In this section, we review the calculation of the gluon self-energy in the imaginary time formalism [63]. We start with the calculation of photon self-energy. This will be done at the one-loop approximation and in the high-temperature limit, which means that the tem-perature is much larger than the electron mass and the external momenta. In the end, we continued these results analytically to Minkowski space. We also generalize these calcula-tions to the QCD case. It turns out the gluonic contribucalcula-tions to the self-energy have the same structure as the quark ones except the trivial color factor. Since the one-loop photon self-energy contains no gauge boson propagator, it is gauge independent. Consequently, the same holds for the gluon self-energy. We also show that the results obtained in the diagrammatic approach is equivalent to those obtained in the transport theory approach as we discussed in Chapter 2.
The one-loop photon self-energy can be expressed as Πµν = e2P Note that in imaginary time formalism, we use Euclidean metric δµν which is a diagonal matrix defined by δµν = diag(1,1,1,1) with µ, ν = 1, . . . ,4. To distinguish the notations in Minkowski space, we denote Euclidean momentum with a tilde. In Eq. (3.16) the four-momentum ˜Pµ = (p4,p) = (−ω,˜ p) and ˜Kµ = (k4,k) = (−ω˜n,k) 2. The unit vector is defined as ˆk = k/k. Our convention for Dirac matrices is that γ-matrices in Euclidean space obey anti-commutation relations: {γµ, γν} = −2δµν with γ4 = iγ0. As a result, we use the HTL approximation. The loop momentum ˜K is assumed at the order of T, while the external momentum ˜P is at the order ofgT. In the weak-coupling limit, we have K˜ ≫P˜.
One important step of the calculation in imaginary time formalism is to perform the Matsubara sums. Here we will encounter the following three sums:
TX
2Note that in imaginary time formalism, the so-calledMatsubarafrequencies have discrete values. The loop momenta for which we should perform the frequency sum, is labeled by (˜ωn,k). For fermions, the Matsubara frequencies are given by ˜ωn = (2n+ 1)πT with integern. We don’t need to perform the frequency sum for the external momentum ˜P, however, we still label its frequency as ˜ωwithout the index n.
3.2 The calculation of gluon self-energy in imaginary time formalism 27
where the subscripts “11” and “22” correspond to the momentumkandk−p, respectively.
E1 = k, E2 = |k−p| and n1, n2 are the equilibrium distribution functions as defined in Eq. (2.43). For example,n1=neq(k).
The calculation of the second term in Eq. (3.16) which we denote as Π2µν, is straightforward once we carry out theMatsubara sums
Π2µν =−4e2P
In fact, this integral is divergent. However, this divergence is temperature independent and it is absorbed into the zero temperature renormalization.
The calculation of the first term in Eq. (3.16) denoted as Π1µν is more complicated than the second term. For simplicity, we will consider the calculation in high temperature limit and use the HTL approximation, E1−E2 ≈ p·kˆ and n1−n2 ≈ dneqdk(k)p·k. Firstly, weˆ consider the spatial components of Π1µν
Π1ij =−e2 above expression, we use Eq. (3.17) for frequency sum and the temperature independent divergence is absorbed into the zero temperature renormalization as before. Note that, at high temperature, the leading contribution to the photon self-energy is proportional toT2, so the subleading terms have been neglected. For example, there should be such a term Rkdkdneqdk(k) in Eq. (3.21). However, this term is only proportional to T which can be neglected in the high temperature limit.
For the integral over the solid angle, it is easy to show that Z
After performing the integral overk, we get Πij = e2T2
where the contribution from the second term in Eq. (3.16) was included.
28 3 Gluon self-energy from finite-temperature field theory
Using the mass-shell constraint for the massless hard particles, ˜ωn2 =−k2, the temporal component of Π1µν can be calculated similarly as the spatial components. The result can be expressed as
For the “4i” components, we have
Π4i = e2T2
where the Eq. (3.18) is used for the Matsubarasums.
Combining all the expressions above, the final result for the photon self-energy is of the form
The symmetry of the self-energy tensor is obvious. It is also easy to show this tensor is transverse, i.e. ˜PµΠµν = e23T2 R dΩ
4π(i˜ωKˆ˜ν−ωδ˜ 4ν) = 03.
As we did in Chapter 2, we can decompose the self-energy with the transverse and longi-tudinal projectors. In Euclidean space, the two projectors read
A44= A4i= 0, Aij =δij −pipj/p2,
The corresponding transverse and longitudinal projections are related to the “xx” and “4z”
components of the self-energy. We can perform the integration over the solid angle by taking p parallel to thez-axis. As a result, we have
Π4z = e2T2
3.2 The calculation of gluon self-energy in imaginary time formalism 29
Figure 3.1: Feynmandiagrams contribute to gluon self-energy at one-loop approximation.
The solid lines: quarks; The curly lines: gluons; The dashed lines: ghosts; The gray solid lines indicate the momentum direction of gluons.
It is easy to show that the corresponding projections in Euclidean space are ΠL=−p
Finally, we should analytically continue Eqs. (3.31) and (3.32) from Euclidean space to Minkowski space. This continuation is performed by making the substitutionsi˜ω →ω+iǫ4 and ˜P2 → −P2. We find that after the continuation Eqs. (3.31) and (3.32) coincide with Eqs. (2.53) and (2.52) which are obtained in the transport theory approach.
Contrary to the photon self-energy, in principle, the gluon self-energy is not gauge-fixing independent. However, we find that in HTL approximation, the gluon self-energy does not depend on the choice of gauge. Here, we compute the gluon self-energy in Euclidean space in Feynman gauge. There are four diagrams contribute at one-loop order as shown in Fig. 3.1. Adding the contributions from gluon and ghost loops, we have
Π(g)µν( ˜P) =−g2NcP Z
K˜
d4K˜
(2π)4(4 ˜KµK˜ν −2 ˜K2δµν) ˜D( ˜K) ˜D( ˜K−P˜), (3.33) where ˜D( ˜K) is related to the gauge boson propagator in Feynman gauge. The inverse D( ˜˜ K) equals to ˜ωn2+k2 with ˜ωn= 2πnT. The fermion loop gives the similar expression as
4Such a substitution corresponds to the retarded solution in Minkowski space.
30 3 Gluon self-energy from finite-temperature field theory
Note that when performing the frequency sums, one goes from a bosonic to a fermionic loop by the substitution N(E) → −n(E). In HTL approximation, we have the following relations where we make use of these equations
Z ∞ Taking the above equations into account, we find the expression for the gluon self-energy has the similar structure as that we have for photon in QED [86]
Πµν( ˜P) = Π(fµν)( ˜P) + Π(g)µν( ˜P) The final result then takes the same form as the photon self-energy except the color factor and coupling constant:
The only modification to the QED result in Eq. (3.26) is a change of the Debye mass.
Again, the gluon self-energy obeys the Ward identity ˜PµΠµν = 0.