Finitesize effects on twoparticle production in continuous and

Finite-size effects on two-particle production in continuous and discrete spectrum Pionium lifetime from DIRAC at CERN R. Lednický, JINR Dubna & IP ASCR Prague • • Physics motivation Pionium production How to measure lifetime ~3 fs Finite-size effects Results from DIRAC Correlation study of particle interaction Conclusions

Physics motivation A 2 2 0 99. 6% 2 0. 4% + A 2 0 f - Bohr radius |a| = 1/( e 2) = 387 fm d~1 fm 0 1/ n 0 ~ | cn 0 (r*=0)|2 |f 00; +- (k*=0) |2 1/( |na|3) (2/9)|a 0 -a 2|2 a. I = 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 2. 12. 2004 R. Lednický Budapest

Spontaneous breaking asymmetric vacuum order parameters: Pion decay constant F 0 F = 93. 3 Me. V Quark condensate 0|qq|0 = -B 0 F 02 Explicit breaking: mq=(mu+md)/2 0 non-zero GB masses: m 2 = 2 B 0 mq + 4 A 0 mq 2 +. . Modification of low-energy theorems: Atree (s, t, u)= (s- m 2)/F 02 (32 /3) m (a 0 - a 2) = -2 mq 0|qq|0 /(m 2 F 2) S PT: 1 (neglect mq 2 term) Gasser, Leutwyler ‘ 84 a 0 - a 2 = = 2. 12. 2004 0. 290 fm + 0. 073 fm + 0. 011 fm tree 1 -loop 2 -loop Weinberg ’ 66 0. 374 fm 2% R. Lednický Budapest 3

G PT: arbitrary from 1 0 (vanishing condensate) Fuchs, Sazdjian, Stern, Knecht, Moussalam ’ 91 -95 Quark mass ratio ms/mq = 26 8 phase transition Tc = 200 150 a 0 - a 2 can be up to 25% higher than 0. 374 fm from S PT Experiment up to 2001: 0. 41 fm 20% Added BNL Kl 4 data: 0. 37 fm 10% DIRAC at CERN aims: 5% 2. 12. 2004 R. Lednický Budapest 4

Pionium production Dominated by FSI due to small binding energy < 3 ke. V Closely related to free + - production at Q=2 k* 0 FSI theory: Fermi -decay Migdal, Watson, Sakharov hadronic processes Continuum: d 6 N/(d 3 p 1 d 3 p 2)= d 6 N 0/(d 3 p 1 d 3 p 2) | -k*(r*)|2 Discrete spectrum: A 2 d 3 N/(d 3 p. A)= (2 )3 Ad 6 N 0/(d 3 p 1 d 3 p 2) | A(r*)|2 p 1 ≈ p 2 x 1 x 2 p 1 + p 2 p. A A 2 r*= x 1*- x 2*= distance between the + and + emitters in pair cms 2. 12. 2004 R. Lednický Budapest

FSI amplitudes wave functions Continuum: e-ikr FSI s-wave strong FSI -k(r) fc Ac (G 0+i. F 0) } Similar to Coulomb distortion of -decay RL, Lyuboshitz (82) condition |t*| r*2 [ e-ikr +f(k)eikr/r ] |a| = Bohr radius Coulomb kr+kr+ … _______ Ac = Point-like Coulomb factor F=1+ ka ei c Ac -k*(r*) Ac [1 - r*/(1+cos *)|a| + f(0)/r* + O(r*/a)] at k* 0 1 at r* p-wave | -k*(r*)| >> |na| Discrete spectrum: A 2 n 0(r*) = cn 0(0) [1+ (n)f(0)/|na|] [1 - r*/|a|+f(0)/r* + O(r*/a)] n 0(r*) exp(-r*/|na|) 0 at r* >> |na| Universal FSI dependence on r* and scattering amplitude f at k* 0 , r*<< |na| and a given orbital angular momentum 6

Assume first only two types of pion sources Nemenov ‘ 85 Short-Lived Sources SLS ( , , . . ) r* |a| = 387 fm both ’s from SLS Long-Lived Sources 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 fc= SLS fraction Ac(Q) 1 | A(r*)|2 = fc | c. A(0)|2 + (1 - fc) | c. A( )|2 A={n, l=0} 1/( |na|3) 0 Universal relation between production of free and bound ’s: d 6 N/(d 3 p 1 d 3 p 2)= g d 6 N 0/(d 3 p 1 d 3 p 2) [fc Ac(Q) + (1 - fc)] d 3 N/(d 3 p. A)= g fc d 6 N 0/(d 3 p 1 d 3 p 2) (2 )3 A/( |na|3), p 1 ≈ p 2 7

How to measure lifetime ~3 fs A 2 fate: decay A 2 0 breakup 0 + - excitation A’ 2 Z Z g. s. A 2 at A ≈ 17 (4. 7 Ge. V/c) flies ~ dec = Ac 10 17 m in the target and encounters ~ 105 atoms br ~ Z-2 Nemenov ‘ 85: choose Z and A so that dec br and measure the g. s. lifetime 10 ~3 fs through Pbr = NAbr/NA = breakup (ionization) probability of A 2 in the target Pbr is unique function of Z, target geometry, A and 10 determined to 1% by kinetic eqs. with initial conditions given by 1/n 3 law: P 10= 83%, P 20= 10. 4%, P 10= 3. 1%, P 10= 1. 3%, . . 2. 12. 2004 R. Lednický Budapest 8

Excitation and break-up of produced atoms (NA) compete with decay Break-up probability is linked to the lifetime (theory of relativistic atomic collisions) + - pairs from break-up provide measurable signal NAbr Pbr is linked to NAbr : Pbr = NAbr/NA The number of produced atoms NA is not directly measurable to be obtained from N( + -) =10% Pbr =4% 9

Access to NA and NAbr NA can be calculated from + - spectrum fitted at Q > 4 Me. V/c NAbr can be directly measured as an excess over the fitted spectrum of the free pairs interpolated to Q < 4 Me. V/c Pions from break-up have very small Q At target exit this feature is smeared by multiple scattering, especially in QT 2. 12. 2004 R. Lednický Budapest 10

Finite-size effects r* ~ 10 fm but ~30 -40 fm and ’ ~900 fm Ur. QMD: p. Ni 2 at 24 Ge. V ~1% ’, ~19% ML ~ r*2/[1+(r*/r 0)2 a]2 b short-distance parametrization ML ’ and contributions well fitted based on exponential decay law ’ 11

DIRAC - - CF: CF=N{ | -k*(r*)|2 SLS +(1 - )}[1+s Q] SLS determined by: N r 0 fm a b f f ’ s f = 17 6% ’, f ’ , f ML(r 0, a, b), f. ML=1 -f -f ’ G 2 fm ML G 3 fm ’ 12

Introduce correction factors: 1+ (k*) = | -k*(r*)/ c-k*(0)|2 SLS = |1 -r*/|a|+f(k*)/r*|2 +O((r*/a)2)+O((k*r*)2) SLS 1+ n = | n 0(r*)/ cn 0(0)|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)) 2. 12. 2004 R. Lednický Budapest 13

G 2 fm G 3 fm (k*)- (0) ML ’ ’ r*= n- (0) ’ r*= G 2 fm, G 3 fm, ML ’ Universality violated by and ’ Universality OK for short-distance part G, ML ’ still not a LLS 14

1% ’ 19% G 2 fm f(0)=+20% 80% G 3 fm f(0) = 0. 186 fm f =+30% f =-30% No impact of short-distance (G) uncertainty on the lifetime Main theor. uncertainty in the lifetime from and f(0)

Fit results for: 1% ’ 19% 80% G 3 fm f(0) = 0. 186 fm Shift (if neglecting 1+ ) and uncertainties ( ) in the lifetime ( ) 30% in f 10% in f(0) comment +14% +11% 7. 5% 6. 0% 1. 8% 1. 4% Q=4 -20 Me. V/c Q=3 -20 Me. V/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% 16

Results from DIRAC Setup features Angle to proton beam: =5. 7 Downstream detectors: DCs, VH, HH, C, PSh, Mu. Channel aperture: =1. 2· 10– 3 sr Magnet: 2. 3 T·m Momentum range: 1. 2 -7 Ge. V/c Resolution in relative momentum: QX= QY=0. 4 Me. V/c QL= 0. 6 Me. V/c 2. 12. 2004 Upstream detectors: MSGCs, Sci. Fi, IH. R. Lednický Budapest 17

Experimental Qtot and Ql distributions (Ni 2001) Fit Monte. Carlo C and n. C background outside the A 2 signal region (Qtot > 4 Me. V, Ql > 2 Me. V) simultaneously to experimental Qtot and Ql spectra 2. 12. 2004 R. Lednický Budapest 18

Atomic breakup signals in Qtot and Ql • Qtot and Ql provide same number of events background consistent • Signal shapes well reproduced • = 2. 8 ± 0. 4 stat fs (15% stat. error for 40% data) 2. 12. 2004 R. Lednický Budapest 19

Correlation study of particle interaction CF=Norm [Purity RQMD(r* Scale r*)+1 -Purity] + scattering length f 0 from NA 49 CF + Fit CF( + ) by RQMD with SI scale: f 0 sisca f 0 input -= 0. 232 fm sisca = 0. 6 0. 1 Compare with ~0. 8 from S PT & BNL E 765 K e 20

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 , , . . 2. 12. 2004 R. Lednický Budapest

NA 49 central Pb+Pb 158 AGe. V 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] RQMD overestimates r* by 10 -20% ? Too much rescatterings already at SPS Scale=0. 76 Scale=0. 92 Scale=0. 83 p 2. 12. 2004 R. Lednický Budapest 22

Tails in RQMD: r* = 21 fm for r* < 50 fm 29 fm for r* < 500 fm =0. 89 r* =16 fm =0. 94 r* =24. 4 fm =0. 93 r* =24 fm > =0. 91 r* =22. 9 fm + + Strong FSI on + =0. 81 r* =18. 4 fm =0. 76 r* =18. 1 fm Strong FSI important for + 1 -G fit: ( + +) 0. 8, r* 25% 2 -G fit: + + + r* QS < r* Coul 2. 12. 2004 R. Lednický Budapest 23

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=2 k* and its effect is mainly absorbed in suppression parameter Coulomb FSI: sensitive to r*-tail up to r* ~ Bohr radius |a|=|z 1 z 2 e 2 |-1 K p KK pp fm 388 249 223 110 58 In Gaussian fits one may expect r 0( + +) < r 0( + ) Use realistic models like transport codes 2. 12. 2004 R. Lednický Budapest 24

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/c r 0=2 fm v=0. 1 OK for heavy particles OK within 5% even for pions if 0 ~r 0 or lower 2. 12. 2004 R. Lednický Budapest 25

Coalescence: deuterons. . WF in continuous pn spectrum -k*(r*) WF in discrete pn spectrum b(r*) Edd 3 N/d 3 pd = B 2 Epd 3 N/d 3 pp End 3 N/d 3 pn pp pn ½pd Coalescence factor: B 2 = (2 )3(mpmn/md)-1 t | b(r*)|2 ~ R-3 Triplet fraction = ¾ unpolarized Ns Lyuboshitz (88). . B 2 Usually: n p Much stronger energy dependence of B 2 ~ R-3 than expected from pion and proton interferometry radii R(pp) ~ 4 fm from AGS to SPS 2. 12. 2004 R. Lednický Budapest 26