Thermodynamic pressure for massless QCD and the trace anomaly

From statistical mechanics the trace of the thermal average of any energy-momentum tensor is $\langle T^{\mu}_{\;\;\mu}\rangle =T\partial P/\partial T-4P$. The renormalization group formula $\langle T^{\mu}_{\;\;\mu}\rangle =\beta(g_{M})\partial P/\partial g_{M}$ for QCD with massless fermions requires the pressure to have the structure \begin{equation} P=T^{4}\sum_{n=0}^{\infty} \phi_{n}(g_{M})\big[\ln\big({M\over 4\pi T}\big)\big]^{n},\end{equation} where the factor $4\pi$ is for later convenience. The functions $\phi_{n}(g_{M})$ for $n\ge 1$ may be calculated from $\phi_{0}(g_{M})$ using the recursion relation $n\,\phi_{n}(g_{M})=-\beta(g_{M})d\phi_{n-1}/dg_{M}$. This is checked against known perturbation theory results by using the terms of order $(g_{M})^{2}, (g_{M})^{3}$, $(g_{M})^{4}$ in $\phi_{0}(g_{M})$ to obtain the known terms of order $(g_{M})^{4}, (g_{M})^{5}$, $(g_{M})^{6}$ in $\phi_{1}(g_{M})$ and the known term of order $(g_{M})^{6}$ in $\phi_{2}(g_{M})$. The above series may be summed and gives the same result as choosing $M=4\pi T$, viz. $T^{4}\phi_{0}(g_{4\pi T})$.


I. INTRODUCTION
For a symmetric energy-momentum tensor T µν the dilation current S µ = T µλ x λ and the four conformal currents K αµ = x 2 T αµ − 2x α T µλ x λ are conserved if the energy-momentum tensor is traceless: The classical energy-momentum tensor for QCD with massless fermions is traceless but quantum corrections introduce a renormalization scale that spoils the conservation of scale and conformal currents and renders the trace nonzero [1]. The trace of the thermally averaged energy-momentum tensor is T µ µ = u − 3P where u = T 0 0 is the energy density and P = − 3 j=1 T j j /3 is the pressure. The relation exp(βP V ) = Z = Tr{e −βH } between the pressure and the partition function implies that or equivalently The trace of the energy-momentum tensor becomes For non-Abelian gauge fields with massless fermions the pressure has the form where M is the renormalization scale. From (1) the trace of the energy-momentum tensor is One would expect the calculation of Φ to be primary and the trace anomaly only an afterthought. However with the theorem of Drummond, Horgan, Landshoff, and Rebhan [2] that the anomaly becomes predictive in that the combination of (3) and (4) gives which is Eq. (3.11) of Drummond et al [2]. Note that (4) is similar to the zero temperature operator identity T µ µ = β(g M )∂L/∂g M . Sec. II shows how Eq. (5) ensures that P is independent of the renormalization scale M and requires P to have the structure shown in the abstract. Sec. III tests the recursion relation using known results for φ 0 (g M ) from perturbation theory to calculate the three known terms in φ 1 (g M ) and the only known term of φ 2 (g M ) and illustrates how to improve perturbation theory.
In the first term use M dg M /dM = β(g M ); in the second, M dr/dM = r and r∂Φ/∂r = −T ∂Φ/∂T so that after using Eq (5). Comment: One can reverse the argument and derive the anomaly relation (4) of Drummond et al [2] by starting with the assertion that P is a physical quantity and must therefore be independent of the renormalization scale.

Origin of
and to introduce variables The running coupling is a function of u determined by β(g M ) = dg M /du; Φ is a function of u − v: after at Taylor series expansion. The definition allows the series to be written The recursion relation n φ n (g M ) = −dφ n−1 /du, which follows from (11), may be expressed as One can confirm directly that the series (12) satisfies dΦ/dM = 0. Comment: If ξ is changed to ξ ′ then The binomial theorem allow the series (12) to be expressed in terms of powers of ln(M/ξ ′ T ) with modified functions φ ′ n (g M ). Comment: From u − 3P = T 5 ∂Φ/∂T it follows that the energy density and entropy density are

III. RESULTS FROM PERTURBATION THEORY
The O(g 2 M ) term in P was calculated by Shuryak [3]; the O(g 3 M ) term by Kapusta [4]; to this order there was no ln(M/T ). The O(g 4 M ) term was calculated by Arnold and Zhai [5]; the O(g 5 M ) by Zhai and Kastening [6]; in both cases ln(M/T ) appeared. The same result was obtained by Braaten and Nieto [7] using hard thermal loop resummation.
At O(g 6 M ) nonperturbative magnetic screening effects arise [8][9][10]. Kajantie et al [11] were able to calculate the O(g 6 M ) perturbative terms and found both ln(M/T ) and ln 2 (M/T ). A convenient reference that discusses all the results is Sec. 8.4 of Kapusta and Gale [12].

A. Checks against known results
For comparison with the published results from perturbation theory it is convenient to insert a prefactor in the the series expression for the pressure and choose ξ = 4π: where d A is the dimension of the adjoint representation.
With the order (g M ) 2 , (g M ) 3 , and (g M ) 4 terms of φ 0 (g M ) the recursion relation (13) gives the first three terms of φ 1 (g M ) and the first term of φ 2 (g M ). Using the notation φ (k) where W = (g M /2π) (C A + S F )/3 and The coefficients R j are given in [5,12] in terms of Riemann zeta functions and the Euler constant. For later comparison with [11] it is convenient to employ the approximate numerical values: The standard notation [12] for SU(N) with n f fermions in the fundamental representation is 4N . The first two terms in the beta function are The predictions of the recursion relation (13) are The result for A, agrees with [5][6][7]11]. The result for B, agrees with [6,7,11].
The result for C is To compare this with [11] it is necessary to evaluate (24) for SU ( In [11] the O(g 6 M ) results are expressed in terms of (α M /π) 3 and ln(M/2πT ). When [11] is reexpressed in terms of (g M /4π) 6 and ln(M/4πT ) it agrees completely with Eq. (26).
The final calculation D gives which is exactly the same as [11].

B. Improving perturbation theory
At order (g M ) 6 nonperturbative effects appear in φ above. The argument of Linde [8,9,12] shows that certain diagrams that appear to be of order (g M ) k with k > 6 are so infrared sensitive that nonperturbative magnetic shielding will render them of order (g M ) 6 . Thus φ (6) 0 (g M ) receives contributions from diagrams with infinitely many loops. Nevertheless φ 0 (g M ) is still a series of the form The k = 1 term vanishes; the k = 2 term is the first to depend on g M . Because the beta function begins with (g M ) 3 the recursion relation (13) implies that φ (k) 0 (g M ) will generate terms of order (g M ) 2n+k [ln(M/4πT )] n . The series (17) for P may be considered a double series: Perturbative calculations through order (g M ) 5 determine φ (2n+k) n (g M ) for 2n + k ≤ 5: (31) The difference between P (k≤5) [n] and P k≤4 is not small [6,7,13].
There is no need to terminate the sum over n; one can easily compute the full sum (32) The input is of the form The n'th order derivatives of φ 0 (g M ) required by Eq. (11) give With v = ln(M/4πT ) Eq. (10) requires the sum By the ratio test this sum converges for |v/u| < 1, which is satisfied provided M > 4πT Λ QCD and 4πT > Λ QCD . The result is Applying d/dk as required in (35) gives The dependence on the renormalization scale M disappears since u − v = ln(4πT /Λ QCD ). When a k , b k are expressed in terms of A k , B k and u − v = (β 0 g 4πT ) −1 the result is or more concisely In short, convergence of the infinite sum on n in (30) is automatic; whether a finite number of φ (k) 0 (g M ) in the series for (29) for φ 0 (g M ) is a good approximation, i.e.whether perturbation theory is reliable, is an open question [13].