The quark–gluon–pion plasma

Physics Letters B 552 (2003) 165–171

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The quark–gluon–pion plasma

Vikram Sonia, Moninder S. Modgilb, Deshdeep Sahdevb

a National Physical Laboratory, New Delhi, Indiab Department of Physics, Indian Institute of Technology, Kanpur 208016, India

Received 21 February 2002; received in revised form 5 November 2002; accepted 2 December 2002

Editor: J.-P. Blaizot

Abstract

While it is commonly believed that there is adirect transition from the hadronic to a quark–gluon phase at high temperature,it would be prejudicial to rule out a sequence of dynamically generated intermediate scales. Using as guide, an effectiveLagrangian with unconfined gluons and constituent quarks, interacting with a chiral multiplet, we examine a scenario in whichthe system undergoes first-order transitions atTcomp, the compositeness scale of the pions, atTχ , the scale for spontaneouschiral symmetry breaking, and atTc, the confinement temperature. We find that at current energies, it is likely that the formationtemperature of the plasma,T0 < Tcomp, and that this is, therefore, a quark–gluon–pion plasma (QGPP) rather than the usualquark–gluon plasma (QGP). We propose some dilepton-related signatures of this scenario. 2002 Elsevier Science B.V. All rights reserved.

We know that quarks and gluons are confined ashadrons and chiral symmetry is spontaneously broken.We do not, however, know if these two phenomena setin at an identical temperature. It is unlikely that thechiral symmetry restoration energy/temperature scaleis lower than that of confinement, because if it were,hadrons would show parity doubling below the con-finement but above the chiral breaking scale. This isnot seen either experimentally or in finite temperaturelattice simulations. Determining whether it ishighereven in the simplest case of QCD with a two flavourSU2(L) × SU2(R) chiral symmetry is not straight-forward. Indeed, while there is a bonafide order pa-

E-mail addresses: [email protected] (V. Soni),[email protected] (M.S. Modgil), [email protected] (D. Sahdev).

rameter for chiral symmetry breaking—namely, themass of the constituent quark—the Wilson loop inthe presence of dynamical quarks, isnot a valid or-der parameter for confinement—and there are no otherknown candidates. However, by looking at energy den-sity or specific heat, we can get a fair idea of thechange in the number of operational degrees of free-dom or particle modes with temperature. The rele-vant lattice calculations indicate that the drop from thelarge number of degrees of freedom in the QGP/QGPPphase to a few degrees of freedom in the hadronic onetakes place in one broad step in temperature, suggest-ing that the two transitions are close but not necessar-ily identical, i.e., thatTχ � Tc .

In the same spirit, we could ask whether pionsnecessarily come apart the moment chiral symmetry isrestored, or equivalently, whether the compositeness

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166 V. Soni et al. / Physics Letters B 552 (2003) 165–171

scale for the pion,Tcomp, coincides withTχ . Thefollowing analogy with superconductivity (SC) leadsus to conclude that it may not:

In the usual BCS theory, the formation of Cooperpairs atTcrit coincides with the appearance of a SCorder parameter. This corresponds toTcomp = Tcrit,as there are no Cooper pairs at scales aboveTcrit.Here, Tcomp is the compositeness scale for Cooperpairs, which are normally described as undergoingmomentum space pairing, the interelectron distancebeing much less than the size of a Cooper pair. Thisis indeed what happens for a weak pairing interaction.For a much stronger interaction, Cooper pairs mayactually form by real space pairing. The Cooper pairsize is then smaller than the interelectron distance andit is possible that, even when SC is lost, i.e.,T > Tcrit,pairs continue to exist simply as bound states.

Evidence for the related precursor phenomena forthe chiral transition had been suggested long ago[14]. Recently, there have been several calculations ofthe QCD counterpart of the non-condensed, paired,pseudogap phase of highTc superconductivity [15],which suggest that the strong pairing situation is likelyfor the strong interactions.

We shall accordingly takeTcomp> Tχ , and arguethat heavy-ion collisions are the ideal platform fortesting this assumption. In doing so, we shall use thefollowing SU2(L) × SU2(R) effective Lagrangian, inwhich the quarks are coupled to a chiral multiplet,[π,σ ] [2–4], to keep track of particle modes at variousscales:

L = −1

4Ga

µvGaµv −

∑ψ̄

(/D + gy

(σ + iγ5�τ �π))

ψ

− 1

2(∂µσ)2 − 1

2

(∂µ �π)2 − 1

2µ2(σ 2 + �π2)

(1)− λ

4

(σ 2 + �π2)2 + const.

This has chiral symmetry-breaking but no con-finement. The masses of the scalar/pseudoscalars andfermions, obtained by minimizing the potentials above,areµ2 = −λ〈σ 〉2 andm2

σ = 2λ〈σ 〉2, respectively. Thistheory reproduces several broad features of the stronginteraction at the mean field level [5,6], indicating thatit is, perhaps, a reasonable guide to the physics at in-

termediate scales. More specifically,

1. It provides a nucleon, which is realized as asoliton with quarks bound in a skyrmion configuration[3,5]. Such a nucleon:

(a) Gives a natural explanation for the ‘proton spinpuzzle’: quarks in the background fields are ina spin, ‘isospin’ singlet state in which the quarkspin operator averages to zero. If we collectivelyquantize the soliton to get states of good spin andisospin, the quark spin operator picks up a small,non-zero expectation value [7].

(b) Seems to naturally produce the Gottfried sum rule[8].

(c) Yields, from first principles (albeit with somedrastic QCD evolution), a set of structure func-tions for the nucleon which are close to the ex-perimental ones [9].

2. As a finite temperature field theory, it yieldsscreening masses that match with those obtained fromthe lattice simulation of finite temperature QCD withdynamical quarks [10].

3. It gives a consistent equation of state for stronglyinteracting matter at all densities [5,11].

This Lagrangian and the above discussion on tem-perature scales lead us to consider the following sce-nario:

1. For T > Tcomp, we have gluons and masslesscurrent quarks, in a pure QGP phase.

2. ForTcomp> T > Tχ , we have a QGPP phase forwhich the particle mode count is (i) 16 modes from8 massless gluons, (ii) 24 from 2 flavors of masslesscurrent quarks and antiquarks, and (iii) 3 degeneratepions, which are present, in this temperature range, inthe form of non-Goldstone-boson bound states, whichwe assume to be massless. (We neglect theσ , whichwould otherwise contribute one more mode.)

The operational degrees of freedom in this phaseare then 40—three more than in the QGP phase. Thisincrease may be visible as a slight bump on an ‘energydensity’ vs. ‘temperature’ plot, in a high resolutionlattice simulation.

3. For Tχ > T > Tc, chiral symmetry is sponta-neously broken and the degrees of freedom are (i) glu-

V. Soni et al. / Physics Letters B 552 (2003) 165–171 167

ons, (ii) constituent quarks, which acquire a con-stituent mass of 300–400 MeV from the spontaneousbreaking ofχ -symmetry and are, as a result, Boltz-mann suppressed given thatTχ � Tc is expected [1]to beO(200 MeV), (iii) pions as Goldstone bosons,and (iv) theσ , which also acquires a mass and may beneglected.

The effective degrees of freedom are thus 19, at thisstage.

4. For Tc > T , quarks are confined, and the onlymassless modes left are the 3 Goldstone bosons/pions.

We can get an estimate ofTcomp by examining thebehaviour of the above Lagrangian as a function ofenergy. We find that at a certain scale, both the wavefunction renormalization constant,Z, for theπ andσ ,and the quartic scalar interaction, vanish, leaving uswith a Yukawa and a scalar mass term. The vanishingof Z, or equivalently, of the kinetic term for the scalarand pseudoscalars, means that these are no longerdynamical degrees of freedom, i.e., they have ceasedto exist as composite entities. Eliminating them, viatheir field equation, leaves behind a four-Fermi termwhich gets weaker with increasing energy. While theseresults, gleaned using perturbative RNG, cannot beused all the way out to whereZ goes to zero (sincethe Yukawa coupling has a Landau singularity there),they can perhaps give a reasonable estimate for thecompositeness scale. This works out to be around 700–800 MeV [12], well above the temperatures attained inthe present-day heavy ion colliders. This implies thatthe initial state produced by these is very probably aQGPP.

To investigate this possibility further, we adaptto our scenario, the considerations of [1], whichgives a very transparent and simple treatment ofthe underlying physics. In particular, following [1],we treat each of our transitions as first order, witheach producing a mixed phase in which the numberof degrees of freedom changes continuously, as onephase gives way to the other.

A plasma which thermalizes at a temperature,T0,where Tcomp < T0 < Tχ , a proper time,t0, afternuclear impact, then evolves as follows:

1. It Bjorken expands and cools [13], reachingTχ

at time,tx1 = (T0/Tχ)3t0.

2. At Tχ , it undergoes a first-order transition fromthe quark–gluon–pion to the chirally broken, gluon–pion phase, fromtx1 to r1tx1, wherer1 = 40/19 is theratio of the degrees of freedom in the initial and finalphases.

N.B. If we assume that chiral restoration anddeconfinement occur simultaneously atTχ = Tc, thequark phase lasts in the mixture from timet1 = tx1to t2 = rt1 (wherer = 37/3), which is substantiallylonger.

3. Having passed to the chirally-broken phase, theplasma undergoes a second Bjorken expansion andthereby cools fromTχ to Tc . SinceTχ � Tc, this partof the evolution may, however, be neglected.

4. At Tc, we get a mixture of the gluon–pion andthe purely hadronic confined phases, which lasts fromr1tx1 to tx2 = r2r1tx1, wherer2 is 19/3.

5. Finally, the pion phase expands fromTc to Tf ,the freeze out temperature [1], at which pions loosethermal contact with one another. This phase begins attime tx2 and ends attf = (Tc/Tf )3tx2.

The basic changes in this scenario when comparedto that of [1] are that, (i) the pionic phase startsmuch earlier, at the initial temperature,T0, and notafter confinement atTc and (ii) the quark phase issuppressed aboveTc and is thus shortened.

To see how these changes are reflected in thenumber of dileptons emitted by the plasma, we notethat these come fromπ+π− andqq̄ annihilations. Thecross-section forqq → l+l−, summed over the spin,flavour and color of all quarks is

σq [M] = Fq4π

3

(α

M

)2√

1− 4

(ml

M

)2

(2)×(

1+ 2

(ml

M

)2).

where the numerical factor,Fq = 20/3, if we includejust theu andd quarks in the flavor sum.

The cross-section forπ+π− → l+l− likewise is,

σπ [M] = Fπ [M](4π/3)(α/M)2√

1− 4(ml/M)2

(3)× (1+ 2(ml/M)2)√1− 4(mπ/M)2,


where the pion form factor is given by,

Fπ [M] = m4ρ/

((m2

ρ − M2)2 + m2ρΓ

2ρ

)(4)+ (1/4)m4

ρ′/((

m2ρ′ − M2)2 + m2

ρ′Γ 2ρ′

).

The masses and decay widths ofρ andρ′ resonancesaremρ = 775 MeV,mρ′ = 1.6 GeV,Γρ = 155 MeV,Γρ′ = 260 MeV, respectively.

The rate for producing pairs, per unit space–time, with invariant mass squared,M2, and transverse

energy,ET =√

p2T + M2, wherepT is the momentum

transverse to the beam axis, is (fora = q,π ),

dNa

d4x dM2dET

= σa[M]M2

4(2π)4

(1− 4

m2a

M2

)ET K0(ET /T )

(5)= Aa[M]ET K0(ET /T ).

While using Bjorken’s model, it is expedient to writed4x = d2xT dy t dt , wheret is the proper time andy,the rapidity, of the fluid element. For central collisionsof equal mass nucleid2xT = πR2

A, whereRA is thenuclear radius. We can further perform thet-integralto get:

dNMTq

dy dM2dET

= πR2AAq(M)

(6)

× {3E−5

T T 60 t2

0

(G[ET /T0] − G[ET /Tc]

)+ ET K0[ET /Tc](1/2)(r1 − 1)t2

x1

},

for the quark channel in the multiple transitions(MT) scenario. The first term in braces comes fromintegrating along the cooling curvet = (T0/T )3t0from T0 to Tc and the second, from the coexistenceline atTχ , which we approximate as being equal toTc.The shortening of the quark phase is reflected in thepresence here ofr1 = 40/19, as opposed tor = 37/3for the single transition (ST) atTχ = Tc case.

The corresponding rate forπ+π−-annihilations islikewise,

dNMTπ

dy dM2dET

= πR2AAπ(M)

(7)

×{3E−5

T T 60 t2

0

(G[ET /T0] − G[ET /Tc]

)+ 3E−5

T T 6c t2

x2

(G[ET /Tc] − G[ET /Tf ])

+ ET K0[ET /Tc](t2x2 − t2

x1

)/2

},

where

(8)G[x] = x3(8+ x2)K3[x]andKi [x] are the modified Bessel functions.

In this case, the first term in braces (absent alto-gether for the ST case) results from the cooling ofpions in the QGPP, fromT0 andTc, and the second,from their cooling in the hadronic phase, fromTc toTf . The latter coincides with the ST result iftx2 → t2.The last term comes from integrating over time, thevolume fraction,f (t), of the pion phase, on the co-existence lineTχ � Tc. Since pions are present bothabove and below each of the coexistence lines atTχ

andTc, f (t) = 1 for both these transitions. In the ap-proximation,Tχ = Tc, the integral is then simply,

(9)

tx2∫tx1

t dt = (t2x2 − t2

x1

)/2.

The analogous integral in the ST case is, by contrast,(r − 1)rt21/2.

We note that these differential rates further inte-grate overET to the closed-form expressions:

dNMTq

dy dM2

= πR2AAq(M)T 2

0 t20

(10)

× {6(T0/M)−4(H [M/T0] − H [M/Tc]

)+ (M/Tc)(T0/M)−4K1[M/Tc](r1 − 1)

},

where

(11)H [x] = x2(8+ x2)K0[x] + 4x(4+ x2)K1[x]


and we have translated ratios of times into ratios oftemperatures. Similarly,

dNMTπ

dy dM2

= πR2AAπ(M)T 2

0 t20

(12)

×{6(T0/M)4(H [M/T0] − H [M/Tc]

)+ 6(r1r2)

2(T0/M)4

× (H [M/Tc] − H [M/Tf ])

+ (M/Tc)(T0/Tc)4K1[M/Tc]

× [(r1r2)

2 − 1]}

.

Finally, the differential rates,dNMTtot /dy dM2dET

anddNMTtot /dy dM2 for the total number of pairs, are

obtained by summing the right-hand sides of Eqs. (6)and (7), Eqs. (10) and (12), respectively.

We can now compare our scenario, in quantitativedetail, with that of Ref. [1]. For ease of comparison,

we use the same parameter-values:t0 = 1 fm/c, andπR2

A = 127 fm2.

• In Fig. 1, we plotdNMTtot /dy dM2dET , dNMT

q /

dy dM2dET anddNMTπ /dy dM2dET vs.M, for the

combinations ofT0 and Tc used in Ref. [1] but atvalues ofET , for which (in the first two cases) theST pion peaks, coming from annihilation via theρandρ′ resonances, are largely extinguished. We notethat in the MT scenario, the rate displayed reduces inmagnitude asET increases, but the pion peaks surviveand stay well above theqq̄ contribution. This resultsfrom pions being created when the plasma forms, asopposed to when it hadronizes. It is consistent withthe observation in [1] that peak extinction at largeET

and the initial temperature of the pions are distinctlycorrelated: the smaller this temperature, the smallertheET at which the peaks get extinguished.

• We note that an enhancement in the production oflargeET dileptons, owing to pions being available athigher temperatures, will inevitably be accompanied

(a)

(b)

(c)

Fig. 1. Plots of dNMTtot /dy dM2 dET vs. M for three

(T0, Tc)-combinations. For the values ofET in (a) and (b), the pionpeaks in the single transition rate are essentially non-existent. Alsoplotted are the quark contributions for both the MT (thick line)and ST (thin line) scenarios. These can be distinguished only forT0 = 250 MeV andTc = 240 MeV. The MT quark contribution islower because of the shortening of the quark phase.


Fig. 2. Plots of the ratio of thedNtot/dy dM2 rate vs.T0 for theMT and ST scenarios. For thelower M-values considered in thesegraphs, the pion form factor is appreciable. It seen that the MT rateis distinctly higher than the ST rate.

by an enhancement in the production of largepT

pions, a feature highlighted by Schucraft [16] longago. We can add that pions present at the boundaryof the initial plasma, will, being colourless, haveno problem escaping from the interaction region.The availability of pions at higher temperatures is,incidentally, supported by the observation that theHBT size of the initial pion source is smaller than ifpions formed only after hadronization [17,18].

• We would intuitively expect the difference be-tween the two scenarios to depend (among other fac-tors) on the time spent by the system in the QGPPphase. This would in turn be determined by the ini-tial temperature,T0. In Fig. 2, we accordingly plot theratio of dNtot/dy dM2 for the MT and ST scenariosvs.T0 over the rangeTχ � T0 � Tcomp, for several val-ues ofM. We find that the MT rate is 20–30% higher,whenM is not too far from theρ, ρ′ resonances, butdrops sharply asM increases and the drooping pionform factor begins to cut down the pion contribution.

• We have noted that quarks acquiring constituentmasses larger thanTχ and decoupling at the chiralphase transition, reduces the lifetime of the quarkphaseabove Tc. This, in turn, brings down theqq̄ →l+l− rate for smallM andET . In Fig. 1, the differenceis visible but only whenTc = 240 MeV andT0 =250 MeV. In the total rate, the effect is swamped by thepion contribution for those values ofM for which thepion form factor is appreciable. However, from Fig. 3,we see that for slightly larger values ofM, the pioncontribution disappears and the total differential rate

Fig. 3. Plots of the ratio of thedNtot/dy dM2 rates for the MT andST scenarios vs.T0. The pion contribution is essentially absent forthe higher M-values chosen. Note that the MT rate is nowlowerthan the ST rate. The dashed lines represent the ratio of just thequark contributions. Quarks contribute fewer dileptons in the MTscenario, where the quark phase is shortened by the occurrence ofthe chiral transition.

dNtot/dy dM2 reduces to just the quark contribution,for both the ST and MT cases. Determining the ratein this M-window would then give an indication ofwhether a chiral transition has occurred close to butdistinctly aboveTc.

• The behaviour of these curves as functions ofT0 can be further understood as follows: in the STscenario, a higherT0 does not change the pion con-tribution to dilepton production. However, it clearlyincreases the time spent by the quarks in the cool-ing phase and thus enhances the quark contributionto dileptons, pushing the crossover to the quark domi-nated regime to smallerM. For the MT scenario, boththe quark and pion contributions to dileptons are en-hanced and so the crossover to the quark-dominatedregime is not expected to change much (though it maystill move slightly towards smallerM, due to steep fallin the pion form factor). For both scenarios, the cool-ing term in the quark contribution increases in mag-nitude with T0 while the Maxwell coexistence term(which accounts for the difference between the twoscenarios) remains unchanged. Thus asT0 increases,the ST and MT quark contributions approach eachother.

• Finally, in Fig. 4, we plot the ratio ofdNtot/

dy dM2 for the MT and the No Transition (NoT)scenarios vs.T0. The latter corresponds to there beingno phase transition, i.e., to the fireball’s cooling in


Fig. 4. The ratios of the MT rate to the rate if no transition occurredat all.

the pure hadronic phase. It is seen that this ratio isenormous.

• We conclude that if the dilepton rates increasedramatically, a phase transition has probably occurred.If the pion peaks indNtot/dy dM2dET vs.M surviveto high values ofET , the plasma has a good chanceof being a QGPP. If the peaks donot survive, butrather reduce to a flat line, we are quite likely seeinga quark phase. The actual value of this constant rate,for the window inM discussed above, will then decidewhether the transition to the confined hadronic phaseoccurred directly or through a chirally broken phase.

In summary, we have argued that thereare rea-sons to believe that a QPG may form in heavy-ioncollisions, as a result of several, as opposed to a sin-gle, phase transition. We have explored this possi-bility on the basis of some simple assumptions. Wehave found that this scenario pushes to higher temper-atures the advent of the QGP, but opens up, in return,the exciting possibillity of a QGPP, containing pionsas non-Goldstone-boson bound states, well aboveTχ .We have further investigated experimental signaturesbased on dilepton production, which would help dis-tinguish the single- from the multiple-transition sce-nario. Making this distinction would shed light notmerely on how the QGP forms but on a range of as-sumptions which lie at the very core of our understand-ing of strong interaction dynamics.

Acknowledgements

V.S. would like to thank many colleagues andcollaborators, George Ripka, Manoj Banerjee, Bojan

Golli, Mike Birse, W. Broniowoski, J.P. Blaizot, N.D.Haridass, G. Baskaran, M. Rho and many others.V.S. thanks R. Rajaraman for suggesting that anexperimental signature of the model used here shouldbe presented in the context of heavy ion physics. Weapologize for missing many relevant references—thelist is too extensive.

References

[1] K. Kajantie, J. Kapusta, L. McLerran, A. Mekjian, Phys. Rev.D 34 (1986) 2746.

[2] V. Soni, Mod. Phys. Lett. A 11 (1996) 331.[3] S. Kahana, G. Ripka, V. Soni, Nucl. Phys. A 415 (1984) 351;

M.C. Birse, M.K. Banerjee, Phys. Lett. B 134 (1984) 284.[4] A. Manohar, H. Georgi, Nucl. Phys. B 234 (1984) 203.[5] V. Soni, The nucleon and strongly interacting matter, Invited

talk at DAE Symposium in Nuclear Physics, Bombay, Decem-ber 1992, and references therein.

[6] M.C. Birse, Soliton Models in Nuclear Physics, Progressin Particle and Nuclear Physics, Vol. 25, 1991, p. 1, andreferences therein.

[7] See, for example, R. Johnson, N.W. Park, J. Schechter, V. Soni,H. Weigel, Phys. Rev. D 42 (1990) 2998;J. Stern, G. Clement, Mod. Phys. Lett. A 3 (1988) 1657.

[8] See, for example, J. Stern, G. Clement, Phys. Lett. B 264(1991) 426;E.J. Eichten, I. Hinchcliffe, C. Quigg, Fermilab-Pub 91/272-T.

[9] D. Diakonov, V. Petrov, P. Pobylitsa, M. Polyakov, C. Weiss,Nucl. Phys. B 480 (1996) 341;D. Diakonov, V. Petrov, P. Pobylitsa, M. Polyakov, C. Weiss,Phys. Rev. D 56 (1997) 4069.

[10] A. Gocksch, Phys. Rev. Lett. 67 (1991) 1701.[11] V. Soni, Phys. Lett. B 125 (1985) 231.[12] V. Soni, unpublished.[13] J.D. Bjorken, Phys. Rev. D 27 (1983) 140.[14] T. Hatsuda, T. Kunihiro, Phys. Rev. Lett. 55 (1985) 158, and

references therein.[15] E. Babaev, Phys. Rev. D 62 (2000) 074020;

E. Babaev, J. Mod. Phys. A 16 (2001) 1175;S.J. Hands, J.B. Kogut, C.G. Strouhouse, Phys. Lett. B 515(2001) 407;K. Zarembo, hep-ph/0104305.

[16] J. Schukraft, Review of transverse momentum distributions inultra-relativistic nucleus–nucleus collisions, Invited talk 1990,CERN-PPE/91-04.

[17] A. Drees, in: B.C. Sinha, D.K. Srivastava, Y.P. Viyogi (Eds.),Physics and Astrophysics of the Quark Gluon Plasma, NarosaPublishing House, 1997, p. 348.

[18] D. Ferenc, Nucl. Phys. A 610 (1996).

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The quark–gluon–pion plasma