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Evolution of QCD Coupling Constant at Finite Temperature in the Background Field Method

The evolution of QCD coupling constant at finite temperature is considered by making use of the finite temperature renormalization group equation up to the one-loop order in the background field method with the Feynman gauge and the imaginary time formalism. The results are compared with the ones obtained in the literature. We point out, in particular, the origin of the discrepancies between different calculations, such as the choice of gauge, the break-down of Lorentz invariance, imaginary versus real time formalism and the applicability of the Ward identities at finite temperature.


Introduction
One of the application of field theory at finite temperature [1][2][3][4] is to find the behaviour of coupling constant as a function of energy, temperature, and the chemical potential using the renormalization group (RG) equation. The knowledge of coupling constant at finite temperature environment then can be used, for instance, in perturbative calculations for quark-gluon plasma created in ion-ion collisions at high energies, in the evaluation of the grand unification scale in a cosmological context and the shift of the energy levels in a hydrogen plasma.
Recently a great deal of attention has been paid to the study of the behaviour of the coupling contant at finite T [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The resulting formula for the temperature and scale dependent part of the coupling constant has proved to be very sensitive to the prescription chosen. In this paper we wish to examine this and clarify the origin of the discrepancies between different calculations in the literature. We employ the background field method [20,21], which is based on a manifestly gauge invariant generating functional, and the Feynman gauge. The background field method provides notorious simplifications since we have to calculate only the renormalization constant of the background field gluon propagator. We discuss this method in section 2.1. Since the coupling constant depends not only on energy but also on temperature, we derive a pair of RG equations -one for energy and one for temperature. This is done in section 2.2 and in section 2.3 we derive its solution. In section 2.4 we discuss the Feynman rules at finite temperature in the imaginary time formalism.
In order to get the renormalization constant of the background field gluon propagator we have to calculate the polarization tensor. At finite temperature due to the lack of the Lorentz invariance the structure of the polarization tensor is not equivalent to the zero temperature one. In section 3.1 we give a prescription how to define the renormalization constant at finite temperature. In section 3.2 we calculate the polarization tensor in the one-loop approximation. In section 3.3 calculating the vacuum part of the polarization tensor we reproduce the well known zero temperature formula, i.e. the standard vacuum QCD one-loop result [22,23], which reads as where Here N is for SU(N), n F is the number of flavours, µ is the energy scale and µ 0 is the reference scale. We calculate the matter part of the polarization tensor in section 4.1 and, in section 4.2, we derive the temperature and scale dependent part of the coupling constant.
In section 5.1 we review the results of the previous works and in 5.2 we compare the asymptotic expansion formulas at a = µ/T ≪ 1 derived in different schemes.
We give a few comments and, in particular, discuss the origin of the discrepancies between different calculations.
In Appendix we collect the relevant integrals which we encounter in the text.
2 Formal methods

Background field method
Explicit gauge invariance, which is present at the classical level in gauge field theories, is normally lost at the quantum level. This can be seen from the generating functional for QCD: where A is the gluon field, Ψ andΨ are the fermion fields, η andη are ghost fields, (G a ) 2 is a gauge fixing term (ξ:gauge parameter), ω a is a SU(3) group parameter, and J µa A µa is a source term. In SU(3) the gauge fields transform in the following way: where U(x) = exp[iω(x) a T a ] is a unitary transformation, and T a is a generator for SU(3). By using transformations (2.2)-(2.4) one can see that the gauge invariance of Z(J) is lost in commonly used gauges such as ∂ µ A µ = 0. The advantage of the background field method [20,21] is that it can maintain the explicit gauge invariance. For this purpose we devide the gluon field A µ into a sum of a classical background field B µ and a quantum field Q µ and choose for the gauge fixing term G a the background field gauge condition where D µ is the covariant derivative: The generating functional reads This functional is gauge invariant, which follows from the transformations: In the background field method the quantum gauge fields and the ghost field need not be renormalized since they appear only inside loops. No vertex functions are to be considered. Thus only renormalizations 14) The explicit gauge invariance of Z(J, B) implies that the perturbation series is gauge invariant in every order in g R and that the background field renormalization factor Z B and the coupling constant renormalization Z g are related with each other. The field tensor F µν in L QCD has to be gauge covariant and is renormalized as Thus we have a relation which enables us to calculate the evolution of coupling constant g R in a simple way. We assume this relation to be valid also at finite temperature.

Derivation of the RG equations
From Eqs. (2.13) and (2.17) we have We use the dimensional regularization [24,23] and perform our calculations in (4 −2ε) dimensions. We notice from the Lagrangian of QCD that the dimension of a gluon (quark) field is µ 1−ε (µ 3/2−ε ), where µ is the scale parameter. We redefine g 0 so that it becomes dimensionless and rewrite Eq. (2.18) as These equations, which are not symmetrical in T and µ, determine the behaviour of coupling constant g R with respect to T and µ changes. Generally Z B has the form [22,23] j /ε j are the divergent contributions independent on T and µ, while f (j) (µ, T ) are convergent temperature and scale dependent functions which vanish in the lowtemperature limit.
From Eqs. (2.21) and (2.22) one can derive the following formula, by taking ε → 0, (from now on we write g = g R , for simplicity) (2.23) This equation is reduced to the ordinary one-loop RG equation in the zero-temperature limit since in this limit the function f

Solution of the coupled RG equations
The solution to Eqs. (2.23) and (2.24) can be found in a straightforward way by integrating Eq. (2.23) from µ 0 to µ and Eq. (2.24) from T 0 to T . We have Putting T equal to T 0 in Eq. (2.25), and using Eq. (2.26) we arrive at the desired solution This equation describes the evolution of the QCD coupling constant as a function of the momentum scale and the arbitrary temperature [5][6][7][8][9]. (µ 0 , T 0 ) denotes the reference point.

Imaginary time formalism
The Feynman rules at finite temperature T (= 1/β) are derived from the generating functional and for the time-component of 4-momentum we have for bosons and ghosts: and for fermions: where µ ch is a chemical potential. The frequency sums for bosons and fermions in Eq.
(2.29) are readily changed to contour integrals [26]: We have for bosons and ghosts and for fermions The first two terms in the right-hand side of Eq. (2.34) correspond to particle and anti-particle contributions respectively and vanish at T = 0. Since we will be only interested in phenomena in an enviroment where the sum of quantum numbers is not conserved, hereafter we put µ ch equals to zero. Also we have omitted the term which gives the finite density contribution at T = 0.

Polarization tensor 3.1 Structure of the polarization tensor
At a zero temperature environment the polarization tensor is Lorentz invariant and can be expressed as [22] Π ab µν = δ ab Π µν , The zero temperature polarization tensor is transverse with respect to k (current conservation): At finite temperature, in the presence of matter, the Lorentz invariance is lost and the polarization tensor can only be O(3) rotational invariant [27,17]. Then the polarization tensor can generally depend only on 4 independent quantities, which we can choose, for example, Π 00 , k i Π 0i and the two scalars Π L and Π T appearing in At finite temperature whether the transversality condition is satisfied or not depends on the gauge used. The polarization tensor is not transversal, e.g., in the Coulomb gauge (∂ i A i = 0), but is transversal at the one-loop level in the temporal axial gauge (A 0 = 0), and in every order of the perturbative calculations in the background field gauge [17]. The transversality condition (3.3) restricts the structure of the polarization tensor. From Eq. (3.3) we have and Using Eqs. (3.4) and (3.6) we obtain For the coefficient of the transversal part of the polarization tensor we get from Eqs. (3.4) and (3.7) an expression Thus in Eqs. (3.4)-(3.9) we have derived the general form of the temperature dependent O(3) symmetric polarization tensor in the background field method. The polarization tensor can be splitted into a sum of a temperature independent (vacuum) part and a temperature and scale dependent (matter) part [4]: (3.10) At zero temperature limit the temperature dependent matter part vanishes. The structure of the polarization tensor at zero temperature was given in Eqs. (3.1) and (3.2). The polarization tensor is related to the gluon propagator, D µν , as The relation between the bare and the renormalized propagators (see Eq. (2.14)) is which together with Eq. (3.11) leads us to Thus the renormalization constant Z B can be obtained from the (background field) gluon self-energy tensor.
In investigating the behaviour of the coupling constant one has to include the temperature dependent parts of the polarization tensor in the renormalization constant Z B via Eq. (3.13). Although the finite renormalization is somehow arbitrary one has to follow some rule or prescription in every order of perturbation to be consistent [23].
At finite temperature we encounter another type of ambiguity which is caused by the lack of the Lorentz invariance. In order to define the polarization tensor at finite temperature we generalize Eq.(3.13) for T = 0 as: where we have either or Naturally at T = 0, Eq.(3.14) reduces to Eq.(3.13). Π 00 and Π T are not connected with each other and in general there is no a priori way to decide which one is more natural.  Fig.1 to get the renormalization factor for Z B up to the one-loop order. The Feynman rules for the interaction vertices are the same as in the zero temperature case, and therefore identical to those given in [21]. Evaluating the one-loop diagrams (1a)-(1d) in the Feynman gauge (i.e., we set ξ = 1 in the gluon propagator) we find for the boson contributions in the polarization tensor

The polarization tensor at the one-loop level
where For the fermion loop, neglecting the quark masses compared to momentum scale and temperature, we have from the diagram (1e): where T F = 1/2 for SU(3). We have also included n F (number of flavours) in Eq. (3.19). Our polarization tensor satisfies the transversality condition This can be shown explicitly as follows: From Eq. (3.17) we have Replacing k + p → p in the second term we have Then by changing k − p → −p in the first term, we immediately come to Eq. (3.20) for the boson contributions. Similarly one can prove the transversality condition for the fermion polarization tensor. In the next subsection we will extract from Π µν the information on the vacuum part of the polarization tensor.

Vacuum part of the polarization tensor
Using a standard technique and introducing the Feynman parametrization in Eqs. (3.17) and (3.19), namely we have Changing variable p → p−kx we get the following formula for the boson contributions from Eq. (3.17): where Similarly we get the formula for the fermion contributions from Eq. (3.19): In these equations for the vacuum parts we have dropped the terms proportional to 1/p 2 and 1/(k + p) 2 , which turn out to be zero after simple calculation [see Eq. (A.1) in the Appendix]. Notice that all of the integrals are ultraviolet divergent and thus have to be regularized. For this purpose we employ the dimensional regularization [24,23], which preserves gauge symmetries explicitly. The integrals for the vacuum parts become Euclidean if we change ip 0 → p 4 and thus can be easily evaluated. We obtain the following results: and  In this subsection we calculate Π 00 and Π µµ , without using the Feynman parametrization. To specify the subtraction point we employ the static limit of zero external energy, which is commonly used in the literature [5]. In this prescription the momentum k is specified to be space-like k = (0, k i )(i = 1, 2, 3) with k 2 = −µ 2 . Such a choice enables us to determine the static properties. Using Eqs. (3.17) and (3.19) and the integrals from the Appendix we obtain for Π 00 with a = βµ. Similarly we derive an expression for Π µµ : where we define the boson functions

Temperature dependent part of the coupling constant
As pointed out in section 3.1 we encounter an ambiguity in determining the renormalization constant Z B as a direct consequence of the lack of Lorentz invariance. Here we write two formulas, one derived from Π 00 and the other one from Π T . From Eq.
(3.15) we have (a = βµ) (4.6) and the other one from Eq.(3.16) and The result for Eq. (4.7) was reported in a short communication in [18](where an overall factor of 1/2 was missing in the right-hand side of formulas (8) and (9)). where we have presented the results using our notations. Kajantie et al. [6] studied the gauge field part of QCD with N colors. They used the A 0 = 0 gauge and defined two renormalization schemes by writing

Review on the results of previous works
where D µν is the gluon propagator and F, G and P L µν come from the polarization tensor Π µν = F P L µν + GP T µν . In the so-called "electric prescription" they derived the coupling constant from F as Notice here the change of sign in the high T behaviour. They concluded that the "magnetic prescription" is more natural than the "electric" one because the former uses the physical part of the gluon propagator. They also argued that one has to choose µ ∼ = 3T in the thermal equilibrium. In that case the effect of the temperature dependent parts becomes negligible. Nakaggawa, Niégawa and Yokota [7] used the real-time formalism and studied the scale-parameter ratios Λ(a)/Λ derived from different vertices in 4-flavour QCD. They used the covariant gauge and found that the ratios derived from the three-gluon vertex and the gluon-ghost vertex show just the opposite behaviour than the one derived from the gluon-quark vertex. Namely in the former case one gets a growing ratio while in the latter case a decreasing ratio.
Fujimoto and Yamada [8] used the real-time formalism and derived the temperature depending coupling constant from the gauge-independent Wilson loop. At a ≪ 1 , it reads as (without quark contributions) The same authors have discussed the finite temperature RG equations [9] in the one-loop approximation, using the real-time formalism, the covariant gauge and the dimensional regularization. From the gluon propagator and three-gluon vertex they obtained: where T f = 1/2, and C = N for SU(N). The coefficient in front of F B0 (a) was originally 4/3. The error was pointed out in Ref. [10]. From the fermion propagator and fermion-gluon vertex: where C f = (N 2 − 1)/(2N) for SU(N).
From ghost propagator and ghost-gluon vertex: Stephens et al. [19] performed a background field one-loop calculation of gauge invariant beta functions at finite temperature, using the retarded/advanced formalism developped by Aurenche and Becherrawy [28]. In terms of our notations, their result reads as: (5.14) The numerical coefficients of the leading terms of the high-temperature expansion and the fermion parts are in complete agreement with our result in Eq.(4.7). In the boson parts, however, there are some numerical discrepancies with our result.

Comparison of asymptotic expansions
In order to compare our results with those mentioned in section 5.1 we derive the asymptotic expansions for −f (1) (µ, T ). The asymptotic expansions for Eqs. (5.10)-(5.12) at a = µ/T ≪ 1 in the high-temperature regime read as follows: Next we derive the asymptotic expansions for our results, i.e., Eqs. (4.6) and (4.7) at a ≪ 1, by using the following high temperature expansions [9]: where ℓ 1 = ln(a/4π) + γ, (5.22) We find the asymptotic formula from Eq. (4.6) The T 2 term coincides with the one of Gendenshtein [5], i.e.,Eq. (5.1). The gluon part in this asymptotic formula is consistent with the result of Elze et al. [17] derived in the background field method, which in our notations reads This coincides with the result of Nadkharni [12], who has the terms up to O(1/a).
The delicate reason why in the background field method we have a factor -1/2 in front of the second term proportional to 1/a, while in some results [see Eqs. (5.7) and (5.9)] it is equal to -1/4 is clarified by Elze et al. in [17]. Next from Eq. (4.7) we find the asymptotic formula: The behaviour of the transverse polarization tensor Π T in the infrared region is known to have a form [17] lim Π T (0, k)/µ 2 = g 2 [−cN/a + O(ln a)]. (5.27) The factor c has been calculated in different gauges. In the covariant ξ-gauge its value is [27] c = (9 + 2ξ + ξ 2 )/64, (5.28) whereas in the temporal axial gauge it is [6] c = 5/16. These results can be read also from the results of Refs. [9] [see Eq. (5.17)] and [12]. The relation (5.31) was also noticed by Elze et al. in [17]. We remark here that our c is negative and hence no spurious pole appears in the transversal propagator in contrast to the covariant and the temporal axial gauge cases. It is a known fact that the T 2 terms are gauge independent and the gauge parameter dependence starts at ∼ T order [13]. Thus the fact that the T 2 terms in Eqs.

Concluding remarks
In concluding the paper we give a few comments: 1) In sections 5.1 and 5.2 we have seen that the temperature dependence of the QCD coupling constant is very sensitive to the prescription chosen. This is not a trivial issue, because all the results obtained hitherto also heavily depends on the vertex chosen (i.e.,the trigluon, the ghost-gluon, or the quark-gluon vertex) to satisfy the renormalization condition of the QCD coupling constant. Furthermore in some gauges, e.g., in the Coulomb gauge, the transversality condition (3.3) on the polarization tensor does not hold and hence the structure of the polarization tensor in such gauges is different from that which satisfies the condition.
One of the reasons why we encounter different results in the literature is that the broken Lorentz invariance has not been treated as we have done in Eqs. (3.15) and (3.16). For example in deriving Eqs. (5.10)-(5.12) the authors of Ref. [9] have extracted the gluon and the vertex renormalization constants in front of Lorentz invariant structures not paying attention to the break down of Lorentz invariance.
Another possible source for discrepancies is that the Ward identities are used at finite temperature for different renormalization constants. Remember that the derivation of the Ward identity is based on the gauge invariance and also on the Lorentz invariance at T = 0. Thus it is not a surprise that results from different gauges or even (within the same gauge) from different vertices are totally different.
The correspondence between the imaginary and real time formalisms has been investigated in detail [14][15][16]. The choice of the formalism can also be the source of the discrepancies under study was pointed out in these references.
To clarify the issue of choosing a suitable renormalization prescription, one would need to compute the two-loop contributions to the coupling constant at finite temperature. It was shown, in particular, in the massive O(N) scalar model that the one-loop result is drastically changed by two-loop contributions at high T and in zero momentum limit [29].
2) As we have seen the function f (1) (µ, T ) shows a power-like T -dependence instead of a logarithmic fall-off as a function of µ. Thus we have In this case we would have a power-like dependence even with the relation T = (const)µ.
3) Since in a thermostat T is a measurable quantity, though µ is not as, stated in 2), we should be able to define the coupling constant as a function of only T . This could be done by calculating an expectation value of the coupling constant. In that case as a probability density function we could use either the Bose or the Fermi distribution. However such an expectation value of the coupling constant could not be used to a scattering process in a thermostat wherein a particle with some definite energy enters.
We would like also to mention that all the machinary for the evolution of the running coupling constant at finite temperature can be analogously applied for the case of a quantum field theory(such as QCD) at finite energy as formulated in [30].
In conclusion, we note in accordance with the previous observations that there is in fact no unique way to define a temperature depending QCD coupling constant and the issue of finding its reasonable prescription is left as a subject of further investigation.