Finite coherent length and multi-pion correlation effects on two-pion interferometry

The effects of multi-pion correlations and finite coherent length on two-pion interferometry are studied. It was shown that as the pion multiplicity and coherent length become larger, the apparent radius and the apparent coherent parameters derived from two-pion interferometry become smaller. The influence of the coherent length on the effective temperature is discussed. PACS number(s): 25.75 Dw, 11.38 Mh, 11.30 Rd Two-pion Bose-Einstein(BE) correlation is widely used in high energy heavy-ion collisions to provide the information of the space-time structure, degree of coherence and dynamics of the region where the pions were produced [1–7]. Ultrarelativistic hadronic and nuclear collisions provide the environment for creating dozens, and in some cases hundreds, of pions [8–10]. The bosonic nature of the pion should affect the single and two-pion spectra and distort the two-pion correlation function [11–20]. There is a kind of coherent length corresponds to the wave packet length scale of the emitter, which causes pions to concentrate at low momenta [18–20]. Thus, it is very interesting to analyse the effects of multi-pion 1 correlation and finite coherent length on two-pion interferometry [18–20]. The general definition of the n pion correlation function Cn(~ p1, · · ·~ pn) is Cn(~ p1, · · ·~ pn) = Pn(~ p1, · · ·~ pn) ∏n i=1 P1(~ pi) , (1) where Pn(~ p1, · · ·~ pn) is the probability of observing n pions with momenta {~ pi} all in the same event. The n-pion momentum probability distribution Pn(p1, · · ·pn) can be expressed as [12,13,16,19] Pn(p1, · · ·, pn) = ∑ σ ρ1,σ(1)ρ2,σ(2)...ρn,σ(n), (2) with ρi,j = ρ(pi, pj) = ∫ dxgw(x, (pi + pj) 2 )eij. (3) σ(i) denotes the ith element of a permutation of the sequence 1, 2, 3, · · ·, n, and the sum over σ denotes the sum over all n! permutations of this sequence. gw(x,K) can be explained as the probability of finding a pion at point x with momentum p which is defined as [21–23] gw(x, k) = ∫ dyj(x+ y/2)j(x− y/2)exp(−iky). (4) Where j(x) is the current of the pion, which can be expressed as [22,17] j(x) = ∫ dxdpj(x, p)ν(x′)exp(−ip(x− x′)). (5) Here j(x, p) is the probability amplitude of finding a pion with momentum p , emitted by the emitter at x. ν(x) is a random phase factor which has been taken away from j(x, p). All emitters are uncorrelated in coordinate space when assuming: < ν∗(x′)ν(x) >= δ(x − x). (6) This is in ideal case. In a more realistic case, each chaotic emitter has a small coherent wave packet length scale and the above equation can be replaced by [18] < ν∗(x′)ν(x) >= 1 δ4 exp{− (x1 − x ′ 1) 2 δ2 − (x2 − x ′ 2) 2 δ2 − (x3 − x ′ 3) 2 δ2 − (x0 − x ′ 0) 2 δ2 }. (7) 2 Here δ is a parameter which determines the coherent length (time) scale of the emitter. For simplicity, the same coherent scale is taken for both spacial and time at the moment. The above formula shows that two-emitters within the range of δ can be seen as one emitter, while two-emitters out of this range are incoherent. For simplicity we also assume that < ν∗(x) >=< ν(x) >= 0, (8) which means that for each emitter the phases are randomly distributed in the range of 0 to 2π. Then we have the following relationship < ν∗(x′)ν∗(x) >=< ν(x)ν(x′) >= 0. (9) Inserting eq.(5) into eq.(4) we have gw(Y, k) = ∫