Gauge Fields

Abstract

After introducing the concept of non-Abelian gauge invariance, associated with the existence of spin-1 gauge fields, the main elements of the canonical quantization of gauge fields are recalled. Consequently, an imaginary-time path integral expression is motivated for the partition function of such fields. The rules for carrying out a weak-coupling expansion of this quantity are formulated, and the corresponding Feynman rules are derived. This machinery is employed for defining and computing a thermal gluon mass, also known as a Debye mass. Finally, the free energy density of non-Abelian black-body radiation is determined up to third order in the coupling constant, revealing a highly non-trivial structure in this asymptotic series.

Appendices

Appendix A: Non-Abelian Black-Body Radiation in the Free Limit

In this appendix, we compute the free energy density f(T) for N _c colours of free gluons and N _f flavours of massless quarks, starting from Eq. (5.44), and use the outcome to deduce a result for the usual electromagnetic blackbody radiation. This is an interesting exercise because, inspite of us being in the free limit, ghosts turn out to play a role at finite temperature.

To start with, we recall from Eqs. (2.51), (2.81), (4.60) and (4.73) that

(5.52)

(5.53)

Our task is to figure out the prefactors of these terms, corresponding to the contributions of gluons, ghosts and quarks.

In the gluonic case, we are faced with the matrix

$$\displaystyle{ M_{\mu \nu } = P^{2}\delta _{ \mu \nu } -{\Bigl ( 1 -\frac{1} {\xi } \Bigr )}P_{\mu }P_{\nu }\;, }$$

(5.54)

which is conveniently handled by introducing two further matrices,

$$\displaystyle{ \mathbb{P}_{\mu \nu }^{\mbox{ T} }\; \equiv \;\delta _{\mu \nu }-\frac{P_{\mu }P_{\nu }} {P^{2}} \;,\quad \mathbb{P}_{\mu \nu }^{\mbox{ L} }\; \equiv \; \frac{P_{\mu }P_{\nu }} {P^{2}} \;. }$$

(5.55)

As matrices, these satisfy $\mathbb{P}^{\mbox{ T}}\mathbb{P}^{\mbox{ T}} = \mathbb{P}^{\mbox{ T}}$, $\mathbb{P}^{\mbox{ L}}\mathbb{P}^{\mbox{ L}} = \mathbb{P}^{\mbox{ L}}$, $\mathbb{P}^{\mbox{ T}}\mathbb{P}^{\mbox{ L}} = 0$, $\mathbb{P}^{\mbox{ T}} + \mathbb{P}^{\mbox{ L}} = \nVdash$, making them projection operators and implying that their eigenvalues are either zero or unity. The numbers of the unit eigenvalues can furthermore be found by taking the appropriate traces: $\mathrm{Tr}\,[\mathbb{P}^{\mbox{ T}}] =\delta _{\mu \mu } - 1 = d$, $\mathrm{Tr}\,[\mathbb{P}^{\mbox{ L}}] = 1$.

We can clearly write

$$\displaystyle{ M_{\mu \nu } = P^{2}\,\mathbb{P}_{\mu \nu }^{\mbox{ T} } + \frac{1} {\xi } P^{2}\,\mathbb{P}_{\mu \nu }^{\mbox{ L} }\;, }$$

(5.56)

from which we see that M has d eigenvalues of P ² and one $P^{2}/\xi$. Also, there are a = 1, …, N _c ² − 1 copies of this structure, so that in total

(5.57)

The first term vanishes in dimensional regularization, because it contains no scales.

For the ghosts, the Gaussian integral yields (cf. Eq. (4.47))

$$\displaystyle{ \int \!\prod _{a}\mathrm{d}\tilde{\bar{c}}^{\,a}\mathrm{d}\tilde{c}^{a}\exp (-\tilde{\bar{c}}^{\,a}P^{2}\tilde{c}^{a}) =\prod _{ a}P^{2} =\exp {\Bigl \{ -{\Bigl [- 2(N_{\mathrm{ c}}^{2} - 1)\frac{1} {2}\ln (P^{2})\Bigr ]}\Bigr \}}\;. }$$

(5.58)

Recalling that ghosts obey periodic boundary conditions, we obtain from here

$$\displaystyle{ \left.f(T)\right \vert _{\mbox{ ghosts}} = -2(N_{\mathrm{c}}^{2} - 1)J(0,T)\;. }$$

(5.59)

Finally, quarks function as in Eq. (4.52), except that they now come in N _c colours and N _f flavours, giving

$$\displaystyle{ \left.f(T)\right \vert _{\mbox{ quarks}} = -4N_{\mathrm{c}}N_{\mathrm{f}}\,\tilde{J} (0,T)\;. }$$

(5.60)

Summing together Eqs. (5.57), (5.59) and (5.60), inserting the values of J and $\tilde{J}$ from Eqs. (5.52) and (5.53), and setting d = 3, we get

$$\displaystyle{ \left.f(T)\right \vert _{\mbox{ QCD}} = -\frac{\pi ^{2}T^{4}} {90} {\Bigl [2(N_{\mathrm{c}}^{2} - 1) + \frac{7} {2}N_{\mathrm{f}}N_{\mathrm{c}}\Bigr ]}\;. }$$

(5.61)

This result is often referred to as (the QCD-version of) the Stefan-Boltzmann law.

It is important to realize that the contribution from the ghosts was essential above: according to Eq. (5.59), it cancels half of the result in Eq. (5.57), thereby yielding the correct number of physical degrees of freedom in a massless gauge field as the multiplier in Eq. (5.61). In addition, the assumption that A ₀ ^a is periodic has played a role: had it also had an antiperiodic part, Eq. (5.61) would have received a further unphysical term.

To finish the section, we finally note that the case of QED can be obtained by setting N _c → 1 and N _c ² − 1 → 1, recalling that the gauge group of QED is U(1). This produces

$$\displaystyle{ \left.f(T)\right \vert _{\mbox{ QED}} = -\frac{\pi ^{2}T^{4}} {90} {\Bigl [2 + \frac{7} {2}N_{\mathrm{f}}\Bigr ]}\;, }$$

(5.62)

where the factor 2 inside the square brackets corresponds to the two photon polarizations, and the factor 4 multiplying $\frac{7} {8}N_{\mathrm{f}}$ to the degrees of freedom of a spin-$\frac{1} {2}$ particle and a spin-$\frac{1} {2}$ antiparticle. If left-handed neutrinos were to be included here, they would contribute an additional term of $2 \times \frac{7} {8}N_{\mathrm{f}} = \frac{7} {4}N_{\mathrm{f}}$. Eq. (5.62) together with the contribution of the neutrinos gives the free energy density determining the expansion rate of the universe for temperatures in the MeV range.

Thermal Gluon Mass

We consider next the gauge field propagator, in particular the Matsubara zero-mode sector thereof. We wish to see whether an effective thermal mass $m_{\mbox{ eff}}$ is generated for this field mode, as was the case for a scalar field (cf. Eq. (3.95)). The observable to consider is the full propagator, i.e. the analogue of Eq. (3.63). Note that we do not consider non-zero Matsubara modes since, like in Eq. (3.94), the thermal mass corrections are parametrically subdominant if we assume the coupling to be weak, g ² T ² ≪ (2π T)². For the same reason, we do not need to consider thermal mass corrections for fermions at the present order.

In order to simplify the task somewhat, we choose to carry out the computation in the so-called Feynman gauge, $\xi \equiv 1$, whereby the free propagator of Eq. (5.45) becomes

(5.63)

Specifically, our goal is to compute the 1-loop gluon self-energy $\Pi _{\mu \nu }^{}$, defined via

(5.64)

where the role of the denominator of the left-hand side is to cancel the disconnected contributions.

At 1-loop level, there are several distinct contributions to $\Pi _{\mu \nu }^{}$: two of these involve gauge loops (via a quartic vertex from $-S_{\mbox{ I}}^{}$ and two cubic vertices from $+S_{\mbox{ I}}^{2}/2$, respectively), one involves a ghost loop (via two cubic vertices from $+S_{\mbox{ I}}^{2}/2$), and one a fermion loop (via two cubic vertices from $+S_{\mbox{ I}}^{2}/2$). If the theory were to contain additional scalar fields, then two additional graphs similar to the gauge loops would be generated. For future purposes we treat the external momentum K of the graphs as a general Euclidean four-momentum, even though for the Matsubara zero modes (that we are ultimately interested in) only the spatial part is non-zero.

Let us begin by considering the gauge loop originating from a quartic vertex. Denoting the structure in Eq. (5.49) by

$$\displaystyle\begin{array}{rcl} C_{\alpha \beta \rho \sigma }^{cdef}& \equiv & f^{gcd}f^{gef}(\delta _{\alpha \rho }\delta _{\beta \sigma } -\delta _{\alpha \sigma }\delta _{\beta \rho }) + f^{gce}f^{gdf}(\delta _{\alpha \beta }\delta _{\rho \sigma } -\delta _{\alpha \sigma }\delta _{\beta \rho }) \\ & & +f^{gcf}f^{gde}(\delta _{\alpha \beta }\delta _{\rho \sigma } -\delta _{\alpha \rho }\delta _{\beta \sigma })\;, {}\end{array}$$

(5.65)

we get

(5.66)

where we made use of the complete symmetry of $C_{\alpha \beta \rho \sigma }^{cdef}$. Inserting here Eq. (5.63), this becomes

(5.67)

The sum-integrals over R, P, U in the above expression are trivially carried out. Moreover, we note that

$$\displaystyle\begin{array}{rcl} \delta ^{ac}\delta ^{bd}\delta ^{ef}\delta _{ \mu \alpha }\delta _{\nu \beta }\delta _{\rho \sigma }C_{\alpha \beta \rho \sigma }^{cdef}& =& \delta ^{ef}\delta _{ \rho \sigma }\Bigl [f^{gae}f^{gbf}(\delta _{\mu \nu }\delta _{\rho \sigma } -\delta _{\mu \sigma }\delta _{\nu \rho }) \\ & & +f^{gaf}f^{gbe}(\delta _{\mu \nu }\delta _{\rho \sigma } -\delta _{\mu \rho }\delta _{\nu \sigma })\Bigr ] \\ & =& 2\,d\,f^{age}f^{bge}\delta _{ \mu \nu }\;,{}\end{array}$$

(5.68)

where we made use of the antisymmetry of the structure constants, as well as of the fact that $\delta _{\sigma \sigma } = d + 1 = 4 - 2\epsilon$. Noting that the structure constants furthermore satisfy f ^age f ^bge = N _c δ ^ab, we get in total

(5.69)

where

, cf. Eqs. (2.54) and (2.56). Note that the δ-functions as well as the colour and spacetime indices appear here just like in Eq. (5.64), allowing us to straightforwardly read off the contribution of this graph to $\Pi _{\mu \nu }^{}(K)$.

Next, we move on to the gluon loop originating from two cubic interaction vertices. Denoting the combination of δ-functions and momenta in Eq. (5.48) by

$$\displaystyle{ D_{\alpha \beta \gamma }(R,P,T)\; \equiv \;\delta _{\alpha \gamma }(R_{\beta } - T_{\beta }) +\delta _{\gamma \beta }(T_{\alpha } - P_{\alpha }) +\delta _{\beta \alpha }(P_{\gamma } - R_{\gamma })\;, }$$

(5.70)

we get for this contribution

(5.71)

where we made use of the complete symmetry of f ^cde D _{α β γ} (R, P, T) in simultaneous interchanges of all indices labelling a particular gauge field (for instance $c,\alpha,R \leftrightarrow d,\beta,P$).

Inserting Eq. (5.63) into the above expression, let us inspect in turn the colour indices, spacetime indices, and momenta. The colour contractions are easily carried out, and result in the overall factor

$$\displaystyle{ \delta ^{ac}\delta ^{bg}\delta ^{dh}\delta ^{ei}f^{cde}f^{ghi} = f^{ade}f^{bde} = N_{\mathrm{ c}}\,\delta ^{ab}\;. }$$

(5.72)

The spacetime contractions can all be transported to the D-functions, noting that the effect can be summarized with the substitution rules α → μ, ζ → ν, η → β, ρ → γ. Taking advantage of this, the momentum dependence of the result can be deduced from

(5.73)

We can now easily integrate over R, U, V, X and T, which gives us

(5.74)

Finally, we are faced with the tedious task of inserting Eq. (5.70) into the above expression and carrying out all contractions—a task most conveniently handled using programming languages intended for carrying out symbolic manipulations, such as FORM [5]. Here we perform the contractions by hand, obtaining first

$$\displaystyle\begin{array}{rcl} & & D_{\mu \beta \gamma }(-K,P,K - P)\;D_{\nu \beta \gamma }(K,-P,-K + P) \\ & & \quad = -[\delta _{\mu \gamma }(-2K_{\beta } + P_{\beta }) +\delta _{\gamma \beta }(K_{\mu } - 2P_{\mu }) +\delta _{\beta \mu }(K_{\gamma } + P_{\gamma })] \\ & & \qquad \times \, [\delta _{\nu \gamma }(-2K_{\beta } + P_{\beta }) +\delta _{\gamma \beta }(K_{\nu } - 2P_{\nu }) +\delta _{\beta \nu }(K_{\gamma } + P_{\gamma })] \\ & & \quad = -\delta _{\mu \nu }(4K^{2} - 4K \cdot P + P^{2} + K^{2} + 2K \cdot P + P^{2}) \\ & & \qquad -\, (d + 1)(K_{\mu }K_{\nu } - 2K_{\mu }P_{\nu } - 2K_{\nu }P_{\mu } + 4P_{\mu }P_{\nu }) \\ & & \qquad -\, [(-2K_{\mu } + P_{\mu })(K_{\nu } - 2P_{\nu }) + (-2K_{\mu } + P_{\mu })(K_{\nu } + P_{\nu }) \\ & & \qquad + (K_{\mu } - 2P_{\mu })(K_{\nu } + P_{\nu }) + (\mu \leftrightarrow \nu )] \\ & & \quad = -\delta _{\mu \nu }[4K^{2} + (K - P)^{2} + P^{2}] - (d - 5)K_{\mu }K_{\nu } + (2d - 1)(K_{\mu }P_{\nu } + K_{\nu }P_{\mu }) \\ & & \qquad - (4d - 2)P_{\mu }P_{\nu }\;. {}\end{array}$$

(5.75)

Because the propagators in Eq. (5.74) are identical, we can furthermore simplify the structure K _μ P _ν + K _ν P _μ by renaming one of the integration variables as P → K − P in “one half of this term”, i.e. by writing

$$\displaystyle\begin{array}{rcl} K_{\mu }P_{\nu } + K_{\nu }P_{\mu }& \rightarrow & \frac{1} {2}{\Bigl [K_{\mu }P_{\nu } + K_{\nu }P_{\mu } + K_{\mu }(K_{\nu } - P_{\nu }) + K_{\nu }(K_{\mu } - P_{\mu })\Bigr ]} \\ & =& K_{\mu }K_{\nu }\;. {}\end{array}$$

(5.76)

Therefore a representation equivalent to Eq. (5.75) is

$$\displaystyle\begin{array}{rcl} & & D_{\mu \beta \gamma }(-K,P,K - P)\;D_{\nu \beta \gamma }(K,-P,-K + P) \\ & \rightarrow & -\delta _{\mu \nu }[4K^{2} + (K - P)^{2} + P^{2}] + (d + 4)K_{\mu }K_{\nu } - (4d - 2)P_{\mu }P_{\nu }\;.{}\end{array}$$

(5.77)

Inserting now Eq. (5.77) into Eq. (5.74), we observe that the result depends in a non-trivial way on the “external” momentum K. This is an important fact that plays a role later on. For the moment, we however note that since the tree-level gluon propagator of Eq. (5.63) is massless, the leading order pole position lies at K ² = 0. This may get shifted by the loop corrections that we are currently investigating, like in the case of a scalar field theory (cf. Eq. (3.95)). Since this correction is suppressed by a factor of $\mathcal{O}(g^{2})$, in our perturbative calculation we may insert K = 0 in Eq. (5.77), making only an error of $\mathcal{O}(g^{4})$. Proceeding this way, we get

(5.78)

Now, symmetries tell us that the integral in Eq. (5.78) can only depend on two second rank tensors, δ _{μ ν} and δ _μ0 δ _ν0, of which the latter originates from the breaking of Lorentz symmetry by the rest frame of the heat bath. Denoting P = ( p _n , p), this allows us to split the latter term into two parts according to (note that δ _{μ i} δ _{ν i}  = δ _{μ ν} −δ _μ0 δ _ν0)

(5.79)

At this point, let us inspect the familiar sum-integral (cf. Eq. (2.92))

(5.80)

Taking the derivative $T^{2} \frac{\mathrm{d}} {\mathrm{d}T^{2}} = \frac{T} {2} \, \frac{\mathrm{d}} {\mathrm{d}T}$ on both sides, we find

$$\displaystyle{ \frac{T} {2} \sum _{n=-\infty }^{\infty }\int _{ \mathbf{p}} \frac{1} {(2\pi nT)^{2} + p^{2}} - T\sum _{n=-\infty }^{\infty }\int _{ \mathbf{p}} \frac{(2\pi nT)^{2}} {[(2\pi nT)^{2} + \mathbf{p}^{2}]^{2}} = \frac{T^{2}} {12} + \mathcal{O}(\epsilon )\;, }$$

(5.81)

which can be used in order to solve for the only unknown sum-integral in Eq. (5.79),

(5.82)

Inserting this result into Eq. (5.79), we thereby obtain in d = 3 − 2ε dimensions

(5.83)

which turns Eq. (5.78) finally into

(5.84)

Moving on to the ghost loop, we apply the vertex of Eq. (5.50) but otherwise proceed as in Eq. (5.71). This produces

(5.85)

where the Grassmann nature of the ghosts induced a minus sign at the second equality sign.

Inserting now the gluon propagator from Eq. (5.63) and the ghost propagator from Eq. (5.46), we inspect in turn the colour indices, spacetime indices, and momenta. The colour contractions result in the familiar factor

$$\displaystyle{ \delta ^{ad}\delta ^{bh}\delta ^{eg}\delta ^{ic}f^{cde}f^{ghi} = f^{cae}f^{ebc} = -N_{\mathrm{ c}}\,\delta ^{ab}\;, }$$

(5.86)

whereas the spacetime indices can be directly transported to the momenta: δ _{μ α} δ _{ν β} R _α U _β  = R _μ U _ν . The momentum dependence of the full expression can then be written as

(5.87)

where we can this time integrate over P, V, U, R and X. Thereby, we obtain

(5.88)

where we renamed T → P. Repeating the trick of Eq. (5.76), this can be turned into

(5.89)

which in the K → 0 limit produces, upon setting d → 3 and using Eq. (5.83),

(5.90)

Finally, we consider the fermion loop, originating from the vertex of Eq. (5.51). Proceeding as above, we obtain

(5.91)

where the Grassmann nature of the fermions induced a minus sign. As noted earlier, the capital indices originating from the quark spinors stand both for colour and flavour quantum numbers.

Inserting next the gluon propagator from Eq. (5.63) and the fermion propagator from Eq. (5.47), let us once more inspect in turn the colour and flavour indices, Lorentz indices, and momenta. The colour and flavour contractions result this time in the factor

$$\displaystyle{ \delta ^{ac}\delta ^{bd}\delta _{ DA}^{}\delta _{BC}^{}T_{AB}^{c}T_{ CD}^{d} = \mathrm{Tr}\,[T^{a}T^{b}] = \frac{N_{\mathrm{f}}} {2} \;, }$$

(5.92)

where we assumed the flavours to be degenerate in mass and in addition took advantage of the assumed normalization of the fundamental representation generators T ^a. The spacetime indices yield on the other hand

$$\displaystyle\begin{array}{rcl} \delta _{\mu \alpha }\delta _{\nu \beta }\mathrm{Tr}\,[(-i\,/\!\!\!\!R\,\! + m)\gamma _{\alpha }^{}(-i\,/\!\!\!\!U\,\! + m)\gamma _{\beta }^{}]& =& 4[-R_{\sigma }U_{\rho }(\delta _{\sigma \mu }\delta _{\rho \nu } -\delta _{\sigma \rho }\delta _{\mu \nu } +\delta _{\sigma \nu }\delta _{\rho \mu }) \\ & & +m^{2}\delta _{ \mu \nu }] \\ & =& 4[\delta _{\mu \nu }(R \cdot U + m^{2}) - R_{\mu }U_{\nu } - R_{\nu }U_{\mu }],{}\end{array}$$

(5.93)

where we used standard results for the traces of Euclidean γ-matrices. The momentum dependence of the full expression can finally be written in the form

(5.94)

where integrations can now be performed over P, V, R, U and X.

Assembling everything, we obtain as the contribution of the quark loop diagram to the self-energy

(5.95)

where we again renamed T → P. For vanishing chemical potential, a shift like in Eq. (5.76) works also with fermionic four-momenta, so that this expression further simplifies to

(5.96)

The structure in the numerator of Eq. (5.96) is similar to that in Eq. (5.77), except that the Matsubara frequencies are fermionic. In particular, if we again set the external momentum to zero, and for simplicity also consider the limit T ≫ m, so that quark masses can be ignored, the entire term becomes proportional to

(5.97)

The relation in Eq. (5.82) continues to hold in the fermionic case, so setting d → 3, we get

(5.98)

Inserting here finally $\tilde{I} _{T}^{}(0) = -T^{2}/24$ from Eq. (4.75), we arrive at the final result for the fermionic contribution,

(5.99)

Summing together Eqs. (5.69), (5.84), (5.90) and (5.99) and omitting the terms of $\mathcal{O}(\epsilon )$, we find the surprisingly compact expression

(5.100)

It is important to note that all corrections have cancelled from the spatial part.⁵ Due to Ward-Takahashi identities (or more properly their non-Abelian generalizations, Slavnov-Taylor identities), the gauge field self-energy must be transverse with respect to the external four-momentum, which in the case of the Matsubara zero mode takes the form K = (0, k). Since we computed the self-energy with k = 0, the transverse structure $\delta _{ij}^{}k^{2} - k_{i}k_{j}$ cannot appear, and the spatial part must vanish altogether.

The result obtained above has a direct physical meaning. Indeed, we recall from the discussion of scalar field theory, Eq. (3.70), that Eqs. (5.64) and (5.100) can be interpreted as a (resummed) full propagator of the form

(5.101)

where

$$\displaystyle{ m_{\mbox{ E}}^{2}\; \equiv \; g^{2}T^{2}{\biggl (\frac{N_{\mathrm{c}}} {3} + \frac{N_{\mathrm{f}}} {6} \biggr )} }$$

(5.102)

is called the Debye mass parameter. Its existence corresponds to the fact the colour-electric field A ₀ gets exponentially screened in a thermal plasma like the scalar field propagator in Eq. (3.46). In contrast, the colour-magnetic field A _i does not get screened, at least at this order.

We conclude with two remarks:

• If we consider the full Standard Model rather than QCD (the corresponding Euclidean Lagrangian is given on p. 286), then there is a separate thermal mass for the zero components of all three gauge fields, and for the Matsubara zero mode of the Higgs field. These can be found in [6].

• The definition of a Debye mass becomes ambiguous at higher orders. One possibility is to define it as a “matching coefficient” in a certain “effective theory”; this is discussed in more detail in Sect. 6.2, cf. Eq. (6.37). In that case higher-order corrections to the expression in Eq. (5.102) can be computed [7]. On the other hand, if we want to define the Debye mass as a physical quantity, the result becomes non-perturbative already at the next-to-leading order [8], and a proper definition and extraction requires a lattice approach [9].

Free Energy Density to $\mathcal{O}(g^{3})$

As an application of the results of the previous section, we now compute the free energy density of QCD up to $\mathcal{O}(g^{3})$, parallelling the method introduced for scalar field theory around Eq. (3.94). We recall that the essential insight in this treatment was to supplement the quadratic part of the Lagrangian for the Matsubara zero modes by an effective thermal mass computed from the full propagator, and to treat minus the same term as part of the interaction Lagrangian. The “non-interacting” free energy density computed with the corrected propagator then yields the result for the ring sum, whereas the bilinear interaction term cancels the corresponding, infrared (IR) divergent contributions order by order in a loop expansion.⁶ In the present case, given the result of Eq. (5.101), we see that only the temporal components of the gauge fields need to be corrected with a mass term. This is in accordance with the gauge transformation properties of static colour-electric and colour-magnetic fields, which forbid the spatial components from having a mass; we return to this in Sect. 6.2.

With the above considerations in mind, the correction of $\mathcal{O}(g^{3})$ [10] to the tree-level result in Eq. (5.61) can immediately be written down, if we employ Eq. (2.87) and take into account that there are N _c ² − 1 copies of the gauge field. This produces

$$\displaystyle\begin{array}{rcl} \left.f_{(\frac{3} {2} )}(T)\right \vert _{\mbox{ QCD}}& =& (N_{\mathrm{c}}^{2} - 1){\biggl ( -\frac{Tm_{\mbox{ E}}^{3}} {12\pi } \biggr )} \\ & =& (N_{\mathrm{c}}^{2} - 1)T^{4}g^{3}{\biggl ( - \frac{1} {12\pi }\biggr )}{\biggl (\frac{N_{\mathrm{c}}} {3} + \frac{N_{\mathrm{f}}} {6} \biggr )}^{\frac{3} {2} } \\ & =& -\frac{\pi ^{2}T^{4}} {3} 2(N_{\mathrm{c}}^{2} - 1){\biggl (\frac{g^{2}} {4\pi ^{2}}\biggr )}^{\frac{3} {2} }{\biggl ( \frac{N_{\mathrm{c}}} {3} + \frac{N_{\mathrm{f}}} {6} \biggr )}^{\frac{3} {2} }\;,{}\end{array}$$

(5.103)

where the effective mass m _E was taken from Eq. (5.102).

Next, we consider the contributions of $\mathcal{O}(g^{2})$. In analogy with Eq. (3.99), these terms [11, 12] come from the non-zero mode contributions to the 2-loop “vacuum”-type graphs in [cf. Eq. (3.12)]

$$\displaystyle{ \left.f_{(1)}(T)\right \vert _{\mbox{ QCD}} ={\Bigl \langle S_{\mbox{ I}}^{} -\frac{1} {2}S_{\mbox{ I}}^{2} +\ldots \Bigr \rangle} _{ 0,\mbox{ c},\,\mbox{ drop overall $\int _{X}$}}\;. }$$

(5.104)

It is useful to compare this expression with the computation of the full propagator in the previous section, Eq. (5.100). We note that, apart from an overall minus sign, the two computations are quite similar at the present order. In fact, we claim that we only need to “close” the gluon line in the results of the previous section and simultaneously divide the graphs by − 1∕2n, where n is the number of gluon lines in the vacuum graph in question. Let us prove this by direct inspection.

Consider first the $\mathcal{O}(g^{2})$ contribution from the 4-gluon vertex. In vacuum graphs, this leads to the combinatorial factor

$$\displaystyle{ \langle \tilde{A}\,\tilde{A}\,\tilde{A}\,\tilde{A}\rangle _{0,\mbox{ c}}^{} = 3\,\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;, }$$

(5.105)

whereas in the propagator calculation we arrived at

$$\displaystyle{ -\langle \tilde{A}\,\tilde{A}\;\;\tilde{A}\,\tilde{A}\,\tilde{A}\,\tilde{A}\rangle _{0,\mbox{ c}}^{} = -4 \times 3\,\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;. }$$

(5.106)

The difference is − 4 = −2 × n, with n = 2 being the number of contractions in Eq. (5.105). Similarly, with the contribution from two 3-gluon vertices, the vacuum graphs lead to the combinatorial factor

$$\displaystyle{ -\langle \tilde{A}\,\tilde{A}\,\tilde{A}\;\;\tilde{A}\,\tilde{A}\,\tilde{A}\rangle _{0,\mbox{ c}}^{} = -3 \times 2\,\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;, }$$

(5.107)

whereas when considering the propagator we got

$$\displaystyle{ \langle \tilde{A}\,\tilde{A}\;\;\tilde{A}\,\tilde{A}\,\tilde{A}\;\;\tilde{A}\,\tilde{A}\,\tilde{A}\rangle _{0,\mbox{ c}}^{} = 6 \times 3 \times 2\,\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;. }$$

(5.108)

There is evidently a difference of − 6 = −2 × n, with n = 3 the number of contractions in Eq. (5.107). Finally, the ghost and fermion contributions to the vacuum graphs lead to the combinatorial factor

$$\displaystyle{ -\langle \;\tilde{\!\bar{c}}\,\tilde{A}\,\tilde{c}\;\;\tilde{\!\bar{c}}\,\tilde{A}\,\tilde{c}\rangle _{0,\mbox{ c}}^{} =\langle \tilde{ A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{c}\;\tilde{\!\bar{c}}\rangle _{0}^{}\;\langle \tilde{c}\;\tilde{\!\bar{c}}\rangle _{0}^{}\;, }$$

(5.109)

whereas in the propagator computation we obtained

$$\displaystyle{ \langle \tilde{A}\,\tilde{A}\;\;\tilde{\!\bar{c}}\,\tilde{A}\,\tilde{c}\;\;\tilde{\!\bar{c}}\,\tilde{A}\,\tilde{c}\rangle _{0,\mbox{ c}}^{} = -2\,\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{A}\,\tilde{A}\rangle _{0}^{}\;\langle \tilde{c}\;\tilde{\!\bar{c}}\rangle _{0}^{}\;\langle \tilde{c}\;\tilde{\!\bar{c}}\rangle _{0}^{}\;. }$$

(5.110)

So once more a difference of − 2 × n, with n = 1 the number of gluon contractions in Eq. (5.109).

With the insights gained, the contribution of the 4-gluon vertex to the free energy density of QCD can be extracted directly from Eq. (5.69):

(5.111)

The contribution of the 3-gluon vertices is similarly obtained from Eq. (5.74): noting from Eq. (5.75) that

$$\displaystyle\begin{array}{rcl} & & \delta _{\mu \nu }\;D_{\mu \beta \gamma }(-K,P,K - P)\;D_{\nu \beta \gamma }(K,-P,-K + P) \\ & & \quad = -(d + 1)[4K^{2} + (K - P)^{2} + P^{2}] - (d - 5)K^{2} + 2(2d - 1)K \cdot P - (4d - 2)P^{2} \\ & & \quad = -\{K^{2}[5d + 5 + d - 5] + K \cdot P[-2d - 2 - 4d + 2] + P^{2}[2d + 2 + 4d - 2]\} \\ & & \quad = -3d\{K^{2} + (K - P)^{2} + P^{2}\}\;, {}\end{array}$$

(5.112)

we get from Eq. (5.74)

(5.113)

Note that unlike in Eq. (5.78), for the present calculation it was crucial to keep the full K-dependence in the two-point function, because all values of K are now integrated over.

Similarly, the contribution of the ghost loop can be extracted from Eq. (5.88), producing

(5.114)

whereas the contribution of the fermion loop is obtained from Eq. (5.95):

(5.115)

Simplifying the last expression by setting m∕T → 0, we get

(5.116)

Here careful attention needed to be paid to the nature of the Matsubara frequencies appearing in the propagators.

Adding together the terms from Eqs. (5.111), (5.113), (5.114) and (5.116), setting d = 3 (note the absence of divergences), and using I _T (0) = T ²∕12, $\tilde{I}_{T}^{}(0) = -T^{2}/24$, we get as the full $\mathcal{O}(g^{2})$ contribution to the free energy density

$$\displaystyle\begin{array}{rcl} \left.f_{(1)}(T)\right \vert _{\mbox{ QCD}}& =& g^{2}(N_{\mathrm{ c}}^{2} - 1) \frac{T^{4}} {144}{\biggl [{\biggl (3 -\frac{9} {4} + \frac{1} {4}\biggr )}N_{\mathrm{c}} -{\biggl (-2 \times \frac{1} {2} -\frac{1} {4}\biggr )}N_{\mathrm{f}}\biggr ]} \\ & =& g^{2}(N_{\mathrm{ c}}^{2} - 1) \frac{T^{4}} {144}{\biggl (N_{\mathrm{c}} + \frac{5} {4}N_{\mathrm{f}}\biggr )} \\ & =& -\frac{\pi ^{2}T^{4}} {90} (N_{\mathrm{c}}^{2} - 1){\biggl ( -\frac{5} {2} \frac{g^{2}} {4\pi ^{2}}\biggr )}{\biggl (N_{\mathrm{c}} + \frac{5} {4}N_{\mathrm{f}}\biggr )}\;. {}\end{array}$$

(5.117)

Adding to this the effects of Eqs. (5.61) and (5.103), the final result reads

$$\displaystyle\begin{array}{rcl} \left.f(T)\right \vert _{\mbox{ QCD}}& =& -\frac{\pi ^{2}T^{4}} {45} (N_{\mathrm{c}}^{2} - 1)\biggl \{1 + \frac{7} {4} \frac{N_{\mathrm{f}}N_{\mathrm{c}}} {N_{\mathrm{c}}^{2} - 1} -\frac{5} {4}\biggl (N_{\mathrm{c}} + \frac{5} {4}N_{\mathrm{f}}\biggr )\frac{\alpha _{s}} {\pi } \\ & & +\,30\biggl (\frac{N_{\mathrm{c}}} {3} + \frac{N_{\mathrm{f}}} {6} \biggr )^{\frac{3} {2} }\biggl ( \frac{\alpha _{s}} {\pi } \biggr )^{\frac{3} {2} } + \mathcal{O}(\alpha _{s}^{2})\biggr \}\;, {}\end{array}$$

(5.118)

were we have denoted α _s  ≡ g ²∕4π.

A few remarks are in order:

• The result in Eq. (5.118) can be compared with that for a scalar field theory in Eq. (3.93). The general structure is identical, and in particular the first relative correction is negative in both cases. This means that the interactions between the particles in a plasma tend to decrease the pressure that the plasma exerts.

• The second correction to the pressure turns out to be positive. Such an alternating structure indicates that it may be difficult to quantitatively estimate the magnitude of radiative corrections to the non-interacting result. We may recall, however, that $1 -\frac{1} {2} + \frac{1} {3} -\frac{1} {4}\ldots =\ln 2 = 0.693\ldots$, whereas $1 -\frac{1} {2} -\frac{1} {3} -\frac{1} {4}\ldots = -\infty$; in principle an alternating structure is beneficial as far as (asymptotic) convergence goes.

• The coefficients of the four subsequent terms, of orders $\mathcal{O}(\alpha _{s}^{2}\ln \alpha _{s})$, $\mathcal{O}(\alpha _{s}^{2})$, $\mathcal{O}(\alpha _{s}^{5/2})$, and $\mathcal{O}(\alpha _{s}^{3}\ln \alpha _{s})$, are also known [13–17]. Like for scalar field theory, this progress is possible thanks to the use of effective field theory methods that we discuss in the next chapter.

Appendix A: Do Ghosts Develop a Thermal Mass?

In the computation of the present section, we have assumed that only the Matsubara zero modes of the fields A ₀ ^a need to be resummed, i.e. get an effective thermal mass. The fact that fermions do not need to be resummed is clear, but the case of ghosts is less obvious. To this end, let us finish the section by demonstrating that ghosts do not get any thermal mass, and thus behave like the spatial components of the gauge fields.

The tree-level ghost propagator is given in Eq. (5.46), and we now consider corrections to this expression. The relevant vertex is the one in Eq. (5.50), yielding for the only correction of $\mathcal{O}(g^{2})$

(5.119)

where an even number of minus signs originated from the commutations of Grassmann fields. Inserting here the gluon propagator from Eq. (5.63) as well as the free ghost propagator from Eq. (5.46), we again end up inspecting colour indices, Lorentz indices, and momenta in the resulting expression. The colour contractions are seen to result in the factor

$$\displaystyle{ \delta ^{ac}\delta ^{eg}\delta ^{ib}\delta ^{dh}f^{cde}f^{ghi} = f^{ade}f^{edb} = -N_{\mathrm{ c}}\,\delta ^{ab}\;, }$$

(5.120)

whereas the spacetime indices yield simply δ _{α β} . Finally, the momentum dependence can be written in the form

(5.121)

where we can integrate over R, U, X, V and P. Thereby we obtain as the final result

(5.122)

where we again renamed T → P.

The expression in Eq. (5.122) is proportional to the external momentum K. Therefore, it does not represent an effective mass correction, but is rather a “wave function (re)normalization” contribution, as can be made explicit through a shift like in Eq. (5.76).