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Dynamical properties of heavy-ion collisions from the

arXiv:hep-ph/9308245v1 8 Aug 1993DMR-THEP-93-2/Whep-ph/9308245May 1993Dynamical properties of heavy-ion collisions from thePhoton-Photon intensity correlationsL.V. Razumov∗,R.M.

Weiner†Physics Department, University of Marburg, Marburg, F.R. GermanyAbstractWe consider here the bremsstrahlung emission of photons at low and intermediate energiesElab ≤1000MeV/u of the projectile.

and derive expressions more general than previousresults obtained by Neuhauser which were limited to the case of isotropic systems.Wefind that the two-photon correlation function strongly depends not only on the space-timeproperties of the collision region but also on the dynamics of the proton-neutron scatteringprocess in nuclear matter. As a consequence of polarisation correlations it turns out thatfor a purely chaotic source the intercept of the correlation function of photons can reach thevalue 3 (as compared with the maximum value 2 for isotropic systems).

Furthermore evenfor “hard” photons (Eγ > 25MeV ) the maximum of the correlation function can reach thevalue of 2 in contrast with the value of 1.5 derived by Neuhauser for this case. The formulaeobtained in this paper which include also the possible presence of a coherent component canbe used as a basis for a systematic analysis of photon intensity-interferometry experiments.∗E.Mail: razumov@convex.hrz.Uni-Marburg.de†E.

Mail: WEINER@vax.hrz.Uni-Marburg.de

1IntroductionThe Hanbury-Brown and Twiss intensity interferometry [1] plays at present an importantrole in particle and nuclear physics in a wide range of c.m. energies, because this methodcan provide information about the space-time properties of the sources.

From hadron inter-ferometry one obtains information mainly about the properties of the fireball just near thefreeze-out. Moreover, in this case we are faced with some dynamical problems like the finalstate interaction and contribution from resonances which mask the real space-time size of thecollision region.

In contrast, photons and leptons have a large mean free path and leave thecollision region just after their production. Additionally we can assume that photons haveno final state interaction (light-light scattering can be neglected).

Therefore direct photons(i.e. photons not originating from π0 decay) and dileptons are good probes for the earlieststages of the reaction.

Another advantage of electro-magnetic probes is that their emissionis controlled by QED rather than by strong interaction dynamics for which no comparabletheory exists yet.In this paper we consider mostly the emission of “hard” photons (their definition will begiven below) in heavy ion reactions in the energy range up to 1 Gev/u. We will show thatthe photon-photon correlation function can provide there not only space-time informationbut also some dynamical information about the proton-neutron scattering in nuclear matter.It is known [2, 3] that at intermediate energies hard photons are produced mainly throughbremsstrahlung and π0 decay.

Since the photon pairs from the π0 decay can be eliminatedto a great extent by measuring the photon pair invariant mass, we concentrate ourselves onbremsstrahlung photons.2Bremsstrahlung photons and their correlationsThe photon spectrum is usually described by a superposition of two contributions, bothparametrized by exponentials [2, 3] and referring to different photon energies respectively:the low energy part is assumed to be due to thermal photons and the slope is just given bythe temperature. The second (high energy) part can be considered [2] as due to a mixtureof bremsstrahlung (see Chapter 5) and photons originating from decays (mostly π0 →γγ).Measurements of the angular distribution of radiation, the photon-source velocity [2, 3] aswell as the impact-parameter dependence of the photon multiplicity [4] suggest that thedirect hard photons mainly originate from bremsstrahlung in independent proton-neutroncollisions1.As was shown in [3] the non-relativistic recoilless bremsstrahlung formula for the current1The photon emission from proton-proton collisions is of quadrupole form and therefore highly suppressedas compared with the dipole radiation from proton-neutron collisions.

in a p −n collision2jλ(k) =iemk0p · ǫλ(k)(1)(where p = pi −pf is the difference between the initial and final momentum of the proton,ǫλ(k) is the vector of linear polarisation and k the 4-momentum) works well even in therelativistic case due to the fact that relativistic and recoil corrections to some extent com-pensate each other. If we have N such sources we can write down the transition current asfollows:Jλ(k) =NXn=1eikynjλn(k)(2)The index n here labels the independent p −n collisions taking place at different space-time points yn.

Formulas (1), (2) are examples of radiation by classical currents, where theinfluence of the emitted photons on the p −n collision process is negligible.For an arbitrary classical current Jµ(x) coupling with the photon fieldAµ(x) =Z˜dkaλ(k)ǫµλ(k)e−ikx + a+λ(k)ǫ∗µλ (k)eikx,(3)˜dk ≡d3k/[(2π)32k0] ,haλ1(k1) , a+λ2(k2)i≡δλ1λ2(2π)3 · 2k0δ3(k1 −k2)through the conventional interaction lagrangianLint = −Jµ(x)Aµ(x)(4)we can easily find the exact S-matrixS = exp"−122Xλ=1Z˜dkJλ(k)2#: exp−iZ˜dkJλ(k)a+λ(k) + J+λ(k)aλ(k):(5)which contains all the information on the photon production and absorption. We use in (5)the notation:Jλ(k) ≡ǫ∗λµ (k) ˜Jµ(k) .

(6)where ˜Jµ(k) is the Fourier transform of the current Jµ(x). Before the collision there are nophotons, i.e.

we have the photon vacuum in the initial state.The amplitude to produce n-photons from the photon vacuum by the classical current Jfollows directly from (5):< 0|nYi=1aλi(ki)S|0 > = < 0|S|0 >nYi=1−iJλi(ki). (7)2The experimently found one-photon spectra are of exponential form [2, 3].

This property can be alsoincorporated into this model, see Chapter 5.

The corresponding cross-section can be written in a condensed and convenient form bymeans of the generating functional for the exclusive processes. Let us introduce the notationq ≡(k, λ) for the momentum k and polarisation λ degrees of freedom of photon.

Integrationover q means integration over momentum (with the invariant measure˜dk ) and summationover polarisation λ . We define the exclusive generating functional gex[Z] as follows:gex[Z] ≡∞Xn=01n!ZnYi=1dqiZ(qi) 1σex ·dnσexdq1 × .

. .

× dqn(8)The conservation of probability gives the normalisation gex[Z = 1] = 1 . The method ofthe generating functionals allows one to avoid the complicated combinatorics which usuallyappears when one consider inclusive processes.

The generating functional for the inclusivecross-section can be introduced in a similar waygin[Z] ≡∞Xn=01n!ZnYi=1dqiZ(qi) 1σin ·dnσindq1 × . .

. × dqn(9)(with the normalisation condition gin[Z = 0] = 1 ) and can be very simple expressed throughthe exclusive onegin[Z] = gex[Z + 1] .

(10)In our case (see eq. (7)) the summation over n can be performed explicitly and we are leftwith the simple expressionsgex[Z]=exp" 2Xλ=1Z˜dk|Jλ(k)|2 (Zλ(k) −1)#(11)gin[Z]=gex[Z + 1] = exp" 2Xλ=1Z˜dk|Jλ(k)|2Zλ(k)#(12)If the classical current obeys some random behaviour which is the present case, as we considerchaotic sources, the generating functional should be subjected to the averaging over thecurrent distributionGin =Dgin[Z]E.(13)Taking the variational derivatives of Gin[Z] with respect to Z one gets the inclusive spectra.For example single and double inclusive spectra read as follows:ρλ1(k)= δGin[Z]δZλ(k)Z=0=D|Jλ(k)|2E(14)ρλ1λ22(k)=δ2Gin[Z]δZλ1(k1)δZλ2(k2)Z=0=D|Jλ1(k1)Jλ2(k2)|2E(15)An essential ingredient in eqs.

(13) - (15) is the average prescription < ... >.For the model considered here, where hard bremsstrahlung photons originate mostlyfrom independent proton-neutron collisions at the early stage of the heavy-ion reaction, theaveraging prescription < ... > appears quite naturally and consists of two parts:

1. The pointsyn where the independentp −n collisions take place (see eq.

(2)) areconsidered to be randomly distributed in the space-time volume of the source (e.g.fireball) with a distribution functionf(y) for eachp −n collision.This kind ofaveraging is typical for Bose-Einstein correlation studies.2. The amplitude of photon production (1) is very sensitive to the momentum transfer ofthe proton in the proton-neutron collision.

That is why we have to take into accountalso the fluctuations of the initial momentum of the proton: pi = p0 + ∆pF . Here p0is the momentum of the nucleus as a whole and ∆pF is the Fermi-motion of a nucleon.The final momentum distribution of the proton (pf) can be in principle determinedby dynamical models of nucleus-nucleus collisions, e.g.

BUU [3], etc.Since in practice we need only the first two moments of this distribution we keep them as afree phenomenological parameters to be determined in the experiment. The comparison ofthe experimental values with model predictions can serve as an indirect test of the model.The fundamental quantity in our approach is the two-current correlator in momentumspace:< Jλ1(k1)J∗λ2(k2) >=< Jλ1(k1)Jλ2(−k2) >=NXn,m=1ZNYl=1d4ylf(yl) ×exp(ik1yn −ik2ym) < jλ1n (k1)jλ2m (−k2) >=NXn=1h ˜f(k1 −k2) < jλ1n (k1)jλ2n (−k2) >−˜f(k1) ˜f(−k2) < jλ1n (k1) >< jλ2n (−k2) >i+ < Jλ1(k1) >< Jλ2(−k2) >.

(16)Here we use the properties (Jλ(k))∗= Jλ(−k), < jλ1n (k1)jλ2m (−k2) >=< jλ1n (k1) >< jλ2m (−k2) >for n ̸= m (that is the hypothesis of independent proton-neutron collisions), and˜f(k) de-notes the Fourier transform of f(y) with the normalisation˜f(k = 0) = 1 .Usually the function˜f(k) has a steep maximum around k = 0 with a width of the order ofthe size of the source R . In the region of the Bose-Einstein peak for hard photons in partic-ular for 2kR ≫1 we can therefore neglect˜f(k1) ˜f(−k2) as compared to˜f(k1 −k2) .

Thisapproximation strongly simplifies the algebra without influencing significantly the accuracyof the calculations.3Chaotic sourcesLet us consider the case of a chaotic source for which < Jλch(k) >= 0 . The current correlator(16) is now given by the simple formula:< Jλ1(k1)Jλ2(−k2) >=F λ1λ2(k1, k2) ≡˜f(k1 −k2)NXn=1< jλ1n (k1)jλ2n (−k2) >

= ˜f(k1 −k2)e2/m2k01k02ǫiλ1(k1) NXn=1< pinpjn >!ǫjλ2(k2). (17)< pinpjn > denotes here the averaging with respect to the distribution of the momentumtransfer of the proton in the collision n andPNn=1 goes through all the relevant proton-neutron collisions.

As mentioned above the quantity< pinpjn >can be extracted fromdynamical models of heavy-ion collisions. But one can reach important and general conclu-sions without specifying this quantity as follows.

We use the axial symmetry around thebeam direction. The tensor decomposition of < pinpjn > gives then:< pinpjn > = σn3 δij + δnℓiℓj,(18)where ℓis the unit vector in the beam direction and σn , δn are real positive constants.

Notethat this expression is more general than the corresponding one used in [5] where becauseof the isotropy assumption δn was assumed to vanish. This generalization has importantconsequences to be exhibited below.

Now let us separate the average value of the protonmomentum transfer < pn > :pn = ∆pn + < pn >(19)We have then:< pinpjn >=< ∆pin∆pjn > + < pin >< pjn >=< ∆pin∆pjn > + < p >2 ℓiℓj(20)where the tensor < ∆pin∆pjn > can be represented (due to axial symmetry) again as thesum of the two terms:< ∆pin∆pjn > = σn3 δij + ξnℓiℓj. (21)In order to find the coefficients σn, ξn, δn we split the 3-vectors in the transverse and thelongitudinal parts:∆pln = ℓ· (ℓ· ∆pn);∆ptn = ∆pn −∆pln.

(22)The coefficients in (18), (21) read :σn3 = 12 < (∆ptn)2 > ;ξn = < (∆pln)2 > −σn3 ;δn = < (∆pln)2 > −σn3 + < pn >2. (23)With the help of (18), (23) one can express the current correlator (17)F λ1λ2(k1, k2) = ˜f(k1 −k2)N · e2/m2k01k02ǫiλ1(k1)σ3 δij + δℓiℓjǫjλ2(k2) ;(24)

through two parameters σ and δσ3 = 1NNXn=112 < (∆ptn)2 >(25)δ = 1NNXn=1[< (∆pln)2 > + < pn >2] −σ3,(26)which absorb the relevant dynamical information about the proton-neutron scattering in themedium. The parameters σ and δ can be extracted from experimental data on photon-photon Bose-Einstein Correlations (cf.

below) and/or calculated from dynamical models forheavy-ion collisions.The single inclusive cross-section for a detector, which does not measure polarizationsfollows directly from (14), (24)ρ1(k) =2Xλ=1F λλ(k,k) = N e2/m2(k0)223σ2 + δ2 · sin2θ(27)where θ is the angle between the photon and the beam directions and the polarization sumis calculated using the well-known identity:2Xλ=1ǫiλ(k)ǫjλ(k) = δij −ninj(28)with n = k/|k| .To calculate the double inclusive cross-section (as well as the higher order inclusive spec-tra) one has to know higher order current correlators. In our case when all proton-neutroncollisions are considered to be independent from each other and the number of the partic-ipating protons is sufficiently large N > 10 , we can apply the central limit theorem andexpress the higher order current correlators through the first and second ones.

Assuming< Jλ(k) >= 0 (no coherence) the double inclusive cross-section is represented through thesum of products of the two-current correlatorsρ2(k1, k2) =2Xλ1,λ2=1< Jλ1(k1)Jλ1(−k1)Jλ2(k2)Jλ2(−k2) >=2Xλ1,λ2nF λ1λ1(k1, k1)F λ2λ2(k2, k2) + F λ1λ2(k1, k2)F λ2λ1(k2, k1) + F λ1λ2(k1, −k2)F λ2λ1(−k2, k1)o= ρ1(k1)ρ1(k2) +2Xλ1,λ2=1F λ1λ2(k1, k2)F λ2λ1(k2, k1) + (k2 ↔−k2)(29)and the polarization sum2Xλ1λ2=1F λ1λ2(k1, k2)F λ2λ1(k2, k1) = | ˜f(k1 −k2)|2 · N2 e4/m4(k01k02)2 ×(30)

(σ29 (1 + cos2 ψ) + δ2 sin2 θ1 sin2 θ2 + 23σδ[1 −cos2θ1 −cos2 θ2 + cos ψ cos θ1 cos θ2])is performed with the help of (24), (28).The general expression for the second order correlation function is defined byC2(k1, k2) =ρ2(k1, k2)ρ1(k1)ρ1(k2)(31)and has a complicated angular dependence, but the two limiting cases 1) σ ≫δ and 2)σ ≪δ lead to very simple expressions:1. For the case σ ≫δ we have:C2(k1, k2|σ ̸= 0, δ = 0) = 1 + 14(1 + cos2 ψ)h| ˜f(k1 −k2)|2 + | ˜f(k1 + k2)|2i(32)which is the result derived in [5] and which gives for the interceptC2(k, k) = 32 + 12| ˜f(2k)|2 .

(33)2. The opposite case σ ≪δ leads to another formula:C2(k1, k2|σ = 0, δ ̸= 0) = 1 + | ˜f(k1 −k2)|2 + | ˜f(k1 + k2)|2 ;(34)with the intercept exceeding 2:C2(k, k) = 2 + | ˜f(2k)|2 .

(35)For hard photons| ˜f(2k)|2 ≪1(cf. [5]) and one can neglect this contribution to theintercept while for soft photons ˜f(2k) is non-neglegible and in the limit k = 0, ˜f(0) = 1 sothat C2(k, k) = 3.

The real situation (when both σ and δ contribute) is between (32) and(34) and exhibits a more complicated angular behaviour than the considered above limitingcases [6]. For instance the general expression for the intercept which follows directly from(27), (29), (30)C2(k, k) =R dΩρ2(k, k)R dΩρ21(k)= 1 + 12[1 + | ˜f(2k)|2] ·"1 +1.2δ2σ(σ + 2δ) + 1.2δ2#(36)depends on both parameters σ and δ and varies in the range between 3/2 and 3.

The solidangle integration over all possible orientations of the photon momentum k corresponds to a4π detector.The function f(x) and its Fourier transform ˜f(k)) reflects the space-time properties of thephoton source. It depends in principle on three constants: the time duration R0, the longi-tudinal radius Rl and the transverse radius Rt.

One can propose a concrete parametrizationfor the source geometry ˜f(k) and then fit the inclusive data using (31). However, becauseof insufficient statistics one has usually to limit oneself to a smaller number of parameters.

In particular the choice of two parameters T = R0 and R = Rl = Rt is good enough forthe present state of the art. With this assumption we can perform analytically the angularintegration over θ1 and θ2 in (27) and (29), (30) keeping constant the angle between the twophotons n1 · n2 = cosψ = const.

The corresponding integrationZdµ = 2ππZ0sinθ1dθ12πZ0dϕ(37)extends over the rotations of n2 around n1 (dϕ) and over all orientations of n1 around thebeam direction. After some algebra we are left with the general expression for the Bose-Einstein correlation functionC2(k1, k2) ≡R dµρ2(k1, k2)R dµρ1(k1)ρ1(k2) = 1 +(38)14(| ˜f(k1 −k2)|2 + | ˜f(k1 + k2)|2·"1 + cos2ψ + 0.3δ2(3 + cos2ψ)(3 −cos2ψ)σ(σ + 2δ) + 0.3δ2(3 + cos2ψ)#)which has a pronounced dependence not only on the space-time characteristics R0, R butalso on the angle ψ and the “dynamical” constants σ, δ.

The limiting cases (32) and (34)as well the intercept formula (36) can be rederived directly from (38).4Partially coherent sourcesIn general, photon sources are not totally chaotic but may contain also a coherent compo-nent < Jλ(k) >≡Iλ(k) ̸= 0 so that the total current JλT = Jλch + Iλ leads to a partiallycoherent field. There are several mechanisms which are responsible for coherence: collectivedeacceleration of the initial nuclei, collective flow, coherent radiation from nuclear fragments,etc.

The possibility to investigate such collective phenomena is an interesting and importantpart of heavy-ion physics. In this chapter we discuss phenomenologically the influence ofthe coherent part of the electric current on the photon correlation function.

The single anddouble inclusive cross-sections (see (27) and (29) ) read:ρ1(k) =2Xλ=1F λλ(k, k) + |Iλ(k)|2,(39)ρ2(k1, k2) =2Xλ1,λ2=1< Jλ1(k1)Jλ1(−k1)Jλ2(k2)Jλ2(−k2) >= ρ1(k1)ρ1(k2) +(40)+2Xλ1,λ2=1n|F λ1λ2(k1, k2)|2 + 2Re[I∗λ1(k1)F λ1λ2(k1, k2)Iλ2(k2)] + (k2 ↔−k2)o.As before we assume axial symmetry around the beam direction and parametrise Iλ(k) asfollows:Iλ(k) =iemk0√NS(k0)ℓ· ǫλ(k). (41)

Using (24), (28) and (41) we get2Xλ=1|Iλ(k)|2 = N e/mk0!2|S(k0)|2sin2θ,(42)2Xλ1,λ2=1I∗λ1(k1)F λ1λ2(k1, k2)Iλ2(k2) = ˜f(k1 −k2)N2 e2/m2k01k02!2S∗(k01)S(k02) ×σ3 (1 −cos2θ1 −cos2θ2 + cosψcosθ1cosθ1) + δsin2θ1sin2θ2(43)which together with F λλ(k, k) and F λ1λ2(k1, k2)F λ2λ1(k2, k1) (see (27),(30)) determine com-pletely the single and double inclusive spectra (39,40).As compared to the completelychaotic case there is now one more function S(k0) entering in ρ1(k) and ρ2(k1, k2).Significant simplifications can be achieved again assuming Rl = Rt = R.The timedependence of the photon source is still arbitrary.It is then possible again to performexplicitly the integrations over all the angles except for ψ and write the photon-photoncorrelation function in the compact formC2(k1, k2) ≡R dµρ2(k1, k2)R dµρ1(k1)ρ1(k2) =(44)= 1 +nλ2(k0)A| ˜f(k1 −k2)|2 + 2λ(k0)(1 −λ(k0))BRe ˜f(k1 −k2) + (k2 ↔−k2)owhere we have introduced a new set of parameters:λ(k0)=σ + δσ + δ + |S(k0)|2 ,x =σσ + δ ,(45)A=14 · 1 + cos2ψ + (1 −x)2(13 + cos2ψ)/51 + (1 −xλ)2(3cos2ψ −1)/10,B=14 · 1 + cos2ψ + (1 −x)(13 + cos2ψ)/51 + (1 −xλ)2(3cos2ψ −1)/10.Here λ(k0) is of the chaoticity parameter which is the ratio of the number of chaoticallyproduced photons with energy k0 and their total number λ(k0) =< n(k0) >ch / < n(k0) > .We assume also that on the scale important for the Bose-Einstein study (|k1 −k2| ∼1/R)the function λ(k0) does not vary significantly: |(k01 −k02)dλ(k0)/dk0| ≪(λ(k01) + λ(k02))/2 .The other parameter which influences strongly the behaviour of the correlation functionC2(k1, k2) is the isotropy parameter x (see (45)). Both these parameters λ and x can beextracted from the analysis of the angular distribution of the radiation and the value of theintercept C2(k, k):ρ1(k)/ < n(k0) > =14π[xλ(k0) + 32(1 −xλ(k0))sin2θ],(46)C2(k, k) = 1 + (1 + | ˜f(2k)|2)/25 + (1 −xλ)2nλ2(k0)[5 + 7(1 −x)2] + 2λ(1 −λ)[5 + 7(1 −x)]o(47)

(for the case of hard photons | ˜f(2k)|2 ≪1 and one can neglect this contribution in (47)).After this has been done the only unknown function in (44) is the space-time distributionfunction of the source f(x) (or ˜f(k)) which can now be obtained by fitting the experimentaldata. This function has the physical meaning of the space-time distribution of the radiatingregion and reflects the geometry of the early stages of the collision.We would like to stress once again that both the parameters λ and x strongly influenceC2(k1, k2) and the knowledge of only the one of them (e.g.

λ) is not enough to determine ina unique way the two-photon correlation function. For instance, one can check that all theλ, x pairs, for which λ = (√2 −1)/(√2 −x) and 0 ≤x ≤1 , lead to the same interceptC2(k, k) = 3/2 .5Exponential fall of the bremsstrahlung amplitudeThe experimental observations [2, 3] show that the one-particle inclusive spectrum of brems-strahlung photons has exponential form suggesting that the underlying proton current reads:jλ(k) =iemk0p · ǫλ(k) · exp[−k0/(2E0)](48)rather than (1).

The single and double inclusive cross-sections calculated with (48) insteadof (1) can be obtained from the previous results multiplying them by the correspondingpowers of exp[−|k01|/(2E0)] and exp[−|k02|/(2E0)] .The homogeneous functions like the two-photon correlation function (31) and the angulardistribution of radiation (46) remain unchanged and all the conclusions about the Bose-Einstein correlations obtained above hold.In the following we shall derive the non-relativistic classical trajectory of a proton whichleads to the mentioned above current (48).The current in momentum space is defined through the proton trajectory as follows:jλ(k) = ǫλµ(k)˜jµ(k) = −e+∞Z−∞dt exp{i[k0t −kr(t)]} (ǫλ · v(t)). (49)In non-relativistic case |v(t)| ≪1 and as a consequence k0t −kr(t) ∼= k0t .

Therefore jλ(k)reduces to the time Fourier transform of the velocity. Using the identity+∞Z−∞dteiωt121 −2πarctg(2E0t)=−iω −iǫexp[−|ω|/(2E0)](50)one finds the trajectoryv(t) = v021 −2πarctg(2E0t)(51)leading to the proton current (48).

The standard formula (1) corresponds to the special caseE0 →+∞when the proton trajectory is described by the step function v(t) = v0Θ(−t) .

The finite value of E0 reflects more smooth then step-like deacceleration of the proton withthe characteristic stopping time τ ∼1/(2E0) and stopping length l ∼v0τ. For instance, forthe projectile energy 45 MeV/u (slop-parameter E0 = 18 MeV [3]) one gets τ = 5.6 fm/cand l = 1.7 fm which seem to be quite reasonable.6SummaryIn this paper the production of photons is analysed in the framework of quasi-classicalapproximation.

Our consideration is valid in the region of low and intermediate collidingenergies up to 1000 MeV/u. We assume that the hard photons (Eγ ≥25MeV ) are producedin independent proton-neutron collisions3 (see Chapter 2) and the whole system has theaxial symmetry with respect to the beam direction.

Then we derive the expressions for thesingle and double inclusive cross-sections and for the two-photon correlation function as well(see Chapters 3,4). We show that the behaviour of the photon-photon correlation functiondepends not only on the space-time properties of the collision region (function ˜f(k1 −k2))but also on dynamics of the proton-neutron scattering in matter (parameters σ, δ and theamount of chaoticity λ).

So far the photon intensity interferometry can be considered as anindirect way to check the dynamical properties of the heavy-ion system as well as to obtainthe space-time information about a collision.It turns out that the maximum value of the correlation function (the intercept C2(k, k)) isvery sensitive to the details of the proton-neutron scattering and varies generally speaking inthe interval 1 ≤C2(k, k) ≤3 (see (47)). If we deal with so hard photons that | ˜f(2k)|2 ≪1the intercept finds itself in more narrow range 1 ≤C2(k, k) ≤2 (which is nevertheless widerthen the one obtained in [5] 1 ≤C2(k, k) ≤1.5 ).

The reason why the intercept can exceedthe value 1.5 (see [5]) is strongly connected with the beam-direction anisotropy specific forheavy-ion collisions.The photons produced with the close momenta can have the correlations in their polarisa-tions. As far as the photons with the same polarisation obey the Bose-Einstein effect likethe scalar bosons (e.x.

π0) and ones with the perpendicular polarisations behave like non-identical particles, these correlations in polarisations affect positivly on the Bose-Einsteinpeak increasing intercept. The presence of coherence leads to decreasing of the correlationeffect.

In order to study the two-photon correlation function including the influence of theanisotropy and the coherent contribution as well (44) one needs additional information whichcan be obtained from the angular distribution of the radiation (46) and the intercept value(47) analysed together in order to extract the values λ and x. After it has been done theonly unknown quantity in (44) is the Fourier transform ˜f(k) of the space-time probabilitydistribution f(x) of the photon source which reflects the geometry of the early stages of the3the photon emission from the proton-proton collisions has the quadrupole nature and therefore is highlysuppressed with respect to dipole radiation from the proton-neutron collisions.

heavy-ion collision.We would like to acknowledge fruitful comments by Y.Schutz, I.V.Andreev, G.R¨opke,T.Alm, J.Clark.This work was supported in part by the Gesellschaft f¨ur Schwerionenforschung, Darmstadt.References[1] R.Hanbury-Brown, The Intensity Interferometer , edited by Taylor & Francis Ltd (Lon-don 1974). Y.Schutz et al., GANIL-Preprint 92-05;[2] E.Grosse, in Fundamental problems in heavy-ion collisions, edited by N.Cindro,W.Greiner and R.Caplar, 1984, P.347.

[3] W.Cassing et al., Phys.Rep., 188(1990), 363[4] E.Migneco et al., Phys.Lett. B298 (1993), 46.

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