Quark mass dependent collective excitations and quark number susceptibilities within the hard thermal loop approximation

# Quark mass dependent collective excitations and quark number susceptibilities within the hard thermal loop approximation

Najmul Haque Institut für Theoretische Physik, Justus-Liebig–Universität Giessen, 35392 Giessen, Germany
###### Abstract

We calculate all those QCD -point functions which are relevant for three-loop QCD thermodynamics calculation with finite quark masses within the hard the thermal loop approximation. Using the effective quark propagator, we also calculate second order quark and baryon number susceptibilities within the hard thermal loop approximation and compare the results with available lattice data.

## I Introduction

It is an experimental fact that colored quarks and gluons are confined to hadrons by the strong interaction. The theory, which correctly describes this interaction, is known as Quantum Chromodynamics (QCD). In extreme conditions, such as at very high temperatures and/or densities, hadronic states are believed to undergo a (partially crossover) transition into a deconfined state of quarks and gluons, known as Quark-Gluon Plasma (QGP). Such extreme conditions existed in the very early universe and can also be generated in ultrarelativistic heavy-ion collision experiments at the Relativistic Heavy Ion Collider (RHIC), the Large Hadron Collider (LHC) and in the future also at the Facility for Antiproton and Ion Research (FAIR). In addition, the cores of the densest astrophysical objects in existence, neutron stars, may contain cold deconfined matter, commonly referred to as quark matter. The reason why deconfined matter is expected to be encountered at high energy densities is related to the asymptotic freedom of QCD, i.e. the fact that the value of the strong coupling constant decreases logarithmically as a function of the energy scale. Proceeding to higher energies, nonperturbative effects are also expected to diminish in importance, and calculations based on a weak coupling expansion should eventually become feasible at such extreme conditions. This is very important for a successful quantitative description of the system especially at nonzero chemical potentials, as no nonperturbative first principles method applicable to finite-density QCD exists due to the Sign Problem of lattice QCD. Unfortunately, at energy densities of phenomenological relevance for most practical applications, the value of the strong coupling constant is not small, and it in fact turns out that e.g. for bulk thermodynamic quantities a strict expansion in the QCD coupling constant converges only at astronomically high temperatures and chemical potentials. The source of this problem has been readily identified as the infrared sector of the theory, i.e. contributions from soft gluonic momenta of the order of the Debye mass or smaller. This suggests that to improve the situation, one needs a way of reorganizing the perturbative series that allows for the soft contributions to be included in the result in a physically consistent way. To date, two successful variations of resummed perturbation theory have been introduced as remedy, namely Hard Thermal Loop perturbation theory (HTLpt) Braaten:1989mz (); Andersen:1999fw (); Haque:2013sja (); Haque:2014rua (); Haque:2012my (); Haque:2013qta (); Andersen:2015eoa () and Dimensionally Reduced (DR) effective field theory Kajantie:2002wa (); Vuorinen:2003fs (), of which the latter is only applicable to high temperatures but the former in principle covers also the cold and dense part of the QCD phase diagram. Both approaches have been shown to lead to results which are in quantitative agreement with lattice QCD in the high-temperature and zero (or small-) density limit, down to temperatures a few times the pseudocritical temperature of the deconfinement transition.

In nearly all thermodynamic calculations applying perturbation theory, either the temperature or the chemical potentials are assumed to be the dominant energy scale in the system and in particular much larger than the QCD scale or any quark masses. It is, however, questionable, whether the latter is necessarily a good approximation at the lowest temperatures and densities where the HTLpt results are typically applied; the strange quark mass is after all of the order of 100 MeV, which is certainly not negligible at the temperatures reached in the RHIC and LHC heavy ion experiments. There have been very few high-order perturbative calculations incorporating finite quark masses, and in the context of high temperatures, the state-of-the-art result is from a two-loop perturbative calculation performed long ago Kapusta:1979fh () in a renormalization scheme for quark masses. Later, two-loop perturbative calculation for the thermodynamical quantities has been extended in Refs. Laine:2006cp (); Graf:2015tda () using the renormalization scheme. Additionally, the current state-of-the-art result for the QCD pressure at zero temperature and finite quark mass is from a three-loop bare perturbation theory Kurkela:2009gj (). On the other hand, finite quark masses have not been considered at all in any resummed pertutbative framework for thermodynamic calculations.

Collective excitations of heavy-fermion have been studied long ago in Ref. Petitgirard:1991mf () calculating heavy-fermion propagator. Later, in Ref. Seipt:2008wx (), the authors have studied quark mass dependent thermal excitations calculating both quark and gluon propagators. In this article we calculate all the QCD -point functions, such as gluon propagator, quark propagator, three- and four-point quark-gluon vertices considering HTL approximation. These -point functions are relevant for higher order thermodynamics calculation within the HTLpt framework. In this direction, we calculate quark number susceptibility (QNS) and baryon number susceptibility (BNS) using two-loop -derivable self-consistent approximation within HTL scheme, as described in Blaizot:2000fc (); Blaizot:2001vr () for massless quarks.

The paper is organized as follows. In Sec. II we calculate the HTL effective gluon-propagator and gluon dispersion relations including finite quark masses. In Sec. III we calculate the HTL effective quark-propagator with finite quark mass. In Sec. IV, we discuss about the three- and four-point quark-gluon vertices with finite quark mass. In the next section (Sec. V), we calculate the second-order light quark- as well as strange quark-number susceptibility which lead to the computation of the second order BNS. In Sec. VI, we conclude our results.

## Ii Gluon propagator

One loop Feynman diagram for the gluon self-energy consisting the quark loop is illustrated in the Fig. (1).

The contribution from the Feynman diagram (1) to the gluon self-energy in Minkowski space can be written as

 Πab,fμν(P,m) (1) = (−1)(ig)2(ta)ij(tb)ji∑f∫d4K(2π)4 × Tr[γμS0(K)γνS0(K−P)] = g2Tr[tatb]∑f∫d4K(2π)4Tr[γμS0(K)γνS0(Q)] = i Πfμν(P,m)δab,

with , and . So, the gluon self-energy tensor is

 Πfμν(P,m) (2) = −ig22∑f∫d4K(2π)4Tr[γμS0(K)γνS0(Q)] ≃ ig22∑f∫d4K(2π)4[8KμKν−4gμν(K2−m2f)] × ~Δ(K)~Δ(Q),

where ‘tilde’ in represents fermionic propagator and can be written as

 ~Δ(K)=1K2−m2f=1k20−k2−m2f. (3)

Now, the time-time component of the self-energy tensor becomes

 Πf00(P,m) (4) = 4ig22∑f∫d4K(2π)4(2k20−K2+m2f)~Δ(K)~Δ(Q) = 4ig22∑f∫d4K(2π)4[~Δ(Q)+2(k2+m2f)~Δ(K)~Δ(Q)] ≃ 2ig2∑f∫d4K(2π)4[~Δ(K)+2(k2+m2f)~Δ(K)~Δ(Q)].

Similarly, the trace of the gluon self-energy tensor is

 Πμμf(P,m) (5) ≃ −4ig2∑f∫d4K(2π)4 [~Δ(K)−m2f~Δ(K)~Δ(Q)].

The frequency sum for the first term in Eqs. (4) and  (5) can be evaluated easily as

 i∫d4K(2π)4~Δ(K) (6) = −T∑n∫d3k(2π)31k20−k2−m2f = −∫k2dk2π2n+F(Ek)+n−F(Ek)2Ek,

where

 Ek = √k2+m2f, (7) n±F(Ek) = 1exp[β(Ek∓μf)]+1. (8)

We calculate the second Matsubara sum of Eqs. (4) and (5) using the mixed representation prescribed by Pisarski Pisarski:1987wc ().

 ~Δ(K) = 1k20−E2k=−β∫0dτek0τΔF(τ,Ek), (9) ~Δ(P−K) = 1(p0−k0)2−E2pk (10) = −β∫0dτe(p0−k0)τΔF(τ,Epk),

with

 ΔF(τ,Ek) (11) = 12Ek[(1−n−F(Ek))e−Ekτ−n+F(Ek)eEkτ].

Using the relations in Eqs. (9)-(11), the following Matsubara sum can be performed as

 T∑k0~Δ(K)~Δ(P−K)=β∫0dτep0τΔF(τ,Ek)ΔF(τ,Epk) (12) = −∑s1,s2=±1s1s2[1−n−F(s1Ek)−n+F(s2Epk)]4EkEpk(p0−s1Ek−s2Epk) = 14EkEpk[n+F(Ek)−n+F(Epk)p0+Ek−Epk−n−F(Ek)−n−F(Epk)p0−Ek+Epk−1−n−F(Ek)−n+F(Epk)p0−Ek−Epk+1−n+F(Ek)−n−F(Epk)p0+Ek+Epk].

where

 Ek=√k2+m2f , (13) Epk=√|p−k|2+m2f . (14)

In HTL approximation,

 Epk=√|p−k|2+m2f≈Ek−v^k⋅p, n±F(Epk)≈n±F(Ek)−^k⋅pdn±F(Ek)dk, (15)

with .

Using the HTL approximations (Eq. (15)), the Matsubara sum in Eq. (12) can be written as

 T∑k0~Δ(K)~Δ(P−K)=14E2k[−n+F(Ek)+n−F(Ek)Ek (16) + dn+F(Ek)dk^k⋅pp0+v^k⋅p−dn−F(Ek)dk^k⋅pp0−v^k⋅p].

So, Eq. (4) becomes

 Πf00(p0,p,m) (17) = 2ig2∑f∫d4K(2π)4 [~Δ(K)+2E2k~Δ(K)~Δ(Q)] = −2g2∑f∫k2dk4π2dΩ4π ×ddk[n+F(Ek)+n−F(Ek)]^k⋅pp0+v^k⋅p = −g22π2∫k2dkEkkddk[n+F(Ek)+n−F(Ek)] ×[1−p02pvlogp0+pvp0−pv].

Eq. (17) can also be simplified as

 Πf00(p0,p,m) (18) = ×[1+p20−p2p20−p2v2−p0pvlogp0+pvp0−pv].

Similarly, the trace of the gluon self-energy tensor in Eq. (5) can be written using the Matsubara sums in Eqs. (6) and (12) as

 Πμμf(p0,p,m) (19) = −4ig2∑f∫d4K(2π)4 [~Δ(K)−m2f~Δ(K)~Δ(Q)] = g2π2∑f∫k2dkdΩ4π[n+F(Ek)+n−F(Ek)Ek + m2f2E2k{n+F(Ek)+n−F(Ek)Ek − Ekkddk[n+F(Ek)+n−F(Ek)](1−p0p0+pvx)} = g2π2∑f∫k2dk[n+F(Ek)+n−F(Ek)Ek⎛⎝1+m2f2E2k⎞⎠ − m2f2kEkddk[n+F(Ek)+n−F(Ek)] ×(1−p02pvlogp0+pvp0−pv)].

Eq. (19) can also be simplified as

 Πμμf(p0,p,m) = (20) × [1+m2f2E2kp20−p2p20−p2v2].

Pure Yang-Mills contributions to the gluon self-energy do not get affected by the inclusion of finite quark masses. So, the time-time and component of the total gluon self-energy tensor from fermion as well as pure Yang-Mills diagrams is

 Π00(p0,p,m) (21) = g2T2CA3[1−p02plogp0+pp0−p] +g22π2∑f∫k2dkEk[n+F(Ek)+n−F(Ek)] ×[1+p20−p2p20−p2v2−p0pvlogp0+pvp0−pv],

and

 Πμμ(p0,p,m) (22) = g2T2CA3+g2π2∑f∫k2dkEk[n+F(Ek)+n−F(Ek)] × ⎡⎣1+m2f2E2k p20−p2p20−p2v2⎤⎦.

Note that, unlike massless case, gluonic and quark contributions can not combined together.

Now one-loop gluon self-energy can be decomposed in terms of two independent and mutually transverse second rank projection tensors as

 Πμν=ΠTAμν+ΠLBμν, (23)

with

 Aμν = gμν−PμPνP2−Bμν, (24) Bμν = −P2p2(uμ−p0PμP2)(uν−p0PνP2). (25)

So, the longitudinal and transverse parts of gluon self-energy tensor can be obtained from Eqs. (21) and (22) as

 ΠL(p0,p,m) (26) = −p20−p2p2Π00(p0,p,m) = −p20−p2p2[g2T2CA3(1−p02plogp0+pp0−p) + g22π2∑f∫k2dkEk(n+F(Ek)+n−F(Ek)) × (1+p20−p2p20−p2v2−p0pvlogp0+pvp0−pv)],

and

 ΠT(p0,p,m) (27) = 12[Πμμ(p0,p,m)−ΠL(p0,p,m)] = g2T2CA3[1+p20−p2p2(1−p02plogp0+pp0−p)] ×(2p20p2−p20−p2p2p0pvlogp0+pvp0−pv).

### ii.1 Debye mass

Debye mass in QCD is obtained as

 m2D=ΠL(p0=0,p→0,m). (28)

At vanishing quark masses, Debye mass is

 m2D(T,μ)=g2T23[CA+Nf2(1+3^μ2π2T2)], (29)

where is common chemical potential for all the quark-flavors. Now, the Debye mass with finite quark masses can be obtained from Eq. (26) as

 m2D(T,μ,m)=ΠL(p0=0,p→0,m) (30) = g2T2CA3+g22π2Nf∑f=1 ×

Plasma frequency is the oscillation frequency for vanishing wave vectors, namely, spatially uniform oscillations Bellac:2011kqa () and can be expressed as

 ω2p = g2T2CA9+g22π2Nf∑f=1 (31) × ∞∫0dkkv(1−v23)[n+F(Ek)+n−F(Ek)].

Unlike massless case, Debye mass and plasma frequency are not proportional to each other.

In Fig. (2) we plot quark mass dependent Debye mass and plasma frequency, scaled with corresponding massless values, with temperature. In this plot and also in the rest of the plots, we use MeV and .

### ii.2 Dispersion relation

In-medium gluon propagator can be written as

 Dμν = ξPμPνP4+AμνP2−ΠT+BμνP2−ΠL (32) = ξPμPνP4+AμνDT+BμνDL.

The zeros of the denominators give the dispersion laws as and as plotted in Fig. (3).

In Fig. 3 we plot longitudinal and transverse dispersion relations scaled with massless plasma frequency.

At small momentum , it is possible to find approximate analytic solution of as

 ω2L = ω2p+p2ω2paL+p4ω4pbL+O(p6), (33) ω2T = ω2p+p2ω2paT+p4ω4pbT+O(p6), (34)

where

 aL = g2T2CA15 +g22π2Nf∑f=1 (35) × ∞∫0dkkv3(1−3v25)[n+F(Ek)+n−F(Ek)], bL = g2T2CA21−a2Lω2p+g22π2Nf∑f=1 (36) × ∞∫0dkkv5(1−5v27)[n+F(Ek)+n−F(Ek)], aT = ω2p+aL3, (37) bT = bL5−aL3+445a2Lω2p. (38)

From Figs. (2) and (3), we conclude that the finite mass effect to the gluon collective excitations is negligible and one may consider effective gluon propagator is same as in massless case to calculate various physical observables.

## Iii Quark propagator

Free massive quark propagator is

 S0(P) = i⧸P−mf. (39)

The inverse of the propagator in Eq. (39) can be written as

 iS−10(P) = ⧸P−mf. (40)

Now, the inverse of in-medium quark propagator is

 iS−1(P) = iS−10(P)−Σ(P), (41)

where is the one-loop quark self-energy and can be decomposed as

 Σ(P)=−a⧸P−b⧸u+cmf, (42)

The coefficients can be obtained as

 a = −14pTr[γ⋅^p Σ(P)], b = −14Tr[γ0 Σ(P)]+p04pTr[γ⋅^p Σ(P)], c = 14mfTr[Σ(P)]. (43)

So, Eq. (41) becomes

 iS−1(P) = (1+a)⧸P+b⧸u−(1+c)mf, −iS(P) = (1+a)⧸P+b⧸u+(1+c)mfD(p0,p), (44)

where

 D(p0,p) (45) = (1+a)2P2+b2−(1+c)2m2f+2b(1+a)p0 = D+(p0,p)D−(p0,p),

with

 D±(p0,p) (46) = (1+a)p0+b∓√(1+a)2p2+(1+c)2m2f .

Now, the numerator of Eq. (44) can be written as

 (1+a)⧸P+b⧸u+(1+c)mf (47) = D+(p0,p)2⎡⎢ ⎢⎣γ0+γ⋅^p(1+a)p+(1+c)mf√(1+a)2p2+(1+c)2m2f⎤⎥ ⎥⎦ + D−(p0,p)2⎡⎢ ⎢⎣γ0−γ⋅^p(1+a)p+(1+c)mf√(1+a)2p2+(1+c)2m2f⎤⎥ ⎥⎦.

So, Eq. (44) becomes

 −iS(P) (48) = 12D+(p0,p)⎡⎢ ⎢⎣γ0−γ⋅^p(1+a)p+(1+c)mf√(1+a)2p2+(1+c)2m2f⎤⎥ ⎥⎦ + 12D−(p0,p)⎡⎢ ⎢⎣γ0+γ⋅^p(1+a)p+(1+c)mf√(1+a)2p2+(1+c)2m2f⎤⎥ ⎥⎦.

Zeros of the denominator of Eq. (48) give the dispersion relation. Now,

 iS−1(P) (49) = (1+a)⧸P+b⧸u−(1+c)mf = [(1+a)p0+b]γ0−(1+a)γ⋅^p p−(1+c)mf = A0(p0,p)γ0−Asγ⋅^p−Am,

with

 A0(p0,p) = (1+a)p0+b, As(p0,p) = (1+a)p, Am(p0,p) = (1+c)mf. (50)

Alternatively,

 D±(p0,p) = A0(p0,p) (51) ∓√A0(p0,p)2+Am(p0,p)2.

Self-energy can also be written using Eqs. (42) and (50) as

 Σ(P) = −a⧸P−b⧸u+cmf (52) = −(ap0+b)γ0+apγ⋅^p+cmf = [p0−A0(p0,p)]γ0+[As(p0,p)−p]γ⋅^p +[Am(p0,p)−mf].

Eq. (48) can also be written using Eq. (50) as

 −iS(P) (53) = 12D+(p0,p)[γ0−As(p0,p)γ⋅^p+Am(p0,p)√As(p0,p)2+Am(p0,p)2] + 12D−(p0,p)[γ0+As(p0,p)γ⋅^p+Am(p0,p)√As(p0,p)2+Am(p0,p)2].

Now, the quark self-energy in Feynman gauge can be obtained from the Feynman diagram in Fig. (4) as

 −iΣ(P) = (ig)2CF∫d4K(2π)4 γμi⧸P−⧸K−mfγν−igμνK2 Σ(P) = 2g2CFT∑n∫d3k(2π)3 [⧸K+2mf] (54) × Δ(K)~Δ(P−K).

To compute the quark self-energy in Eq. (54), we need to compute the following Matsubara sums: and . We evaluate them following the same procedure as in gluonic case using the mixed representation. Fermionic propagator is given in Eq. (10) and the gluonic propagator can be represented as

 Δ(K) = 1k20−k2