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this is achieved at sufficiently small parameters @xmath93 and @xmath156 .
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in contrast , @xmath152 and @xmath144 depend on two wave - correlations , both waves being given at the same instant .
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therefore , the values of @xmath152 and @xmath144 do not depend on the rate of phase variations [ @xmath105 does not enter the factor ( [ seventeen ] ) describing the effect of phase diffuser ] .
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moreover , these quantities become independent of @xmath93 at long propagation paths because light scattering on atmospheric inhomogeneities prevails in this case .
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the plots in figs .
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3 anf 4 show that the finite - time effect is quite sizable even for very slow " detectors ( @xmath157 ) .
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our paper makes it possible to estimate the actual utility of phase diffusers in several physical regimes .
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we have analyzed the effects of a diffuser on scintillations for the case of large - amplitude phase fluctuations .
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this specific case is very convenient for theoretial analysis because only two parameters are required to describe the effects of the diffuser .
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phase fluctuations may occur independently in space as well as in time .
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also , our formalism can be applied for the physical situation in which a spatially random phase distribution drifts across the beam .
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[ twtw ] . )
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our results show the importance of both parameters , @xmath93 and @xmath142 , on the ability of a phase diffuser to suppress scintillations .
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this work was carried out under the auspices of the national nuclear security administration of the u.s .
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department of energy at los alamos national laboratory under contract no .
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de - ac52 - 06na25396 .
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we thank onr for supporting this research .
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the so - called `` nucleon spin crisis '' raised by the european muon collaboration ( emc ) measurement in 1988 is one of the most outstanding findings in the field of hadron physics @xcite,@xcite .
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the renaissance of the physics of high energy deep inelastic scatterings is greatly indebted to this epoch - making finding .
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probably , one of the most outstanding progresses achieved recently in this field of physics is the discovery and the subsequent research of completely new observables called generalized parton distribution functions ( gpd ) .
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it has been revealed that the gpds , which can be measured through the so - called deeply - virtual compton scatterings ( dvcs ) or the deeply - virtual meson productions ( dvmp ) , contain surprisingly richer information than the standard parton distribution functions @xcite@xcite .
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roughly speaking , the gpds are generalization of ordinary parton distributions and the elastic form factors of the nucleon .
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the gpds in the most general form are functions of three kinematical variables : the average longitudinal momentum fraction @xmath1 of the struck parton in the initial and final states , a skewdness parameter @xmath3 which measures the difference between two momentum fractions , and the four - momentum - transfer square @xmath4 of the initial and final nucleon . in the forward limit @xmath5 ,
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some of the gpds reduce to the usual quark , antiquark and gluon distributions .
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on the other hand , taking the @xmath0-th moment of the gpds with respect to the variable @xmath1 , one obtains the generalizations of the electromagnetic form factors of the nucleon , which are called the generalized form factors of the nucleon .
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the complex nature of the gpds , i.e. the fact that they are functions of three variable , makes it quite difficult to grasp their full characteristics both experimentally and theoretically . from the theoretical viewpoint
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, it may be practical to begin studies with the two limiting cases .
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the one is the forward limit of zero momentum transfer .
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we have mentioned that , in this limit , some of the gpds reduce to the ordinary parton distribution function depending on one variable @xmath1 .
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however , it turns out that , even in this limit , there appear some completely new distribution functions , which can not be accessed by the ordinary inclusive deep - inelastic scattering measurements .
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very interestingly , it was shown by ji that one of such distributions contains valuable information on the total angular momentum carried by the quark fields in the nucleon @xcite@xcite .
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this information , combined with the known knowledge on the longitudinal quark polarization , makes it possible to determine the quark orbital angular momentum contribution to the total nucleon spin purely experimentally .
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another relatively - easy - to - handle quantities are the generalized form factors of the nucleon @xcite,@xcite , which are given as the non - forward nucleon matrix elements of the spin-@xmath0 , twist - two quark and gluon operators .
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since these latter quantities are given as the nucleon matrix elements of local operators , they can be objects of lattice qcd simulations .
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( it should be compared with parton distributions .
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the direct calculation of parton distributions is beyond the scope of lattice qcd simulations , since it needs to treat the nucleon matrix elements of quark bilinears , which are _ nonlocal in time_. ) in fact , two groups , the lhpc collaboration and the qcdsf collaboration independently investigated the generalized form factors of the nucleon , and gave several interesting predictions , which can in principle be tested by the measurement of gpds in the near future @xcite@xcite . although interesting , there is no _ a priori _ reason to believe that the predictions of these lattice simulations are realistic enough .
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the reason is mainly that the above mentioned lattice simulation were carried out in the heavy pion regime around @xmath6 with neglect of the so - called disconnected diagrams .
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our real world is rather close to the chiral limit with vanishing pion mass , and we know that , in this limit , the goldstone pion plays very important roles in some intrinsic properties of the nucleon .
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the lattice simulation carried out in the heavy pion region is in danger of missing some important role of chiral dynamics . on the other hand ,
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the chiral quark soliton model ( cqsm ) is an effective model of baryons , which maximally incorporates the chiral symmetry of qcd and its spontaneous breakdown @xcite,@xcite .
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( see @xcite@xcite for early reviews . )
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it was already applied to the physics of ordinary parton distribution functions with remarkable success @xcite@xcite . for instance , an indispensable role of pion - like quark - antiquark correlation was shown to be essential to understand the famous nmc measurement , which revealed the dominance of the @xmath7-quark over the @xmath8-quark inside the proton @xcite,@xcite,@xcite .
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then , it would be interesting to see what predictions the cqsm would give for the quantities mentioned above .
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now , the main purpose of the present study is to study the generalized form factors of the nucleon within the framework of the cqsm and compare its predictions with those of the lattice qcd simulations . of our particular interest here
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is to see the change of final theoretical predictions against the variation of the pion mass .
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such an analysis is expected to give some hints for judging the reliability of the lattice qcd predictions at the present level for the observables in question .
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the plan of the paper is as follows . in sect.ii
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, we shall briefly explain how to introduce the nonzero pion mass into the scheme of the cqsm with pauli - villars regularization . in sect.iii
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, we derive the theoretical expressions for the generalized form factors of the nucleon .
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sect.iv is devoted to the discussion of the results of the numerical calculations .
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some concluding remarks are then given in sect.v .
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we start with the basic effective lagrangian of the chiral quark soliton model in the chiral limit given as @xmath9 with @xmath10 which describes the effective quark fields , with a dynamically generated mass @xmath11 , strongly interacting with pions @xcite,@xcite .
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since one of the main purposes of the present study is to see how the relevant observables depend on pion mass , we add to @xmath12 an explicit chiral symmetry breaking term @xmath13 given by @xcite @xmath14 .
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\label{eq : lsb}\ ] ] here the trace in ( [ eq : lsb ] ) is to be taken with respect to flavor indices .
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the total model lagrangian is therefore given by @xmath15 naturally , one could have taken an alternative choice in which the explicit chiral - symmetry - breaking effect is introduced in the form of current quark mass term as @xmath16 .
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we did not do so , because we do not know any consistent regularization of such effective lagrangian with finite current quark mass within the framework of the pauli - villars subtraction scheme , as explained in appendix of @xcite .
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the effective action corresponding to the above lagrangian is given as @xmath17 = s_f [ u ] + s_m [ u ] , \label{eq : energsol}\ ] ] with @xmath18 = - \,i \,n_c \,\mbox{sp } \
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, \ln ( i \not\!\partial - m u^{\gamma_5 } ) , \ ] ] and @xmath19 = \int \frac{1}{4 } \,f_{\pi}^2 \,m_{\pi}^2 \ , \mbox{tr}_f \ , [ u ( x ) + u^{\dagger } ( x ) - 2 ] \,d^4 x .\ ] ] here @xmath20 with @xmath21 and @xmath22 representing the trace of the dirac gamma matrices and the flavors ( isospins ) , respectively .
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the fermion ( quark ) part of the above action contains ultra - violet divergences . to remove these divergences , we must introduce physical cutoffs . for the purpose of regularization , here we use the pauli - villars subtraction scheme . as explained in @xcite ,
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we must eliminate not only the logarithmic divergence contained in @xmath23 $ ] but also the quadratic and logarithmic divergence contained in the equation of motion shown below .
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to get rid of all these troublesome divergence , we need at least two subtraction terms .
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the regularized action is thus defined as @xmath24 = s_f^{reg } [ u ] + s_m [ u ] , \ ] ] where @xmath25 = s_f [ u ] - \sum_{i = 1}^2 \,c_i \,s_f^{\lambda _ i } [ u ] .\ ] ] here @xmath26 is obtained from @xmath27 $ ] with @xmath11 replaced by the pauli - villars regulator mass @xmath28 .
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these parameters are fixed as follows .
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first , the quadratic and logarithmic divergence contained in the equation of motion ( or in the expression of the vacuum quark condensate ) can , respectively , removed if the subtraction constants satisfy the following two conditions : @xmath29 ( we recall that the condition which removes the logarithmic divergence in @xmath23 $ ] just coincides with the 1st of the above conditions . ) by solving the above equations for @xmath30 and @xmath31 , we obtain @xmath32 which constrains the values of @xmath30 and @xmath31 , once @xmath33 and @xmath34 are given . for determining @xmath33 and @xmath34
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, we use two conditions @xmath35 and @xmath36 which amounts to reproducing the correct normalization of the pion kinetic term in the effective meson lagrangian and also the empirical value of the vacuum quark condensate . to derive soliton equation of motion , we must first write down a regularized expression for the static soliton energy . under the hedgehog ansatz @xmath37 for the background pion fields ,
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it is obtained in the form : @xmath38 = e_f^{reg } [ f ( r ) ] + e_m [ f ( r ) ] , \ ] ] where the meson ( pion ) part is given by @xmath39 = - \,f_{\pi}^2 \,m_{\pi}^2 \int d^3 x \,[\cos f ( r ) - 1 ] , \ ] ] while the fermion ( quark ) part is given as @xmath40 = e_{val } + e_{v p}^{reg } , \label{eq : estatic}\ ] ] with @xmath41 here @xmath42 are the quark single - particle energies , given as the eigenvalues of the static dirac hamiltonian in the background pion fields : @xmath43 with @xmath44 , \label{eq : dirach}\ ] ] while @xmath45 denote energy eigenvalues of the vacuum hamiltonian given by ( [ eq : dirach ] ) with @xmath46 ( or @xmath47 ) . the particular state @xmath48 , which is a discrete bound - state orbital coming from the upper dirac continuum under the influence of the hedgehog mean field , is called the valence level .
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the symbol @xmath49 in ( [ eq : regenerg ] ) denotes the summation over all the negative energy eigenstates of @xmath50 , i.e. the negative energy dirac continuum .
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the soliton equation of motion is obtained from the stationary condition of @xmath51 $ ] with respect to the variation of the profile function @xmath52 : @xmath53 \nonumber \\ & = & 4 \pi r^2 \,\left\ { - \,m \,[s ( r ) \sin f ( r ) - p ( r ) \cos f ( r ) ] + f_{\pi}^2 m_{\pi}^2 \sin f ( r ) \right\ } , \end{aligned}\ ] ] which gives @xmath54 here @xmath55 and @xmath56 are regularized scalar and pseudoscalar quark densities given as @xmath57 with @xmath58 while @xmath59 and @xmath60 are the corresponding densities evaluated with the regulator mass @xmath28 instead of the dynamical quark mass @xmath11 .
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we also note that @xmath61 and @xmath62 . as usual
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, a self - consistent soliton solution is obtained as follows with use of kahana and ripka s discretized momentum basis @xcite,@xcite .
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first by assuming an appropriate ( though arbitrary ) soliton profile @xmath52 , the eigenvalue problem of the dirac hamiltonian is solved . using the resultant eigenfunctions and their associated eigenenergies
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, one can calculate the regularized scalar and pseudoscalar densities @xmath55 and @xmath56 .
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with use of these @xmath63 and @xmath64 , eq.([eq : profile ] ) can then be used to obtain a new soliton profile @xmath52 .
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the whole procedure above is repeated with this new profile @xmath52 until the self - consistency is attained .
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since the generalized form factors of the nucleon are given as moments of generalized parton distributions ( gpds ) , it is convenient to start with the theoretical expressions of the unpolarized gpds @xmath65 and @xmath66 within the cqsm . following the notation in @xcite,@xcite
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, we introduce the quantities @xmath67 and @xmath68 here , the isoscalar and isovector combinations respectively correspond to the sum and the difference of the quark flavors @xmath69 and @xmath70 . the relation between these quantities and the generalized parton distribution functions @xmath71 and @xmath72
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are obtained most conveniently in the so - called breit frame .
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they are given by @xmath73 where @xmath74 these two independent combinations of @xmath65 and @xmath66 can be extracted through the spin projection of @xmath75 as @xmath76 where tr " denotes the trace over spin indices , while @xmath77 .
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now , within the cqsm , it is possible to evaluate the right - hand side ( rhs ) of ( [ eq : hetrace ] ) and ( [ eq : emtrace ] ) .
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since the answers are already given in several previous papers @xcite@xcite , we do not repeat the derivation . here
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we comment only on the following general structure of the theoretical expressions for relevant observables in the cqsm .
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the leading contribution just corresponds to the mean field prediction , which is independent of the collective rotational velocity @xmath78 of the hedgehog soliton .
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the next - to - leading order term takes account of the linear response of the internal quark motion to the rotational motion as an external perturbation , and consequently it is proportional to @xmath78 .
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it is known that the leading - order term contributes to the isoscalar combination of @xmath79 and to the isovector combination of @xmath80 , while the isoscalar part of @xmath79 and the isovector part of @xmath80 survived only at the next - to - leading order of @xmath78 ( or of @xmath81 ) .
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the leading - order gpds are then given as @xmath82 here the symbol @xmath83 denotes the summation over the occupied ( the valence plus negative - energy dirac sea ) quark orbitals in the hedgehog mean field . on the other hand ,
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the theoretical expressions for the isovector part of @xmath84 and the isoscalar part of @xmath85 , which survive at the next - to - leading order , are a little more complicated .
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they are given as double sums over the single quark orbitals as @xmath86 \
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, e^{i x m_n z^0 } \nonumber \\ & \ , & \hspace{30 mm } \times \ \langle n | \tau^a | m \rangle \langle m | \,\tau^a \,(1 + \gamma^0 \gamma^3 ) \ , e^{-i ( z^0/2 ) \hat{p}_3 } \ , e^{i { \mbox{\boldmath$\delta$ } } \cdot { \mbox{\boldmath$x$ } } } \,e^{-i ( z^0/2 ) \hat{p}_3 } | n \rangle .
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\label{eq : gpdheiv}\end{aligned}\ ] ] and @xmath87 e^{i x m_n z^0 } \nonumber \\ & \ , & \hspace{25 mm } \times \ \langle n | \tau^b | m \rangle \langle m | \,(1 + \gamma^0 \gamma^3 ) \,e^{-i ( z^0/2 ) \hat{p}_3 } \ , \frac{\varepsilon^{3 a b }
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\delta^a}{{\mbox{\boldmath$\delta$}}_{\perp}^2 } \ , e^{i { \mbox{\boldmath$\delta$ } } \cdot { \mbox{\boldmath$x$ } } } \,e^{-i ( z^0/2 ) \hat{p}_3 } | n \rangle .
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\label{eq : gpdemiv}\end{aligned}\ ] ] these four expressions for the unpolarized gpds , i.e. ( [ eq : gpdheis ] ) @xmath88 ( [ eq : gpdemiv ] ) , are the basic starting equations for our present study of the generalized form factors of the nucleon within the cqsm .
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there are infinite tower of generalized form factors , which are defined as the @xmath0-th moments of gpds . in the present study ,
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we confine ourselves to the 1st and the 2nd moments , which respectively corresponds to the standard electromagnetic form factors of the nucleon and the so - called gravitational form factors .
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we are especially interested in the second one , since they are believed to contain valuable information on the spin contents of the nucleon through ji s angular momentum sum rule @xcite,@xcite . for each isospin channel , the 1st and the 2nd moments of @xmath89 define the sachs - electric and gravito - electric form factors as @xmath90 and @xmath91 on the other hand , the 1st and the 2nd moments of @xmath92 respectively define the sachs - magnetic and gravito - magnetic form factors as @xmath93 and @xmath94 in the following , we shall explain how we can calculate the generalized form factors based on the theoretical expressions of corresponding gpds , by taking @xmath95 and @xmath96 as examples .
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setting @xmath97 and integrating over @xmath98 in ( [ eq : gpdheis ] ) , we obtain @xmath99 putting this expression into ( [ eq : ge10def ] ) , we have @xmath100 it is easy to see that , using the generalized spherical symmetry of the hedgehog configuration , the term containing the factor @xmath101 identically vanishes , so that @xmath102 is reduced to a simple form as follows : @xmath103 aside from the factor @xmath104 , this is nothing but the known expression for the isoscalar sachs - electric form factor of the nucleon in the cqsm @xcite .
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a less trivial example is @xmath96 , which is defined as the 2nd moment of @xmath105 . inserting ( [ eq : heis ] ) into ( [ eq : ge10def ] ) and carrying out the integration over @xmath1 ,
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we obtain @xmath106 using the partial - wave expansion of @xmath107 , this can be written as @xmath108 this can further be divided into four pieces as @xmath109 where @xmath110 with @xmath111 to proceed further , we first notice that , by using the generalized spherical symmetry , @xmath112 survives only when @xmath113 , i.e. @xmath114 which leads to the result : @xmath115 to evaluate @xmath116 , we first note that @xmath117^{(\lambda ) } .\ ] ] here , the generalized spherical symmetry dictates that @xmath118 must be zero , so that the rhs of the above equation is effectively reduced to @xmath119^{(0 ) } .\ ] ] this then gives @xmath120^{(0 ) } \ , \phi_n ( { \mbox{\boldmath$x$ } } ) .\end{aligned}\ ] ] owing to the identity @xmath121 we therefore find that @xmath122 next we investigate the third term @xmath123 . using @xmath124^{(\lambda ) }
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\ \sim \ - \,\frac{1}{\sqrt{3 } } \,\delta_{l,1 } \,\delta_{m,0 } \ , [ y_1 ( \hat{x } ) \times \hat{{\mbox{\boldmath$p$ } } } ] ^{(0 ) } , \end{aligned}\ ] ] we obtain @xmath125^{(0 ) } \ , \phi_n ( { \mbox{\boldmath$x$ } } ) .\end{aligned}\ ] ] this term vanishes by the same reason as @xmath116 does .
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the last term @xmath126 is a little more complicated .
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we first notice that @xmath127^{(\lambda)}_0 \nonumber \\ & \sim & \sum_{\lambda } \ , \langle 1 0 1 0 | \lambda 0 \rangle \ , \langle l m \lambda 0 | 00 \rangle \ , [ y_l ( \hat{x } ) \times [ { \mbox{\boldmath$\alpha$ } } \times \hat{{\mbox{\boldmath$p$ } } } ] ^{(\lambda ) } ] ^{(0 ) } \nonumber \\ & = & \delta_{m , 0 } \,\langle 1 0 1 0 | l 0 \rangle \
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