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ON SEMI-CLASSICAL PION PRODUCTION IN HEAVY

arXiv:hep-ph/9306301v1 22 Jun 1993ON SEMI-CLASSICAL PION PRODUCTION IN HEAVYION COLLISIONSJean-Paul BlaizotService de Physique Th´eorique,Centre d’Etudes de Saclay,F-91191 Gif-sur-Yvette, FranceandDmitri Diakonov 1St.Petersburg Nuclear Physics Institute,Gatchina, St.Petersburg 188350, RussiaAbstractIn high energy heavy ion collisions the pion multiplicity is large, and onemight expect that pions are radiated semi-classically. The axial symmetry ofthe collision and approximately zero isotopic spin of the colliding nuclei resultthen in peculiar isotopic spin – azimuth correlations of the produced pions.These correlations are easy to test – and should be tested1Alexander von Humboldt Forschungspreistr¨ager

Central collisions of heavy ions at high energies are expected to producemany pions (see for instance [1]), so many that the typical number of pionquanta per unit cell of phase space may substantially exceed unity. Under suchcircumstances, one may speculate about the possibility that, in the early stagesof the collisions, a classical pion field develops, before subsequently decayinginto pions [2, 3, 4].

One interesting possibility showing how this could happenfollowing chiral symmetry restoration has been investigated recently [5]. Inde-pendently of the detailed dynamics, the formation of an intermediate classicalpion field could result in certain correlationsin the production of pions withdefinite isospin components.The main point of this letter is to show that,indeed, rather peculiar correlations can be indicated from symmetry consider-ations only – the role of dynamics is to guarantee that the pion radiation isso strong that it can be treated semi-classically; testing the correlations is away to check that hypothesis.

These correlations, quite distinct from the muchstudied Bose Einstein correlations, find their origin in the special correlationswhich develop between spatial and isospin coordinates in the solution of thenon linear field equations obeyed by classical pion fields.Mathematically, one can write the amplitude of N pions’ production withthe help of the Lehmann–Symanzik–Zimmermann formula, where the N-pointpion Green function is presented through the functional integral over the pionfields:Aa1...aN (k1 . .

. kN) =limk2n→m2πZDπaZDJa W[J] expiS[π] + iZd4xπaJa·NYnZd4xneiknxn(−∂2xn −m2π)πan(xn).

(1)Here S[π] is the effective pion action and J is the source formed by the col-liding nuclei; one has to integrate over the sources with some weight functionalW[J] which summarizes the dynamics of the collision. If the source J is in asense “large”, the functional integral can be evaluated in the saddle-point ap-proximation, where the saddle-point pion field is the solution of the equationsof motion,δS[π]δπa(x) + Ja(x) = 0,(2)supplemented with the radiation condition at large distances, which at mπ →0reads2

∂πa∂t + ∂πa∂r = 0. (3)Let us denote the solution of these eqs.

as πa(r, t). In the leading WKBapproximation one replaces the pion fields everywhere in eq.

(1) by this saddle-point field. In the next approximation quantum fluctuations about the classicalfield πa(r, t) should be also taken into account.

We do not discuss dynamicalquestions in this letter but concentrate solely on general symmetry considera-tions. We use two assumptions:1) the collision is axially-symmetrical;2) the collision is isotopically-symmetrical; that would be an exact state-ment in case of the zero isospin of the colliding nuclei and is anyhow a goodapproximation for pions produced in the central rapidity region since a transferof any quantum numbers over a long rapidity range ∆y is suppressed at leastas (∆y)−1.The first assumption means that the saddle-point pion radiation field canbe sought in the axially-symmetrical “hedgehog” form:πa(r, t) = Oai n⊥iP⊥(ρ, z, t) + n∥iP∥(ρ, z, t)(4)where P⊥,∥are functions of the distance ρ from the beam axis, of the dis-tance z from the collison point and of time t; n∥(⊥) are unit vectors parallel(perpendicular) to the beam axis and Oai is an arbitrary 3 × 3 orthogonal ma-trix.

(In principle, axial symmetry does not contradict higher harmonics inthe transverse plane with n⊥replaced by a unit vector with the components(cos mφ, sin mφ) where φ is the azimuthal angle and m is an integer.Weshall work with m = 1, i.e. assume that n⊥points in the radial direction, andintroduce the arbitrariness in the choice of m only in our final results).

Thetype of correlations between space and isospin degrees of freedom is a familiarfeature of solutions of non linear field equations satisfied by the pion fields, asillustrated for example in the Skyrme model (see, e.g. [6]).

The second assump-tion means that the saddle point is degenerate in the global isospin rotationOai, so that the functional integral in eq. (1) reduces to the integration overall possible orientations of the pion field in the isospin space.

The action andthe source-weight functionals are taken at the saddle-point values of Ja and πaand provide an overall normalization factor √N which may be a function ofthe total 4-momentum of the pions Pµ, but now we are not interested in thisfactor.3

At large distances / time the isospin source Ja dies out and one can alsoneglect the non-linearity of the pion effective action. Therefore, eq.

(2) reducesto the free Klein–Gordon eq. at large distances, and we are guaranteed thatthe Fourier transform of eq.

(4) has a pole at k20 −k2z −k2⊥= m2π, which cancelsout in the LSZ leg amputation procedure (see eq. (1)).

We have therefore:limk2→m2πZd4xeikx(−∂2x −m2π)πa(x) = Oai k⊥iF⊥(k⊥, k∥) + k∥iF∥(k⊥, k∥)≡OaiFi(k)(5)where F⊥,∥(k⊥, k∥) are related through Fourier transformation to P⊥,∥of eq. (4).Since P⊥,∥are real the functions Fi(k) are purely imaginary, with F ⋆i (k) =Fi(−k) = −Fi(k).Squaring the amplitudes, summing over the isospin a = 1, 2, 3 of the pionsand multiplying by the phase space factor, one gets for the N pion productioncross-section:σ(N) = NN!ZdO1dO2ZNYn=1d4kn(2π)4 2πδ+(k2n −m2π)(2π)4· Oai1 Fi(kn)Oaj2 F ⋆j (kn)δ(4)(Pµ −Σknµ),(6)where knµ are individual 4-momenta of the produced pions, O1,2 are isospinorientation of the pions in the amplitude and the conjugate amplitude, respec-tively, P is the total 4-momentum of the produced pions; the factorial accountsfor the identical particles and δ+(k2n −m2π) = δ(k2n −m2π)θ(k0).Writing the 4-momentum conservation restriction as(2π)4δ(4)(P −Xkn) =Zd4R ei(P ·R)−iP(kn·R),(7)we get factorized integrals over the momenta of produced pions:Zd4k(2π)4 2πδ+(k2 −m2π)e−i(k·R)Fi(k)F ⋆j (k) = Fij(R0, R).

(8)This tensor is further on contracted with the relative orientation matrixOij12 ≡Oai1 Oaj2= (OT1 O2)ij. We note that in integrating over the SO(3) rota-tions one can use the Haar measure property, dO = d(CO) = d(OC), where Cis an arbitrary orthogonal matrix, so thatZ ZdO1dO2 =Z ZdO1dO12 ;ZdO = 1.

(9)4

The total cross section being a sum of σ(N) over N becomes thus a seriesfor an exponent, and we getσtot = NZdO12Zd4R exphi(P · R) + Fij(R0, R)Oij12i. (10)Let us use the frame in which the total momentum of the produced pionsis zero, P = 0, P0 = E where E is the total energy of the pions.

Since both Eand the function F, proportional to the probability of the pion production, arepresumably large, one can integrate over R0, R and O12 by the saddle-pointmethod. Let us parametrize the relative orientation matrix O12 in terms of aunit 4-vector uµu2µ = 1, τµ = (1, −iτ ),Oij12 = 12Tr(uµτµτ iuντ †ντ j) = (1 −2u2)δij + 2uiuj + 2u0ukεijk.

(11)We expect the saddle point to be atu ≈0,R ≈0,R0 ∼1E . (12)Expanding the exponent in eq.

(10) around this point, and using the explicitform of F given by eq. (8), it can be shown that, at small R0, the saddle-point condition is indeed satisfied.

Without knowing the explicit form of thefunctions Fi(k) we cannot prove that there are no other saddle-points, but weshall disregard such possibilities. Note that the maximum at u = 0 correspondsto Oij12 = δij, i.e.

to the case when the pion isospin orientation in the conjugateamplitude is the same as in the direct amplitude – not an unnatural result.We thus write the total cross section asσtot ≈NZdR0 exp [i(ER0) + Fii(R0, 0)] . (13)In what follows we shall use this relation to remove the unknown normalizationfactor N.We next turn to the 1,2,... particle inclusive cross sections.

All of them aredirectly derived from eq. (6) where one skips integration over 1,2,... momentaand summation over the 1,2,... isotopic subscripts.

Thus, the 1-particle inclu-sive cross section is given by (we use the abbreviation (dk) = d3k/2k0(2π)3):dσ(A)(dk) = u(A)bu⋆(A)b′ZdO1dO2Obi1 Ob′j2 Fi(k)F ⋆j (k)·N∞XN=11(N −1)!Z N−1Yn(dkn) (F(kn)O12F ⋆(kn)) (2π)4δ(4)(P −k−Xkn). (14)5

Here u(A)bare the isospin ”polarization” vectors,u(π0)b=001b,u(π+)b=1√21i0b,u(π−)b=1√21−i0b,(15)to be contracted with the isospin orientation matrices O1,2. The superscript Arefers to the isospin component of the observed pion; there is no summation onA.The 2-particle inclusive cross section for production of the pion of sort A1with momentum k1 and of the pion of sort A2 with momentum k2 isdσ(A1A2)(dk1)(dk2) = u(A1)bu⋆(A1)b′u(A2)cu⋆(A2)c′·ZdO1dO2Obi1 Ob′j2 Fi(k1)F ⋆j (k1)Ock1 Oc′l2 Fk(k2)F ⋆l (k2)N∞XN=21(N −2)!·Z N−2Yn(dkn) (F(kn)O12F ⋆(kn)) (2π)4δ(4)(P −k1 −k2 −Xkn),(16)and so on.

Writing the 4-momentum conservation δ function with the help ofan auxiliary integral as in eq. (7), we again obtain the exponential series, sothatdσ(A)(dk) = u(A)bu⋆(A)b′ZdO1dO12Obi1 Ob′k1 Okj12Fi(k)F ⋆j (k)· NZd4R exphi(P · R) −i(k · R) + Fij(R)Oij12i(17)where we used Ob′j2= Ob′k1 Okj12.

If the momentum of the observed pion is negli-gible as compared to the total momentum P of the pions and if the integrationover O12 and R is performed about the presumably steep saddle point given byeq. (12), we obtain:dσ(A)(dk) ≃u(A)bu⋆(A)b′ZdO1Obi1 Ob′j1 Fi(k)F ⋆j (k)σtot= u(A)bu⋆(A)b′13δbb′δijFi(k)F ⋆j (k)σtot = 13Fi(k)F ⋆i (k)σtot.

(18)We see that the inclusive cross section is directly related to the square of theFourier transform of the classical pion radiation field, – a most natural result.Also naturally, we find identical cross sections for π+, π−and π0 production.6

The average multiplicity is obtained by integrating the inclusive cross sectionover k and summing over A = π+, π−, π0:⟨N⟩≈Z(dk)Fi(k)F ⋆i (k) ≈Fii(Rµ = 0). (19)It should be kept in mind that if this integral is not convergent by itself at largek it should be cut at least at P in accordance with a more precise eq.

(17).In eq. (17) we used the following formula for averaging over isospin rotations:ZdOObiOb′k = 13δbiδb′k.

(20)To calculate the 2-particle inclusive cross section we need a formula for averag-ing over four matrices:ZdOObiOb′jOckOc′l = 130δijδkl(4δbb′δcc′ −δbcδb′c′ −δbc′δb′c)+ 130δikδjl(−δbb′δcc′ + 4δbcδb′c′ −δbc′δb′c)+ 130δilδjk(−δbb′δcc′ −δbcδb′c′ + 4δbc′δb′c). (21)One can check this formula by applying various contractions and reducing it toeq.

(20); an alternative method is to note that Obi is a Wigner D function forisospin 1, and using the Clebsch–Gordan machinery.Starting from eq. (16) and repeating the same steps as above we get for the2-particle inclusive cross section:dσA1A2(dk1)(dk2) = u(A1)bu⋆(A1)b′u(A2)cu⋆(A2)c′· 130h2δbb′δcc′(2V −W) + (δbcδb′c′ + δbc′δb′c)(−V + 3W)iσtot= σtot303V + W4V −2W2V + 4W=σ(1)σ(2)σ(3)(22)where we have introduced the abbreviation:V = |Fi(k1)|2|Fj(k2)|2,W = |Fi(k1)Fi(k2)|2.

(23)The upper line in eq. (22) (case 1) refers to the π+π−, π−π+, π+π+ or π−π−pro-duction, the second line (case 2) refers to four other combinations, π+π0, π−π0,π0π+, π0π−, while the last line (case 3) corresponds to the π0π0 production.7

It should be noted that, if the two observed pions are not identical, they aredistinguished by the momenta k1, k2. The three possibilities in eq.

(22) reflectthree possible isospin states (T = 0, 1, 2) which can be formed by a pair ofpions. Since, however, the three cross sections are expressed through only twofunctions, we get a relation:σ(2) + σ(3) = 2σ(1).

(24)Let us now investigate eq. (22).

If one sums up all 9 possible combinationsof pion pairs, one getsdσall(dk1)(dk2) = |Fi(k1)|2|Fj(k2)|2σtot(25)which is independent of the angle between the two pions. Further on, inte-grating eq.

(25) over the momenta k1,2 and recalling eq. (19) for the averagemultiplicity, one finds⟨N(N −1)⟩=1σtotZ(dk1)(dk2)dσall(dk1)(dk2) = ⟨N⟩2,(26)which is the dispersion law of the Poisson distribution.

It should be stressedthough that for charged or neutral pions separately there is no Poisson distru-bution! Also, we would expect a deviation from the Poisson distribution atthe end point of the spectrum where the 4-momentum conservation law from amore accurate eq.

(16) imposes additional correlations.The structure denoted as W depends on the azimuthal angle between thetwo pions,W =hcos(φ1 −φ2)k⊥1 k⊥2 F⊥(k1)F⊥(k2) + k∥1k∥2F∥(k1)F∥(k2)i2 ,(27)whileV =h(k⊥1 )2F 2⊥(k1)) + (k∥1)2F 2∥(k1)i h(k⊥2 )2F 2⊥(k2)) + (k∥2)2F 2∥(k2)i. (28)Out of the three cross sections mentioned in eq.

(22) one can construct angle-dependent and -independent combinations:2σ(3) −σ(2) = 4σ(1) −3σ(2) = σtot3 W(k1, k2),2σ(1) + σ(2) = 2σ(2) + σ(3) = σtot3 V (k1, k2). (29)8

Eqs. (22–29) summarize our result for double-inclusive pion production.

How-ever, it might be useful to make a prediction which is independent of the dy-namics hidden in the Fourier-transformed pion fields F⊥,∥.To this end werestrict ourselves to pions with zero rapidity, i.e. to the case k∥1 = k∥2 = 0, sothat W/V = cos2(φ1 −φ2).

This quantity is obtained by taking the ratio ofdifferential double-inclusive cross sections, say,4σ(1) −3σ(2)2σ(2) + σ(3) = cos2(φ1 −φ2). (30)Another way to isolate the azimuthal angle dependence is to normalize to thesingle-inclusive cross sections.

For example, we predict at k∥1,2 = 0: σtotdσπ+π−(dk1)(dk2) −910dσπ+(dk1)dσπ−(dk2)!/ dσπ+(dk1)dσπ−(dk2)!= 310 cos2(φ1−φ2). (31)At this point an experimentalist may derive correlations for his own favourite(charged or neutral) pairs of pions.

Let us recall finally that the axial symmetrydoes not contradict higher harmonics in the transverse plane, with the replace-ment cos(φ1 −φ2) →cos m(φ1 −φ2) where m is an integer.D.D. would like to thank Maxim Polyakov and Michal Praszalowicz for ahelpful discussion, the Institute for Theoretical Physics-II of the Ruhr Univer-sity at Bochum for hospitality during the completion of this paper and theA.v.Humboldt - Stiftung for a support.References[1] H.Satz, Nucl.

Phys. A544 (1992) 371c[2] A.A. Anselm, Phys.

Lett. 217B (1989) 169; A.A. Anselm and M. Ryskin,Phys.

Lett. 266B (1991) 482[3] J.D.

Bjorken, Acta Physica Polonica B23 (1992) 561[4] J.P. Blaizot and A. Krzywicki, Phys. Rev.

D46 (1992) 246[5] K. Rajagopal and F. Wilczek, “Emergence of Coherent Long WavelengthOscillations After a Quench: Application to QCD” PUPT-1389, IASSNS-HEP-93/16 , March 93[6] A.P. Balachandran, G. Marmo, B.S.

Skagerstam and A. Stern, “ClassicalTopology and Quantum States”, World Scientific (1991)9

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