A critical point of second order, belonging to the universality class of the
3d Ising model, has recently been advocated as a strong candidate for the
critical behaviour (at high temperatures) of QCD with non-zero quark masses.
The implications of this conjecture are investigated in the multiparticle
environment of high-energy collisions. A universal intermittency pattern of
pion-density fluctuations is found, at the critical point, and its association
to the critical exponents is discussed. A Monte Carlo simulation of critical
events, in heavy-ion collisions, reveals the detailed structure of these
fluctuations, suggesting a framework of (event-by-event) measurements in which
the critical theory of QCD may become falsifiable.Comment: 8 pages, 3 figures (ps

Antoniou, N. G.

Contoyiannis, Y. F.

Diakonos, F. K.

English

arXiv

A critical point of second order, belonging to the universality class of the 3d Ising model, has recently been advocated as a strong candidate for the critical behaviour (at high temperatures) of QCD with non-zero quark masses. The implications of this conjecture are investigated in the multiparticle environment of high-energy collisions. A universal intermittency pattern of pion-density fluctuations is found, at the critical point, and its association to the critical exponents is discussed. A Monte Carlo simulation of critical events, in heavy-ion collisions, reveals the detailed structure of these fluctuations, suggesting a framework of (event-by-event) measurements in which the critical theory of QCD may become falsifiable.A critical point of second order, belonging to the universality class of the 3d Ising model, has recently been advocated as a strong candidate for the critical behaviour (at high temperatures) of QCD with non-zero quark masses. The implications of this conjecture are investigated in the multiparticle environment of high-energy collisions. A universal intermittency pattern of pion-density fluctuations is found, at the critical point, and its association to the critical exponents is discussed. A Monte Carlo simulation of critical events, in heavy-ion collisions, reveals the detailed structure of these fluctuations, suggesting a framework of (event-by-event) measurements in which the critical theory of QCD may become falsifiable

Antoniou, N.G.

Contoyiannis, Y.F.

Diakonos, F.K.

CERN Document Server

CERN-TH/99-44hep-ph/9902455Pion Fluctuations near the QCD Critical PointN.G. Antonioua,b 1, Y.F. Contoyiannisb and F.K. Diakonosba Theory Division, CERN, CH-1211 Geneva 23, Switzerlandb Department of Physics, University of Athens, GR-15771, Athens, GreeceAbstractA critical point of second order, belonging to the universality classof the 3d Ising model, has recently been advocated as a strong can-didate for the critical behaviour (at high temperatures) of QCD withnon-zero quark masses. The implications of this conjecture are inves-tigated in the multiparticle environment of high-energy collisions. Auniversal intermittency pattern of pion-density fluctuations is found,at the critical point, and its association to the critical exponents isdiscussed. A Monte Carlo simulation of critical events, in heavy-ioncollisions, reveals the detailed structure of these fluctuations, suggest-ing a framework of (event-by-event) measurements in which the criticaltheory of QCD may become falsiable.1e-mail address: nantonio@atlas.uoa.grCERN-TH/99-44February 1999The nature of the QCD critical behaviour, associated with chiral phasetransition, has gradually been better understood, during the last few years,in terms of the number of flavours and the primordial mass spectrum ofthe light quarks. In an approximate world of two flavours (u, d) with zeroquark masses (mu  md  0), the high-temperature QCD phase transition(Tc  150MeV) is of second order [1{3]; it belongs to the O(4) universalityclass of a 3d Heisenberg magnet [3] and the order parameter is associatedwith pion and sigma eld condensates φ = (~pi, σ). In a more realistic worldwith three flavours (u, d, s) and a nite mass (ms) of the strange quark, thesystem develops a tricritical point (ms, Tc) at which the transition changesfrom rst (ms < ms) to second order (ms > ms) [2, 3]. In the real world,however, with non-zero quark masses (mu, md 6= 0) the second-order branchdisappears from the phase diagram: the corresponding transition becomes asmooth crossover [4] and the tricritical point is transformed into an endpointof the rst-order phase transition line (critical point of second order). Ithas also been argued [5, 6] that the behaviour of the system near the QCDcritical point (Tc, µc) belongs to the universality class of the Ising model in3d (gas{liquid transition); in this case, the order parameter is associatedwith the σ-eld condensate alone, σ  h ψψi. The above over-all pictureof critical QCD has been advocated recently by a number of authors [3{8] and despite the fact that one cannot (yet) verify these ideas completelyin a detailed ab initio study on the lattice, owing to the involvement ofa non-zero chemical potential µc at the critical point, this picture remainsattractive, as it is based on rather general and universal properties of stronginteractions, near criticality. In this Letter we claim that it can also befalsiable, because it may lead to quantitative and sharp predictions forpion-density fluctuations, near the critical point. Such critical fluctuationscan, in principle, be measured in relativistic heavy-ion collisions, by selectingrare events with strong and universal intermittency pattern [9] in momentumspace.To this end, consider the 3d eective action Γc[σ], which eectively de-scribes the system of QCD at the critical point (T = Tc, µ = µc) :Γc[σ] = T−1c∫d3~x[12(rσ)2 +GT 4c (T−1c σ)δ+1]. (1)In eq.(1) the macroscopic eld σ has a dimension σ  (length)−1, δ is theisothermal critical exponent, and G, a dimensionless coupling, species the1critical equation of state: δΓcδφ Gφδ. Both parameters (G, δ) are universal; inthe 3d Ising class one may x the exponent at the mean eld value δ  5, be-cause of the smallness of the anomalous dimension [10]. The coupling G, forthe same universality class, has been estimated recently, in non-perturbativestudies of the critical equation of state, and its actual value is xed, to a goodapproximation, in the range G  1.5{2.5 [10]. Hence the only free parameterin the eective theory (1) is essentially the critical temperature Tc, whichdetermines the length scale (in heavy-ion physics, βc = T−1c  1{2 fm).In order to test the theory (1) in the environment of a high-energy colli-sion, we have to rst adapt the longitudinal geometry to the rapidity space(ξ) by introducing the proper time-scale (τ) of the collision (A + A):Γc[σ] =1CA∫d2~x?dξ12(∂σ∂ξ)2+τ 22(r?σ)2 +GC2Aβ4cσ6 , (2)where CA =τβc. The eective eld σ(ξ, ~x?), being macroscopic (classi-cal), is naturally associated with the density of σ-particles in the 3d space(ξ, ~x?): σ2 = Nσ(volume)−1  (length)−2. Technically, this interpretationfollows from the requirement that the density matrix, associated with thepartition function Z =∫[Dσ]e−Γc of the critical theory (2), be diagonal inthe coherent-state representation of the particles associated with the σ-eld[11]. As a consequence, one nds for the average multiplicity hNσi the form:hNσi = Z−1 ∫ [Dσ]e−Γc(∫ d2~x?dξ σ2(ξ, ~x?)), suggesting that the appropriateorder parameter for the study of the density fluctuations of the isoscalarparticles at the critical point, is σ2(ξ, ~x?) [12].In the cylindrical geometry of the collision, the theory (2) can be solvedby projecting out the eective action onto rapidity and transverse space.Following the results in ref. [13], the critical fluctuations of the order pa-rameter in these two spaces (1d and 2d) are developed within clusters of sizeδc =pi1/2R?2τand r?c = pi3/2R?32respectively (R? is the transverse radius of theentire system at the critical point). These fluctuations are properly described(with the expected fractal dimension) by the contribution to the partitionfunction of instanton-like congurations, which, within these clusters, areslowly varying functions and contribute equally to the eective potential andto the derivative terms in eq. (2) [13]. In this framework, the canonicalpartition function, at T = Tc, for scalar particles (σ) in a cylindrical volume2V  piδcr2?c, is written:Z(Nσ, V, Tc) = N−1/2σ exp[−(VoV)2N3σ], (3)where Vo = β2cp2GCA.The average multiplicity hNσi of scalars, in each 3d cluster, as a functionof the volume V , now follows from (3):hNσi = Γ(1/2)Γ(1/6)(VVo)2/3 ; V  Vo. (4)Equation (4) shows that the critical clusters are local, cylindrical fractalswith a minimal volume-scale V = Vo where self-similarity breaks down. Thefractal dimensions in rapidity and transverse space are dF,1 =23and dF,2 =43,reflecting the actual value of the isothermal critical exponent δ  5 andalso the choice of the order parameter σ2 (dF,1 =δ−1δ+1, dF,2 =2(δ−1)δ+1). Thecorresponding density-density correlation function obeys for each cluster acharacteristic power-law:hρ(ξ, ~x?) ρ(0, 0)i =(32piGCA)1/3β−2cΓ(1/2)Γ(1/6)jξj−1/3j~x?βcj−2/3. (5)Since the size of the critical clusters species also the correlation length ofthe nite system [13], both in rapidity (δc =ppiR?2τ(G2)−1/2) and transversespace, (r?,c = pi3/232(G2)−3/4R?) we may safely assume that in the global volumeVg = piR2? ( is the total rapidity available in the experiment) the criticalclusters interact weakly, so that they form an ideal classical gas with a Poissonmultiplicity distribution [14]. The average multiplicity of clusters, in thisapproximation, is hNcli (R?r?,c)2 (δc)and, using eq. (4), an upper boundof the excess multiplicity of pions produced when the system passes throughthe critical point, can be extracted (Npi  2Nσ) Npi  16ppi(R?βc) (G2)7/12.On the other hand, the local fractals, associated with the critical clustersnally build-up a global fractal with the same dimension [15]. In fact, inthe transverse momentum space, the fractal ~F2 is even stronger, since thecorresponding Fourier transform of the power-law (5) gives ~dF,2 =23( ~dF,2 =2 − dF,2). On the basis of these structures, the theory predicts a universal3pattern of strong, multidimensional intermittency (in rapidity, transversemomentum and 3d momentum space) of the form: F (1)p  Mp−13 , F (2)p (M2)2(p−1)3 , F (3)p  (M3)5(p−1)9 (p = 2, 3, .) where Fp is a scaled factorialmoment of order p [9] as a function of M , the number of one-dimensional cellsin momentum space. Other power-laws coming from the 1d projections ofthe fractal ~F2 or from the Cartesian product of two fractals in the cylindricalgeometry of the system, can also be written [12]. The predicted intermittencypower-laws describe the density fluctuations of scalars produced at the criticalpoint quantitatively, relating these fluctuations to the critical exponents ofthe 3d universality class.We have performed a Monte Carlo simulation of the critical events inPb+Pb collisions in the SPS (  6), corresponding to the choice of theparameters: Tc = 120 MeV,R? = 30 fm, τc = 25 fm. We have also xedthe coupling G at the value G = 2 [10]. In momentum space, the isoscalarparticles (σ) are grouped in critical clusters (cylindrical) with appropriatepower-law correlations, dictated by the Fourier transform of eq. (5) in trans-verse space: h~ρ(ξ, ~p?)~ρ(0, 0)i  jξj−1/3jβc~p?j−4/3. The distribution of theclusters in rapidity (projected) corresponds to an ideal 1d gas, whereas intransverse momentum space the following extra constraints are imposed: (a)the clusters (their centers) are exponentially distributed according to thetemperature Tc, dnc  exp(−p?,cTc ) and (b) the total transverse momentumcarried by the clusters is zero,∑i ~p(i)?,c = 0. Moreover the size of the criti-cal clusters in transverse momentum space, as determined by the minimumlength scale in transverse conguration space, is given by the expressionj~p?j  Tc2(R?τc)1/2 (G2)−3/8 (pi3cA)1/4and for the above set of parameters weestimate j~p?j  70 MeV. With these ingredients and the overall constraintson the formation of sigmas at T = Tc, a typical, simulated event is obtainedwith characteristics shown in Figs.1{3: the overall picture is consistent witha 3d cylindrical fractal in momentum space (Fig. 1), which is decomposedinto a 2d self-similar structure in transverse momentum space (Fig. 2) and a1d self-similar structure in rapidity space (Fig. 3). The multidimensional in-termittency pattern clearly reveals a linear spectrum of indices in each case,consistent with the critical exponents of the theory [13].In the process of a phase transition in thermal equilibrium, the fluctuatingsystem is very likely to get frozen immediately after crossing the critical4point of second order (µc, Tc) [7]; therefore, the fluctuations predicted atT = Tc may enter the horizon of observation (in the corresponding events)without any distortion (Tf  Tc). It remains to guarantee that the decay ofσ may proceed near the pipi threshold, rather rapidly transferring the criticalfluctuations found at T = Tc automatically to the pi− spectra (σ ! pi+pi−)(Figs. 1{3). The isoscalar σ has a very small mass at T = Tf ; therefore,the system of sigmas remains frozen and stable until the mass mσ, as afunction of temperature, rises to the pipi threshold value, mσ = 2mpi. If thecorresponding time scale τth is suciently long (in particular events), theexpansion rate j _TTj  13τth, for τ  τth, becomes very small with respect tothe decay rate of σ (even near the pipi threshold), also because the isoscalaris very strongly coupled to the pipi system. Under these circumstances, thesigmas withmσ> 2mpi have enough time to decay, before their mass increasesfurther, leading to charged pions (pi−) with the same fluctuation pattern inmomentum space. We have veried numerically this decay mechanism, nearthe pipi threshold, in the case of the typical event shown in Figs. 1{3. Wehave used for the parameters of the σ-meson in vacuum (T = 0) the averagevalues m(0)σ  Γ(0)σ  0.8 GeV of the fo(400− −1200) pipi resonance [16]. Itturns out that, if τth  10 fm, the decay σ ! pipi proceeds near pipi thresholdand therefore the intermittency pattern found at T = Tc for sigmas (Figs.1{3) is also a prediction for the density fluctuations of the abundant chargedpions (pi−) associated with the critical point.In the framework of heavy-ion physics, we have derived a pattern of lo-cal dynamical fluctuations of the classical σ-eld associated with the criticalpoint in QCD with non-zero quark masses. This critical point belongs tothe 3d Ising universality class, but the same pattern is valid in any univer-sality class associated with a 3d scalar theory. The reason is that, in thesetheories, the anomalous dimension η is very small [10], leading to the same(mean eld) value of the exponent: δ  5. Hence a similar pattern of fluctu-ations is expected, for instance, in the O(4) 3d theory [14], which describesthe chiral QCD phase transition in the limit of zero quark masses [3]. Atthe level of observed pions, however, these theories are expected to lead todierent fluctuation patterns since the in-medium evolution (and decay) ofscalars varies from one universality class to the other. In the case of realQCD with non-zero quark masses, we have argued that in a particular classof events, in which the in-medium evolution of σ is rather slow, τth  τc,5the fluctuation pattern found at T = Tc (Figs. 1{3) is inherited by thecharged decay products of σ near the pipi threshold (pi−) and manifests itselfas a clear signature of the quark-hadron phase transition in nuclear collisions.We thank A.L.S. Angelis and C.G. Papadopoulos for their comments andsuggestions on the present work.References[1] R. Pisarski and F. Wilczek, Phys. Rev. D29, 338 (1984).[2] Columbia Group, Phys. Rev. Lett. 65, 2491 (1990).[3] F. Wilczek, Int. J. Mod. Phys. A7, 3911 (1992);K. Rajagopal and F. Wilczek, Nucl. Phys. B399, 395 (1993).[4] J. Berges, D.-V. Jungnickel and C. Wetterich, hep-ph/9705474.[5] J. Berges and K. Rajagopal, hep-ph/9804233.[6] M. A. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov andJ. J. M. Verbaarschot, Phys. Rev. D58, 096007 (1998).[7] M. A. Stephanov, K. Rajagopal and E. Shuryak, Phys. Rev. Lett. 81,4816 (1998).[8] F. Wilczek, Nature 395, 220-221 (1998).[9] A. Bialas and R. Peschanski, Nucl. Phys. B273, 703 (1986); B308, 857(1988).[10] M. M. Tsypin, Phys. Rev. Lett. 73, 2015 (1994); J. Berges, N. Tetradisand C. Wetterich, Phys. Rev. Lett. 77, 873 (1996).[11] J. C. Botke, D. J. Scalapino and R. L. Sugar, Phys. Rev. D9, 813 (1974).[12] N. G. Antoniou, C. N. Ktorides, I. S. Mistakidis and F. K. Diakonos,Eur. Phys. J. C4, 513 (1998).6[13] N. G. Antoniou, Y. F. Contoyiannis, F. K. Diakonos and C. G. Pa-padopoulos, Phys. Rev. Lett. 81, 4289 (1998).[14] N. G. Antoniou, F. K. Diakonos, C. N. Ktorides and M. Lahanas, Phys.Lett. B432, 8 (1998).[15] A. Lesne, “Renormalization Methods: Critical Phenomena, Chaos, Frac-tal Structures”, (John Wiley and Sons Ltd., Chichester, 1998).[16] Meson Particle Listings, Eur. Phys. J. C3, 363 (1998).Figure CaptionsFigure 1: (a) The distribution of the σ-particles of a typical critical eventin 3d momentum space. The multiplicity of σ’s is 138. (b) The corresponding(3d) factorial moments of order p = 2, 3, 4 in a log-log plot. A linear t(dashed line) indicating the intermittency pattern for this event is included.Figure 2: (a) The distribution of the σ-particles in transverse-momentumspace for the event of Fig. 1. (b) The corresponding (2d) factorial momentsof order p = 2, 3, 4 in a log-log plot. A linear t (dashed line) indicating theresulting intermittency pattern is also shown.Figure 3: (a) The distribution of the σ-particles in rapidity for the eventof Fig. 1. (b) The rapidity factorial moments of order p = 2, 3, 4 in a log-log plot together with a linear t (dashed line) indicating the correspondingintermittency pattern.7-3-2-1 0 1 2 3 -4000400-4000400(a)p T,y (MeV)p T,x (MeV)ξ0,5 1,0 1,5 2,0 2,5024681012(b)Figure 1ln(M)ln(Fp(M)) p=2, slope=1.53 p=3, slope=3.51 p=4, slope=5.53-400 -200 0 200 400-800-600-400-2000200400600Figure 2(a)p T,y (MeV)pT,x (MeV)0,5 1,0 1,5 2,0 2,50246810(b)ln(M)ln(Fp(M)) p=2, slope=1.28 p=3, slope=2.9 p=4, slope=4.55-3 -2 -1 0 1 2 3051015(a)ξn( ξ)0 1 2 3 4 5 60246Figure 3(b)ln(Fp(M))ln(M) p=2, slope=0.3 p=3, slope=0.7 p=4, slope=1.2

Pion Fluctuations near the QCD Critical Point

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