aa r X i v : . [ h e p - ph ] N ov Phase diagram in Quantum Chromodynamics
M. ApostolDepartment of Theoretical Physics, Institute of Atomic Physics,Magurele-Bucharest MG-6, POBox MG-35, Romaniaemail: [email protected]
Abstract
It is suggested that the hadronization of the quark-gluon plasmais a first-order phase transition described by a critical curve in thetemperature-(quark) density plane which terminates in a critical point.Such a critical curve is derived from the van der Waals equation andits parameters are estimated by using the theoretical approach givenin M. Apostol, Roum. Reps. Phys.
249 (2007); Mod. Phys.Lett.
B21
893 (2007). The main assumption is that quark-gluonplasma created by high-energy nucleus-nucleus collisions is a gas ofultrarelativistic quarks in equilibrium with gluons (vanishing chemicalpotential, indefinite number of quarks). This plasma expands, getscool and dilute and hadronizes at a certain transition temperatureand transition density. The transition density is very close to thesaturation density of the nuclear matter and, it is suggested that boththese points are very close to the critical point n ≃ f m − (quarkdensity) and T ≃ M eV (temperature).
PACS: 12.38.Mh; 25.75.Nq; 21.65.Qr
Keywords : quark-gluon plasma; phase diagram; hadronization; van der WaalsequationAs it is well known, the Quantum Chromodynamics (QCD) developed inthe past 50 years describes the quark-gluon strong interaction. The QCDlagragian is L = − G aµν G µνa + ψ fα (cid:2) iγ µ ( ∂ µ δ αβ + igt αβa A aµ ) − m f δ αβ (cid:3) ψ fβ , (1)where G aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν (2)1nd [ t a , t b ] = if cab t c . (3) A aµ are gluon potentials, with µ the Lorentz label and a = 1 ... the gluonlabel; G aµν are the gluon fields, g is the coupling constant and f abc is thestructure factor of the SU(3) group. The eight × matrices t a ( α, β = 1 , , )are the SU(3) generators. ψ fα are the quark fields (bispinors), labelled byflavour f = 1 ... and color α = 1 , , , with mass m f ; γ µ are the Diracmatrices. (For a review of QCD the reader can consult the recent Refs.[1]-[3]).The lagrangian given by equation (1) is constructed by close analogy withthe Quantum Electrodynamics (QED), with two major differences: the Yang-Mills fields (quadratic term in the gluon fields in equation (2)) and the un-derlying SU(3) symmetry (color group). Perturbation-theory calculationsindicate that effective (renormalized) coupling strength becomes weak forhigh-energy processes (and short distances), a phenomenon known as quarkde-confinement; the quarks and gluons are free at high energy and get con-fined at low energy; for instance, they are bound in hadrons (mesons andbaryons), which are color-singlet states (zero color charge).[4, 5] The confine-ment phenomenon, which is opposite to QED, is due to the non-linear Yang-Mills contribution. The usual type of calculations in QCD is the lattice-gaugetheory calculations,[6] the results being often of a more qualitative nature.Various simplifications are customary in calculations, in particular the limi-tation to only (the lightest) u and d quarks, whose mass is set equal to zero.In this case, the chiral symmetry (handedness) of the theory should be bro-ken at low energies. Broken symmetries, associated phase transitions and,in general, methods borrowed from condensed matter physics, are employedwith the hope of getting more quantitative results in QCD.[2]In nucleus-nucleus collisions high energy can be transferred to the internalstructure of the nucleons ( e.g. , T eV per nucleon as compared with the nu-cleon binding energy GeV ), such that we may expect the liberation of quarksand gluons for a short while, followed by a quick hadronization.[7]-[11] An ul-trarelativistic gas of quarks can be formed in such collisions, reaching quicklythe thermal equilibrium at a temperature produced by the collision energy;we may speak of a quark-gluon plasma, with a threshold (ignition) tempera-ture (of cca − M eV ), and a hadron-quark-gluon plasma transition.[12]-[14] In a high-energy collision the plasma expands, its volume and numberof quarks and gluons (at local equilibrium) increase, the quark density andtemperature decrease, and the quarks in the outer shell hadronize.[15] Itis tempting to assign a second kind to such a transition, corresponding tothe broken chiral symmetry with zero-mass quarks, but the real situation2nvolves non-vanishing masses. We may only assume that the hadron-quark-gluon plasma is first order, involving a hadron binding energy, very similarwith the van der Waals liquid (solid)-gas transition.[16, 17] As it is wellknown, the van der Waals isotherms are given by ( p + an )(1 − bn ) = nT , (4)where p is the presssure, n is the density, T is the temperature and a, b areconstants. Since the van der Waals characteristic pressure is ∼ an , we mayalso assume p = cn at transition, where c is a constant. The van de Wallsisotherms become ( a + c ) n (1 − bn ) = T , (5)or T = − Bn + An , (6)where A and B > are constants . We can see that equation (6) is equivalentwith the van der Waals equation (4) for zero pressure. Since ∂T /∂n < for a physical transition, we can see that we should consider the aboveequation from n = A/ B up to n = A/B ( T > ), so we should have A > (descending branch of the second-order trinomial in equation (6)).Under these conditions the above equation (6) gives the curve correspondingto the hadron-quark-gluon plasma transition in the ( n, T ) plane, the point n c = A/ B , T c = A / B being the critical point. Equation (6) can alsobe written as T = An [1 − ( B/A ) n ] , where we can see that the ratio B/A is a limiting volume, which may be viewed as corresponding to a nominal"volume" v n of the quarks, B/A = v n ; the quark density n can be writtenas n = N/V = 1 /v q , where v q is the mean volume assigned to a quark in thevolume V occupied by N quarks. For a mixture of various quark species, thedensity n can be generalized to the mean density involving partial densities.Now we describe briefly the theoretical approach given in Ref. [15], becauseit gives us access to the parameters A and B in equation (6). We consider anucleus with N n nucleons in a volume V = R , where R is, approximately,the radius of the nucleus (for simplicity, we leave aside the numerical factor π/ ); the nucleus is subjected to a high-energy collision, with an energy(per nucleon) E/N n = 1 T eV , for instance. We limit ourselves to the lightestquarks u and d, for which we may neglect their mass at these energy values( m u ≃ M eV , m d ≃ M eV ). In general, the energy is dominated by gluons,except for assuming an ultrarelativistic gas of an indefinite number of quarks(vanishing chemical potential) in equilibrium with gluons, i.e. a quark-gluonplasma. In this case, the plasma energy (quarks plus gluons, almost equalenergy) is given by E = V T / ( ~ c ) (7)3nd the mean number of quarks is N = V T / ( ~ c ) (8)(up to some immaterial numerical factors). (The pressure is p = E/ , theentropy is S ≃ E/ T ≃ N and the density is given by T = ~ cn / ; ~ isPlanck’s constant ad c is the speed of light). At the initial moment we have E = V T / ( ~ c ) and N = V T / ( ~ c ) ; for an energy E/N n = 1 T eV weget T = 1 GeV and N = 10 N n ( N n ≃ ), assuming a nucleon radius a = 2 f m and R = aN / n ( ~ c = 200 M eV · f m ). This plasma expands intime according to the laws R = R (1 + ct/R ) , V = V (1 + ct/R ) ,T = T (1 + ct/R ) − / , N = N (1 + ct/R ) / ; (9)its density goes like n = N/V = n (1 + ct/R ) − / = n ( T /T ) (10)or, using N = V T / ( ~ c ) ( i.e. n = ( T / ~ c ) ), T = ~ cn / ; (11)if we put here the quark density in the cold nucleus ( n ≃ N n /V , or n ≃ N n /V ) we get the threshold (ignition) temperature − M eV (for a = 2 f m ; the values − M eV given above are obtained for a = 1 . f m ).We can see that, during expansion, plasma gets cool and the quark densitydecreases according to equations (9); at the same time, the energy is con-served and the entropy increases. Equation (11) defines also the chemicalpotential of a degenerate ultrarelativistic gas of quarks ( µ = ~ cn / ).Further on, a mechanism of condensation (hadronization) has been put for-ward in Refs. [15, 22] (a first-order phase transition). The transition tem-perature is given by T t ≃ T q ( T q /T m ) / , (12)where T q is a characteristic quark temperature and T m ≃ m c is a character-istic temperature given by the average mass m of the condensed quarks (upto some immaterial numerical factors). It is shown in Ref. [15] that only afraction f of the quark number is affected by hadronization (and dominates Compare with the hydrodynamical model of particle production, Refs. [18]-[21]. T c T t n c ≃ n t ≃ ≃ n = 125 T ( M eV ) O Ot h qg ¯ hcn / n ( f m − ) Figure 1: Hadronization of the quark-gluon plasma. Phase diagram temper-ature ( T ) vs quark density ( n ). Note the hadronization curve T = ~ cn / .the hadronization process, with a classical Boltzmann statistics), so that wehave in fact T t ≃ f / T q ( T q /T m ) / . (13)At transition T t = T q and T t = f − T m , ~ cn / t ≃ f − m c , (14)an expected and plausible result. We can see that the transition density n t = f − (cid:16) m c ~ (cid:17) (15)is related to the Compton wavelength ~ /m c of the "average" condensedquarks. We can take m = 4 M eV (mean mass of the u and d quarks),and get n t = f − (50 f m ) − . In Ref. [15] it is suggested that fraction f isgiven approximately by f = 1 /N / ≃ × − (an argument derived from thesaturation of the nuclear forces), so we get the transition density n t ≃ f m − .It corresponds to a transition temperature T t = f − T m ≃ M eV (slightlyabove the ignition threshold) and a transition radius R t = R ( n t /n ) − / = R ( T /T t ) / ≃ R . This hadronization happens after t = 5 × − s fromthe collision and involves f N t = f N ( R t /R ) / ≃ N n hadronized quarks.(The transition implies a latent heat, discontinuities of the thermodynamicpotentials, etc, as for a first-order (van der Waals) transition). We can seethat the transition density n t = 1 f m − is very close to the saturation densityof the nuclear matter. 5he transition temperature and density must obey the critical curve give byequation (6). With temperature measured in M eV and density measured in f m − we get
200 = A − B , (16)which gives a relation between the two parameters A and B of the criticalcurve. Since the transition density is very close to the saturation densityof the nuclear matter, we may take tentatively the critical density n c = A/ B = n t = 1 f m − ; we get A = 400 and B = 200 (and the criticaltemperature T c = A / B = T t = 200 M eV ); the density where the criticalcurve crosses the n -axis is A/B = 2 f m − . It is worth noting that the criticalpoint n c = 1 f m − , T c = 200 M eV derived here can have a universal character;indeed, it lies on the curve T = ~ cn / for n equal to the saturation densityof the nuclear matter ( e.g. , n = 1 f m − ).The hadronization of the quark-gluon plasma is shown in Fig. 1. Accord-ing to the description given above the hadronization process starts withthe creation of a quark-gluon plasma at the initial quark density n (e.g., ≃ N n /V = 125 f m − ) and temperature T ( e.g. , GeV ), followed by acooling of the plasma along the curve T = ~ cn / untill it encounters thecritical curve at the transition point n t ( e.g. , f m − ) and T t ( e.g. , M eV )where the hadronization occurs. With our units (
M eV and f m ) the curve T = ~ cn / reads T = 200 n / . The transition temperature is very close tothe saturation density of the nuclear matter and, very likely, it is very closeto the critical density.[23] Acknowledgments.
The author is indebted to MPD-NICA JINR Dubna(Russia) Project, especially to A. G. Litvinenko and I. Cruceru, for manyuseful discussions, and to the members of the Seminar of the Laboratory ofHigh Energy Physics, JINR Dubna, for a thorough analysis of this work.
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