Interferometry radii for expanding hadron resonance gas

NUCLEAR PHYSICS A ELSEVIER

N u c l e a r Physics A 6 1 0 (1996) 2 7 8 c - 2 8 5 c

Interferometry radii for expanding hadron resonance gas Yu.M. Sinyukova, S.V. Akkelina, A.Yu. Tolstykhb alnstitute for Theoretical Physics, Kiev 252143 Ukraine bphysics Department, Stanford University, Stanford, CA 94305, USA

We discuss the influence of the different physical factors on the interferometry radii of thermalized sources in A+A collisions. They are, first, non-Gaussian form of Bose-Einstein correlation function for direct pions at small Pr, second, influence of the resonances, third, modification of the Gamov correction method for large expanding systems, including, particularly, specific role of long-lived resonances r/, 7/'. Our results are based on the conception of the lengths of homogeneity.

1. LENGTHS OF HOMOGENEITY IN EXPANDING THERMALIZED SYSTEMS

The main properties of hadron and quark-gluon matter such as the order and scenario of the phase transition are reflected in space-time dynamics of the systems formed in A+A collisions. The boson interferometry allows to extract the important features of a system evolution: proper time of the expansion % the intensity of transverse flow ~, transverse radius ff~r, inverse of freeze-out temperature t , etc. [1,2]. So, in this non-direct way the interferometry allows us to study the properties of hadronic matter at very high temperatures and densities. Typically, the results of HBT measurements are presented by the interferometry radii Ri(P): Ro<,,), Rs(ide) , RL(ong) in the different momentum regions (y, PT). They appear as the result of the Gaussian fit of the Bose-Einstein correlation function. At high PT the interferometry radii for thermalized systems are associated directly with the system's length of homogeneity 2, [2,3].

-2

A, ( p ) =

82 f ( x ' P ) /'~ 82 x i i/ ~ f ( p , x )

at d f ( x , p ) ~(p)

8xi

=0

(1)

~(p)

wheref(p,x) is the distribution function in the radiating system. In the expanding systems where the correlation between a position and momentum of radiating bosons takes place, the lengths of homogeneity depend explicitly on the above-mentioned parameters of evolution: 0375-9474(96)$15.00© 1996 - Elsevier Science B.V. A]I rights reserved. PII: S0375-9474(96)00362-4

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x, ~, etc. In the c.m.s, of the pairs (marked by asterisk) the interferometry radii are coincided with ~ at large P r in LCMS: R*i(p) = 2.*i(p; r, a,,fl, Rr)

(2)

In other reference systems the kinematics factors and so-called crossing term appears [2, 4]. The boson Wigner function in thermalized and transversally finite-size (Rr)systems is proportional to the Bose-Einstein distribution [2]: (3)

f ( p , x) = fs.e.(P, u(x),fl, lt~,Its)p(-Rr, ur (x))

At small r Yr °CVr °cr/-Rv and pocexp(-r2 /2ff~) For the boost-invariant longitudinal flows and transversal flows with intensity a = ff2/ff~ we have [2] In LCMS

for non-relativistic transverse flows

//pr~ 2 2~2 ~s - flPr sinh fir

( timr << a )

t, R~ )

1 ~-1 2 (tim / ~ 2 -T + -~I k. R, Rr )

2~"2

2g 2

,Sin-r cosh.vr

flmr

(4)

where Yr, r, r are defined at the point of the maximum of Wigner function: tanh~r-F r =flpr/(flmr+at)

andforlinearflows, y r = r / ~ ,

Rv i n ( l + v r / F= 2 \l-~r)"

We suppose that the freeze-out hypersurface ~-(r) ~ const = r. 1. For developed (o~ < 1) transversal flows with the monotonous growing velocity the longitudinal length of homogeneity decreases more quickly then 1 / ~-mr at large Pr. The same behavior is for the side-length. The decrease of the side-length is more fast that the out-length. 2. If the transverse velocity has the maximum value v r . . . . the behavior of all radii is universal, Ri oc 1/ x/mr, at rather large Pr : vr = ,SPr/(18mr + c0 > Vr,max" 2. CORRELATION FUNCTIONS AND INTERFEROMETRY RADII The Bose-Einstein correlation functions are defined in the standard manner using the creations (annihilation) operators and the Wigner function:

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C(p,,p2)=I +(a+~ap~)(a+peap~)l(a+~ap,)(a+pa~ )

(5) It

(a+(p~)a(p2))~ = ~dcrItp e

iq.x

f(x,p);

P=(Pl +Pz)/2, q = P 2 - P ,

The correlation function can be expressed with good accuracy through the lengths of homogeneity. First, let us demonstrate it using the analytically solvable model that neglects by the transversal expansion ( a = oo). Then using the boost-invariant approach and mass-shell approximation f o r p one can evaluate the integral in Eqs. (5) and find the correlation function:

[£ C(p, qL,qT = 0) = 1 q

II (.pm,)2

L.=l

+

q~,'t'2 "]( , , 14 cosh ~

O) ~.

qL, t.22

0)-1/212

(n~m~) 2 c o s h ~ (6)

[~=K,(n,Bmr)] 2 The 0 is the half-sum of the rapidities of particles. The sum in Eq. (6) is the result of the expansion of the Bose-Einstein distribution in the series. The Boltzmann approximation corresponds to n = l in Eq.(6) and can be used at tim r > 1. Within this approximation we apply the saddle-point method to calculate integral in Eqs. (5). Renormalizing the homogeneity lengths (4) (in particularly, passing to the longitudinal length of homogeneity in rapidity space) )[o2s = 2~, s / 2,

~ = ,~ / 2r 2

(7)

we have in LCMS:

Pr

C(p, qL;qT = O) = 1 +

--+oo )

[ 2 ^ 22~q 2 ]L

l+exp-r

(9)

C(p, qs;qo = qL = O) = 1 + exp - 2,sq s

[

C(p, qo;qs =qL = 0 ) = l + e x p - ~ 2 q Z o

(8)

l+r

Pr/~Lq°2

mT

P~-~ ) l + e x p - ~ . ~ q o

(10)

The results demonstrate, in general, non-Gaussian behavior of the correlation function at small p r . First this fact have been found in Ref. [1] for long-correlation function. At large P r Eqs. (8)-(10) give the Gaussian results [1,2]. The most serious deviation from the Ganssian form has the correlation function in longitudinal direction. At Fig. 1 we demonstrate the behavior of the correlation function C(p, qL ) at p=0. Let us note, first, the Gaussian radius ob-

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tained on the base of Z 2 -method leads to 15% overestimation for Boltzmann approximation in comparison with radius calculated using exact 1.8 ' ~ k °*° Bose-Einstein distribution. Second, the extrapolation to PT"=0 of the simple asymptotic result in 1.6 ~ * * * , fight-hand part of Eq. (8) demonstrates the no~+ ',+,~ °o ticeable deviation of the correlation function c(q) 1.4 \~,~ ,° from the Gaussian form as well as the underestimation of the Gaussian radius in Z 2 -fit by the 22%. Even at PT =0.35 GeV/c the underestima',t+ ** tion is 15%. These deviations are essentially smaller that it was demonstrated in Ref. [5,6]. I0 0.0i 0.02 o.03 0.04 0.05 0.06 0.07 The reasons are that we use the exact Boseq Einstein distribution instead of Boltzmann ap- Fig.1. Behavior of longitudinal correlation proximation and define the interferometry radii function at PT =0" -Rr = r = 7 frn, using Z 2 -procedure (not at the limit q--+ 0 [6]) T=0.14 GeV. Solid line - exact curve, as experimentalists do it usually. So, the use of dash line Boltzmaun approximation, asymptotic Gaussian results at small Pr are dotted line corresponds to Eq. (8), diapossible but rough enough. They lead to 10-20% mond - asymptotic Gaussian result. of underestimation for longitudinal interferometry radii in currently observable pr-region in comparison with numerical results. More precise non-Gaussian fit is presented by the Eqs. (8)-(10). At large PT > 0.25 GeV/c the Boltzrnann approximation works good within a few percents accuracy for all correlation functions (8)-(10). In all PT-regi°n the representation (8)-(10) for correlation functions coincides with results of Boltzmann approximation with accuracy better then 1%. Note that correlation function in the side-direction is approximated good by the Gaussian asymptotic form. The main analytical results we are presented here is that the non-Gaussian approximation (8)-(10) for correlation function in the LCMS works with the same 1% accuracy for 3Dexpanding system as it was described above for one-dimeusional expansion (c~ = oo). The lengths of homogeneity are defined now by the Eqs.(4), (7) for any ~z. 3. T H E INFLUENCE OF T H E RESONANCE DECAYS TO I N T E R F E R O M E T R Y RADII The contribution from the resonance decays can be taken into account in the following manner (see, e.g., [7]) (a+(pl)a(P2)) =

tPl)

tP2)),r

E(a i

(pl)a(P2))i.,_

(11)

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The quasi-classical approximation is used to evaluate the operator averages: co+

3

(a + (pl)a(p2)),~" ~ _ = -~dco~ g(co~) f ~ p' co-

a

2m, t~((pi -

Ei

+1 p)2 _ co2) 23, (2~)3 (12)

x

.... 1 exp(iquxU) dtru(x')P~U exp[_ct(cosh YrtrD_l)l.l + ~ _ / _ ~ i F , ~ exp(flp~ u,,,( x , ) - flu, ) -T-1 In Eq. (12) we consider a resonance with the width Fi, that is created at a space-time point

x~ from the hydrodynamic tube and decays into ~r- + X after a mean proper time 1/F i. The probability to produce a pion of momentum pU and other particles X with invariant mass co from a resonance of momentum pU is described by the function g(coz)[8]. The values E0 (co 2) and P0 (co z) are the pion energy and momentum in the rest system o f the resonance. It is possible to fred the resonance contribution to the correlation function using the saddle point method. Because of relatively large resonance masses tim, >> 1 this method gives a good accuracy even at small PT" We take into account the following resonances here: shortlived resonances p, A ; medium-lived resonances K*, co ; long-lived resonances r/, r/'. The last group reduces the intercept, A < 1, of the correlation function, C(p,O) = 1 + A ( p ) , since the corresponding width of the interferometry peak is much smaller than momentum resoluP0 tion qml, of the detectors: qmin ~ >> 1. As the result we have

A,(p) = 1

%(P)+n--~'(P)I2 n(p) .]

(13)

where n(p) = (a +(p)a(p)) = p ° d3 N / d3 p. The corresponding plot is presented at the Fig.2. All calculations are given for the following values of the parameters: Rr = r = 7 f r n / c ,

tz=3, T = I / f l = O . 1 4 G e V ,

UB =0.363 GeV, Us =0.069 GeV

(14)

The values for chemical potential for Pb+Pb CERN SPS collisions are taken from the Ref. [9]. At the Fig.3 we demonstrate the influence o f the resonance decays on the correlation function at Pr = 0.1 GeV/c. It is the most essential for longitudinal direction. We compare the complete calculation with the only "direct" pion contribution in Boltzmann approximation and with asymptotic Gauss,an results (see (8)). The ratio for corresponding Gauss,an radii obtained by the ,~2 -method is Rcompt,~e:Roi,~c,:R~y,,p,oac = 1A8:1.26:1. It means that the ignoring of resonance decays in the theoretical description of the interferometry radii in A+A collisions leads to the loosing of the 15% accuracy at small Pr-

Yu.M. Sinyukov et al./Nuclear Physics A610 (1996) 278c-285c

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2l'*t;°. A tl

÷÷o** '1

O.C.

÷

** +

, +

*

0.~

C(q) 0.7 I

+÷ +÷

~* *~o ÷+÷

0.~ 0

o.b~

0'.1

0.~5

0'.2

o.~5

0.3

Pr [GeV l c] Fig.2. The dependence A~ ( P r ) .

10

0,01

0.02

0.03

q

0.04

o~,

0.05

0.06

0.07

Fig.3. Correlation function in longitudinal direction. Solid line - with resonances, cross -direct pions only, diamond - asymptotic Gaussian.

4. COULOMB CORRECTION FOR LARGE EXPANDING SYSTEMS

The standard method that takes into account the Coulomb interactions of z-charged bosons is the so-called Gamov correction: Coxp (p, q) = G(q* )CB.g.(p, q) 2~rtc where G(q*) - exp(2n-t¢)-l' tc = z l z 2 a m / q* , q* = qi,,,,. The pions from the long-lived resonance group (rl,rl/-mesons) are created far away from other pions and so one waits no Coulomb correction for them. Taking this into account we get from Eqs. (5), (11), (13): CB.E.(p,q)=[C~xp(p,q)-(1-~]A~(p+q/2)A~(p-q/2))(1-G(q*))]/G(q*)

(15)

Let us emphasize that subtraction factor in (15) depends o n q and so the final procedure of an experimental extraction of the Bose-Einstein correlation function will depend on behavior of A(p) (see, e.g., Fig.2). The second serious problem with Coulomb correction is the large sizes o f the main source of charges pions. The effective sizes of the system in SPS CERN Pb + Pb collisions are large enough and will continue to grow if the tendency of the increase o f the interferomertry radii R ~ 2 o c ( d n / d y ) 1/3 will preserve. The average distances between any two particles < r* > oc 2 will grow, so the Coulomb interaction will be weaker and the Gamov correction corresponding to small < r* > has to be modified. Even for static sources (A = const) the complete Coulomb correction depends on mo,

m T

mentum region where the pairs are registered (e.g., 2 o =-m-2O). For expanding systems

Yu.M. Sinyukou et al./Nuclear Physics A610 (1996) 278c-285c

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2~ = 2~(pr ) . Using the first approximation to the Coulomb function we get in the Gaussian approach in LCMS [10]

'2)1

i= O,S,L

(16)

where the <) A a that extracted from the of standard Gamov correction method is tied with <> A and the same concerns of the interferometry radii Rv and "true" one R :

A=

Aa

"

2 < r* >'1(" 2 < r* >'~-1 -1|1+-----7-7---/ , /\ lal /

>--d'I l lexP-2

AAU~-~

Ri = RG'iV A

Xs =7

where

1

a=2

and

(17)

O~/IZ1Z 2

2Fc)3/2R°RsR*L

(18)

In the left-hand side of Eq. (18) we used the equality 2,7 = R~* that takes place in the asymptotic Gaussian approach applied to the Coulomb final state interaction problem. We preserve the form of Eq. (18) as physically reasonable approximation also for all sources that allow to use the Gaussian radii for interpretation of interferometry data. The first step o f the iteration procedure is to use the values of the interferometry radii RG (the result of naive Gamov carrection according to Eq.(16)) in the Eq. (18), then find R according to Eq.(17). The more exact value of (r*) one can obtain in the next iteration step. The latter result can be used for more precise method of Coulomb correction. It is the following.

~ 2.2 c)

2

.=_ . ¢ . ~--41- .9

Gomov correction

*

•O-

o Combined correc[ion

1.8

At small q* (r*) / 2 < 1 the quantum effects dominate in the Coulomb interaction and modified at small q* Gamov factor is

G(q*) ~ Ga(q*)= G(q*)(1 + 2(r*)/lal)

• Quantum Stotlsties

1.4

[10]. At large q*(r*) > 1 the Coulomb carrelation function can be describe in the quasi-classical approach [11],

1.2

1 d/(r*)aq , 2 . . The coefficient can be defined from the equality of the both

0.8

Q(q*)

C"

1.6

1

r

,

,

I

0.02

.

,

,

I

0.04

.

,

,

I

0.06

,

,

,

I

.

O.OB

,

,

I

0,I

,

,

,

0.1

q~,GeV/c

functions at the point q* = l/(r* ) : d : a [ 1 - G(q; )(1 + 2@*} / a)]/(r*)

Fig.4. The comparison of the combined correction fit with complete and Camov corrected data. Rr = "r = 7fm, p r = 0.15 GeV / c.

Yu.M. Sinyukov et al./Nuclear Physics A610 (1996) 278c-285c

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Finally, we propose to use the combined correction function G(q*) = G 1(q*)O(q* - qo) + Q(q" )O(qo - q*)

(19)

instead of the Gamov factor. As we can see from the Fig. 4 it works good in whole q* - q~,,region. The results are obtained on the base of the Monte-Carlo simulation with faking into account the final state Coulomb interaction [10]. The similar result takes place for K + K +pairs. This allows to extract the interferometry radii in the case when impossible to use K+K-Coulomb correlation (it suppressed by the strong final state interaction and G ÷- ~ (G++} -1 at rather large q~,) for this aim. 5. CONCLUSION There are a few main points that complicate the interpretation of the interferometry data for ultra-relativistic A+A collisions. These factors are: non-Gaussian behavior of the correlation function for the 3D-expanding hadronic systems at small Pr ; the contribution of the resonance decays that is most essential (15% at small P r ) for longitudinal interferometry radii; the problem of Coulomb final state interaction for large expanding systems. The latter include also the problem of the q-dependent subtraction of (r/, r/' )- contribution from the complete correlation function. We demonstrate here that most of these problems can be solved analytically on the basis of the lengths of homogeneity. They contain the basic information concerning the evolution of the thermalized systems. REFERENCES

1. V.A. Averchenkov, A.N. Makhlin,_Yu.M. Sinyukov, Sov. J. Nucl.Phys. 46 (1987) 905. 2. S.V. Akkelin, Yu.M. Sinyukov, Phys. Lett. B356 (1995) 525. 3. Yu.M. Sinyukov, Nucl. Phys. A566 (1994) 589c; Yu.M.Sinyukov, In Hot Hadronic Matter:Theory and Experiment, (J.Letessier, H.Guthrod, J.Rafelski, eds.) p. 309, Plenum Publ. Corp., 1995. 4. S.Chapman, P.Scotto and U.Heinz, Phys. Rev. Lett. 74 (1995) 4400. 5. M.Herrmann and G.F.Bertsch, Phys. Rev. C51 (1995) 328. 6. U.A.Weidemann, P.Scotto and U.Heinz. Phys. Rev. C53 (1996) 918. 7. J.Bolz et al., Phys.Rev. D47 (1993) 3860. 8. V.V. Anisovich et.al., "Quark Model and High Energy Collisions", Word Scientific Publ., Singapure, 1985. 9. U.Omik et.al., Preprint LAUR-96-1298, 1996. 10. Yu.M.Sinyukov, R.Lednicky, S.V.Akkelin, J.Pluta and B.Erazmus (to be published). 11. Y.D.Kim et.al., Phys.Rev. C45 (1992) 387; G.Baym, P.Braun-Munzinger, the talk at QM'96.