# Presentation: 594558

## Finite-size effects on two-particle production in continuous and discrete spectrum Pionium lifetime from DIRAC at CERN

- Physics motivation
- Pionium production
- How to measure lifetime ~3 fs
- Finite-size effects
- Results from DIRAC
- Correlation study of particle interaction
- Conclusions

- R. Lednický, JINR Dubna & IP ASCR Prague

## Physics motivation

- A2  20   99.6%
-        2       0.4%

- Bohr radius |a| = 1/(e2) = 387 fm

- f

- +

- -

- 0

- 0

- d~1 fm

- A2

- 1/n0 ~ |cn0 (r*=0)|2 |f00;+- (k*=0) |2

- 1/(|na|3)

- (2/9)|a0-a2|2

- aI = s-wave isospin-I
-  scattering length

****** **Access to low-energy** ** scattering

****** **Testing mechanism of SBS** **of QCD

- SU(Nf)L  SU(Nf)R  SU(Nf)L+R

- R. Lednický       Budapest

- 2.12.2004

## Spontaneous breaking  asymmetric vacuum  order parameters:

- Spontaneous breaking  asymmetric vacuum  order parameters:

- Explicit breaking: mq=(mu+md)/2  0

-  non-zero GB masses: m2 = 2B0 mq + 4A0 mq2 + ..

- Pion decay constant F0  F = 93.3 MeV

- Quark condensate 0|qq|0 = -B0F02

-  Modification of low-energy theorems:

- Atree (s,t,u)= (s-****m2)/F02 (32/3) m (a0 - a2)

****** **= -2 mq 0|qq|0 /(m2 F2)

- SPT: ****** **** ** 1 (neglect mq2 term)  Gasser, Leutwyler ‘84

- a0 - a2 =      0.290 fm                        tree

-               + 0.073 fm                     1-loop

-               + 0.011 fm                     2-loop

-               =      0.374 fm       2%

- Weinberg ’66

- R. Lednický       Budapest

- 2.12.2004

## GPT:  arbitrary from 1  0 (vanishing condensate)

- GPT: ****** **arbitrary from 1  0 (vanishing condensate)

- Fuchs, Sazdjian, Stern, Knecht, Moussalam ’91-95

****** **a0 - a2 can be up to 25% higher than  0.374 fm from SPT

- Experiment up to 2001: 0.41 fm   20%

- Added BNL Kl4 data: 0.37 fm   10%

****** **Quark mass ratio ms/mq = 26 8

****** ** phase transition Tc = 200  150

- DIRAC at CERN aims:    5%

- R. Lednický       Budapest

- 2.12.2004

## Pionium production

- Migdal, Watson, Sakharov    hadronic processes

-  Dominated by **FSI **due to small binding energy < 3 keV

-  Closely related to free +- production at **Q=2k***  0

**FSI** theory: Fermi    -decay

-  Continuum:

- d6N/(d3**p**1 d3**p**2)= d6N0/(d3**p**1 d3**p**2)|-**k***(**r***)|2

- d3N/(d3**p**A)= (2)3Ad6N0/(d3**p**1 d3**p**2)|A(**r***)|2

-  Discrete spectrum: A2

- +

- x1

- x2

- x2

- x1

- p1

- p2

- p1

- p2

- pA

**r***= **x**1*- **x**2*= distance between the + and + emitters in pair cms

- +

- -

- A2

**p**1 ≈  **p**2

- R. Lednický       Budapest

- 2.12.2004

## FSI amplitudes  wave functions 

- Similar to Coulomb distortion of -decay
- RL, Lyuboshitz (82)  condition |t*|  r*2

- e-i**kr   **** **   -**k**(**r**) **  **    [  e-i**kr **+f(k)eikr/r ]

- eicAc

- F=1+ _______ + ...

**kr**+kr

- ka

- Coulomb

- s-wave
- strong FSI

- FSI

- fcAc(G0+iF0)

- }

- |a| = Bohr radius

- Ac = Point-like Coulomb factor

-  **Continuum**:

-  **Discrete spectrum**: A2

- n0(r*) = cn0(0) [1+(n)f(0)/|na|] [1- r*/|a|+f(0)/r* + O(r*/a)]

- n0(r*)  exp(-r*/|na|)  0   at  r* >> |na|

- -**k***(**r***)  Ac [1 - r*/(1+cos*)|a|** **+ f(0)/r* + O(r*/a)]   at   k*  0

-  **Universal** FSI dependence on r* and scattering amplitude f

- at k*  0 , r*<< |na| and a given orbital angular momentum

- 

- p-wave

- |-**k***(**r***)|  1  at  r* >> |na|

##  Then, neglecting FSI

**S**hort-**L**ived **S**ources  **SLS** (, , ..)  r* |a| = 387 fm both ’s from **SLS**

**L**ong-**L**ived **S**ources  **LLS** (, Ks, , ..)  r* |a| one or both ’s from **LLS**

****** **Then, neglecting **FSI**

- |-**k***(**r***)|2 = fc |c-**k***(**0**)|2  + (1- fc) |c-**k***(****)|2

- |**A**(**r***)|2 = fc |c**A**(**0**)|2  + (1- fc) |c**A**(****)|2

- 1

- Ac(Q)

- 0

- 1/(|na|3)

- A={n,l=0}

****** **Universal relation between production of free and bound ’s:

- d6N/(d3**p**1 d3**p**2)= g d6N0/(d3**p**1 d3**p**2) [fc Ac(Q) + (1- fc)]

- d3N/(d3**p**A)= g fc d6N0/(d3**p**1 d3**p**2) (2)3A/(|na|3),   **p**1 ≈  **p**2

- fc= **SLS** fraction

## How to measure lifetime ~3fs

- Z

- Z

- +

- -

- 0

- 0

- A2

- A’2

- Nemenov ‘85: choose Z and A so that dec  br  and

- = breakup (ionization) probability of A2 in the target

**P****br**** ****= N****A****br****/N****A**

**P****br** is unique function of Z, target geometry, A and 10

- determined to 1% by kinetic eqs. with initial conditions given by 1/n3 law:

- P10= 83%, P20= 10.4%, P10= 3.1%, P10= 1.3%, ..

- g. s. A2 at A≈ 17 (4.7 GeV/c) flies ~ dec = Ac 10  17 m in

- the target and encounters ~ 105 atoms  br ~ Z-2

- measure the g.s. lifetime 10 ~3 fs through

- R. Lednický       Budapest

- 2.12.2004

## Excitation and break-up of produced atoms (NA) compete with decay

- pairs from
- break-up provide
- measurable signal **N****A****br**

**P****br**** ** is linked to **N****A****br **: **P****br**** ****= N****A****br****/N****A**

- The number of produced atoms **N****A** is not directly measurable  to be
- obtained from **N**()

-   Break-up probability is linked to the lifetime
-                                (theory of relativistic atomic collisions)

- =10%  Pbr =4%

## Access to NA and NAbr

- Pions from break-up have very small Q
- At target exit this feature is smeared by
- multiple scattering, especially in QT

- Access to **N****A **and **N****A****br**

****** ****N****A****br** can be directly measured as an excess over the fitted

****** ****N****A **can be calculated from spectrum fitted at Q > 4 MeV/c

- spectrum of the free pairs interpolated to Q < 4 MeV/c

- R. Lednický       Budapest

- 2.12.2004

## Finite-size effects

- r* ~ 10 fm  but    ~30-40 fm   and   ’  ~900 fm

- UrQMD:  pNi  2 at 24 GeV    

- ~1% ’, ~19% 

**ML**

******’**

****

**ML ~ r*****2****/[1+(r*********/r****0****)****2a****]****2b**

-  short-distance parametrization

****

- ’ and 
- contributions
- well fitted based
- on exponential
- decay law

## DIRAC CF:

- DIRAC CF:

- CF=N{ |-**k***(**r***)|2SLS +(1- )}[1+s Q]

- SLS determined by:

- ML(r0,a,b),  fML=1-f-f’

- ’,  f’

- ,  f

- N

****

**r****0  **fm

**a**

**b**

**f******

**f********’**

**s**

**f******** ****= 17******** 6%**** **

**ML**

****

**G****2fm**

**G****3fm**

******’**

## Introduce correction factors:

- 1+(k*) = |-**k***(**r***)/c-**k***(**0**)|2 SLS

- 1+n = |n0(r*)/cn0(**0**)|2 SLS

- = |1-r*/|a|+f(k*)/r*|2 +O((r*/a)2)+O((k*r*)2)SLS

- = (1+2(n)f(0)/|na|)|1-r*/|a|+f(0)/r*|2 +O((r*/a)2)SLS

- f(k*) = s-wave scattering amplitude,  (n) ≈ 3

**Universality** slightly violated even at k* = 0:

- 1+n ≈ [1+6 f(0)/|na| + O((r*/a)2)SLS ] (1+(0))

- R. Lednický       Budapest

- 2.12.2004

##  Universality violated by  and ’ 

****

****

**G****2fm****, G****3fm****, ML**

**G****2fm**

**G****3fm**

******’**

******’**

******’**

******’**

****

****

**ML**

- (k*)-(0)

- n-(0)

**r*********=******

**r*********=******

-  Universality violated by ****** **and ******’**

-  ******’** still not a LLS

-  Universality OK for short-distance part G, ML

## 1% ’      19%        80% G3fm       f(0) = 0.186 fm 

- 1% ******’**      19% ******       **80% G3fm       f(0) = 0.186 fm** **

-  No impact of short-distance (G) uncertainty on the lifetime

-  Main theor. uncertainty in the lifetime from ****** **and f(0)

**G****2fm**

******f(0)=+20%**

******f********=-30%**

******f********=+30%**

## 1% ’      19%        80% G3fm       f(0) = 0.186 fm 

- 1% ******’**      19% ******       **80% G3fm       f(0) = 0.186 fm** **

- Shift ****** **(if neglecting 1+) and uncertainties ******(********)** in the lifetime

****

******(********)**

- 30% in f****

- 10% in f(0)

- +14%            7.5%             1.8%           Q=4-20 MeV/c

- comment

- +11%            6.0%             1.4%           Q=3-20 MeV/c

- +2.5%            0.8%             0.3%         multi-layer target

-  Systematic uncertainty ******(********)** from finite-size effect < 10%

- < present DIRAC statistical error in ****

-  Future DIRAC with multi-layer target: ******(********)** < 1%

## Results from DIRAC

- Downstream detectors: DCs, VH, HH, C, PSh, Mu.

- Upstream detectors: MSGCs, SciFi, IH.

- Setup features
- Angle to proton
- beam:                =5.7
- Channel
- aperture:  =1.2·10–3 sr
- Magnet:            2.3 T·m
- Momentum
- range:        1.2-7 GeV/*c*
- Resolution in
- relative momentum:
-   QX= QY=0.4 MeV/*c*
-    QL= 0.6 MeV/*c*

- R. Lednický       Budapest

- 2.12.2004

## Experimental Qtot  and Ql distributions (Ni2001)

- Fit MonteCarlo C and nC background

- simultaneously to experimental Qtot and Ql  spectra

- outside the A signal region (Qtot > 4 MeV, Ql > 2 MeV)

- R. Lednický       Budapest

- 2.12.2004

## Atomic breakup signals in Qtot  and Ql

** ****Q****tot**** and Q****l**** provide same number of events ******** background consistent**

** ******** ****= 2.8 ± 0.4****stat**** fs (15% stat. error for 40% data)**

** ****Signal shapes well reproduced**

- R. Lednický       Budapest

- 2.12.2004

## Correlation study of particle interaction

**-**

******+************ ****scattering length**** **
**f****0**** from NA49 CF**

**Fit  CF(********+************) by RQMD  **
**with SI scale:**** **
**f****0**** ******** ****sisca**** ****f****0****input**** **
**f****0****input**** = ****0.232 fm**

**sisca**** = ****0.6********0.1 **
**Compare with**** **
**~0.8 ****from S********PT **
**& BNL E765 **
**K ******** e******** **

******+**********

**CF=****Norm**** ****[****Purity**** RQMD(****r***** ******** ****Scale********r*****)+1-****Purity****]**

## Conclusions

- Multiparticle production can serve as a lab to study the two-particle strong interaction through hadronic atoms and particle correlations
- Experiment DIRAC accumulated data allowing to measure pionium lifetime with the planned 10% statistical error
- Systematic lifetime error due to the finite-size effect is under control and < stat. error
- The FSI effect on particle correlations can be used to extract the information on two-particle strong interaction sometimes hardly accessible by other means , K, KK, , p, , ..

- R. Lednický       Budapest

- 2.12.2004

## NA49 central Pb+Pb 158 AGeV vs RQMD

**Long **** ****tails in RQMD:  ******r***** = **21** fm  for r* < **50** fm

**29** fm  for r* < **500** fm

**Fit   ****CF=****Norm**** ****[****Purity**** RQMD(****r***** ******** ****Scale********r*****)+1-****Purity****]**

**Scale=0.76**

**Scale=0.92**

**Scale=0.83**

****** ****RQMD overestimates**** ****r* ****by 10-20%  ? Too much rescatterings **

**already at SPS**

******p**********

- R. Lednický       Budapest

- 2.12.2004

## >

******=0.89**
******r*********=16 fm**

******=0.93**
******r*********=24 fm**

******=0.81**
******r*********=18.4 fm**

******=0.76**
******r*********=18.1 fm**

******=0.94**
******r*********=24.4 fm**

******=0.91**
******r*********=22.9 fm**

**Tails in RQMD****:  **
****r***** = **21** fm  for r* < **50** fm

**29** fm  for r* < **500** fm

**Strong FSI on**

**>**

******+********+**

******+********-**

****** ****Strong FSI important for**** ********+********-**

****** ****1-G fit:****  ********(********+********+****) ******** 0.8,  ********r********* 25% ******** **

****** ****2-G fit:**** ********+********+**** ******** ********+********- **

******r*********QS**** ****< ********r*********Coul**** **

- R. Lednický       Budapest

- 2.12.2004

## Femtoscopy with nonidentical particles

- CF = ****|-**k***** **(**r***)|2****

**Be careful when comparing **
**QS**** (********+********+**** ****..)**** ****and ****FSI**** correlations (********+************..) ******
**different sensitivity to ****r*****-distribution ****tails **

****** ****QS & strong FSI****:  non-Gaussian ****r*****-tail influences only **
**first few bins in ****Q=2k***** and its effect is mainly **
**absorbed in suppression parameter ******** **

** ******** ****Coulomb FSI****: sensitive to ****r*****-tail up to ****r***** ~ Bohr radius**

- |a|=|z1z2e2|-1
**fm**

******      ********K      ********p      KK     pp**
**388     249     223     110     58**

****** ****In Gaussian fits one may expect  ****r****0****(********+********+****) <  r****0****(********+************)**** **

****** ****Use realistic models like transport codes**

- R. Lednický       Budapest

- 2.12.2004

## Effect of nonequal times in pair cms

- RL, Lyuboshitz SJNP 35 (82) 770

**Applicability condition of ****equal-time approximation****: ** **|t*| ******** ********r*****2**

**r****0****=2 fm **** ********0****=2 fm/c**

**r****0****=2 fm **** ****v=0.1**

****
**OK for heavy **
**particles**
****** ****OK within 5%**
**even for ****pions**** if**
**************0 ****~r****0 ****or lower**

- R. Lednický       Budapest

- 2.12.2004

## Coalescence: deuterons ..

**   ****E****d****d****3****N/d****3****p****d**** ****= ****B****2**** E****p****d****3****N/d****3****p****p**** ****E****n****d****3****N/d****3****p****n**** **  **p****p******** p****n******** ****1⁄2****p****d**** **

- WF in **continuous** **pn** spectrum -**k***(**r***)  WF in **discrete ****pn** spectrum **b**(**r***)

**Coalescence** factor: **B****2 ****= (2********)****3****(m****p****m****n****/m****d****)****-1********t******|**b**(**r***)|2****** ~  ****R****-3**

**Triplet fraction** = **3⁄4**** ******** ****unpolarized Ns**

- Usually: **n**** ** **p**

- Much **stronger energy dependence**
- of  **B****2 **** ~  ****R****-3  **than expected from
- pion and proton interferometry radii

**B****2**

**R****(****pp****) ~**** 4 fm ****from AGS to SPS**

- Lyuboshitz (88) ..

- R. Lednický       Budapest

- 2.12.2004