath568 be defined by @xmath569 then for all @xmath49 and @xmath38-almost every @xmath25 , the law of @xmath570 on @xmath50,{\mathscr{w } } _ t)$ ] converges weakly to the law of an isotropic brownian motion @xmath56 with diffusion constant @xmath72 , i.e. , @xmath571 , where @xmath572 proof without much loss of generality , we may confine ourselves to the case when @xmath573 . let @xmath574 and fix a vector @xmath144 . we will show that ( the piece - wise linearization ) of @xmath575 scales to one - dimensional brownian motion . for @xmath576 , consider the random variable @xmath577 ^ 2{\operatorname{\sf 1}}_{\{|a\cdot(m_{k+1}^{(\omega)}-m_k^{(\omega)})|\ge\epsilon\sqrt n\}}\big|{\mathscr{f}}_k\bigr).\ ] ] in order to apply the lindeberg - feller functional clt for martingales ( theorem 7.7.3 of durrett @xcite ) , we need to verify that for @xmath38-almost every @xmath25 , 1 . @xmath578 in @xmath44-probability for all @xmath579 $ ] and some @xmath580 . 2 . @xmath581 in @xmath44-probability for all @xmath122 . both of these conditions will be implied by theorem [ thm3.1 ] . indeed , by the last conclusion of lemma [ lemma6.2 ] we may write @xmath582 where @xmath583 ^ 2{\operatorname{\sf 1}}_{\{|a\cdot m_1^{(\omega)}|\ge k\}}\bigr).\ ] ] now if @xmath584 , theorem [ thm3.1 ] tells us that , for @xmath38-almost every @xmath25 , @xmath585 ^ 2\bigr)\bigr)= \frac1d d|a|^2,\ ] ] where we used the symmetry of the joint expectations under rotations by @xmath586 . from here condition ( 1 ) follows by scaling out the @xmath587-dependence first and working with @xmath588 instead of @xmath80 . on the other hand , when @xmath122 , we have @xmath589 once @xmath80 is sufficiently large and so , @xmath38-almost surely , @xmath590 ^ 2{\operatorname{\sf 1}}_{\{|a\cdot m_1^{(\omega)}|\ge k\}}\bigr)\bigr ) \,\underset{k\to\infty}\longrightarrow\,0,\ ] ] where to apply dominated convergence we used that @xmath591 . hence , the above conditions ( 1 ) and ( 2 ) hold in fact , even with limits taken @xmath44-almost surely . applying the martingale functional clt and the cramr - wold device ( theorem 2.9.2 of @xcite ) , we conclude that , for @xmath38-almost every @xmath25 , the linear interpolation of the sequence @xmath592 converges to isotropic brownian motion with covariance matrix @xmath593 . to make the proof complete , we need to show that @xmath594 . here the finiteness is immediate by the square - integrability of @xmath304 . the positivity can be shown in many ways : either by a direct computation from using that @xmath595 [ which in turn is implied by @xmath596 for every coordinate vector @xmath13 ] or by invoking the sublinearity of the corrector proved in theorems [ thm5.1][thm5.4 ] , or by an appeal to the lower ( or , alternatively , upper ) bound in ( * ? ? ? * theorem 1 ) . it remains to estimate the influence of the harmonic deformation on the path of the walk . as already mentioned , while our proof in @xmath85 is completely self - contained , for @xmath79 we rely heavily on ( a discrete version of ) the sophisticated theorem 1 of barlow @xcite . let us first dismiss the two - dimensional case of theorem [ thm : mainthm ] : proof of theorem [ thm : mainthm ] ( @xmath85 ) we need to extend the conclusion of theorem [ thm6.1 ] to the linear interpolation of @xmath48 . since the corrector is an additive perturbation of @xmath564 , it clearly suffices to show that , for @xmath38-almost every @xmath25 , @xmath598 by theorem [ thm5.1 ] we know that for every @xmath122 there exists a @xmath599 such that @xmath600 if @xmath601 , then this implies @xmath602 but the above clt for @xmath603 tells us that @xmath604 converges in law to the maximum of a brownian motion @xmath605 over @xmath579 $ ] . hence , if @xmath606 denotes the probability law of the brownian motion , the portmanteau theorem ( theorem 2.1 of @xcite ) allows us to conclude @xmath607 the right - hand side tends to zero as @xmath166 for all @xmath504 . in order to prove the same result in @xmath79 , we will need the following upper bounds on the transition probability of our random walk : [ thm6.3a ] ( 1 ) there is a random variable @xmath608 with @xmath609 such that for all @xmath41 and all @xmath403 , @xmath610 \(2 ) there are constants @xmath611 and random variables @xmath612 such that for all @xmath41 , all @xmath403 , all @xmath613 , and all @xmath614 , @xmath615 moreover , the random variables @xmath616 have stretched - exponential tails , i.e. , there exist constants @xmath617 and @xmath618 such that for all @xmath29 , @xmath619 for a continuous - time version of our walk , these bounds are the content of theorem 1 of barlow @xcite . ( in fact , the continuous - time version of the bound was obtained already by mathieu and remy @xcite . ) unfortunately , to derive theorem [ thm6.3a ] from barlow s theorem 1 , one needs to invoke various non - trivial facts about percolation and/or mixing of markov chains . in appendix [ appa ] we list these facts and show how to assemble all ingredients together to establish the above upper bounds . proof of theorem [ thm : mainthm ] ( @xmath79 ) we will adapt ( the easier part of ) the proof of theorem 1.1 in sidoravicius and sznitman @xcite . first we show that the laws of @xmath620 on @xmath50,{\mathscr{w } } _ t)$ ] are tight . to that end it suffices to show ( e.g. , by theorem 8.6 of ethier - kurtz @xcite ) that if @xmath621 is the class of all stopping times of the filtration @xmath622 , then @xmath623 as in @xcite , we replace @xmath624 by its integer - valued approximation . explicitly , let @xmath625 and let @xmath510 be a number such that @xmath626 . since @xmath627 differs from @xmath628 by a constant of order unity , and similarly for @xmath629 and @xmath630 , we have @xmath631 for some constant @xmath632 . this allows us to estimate by means of the second moment of @xmath633 . recalling that @xmath634 , we may assume that @xmath635 . by we know that there exists an almost - surely finite random variable @xmath636 such that @xmath637 once @xmath638 , where @xmath639 . since @xmath640 , this implies that @xmath641^\zeta$ ] . theorem [ thm6.3a](2 ) and the strong markov property@xmath627 is a stopping time of the random walk tell us that , for some constant @xmath642 ( depending only on @xmath643 and the dimension ) , @xmath644 here we used @xmath645 and let @xmath646 be such that @xmath647^\zeta$ ] for all @xmath648 . the bound is now proved by combining ( [ e:6.18a][e:6.19a ] ) and taking the required limits . once we know that the laws of @xmath620 are tight , it suffices to show the convergence of finite - dimensional distributions . in light of theorem [ thm6.1 ] ( and the markov property of the walk ) , for that it is enough to prove that for all @xmath649 and @xmath38-almost every @xmath25 , @xmath650 without loss of generality , we need to do this only for @xmath651 . by theorem [ thm6.3a ] , the random variable @xmath562 lies with probability @xmath652 in the block @xmath653^d\cap{\mathbb z}^d$ ] , provided @xmath654 sufficiently large ( with `` large '' depending possibly on @xmath25 ) . using theorem [ thm6.3a](1 ) to estimate @xmath655 for @xmath11 inside this block , we have @xmath656 but theorem [ thm5.4 ] tells us that , for all @xmath657 and @xmath38-almost every @xmath25 , the second term tends to zero as @xmath658 . this proves and the whole claim . it remains to prove theorem [ thm:2ndmainthm ] for the `` agile '' version of simple random walk on @xmath28 . since the proof is based entirely on the statement of theorem [ thm : mainthm ] , we will resume a unified treatment of all @xmath1 . first we will make the observation that the times of the two walks run proportionally to each other : [ lemma6.3 ] let @xmath58 be the stopping times defined in . then for all @xmath659 and @xmath38-almost every @xmath25 , @xmath660 where @xmath661 proof this is an easy consequence of the second part of theorem [ thm3.1 ] and the fact that for @xmath38-almost every @xmath25 we have @xmath662 once @xmath663 . indeed , let @xmath664 . for @xmath665 the statement holds trivially so let us assume that @xmath649 . if @xmath80 is so large that @xmath666 , we have @xmath667 since @xmath668 as @xmath658 , by theorem [ thm3.1 ] the right hand side converges to the expectation of @xmath669 in the annealed measure @xmath274 . a direct calculation shows that this expectation equals @xmath670 . proof of theorem [ thm:2ndmainthm ] the proof is based on a standard approximation argument for stochastic processes . let @xmath671 be as in theorem [ thm : mainthm ] and recall that @xmath65 is a linear interpolation of the values @xmath672 for @xmath673 . the path - continuity of the processes @xmath671 as well as the limiting brownian motion implies that for every @xmath122 there is a @xmath504 such that @xmath674 once @xmath80 is sufficiently large . similarly , lemma [ lemma6.3 ] , the continuity of @xmath675 and the monotonicity of @xmath676 imply that for @xmath80 sufficiently large , @xmath677 on the intersection of these events , the equality @xmath678 yields @xmath679 in light of piece - wise linearity this shows that , with probability at least @xmath680 , the paths @xmath681 and @xmath682 are within a multiple of @xmath356 in the supremum norm of each other . in particular , if @xmath683 denotes the weak limit of the process @xmath684 , then @xmath685 converges in law to @xmath686 . the latter is an isotropic brownian motion with diffusion constant @xmath74 . let @xmath687 denote the continuous - time random walk which attempts a jump to one of its nearest - neighbors at rate one ( regardless of the number of accessible neighbors ) . let @xmath688 denote the probability that @xmath689 started at @xmath11 is at @xmath12 at time @xmath587 . in his paper @xcite , barlow proved the following statement : there exist constants @xmath690 and , for each @xmath29 , a random variable @xmath691 such that for all @xmath692 and all @xmath693 , @xmath694 moreover , @xmath695 has uniformly stretched - exponential tails , i.e. , @xmath696 barlow provides also a corresponding , and significantly harder - to - prove lower bound which requires the additional condition @xmath697 . however , for , this condition is redundant . in the remarks after his theorem 1 , barlow mentions that appropriate modifications to his arguments yield the corresponding discrete time estimates . here we present the details of these modifications which are needed to make our proof of the invariance principles in theorems [ thm : mainthm ] and [ thm:2ndmainthm ] complete . notice that we do not re - prove barlow s bounds in their full generality , just the absolute minimum necessary for our purposes . there will be two kinds of bounds on the heat - kernel as a function of the terminal position of the walk after @xmath80 steps : a uniform bound by a constant times @xmath698 and a non - uniform , gaussian bound on the tails . we begin with the statement of the uniform upper bound : [ propa.1 ] let @xmath1 and let @xmath53 . there exists a random variable @xmath608 with @xmath699 such that for all @xmath41 and all @xmath82 , @xmath700 the proof will invoke the isoperimetric bound from barlow @xcite : [ lemmaa.2 ] there exists a constant @xmath701 such that for @xmath38-almost every @xmath25 and all @xmath702 sufficiently large , @xmath703 for all @xmath704^d$ ] such that @xmath705 . proof this is a consequence of proposition 2.11 on page 3042 , and lemma 2.13 on page 3045 of barlow s paper @xcite . this isoperimetric bound will be combined with the technique of _ evolving sets _ , developed by morris and peres @xcite , whose salient features we will now recall . consider a markov chain on a countable state - space @xmath120 , let @xmath706 be the transition kernel and let @xmath707 be a stationary measure . let @xmath708 and for each @xmath709 , let @xmath710 . for each set @xmath711 with finite non - zero total measure @xmath712 we define the _ conductance _ @xmath713 by @xmath714 for sufficiently large @xmath371 , we also define the function @xmath715 the following is the content of theorem 2 in morris and peres @xcite : suppose that @xmath716 for some @xmath717 $ ] and all @xmath718 . let @xmath122 and @xmath719 . if @xmath80 is so large that @xmath720}^{4/\epsilon}\frac4{u\phi(u)^2}{\text{\rm d}\mkern0.5mu}u,\ ] ] then @xmath721 equipped with this powerful result , we are now ready to complete the proof of proposition [ propa.1 ] : proof of proposition [ propa.1 ] first we will prove the desired bound for even times . fix @xmath198 and let @xmath722 be the random walk on @xmath43 observed only at even times . for each @xmath131 , let us use @xmath706 to denote the transition probability @xmath723 . let @xmath724 denote the degree of @xmath11 on @xmath43 . then @xmath707 is an invariant measure of this chain . moreover , by our restriction to even times we have @xmath725 and so ( [ a.5][a.6 ] ) can be applied . by lemma [ lemmaa.2 ] we have that @xmath726 for some @xmath727 and all sets @xmath220 of the form @xmath728^d$ ] for @xmath729 . hence @xmath730 for some finite @xmath731 . plugging into the integral and using that @xmath707 is bounded , we find that if @xmath732 , then holds . here @xmath733 is a positive constant that may depend on @xmath25 . choosing the minimal @xmath80 possible , and applying @xmath734 , the bound proves the desired claim for all even times . to extend the result to odd times , we apply the markov property at time one . next we will attend to the gaussian - tail bound . given the random variables @xmath735 from ( [ a.10][sxtail ] ) , define random variables @xmath612 by @xmath736 here is a restatement of the corresponding bound from theorem [ thm6.3a ] : [ propa.3 ] let @xmath1 and @xmath53 . there exist constants @xmath611 such that for all @xmath41 , all @xmath403 , all @xmath613 and all @xmath737 , @xmath738 proof the proof is an adaptation of barlow s theorem 1 to the discrete setting . let @xmath48 be the discrete time random walk , and let @xmath687 be the continuous time random walk with jumps occurring at rate @xmath739 , both started at @xmath11 . we consider the coupling of the two walks such that they make the same moves . we will use @xmath606 and @xmath740 to denote the coupling measure and the corresponding expectation , respectively . let @xmath741 and let @xmath742 be the event that @xmath743 . pick @xmath744 and let @xmath745 be the amount of time in @xmath746 $ ] that the walk @xmath747 spends at distance larger than @xmath748 from @xmath11 . by the inequality @xmath749 it suffices to derive an appropriate upper bound on @xmath750 and a matching lower bound on @xmath751 . note that we may assume that @xmath752 because otherwise we have @xmath753 and there is nothing to prove . to derive an upper bound on @xmath750 , we note that for @xmath754 , our choice @xmath741 implies @xmath693 . the expectation can then be bounded using : @xmath755 where @xmath756 and @xmath757 are constants ( possibly depending on @xmath412 ) . it thus remains to prove that , for some constant @xmath758 , @xmath759 to derive this inequality , let us recall that the transitions of @xmath689 happen at rate one , and they are independent of the path of the walk . hence , if @xmath760 is the event that @xmath689 attempted at least @xmath80 jumps by time @xmath761 , then @xmath762 is bounded away from zero for all @xmath245 . therefore , it suffices to prove that @xmath763 . let @xmath227 be the first time when the walk @xmath747 is farther from @xmath11 than @xmath702 . on @xmath764 , this happens before time @xmath761 , i.e. , @xmath765 . let @xmath766^d\cap{\mathbb z}^d$ ] and @xmath767^d\cap{\mathbb z}^d$ ] . then for values @xmath768 on the external boundary of @xmath769which are those that @xmath770 can take the bound tells us @xmath771 provided that @xmath772 . but our assumptions @xmath741 and @xmath752 imply @xmath773 , and so in light of the fact that @xmath774 on @xmath764 , actually holds for all @xmath587 such that @xmath775 $ ] . plugging @xmath770 for @xmath768 on the left - hand side and taking expectation gets us an upper bound on @xmath776with @xmath587 now playing the role of @xmath777 . hence , @xmath778 choosing @xmath412 sufficiently large , the right - hand side grows linearly in @xmath80 . proof of theorem [ thm6.3a ] part ( 1 ) is a direct consequence of proposition [ propa.1 ] , while part ( 2 ) follows from proposition [ propa.3 ] and the fact that if the @xmath695 have stretched exponential tails ( uniformly in @xmath11 ) , then so do the @xmath779 s . while our control of the corrector in @xmath79 is sufficient to push the proof of the functional clt through , it is not sufficient to provide the _ conceptually correct _ proof of the kind we have constructed for @xmath85 . however , we do not see any reason why @xmath79 should be different from @xmath85 , so our first conjecture is : [ conj : cor_small ] theorem [ thm5.1 ] is true in all @xmath1 . our proof of theorem [ thm5.1 ] in @xmath85 hinged on the fact that the corrector plus the position is a harmonic function on the percolation cluster . of interest is the question whether harmonicity is an essential ingredient or just mere convenience . yuval peres suggested the following generalization of conjecture [ conj : cor_small ] : [ ques : yuval ] let @xmath780 be a shift invariant , ergodic process on @xmath0 whose gradients are in @xmath392 and have expectation zero . is it true that @xmath781^d}\,\bigl|f(x)\bigr| = 0\ ] ] almost surely ? _ update _ : the above question , while obviously true in @xmath782 , has a negative answer in all @xmath1 . the first counterexample , based on constructions in @xcite and @xcite , was provided to us by martin zerner . later tom liggett pointed out the following , embarrassingly simple , counterexample : let @xmath783 be i.i.d . with distribution function @xmath784 for @xmath785 . then @xmath786 is shift - invariant , ergodic , with @xmath397 and the gradients of @xmath111 having zero mean , yet @xmath787 has a non - trivial distributional limit as @xmath658 . the harmonic embedding of @xmath28 has been indispensable for our proofs , but it also appears to be a very interesting object in its own right . this motivates many questions about the corrector @xmath83 . unfortunately , at the moment it is not even clear what properties make the corrector unique . the following question has been asked by scott sheffield : is it true that , for a.e . @xmath41 , there exists only one vector - valued function @xmath555 on @xmath43 such that @xmath187 is harmonic on @xmath43 , @xmath788 and @xmath789 as @xmath790 ? if this question is answered in the affirmative , we could generate the corrector by its finite - volume approximations ( this would also fully justify fig . [ fig1 ] ) . if we restrict ourselves to functions that have the shift - invariance property , uniqueness can presumably be shown using the `` electrostatic methods '' from , e.g. , @xcite . however , it is not clear whether holds for the corrector defined by the thermodynamic limit from finite boxes . as to the more detailed properties of the corrector , for the purposes of the present work one would like to know how @xmath83 scales with @xmath11 and whether it has a well - defined scaling limit . we believe that , in sufficiently high dimension , the corrector is actually tight : [ conj4 ] let @xmath791 . then for each @xmath122 there exists @xmath409 such that @xmath792 for all @xmath29 . it appears that one might be able to prove conjecture [ conj4 ] by using barlow s heat - kernel estimates . to capture the behavior in low dimensions , we make a somewhat wilder guess : [ conj5 ] let @xmath78 . then the law of @xmath793 on compact subsets of @xmath81 converges weakly ( as @xmath166 ) to gaussian free field , i.e. , a multivariate gaussian field with covariance proportional to @xmath794 , where @xmath483 is the dirichlet laplacian on @xmath81 and @xmath795 is the @xmath7-dimensional unit matrix . here is a heuristic reasoning that led us to these conjectures : consider the problem of random conductances to avoid problems with conditioning on containment in the infinite cluster . to show the above convergence , we need that for any smooth @xmath796 with compact support , @xmath797 where @xmath798 and @xmath483 denote the ( continuous ) gradient and laplacian , respectively , and where @xmath799 is a mean - zero , covariance-@xmath370 multivariate normal random variable . next we note that the corrector is defined , more or less , as the solution to the equation @xmath800 , where @xmath120 is the local drift and @xmath801 is the relevant generator , which is basically a discrete laplacian on @xmath0 . thus , if @xmath802 is smooth with compact support and @xmath803 , then @xmath804 the convergence statement would then follow from provided we can replace the `` discretized '' laplacian @xmath805 by its continuous counterpart @xmath806 . note that for @xmath782 and conductances bounded away from zero , conjecture [ conj5 ] is actually a theorem . indeed , the corrector is a random walk with increments given by reciprocal conductances and so the convergence follows by the invariance principle for random walks . conjecture [ conj5 ] suggests that conjecture [ conj4 ] applies for @xmath79 . despite the emphasis on the harmonic embedding of @xmath28 , our proofs used , quite significantly , the underlying group structure of @xmath0 ; e.g. , in sect . [ sec : sec4 ] . presumably this will not prevent application of our method to other regular lattices , but for more irregular graphs , e.g. , voronoi percolation in @xmath81 , significant changes may be necessary . a similar discussion applies to various natural subdomains of @xmath0 ; for instance , it is not clear how to adapt our proof to random walk on the infinite percolation cluster in the half - space @xmath807 . a different direction of generalizations are the models of _ long - range _ percolation with power - law decay of bond probabilities . here we conjecture : [ conj : stable ] let @xmath78 and consider long - range percolation obtained by adding to @xmath0 a bond between every two distinct sites @xmath808 independently with probability proportional to @xmath809 . if @xmath810 , then the corresponding random walk scales to a symmetric @xmath811-stable levy process in @xmath81 . note that , according to this conjecture , in @xmath782 , the interval @xmath810 of `` interesting '' exponents is larger than the interval for which an infinite connected component may occur even without the `` help '' of nearest neighbor connections . on the other hand , in dimensions @xmath79 , the interval conjectured for stable convergence is strictly smaller than that of `` genuine '' long - range percolation behavior , as defined , e.g. , in terms of the scaling of graph distance with euclidean distance ; cf @xcite . the research of m.b . was supported by the nsf grant dms-0306167 . part of the research was performed while n.b . visited eth - fim and m.b . visited microsoft research in redmond . we wish to thank these institutions for their hospitality and financial support . we are also grateful to g.y . amir , a. de masi , a. dembo , p. ferrari , t. liggett , s. olla , y. peres , o. schramm , s. sheffield , v. sidoravicius , a .- s . sznitman and m.p.w . zerner for interesting and useful discussions at various stages of this project . a. de masi , p.a . ferrari , s. goldstein and w.d . wick ( 1985 ) . invariance principle for reversible markov processes with application to diffusion in the percolation regime . in : _ particle systems , random media and large deviations ( brunswick , maine ) _ , pp . 7185 , _ contemp . math . _ , * 41 * , amer . soc . , providence , ri . a. de masi , p.a . ferrari , s. goldstein and w.d . wick ( 1989 ) . an invariance principle for reversible markov processes . applications to random motions in random environments . _ j. statist . * 55 * , no . 3 - 4 , 787855 . r. durrett ( 2005 ) . _ probability : theory and examples _ ( third edition ) , brooks / cole thomson learning , belmont , ca . ethier and t.g . kurtz ( 1986 ) . _ markov processes . characterization and convergence_. wiley series in probability and mathematical statistics . john wiley & sons , inc . , new york . d. levin and y. peres ( 1999 ) . energy and cutsets in infinite percolation clusters . in : _ random walks and discrete potential theory ( cortona , 1997 ) _ , pp . 265278 , sympos . xxxix , cambridge univ . press , cambridge . g. papanicolau and s.r.s . varadhan ( 1979 ) . _ boundary value problems with rapidly oscillating random coefficients_. colloquia mathematica sociatatis jnos bolay , vol random fields , esztergom ( hungary ) , pp . 835 - 873 .
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we consider the simple random walk on the ( unique ) infinite cluster of super - critical bond percolation in @xmath0 with @xmath1 .
we prove that , for almost every percolation configuration , the path distribution of the walk converges weakly to that of non - degenerate , isotropic brownian motion .
our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square - integrable martingale .
the size of the deformation , expressed by the so called corrector , is estimated by means of ergodicity arguments .
= 1
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the validity of the optically thin approximation for coronal plasma has been discussed in the literature since the beginnings of solar x - ray and euv spectroscopy ( pottasch 1963 , and references therein ) . resonance scattering of @xmath015.01 ( @xmath1 , known as `` 3c '' ) , in particular , has been the subject of a long - standing controversy . observations of the ratio of the 3c flux to that of @xmath015.26 ( @xmath2 , known as `` 3d '' ) gave ratios in the range 1.6 to 2.3 ( rugge & mckenzie 1985 ; schmelz 1997 ; saba 1999 ) , whereas collisional radiative models predicted a ratio of about four ( smith 1985 ; loulergue & nussbaumer 1973 ; bhatia & doschek 1992 ; cornille 1994 ) . schmelz and saba also found , using the flat crystal spectrometer ( fcs ) on the _ solar maximum mission _ , that the lowest values were preferentially at the solar limb . this center - to - limb effect bolstered the argument for resonance scattering of @xmath015.01 , given that the limb photons traverse a longer path ( _ cf , _ phillips 1996 ) . significant new results have recently come to light : laboratory measurements from the electron beam ion trap ( ebit ) at lawrence livermore national laboratory found the 3c/3d flux ratio to be 3.04 @xmath3 0.12 ( brown et al . 1998 ) . using the national institute of standards and technology ebit , laming et al . ( 2000 ) confirmed the value to be close to three . individual measurements at different beam energies from the two groups span a range from 2.50 to 3.15 . these measurements imply that the amount of resonance scattering was overestimated in the solar analyses described above . furthermore , new theoretical models are converging toward a ratio closer to three ( e.g. doron & behar 2002 ; chen & pradhan 2002 ; gu 2003 ) , though all published models continue to exceed the measurements by at least 10% . new dirac r - matrix calculation show excellent agreement with the ebit measurements ( chen 2005 ; chen & pradhan 2005 ) . brown ( 2001 ) have also reported on experiments in which a steady stream of neutral iron was injected into the ebit , producing an underionized plasma with both and . they found 3c/3d flux ratios as low as 1.9 @xmath3 0.11 , and argued that contamination of @xmath015.26 by the inner - shell line ( @xmath4 ) could account for the discrepancy between the laboratory ratio for a pure fe xvii plasma and the solar spectra . blending as an explanation for the solar results presumably implies that the center - to - limb effect in the solar data is due to chance . new observations of stellar coronae have not confirmed the solar line ratios . spectra for many stars observed with _ chandra _ and _ xmm - newton _ yield a 3c/3d ratio of about three ( ness et al . 2003 ; audard et al . 2004 ; gdel et al . 2004 ) , suggesting that stellar coronae do _ not _ generally show resonance scattering ( but see testa 2004 and matranga 2005 ) . full - star observations can not rule out resonance scattering in individual active regions , however , since the number of photons scattered out of the line of sight could be offset by a similar number of photons scattered into the line of sight . furthermore , the sample of stars does not include stars with coronae in the solar coronal temperature range ( 24 mk ) , such as @xmath5 centauri or procyon , for which blending with might be expected . these new experimental , theoretical , and observational results motivate us to re - investigate solar observations from the flat crystal spectrometer on the _ solar maximum mission . _ the reanalysis of these data , using new atomic data , is presented in 2 . in 3 we consider the implications of our results . in particular , we find that resonance scattering is not responsible for the fuzziness seen in the solar images obtained with the 284 passband of the _ transition region and coronal explorer _ ( ) and the euv imaging telescope ( eit ) on the _ solar and heliospheric observatory _ ( ) . the data analyzed here were obtained with the fcs ( acton 1980 ) and are discussed in detail in earlier papers ( schmelz 1997 ; saba 1999 ) . the instrument had a 15 arcsec field - of - view and could scan the soft x - ray resonance lines of prominent ions in the range of 1.5 to 20.0 with a spectral resolution of 0.015 at 15 . in this letter , we reanalyze the lines from 31 of the 33 spectral scans from quiescent active regions ( i @xmath014.21 could not be measured in two of these spectra see below ) . figure 1 shows the portion of a typical fcs spectrum containing the lines of interest . spectra where plasma conditions were changing significantly with time were excluded from the sample . the top panel of figure 2 shows the observed 3c/3d line ratio as a function of temperature . the flux ratio of i @xmath014.21 to @xmath016.78 provides a good temperature diagnostic , with its high signal - to - noise ratio and abundance - insensitivity . calculations using the astrophysical plasma emission code ( apec ) version 1.3 ( smith et al . 2001 ) give the temperatures for each of the individual measured flux ratios . the apec emissivities incorporate the ionization balance models of mazzotta et al . models for the other strong lines , @xmath017.05 and @xmath017.10 , are less certain than for @xmath016.78 , since these lines have a larger contribution from dielectronic recombination and resulting cascades ( see gu 2003 ) , and hence are more dependent on the ionization state model . the and g - ratios ( i.e. the ratios of the sum of the forbidden plus intercombination line fluxes to the flux of the resonance line ) are also temperature - dependent , but are of lower signal - to - noise ratio due to the weakness of the intercombination lines . most of the observed 3c/3d ratios are clustered , with significantly less than the average laboratory value of 2.9 ; however , two of the three highest temperature measurements also give higher 3c/3d ratios , within 1@xmath6 of 2.9 . moreover , the best - fit line to these data shows a strong temperature - dependence , inconsistent with calculations ( gu 2003 ) , while the flux ratio of 3c to @xmath016.78 ( middle panel ) shows only modest temperature - dependence as expected from calculations . these results strongly suggest blending of @xmath015.26 . in the underionized ebit plasma , brown et al . ( 2001 ) also measure a second inner - shell line at 15.21 ( @xmath7 ) . we identify this line for the first time in an active region spectrum , and use it to estimate the strength of the blend at 15.26 . calculations from the hebrew university lawrence livermore atomic code ( hullac ) , which will be available in the next apec release , give a scaling factor of 0.83 ( d. liedahl , private communication ) . we note that this scaling factor is significantly higher than the factor of 0.5 recommended by brown et al . ( 2001 ) based on multiconfigurational dirac - fock calculations , primarily due to a large difference in the branching ratio between radiative decay and autoionization for @xmath015.21 . the bottom panel of figure 2 shows the line ratios that result after subtracting the @xmath015.26 blend , using @xmath015.21 as a proxy to determine its flux . the weighted mean of this distribution is 2.76 @xmath3 0.23 , statistically indistinguishable from the laboratory ratio . while the @xmath015.21 proxy is , to our knowledge , unblended at temperatures below 5 @xmath8 k , its use for hotter plasmas is complicated by the presence of @xmath015.20 ( e.g. the _ chandra _ spectrum of capella , desai et al . 2005 ) . it seems likely that the line tentatively identified in the flare spectrum by phillips et al . ( 1982 ) as may be dominated by as well . in light of this new analysis , we reconsider other examples of full - sun x - ray spectra which include @xmath015.01 and @xmath015.26 . in particular , we expect active region measurements to show lower line ratios and flare spectra to approach the laboratory value . indeed active region measurements show lower 3c/3d values ( blake 1965 ; evans & pounds 1968 ; walker 1974 ) , whereas flare spectra ( e.g. neupert 1973 ) give higher values . these results are thus consistent with blending at 15.26 in the low - temperature active region spectra . they also suggest that a reduced ratio might be observed also in high signal - to - noise ratio , high resolution spectra of stars with cooler coronae . resonance scattering has been suggested as one of the reasons for the fuzzy appearance of the images obtained from the 284 passband of and -eit , illustrated in figure 3 . in this section , we discuss possible explanations of the 284-fuzziness and estimate the upper limit for the resonance scattering contribution to this fuzziness . a common explanation for the fuzzy appearance of the and eit 284 images is instrument scattering ; however , measurements of the 284 point spread function are identical to those of the 171- and 195 passbands , which show much cleaner images ( golub et al . furthermore , prominences seen in absorption have sharp edges in the 284 images ( fig . 3 , lower panel ) , demonstrating that the instrument can resolve fine structures . a second candidate explanation is contamination of @xmath0304 photons in the passband . this contamination certainly exists for coronal holes , which do not appear dark ( as they do , for example , in the thin aluminum images from the soft x - ray telescope on ) ; however , there should be no significant @xmath0304-contribution in active region areas because of the higher temperatures , or above the solar limb where the scale height of @xmath0304 is too small . resonance scattering has also been suggested as the cause of the 284 fuzziness . the @xmath015.01 result allows us to estimate the contribution of resonance scattering to @xmath0284.2 . using chen s ( 2005 ) calculated 3c/3d ratio of 2.85 at @xmath9 6.4 , and the observed fcs ratio of 2.76 @xmath3 0.21 gives an escape probability of 2.76/2.85 , corresponding to @xmath10 . following acton ( 1978 ) , @xmath11 , where @xmath12 is the oscillator strength , @xmath13 is the ionization fraction ( which is a function of temperature ) , and @xmath0 is the wavelength . solving for @xmath14 , with the ionization balance at @xmath9 6.4 , gives this corresponds to an escape probability of 92% , which indicates that very few photons are available to contribute to the 284 - image fuzziness . for possible uncertainties of 10 to 15 % in experimental and theoretical 3c/3d ratios , the escape probability ranges from 0.84 to 1.0 , for @xmath17 . using the same argument , with the relative coronal abundances of fludra & schmelz ( 1999 ) , we find that the @xmath018.97 opacity is comparable to that of @xmath015.01 , while the opacities of @xmath013.46 and @xmath09.17 are 2 to 3 times lower , confirming the findings of schmelz et al . ( 1997 ) that these important diagnostics are not contaminated for use in emission measure distribution analysis . the assumed temperature @xmath18 , estimated from the measured ratios of @xmath014.21 to @xmath016.78 , can be further constrained by the laboratory measurements . the experimental 3c/3d ratio was measured as a function of the abundance ratio of to , providing a good fit to the theoretical predictions . the experimental plasma is not in collisional ionization equilibrium , due to the continuous ionization from neutral iron up to the charge states of interest , and moreover , the collisional processes are excited by a narrow beam rather than a maxwellian distribution as assumed for the solar coronal plasma . thus , an interpretation of the experimental line ratio in terms of an ionization equilibrium temperature is not strictly valid , since the resonance excitation contributions will not be identical . nevertheless , this estimate can give a consistency check on the population . for the laboratory value where the populations of and are equal , the 3c/3d ratio is 1.90 @xmath3 0.11 ( brown et al . 2001 ) , slightly below the average of the cluster . thus the diagnostics all appear consistent with each other in the blending scenario . in collisional ionization equilibrium , such a ratio corresponds to @xmath19 6.32 . for this lower temperature , @xmath20 is about 3 times larger ; however , chen s ( 2005 ) predicted line ratio is about 5% less , such that our average value puts @xmath21 essentially at zero , with an upper limit below 0.2 . meanwhile , loop differential emission measure ( dem ) distributions tend to be fairly flat - topped between @xmath9 6.3 and 6.4 ( e.g. schmelz et al . 2001 ) , such that an intermediate value seems most reliable for estimating the @xmath20 ratio . we then find @xmath22 0.12 , with a plausible range from 0.0 to @xmath23 0.2 . for opacities in this range , the model images of wood & raymond ( 2000 ) still retain the appearance of loops with sharp rather than blurred boundaries . we refer to the last candidate explanation for the 284 - fuzziness as the `` filling - factor '' model , which is related to the observational result that coronal structures appear fuzzy if they are not resolved by the instrument ( see fig . 3 of deluca et al . 2005 ) . it is well established that the dem of active region and quiet sun plasma peaks between 2 and 3 mk . in other words , more higher temperature ( 284 ) plasma exists along the line of sight than cooler ( 171 ) plasma , such that the piling up of structures along the line of sight may contribute to the fuzziness factor . in the 171- band ( @xmath241 mk ) , the 0.5-arcsec resolution of revealed substantial substructure that had never been seen before ; in the 284- band ( @xmath2423 mk ) , the spatial resolution does not appear to be quite sufficient . an additional factor of two improvement in spatial resolution may be adequate , but 0.1-arcsec pixel size is technically feasible and seems preferable . we conclude , therefore , that the filling - factor model provides the most likely explanation for the 284-image fuzziness , and thus future instruments with higher spatial resolution may be able to resolve the active region plasma into its component structures . efforts to understand the coronal heating process should benefit from resolved images closer to the temperature at which the dominant heating occurs . we would like to thank j. saba , k. nasraoui , d. liedahl , l. golub , and j. cirtain . solar physics research at the university of memphis is supported by nsf atm-0402729 and nasa nng05ge68 g . n. b. was supported by nasa contract nas8 - 39073 to the smithsonian astrophysical observatory for the chandra x - ray center . acton , l. w. 1978 , , 225 , 1069 acton , l. w. 1980 , , 65 , 53 audard , m. , telleschi , a. , gdel , m. , skinner , s. l. , pallavicini , r. , & mitra - kraev , u. 2004 , , 617 , 531 bhatia , a. k. & doschek , g. a. 1992 , atomic data nuclear data , 52 , 1 blake , r. l. , chubb , t. a. , friedman , h. , unzicker , a. e. 1965 , , 142 , 1 brown , g. v. , beiersdorfer , p. , chen , h. , chen , m. h. , reed , k. j. 2001 , , 557 , l75 brown , g. v. , beiersdorfer , p. , liedahl , d. a. , widmann , k. , kahn , s. m. 1998 , , 502 , 1015 chen , g .- x . 2005 , submitted chen , g. x. , & pradhan , a. k. 2002 , , 89 , 13202 chen , g. x. , & pradhan , a. k. 2005 , mnras , submitted cornille , m. , dubau , j. , faucher , p. , bely - dubau , f. , & blancard , c. 1994 , , 1 , 1 deluca , e.e . , weber , m.a . , sette , a.l . , golub , l. , shibasaki , k. , sakao , t. & kano , r. 2005 , adv . space res . , in press desai , p. et al . 2005 , , 625 , l59 doron , r. & behar , e. 2002 , , 574 , 518 evans , k. & pounds , k. a. 1968 , , 152 , 319 fludra , a. & schmelz , j. t. ( 1999 ) , a&a , 348 , 286 golub , l. et al . 1999 , phys . plasmas , 6 , 2205 gu , m .- f . 2003 , , 582 , 1241 gdel , m. , audard , m. , reale , f. , skinner , s. l. , linsky , j. l. 2004 , , 416 , 713 laming , j. m. 2000 , , 545 , l161 loulergue , m. & nussbaumer , h. 1973 , , 45 , 125 matranga , m. , mathioudakis , m. , kay , h. r. m. , & keenan , f. p. 2005 , , 621 , l125 mazzotta , p. , mazzitelli , g. , colafrancesco , s. , & vittorio , n. 1998 , , 133 , 403 ness , j .- u . , schmitt , j. h. m. m. , audard , m. j. , gdel , m. , & mewe , r. 2003 , , 407 , 347 neupert , w.m . , swartz , m. , kastner , s o. 1973 , , 31 , 171 phillips , k. j. h. , greer , c. j. , bhatia , a. k. , keenan , f. p. 1996 , , 469 , l57 phillips , k. j. h. et al . 1982 , , 256 , 774 pottasch , s. r. 1963 , , 137 , 945 rugge , h. r. & mckenzie , d. l. 1985 , , 297 , 338 saba , j. l. r. , schmelz , j. t. , bhatia , a. k. , & strong , k. t. 1999 , , 510 , 1064 schmelz , j. t. , saba , j. l. r. , chauvin , j. c. , & strong , k. t. 1997 , , 477 , 509 schmelz , j. t. , scopes , r. t. , cirtain , j. w. , winter , h. d. , & allen , j. d. 2001 , , 556 , 896 smith , b. w. , raymond , j. c. , mann , j. b. , & cowan , r. d. 1985 , , 298 , 898 smith , r. k. , brickhouse , n. s. , liedahl , d. a. , & raymond , j. c. 2001 , , 556 , l91 testa , p. , drake , j. j. , peres , g. , & deluca , e. e. 2004 , , 609 , l79 walker , a. b. c. , rugge , h. r. , weiss , k. 1974 , , 194 , 471 wood , k. , & raymond , j. c. 2000 , , 540 , 563 14.21 to @xmath016.78 ) . the dotted line shows the best flat fit and the dashed line shows the best linear fit ; ( _ upper _ ) 3c/3d ratio with flat ( @xmath25 ) and linear ( @xmath26 ) fits ; ( _ middle _ ) @xmath015.01-to-@xmath016.78 ratio with flat ( @xmath27 ) and linear ( @xmath28 ) fits ; ( _ lower _ ) same as ( _ upper _ ) except that 0.83 times the @xmath015.21 flux has been subtracted from @xmath015.26 flux for all but the three hottest spectra with flat ( @xmath29 ) and linear ( @xmath30 ) fits.,width=288 ]
|
resonance scattering has often been invoked to explain the disagreement between the observed and predicted line ratios of @xmath015.01 to @xmath015.26 ( the `` 3c/3d '' ratio ) . in this
process photons of @xmath015.01 , with its much higher oscillator strength , are preferentially scattered out of the line of sight , thus reducing the observed line ratio .
recent laboratory measurements , however , have found significant inner - shell lines at 15.21 and 15.26 , suggesting that the observed 3c/3d ratio results from blending . given our new understanding of the fundamental spectroscopy
, we have re - examined the original solar spectra , identifying the @xmath015.21 line and measuring its flux to account for the contribution of to the @xmath015.26 flux .
deblending brings the 3c/3d ratio into good agreement with the experimental ratio ; hence , we find no need to invoke resonance scattering .
low opacity in @xmath015.01 also implies low opacity for fe xv @xmath0284.2 , ruling out resonance scattering as the cause of the fuzziness of and -eit 284 images .
the images must , instead , be unresolved due to the large number of structures at this temperature .
insignificant resonance scattering implies that future instruments with higher spatial resolution could resolve the active region plasma into its component loop structures .
| 6,109 | 346 |
hadronic properties at finite temperature and baryon density are of great importance in the phenomenology of heavy ions collisions , star interior and the early universe . moreover , the theoretical expectation of transitions to a chirally symmetric phase and , perhaps , to a quark - gluon plasma phase contributes to the interest in studying the effect of matter and temperature on the quantum chromodynamics ( qcd ) vacuum . our present understanding of qcd at finite temperature ( @xmath0 ) and baryon density ( or chemical potential @xmath1 ) is mainly limited in the euclidean realm , due to the lack of non - perturbative and systematic calculating tools directly in the minkowski space . typical methods , with qcd lagrangian as the starting point , are the ope and lattice simulations . because these two formulations are intrinsically euclidean , only static quantities are conveniently studied . in order to gain dynamical informations , which are more accessible experimentally , the analytic structure implemented through dispersion relations often have to be invoked within the theory of linear response . in principle , dispersion relations allow the determination of the spectral function ( sf ) , which carries all the real - time information , from the corresponding correlator in the euclidean region . in practice , realistic calculations , e.g. ope or lattice simulations , yield only partial information on the correlators , making impossible a direct inversion of the dispersion relation . therefore , the standard approach assumes a phenomenological motivated functional form with several parameters for the sf , and uses the information from the approximate correlator , mediated by the dispersion relation , only to determine the value of parameters by a fit . this approach has been quite successful at zero temperature and density , thanks to the fortuitous situation that we roughly know how to parameterize the sf s in many cases . two important examples are the qcd sum rules pioneered by shifman , vainshtein and zakharov @xcite , and the analysis of lattice qcd data @xcite . so far , standard parameterizations have included poles plus perturbative continuum @xcite . the success of such approaches heavily rests on our good understanding of the qualitative behavior of sf s at zero @xmath2 . we can find other such favorable examples in the low-@xmath0 regime @xcite , where the shape of the sf has the same qualitatively features of the zero @xmath2 case , or even in the high-@xmath0 regime for simple models @xcite , for which the functional form of the sf is known . the qcd sum rules approach has been extended also to systems at finite temperature @xcite . the lack of experimental data , and of reliable nonperturbative calculations has prompted people to use the same kind of parameterizations that have worked so well at zero temperature with , at most , perturbative corrections . we believe that physical results at finite @xmath2 can be strongly biased by this assumption . in fact , recent interpretations of lattice simulation data @xcite appear to indicate the existence of such problems . the purpose of this work is to derive exact sum rules that constrain the variation of sf s with @xmath2 . in addition , we apply these sum rules to the chiral phase transition , and demonstrate that sf s in some channels are drastically modified compared to both their zero @xmath2 and their perturbative shapes . this result confirm our worries about non - trivial effect of finite @xmath0 or baryon density on the shape of the sf . our derivation of these exact sum rules , based on the ope and the rge , has a closer relation with the derivation of sum rules for deep inelastic scatterings than with the qcd sum rule approach of svz @xcite . in fact , we establish relationships between moments of the sf and corresponding condensates as functions of @xmath2 , without assuming any functional form of the sf . in the derivation process , we find that the logarithmic corrections are essential to establish the exact sum rules . in contrast , the qcd logarithmic corrections are only marginally relevant in the finite energy sum rules , and hence are rarely discussed in the literature . to properly take into account the logarithmic corrections , a repeated partial integration method is used to match the relevant asymptotic expansions . since no further assumptions other than the validity of the ope and rge are involved in the derivation , our sum rules are very general and can be applied anywhere in the @xmath2-plane , even near or at the boundary of a phase transition . the paper is organized as follows . in section ii we present the general derivation of exact sum rules in asymptotically free theories . the matching of asymptotic behaviors of the correlator and the dispersion integral , including explicitly their logarithmic corrections , are carefully discussed . in section iii we illustrate each single step of the derivation in a soluble model , the gross - neveu model in the large-@xmath3 limit . in this model we can calculate exactly all the relevant quantities ( spectral functions , wilson coefficients , condensates , anomalous dimensions and correlators in space - like region , etc . ) , and , therefore , give a concrete example of how our method works . the application of our method to the derivation of exact sum rules for the mesonic channels in qcd is presented in section iv . in the same section , we also discuss the phenomenological consequences of the exact sum rules near chiral restoration phase transitions . finally , in section v we summarize our work , draw some conclusions , and discuss possible future directions . we start this section with a short review of the linear response theory , the ope and the rge . next we introduce a convenient subtraction in the dispersion relation for studying the dependence of sf s on @xmath0 and @xmath1 . then , we present in detail a crucial part of our method : how to match the asymptotic ope expansion with a corresponding asymptotic expansion of the sf and its dispersion integral . this approach is necessary for properly taking into account the logarithmic corrections , and studying the convergence properties of the relevant moments of the sf . more naive approaches not only yield , in general , incorrect sum rules , but might also fail to recognize that a given sum rule does not exist in the first place , since the integral of the sf involved is divergent . finally , a comparison of the two asymptotic expansions leads to the desired exact sum rules . we end this section with some general comments on the derivation and meaning of these sum rules . the real - time linear response @xcite to an external source , @xmath4 , coupled to a renormalized current @xmath5 in the form of @xmath6 is given by the retarded correlator : @xmath7\rangle_{t,\mu}\ , , \label{corr}\ ] ] where the average is on the grand canonical ensemble specified by temperature @xmath0 and chemical potential @xmath1 . disregarding possible subtraction terms , which are @xmath2-independent since they are related to short distance properties of the theory , we can write the following dispersion relation for the frequency dependence of the retarded correlation function : @xmath8 for convenience , we discuss only the uniform limit ( @xmath9 ) and , from now on , drop the momentum label . we wish to emphasize that , even if the derivation of the sum rules requires the analytic continuation of eq . ( [ disp ] ) into the euclidean region ( @xmath10 ) , the dispersion integral is , nevertheless , determined by the singularities of @xmath11 in the minkowski region . one should keep clearly in mind the distinction between these real - time singularities with the ones related to screening phenomena . in asymptotically free theories , the correlator @xmath12 is calculable in terms of the ope @xcite asymptotic series in the limit of @xmath13 . for a given renormalization prescription , the asymptotic expansion reads @xmath14_\kappa\rangle_{t,\mu } \ , , \ ] ] where @xmath15 , @xmath16 , the @xmath17_\kappa$ ] s and the @xmath18 s are , respectively , the current quark mass , the coupling constant , the renormalized composite operators , and their corresponding wilson coefficients at the subtraction scale @xmath19 . here and in the following , we use the symbol @xmath20 to denote asymptotic equality . we explicitly singled out the perturbative term @xmath21 , which corresponds to the identity operator and , being independent of @xmath0 and @xmath1 , is calculated at @xmath22 . this term is the only one that is not suppressed by powers of @xmath23 and can depend on an overall subtraction scale , which we identify , for simplicity , with the same subtraction scale @xmath19 used in the renormalization of the operators . it is important to note that the information of the ensemble average is encoded in the matrix elements of composite operators , while the wilson coefficients , including @xmath24 , are independent of @xmath0 and @xmath1 . although the matrix elements @xmath25_\kappa\rangle_{t,\mu}$ ] can not be determined perturbatively , the @xmath26-dependence of the wilson coefficient @xmath18 is controlled by the following renormalization group equation @xcite @xmath27 c_n(q^2,m^2(\kappa),g^2(\kappa),\kappa)=0 \ , , \ ] ] where @xmath28 [ rge ] and @xmath29 , @xmath30 and @xmath31 are the anomalous dimensions for the current @xmath32 , the operator @xmath33 and the current quark mass @xmath34 respectively . for the purpose of illustration , we are only considering operators that do not mix under renormalization , but the mixing will be properly taken into account when necessary . the standard approach @xcite to the renormalization group equation is the introduction of a running coupling @xmath35 and a running mass @xmath36 . in asymptotically free theories @xmath35 vanishes logarithmically at large @xmath26 . it is therefore meaningful to consider , in this limit , a perturbative expansion of the renormalization group equation functions @xmath37 where @xmath38 , @xmath39 , @xmath34 , while @xmath40 , @xmath41 are pure numbers determined by a one - loop calculation . within this perturbative context , eq . ( [ rge ] ) can be solved @xcite : @xmath42 where @xmath43 is the canonical dimension of the composite operator @xmath33 minus the dimension of @xmath44 in units of a mass , and the @xmath45 s are calculable perturbatively . corrections to this result coming from nonleading powers of @xmath46 in the expansions of the renormalization group equation functions , see eq . ( [ rgef ] ) , are analytic in @xmath47 and hence can be absorbed into @xmath48 . therefore , the leading term in the large-@xmath26 limit is totally characterized by the one - loop structure of the theory . to study the temperature and chemical potential dependence of @xmath44 , we only need to consider the difference @xmath49 : @xmath50 where @xmath51 . this subtraction is crucial to remove @xmath21 , which contains terms not suppressed by powers of @xmath23 and is explicitly dependent on the renormalization point . the ope asymptotic expansion of @xmath49 is then @xmath52_\kappa\rangle \ , \label{dope}\ ] ] where @xmath53_\kappa\rangle$ ] denotes the difference of the expectation values of @xmath17_\kappa$ ] in the ensembles specified by @xmath2 and @xmath54 , respectively . since the current quark mass only runs logarithmically in asymptotically free theories , we can safely ignore the corrections to the @xmath48 s due to their dependence on @xmath55 when in deep euclidean limit . notice that the subtraction has made @xmath49 independent of the renormalization point @xmath19 , i.e. @xmath49 satisfies a homogeneous rge . since @xmath45 is perturbative and hence can be expanded in power of @xmath35 ( and denoting the first non - vanishing power as @xmath56 ) , the left - hand side of eq . ( [ ddisp ] ) can be expressed as a double ( in @xmath23 and @xmath35 ) asymptotic expansion of the form : @xmath57_\kappa\rangle}{q^{d_n } } [ g^2(q)]^{\nu+\eta_n}\ , , \label{asyope}\ ] ] where the exponents @xmath58 and the @xmath26-independent coefficients @xmath59 are again known perturbatively . it is important to emphasize here that the leading term in @xmath35 for a given power of @xmath23 is controlled by the relevant anomalous dimensions . we proceed by making an analogous asymptotic expansion of @xmath60 valid for @xmath61 with @xmath62 suitably large but otherwise arbitrary : @xmath63 with @xmath64^{-\xi_n}}{u^{2(n+1 ) } } \sum_{\nu=0}^\infty a_{n}^{(\nu)}(t,\mu ) [ \ln(u^2)]^{-\nu } \ , , \label{rholn}\ ] ] where we have explicitly isolated in @xmath65 all terms that vanish exponentially when @xmath66 , such as the pole contributions to @xmath60 or terms containing the factor @xmath67 . to simplify the notation we have chosen the units such that the running coupling has the form at one - loop level @xmath68 ( or equivalently , energy scales are measured in units of the relevant @xmath69-parameter ) . this ansatz is sufficient to produce an asymptotic series of the form of eq . ( [ asyope ] ) to one - loop level . more generally , one could replace @xmath70 with the full running coupling @xmath71 in the asymptotic sequence and generalize the method we are going to describe ; the general strategy involved in this generalization can be found , for example , in ref . @xcite . since we are presently only interested in one - loop calculations , we can regard @xmath35 and @xmath72 to be proportional . it is easy to recognize that the sum over `` @xmath39 '' is meant to match the sum over mass dimension in eq . ( [ asyope ] ) , while the sum over `` @xmath73 '' will match the sum over the order in perturbation series . the existence of anomalous dimensions in the ope makes it necessary to introduce @xmath74 in the expansion for the spectral function . in the following we ignore the dependence of @xmath74 on `` @xmath39 '' , as we have already ignored the fact that there generally exist more than one operator at a given dimension , and write @xmath75 to avoid a too cumbersome notation . the complete notation will be restored when necessary . before we proceed further , we wish to discuss whether it is possible that additional terms might appear in the expansion of @xmath76 in eq . ( [ rholn ] ) . in general , our asymptotic expansion procedure is powerful enough to exclude this possibility within the framework of the ope . in fact , it allows to verify unambiguously that terms different from the ones already present generate , when substituted in the dispersion integral , terms that are missing in the ope . let us examine two specific examples that might be suspected to exist otherwise . first , dimensional arguments could suggest terms like @xmath77 . our procedure shows that any @xmath78 term in @xmath60 would generate a @xmath79 term in @xmath49 , which does not correspond to any known condensate , and it is therefore excluded . the second example is given by terms such as @xmath80 , which are naturally produced by elementary perturbative calculations of the spectral function : are such terms present in eq . ( [ rholn ] ) ? it is indeed true that such terms appear in the high-@xmath0 expansion of the spectral function at fixed @xmath81 . however , the expansion of the spectral function that is relevant for comparing to the ope is a high-@xmath81 expansion at fixed @xmath0 . in the next section an explicit calculation in the gross - neveu model will illustrate the general fact that , contrary to the high-@xmath0 expansion , the high-@xmath81 expansion does not generate terms like @xmath80 . at this stage , we can already recognize a fundamental , and often overlooked , characteristic of the spectral function . if we insert the term @xmath65 in the dispersion integral , we only obtain pure powers of @xmath23 , since the exponential convergence allows a naive expansion of the factor @xmath82 ( watson s lemma @xcite ) . therefore logarithmic corrections , i.e. powers of @xmath35 , come solely from the @xmath76 term . because we know that the running coupling @xmath35 is always present in the ope series , the term @xmath76 must be present in the subtracted sf , and it obviously dominates the asymptotic regime ( @xmath66 ) . this fact immediately implies that only a finite number of moments of the subtracted sf can possibly be finite , i.e. the naive expansion of the factor @xmath82 is generally wrong , and that logarithmic corrections play a important role . a standard method to tackle the dispersion integral in the large-@xmath26 limit is the mellin transform . however , the use of mellin transform methods @xcite is extremely cumbersome when logarithms appear in the denominator . since inverse logarithms can not be avoided in the spectral function , we need to resort to other means . we carry out the dispersion integral of the @xmath76 term by splitting the integral in eq . ( [ ddisp ] ) into three intervals : @xmath83 , @xmath84 and @xmath85 . the integral over the first interval can be naively expanded in powers of @xmath23 , since there are no convergence problems : @xmath86 in particular , the leading term in @xmath23 is @xmath87 in the second interval , we use the asymptotic form of @xmath88 given in eq . ( [ rholn ] ) and obtain ( for instance by repeatedly integrating by parts ) @xmath89 where @xmath90 , and @xmath91 $ ] is the incomplete gamma function . notice that when @xmath92 the correct result for the term with @xmath93 is @xmath94 , which also corresponds to its limiting value : @xmath95/\alpha = \ln\frac{\ln q^2}{\ln\lambda^2}$ ] . we then substitute in the second incomplete gamma function that appears in eq . ( [ int2 ] ) the following asymptotic expression which is valid for @xmath96 and @xmath97 : @xmath98\equiv\int_z^\infty dx\ , x^{\alpha-1 } e^{-x } \sim\ , z^{\alpha-1 } e^{-z}\sum_{m=0}^\infty { ( -1)^m\gamma(1-\alpha+m)\over z^m\gamma(1-\alpha)}\ , . \label{asygamma}\ ] ] the resulting expression for @xmath99 is @xmath100 \label{dk2asym } \\ & -&{1\over q^2}\sum_{n,\nu=0}^\infty { a_n^{(\nu)}\over q^{2n}}\biggl [ \sum_{l=0\atop l\neq n}^\infty \sum_{m=0}^\infty \frac{(-1)^{l+m } } { ( n - l)^{m+1}[\ln q^2]^{m+1-\alpha } } \frac{\gamma(m+1-\alpha)}{\gamma(1-\alpha)}- \frac{(-1)^n [ \ln q^2]^{\alpha}}{\alpha } \biggr]\ , , \nonumber\end{aligned}\ ] ] where again @xmath101 and @xmath102 should be understood as @xmath103 and @xmath104 when @xmath105 . let us notice that in the first line of eq . ( [ dk2asym ] ) any term with @xmath106 such that the integral @xmath107 is finite can be formally identified as @xmath108= \int_{\lambda^2}^\infty du^2\,u^{2l}\delta\rho_{\text{power}}(u)\ , . \label{formom}\ ] ] in particular , the leading @xmath23 term in eq . ( [ dk2asym ] ) is @xmath109\\ & + & { 1\over q^2}\sum_{\nu=0}^\infty a_0^{(\nu)}\biggl [ \sum_{l=1}^\infty \sum_{m=0}^\infty \frac{(-1)^{l } } { l^{m+1}[\ln q^2]^{m+1-\alpha } } \frac{\gamma(m+1-\alpha)}{\gamma(1-\alpha)}+ \frac { [ \ln q^2]^{\alpha}}{\alpha } \biggr ] \label{int2asym } + { \cal o}\left(\frac{1}{q^4}\right ) \ , . \nonumber\end{aligned}\ ] ] we can similarly expand the integral over the third interval @xmath110^{m+1-\alpha } } { \gamma(m+1-\alpha)\over\gamma(1-\alpha ) } \nonumber \\ & \sim&\mbox{}-{1\over q^2 } \sum_{\nu=0}^\infty a_0^{(\nu ) } \sum_{l=1}^\infty \sum_{m=0}^\infty \frac{(-1)^{l+m } } { l^{m+1 } [ \ln q^2]^{m+1-\alpha } } { \gamma(m+1-\alpha)\over\gamma(1-\alpha ) } + { \cal o}\left(\frac{1}{q^4}\right ) \ , . \label{int3}\end{aligned}\ ] ] in the end we add the leading @xmath23 contributions from eqs . ( [ int1asym ] ) , ( [ int2asym ] ) and ( [ int3 ] ) to the corresponding contribution from the naive expansion of @xmath111 and obtain @xmath112^{1-\xi}}{q^2 } \sum_{\nu=0}^\infty \frac{a_0^{(\nu)}}{[\ln q^2]^{\nu}}\biggl [ \frac{1}{1-\xi-\nu } \nonumber \\ & & \,\,\,\,\ , + 2\sum_{l=1\atop m=0}^\infty\frac{(-1)^l } { [ l\ , \ln q^2]^{2m+2 } } \frac{\gamma(2m+1+\nu+\xi)}{\gamma(\nu+\xi)}\biggr ] + { \cal o}\left(\frac{1}{q^4}\right)\ , , \label{leadqrho}\end{aligned}\ ] ] where @xmath113 is defined by @xmath114 since the integral @xmath115 can be finite only if @xmath116 , we can use eq . ( [ formom ] ) to identify @xmath113 as the zeroth moment of the subtracted spectral function , only when the asymptotic expansion of the subtracted sf in eq . ( [ rholn ] ) yields @xmath116 . otherwise , if @xmath117 , the zeroth moment is infinite , though @xmath113 as defined by eq . ( [ drhobar ] ) still exists . we derive the sum rules by comparing the coefficient of @xmath23 in eq . ( [ leadqrho ] ) @xmath118^{1-\xi } } { \textstyle ( 1-\xi ) } \\[0.6 cm ] \ln(\ln q^2 ) \end{array } + { \cal o}\left({1\over q^2},{1\over[\ln q^2]^\xi}\right ) \quad \begin{array}{l } \mbox{if $ \xi\neq 1 $ } \\[0.6 cm ] \mbox{if $ \xi=1 $ } \end{array}\right . \label{leadingrh}\ ] ] and the corresponding coefficient in eqs . ( [ asyope ] ) @xmath119_\kappa\rangle [ g^2(q)]^{\eta } + { \cal o}\left(\frac{1}{q^2},[g^2(q)]^{\eta+1}\right ) \ , , \ ] ] where we are only considering cases with @xmath120 and @xmath121 . furthermore , we are presently interested in exact sum rules that can be derived with one - loop calculations and , therefore , we only compare leading orders in @xmath72 . then there exist three possibilities , depending on the value of @xmath122 , which can be calculated using the ope . \(1 ) if @xmath123 , then @xmath124 ( or a bigger integer ) and @xmath125_\kappa\rangle \ , . \label{cons}\ ] ] this is the result one would get by naively expanding @xmath126 without worrying about the convergence of the moments and disregarding logarithmic corrections in the ope . \(2 ) if @xmath127 , then @xmath128 and @xmath129 \(3 ) if @xmath130 ( positive powers of @xmath131 in eq . ( [ dope ] ) ) or if the term @xmath132 appears in eq . ( [ dope ] ) , then @xmath133 or @xmath134 , respectively . this implies that @xmath135 we remark that , even in this case when the moment is not finite , the asymptotic expansion is still well defined . it is nice to see that whether the zeroth moment of the subtracted sf exists is reflected directly through the leading power of @xmath35 in the ope series . our main results , eq . ( [ cons ] ) and eq . ( [ nonc ] ) , can be expressed in physical terms as follows . the zeroth moment of a sf for a current @xmath32 whose ope series yields @xmath136 is independent of @xmath0 and @xmath1 , while the same moment for a current with @xmath123 changes with @xmath0 and @xmath1 proportionally to the corresponding change(s ) of the condensate(s ) of the leading operator(s ) . although @xmath137 and @xmath138_\kappa\rangle$ ] can separately depend on @xmath19 , their product must be independent of @xmath19 , since the zeroth moment is independent of @xmath19 . at this point several general comments are appropriate : \(1 ) our derivation relies on the fact that an asymptotically free theory allows a perturbative expansion at short distances , making practical the use of the ope and of the rge . we understand why only short distance physics is involved if we realize that the integral over frequencies reduces eq . ( [ corr ] ) to the ensemble average of the equal - time commutator of the currents . therefore , results such as eq . ( [ cons ] ) and eq . ( [ nonc ] ) are completely determined by the one - loop structure of the theory and the particular current under exam . \(2 ) flavor , or other non - dynamical quantum numbers , does not change the expansions at the one - loop level in an essential way . therefore , one can derive analogous sum rules by using other kind of subtractions , instead of the one we adopted . one such example is given by the exact weinberg sum rules @xmath139 in the chiral limit @xcite . \(3 ) the derivation of sum rules for higher moments of the sf requires the complete cancelation of all the lower dimensional operator terms , not just the leading @xmath35 . in particular , one also needs current quark mass corrections to the wilson coefficients . without appropriate subtractions , higher moments do not even converge @xcite . \(4 ) it is essential to properly take into account the logarithmic corrections when deriving exact sum rules , since the logarithmic corrections not only dictate whether @xmath113 satisfy eq . ( [ cons ] ) or eq . ( [ nonc ] ) but they also control the very existence of @xmath113 @xcite . this procedure is in sharp contrast with the usual qcd sum rule approach , where the convergence issue is by - passed by applying the borel improvement by explicitly introducing a cut - off parameter ( the borel mass ) . \(5 ) we believe that the @xmath2-dependent part of the leading condensate appearing in eq . ( [ cons ] ) does not suffer from the infrared renormalon ambiguity . in fact , only the perturbative term @xmath24 can generate contributions to the leading condensate that are dependent on the prescription used to regularize these renormalons . but @xmath24 is independent of @xmath0 and @xmath1 : any prescription dependence cancels out when we make the subtraction in eq . ( [ ddisp ] ) . on the contrary , unless we generalize eq . ( [ ddisp ] ) and make other subtractions , sum rules that involve non - leading condensates are , in principle , ambiguous . \(6 ) it is well - known that conserved operators are not renormalized and , barring anomalous violations , verify the same `` classical '' identities that can be derived at the tree level . this fact is also verified in the sum rules . in fact when both the currents and the operators are conserved ( @xmath123 ) and we obtain the result of eq . ( [ cons ] ) , i.e. one can use the `` naive asymptotic expansion '' to derive the sum rule . conservation of the current alone is not enough to warrant a `` classical '' identity . in the preceding section we have derived sum rules valid for any asymptotically free theory . in this section we illustrate the procedure in the 1 + 1 dimensional gross - neveu model @xcite in the infinite-@xmath3 limit . on one hand , we can derive the sum rules in eqs . ( [ cons ] ) and ( [ nonc ] ) by explicitly calculating wilson coefficients , @xmath140- and @xmath141-functions in the vector and pseudoscalar channels , following the general procedure discussed in section ii . on the other hand , since this model is soluble , we can obtain the exact spectral function at arbitrary @xmath2 , and then explicitly verify both that the sum rules are satisfied and that the asymptotic expansion of the spectral function has the form given in eq . ( [ rholn ] ) . moreover , we can also calculate the condensates , and therefore explicitly check that the ope really matches the asymptotic expansion of the exact spectral integral . the lagrangian of the gross - neveu model is @xmath142\ , , \ ] ] where @xmath143 is a two - component dirac spinor and has @xmath3-component in the internal space . equivalently , we can write @xmath144 where @xmath145 and @xmath146 are auxiliary fields . the coupling constant @xmath46 is independent of @xmath3 and held fixed in the limit @xmath147 . this model is asymptotically free when @xmath148 , and the chiral symmetry is dynamically broken at @xmath149 and @xmath150 to the leading order in @xmath151 @xcite . in the following we give the exact solution at @xmath152 with finite @xmath2 . although formulas are explicitly considered in the symmetry breaking phase , they are also valid in the symmetric phase provided that the vanishing limit of certain condensates ( such as the dynamical fermion mass ) is properly taken . we wish to remark on the well - known fact that the limit @xmath147 here should not be interpreted as a starting point for an expansion of the model at finite @xmath3 , but rather as different model in itself , which is in fact the model we decided to use for the purpose of illustration . moreover , there exist arguments @xcite suggesting that this model in the limit @xmath147 is actually more relevant to @xmath153 phenomenology than the model with finite @xmath3 . since the procedure to obtain the exact solution in the large-@xmath3 limit is rather standard and some of the intermediate steps can be found in the literature , see for instance refs . @xcite , we only give the definitions and final results . the gap equation and the phase diagram can be derived from the first derivative of the effective potential @xmath154 we introduce a momentum cutoff @xmath69 and add the counterterm @xmath155 , where @xmath19 is the subtraction point . then the renormalized effective potential becomes @xmath156 where we have defined @xmath157 and , in the second equality , we have eliminated the subtraction point by introducing the dynamical fermion mass at @xmath22 , i.e. @xmath158 . the effective potential is then obtained by integrating eq . ( [ effpotp ] ) : @xmath159\right\}\ , . \label{effpot}\ ] ] the gap equation is simply given by @xmath160 the system possesses `` metastable states '' at those values of @xmath2 for which the gap equation has more than one minimum solutions : the solution with lowest @xmath161 selects the true ground state . when @xmath2 s are such that @xmath162 is the only minimum solution to the gap equation , these points in the @xmath2-plane defines a second order phase transition line , which separate the symmetry - broken phase , i.e. @xmath163 , from the symmetric phase , i.e. @xmath162 . taking the limit @xmath164 in eq . ( [ effpotp ] ) , after some suitable rewriting , this critical line obeys the following equation @xmath165 \cosh[(k-\mu)/2 t ] } - \int_m^\infty { dk\over k}\ , f(k;t,\mu)\ , , \label{gap0}\ ] ] whose numerical solutions are plotted in fig . [ fig1 ] in full line . the full line turns into a dotted line at the heavy dot , where another minimum ( dash line ) appears and becomes lower than the minimum given by eq . ( [ gap0 ] ) . since the mass gap is finite on the dash line ( given by the solution of @xmath166 with @xmath167 , where prime denote a derivative with respect to @xmath168 ) , the dash line is a first order phase boundary . the critical temperature at @xmath150 is given by @xmath169 . the `` tricritical point '' ( the heavy dot in fig . [ fig1 ] ) can be found by imposing that @xmath170 : @xmath171 @xcite . when @xmath149 the chemical potential at which the first order phase transition takes place is @xmath172 @xcite . the basic ingredient to build up the correlator is the fermion bubble graph . in the euclidean region the pseudoscalar bubble , i.e. the free bubble graph between two @xmath173 currents , can be conveniently expressed as @xmath174 where @xmath175 and @xmath176 here and in the following @xmath19 is always the renormalization subtraction point , @xmath177 is the dynamically generated fermion mass , and @xmath178 . we shall also use the short - hand notation @xmath179 for instance , the use of this notation makes possible to write the gap equation as @xmath180 in the large-@xmath3 limit , the correlator in pseudoscalar channel is given by the geometric sum of the bubble graph : @xmath181 in the euclidean region , we can then expand this correlator in the @xmath13 limit as @xmath182 where we have introduced the running coupling constant @xmath183 and used the gap equation eq . ( [ gapsh ] ) . we can recognize the first term in eq . ( [ corrq ] ) as the perturbative contribution at @xmath22 , @xmath184 , which corresponds to the term @xmath185 in the generic ope expansion shown in section [ general ] . the other two terms in eq . ( [ corrq ] ) can be interpreted as contributions from the condensates . as we will verify later , in fact , @xmath186\rangle_{t,\mu}$ ] , while @xmath187\rangle_{t,\mu}$ ] is the condensate of the energy - momentum tensor . we shall see that the perturbative contribution and the coefficients of the condensates are indeed the ones obtained by the ope of @xmath188 , and therefore the expansion of the correlator shown in eq . ( [ corrq ] ) is of the form of the ope . note that in defining the condensates we have absorbed factors of @xmath3 in order to simplify the notation . to obtain the spectral function we need the bubble graph in the time - like region @xmath189 : @xmath190 and @xmath191 where @xmath192 stands for the principle value of the integral . the complete spectral function has the form @xmath193 the pole contribution comes from the bound state ( the pion ) , whose mass is the solution to the equation @xmath194 in the region @xmath195 . this equation has in fact solution @xmath196 , and the coupling constant of this massless pion to its constituents is given by @xmath197 the continuum part of the spectral function is related to the bubble @xmath198 in the time - like region through @xmath199 in order to get an idea of the dependence of the spectral function on @xmath2 , we plot @xmath200 for several typical values of @xmath2 , which are indicated in the phase diagram in fig . [ fig1 ] . in fig . [ fig2 ] we show @xmath200 for values of @xmath2 that go from the symmetry - broken phase to the symmetric phase through the second order line , while in fig . [ fig3 ] we show @xmath200 as @xmath2 go through the first order boundary . we immediately notice that the behavior of the spectral function changes drastically near the second order phase boundary . this is a clear reflection of the low energy critical phenomena related to the continuous phase transition . in fact , the pseudoscalar correlator ( in the symmetric phase only , since it is then degenerate with the scalar correlator ) at @xmath201 can be regarded as the chiral susceptibility , and should diverge as @xmath202 ( with @xmath203 in this model ) . at the boundary of the first order phase transition ( fig . [ fig3 ] ) , but still relatively close to the `` tricritical point '' , the same qualitative deformation is present , since a weak first order transition shares certain qualitative features of a second order transition , even if we do not expect any real divergence . it is this kind of dramatic behavior of the spectral function near phase transitions that makes practical parameterizations nearly impossible . the chiral restoration in qcd is expected to be either continuous , or at least a smooth crossover , and the spectral functions in scalar and pseudoscalar channels should show a behavior qualitatively similar to the gross - neveu model , i.e. strong peak right above the origin near the phase transition . we shall comment more on this issue after we derive the sum rules for qcd in section [ qcd ] . it is instructive to expand @xmath200 in the limit @xmath204 . we shall see that this asymptotic expansion is of the form assumed in section [ general ] . furthermore , once substituted in the dispersion integral , it generates the ope series of eq . ( [ corrq ] ) according to the procedure developed in section [ general ] . to the leading order in @xmath205 , we find @xmath206 and @xmath207\rangle_{t,\mu}\bigr ] + { \cal o}\bigl({1\over\omega^4}\bigr)\biggr\ } \ , , \ ] ] which leads to the following asymptotic form for the continuum part of the spectral function @xmath208\rangle_{t,\mu } \nonumber \\ & + & { 4g^2(\omega)\over\omega^2}\bigl({g^2(\omega)\over g^2}\bigr)^2 \langle[\theta_{00}]\rangle_{t,\mu } \bigl\{1+{\cal o}\bigl(g^2(q)\bigr)\bigr\}\ , . \label{rho_asym}\end{aligned}\ ] ] here we only concern with the @xmath35-dependent terms and postpone the pure @xmath23 terms ( related to @xmath111 ) to a later subsection . we recognize again the first term in eq . ( [ rho_asym ] as the asymptotic perturbative spectral function . moreover , the @xmath2-dependent part of eq . ( [ rho_asym ] ) has indeed the general form of eq . ( [ rholn ] ) . in connection with the comments we made after eq . ( [ rholn ] ) , we point out that eq . ( [ rho_asym ] ) has been obtained by expanding the spectral function in the limit @xmath209 at fixed @xmath0 . had we made instead a high-@xmath0 expansion ( @xmath210 ) at fixed @xmath26 , we would have obtained , for instance , @xmath211 from the third term of eq . ( [ repi ] ) . upon substituting eq . ( [ rho_asym ] ) in the dispersion integral , and using the identity @xmath212 where @xmath213 and we have kept only the @xmath35-dependent part , we find the following series to the leading order in @xmath35 , @xmath214\rangle_{t,\mu}\over q^2 } + 4\bigl({g^2(q)\over g^2}\bigr)^2 { \langle[\theta_{00}]\rangle_{t,\mu}\over q^2}+\cdots \label{gndisint}\ ] ] this result is in agreement with the ope series of eq . ( [ corrq ] ) , which has been directly expanded from the correlator in the euclidean region . the fact that no pure @xmath23 term appears in eq . ( [ gndisint ] ) implies that the zeroth moment of the subtracted spectral function vanishes : @xmath215 . the zeroth moment of the spectral function without the subtraction is not convergent , because of the contribution from the perturbative term , even if one can define it through a proper analytic continuation that yields @xmath216 . we have just verified that in the gross - neveu model the exact correlation function can indeed be expanded in an asymptotic series precisely in the form of the ope . now we shall calculate the wilson coefficients , @xmath140-function and appropriate anomalous dimensions for the ope , and verify that they match the coefficients and exponents of this asymptotic series . furthermore , though it is beyond the scope of the ope and rge , we shall also calculate the condensates . in this way , we explicitly verify that the asymptotic expansion of the exact correlation function is identical to the ope . the one - loop @xmath140-function of the 1 + 1 dimensional gross - neveu model has already been calculated in the original paper @xcite : @xmath217 , i.e. according to our notation @xmath218 . the anomalous dimensions for pseudoscalar current @xmath173 and four - quark operator @xmath219 can be calculated using the feynman diagrams shown in fig . [ fig4]a and fig . [ fig4]b , respectively . according to the notation of eq . ( [ rgef ] ) we find @xmath220 , and @xmath221 . the energy - momentum conservation makes the anomalous dimension for @xmath222 vanish , i.e. @xmath223 . the relevant wilson coefficients corresponding to operators @xmath224 and @xmath222 can be calculated , to leading order in @xmath35 , using the feynman diagrams shown in fig . [ fig5]a and fig . [ fig5]b , respectively . the resulting coefficient are @xmath225 ( @xmath226 according to notation in eq . ( [ asyope ] ) ) and @xmath227 ( @xmath228 ) . these results lead to the exponents @xmath229 and @xmath230 , which are precisely what we found in eq . ( [ gndisint ] ) . these explicit one - loop calculations exactly match the solution in eq . ( [ gndisint ] ) . at last a comment on the one - loop calculation : this result is exact to the leading order in @xmath151 and only involves insertion of the quark - bubble chain . since the quark - bubble behaves like @xmath231 , this insertion is equivalent to substituting the coupling constant @xmath46 with the running coupling constant @xmath35 . so one can easily identify those sets of feynman graphs whose sum leads to the solution of the rge in eq . ( [ rge ] ) . the ope itself does not specify how the relevant condensates are calculated , since the ope is only a rge - improved perturbative procedure . the calculation of the condensates can be done only in a non - perturbative context . here , we carry out this calculation using the @xmath151 expansion . the bare quark condensate @xmath232 is given by the dynamical quark tadpole graph @xmath233 using the standard contour integral technique to carry out the matsubara frequency sum , we obtain @xmath234 the renormalized quark condensate is obtained by replacing @xmath235 with the subtraction point @xmath19 in the above equation @xmath236\rangle_{t,\mu } = m\langle\!\langle e_k^0\rangle\!\rangle -{m\over\pi}\ln\bigl({\kappa\over m}\bigr)=-{m\over g^2}\ , .\ ] ] in the last step we have used the gap equation eq . ( [ gapsh ] ) . notice that the renormalization of this condensate is not independent of the one carried out for the effective potential . in fact , the @xmath145-field is just an auxiliary field : @xmath237 . therefore , the gap equation gives us not only the expectation value of the @xmath145-field , but also of @xmath238 : @xmath239\rangle_{t,\mu}$ ] . so the renormalizations of @xmath238 and @xmath46 are related . the four - quark condensate factorizes in the large-@xmath3 limit , and we find @xmath240\rangle_{t,\mu}= \langle[\bar{\psi}\psi]\rangle_{t,\mu}^2=m^2/g^4 $ ] . the bare kinetic energy expectation value is given by @xmath241 after removing the @xmath2-independent volume and quadratic divergences , and introduced a subtraction point for the logarithmic divergence the renormalized kinetic energy becomes @xmath242\rangle_{t,\mu } = \langle\!\langle e_k^2 \rangle\!\rangle -{m^2\over 2\pi}\ln\bigl({\kappa\over m}\bigr)\ , .\ ] ] after subtracting the trace term from the kinetic energy , we finally obtain the expectation value of the traceless energy - momentum tensor @xmath243\rangle_{t,\mu } \equiv \langle[\bar{\psi}i\gamma_0\partial_0\psi -{1\over 2}m\bar{\psi}\psi]\rangle_{t,\mu } = \langle\!\langle e_k^2\rangle\!\rangle -{m^2\over 2}\langle\!\langle e_k^0\rangle\!\rangle\ , .\ ] ] in the symmetric phase ( @xmath244 ) , the expectation value of the energy - momentum tensor has the very simple form at @xmath150 @xmath243\rangle_{t,\mu=0}={\pi\over 6}t^2\ , .\ ] ] one can verify that inserting these condensates in the ope , whose coefficients and exponents we have already calculated , reproduces the asymptotic expansion of the exact correlator in eq . ( [ corrq ] ) . let us first notice that the sum rule @xmath245 converges very slowly . in fact , the subtracted spectral function has the leading behavior @xmath246 where @xmath247\rangle_{t,\mu})\over g^4(\omega)+4}\biggr\}\ , . \label{drho2}\ ] ] since @xmath248 , the leading behavior in eq . ( [ drho2 ] ) is @xmath249^{-1}$ ] , and it is clear that the sum rule only converges logarithmically , i.e. @xmath250 . moreover , it is only thanks to the the logarithmic corrections that the sum rule is finite . had we disregarded this corrections , we would have found @xmath251 , and the zeroth moment of the subtracted spectral function would not exist . since the exact spectral function can not be integrated analytically , we verify the exact sum rule numerically . we take advantage of the fact that the integral of @xmath252 in eq . ( [ drho2 ] ) can be done analytically to improve the numerical convergence of the integral . we rewrite the sum rule as @xmath253 where the integral of @xmath252 can be done analytically and we obtain @xmath254\rangle_{t,\mu})\over ( \ln a)^2+\pi^2/4}\biggr\}+{\cal o}\bigl({1\over a^2}\bigr)\ , .\ ] ] this formula can be easily verified numerically with high accuracy , since the error is now @xmath255 , rather than @xmath256 . it is reminded that @xmath257 in the above equation should be obtained from the complete sf in eq . ( [ rho_full ] ) , including both @xmath88 and @xmath111 . there are several reasons to study the correlator in the vector channel in the gross - neveu model . first , the vector channel gives us an example where the zeroth moment of the subtracted sf does not exist , but the asymptotic procedure can still be carried out . moreover , in connection with the fact that the vector current is conserved , this spectral function is not affected by logarithmic corrections , and we can use it to illustrate the use of the mellin transform method to obtain the asymptotic expansion . in addition , very much like in the qcd case , we shall see that the baryon number susceptibility changes drastically near the chiral restoration transition , even if we do not expect any real singularity associated with critical phenomena in the vector channel . for convenience , we only consider the case of @xmath150 . the feynman graphs that contribute to the vector correlator are depicted in fig . note that if there were no mixing between vector and pseudoscalar channel , which in fact is only present in 1 + 1 dimensions , the sole contribution would come from the first graph . the total contribution from all the graphs in fig . [ fig6 ] is @xmath258 we define the spectral function in vector channel as @xmath259 then we find the following explicit expression for @xmath260 @xmath261 it is clear that leading asymptotic behavior of the spectral function is @xmath262 , which implies that the zeroth moment of the spectral function does not exist ( it is divergent ) . in the section [ general ] we have shown that , corresponding to this divergence , we must find negative powers of @xmath35 ( positive power of @xmath263 ) in the ope series . when there are no explicit logarithms in @xmath264 , it is easier to make the asymptotic expansion by using the mellin transform method , and in particular the convolution property of the mellin transform @xcite , @xmath265m[f;1-z]\ , , \label{convolut}\ ] ] where @xmath266 $ ] is the mellin transform of @xmath267 , defined by @xmath268\equiv\int_0^\infty dt\ , t^{z-1}h(t)\ , .\ ] ] the value of @xmath269 in eq.([convolut ] ) depends on the asymptotic behaviors of @xmath267 at @xmath270 and @xmath271 at @xmath272 respectively ; details can be found , for instance , in chapter 4 of ref . @xcite . now let us illustrate how to use this method to expand the power part of the spectral function ( the part containing exponential dumping factors can be expanded naively ) @xmath273 whose mellin transform is @xmath274= ( 4m^2)^z\sqrt{\pi}\gamma(1-z)/\gamma(3/2-z ) \ , .\ ] ] using the convolution , eq . ( [ convolut ] ) , and the mellin transform of @xmath82 @xmath275=\pi(q^2)^{-z}/\sin\pi(1-z ) \ , , \ ] ] we find ( in this case @xmath269 in eq.([convolut ] ) obeys @xmath276 ) @xmath277_{z = n } \nonumber \\ & = & { 4m^2\over q^2}\ln\bigl({q^2\over m^2}\bigr ) -{8m^4\over q^4}\bigl[1+\ln\bigl({q^2\over m^2}\bigr)\bigr ] + { \cal o}\bigl({1\over q^6}\bigr)\ , . \label{asymmell}\end{aligned}\ ] ] similarly to the pseudoscalar correlator , it is possible to check this result in two ways . we can directly expand the power part ( temperature independent part ) of eq . ( [ vvceucl ] ) , and we can derive the ope ( @xmath278 and @xmath279 for the @xmath23 term ) . as promised , there indeed appears a positive power of @xmath263 in the ope series . at the same time , the singularities of @xmath280 $ ] at @xmath281 positive integer hints the fact that all the non - negative integer moments of @xmath282 do not exist . on the other hand , the asymptotic expansion in eq . ( [ asymmell ] ) is well defined . even though the vector current is conserved , due to the fact that the anomalous dimension of @xmath224 is not zero , the correct result can not be obtained by a naive asymptotic expansion in this case , in accordance with the remark number ( 6 ) at the end of section ii . this same result could have been obtained also by the general method developed in section [ general ] , but it should be clear by now that the use of the mellin transform , when possible , is more straightforward and require less labour . the baryon number susceptibility ( @xmath283 ) is defined as the vector correlator in the limit @xmath284 . from eq . ( [ vvceucl ] ) we find @xmath285 notice that since we have contracted the lorentz indices when defining our vector correlator , @xmath283 is nonpositive definite . it is also possible that our definition of @xmath283 differs from others by an additive constant , which is irrelevant to the temperature dependence we are considering . in the symmetric phase , the limit @xmath286 should be take with some care : the result is @xmath287 , when @xmath288 . the complete temperature dependence of @xmath283 is displayed in fig . as expected , there is no divergence or critical phenomena in this channel . nevertheless , we do see a sudden rising of @xmath283 in correspondence to the relatively rapid drop of the dynamical quark mass , which we show in fig . [ fig7]a for comparison , near the phase transition region . after a lengthy detour to the gross - neveu model let us now back to the main interest of our work , to derive exact sum rules and to explore the corresponding phenomenological consequences in qcd . as we will see , qcd shares many of the qualitative features observed in the gross - neveu model , implying the simple fact that any drastic change induced by a phase transition or rapid crossover would necessarily be reflected through the spectral function in proper channels at low frequencies , independent of the details of the model . let us consider four correlators of mesonic currents in qcd . the correlation between scalar currents and the one between pseudoscalar currents involve non - conserved operators , @xmath289 and @xmath290 , whose anomalous dimensions are @xmath291 . one the other end , correlators between vector currents and axial - vector currents involve conserved quantities , @xmath292 and @xmath293 , whose anomalous dimensions vanish ( @xmath294 ) . in qcd there exist several dimension - four operators , but all of them have non - positive anomalous dimensions , therefore the two correlators between nonconserved currents have @xmath295 and eq . ( [ nonc ] ) applies , _ i.e. _ the zeroth moments of their sf s are independent of @xmath0 and @xmath1 . on the other hand , the two conserved currents have , in correspondence with conserved operators @xmath296 and a generalization of eq . ( [ cons ] ) applies . in this case in fact there are three dimension - four operators with zero anomalous dimension . two of these operators are lorentz scalars : @xmath297 and @xmath298 , while the third is the energy - momentum tensor : @xmath299 the sum rules for the vector and axial - vector currents ( with the lorentz indices contracted ) are : @xmath300\rangle + { \delta\langle[\alpha_s g^2]\rangle\over 2\pi } + 8\delta\langle [ \theta_{00}]\rangle\ , , \label{va}\ ] ] with @xmath301 and @xmath302 for vector and axial - vector cases , respectively . obviously , the general remarks at the end of section ii apply to the sum rules derived above . however , there is a new feature of qcd which may not be shared by all asymptotically free theories . it is known that qcd ope series has explicit instanton induced corrections that can not be related to condensates of some local operators . but these exact sum rules are not explicitly affected by this problem , although the value of the condensates certainly have instanton contributions . the reason is that the instanton singularities in the borel - plane are located on the positive axis starting at @xmath303 , which in turn implies that the explicit contribution of instantons to correlation functions are of higher order than @xmath23 @xcite in the ope series . finally , let us discuss some of the phenomenological consequences of these exact sum rules . the power of these exact sum rules rests on their generality , in the sense that there is no assumptions on the functional form of the sf s and they can be applied anywhere in the @xmath2-plane . first let us consider the pseudoscalar channel . the sum rule @xmath215 implies that , in the broken - chiral - symmetry phase , the change of the pion pole induced by @xmath0 or @xmath1 is exactly compensated by a corresponding change of the continuum part of the sf . next let us consider the scalar correlation function at zero frequency , i.e. the chiral susceptibility ( a measure of the fluctuation of the chiral order parameter ) , @xmath304\rangle_{t,\mu } = \int_0^\infty\!\ ! du^2\ , { \rho(u;t,\mu)\over u^2}\ , , \label{chisus}\ ] ] which diverges when @xmath2 approaches the phase boundary , provided the chiral restoration is a continuous transition . the divergence of the chiral susceptibility near the phase transition can only be produced in eq . ( [ chisus ] ) by singularities very close to the origin ; singularities not near the origin are not compatible with the sum rule @xmath215 . thus , when approaching the phase boundary in the @xmath2 plane , the threshold of the spectral function vanishes ( since there is no massless pole in the chirally symmetric phase ) , and a strong peak develops right above threshold . in the chirally symmetric phase the pseudoscalar and scalar channels are degenerate , therefore we expect the same behavior for the pseudoscalar sf . this strong peak in the pseudoscalar and scalar sf s , which is intimately connected with critical phenomena of a diverging susceptibility and correlation length near the phase transition , can be interpreted as some kind of quasi - particle , thus confirming the qualitative picture , originally proposed in the context of the nambu - jona - lasinio model @xcite , of the appearance of soft modes near the chiral phase transition . we wish to remark that the sum rule has been used only to exclude the logical possibility of the sf developing singularities at finite energies . but the sum rule is not a necessary condition for the appearance of a peak . the presence of a second order phase transition ( infinite correlation length ) is the real physical cause of the peak . nevertheless , one should not assume that if the phase transition is not second order the zero-@xmath0 or perturbative spectral function is a good approximation to the real spectral function . even if the chiral restoration turns out to be a cross - over or weak first order transition ( finite but large correlation ) , as the lattice data seem to indicate @xcite , we still expect the same qualitative features , though less pronounced : a peak develops but it does not actually diverge at the transition . our calculation in the gross - neveu model is a clear illustration of this situation . while fig . ( [ fig2 ] ) shows the divergent peak of the spectral function near the second order boundary , we see in fig . ( [ fig3 ] ) that the spectral function is also strongly peaked near the first order transition as long as we are not too far away from the `` tricritical point '' . we believe that similar results hold in the vector and axial - vector channels , even if our conclusions can not be as strong as in the previous case , because of two main differences . the first is that now @xmath305 . this problem is not very serious , since we only need that @xmath113 is not singular crossing the phase boundary . and this can still be argued by means of the sum rule in eq . ( [ va ] ) and the fact that , for a continuous phase transition , the changes of the thermal energy @xmath306\rangle$ ] , and of both condensates @xmath307\rangle$ ] and @xmath308\rangle$ ] should behave smoothly across the critical line . the most serious difference is that , in these channels , we can not argue on physical grounds that the corresponding susceptibility diverges . nonetheless , there exist lattice simulations @xcite showing that the so - called quark number susceptibility rapidly increases in the transition . thus it is plausible that also in the vector channel the sf has a vanishing threshold , and accumulates strength just above it . a similar argument in the context of the nambu - jona - lasinio model can be found in ref . we used the operator product expansion and the renormalization group equation to derive exact sum rules at finite @xmath0 and @xmath1 valid for asymptotically free theories . our derivation explicitly show that logarithmic corrections can not be neglected . we found that , depending on the theory and on the current under study , the zeroth moment of a spectral function is either independent of @xmath0 and @xmath1 , or its change is related to the corresponding changes of the condensates of operators of lowest dimension . in particular the zeroth moment of the scalar and pseudoscalar mesonic currents in qcd are independent of @xmath0 and @xmath1 . as a consequence any change in the strength of the pion pole must be exactly compensated by a change in the continuum contribution . we also infer that spectral functions in the scalar and pseudoscalar channel should dramatically change near phase transition . we also find that the zeroth moment of the vector and axial - vector mesonic currents in qcd changes with @xmath0 and @xmath1 , and these changes are related to the corresponding changes of the condensates . due to their generality , these exact sum rules strongly constrain the qualitative shape of sf s , in particular , near phase transitions . since it appears that a strong deformation of the spectral function from its counter part at @xmath22 and perturbative cases is the most likely scenario near phase transitions , the information carried by these exact sum rules is particularly welcome . we urge whoever parameterizes a spectral function , e.g. in qcd sum rule type of calculations or to interpret lattice simulation data , to incorporate these exact constraints . we have also illustrated in great detail the derivation of the sum rules , and confirmed their validity , in the gross - neveu model , where we can also calculate the exact spectral function . even if this model calculation is meant mostly as an illustration of many delicate issues of the derivation ( such as convergence , asymptotic expansion , mellin transform , high momentum expansion vs. high temperature expansion , etc . ) , it is nevertheless comforting to find that all our general expectations about the spectral function are explicitly verified in the model .
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within the framework of the operator product expansion ( ope ) and the renormalization group equation ( rge ) , we show that the temperature and chemical potential dependence of the zeroth moment of a spectral function ( sf ) is completely determined by the one - loop structure in an asymptotically free theory , and in particular in qcd .
logarithmic corrections are found to play an essential role in the derivation .
this exact result constrains the shape of sf s , and implies striking effects near phase transitions .
phenomenological parameterizations of the sf , often used in applications such as the analysis of lattice qcd data or qcd sum rule calculations at finite temperature and baryon density must satisfy these constraints .
we also explicitly illustrate in detail the exact sum rule in the gross - neveu model .
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the last decade brought a dynamic evolution of the computing capabilities of graphics processing units ( gpus ) . in that time , the performance of a single card increased from tens of gflops in nvxx to tflops in the newest kepler / maxwell nvidia chips @xcite . this raw processing power did not go unnoticed by the engineering and science communities , which started applying gpus to accelerate a wide array of calculations in what became known as gpgpu general - purpose computing on gpus . this led to the development of special gpu variants optimized for high performance computing ( e.g. the nvidia tesla line ) , but it should be noted that even commodity graphics cards , such as those from the nvidia geforce series , still provide enormous computational power and can be a very economical ( both from the monetary and energy consumption point of view ) alternative to large cpu clusters . the spread of gpgpu techniques was further facilitated by the development of cuda and opencl parallel programming paradigms allowing efficient exploitation of the available gpu compute power without exposing the programmer to too many low - level details of the underlying hardware . gpus were used successfully to accelerate many problems , e.g. the numerical solution of stochastic differential equations @xcite , fluid simulations with the lattice boltzmann method @xcite , molecular dynamics simulations @xcite , classical @xcite and quantum monte carlo @xcite simulations , exact diagonalization of the hubbard model @xcite , _ etc_. parallel computing in general , and its realization in gpus in particular , can also be extremely useful in many fields of solid state physics . for a large number of problems , the ground state of the system and its free energy are of special interest . for instance , in order to determine the phase diagram of a model , free energy has to be calculated for a large number of points in the parameter space . in this paper , we address this very issue and illustrate it on a concrete example of a superconducting system with an oscillating order parameter ( op ) , specifically an iron - based multi - band superconductor ( fesc ) . our algorithm is not limited to systems of this type and can also be used for systems in the homogeneous superconducting state ( bcs ) . the discovery of high temperature superconductivity in fesc @xcite began a period of intense experimental and theoretical research . @xcite all fesc include a two - dimensional structure which is shown in fig . [ fig.feas].a . the fermi surfaces ( fs ) in fesc are composed of hole - like fermi pockets ( around the @xmath1 point ) and electron - like fermi pockets ( around the @xmath2 point ) fig . [ fig.feas].b . moreover , in fesc we expect the presence of @xmath3 symmetry of the superconducting op . @xcite in this case the op exhibits a sign reversal between the hole pockets and electron pockets . for one @xmath4 ion in the unit cell , the op is proportional to @xmath5 . layers in fesc are built by @xmath4 ions ( red dots ) forming a square lattice surrounded by @xmath6 ions ( green dots ) which also form a square lattice . @xmath6 ions are placed above or under the centers of the squares formed by @xmath4 . this leads to two inequivalent positions of @xmath4 atoms , so that there are two ions of @xmath4 and @xmath6 in an elementary cell . ( panel b ) true ( folded ) fermi surface in the first brillouin zone for two @xmath4 ions in unit cell . the colors blue , red and green correspond to the fs for the 1st , 2nd , and 3rd band , respectively . ] fesc systems show complex low - energy band structures , which have been extensively studied . @xcite a consequence of this is a more sensitive dependence of the fs to doping . @xcite in the superconducting state , the gap is found to be on the order of 10 mev , small relative to the breadth of the band . @xcite this increases the required accuracy of calculated physical quantities needed to determine the phase diagram of the superconducting state , such as free energy . @xcite in this paper we show how the increased computational cost of obtaining thermodynamically reliable results can be offset by parallelizing the most demanding routines using cuda , after a suitable transformation of variables to decouple the interacting degrees of freedom . in section [ sec.theory_ph ] we discuss the theoretical background of numerical calculations . in section [ sec.algorithm ] we describe the implementation of the algorithm and compare its performance when executed on the cpu and gpu . we summarize the results in section [ sec.summary ] . many theoretical models of fesc systems have been proposed , with two @xcite , three @xcite , four @xcite and five bands @xcite . most of the models mentioned describe one ` fe ` unit cell and closely approximate the band and fs structure ( fig [ fig.feas].b ) obtained by lda calculations . @xcite in every model the non - interacting tight - binding hamiltonian of fesc in momentum space can be described by : @xmath7 where @xmath8 is the creation ( annihilation ) operator for a spin @xmath9 electron of momentum @xmath10 in the orbital @xmath11 ( the set of orbitals is model dependent ) . the hopping matrix elements @xmath12 determine the model of fesc . here , @xmath13 is the chemical potential and @xmath14 is an external magnetic field parallel to the ` feas ` layers . for our analysis we have chosen the minimal two - band model proposed by raghu _ et al . _ @xcite and the three - band model proposed by daghofer _ _ @xcite ( described in [ app.twoband ] and [ app.threeband ] respectively ) . the band structure and fs of the fesc system can be reconstructed by diagonalizing the hamiltonian @xmath15 : @xmath16 where @xmath17 is the creation ( annihilation ) operator for a spin @xmath9 electron of momentum @xmath18 in the band @xmath19 . [ [ superconductivity - in - multi - band - iron - base - systems - in - high - magnetic - fields ] ] superconductivity in multi - band iron - base systems in high magnetic fields + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + fesc superconductors are layered @xcite , clean @xcite materials with a relatively high maki parameter @xmath20 . @xcite all of the features are shared with heavy fermion systems , in which strong indications exist to observe the fulde ovchinnikov ( fflo ) phase @xcite a superconducting phase with an oscillating order parameter in real space , caused by the non - zero value of the total momentum of cooper pairs . in contrast to the bcs state where cooper pairs form a singlet state @xmath21 , the fflo phase is formed by pairing states @xmath22 . these states can occur between the zeeman - split parts of the fermi surface in a high external magnetic field ( when the paramagnetic pair - breaking effects are smaller than the diamagnetic pair - breaking effects ) . @xcite in one - band materials , the fflo can be stabilized by anisotropies of the fermi - surface and of the unconventional gap function , @xcite by pair hopping interaction @xcite or , in systems with nonstandard quasiparticles , with spin - dependent mass . @xcite this phase can be also realized in inhomogeneous systems in the presence of impurities @xcite or spin density waves @xcite . in some situations , the fflo can be also stable in the absence of an external magnetic field . @xcite in multi - band systems , the experimental @xcite and theoretical @xcite works point to the existence of the fflo phase in fesc . through the analysis of the cooper pair susceptibility in the minimal two - band model of fesc , such systems are shown to support the existence of an fflo phase , regardless of the exhibited op symmetry . it should be noted that the state with nonzero cooper pair momentum , in fesc superconductors with the @xmath3 symmetry , is the ground state of the system near the pauli limit . @xcite this holds true also for the three - band model ( e.g. [ app.suscept ] and ref . @xcite ) . [ [ free - energy - for - intra - band - superconducting - phase ] ] free energy for intra - band superconducting phase + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in absence of inter - band interactions , the bcs and the fflo phase ( with cooper pairs with total momentum @xmath23 equal zero and non - zero respectively ) can be described by the effective hamiltonian : @xmath24 where @xmath25 is the amplitude of the op for cooper pairs with total momentum @xmath26 ( in band @xmath19 with symmetry described by the form factor @xmath27 for more details see ref . @xcite ) . using the bogoliubov transformation we can find the eigenvalues of the full hamiltonian @xmath28 : @xmath29 in this case we formally describe two independent bands . the total free energy for the system is given by @xmath30 , where latexmath:[\[\begin{aligned } \label{eq.freeene } \omega_{\varepsilon } = - \frac{1}{\beta } \sum_{\alpha \in \{+ , -\ } } \sum_{\bm k } \ln \left ( 1 + \exp ( - \beta \mathcal{e}_{\varepsilon{\bm k}}^{\alpha } ) \right ) + \sum_{\bm k } \left ( e_{\varepsilon{\bm k } \downarrow } - \frac { 2 | \delta_{\varepsilon } the free energy in @xmath19-th band , where @xmath32 is the respective interaction intensity and @xmath33 . for different parameters @xmath32 in @xmath34 results for @xmath35 ( panels a , c and e ) and @xmath36 ( panels b , d and f ) . ] [ [ historical - and - technical - note ] ] historical and technical note + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the historically basic concept of the fflo phase was simultaneously proposed by two independent groups , fulde - ferrell @xcite and larkin - ovchinnikov @xcite in 1964 . the first group proposed a superconducting phase where cooper pairs have only one non - zero total momentum @xmath37 , and the superconducting order parameter in real space @xmath38 . in the second case , cooper pairs have two possible momenta : @xmath37 and the opposite @xmath39 , with an equal amplitude of the order parameter . thus in real space the superconducting order parameter is given by @xmath40 . however , the most general case of fflo is a superconducting order parameter given by a sum of plane waves , where the cooper pairs have all compatible values of the momentum @xmath41 in the system : @xmath42 where @xmath43 is the cardinality of the first brillouin zone ( in the square lattice it is equal to @xmath44 ) . for the historical reasons described above , whenever @xmath45 ( @xmath46 and @xmath47 ) we can speak about the fulde ferrell ( ff ) phase , whereas for @xmath48 ( and @xmath49 , @xmath50 ) about the larkin ovchinnikov ( lo ) phase . larger @xmath43 impose a more demanding spatial decomposition of the order parameter , both in the theoretical and computational sense . however , every time it can be reduced to the diagonalization of the ( block ) matrix representation of the hamiltonian . using the translational symmetry of the lattice , the problem for the ff phase ( @xmath51 ) in one - band systems corresponds to the independent diagonalization of @xmath52 matrices ( with eigenvalues given like in eq . [ eq.enequasiparticle ] with the number of bands @xmath53 ) for each of the @xmath44 different momentum sectors . in case of the lo phase ( @xmath54 ) , the calculation can be similarly decomposed in momentum space or using other spatial symmetries of the system ( an example of this procedure can be found in ref . @xcite ) , with a much greater computational effort due to the lower degree of symmetry , leading to @xmath55 independent diagonalization problems of size @xmath56 . in the _ full _ fflo phase ( i.e. in a system with impurities @xcite or a vortex lattice @xcite ) , the spatial decomposition is determined in real space using the self - consistent bogoliubov - de gennes equations , which require the full diagonalization of a hamiltonian of maximal rank @xmath57 @xcite at every self - consistent step . to work around these limitations , iterative methods @xcite or the kernel polynomial method @xcite can be used . these methods are based on the idea of expressing functions of the energy spectrum in an orthogonal basis , e.g. chebyshev polynomial expansion . @xcite by doing so , it becomes possible to conduct self - consistent calculations in the superconducting state without performing the diagonalization procedure . the time expense of iterative methods can also be reduced by a careful gpu implementation , which is currently a work in progress . in the present work , we describe how the calculation of the free energy can be accelerated in the ff phase , which due to its greater symmetry allows optimal parallelization on a gpu architecture . parallel programing can be realized in cpus and gpus in many different ways . in this section we compare the performance of the same algorithm implemented using openmp @xcite , pgi cuda / openacc fortran @xcite , and directly in cuda c @xcite . the first two are generic extensions of fortran / c++ that make it easy to , respectively , use multiple cpu cores , and compile a subset of existing fortran / c++ code for a gpu . they take the form of annotations which can be added to existing code , and as such , enable the use of additional computational power with very little overhead by the programmer . typically , much better efficiency can be achieved by the third option i.e. a specifically optimized implementation targeting the gpu architecture directly . this requires more work on the part of the programmer , both in adjusting the algorithms and in rewriting the code , but it makes it possible to fully utilize the available resources . the global ground state for a fixed magnetic field strength @xmath14 and temperature @xmath58 is found by minimizing the free energy over the set of @xmath59 and @xmath23 . in case of @xmath60 independent bands this corresponds to global minimization of the free energy @xmath61 in every band separately , for every @xmath26 in the first brillouin zone ( fbz ) algorithm [ alg.1 ] . for the calculation of the free energy @xmath61 , we must know the eigenvalues @xmath62 reconstructing the band structure of our systems . in the case of the two - band model , it can simply be found analytically ( see [ app.twoband ] ) . however , for models with more bands ( such as the three - band model [ app.threeband ] ) the band structure has to be determined numerically ( e.g. using a linear algebra library , such as lapack ( cpu ) or magma ( gpu ) @xcite ) . with this approach , the calculation of @xmath63 and @xmath64 becomes a computationally costly procedure , and if it were to be repeated inside the inner loop of algorithm [ alg.1 ] , it would significantly impact the execution time . for this reason , we propose to precalculate the eigenvalues for every momentum vector @xmath65 and store them in memory for models with more than two bands . the main downside of this approach is the large increase in memory usage . while algorithm [ alg.1 ] is simple to realize on a cpu , its execution time is proportional to the system size @xmath44 , and as such scales quadratically with @xmath55 for a square lattice ( @xmath55 and @xmath66 are the number of lattice sites in the @xmath67 and @xmath68 direction , respectively ) . generate matrices @xmath63 and @xmath69 for @xmath65 calculate matrices @xmath70 for @xmath65 eq . [ eq.enequasiparticle ] calculate @xmath61 find and save @xmath59 corresponding to a fixed @xmath23 and minimal value @xmath61 find and save @xmath23 and @xmath59 corresponding to minimum of @xmath61 sometimes the physical properties of the system make it possible to reduce the amount of computation for instance when it is known that the minimum of the energy is attained for values of momentum @xmath23 in specific directions fig . [ fig.minene ] . @xcite in this case , the outer loop of algorithm [ alg.1 ] can be restricted to @xmath71 , where @xmath72 is a set of @xmath73 vectors . such reductions are not unique to linear systems with translational symmetry but are also the case for systems with rotational symmetry . @xcite in the case of bcs - type superconductivity where cooper pairs have zero total momentum ( @xmath74 ) , algorithm [ alg.1 ] can be further simplified by taking into account the following property of the dispersion relation : @xmath75 in eq . [ eq.enequasiparticle ] . this can be particularly useful in determining the system energy in the presence of the bcs phase i.e. either in complete absence of external magnetic fields or when only weak fields are present . a more general approach to the reduction of the execution time of our algorithm is to exploit the large degree of parallelism inherent in the problem . in fact , algorithm [ alg.1 ] can be classified as , , embarrassingly parallel since the vast majority of computation can be carried out independently for all combinations of @xmath76 . for simplicity , in this paper we concentrate on optimizing the inner loop , as all the presented methods apply to the outer loop in a similar fashion . we present two approaches to this problem . the first is to parallelize the execution of the serial loop over @xmath59 with openmp to fully utilize all available cpu cores . this has the advantage of simplicity , as the implementation requires minimal changes to the original ( serial ) code . the second approach is to implement algorithm [ alg.1 ] on a gpu using the cuda environment . modern gpus are capable of simultaneously executing thousands of threads in simt ( same instruction , multiple threads ) mode . from a programmer s point of view , all the threads are laid out in a 1- , 2- or 3-dimensional grid and are executing a _ kernel function_. the grid is further subdivided into blocks ( groups of threads ) , which are handled by a physical computational subunit of the gpu ( the so - called streaming multiprocessor ) . threads within a block can exchange data efficiently during execution , but cross - block communication can only take place through global gpu memory , which is significantly slower . mapped to gpu hardware . ] to fully utilize the gpu hardware , we split algorithm [ alg.1 ] into three steps . in the first step , we execute the ` computefreeenergy ` kernel ( algorithm [ alg.2 ] ) on a 3d grid @xmath77 . to take advantage of the efficient intra - block communication , we also carry out partial sums within the block ( corresponding to a subset of values spanning @xmath78 ) using the parallel sum - reduction algorithm . @xcite in the second step , we execute the sum - reduction algorithm again on the partial sums that were generated by algorithm [ alg.2 ] . in the third and last step , we copy the output of step 2 from gpu memory to host memory , and look for the value of @xmath79 corresponding to the lowest free energy with a linear search . depending on the exact configuration of the kernels in step 1 and 2 , the summation might not be complete at the beginning of step 3 . if this is the case , we carry out the remaining summation within the serial loop computing @xmath79 . with block sizes of 128 and 1024 used for the kernels in steps 1 and 2 , we can sum up to @xmath80 terms in parallel on the gpu . we found that the remaining summation was not worth the overhead of carrying it out on the gpu . should this not be the case for some larger problems , further parallel execution can be trivially achieved by repeating step 2 one more time . compute @xmath81 and @xmath18 corresponding to the current thread load @xmath82 and @xmath83 from global memory ( precomputed by a separate kernel ) compute @xmath84 and @xmath85 sum @xmath85 for a range of @xmath18 corresponding to one block of threads save the partial sum from the previous step in global gpu memory to test our approach , we executed algorithms [ alg.1 ] and [ alg.2 ] on linux machines with the following hardware : * cpu : intel(r ) core(tm ) i7 - 3960x cpu @ 3.30ghz 6 cores / 12 threads , * gpu : nvidia tesla k40 ( gk180 ) with the sm clock set to 875 mhz . the programs were run for a single value of @xmath86 and 200 values of @xmath79 . calculations were done for a square lattice of size @xmath87 for various values of @xmath88 . the execution times ( including only the computation part of the code , and excluding any time spent on startup or input / output ) are presented in figure [ fig.scaling ] . and [ alg.2 ] for one vector @xmath89 . right panel : speedup factors for all configurations at @xmath90 . the last 3 case names correspond to runs of the same cuda c code in double precision ( dp ) , single precision ( sp ) , and single precision with fast intrinsic functions ( spfm ) . all versions of the fortran code used double precision calculations . ] comparing the best cpu execution time ( with openmp ) to the gpu fortran code using openacc , we find a speedup factor of @xmath91 in the limit of large lattices . the custom gpu code shows slightly better performance , with a @xmath92x speedup for the double precision version , and additional speedup factors of @xmath93 for single precision , and @xmath94 for intrinsic functions . when taken together , the fastest gpu version is @xmath95 times faster than the openmp code and @xmath96 times faster than the serial cpu code utilizing only a single core . it is remarkable that the original fortran code enhanced with openacc annotations provides performance comparable to a manual implementation in cuda c. this result shows the power of appropriately used annotations marking parallelizable regions of the code . while still requiring explicit input from the programmer and a good understanding of the structure of the code , this approach is in practice significantly faster than writing the program from scratch in cuda c and dealing with low level details of gpu programming and resource allocation . this conclusion however only applies in the limit of large lattices ( see the left panel in figure [ fig.scaling ] ) . for smaller ones , the cuda c code can be seen to be noticeably faster than openacc , which is likely caused by the automatically generated gpu code introducing unnecessary overhead . it should be noted that the last two speedup factors were achieved by trading off precision of calculations for performance e.g. intrinsic functions are faster , but less precise implementations of transcendental functions . in our tests , we obtained the same results with all three approaches . this might not be true for some other systems though , so we advise careful experimentation . with a factor of 2.8x between the most and least precise method , it might also be worthwhile to run larger parameter scans at lower precision and then selectively verify with double precision calculations . the rich phenomenology and the subtle competing and interplaying phenomena of high-@xmath97 materials such as fesc ( section [ sec.intro ] ) , require us to probe fine regimes and precisely determine possible experimental signatures of exotic phases such as fflo ( section [ sec.theory_ph ] ) . by conducting our calculations in momentum space , and by fully exploiting the symmetries of the system , we are able to increase the size of the studied system by two orders of magnitude compared to previously reported results and practically eliminate finite size effects . the cost is borne by the increased complexity of the efficient custom - tailored gpu implementation , described in section [ sec.algorithm ] . our method shown here on the example of an iron - based multi - band superconductor exhibiting a fflo phase , can also be used in calculations of the ground state in standard bcs - type superconductors . overall , we achieved a 19x speedup compared to the cpu implementation ( 119x compared a single cpu core ) . in the spectrum of gpu - accelerated results in physics , this puts us towards the higher end , with the highest speedups being @xmath98x for compute - bound problems with large inherent parallelism . d.c . is supported by the forszt phd fellowship , co - funded by the european social fund . is supported by the ncn project dec-2011/01/n / st3/02473 . the authors would like to thank nvidia for providing hardware resources for development and benchmarking . the model of fesc proposed by raghu _ et al . _ in ref . @xcite , is a minimal two - band model of iron - base pnictides describing the @xmath99 and @xmath100 orbitals with hybridization : @xmath101 where @xmath102 , @xmath103 , @xmath104 , @xmath105 . @xmath106 is the energy unit . half - filling , a configuration with two electrons per site requires @xmath107 . the model is exactly diagonalizable , with eigenvalues : @xmath108 the spectrum @xmath109 reproduces the band structure and fermi surface of fesc for @xmath110 we get the electron - like ( hole - like ) band . this model of fesc was proposed by daghofer _ et al . _ in ref . @xcite and improved in ref . @xcite . beyond the @xmath99 and @xmath100 orbitals , the model also accounts for the @xmath111 orbital : @xmath112 in ref . @xcite the hopping parameters in electron volts are given as : @xmath113 , @xmath114 , @xmath115 , @xmath116 , @xmath117 , @xmath118 , @xmath119 , @xmath120 , @xmath121 , @xmath122 , @xmath123 , @xmath124 and @xmath125 . the average number of particles in the system @xmath126 is attained for @xmath127 . the fs for this model is shown in fig . [ fig.fsdag ] . the static cooper pair susceptibility indicates the possible formation of the fflo phase : @xcite @xmath128 where @xmath129 is the retarded green s function and @xmath130 is the op in band @xmath19 . the operator @xmath131 in real space corresponds to the operator @xmath132 in momentum space . the factor @xmath133 defines the op symmetries for @xmath3 pairing , @xmath134 is equal to @xmath135 for next nearest neighbors and zero otherwise . @xcite in momentum space : @xmath136 @xmath137 where @xmath138 is the structure factor corresponding to the @xmath3-wave symmetry , and @xmath139 is the fermi function . this quantity can be calculated numerically similarly to the procedure used for free energy in section [ sec.theory_ph ] . j. a. anderson , c. d. lorenz , a. travesset , general purpose molecular dynamics simulations fully implemented on graphics processing units , http://dx.doi.org/10.1016/j.jcp.2008.01.047[j . * 227 * ( 2008 ) 5342 ] t. preis , p. virnau , p. wolfgang and j. j. schneider , gpu accelerated monte carlo simulation of the 2d and 3d ising model , http://dx.doi.org/10.1016/j.jcp.2009.03.018[j . comput . phys . * 228 * ( 2009 ) 4468 ] y. kamihara , t. watanabe , m. hirano , h. hosono , iron - 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obtaining a thermodynamically accurate phase diagram through numerical calculations is a computationally expensive problem that is crucially important to understanding the complex phenomena of solid state physics , such as superconductivity . in this work
we show how this type of analysis can be significantly accelerated through the use of modern gpus .
we illustrate this with a concrete example of free energy calculation in multi - band iron - based superconductors , known to exhibit a superconducting state with oscillating order parameter ( op ) .
our approach can also be used for classical bcs - type superconductors . with a customized algorithm and compiler tuning
we are able to achieve a 19x speedup compared to the cpu ( 119x compared to a single cpu core ) , reducing calculation time from minutes to mere seconds , enabling the analysis of larger systems and the elimination of finite size effects .
fflo , pnictides , nvidia cuda , pgi cuda fortran , superconductivity * program summary * _ manuscript title : _ gpu - based acceleration of free energy calculations in solid state physics + _ authors : _ micha januszewski , andrzej ptok , dawid crivelli , bartomiej gardas + _ journal reference : _ + _ catalogue identifier : _
+ _ licensing provisions : lgplv3 _ + _ programming language : _ fortran ,
cuda c + _ computer : _ any with a cuda - compliant gpu + _ operating system : _ no limits ( tested on linux ) + _ ram : _ typically tens of megabytes .
+ _ keywords : _ superconductivity , fflo , cuda , openmp , openacc , free energy + _ classification : _ 7 , 6.5 + _ nature of problem : _ gpu - accelerated free energy calculations in multi - band iron - based superconductor models .
+ _ solution method : _ parallel parameter space search for a global minimum of free energy .
+ _ unusual features : _
+ the same core algorithm is implemented in fortran with openmp and openacc compiler annotations , as well as in cuda c. the original fortran implementation targets the cpu architecture , while the cuda c version is hand - optimized for modern gpus . + _ running time : _ problem - dependent , up to several seconds for a single value of momentum and a linear lattice size on the order of @xmath0 .
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galactic winds are observed to be ubiquitous in galaxies that have recently experienced significant amounts star formation ( see e.g. , * ? ? ? * for a review ) . these outflows represent a fundamental part of galaxy formation models , because the absence of outflows galaxy star formation rates ( sfrs ) are much higher than those observed ( e.g. , * ? ? ? * ) and baryon fractions in the disk are close to the universal value ( e.g. , * ? ? ? * ) , much higher than inferred from observations . in contrast , models that include a variety of feedback effects predict much lower sfrs and baryon fractions . additionally , outflows are required to drive metal - enriched gas out of galaxies , as suggested by both observational ( e.g. * ? ? ? * ) and theoretical ( e.g. * ? ? ? * ) work . however , despite their key role in galaxy formation , the exact processes driving winds remain an open question . plausible driving mechanisms include core collapse supernovae ( sn , * ? ? ? * ) and radiation pressure @xcite . sn - driven winds are now routinely included in semi - analytic and numerical simulations . however , it has long been known that in the disk of the galaxy there is a rough equipartition of the magnetic and cosmic ray ( cr ) energy densities ( e.g * ? ? ? this indicates that crs play a significant role in dynamics of interstellar medium . only relatively recently have the effects of crs have been considered in the context of galaxy formation ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and galaxy cluster @xcite simulations . a tight link between crs and star formation is evidenced by the correlation between a galaxy s infra - red luminosity , closely related to its sfr , and the luminosity of its radio halo ( e.g. * ? ? ? * ; * ? ? ? the relationship is almost linear , has very little scatter , and does not evolve with redshift @xcite , indicating that the coupling between star formation and crs is robust over a wide variety of conditions . although the energy injection rate of crs is small when compared to the other sources of energy from star formation , the rate at which they inject momentum is not @xcite . this is because the crs that supply most of the pressure in the galaxy generate alfvn waves in the ism @xcite , which then scatter the crs with a mean free path of @xmath3pc . crs are self - confined ( e.g. * ? ? ? * ) , and it takes @xmath4myr for a typical cr to escape its parent galaxy . theoretical models of dynamical haloes in which crs diffuse and are advected out in a galactic wind predict steady , supersonic galaxy - scale outflows driven by a combination of cr and thermal pressure @xcite . in this letter we present high - resolution hydrodynamical simulations of isolated disk galaxies , including a model for the injection , transport and decay of crs , to investigate how outflows are driven by crs and the properties of the outflowing gas . our simulations are performed with the adaptive - mesh - refinement ( amr ) code ramses , described in @xcite . the detailed description of physical processes included in our simulations star formation , radiative cooling , and metal enrichment from type ia sne , type ii sne and intermediate mass stars can be found in @xcite . sn feedback is modelled by injecting a total of @xmath5ergs of thermal energy per sn into the cells neighbouring the star particle . we do not employ any delay of dissipation for the injected energy in these runs ( the runs are equivalent to the energy only run in * ? ? ? a full description of the cr field would require modelling the distribution function of crs as a function of position , momentum and time . however , if the cr mean free path is shorter than the length scale of the problem , the cr field can be described as a fluid @xcite . we thus take the approach of modelling the cr energy density , @xmath6 , as an additional energy field that advects passively with the gas density ( e.g. * ? ? ? * ) and exerts a pressure @xmath7 . thus , the total pressure entering the momentum and energy equations governing gas evolution is @xmath8 . we assume throughout that the cr fluid is an ultra relatavistic ideal gas with @xmath9 . as described above , crs undergo a random walk through the ism after their injection . their evolution is thus a combination of advection with the ambient gas flow and diffusion , which we parametrize by the diffusion coefficient , @xmath10 . the evolution of baryon and cr fluids is thus governed by the standard continuity and momentum equations and the following energy equations : @xmath11 @xmath12 where @xmath13 is gas velocity , @xmath14 , @xmath15 and @xmath16 , @xmath17 are the pressure and internal energy of gas and crs , respectively . the @xmath18 indicates energy injection by sn , and @xmath19 is the fraction of this energy that is injected in the form of crs . @xmath20 indicates radiative cooling of gas , while @xmath21 indicates the heating of gas by both crs and uv radiation . finally , @xmath22 corresponds to energy losses by crs both due to decays and coulomb interactions with gas mediated by magnetic fields ( e.g. * ? ? ? * ; * ? ? ? following @xcite we assume that the cr cooling rate is : @xmath23 where @xmath24 is the local electron number density . the ratio of the catastrophic cooling rate to the couloumb cooling rate for our cr population is 3.55 . some fraction of the energy lost by the cr population heats the thermal gas ( e.g. * ? ? ? * ) at a rate given by @xcite @xmath25 equations [ eq : crcool ] and [ eq : crheat ] are solved on every timestep to calculate the rate of decay of the cr energy density along with the corresponding gain in the gas thermal energy . we have tested our cr implementation using a standard shock - tube test involving gas and cr fluids @xcite and found that results accurately match the analytic solution . results of this and other tests will be presented in a forthcoming paper . strong shock waves associated with sn explosions have long been recognized as a likely source of galactic crs ( e.g. * ? ? ? empirically , in order to match the galactic energy density in crs , sne must be capable of transferring a fraction @xmath26 of the explosion kinetic energy into the form of cr energy @xcite . in our models we make the assumption that a certain fraction , @xmath27 , of the sn energy is injected to the cr fluid energy density . the remaining fraction @xmath28 is injected thermally into the gas field . we note that the assumptions that the diffusion of crs is isotropic and that the diffusion coefficient is a constant are necessary simplification in our models , which track neither the direction nor the strength of the magnetic field . on small scales ( @xmath29100 pc ) , the strength of the random component of the galactic magnetic field is several times higher than the average field strength ( e.g. * ? ? ? * ) because galaxy formation processes ( e.g. supernovae and hydrodynamical turbulence ) in the disk @xcite and the turbulent dynamo effect and cr buoyancy in the halo @xcite tangle the magnetic field to the extent that isotropic diffusion is a good approximation(e.g . codes that assume isotropic diffusion are able to predict cr - emitted spectral data down to the few percent level ( e.g * ? ? ? for the purposes of this exploratory work we employ the isotropic diffusion model , but note that investigation of complex models represents an interesting future direction for this work . we simulate isolated , model galaxies of two different masses representing an smc - sized dwarf galaxy and mw - sized disk galaxy with three different feedback models : no feedback , thermal feedback only , and thermal plus cr feedback . the thermal feedback runs inject @xmath30 of the energy released by each sn blast into the gas thermal energy . the cr feedback runs inject @xmath31 of the sn energy into the gas thermal energy and the remaining @xmath32 into the cr energy density field . every simulation models radiative cooling , star - formation and metal enrichment . all runs are evolved for 0.5gyr and throughout this letter we report results for this time . following @xcite and @xcite the galaxy model consists of a dark matter halo , a stellar bulge and an exponential disc of stars and gas . the dark matter halo is modelled as an nfw halo @xcite . the gas and stars are then initialized into an exponential disk , and the bulge is assumed to have a @xcite profile with a scale length that is 10% of the disk scale length . the relevant parameters for each set of initial conditions are given in table [ tab : ics ] . each simulation is run with a maximum spatial resolution of 75pc ( 37.5pc ) for the mw ( smc ) runs . [ cols= " > , < , > , > , > , > , < , < , < , > , > " , ] + notes : from left to right the columns contain : ( 1 ) simulation set name ; ( 2 ) spherical overdensity dm halo mass defined relative to the 200 times the critical density at @xmath33 ; ( 3 ) circular velocity at the virial radius ; ( 4 ) concentration of nfw halo ; ( 5 ) halo spin parameter ; ( 6 ) disk gas fraction ; ( 7 ) mass of gas in the disk ; ( 8) mass of stars in the disk ; ( 9 ) mass of stars in the bulge ; ( 10 ) scale length of exponential disk ; ( 11 ) scale height of gas disk . + [ tab : ics ] the solid curves show the mass loading factor , @xmath34 , of the galactic wind , defined as the ratio of the sfr to the gas outflow rate , as a function of time ( left - hand axis ) . the dotted curves show the galaxy sfr ( right - hand axis ) . the color of each curve denotes the feedback model and the top ( bottom ) panel shows results for the smc ( mw ) simulation . the no - feedback model ( black curves ) is not shown on the mass - loading plot because there is a net inflow of gas at all times . both feedback models predict mass loadings of @xmath35 for the mw galaxy , but the cr feedback is capable of suppressing the sfr by a larger fraction than the thermal feedback model . in the smc galaxy the cr feedback model is capable of driving galactic winds with large ( @xmath36 ) mass loadings and suppresses the sfr significantly more than thermal feedback alone . + ] we begin by considering the sfrs of the simulated galaxies in fig . [ fig : ml ] . the sfr in simulations without feedback is higher than in simulations with feedback and is higher than typically observed sfrs of galaxies of these sizes . simulations with crs suppress sfr compared to simulations with thermal sn feedback only , especially in the smc - sized galaxy . this is because crs act as a source of pressure in the galaxy disk . this significantly changes the density pdf of the gas in the disk reducing the fraction of mass in star forming regions . outflow efficiency can be parametrized by the mass loading factor , @xmath34 , defined as the ratio of the gas outflow rate to the sfr . the solid curves in fig . [ fig : ml ] show @xmath34 as a function of time for different simulations . outflow rates are measured as the instantaneous mass flux through the plane parallel to the galactic disk at a height of 20kpc . in the mw simulation the mass loading is approximately 0.5 in both simulations , whereas in the smc simulation the mass loading is @xmath37 in the simulation with crs and only @xmath38 in the simulation with thermal feedback only . this indicates that crs greatly enhance efficiency of outflows from dwarf galaxies . velocity of the outflowing gas ( @xmath39 ) as a function of halo circular velocity . the gray points show the observations of @xcite ( downward pointing triangles ) and @xcite ( upward pointing triangles ) . the solid points show simulation predictions . the squares ( circles ) show the mw ( smc ) simulations and the colors denote the feedback model . in both galaxies , the outflows in the cr feedback models ( blue points ) have velocities comparable to the obserations , whereas the thermal feedback models ( red points ) overestimate the wind velocity by a large factor . + ] figure [ fig : vz ] shows velocity of the outflowing gas , @xmath40 , as a function of the circular velocity of the halo , @xmath41 , compared to observations of cool wind gas around dwarf galaxies @xcite and @xmath42 starburst dominated galaxies @xcite . we measure outflow velocities by projecting the gas field perpendicular to the disk and calculating the velocity that contains 90% of the cool ( @xmath43k ) gas . in each galaxy the thermal feedback simulation predicts outflow velocities that are significantly larger than those observed whereas the cr runs are comparable to the observations . finally , fig . [ fig : im ] shows the temperature of the outflowing gas in a thin slice through the centre of the simulated galaxies ( left ) . the notable difference between simulations is that wind in the cr simulation is considerably cooler , especially in the smc simulation . the panels to the right of this figure show the profiles of velocity and outward pressure gradient . the thermal feedback run has winds that accelerate abruptly from the galactic disk up to @xmath44km / s and thereafter have a constant velocity . the cr simulations , however show a wind that accelerates smoothly into the halo . the reason for this is revealed in the right - hand panels , where it is immediately apparent that the pressure gradient in the halo with crs is a factor of 3 - 10 larger in the cr simulation than in the thermal feedback simulation ( the difference is particularly striking in the smc simulation ) . these results illustrate that the wind properties in the simulations with crs are qualitatively different properties to the wind driven by thermal sn feedback . our simulations show that energy injection in the form of crs is a promising feedback process that can substantially aid in driving outflows from star - forming galaxies . first , we find that cr injection can suppress the sfr by providing an extra source of pressure that stabilizes the disk . turbulent and cr pressure are in equipartition in the disk , thus the cr pressure can significantly affect most of the volume of the disk , but will be sub - dominant inside supersonic molecular clouds , where turbulent pressure dominates over both cr and thermal presure , particularly in the dwarf galaxy . the sfrs measured in our galaxies with cr feedback are comparable to observed sfrs for both the mw and the smc . second , we find that addition of the cr feedback increases the mass loading factor , @xmath34 , in the dwarf galaxy by a factor of ten compared to the simulation with sn only feedback . as a result , the smc and mw - sized galaxies ( circular velocities of @xmath45 and @xmath46 km / s , respectively ) have mass loading factors that differ by a factor of @xmath47 , depending on the stage of evolution . this is in rough agreement with expectations from theoretical models based on simulations and semi - analytic models , which show that dependence @xmath48 with @xmath49 is needed to reproduce the observed faint end of the galaxy stellar mass function and other properties of the galaxy population ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . moreover , the wind velocities in the smc and mw - sized simulated galaxies are consistent with the observed trend for galaxies in this mass range @xcite both in normalization and slope . although we have reported only two models , these results are encouraging , especially because simulation parameters have not been tuned to reproduce these observations . perhaps the most intriguing difference of the cr - driven winds compared to the winds driven by thermal sn feedback is that they contain significantly more `` warm '' @xmath2 k gas . this is especially true for the dwarf galaxy , which develops a wind strikingly colder than in the sn - only simulation ( see fig . [ fig : im ] ) . the cr - driven wind has a lower velocity , and is accelerated gradually with vertical distance from the disk . the reason for these differences is that the gas ejected from the disk is accelerated not only near star - forming regions , as is the case in sn - only simulations , but is continuously accelerated by the pressure gradient established by crs diffused outside of the disk ( see fig . [ fig : im ] ) . the diffusion of crs is thus a key factor in ejecting winds and in their resulting colder temperatures . the cooler temperatures of the ejected gas may be one of the most intriguing new features of the cr - driven winds , as this may provide a clue on the origin of ubiquitous warm gas in gaseous halos of galaxies ( e.g. , * ? ? ? * and references therein ) . detailed predictions of cgm properties will require cosmological galaxy formation simulations incorporating cr feedback , which we will pursue in future work . several studies have explored effects of cr injection on galaxies . @xcite found that crs suppress the sfr in dwarf galaxies by an amount comparable to our simulations , but have almost no effect on the sfr of mw - sized systems . we find significant sfr suppression for both masses . additionally , @xcite found that crs did not generate winds with diffusion alone and in a recent study using a similar model @xcite argued that to launch winds cr streaming is crucial . in contrast , we find that cr - driven winds are established with cr diffusion alone . these differences likely arise due to assumption of equilibrium between the sources of crs ( star formation @xmath50 ) and the sinks ( catastrophic losses @xmath51 ) in the subgrid model of @xcite . the subgrid model thus predicts that cr pressure scales as @xmath52 and is subdominant to the thermal ism pressure at densities @xmath53@xmath54 ( see fig . 7 of * ? ? ? this assumption of equilibrium , which is likely true only in the deepest parts of the galaxy potential well ( see e.g. the discussion in * ? ? ? * ) , breaks down in lower density gas . in our simulations we do not assume such equilibrium and we find significant pressure contributions from crs up to much higher densities . our results thus indicate that crs , even in the diffusion limit , not only suppress star formation but also drive outflows efficiently . thus , the effects of cr feedback on the properties of galaxies of different masses should be significantly stronger and span a wider range of masses than simulations that use the @xcite model ( e.g. * ? ? ? while this manuscript was in a late stage of preparation @xcite appeared as a preprint . these authors have presented simulations of a mw - sized galaxy , similar to the model presented here , albeit without accounting for cr cooling losses and with a much larger sfr in their model galaxy ( up to @xmath55 . where our results overlap ( e.g. , mass - loading factor ) with those of @xcite we find remarkably good agreement . these authors also find that outflows are efficiently generated with cr diffusion alone . our study extends the results of @xcite by presenting the differences between wind properties in dwarf and mw - sized systems . the results of @xcite and our study indicate that crs can significantly suppress star formation in galaxies and efficiently drive outflows with significant mass loading factors and velocities comparable to observed outflows . a detailed exploration of the effects of such feedback on the galaxy population in a full cosmological setting is therefore extremely interesting . ng and ak were supported via nsf grant oci-0904482 . ak was supported by nasa atp grant nnh12zda001n and by the kavli institute for cosmological physics at the university of chicago through grants nsf phy-0551142 and phy-1125897 and an endowment from the kavli foundation and its founder fred kavli .
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we present results from high - resolution hydrodynamic simulations of isolated smc- and milky way - sized galaxies that include a model for feedback from galactic cosmic rays ( crs ) .
we find that crs are naturally able to drive winds with mass loading factors of up to @xmath0 in dwarf systems .
the scaling of the mass loading factor with circular velocity between the two simulated systems is consistent with @xmath1 required to reproduce the faint end of the galaxy luminosity function .
in addition , simulations with cr feedback reproduce both the normalization and the slope of the observed trend of wind velocity with galaxy circular velocity .
we find that winds in simulations with cr feedback exhibit qualitatively different properties compared to sn driven winds , where most of acceleration happens violently in situ near star forming sites .
the cr - driven winds are accelerated gently by the large - scale pressure gradient established by crs diffusing from the star - forming galaxy disk out into the halo .
the cr - driven winds also exhibit much cooler temperatures and , in the smc - sized system , warm ( @xmath2 k ) gas dominates the outflow .
the prevalence of warm gas in such outflows may provide a clue as to the origin of ubiquitous warm gas in the gaseous halos of galaxies detected via absorption lines in quasar spectra .
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