A Modified Hard Thermal Loop Perturbation Theory
aa r X i v : . [ h e p - ph ] S e p A MODIFIED HARD THERMAL LOOP PERTURBATIONTHEORY
Najmul Haque and Munshi G. Mustafa
Theory Division, Saha Institute of Nuclear Physics,1/AF, Bidhannagar, Kolkata 700 064, India.
Abstract
Based on the external perturbation that disturbs the system only slightly from its equilibriumposition we make the Taylor expansion of the pressure of a quark gas. It turns out that the firstterm was used in the literature to construct a Hard Thermal Loop perturbation theory (HTLpt)within the variation principle of the lowest order of the thermal mass parameter. Various thermo-dynamic quantities within the 1-loop HTLpt encountered overcounting of the leading order (LO)contribution and also required a separation scale for soft and hard momenta. Using same varia-tional principle we reconstruct the HTLpt at the first derivative level of the pressure that takesinto account the effect of the variation of the external source through the conserved density fluc-tuation. This modification markedly improves those quantities in 1-loop HTLpt in a simple wayinstead of pushing the calculation to a considerably more complicated 2-loop HTLpt. Moreover,the results also agree with those obtained in the 2-loop approximately self-consistent Φ-derivableHard Thermal Loop resummation. We also discuss how this formalism can be extended for thehigher order contributions
PACS numbers: 12.38.Cy, 12.38.Mh, 11.10.WxKeywords: Quark-Gluon Plasma, Hard Thermal Loop Approximation, Quark Number Susceptibility . INTRODUCTION The HTL resummation developed by Braaten and Pisarski [1] has been used to calculatevarious thermodynamic quantities in the literature based on two methods. The 2-loopapproximately Φ-derivable approach was developed by Blaizot et al [2], which produces acorrect LO and plasmon effects for thermodynamic quantities (e.g, entropy density, numberdensity etc.) and also for quark number susceptibility (QNS) [3]. On the other hand theHTLpt using variational principle through the lowest order of the thermal mass parameterwas developed for pressure at the 1-loop level by Andersen et al [4], which is at presentpushed to the 2-loop [5] and 3-loop level [6]. However, the 1-loop HTLpt pressure has a badperturbative LO content, in the sense of severe over-inclusion of the effect of order g as theHTL action is accurate only for soft momenta and for hard ones only in the vicinity of lightcone. Such problem is cured (or at least pushed to higher orders) only after going to 2-looplevel in HTlpt [5], which is indeed a considerably more involved calculation. A very recent1-loop QNS calculation [7] from pressure in HTLpt [4] had, obviously, the problem of over-inclusion of the order g . Moreover, it required an ad hoc separation scale to distinguishbetween hard and soft momenta. On the other hand the calculation of QNS in Ref. [8]dealt with the imaginary part of the charge-charge correlator in the vector channel andrequired to show the charge conservation. It also encountered some technical difficulties andover-inclusion problem in order g as discussed in Ref. [3]. Also the Landau damping (LD)contribution was discussed but ignored.The equation of state (EOS) of strongly interacting matter at nonzero baryon densityand high temperature is a subject of great interest for wide spectrum of physicists. AlsoQNS is a topical quantity in view of the ongoing efforts towards understanding the actualnature of the QGP [9] as QNS plays an important role [10, 11] in locating the critical endpoint in QCD phase diagram. As it stands the LO thermodynamic quantities [2, 4] andQNS [3, 7, 8] in HTL approximation [1] led to different results. This requires a detailedanalysis of the leading order quantities within the HTLpt before extending it to the higherorders. In view of this we do not aim at higher orders calculations, rather we intend inthis article to sort out the problems in 1-loop HTLpt , which produced different results fromthat of the Φ-derivable approach within the HTL resummation [1], and finally arrive at aconsistent result despite the use of different approaches. II. GENERALITIESA. Fluctuation and Susceptibility:
Let O α be a Heisenberg operator where α may be associated with a degree of freedomin the system. In a static and uniform external field F α , the (induced) expectation value ofthe operator O α (0 , −→ x ) is written [12] as φ α ≡ hO α (0 , −→ x ) i F = Tr (cid:2) O α (0 , −→ x ) e − β ( H + H ex ) (cid:3) Tr [ e − β ( H + H ex ) ] = 1 V Z d x hO α (0 , −→ x ) i , (1)where the translational invariance is assumed, V is the volume of the system and H ex isgiven by H ex = − X α Z d x O α (0 , −→ x ) F α . (2)The (static) susceptibility χ ασ is defined as the rate with which the expectation valuechanges in response to an infinitesimal change in external field, χ ασ ( T ) = ∂φ α ∂ F σ (cid:12)(cid:12)(cid:12)(cid:12) F =0 = β Z d x D O α (0 , −→ x ) O σ (0 , −→ E , (3)where hO α (0 , ~x ) O σ (0 , ~ i is the two point correlation function with operators evaluated atequal times. There is no broken symmetry as hO α (0 , −→ x ) i| F→ = D O σ (0 , −→ E(cid:12)(cid:12)(cid:12) F→ = 0 . (4) B. Thermodynamic Relations:
The pressure is defined as P = TV ln Z , (5)3here T is temperature, V is the volume and Z is the partition function of a quark-antiquarkgas. The entropy density is defined as S = ∂ P ∂T . (6)The number density for a given quark flavour can be written as ρ = ∂ P ∂µ = 1 V Tr (cid:2) N e − β ( H− µ N ) (cid:3) Tr [ e − β ( H− µ N ) ] = hN i V , (7)with N is the quark number operator and µ is the chemical potential. If µ →
0, the quarknumber density vanishes due to CP invariance.The QNS is a measure of the response of the quark number density with infinitesimalchange in the quark chemical potential, µ + δµ , at µ →
0. Under such a situation thevariation of the external field, F α , in (2) can be identified as the quark chemical potential µ and the operator O α as the temporal component ( J ) of the external vector current, J σ ( t, ~x ) = ψ Γ σ ψ , where Γ σ is in general a three point function. Then the QNS for a givenquark flavour follows from (3) as χ ( T ) = ∂ρ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ =0 = ∂ P ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ =0 = Z d x D J (0 , ~x ) J (0 , ~ E = − lim p → ReΠ R (0 , p ) , (8)where the number operator, N = R J ( t, ~x ) d x = R ¯ ψ ( x )Γ ψ ( x ) d x and Π R ( ω p , p ) is the re-tarded time-time component of the Fourier transformed vector correlator Π σν ( ω p , ~p ) withan external momentum P = ( ω p , | ~ p | = p ). To write (8) in such a compact form wehave used the fluctuation-dissipation theorem and the quark number conservation [12, 13],lim ~p → ImΠ
R00 ( ω p , p) ∝ δ ( ω p ). III. MODIFICATION ON HARD THERMAL LOOP PERTURBATION THEORY
The HTL Lagrangian density for quark including HTL correction term [14] is written as L HT L = L QCD + δ L HT L = ¯ ψiγ µ D µ ψ + m q ¯ ψγ µ (cid:28) R µ iR · D (cid:29) ψ , (9)where R = (1 , r ) is a light like four-vector, ψ and ¯ ψ are the fermionic fields, D is thecovariant derivative, hi is the average over all possible directions over loop momenta. Thesecond term is gauge invariant, nonlocal and can generate N -point HTL functions [1], which4re inter-related through Ward identities. Now m q , is the quark mass in a hot and densemedium, which depends on the strong coupling g , temperature T and the chemical potential µ . Despite these facts m q is treated in (9) as a parameter much like the rest mass of a quarkand a HTLpt has been developed [4] around this rest mass ( i.e. , m q ) by reorganising theHTL terms where m q was treated as the order of ( gT ) for a hot system. In this way theeffect of m q is taken into account in higher orders much like a variational principle. For a hotand dense system we will also treat the mass parameter as the order of ( gT ) and ( gµ ) , andreorganise the HTL term based on the variation of the external source and setting it zeroat the end. In this way the effect of m q is taken into account to the higher order variationsof the external source and thus to the response of the system.We note that the covariant derivative usually contains background field or any source, j depending upon the physical requirement. To motivate the perspective we define thecovariant derivative D µ as D µ = [ D µ − iδ µ ( j + δj )] = [ ˜ D µ − iδ µ δj ] . (10)We note that D contains gauge coupling and ˜ D µ = D µ − iδ µ j , and δj is an infinitismalchange in external source to which the response of the system can be calculated, as discussedin Sec.II. Later it can be identified with a variation of some physical quantity dependingupon the requirement of the system under consideration.Now, expanding the second term in (9), we can write as L HT L ( j + δj ) = ¯ ψ (cid:16) i/ ˜ D + Σ (cid:17) ψ + δj ¯ ψ Γ ψ + δj ¯ ψ Γ ψ + O ( δj )= L HT L ( j ) + δj ¯ ψ Γ ψ + δj ¯ ψ Γ ψ + O ( δj ) (11)where the various N -point functions in coordinate space are generated asΣ = m q (cid:28) /RiR · ˜ D (cid:29) , Γ = δ µ γ µ − m q (cid:28) /RR µ δ µ ( iR · ˜ D ) (cid:29) , Γ = 2 m q ¯ ψ (cid:28) /RR µ R ν δ µ δ ν ( iR · ˜ D ) (cid:29) , (12)where these functions can easily be transformed into momentum space [15]. We now notethat these N -point HTL functions in (12) are also inter-related by Ward identities. In HTL-approximation the 2-point function, Σ ∼ gT (quark-self energy) is of the same order asthe tree level one, S − ( K ) ∼ K/ ∼ gT (in the weak coupling limit g << i.e. , of the order of gT . The 3-point function is given by g Γ ν = g ( γ ν + δ Γ ν ),5here δ Γ ν is the HTL correction. The 4-point function, g Γ νσ , is higher order and does notexist at the bare perturbation theory and only appears within the HTL approximation [1, 14].Now considering the HTL Lagrangian in (11), we can write the partition function [15] as Z [ β ; j + δj ] = Z D [ ¯ ψ ] D [ ψ ] D [ A ] e i R d x L HTL ( ψ, ¯ ψ ; j + δj ) , (13)where β = 1 /T , is the inverse of the temperature and A is a background gauge field.The pressure can be written as P [ β : j + δj ] = 1 V ln Z [ β : j + δj ] , (14)where the four-volume, V = βV with V is the three-volume.Expanding P in Taylor series around δj one can write P [ β : j + δj ] = P [ β ; j ] + δj P ′ [ β ; j + δj ] | δj → + δj P ′′ [ β ; j + δj ] | δj → + · · · · · · . (15)The first derivative of P w.r.t. j is related to the conserved density in (1) whereas thesecond derivative is related to the conserved density fluctuation in (3). The above expansionin (15) is very important for a resummed perturbation theory. We now note that a HTLptwas developed in Ref. [4] by considering the first term in (15) with j = 0, which caused anover-inclusion of the LO pressure. This was cured by going into two-loop level in HTLpt [5],which is of-course a very involved in nature. As we will see below this could easily becorrected if one constructs a HTLpt at the first derivative level of P in (15) where the effectof the variation of external field is taken into account.Now P ′ can be obtained as ∂ P [ β ; j + δj ] ∂j (cid:12)(cid:12)(cid:12)(cid:12) δj → = i VZ [ β ; j ] Z D [ ¯ ψ ] D [ ψ ] D [ A ] Z d x ¯ ψ ( x )Γ [ j ] ψ ( x ) e ( i R d x L HTL ( ψ, ¯ ψ ; j ) ) , (16)where we have used (12). The full HTL quark propagator in presence of uniform j can bewritten as S ασ [ j ]( x, x ′ ) = R D [ ¯ ψ ] D [ ψ ] D [ A ] ψ α ( x ) ¯ ψ σ ( x ′ ) exp (cid:0) i R d x L HT L ( ψ, ¯ ψ ; j ) (cid:1)R D [ ¯ ψ ] D [ ψ ] D [ A ] exp (cid:0) i R d x L HT L ( ψ, ¯ ψ ; j ) (cid:1) . (17)We now note that this full HTL propagator, S [ j ], is indeed difficult to calculate and wewould approximate it by 1-loop HTL resummed propagator [1, 14], S ⋆ [ j ] and also otherHTL functions below. 6ow using (17) and performing the traces over the colour, flavour, Dirac and coordinateindices in (16) one can write ∂ P [ β ; j + δj ] ∂j (cid:12)(cid:12)(cid:12)(cid:12) δj =0 = − i Z d K (2 π ) tr [ S ⋆ [ j ]( K ) Γ [ j ]( K, − K ; 0)] , (18)where ’tr’ indicates the trace over the colour, flavour and Dirac indices.Similarly, we obtain P ′′ as ∂ P [ β ; j + δj ] ∂j (cid:12)(cid:12)(cid:12)(cid:12) δj → = − N c N f T Z d k (2 π ) × X k Tr [ S ⋆ [ j ]( K ) Γ [ j ]( K, − K ; 0) S ⋆ [ j ]( − K ) Γ [ j ]( K, − K ; 0) − S ⋆ [ j ]( K ) Γ [ j ]( K, − K ; 0 , , (19)where N f is the number of massless flavours, N c is the number of colour and ’Tr’ indicatesthe trace over only the Dirac matrices. We have also used an identity based on unitarity of S ⋆ [ j ] as ∂S ⋆ [ j ]( K ) ∂j = − S ⋆ [ j ]( K ) ∂S ⋆ − [ j ]( K ) ∂j S ⋆ [ j ]( K ) = − S ⋆ [ j ]( K ) Γ [ j ]( K, − K ; 0) S ⋆ [ j ]( K ) . (20)Now, if we identify j as the quark chemical potential µ , and δj as its change δµ , then(19) would represent the QNS as χ ( β ) = ∂ρ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ → = ∂ P ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ → = − N c N f T Z d k (2 π ) × X k =(2 n +1) πiT Tr [ S ⋆ ( K )Γ ( K, − K ; 0) S ⋆ ( K )Γ ( K, − K ; 0) − S ⋆ ( K )Γ ( K, − K ; 0 , , (21)where the temporal correlation functions at the external momentum P = ( ω p , | ~p | ) = 0, isrelated to the thermodynamic derivatives . The first term in the second line of (21) is a1-loop self-energy whereas the second term corresponds to a tadpole in HTLpt with effective N -point HTL functions.Now, (18) represents the LO net quark number density in presence of uniform externalfield µ as ρ ( β, µ ) = ∂ P ∂µ = − i Z d K (2 π ) tr [ S ⋆ [ µ ]( K ) Γ [ µ ]( K, − K ; 0)] As already discussed Ref. [8] dealt with the definition of QNS that involves the static limit of the imaginarypart of the dynamical charge-charge correlator. If one uses the number conservation directly, viz. , theimaginary part of the charge-charge correlator is proportional to δ ( ω p ), then it becomes equal to (21) inwhich charge conservation is in-built by construction. N c N f T Z d k (2 π ) X k =(2 n +1) πiT + µ Tr [ S ⋆ ( K )Γ ( K, − K ; 0)] . (22) ∂∂ µ [ ] ≡ FIG. 1: 1-loop Feynman diagram in HTLpt for quark number density, ρ q that originates with thevariation of µ of the 1-loop HTLpt pressure. The dashed line represents the background field. Thesolid blobs are 1-loop resummed HTL N -point functions. ∂∂ µ [ ] ≡ FIG. 2: Same as Fig.1 but for the lowest order bare perturbation theory. ∂∂ µ [ ] = FIG. 3: The = sign indicates that it is not the correct diagram in the right hand side (rhs) as the µ variation is not taken into account properly. The diagram in rhs actually corresponds to ρ q thatwas obtained in Ref. [7]. When µ →
0, the net quark density in (22) would vanish as there is no broken CPsymmetry, which becomes consistent with (4). Also, (22) constitutes a LO HTLpt in thefirst derivative level of P (see Fig. 1) similar to the usual perturbation theory where thebare N -point functions (see Fig. 2) are automatically replaced by the 1-loop resummed HTL N -point functions. This suggests that HTL resummation technique provides a consistent8erturbative expansion if one goes beyond the lowest order perturbation theory. In contrastRef. [7] did not employ the variation of the external source as done in (15), which leads toFig. 3 with a bare vertex for the calculation of the net quark number density. This resultedin overcounting of the LO QNS. It also required an ad hoc separation scale to distinguishbetween soft and hard momenta.Below we briefly outline some of the essential quantities in HTL resummation [1], whichare required to compute (22). The resummed HTL propagator in 1-loop approximation formomentum K is given as S ⋆ ( K ) = − γ − ~γ · ˆ k D + ( k , k ) − γ + ~γ · ˆ k D − ( k , k ) , (23)with D ± ( k , k ) = − k ± k + m q k (cid:20) (cid:18) ∓ k k (cid:19) ln k + kk − k ± (cid:21) , (24) m q = g (cid:18) T + µ π (cid:19) . (25)where g = 4 πα s , α s is the strong coupling. Now, the zeros of D ± describe [17] the in-medium propagation or quasiparticle (QP) dispersion of a particle excitation with energy ω + having chirality to helicity ratio +1, and of a mode called plasmino with energy ω − havingchirality to helicity ratio −
1. In addition, D ± contains a discontinuous part correspondingto Landau Damping (LD) due to the presence of Logarithmic term in (24). Using thesegeneral properties of the quark propagator one can obtain the in-medium spectral functionfor quarks.The pole part of the spectral function can be written as ̺ ± ( ω, k ) = ω ± − k m q δ ( ω − ω ± ) + ω ∓ − k m q δ ( ω + ω ∓ ) , (26)as D + has poles at ω + and − ω − whereas those of D − are at ω − and − ω + .For k < k , there is a discontinuity in ln k + kk − k as ln y = ln | y | − iπ , which leads to thespectral function, β ± ( ω, k ), corresponding to the discontinuity in D ± ( k , k ) as β ± ( ω, k ) = − π Disc 1 D ± ( k , k ) = − π Im 1 D ± ( k , k ) (cid:12)(cid:12)(cid:12)(cid:12) k → ω + iǫǫ → = m q k (cid:0) ± ωk − (cid:1) Θ( k − ω ) h ω ∓ k − m q k (cid:0) ± − ω ∓ k k ln k + ωk − ω (cid:1)i + h π m q k (cid:0) ∓ ωk (cid:1)i . (27)9he zero momentum limit of the 3-point HTL function can be obtained from the Wardidentity [15, 16] as Γ ( K, − K ; 0) = ∂∂k (cid:0) S ⋆ − ( K ) (cid:1) = aγ + b~γ · ˆ k , (28)where a ± b = − D ′± ( k , k ) , (29)with D ′± = D ± k ∓ k − m q k − k . (30) IV. THERMODYNAMICS AND QUARK NUMBER SUSCEPTIBILITY
We first obtain the net quark density ρ ( T, µ ), which is then used to obtain various ther-modynamic quantities, viz. , pressure, entropy density and QNS.
A. Free case
In free case the number density can be written from (22) as ρ f ( T, µ ) = N c N f T Z d k (2 π ) X k =(2 n +1) πiT + µ Tr[ S f ( K ) γ ] , (31)where the 3-point function is Γ = γ and the 2-point function is the free quark propagatorfor momentum K is given as S f ( K ) = − γ − ~γ · ˆ k d + ( k , k ) − γ + ~γ · ˆ k d − ( k , k ) , (32)with d ± = − k ± k . (33)Using (32) in (31) and performing the trace over Dirac matrices, we get ρ f ( T, µ ) = 2 N c N f T Z d k (2 π ) X k =(2 n +1) πiT + µ (cid:20) k − k + 1 k + k (cid:21) . (34)For evaluating the frequency sum in (34), we use the standard technique of contour integra-tion [16] as12 πi I C (cid:20) k − k + 1 k + k (cid:21) β (cid:18) β ( k − µ )2 (cid:19) dk = β πi × ( − πi ) X Residues . (35)10t can be seen that the first term of (35) has a simple pole at k = k whereas the secondterm has a pole at k = − k . After calculating the residues of those two terms, the numberdensity becomes ρ f ( T, µ ) = − N c N f Z d k (2 π ) (cid:20) tanh β ( k − µ )2 − tanh β ( k + µ )2 (cid:21) = 2 N c N f Z d k (2 π ) [ n ( k − µ ) − n ( k + µ )] , (36)where n ( x ) = 1 / ( e βx + 1), is the Fermi distribution function.Now, the pressure is obtained by integrating the first line of (36) w.r.t. µ as P f ( T, µ ) = 2 N f N c T Z d k (2 π ) (cid:2) βk + ln (cid:0) e − β ( k − µ ) (cid:1) + ln (cid:0) e − β ( k + µ ) (cid:1)(cid:3) , (37)where the first term is the zero-point energy that generates a usual vacuum divergence [16].The entropy density in free case can be written from pressure as S f ( T, µ ) = ∂P f ∂T = 2 N c N f Z d k (2 π ) h ln (cid:0) e − β ( k − µ ) (cid:1) + ln (cid:0) e − β ( k + µ ) (cid:1) + β ( k − µ ) e β ( k − µ ) + 1 + β ( k + µ ) e β ( k + µ ) + 1 (cid:21) = N c N f (cid:18) π T
45 + µ T (cid:19) . (38)The QNS is obtained as χ f ( T ) = ∂∂µ h ρ fI ( T, µ ) i(cid:12)(cid:12)(cid:12)(cid:12) µ =0 = 4 N c N f β Z d k (2 π ) n ( k ) (1 − n ( k )) = N f T . (39) B. HTLpt Case
Using (23), (28) in (22) and then performing the trace over Dirac matrices, the quarknumber density in HTLpt becomes ρ HT L ( T, µ ) = 2 N c N f T Z d k (2 π ) X k =(2 n +1) πiT + µ (cid:20) D ′ + D + + D ′− D − (cid:21) = 2 N c N f T Z d k (2 π ) X k (cid:20) k − k + 1 k + k − m q k − k (cid:18) D + + 1 D − (cid:19)(cid:21) . (40)Apart from the various poles due to QPs in (40) it has LD part as D ± ( k , k ) contain Loga-rithmic terms which generate discontinuity for k < k , as discussed earlier. Equation (40)can be decomposed in individual contribution as ρ HT L ( T, µ ) = ρ QP ( T, µ ) + ρ LD ( T, µ ) . (41)11 . Quasiparticle part (QP) The pole part of the number density can be written as ρ QP ( T, µ ) = 2 N c N f T Z d k (2 π ) πi I C ′ (cid:20) k − k + 1 k + k − m q k − k (cid:18) D + + 1 D − (cid:19)(cid:21) × β β ( k − µ )2 dk . (42)In general residues for various poles in third and fourth terms in (42) can be obtained asRes (cid:26) m q k − k D ± (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) k = ω ± , − ω ∓ = − (cid:26) m q k − k D ± (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) k = ± k = 1 , (43)where D ± ( k = ± k ) = ± m q k .1. First two terms in (42) give the same contribution as free case in (36).2. The third term has four simple poles at k = ω + , − ω − , k, − k . After performing thecontour integration the third term can be written as12 πi I C ′ m q k − k D + β β ( k − µ )2 dk = − β (cid:20) − tanh β ( ω + − µ )2+ tanh β ( ω − + µ )2 + tanh β ( k − µ )2 − tanh β ( k + µ )2 (cid:21) . (44)3. The fourth term has four simple poles at k = ω − , − ω + , − k, k . After performing thecontour integration the fourth term can be written as12 πi I C ′ m q k − k D − β β ( k − µ )2 dk = − β (cid:20) − tanh β ( ω − − µ )2+ tanh β ( ω + + µ )2 + tanh β ( k − µ )2 − tanh β ( k + µ )2 (cid:21) . (45)Using (36), (44) and (45) in (42) one can obtain the HTL quasiparticle contributions tothe quark number density as ρ QP ( T, µ ) = − N c N f Z d k (2 π ) (cid:20) tanh β ( ω + − µ )2 + tanh β ( ω − − µ )2 − tanh β ( k − µ )2 − tanh β ( ω + + µ )2 − tanh β ( ω − + µ )2 + tanh β ( k + µ )2 (cid:21) = 2 N c N f Z d k (2 π ) [ n ( ω + − µ ) + n ( ω − − µ ) − n ( k − µ ) − n ( ω + + µ )12 n ( ω − + µ ) + n ( k + µ )] , (46)which agrees with that of the two-loop approximately self-consistent Φ-derivable HTL re-summation of Blaizot et al [2, 3].Now, the pressure is obtained by integrating the first line of (46) w.r.t. µ as P QP ( T, µ ) = 2 N f N c T Z d k (2 π ) (cid:20) ln (cid:0) e − β ( ω + − µ ) (cid:1) + ln (cid:18) e − β ( ω − − µ ) e − β ( k − µ ) (cid:19) + ln (cid:0) e − β ( ω + + µ ) (cid:1) + ln (cid:18) e − β ( ω − + µ ) e − β ( k + µ ) (cid:19) + βω + + β ( ω − − k ) (cid:21) . (47)This agrees with the form given for quasiparticle contribution by Andersen et al [4] consider-ing the first term in (15) for µ = 0. Both quasiparticles with energies ω + and ω − generate T dependent ultra-violate (UV) divergences in LO HTL pressure, which is an artefact of1-loop HTL approximation [2–4]. At very high T , ω ± → k and (47) reduces to free case asobtained in (37).The corresponding HTL QP entropy density in LO can be obtained as S QP ( T, µ ) = ∂ P QPI ∂T = 2 N c N f Z d k (2 π ) (cid:20) ln (cid:0) e − β ( ω + − µ ) (cid:1) + ln (cid:18) e − β ( ω − − µ ) e − β ( k − µ ) (cid:19) + ln (cid:0) e − β ( ω + + µ ) (cid:1) + ln (cid:18) e − β ( ω − + µ ) e − β ( k + µ ) (cid:19) + β ( ω + − µ ) e β ( ω + − µ ) + 1 + β ( ω − − µ ) e β ( ω − − µ ) + 1 − β ( k − µ ) e β ( k − µ ) + 1 + β ( ω + + µ ) e β ( ω + + µ ) + 1 + β ( ω − + µ ) e β ( ω − + µ ) + 1 − β ( k + µ ) e β ( k + µ ) + 1 (cid:21) , (48)which agrees with that of the 2-loop approximately self-consistent Φ-derivable HTL resum-mation of Blaizot et al [2].The QNS in LO due to HTL QP can also be obtained from (46) as χ QP ( T ) = ∂∂µ h ρ QPI i(cid:12)(cid:12)(cid:12)(cid:12) µ =0 = 4 N c N f β Z d k (2 π ) [ n ( ω + ) (1 − n ( ω + )) + n ( ω − ) (1 − n ( ω − )) − n ( k ) (1 − n ( k ))] , (49)where the µ derivative is performed only to the explicit µ dependence. Obviously (49) agreesexactly with that of the 2-loop approximately self-consistent Φ-derivable HTL resummation We note that the expression for QP pressure in one-loop HTLpt [4] was obtained by adding and subtractingthe free gas pressure. However, in our formalism the correct LO form comes out naturally and no additionand subtraction is required as in Ref. [4]. This is because the fluctuation of the conserved density isappropriately taken into consideration in the present formalism.
13f Blaizot et al [3]. The above thermodynamical quantities in LO due to HTL quasiparticles with excitation energies ω ± are similar in form to those of free case but the hard and softcontributions are clearly separated out and one does not need an ad hoc separating scale asused in Ref. [7].
2. Landau Damping part (LD)
The LD part of the quark number density follows from (40) and (27) as ρ LD ( T, µ ) = N c N f Z d k (2 π ) k Z − k dω π (cid:18) − m q ω − k (cid:19) π [ β + ( ω, k ) + β − ( ω, k )] tanh β ( ω − µ )2= N c N f Z d k (2 π ) k Z − k dω (cid:18) m q ω − k (cid:19) β + ( ω, k ) [ n ( ω − µ ) − n ( ω + µ )] . (50)One can obtain the pressure due to LD contribution by integrating (50) w.r.t. µ as P LD ( T, µ ) = N c N f T Z d k (2 π ) k Z − k dω (cid:18) m q ω − k (cid:19) β + ( ω, k ) (cid:2) ln (cid:0) e − β ( ω − µ ) (cid:1) + ln (cid:0) e − β ( ω + µ ) (cid:1) + βω (cid:3) , (51)which has UV divergence like Andersen et al [4] and can be removed using the appropriateprescription therein.The corresponding LD part of entropy density can be obtained as S LD ( T, µ ) = N c N f Z d k (2 π ) k Z − k dω (cid:18) m q ω − k (cid:19) β + ( ω, k ) h ln (cid:0) e − β ( ω − µ ) (cid:1) + ln (cid:0) e − β ( ω + µ ) (cid:1) + β ( ω − µ ) e β ( ω − µ ) + 1 + β ( ω + µ ) e β ( ω + µ ) + 1 (cid:21) . (52)Also the LD part of the QNS becomes χ LDI ( T ) = ∂∂µ (cid:2) ρ LDI ( T, µ ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) µ =0 = 2 N c N f β Z d k (2 π ) k Z − k dω (cid:18) m q ω − k (cid:19) × β + ( ω, k ) n ( ω ) (1 − n ( ω )) , (53)where the µ derivative is again performed only to the explicit µ dependence. It is alsoto be noted that the LD contribution is of the order of m q . The LD contribution can14ot be compared with that of the 2-loop approximately self-consistent Φ-derivable HTLresummation of Blaizot et al [3] as it does not have any closed form for the final expression.The numerical values of both the QNS agree very well.It is clearly evident that the various LO thermodynamic quantities in 1-loop HTLpt canbe obtained within this modified formalism at ease instead of pushing the calculation to amore involved approaches. Below we demonstrate the correct inclusion of the perturbativecontent of the order g to the QNS in HTLpt in a strict perturbative sense by comparingwith the usual perturbation theory. V. QNS IN PERTURBATIVE LEADING ORDER ( g ) In conventional perturbation theory, for massless QCD the QNS has been calculated [16,18] upto order g log(1 /g ) at µ = 0 as χ p χ f = 1 − (cid:16) gπ (cid:17) + r N f (cid:16) gπ (cid:17) − (cid:16) gπ (cid:17) log (cid:18) g (cid:19) + O ( g ) . (54)We note that for all temperatures of relevance the series decreases with temperature andapproaches the ideal gas value from the above, which is due to the convergence problem ofthe conventional perturbation series.Nevertheless, the perturbative LO, g , contribution is also contained in HTL approxima-tion [1, 14] through the N -point HTL functions. In the left panel of Fig. 4 we display theLO HTL QNS and LO perturbative QNS scaled with free one as a function of m q /T , theeffective strong coupling. In the weak coupling limit both approach unity whereas the HTLcase has a little slower deviation from the ideal gas value than the LO of the conventionalperturbative one. The latter could be termed as an improvement over the conventionalperturbative results. The results are in very good agreement with that of Ref. [3].Next we consider a ratio [3] as R ≡ χ htl − χ f χ p ( g ) − χ f , (55)which measures the deviation of interaction of χ htl from that of pQCD to order g . In theright panel of Fig. 4 we display this ratio as a function of m q /T , which approaches unityin the weak coupling limit indicating the correct inclusion [3] of order g in our approachin a strictly perturbative sense. This comes from the ω + branch [15] of the HTL dispersion15 χ / χ f m q /T HTLpQCD(g ) R m q /T FIG. 4: (Color online)
Left panel:
The ratio of 2-flavour HTL to free quark QNS and also that ofLO perturbative one as a function of m q /T . Right panel:
The interaction measure R as a functionof m q /T . relations, ω + ( k ) ≈ k + m q /k , at hard momentum scale, i.e , k ∼ T . With this one can nowtrivially show by expanding the QP contribution from (49) in Sect. IV that becomes χ QPq = χ f (1 − g / π + · · · ) , (56)which agrees with that in (54). The LD contribution is of the order of m q .We note that the HTL resummation technique provides a consistent perturbative expan-sion for gauge theories at finite temperature and/or density. As discussed going beyond thelowest order bare perturbation theory for quark number density, we use the HTL resummedpropagator and quark-gluon vertices in Fig. 1. The resummed HTL quark propagators cor-respond to static external quarks (valence quark). In 1-loop HTLpt ( viz. , Fig. 1) there isno dynamical quark (no quark loop) and in this sense 1-loop HTlpt is comparable withthe quenched approximation of lattice QCD [19]. The inclusion of dynamical quark loopsrequires one to consider the higher-order diagrams within HTLpt in which HTL resummedgluon propagators (containing quark loops) will show up. This could be taken care through(11) as it contains the covariant derivative with gauge coupling and the calculation is inprogress. 16 I. CONCLUSION
In the literature the HTL resummation have been used through various approaches tocalculate the thermodynamic quantities and also the response of the system, viz. , the QNS toan external perturbation, i.e. , the quark chemical potential. This led to different results inLO indicating the sensitivity of the methods. In this paper we revisited the thermodynamicquantities and in particular the QNS in LO within HTLpt to arrive at similar results withinthe various HTL approaches. For this purpose we modified the existing HTLpt [4] at thefirst derivative level of pressure by incorporating an infinitesimal variation to an externalsource, viz. , the quark chemical potential that disturbs the system only slightly. We showthat the various thermodynamic quantities and the QNS in LO order agree with those ofthe two-loop approximately self-consistent Φ-derivable HTL resummation approach [2, 3]existing in the literature. Our calculation also shows that the soft and hard momenta getseparated out naturally and one does not require any ad hoc separating scale as in [7].All the thermodynamic quantities turned out to be dependent on the chemical potentialautomatically due to the method employed. We also discussed that our formalism can alsobe extended for higher-order calculations.
Acknowledgments
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