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15 December 1994

Physics Letters B 340 (1994) 250-253

PHYSICS LETTERS B

HBT correlators - current formalism

vs. Wigner function formulation

Scott Chapman, Ulrich Heinz lnstitut fiir Theoretische Physik, Universitiit Regensburg, D-93040 Regensburg, Germany

Received 29 July 1994; revised manuscript received 20 September 1994 Editor: P.V. Landshoff

Abstract

We clarify the relationship between the current formalism developed by Gyulassy, Kaufmann and Wilson and the Wigner function formulation suggested by Pratt for the 2-particle correlator in Hanbury-Brown Twiss interferometry. When applied to a hydrodynamical description of the source with a sharp freeze-out hypersurface, our results remove a slight error in the prescription given by Makhlin and Sinyukov which has led to confusion in the literature.

It is widely accepted that if the nuclear matter created in ultra-relativistic heavy-ion collisions attains a high enough energy density, it will undergo a phase transi- tion into a quark-gluon plasma. For this reason, it is of great interest to determine the energy densities actually attained in these collisions. The total interaction energy of a given reaction can be directly measured by particle calorimeters and spectrometers. Although there is no analogous direct measurement for the size of the reac- tion region, Hanbury-Brown Twiss interferometry [ 1 ] provides an indirect measurement in terms of the cor- relations between produced particles.

More than 20 years ago, in two papers [2] which have largely been overlooked, Shuryak derived a covar- iant expression for the two-particle correlation function using the method of classical source currents with ran- dom phases. This method was later worked out in great detail by Gyulassy, Kauffmann and Wilson [4] and has become known as the "covariant current formal- ism". Ten years ago, Pratt [3] used that formalism to

¢~ Work supported by BMFT, DFG, and GSI.

0370-2693/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSDI0370-2693(94)01277-6

show that the correlations between two particles could be expressed in terms of one-particle pseudo-Wigner functions. Although Pratt's derivation was non-relativ- istic, it provided a valuable link between the experi- mental data and many semi-classical event generators whose output came in the form of one-particle distri- butions. Since that time, different methods have been used to relativistically generalize Pratt's result [5-8] , but to our knowledge, the simplest generalization (using the covariant current formalism covariantly) has never been published. The aim of this letter is two- fold: (1) to fill the above void in the literature, and (2) to show that by applying the final result to hydrody- namical models with 3-dimensional freezeout hyper- surfaces, a dispute in the literature about the correct form of the 2-particle correlator in these models can be resolved.

The covariant single- and two-particle distributions for bosons are defined by

E dN P~(p) = ~ p =E<~, + ( p ) d ( p ) ) , (1)

S. Chapman, U. Heinz~Physics Letters B 340 (1994) 250-253 251

dN e2(Pa, Pb) = EaEb d3pad3pb

=EaEb(t~+ (pa)a + (pb)a(pb)a(pa) ) , ( 2 )

where~ + (p) ( a (p ) ) creates (destroys) a particle with momentump. The two particle correlation function is then given by [4]

(N) 2 P2(Pa, Pb) C(pa'Pb) (N(N-- 1 ) ) el(Pa)el(Pb) " ( 3 )

Using the classical covariant current formalism of [4,9] we will show that for a general class of chaotic current ensembles the two particle distribution for bosons obeys a Wick theorem:

( N ( N - 1 ) ) P2(Pa, Pb) = (U)2

X (PI(Pa)PI(Pb) + IS(.Pa,Pb)12) , (4)

where we define the following covariant quantity:

s(t,a,p~) = ~ ( a + (p~)a(t,~)). (5)

We will then show that S is equal to the Fourier trans- form of a kind of Wigner function:

g(P~,Pb) =S(q, K) = ( d4x e-iq'xS(x, K), (6) J

where the off-shell 4-vector K= ½ (Pa -t-pb) is the aver- age of two on-shell (p°i=Ei) 4-momenta, and q =Pa --Pb is the off-shell difference of the same two momenta so that their scalar product vanishes, K"q.=O. For the special case of pa=Pb, K=p. becomes on-shell and

S(pa, Pa) = g ( O, K) = P1 (P~) • (7)

It should be noted that Eq. (6) involves a 4-dimen- sional Wigner transform, in contrast to the 3-dimen- sional expression suggested by Pratt [3] which neglects retardation effects. The occurrance of the off- shell 4-vector K = ½ (p~ +Pa) in the expression for the correlation function was already correctly noted by Shuryak [2].

In [4] it was shown that a classical source current J(x) generates free outgoing pions in a state which satisfies

'~(V) l J ) = / J ( t ' ) I J ) , (8)

where

f exp[i(Ept-p .x) ]J(x) (9) d4x

J(p) = ~/(2~.) 32Ep

is the on-shell Fourier transform of the source J(x), and (J I J) = fd3p ]J(p) 12 = 1. For classical currents, the ensemble expectation values in Eqs. (1), (2), and (5) can then be defined in terms of a density operator 16 involving the state l J ) such that ( ~ ) = t r ( / ~ ) .

Generalizing the result of [9] in order to allow for arbitrary x - p correlations, we consider an ensemble of chaotic source currents at positions x/with momenta

Pi, N

J(x) = ~_, e i~ e-ip"(x-xi)Jo(x-xi) , (10) i=1

where ~b i is a random phase. The momenta Pi of the sources can, but need not be on the boson mass-shell; for example, the source could be a decaying A-reso- nance with 3-momentumpi. The on-shell Fourier trans- form of (10) is

N J(P) = E eioi eip'XiJo(P-Pi) , ( 1 1 )

i=1

where

Jo(P-Pl) = ~ / ( 2 7 r ) 3 2 E p ei(P-Pl)'XJo(x ) ( 1 2 )

is the Fourier transform ofJo(x), andp is on-shell while p~ may be off-shell.

We then choose a density operator such that

tr(16~) = ~ PN I-I d4x~ d4p~ p(xi, p~) N=O i=1

2~"

f dqbi x ~ ( J i l l J ) , (13) 0

where p(x~, p~) is the covariant probability density of the source points (xi, Pi) in phase space, and PN is the probability distribution for the number of sources in the reaction. These probabilities are normalized as fol- lows:

f d4x dap O(x, p) = 1, ~_. PN = 1. (14) " N=O

252 S. Chapman, U. Heinz/Physics Letters B 340 (1994) 250-253

Using (8) and the above definitions, it is straight- forward to show that

PI(P) =ep< ] / (p) 12)

= (N)Ep j d4Xl dap, p(xl, Pl ) I ]o (P- -P l ) [ 2

= (N)Ep f drip1 ~5(pl) [Jo(P-Pl) [2 . (15)

The single particle spectrum is thus obtained by folding the momentum spectrum ]]o(P)]2 of the individual source currents Jo with the 4-momentum distribution of the sources, ~(p) = fdax p(x, p).

Similarly, if one neglects cases in which two parti- cles are emitted from exactly the same point [4], one finds:

P i ~ a , P~)

_ (N(N- 1)) EaEb[< [•(pa)12>< [](,Pb) [2> <N> 2

+ ( ] * (Pa) ](Pb) ) ( ] * (Pb)Y(Pa))1 (16)

which proves Eq. (4) by way of (8). Using Eq. (9), we find the following relationship:

J*(Pa)f(Pb) = d4xl d4xz

(2~-) 32~/-Ea E b

X exp ( - ipa" X 1 + ipb" X2) J* (xl) J(x2)

d4x d4y (2~r) 32~/E~E b exp( - iq . x - iK.y)

× J* (x + ½y)J(x- ½y), (17)

where x = ½ (Xl + x2) and y = x~ - x2. The above relation proves Eq. (6) as long as the following expression for the Wigner function is used (see also [ 8 ] ):

S(x, x') = f d4y

~ i K ° y e

× (J*(x+ ½y)J(x- ½y) ) . (18)

The average on the r.h.s, is defined in the sense of Eq. (13) and can be evaluated with the help of the defini- tion (10) to yield

S(x, K) = (N) j d4z d4q p ( x - z, q)

×So(z, K - q ) , (19)

where

( d4Y e_ip. , So(x, p) = )

X J * ( x + ½Y)Jo(X- ½y) (20)

is the Wigner function associated with an individual source Jo. Thus the one- and two-particle spectra can be constructed from a Wigner function which is obtained by folding the Wigner function for an individ- ual boson source Jo with the Wigner distribution p of the sources. Eq. (19) is useful for the calculation of quantum statistical correlations from classical Monte Carlo event generators for heavy-ion collisions: <N)p(x,p) can be considered as the distribution of the classical phase-space coordinates of the boson emitters (decaying resonances or 2-body collision systems), and So(x, p) as the Wigner function of the free bosons emitted at these points. Replacing the former by a sum of &functions describing the space-time locations of the last interactions and the boson momenta just after- wards, and the latter by a product of two Gaussians with momentum spread Ap and coordinate spread &r such that &rAp>_h/2, we recover the expressions derived in [7].

Using Eqs. (3) to (6), our final result is then

C(pa,Pb ) = 1 +R(q, K ) , (21)

where the "correlator" R is given by

IS(q, K)I 2 R(q, K) = g(0, pa)S(O, Pb) " (22)

Eq. (6) is the starting point for a practical evaluation of the above correlator. It should be noted that due to the on-shell condition of (9), it is impossible to recon- struct S(x, K) from the correlator in a model independ- ent way. Thus any analysis of data on R(q, K) necessarily involves suitable model assumptions for S(x, K), in particular for the x - K correlations in the source distribution. In most practical applications one takes for S(x, K) a classical (on-shell) phase-space distribution. In hydrodynamical models, for example, this phase-space distribution is taken as a local equilib- rium Bose-Einstein distribution localized on a 3- dimensional freeze-out hypersurface £ (x) which separates the thermalized interior of an expanding fire- ball from the free-streaming particles on its exterior [10]:

s. Chapman, U. Heinz~Physics Letters B 340 (1994) 250-253 253

2s~ + 1 S,(x , K) = (2~) 3

f K ~ d 3 ° ' ~ ( x ' ) ~ ( 4 ) ( x - x ' ) (23) × exp{/3(x') [K. u(x ' ) - ~,~(x') ] } - 1"

Here s,, and/z~ denote the spin and chemical potential of the emitted particle species a, while u~(x), ~ (x ) , and d3o-~(x) denote the local hydrodynamic flow velocity, inverse temperature, and normal-pointing freeze-out hypersurface element. Inserting this equa- tion into (7) , one obtains the Cooper-Frye formula [111

~.~(0, p) = PI (p) = fp"d3o'~(x)f(x,p), (24)

manifestly positive definite nature of the correlator (22).

The symmetric form (26) (in contrast to the asym- metric one given in [ 12] ) allows one to replace the exponential by the cosine and to split the expression into two real 3-dimensional integrals:

IS(q, K)12= (Sx(q, K)) 2+ (Sz(q, K)) 2 , (27)

where

$1,2(q, K) = [ K*" d30-/z(x) o ¢

f c o s ( q ' x ) ; (28) ×f(x , K) I, s i n ( q . x ) ) "

This facilitates the numerical evaluation o f the corre- lator.

where we define the distribution function (for clarity we drop the index a for the particle species)

2 s + l f (x , p) = (2q7.) 3

1 × (25)

exp{/3(x) [p. u(x) - Ix(x) ] } - 1"

For the numerator of the correlator,

IS(q, K) 12 = j K j* d3~z , (x )K" d3o' ,(y)

×f(x , K)f (y , K) e x p [ i q - ( x - y ) ] , (26)

we find an expression which is very similar to the one given in [ 12]. There, however, each of the two distri- bution functions under the integral featured on-shell arguments Pa and Pb, respectively, instead of the com- mon (off-shell) average argument K as in (26). This error in [ 12] can be traced back to an inaccurate tran- sition from finite discrete volumes along the freeze-out surface X to the continuum limit [ 13]. As pointed out in [ 14,15 ], taking over this inaccuracy can produce (in particular for very rapidly expanding sources) unphys- ical oscillations of the correlator around zero at large values of q [ 16,17] which are inconsistent with the

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