An introduction to thermal field theory and some of its application

Abstract

In this article an introduction to the thermal field theory within imaginary time vis-a-vis Matsubara formalism has been discussed in detail. The imaginary time formalism has been introduced through both the operatorial and the functional integration method. The prescription to perform frequency sum for boson and fermion has been discussed in detail. Green’s function both in Minkowski time as well as in Euclidean time has been derived. The tadpole diagram in $\lambda \phi ^4$ theory and the self-energy in $\lambda \phi ^3$ theory have been computed and their consequences have also been discussed. The basic features of general two point functions, such as self-energy and propagator, for both fermions and bosons in presence of a heat bath have been discussed. The imaginary time has also been introduced from the relation between the functional integral and the partition function. Then the free partition functions and thermodynamic quantities for scalar, fermion and gauge field, and interacting scalar field have been obtained from first principle calculation. The quantum electrodynamics (QED) and gauge fixing have been discussed in details. The one-loop self-energy for electron and photon in QED have been obtained in hard thermal loop (HTL) approximation. The dispersion properties and collective excitations of both electron and photon in a material medium in presence of a heat bath have been presented. The spectral representation of fermion and gauge boson propagators have been obtained. In HTL approximation, the generalisation of QED results of two point functions to quantum chromodynamics (QCD) have been outlined that mostly involve group theoretical factors. Therefore, one learns about the collective excitations in a QCD plasma from the acquired knowledge of QED plasma excitations. Then, some subtleties of finite temperature field theory have been outlined. As an effective field theory approach the HTL resummation and the HTL perturbation theory (HTLpt) have been introduced. The leading order (LO), next-to-leading order (NLO) and next-to-next-leading order (NNLO) free energy and pressure for deconfined QCD medium created in heavy-ion collisions have been computed within HTLpt. The general features of the deconfined QCD medium have also been outlined with non-perturbative effects like gluon condensate and Gribov–Zwanziger action. The dilepton production rates from quark–gluon plasma with these non-perturbative effects have been computed and discussed in details.

A Appendix

1.1 A.1 One-loop fermionic sum integrals

The dimensionally regularized fermionic sum-integrals are defined as,

$$\begin{aligned} \sum{\kern-15pt}\int\limits _{\{ P\} }= &\, {} \left( \frac{e^{\gamma _E}\Lambda ^2}{4\pi }\right) ^\epsilon T\sum \limits _{\begin{array}{c} \{p_0=i\omega _n\}\\ \omega _n=(2n+1)\pi T-i\mu \end{array}}\int \frac{\text{d}^{d-2\epsilon }p}{(2\pi )^{d-2\epsilon }}, \end{aligned}$$

(A.1)

where $d-2\epsilon$ is the spatial dimension, P is the fermion loop momentum, $\Lambda$ is the $\overline{\text {MS}}$ renormalization scale that introduces the factor $\left( \frac{e^{\gamma _E}}{4\pi }\right) ^\epsilon$ along with it, where $\gamma _E$ being the Euler–Mascheroni constant.

The result of various fermionic sum-integrals are listed below:

$$\begin{aligned} 2\sum{\kern-16pt}\int\limits _{\{P\}} \ln \left( P^2\right)&= \frac{7\pi ^2 T^4}{180} + \frac{\mu ^2T^2}{6} +\frac{\mu ^4}{12\pi ^2} \\ &= \frac{7\pi ^2T^4}{180}\left( 1+\frac{120}{7}\hat{\mu }^2+\frac{240}{7} \hat{\mu }^4\right) \, , \end{aligned}$$

(A.2a)

$$\begin{aligned} \sum{\kern-16pt}\int\limits _{\{P\}}\frac{1}{P^2}&=\frac{T^2}{24}\left( \frac{\Lambda }{4\pi T}\right) ^{2\epsilon }\\ &\left[ 1+12{\hat{\mu }}^2+2\epsilon \left( 1+12{\hat{\mu }}^2 +12\aleph (1,z)\right) \right] , \end{aligned}$$

(A.2b)

$$\begin{aligned} \sum{\kern-16pt}\int\limits _{\{ P\} }\frac{1}{P^4}&=\frac{1}{\left( 4\pi \right) ^2}\left( \frac{\Lambda }{4\pi T}\right) ^{2\epsilon }\Bigg [\frac{1}{\epsilon }-\aleph (z)\Bigg ], \end{aligned}$$

(A.2c)

$$\begin{aligned} \sum{\kern-16pt}\int\limits _{\{P\}}\frac{1}{p^2P^2}&= -\frac{2}{d-2} \sum{\kern-16pt}\int\limits _{\{ P\} }\frac{1}{P^4}, \end{aligned}$$

(A.2d)

$$\begin{aligned} \sum{\kern-16pt}\int\limits _{\{P\}}\frac{1}{p^2P^2}{{\mathcal {T}}}_P&=-\frac{2\Delta _3}{d-2} \sum{\kern-16pt}\int\limits _{\{P\}}\frac{1}{P^4}, \end{aligned}$$

(A.2e)

$$\begin{aligned} \sum{\kern-16pt}\int\limits _{\{P\}}\frac{1}{p^2P^2}{{\mathcal {T}}}_P^2&=-\frac{2\Delta _4''}{d-2} \sum{\kern-16pt}\int\limits _{\{P\}}\frac{1}{P^4}, \end{aligned}$$

(A.2f)

$$\begin{aligned} \sum{\kern-16pt}\int\limits _{\{P\}}\frac{1}{p_0^2P^2}{{\mathcal {T}}}_P^2&=-\frac{2\Delta _3''}{d-2} \sum{\kern-15pt}\int\limits _{\{P\}}\frac{1}{P^4}, \end{aligned}$$

(A.2g)

where the angular integrations are given as

$$\begin{aligned} \Delta _3&= \left\langle \frac{1-c^{4-d}}{1-c^2}\right\rangle c = \ln 2+\left( \frac{\pi ^2}{6}-(2-\ln 2)\ln 2\right) \epsilon \nonumber \\&~~~~+\left\{ \frac{2}{3}(\ln 2)^2(\ln 2 -3)+\frac{\pi ^2}{3}(\ln 2-1)+\zeta (3)\right\} \\ &\epsilon ^2+\mathcal {O}[\epsilon ]^3 \, , \end{aligned}$$

(A.3a)

$$\begin{aligned} \Delta _3''&= \left\langle \frac{1-c_1^{4-d}}{(1-c_1^2)(c_1^2-c_2^2)}+c_1\leftrightarrow c_2\right\rangle\\ & ci = -\frac{\pi ^2}{12} + \left( \frac{\pi ^2}{3} - \frac{\zeta (3)}{2}\right) \epsilon + \mathcal {O}(\epsilon ^2), \end{aligned}$$

(A.3b)

$$\begin{aligned} \Delta _4''&= \left\langle \frac{1-c_1^{6-d}}{(1-c_1^2)(c_1^2-c_2^2)}+c_1\leftrightarrow c_2\right\rangle ci \nonumber \\&= -\frac{\pi ^2}{12} + \ln 4 + \left( \frac{\pi ^2}{3} - \ln 4 (2 - \ln 2) - \frac{\zeta (3)}{2}\right) \epsilon + \mathcal {O}(\epsilon ^2) \, , \end{aligned}$$

(A.3c)

with

$$\begin{aligned} \aleph (z)&=-2\gamma _E-4\ln 2+14\zeta (3)\hat{\mu }^2-62\zeta (5)\hat{\mu }^4\\ &+254\zeta (7)\hat{\mu }^6+\mathcal{O}(\hat{\mu }^8), \end{aligned}$$

(A.4a)

$$\begin{aligned} \aleph (1,z)&=-\frac{1}{12}\left( \ln 2-\frac{\zeta '(-1)}{\zeta (-1)}\right) - \left( 1-2\ln 2-\gamma _E\right) \hat{\mu }^2\\ &-\frac{7}{6}\zeta (3)\hat{\mu }^4 + \frac{31}{15}\zeta (5)\hat{\mu }^6+\mathcal{O}(\hat{\mu }^8)\, . \end{aligned}$$

(A.4b)

1.2 A.2 One-loop bosonic sum integrals

The dimensionally regularized bosonic sum-integrals are defined as,

$$\begin{aligned} \sum{\kern-15pt}\int\limits _{ P}= &\, {} \left( \frac{e^{\gamma _E}\Lambda ^2}{4\pi }\right) ^\epsilon T\sum \limits _{\begin{array}{c} p_0=i\omega _n\\ \omega _n=2n\pi T \end{array}}\int \frac{\text{d}^{d-2\epsilon }p}{(2\pi )^{d-2\epsilon }}, \end{aligned}$$

(A.5)

where $d-2\epsilon$ is the spatial dimension, P is the boson loop momentum, $\Lambda$ is the $\overline{\text {MS}}$ renormalization scale that introduces the factor $\left( \frac{e^{\gamma _E}}{4\pi }\right) ^\epsilon$ along with it, where $\gamma _E$ being the Euler–Mascheroni constant.

Below we list various bosonic sum-integrals:

$$\begin{aligned} \sum{\kern-15pt}\int\limits _P \frac{1}{P^2}&= -\frac{T^2}{12}\left( \frac{\Lambda }{4\pi T}\right) ^{2\epsilon }\\ &\Bigg [1+2\epsilon \left( 1+\frac{\zeta '(-1)}{\zeta (-1)}\right) +\mathcal {O}[\epsilon ]^2\Bigg ], \end{aligned}$$

(A.6a)

$$\begin{aligned} \sum{\kern-15pt}\int\limits _P \frac{1}{p^2P^2}&= -\frac{2}{(4\pi )^2}\left( \frac{\Lambda }{4\pi T}\right) ^{2\epsilon } \\ &\Bigg [\frac{1}{\epsilon }+2\gamma _E+2 +\epsilon \left( 4+4\gamma _E+\frac{\pi ^2}{4}-4\gamma _1\right) +\mathcal {O}[\epsilon ]^2 \Bigg ], \end{aligned}$$

(A.6b)

$$\begin{aligned} \sum{\kern-15pt}\int\limits _P \frac{1}{P^4}&= \frac{1}{(4\pi )^2}\left( \frac{\Lambda }{4\pi T}\right) ^{2\epsilon }\\ &\left[ \frac{1}{\epsilon } +2\gamma _E+\epsilon \left( \frac{\pi ^2}{4}-4\gamma _1\right) +\mathcal {O}[\epsilon ]^2\right] \, , \end{aligned}$$

(A.6c)

$$\begin{aligned} \sum{\kern-15pt}\int\limits _{P} \frac{{{\mathcal {T}}}_P}{p^4}&=-\frac{1}{(4\pi )^2}\left( \frac{\Lambda }{4\pi T}\right) ^{2\epsilon }\\ &\left[ \frac{1}{\epsilon } +2\gamma _E+2\ln 2+\mathcal {O}[\epsilon ] \right] \, , \end{aligned}$$

(A.6d)

$$\begin{aligned} \sum{\kern-15pt}\int\limits _{P} \frac{{{\mathcal {T}}}_P}{p^2P^2}&=-\frac{1}{(4\pi )^2}\left( \frac{\Lambda }{4\pi T}\right) ^{2\epsilon }\\ & \left[ 2\ln 2 \left( \frac{1}{\epsilon }+2\gamma _E\right) + 2\ln ^2 2 +\frac{\pi ^2}{3}+\mathcal {O}[\epsilon ] \right] \, , \end{aligned}$$

(A.6e)

$$\begin{aligned} &\sum{\kern-15pt}\int\limits _{P} \frac{{{\mathcal {T}}}^2_P}{p^4}=-\frac{2}{3}\frac{1}{(4\pi )^2}\left( \frac{\Lambda }{4\pi T}\right) ^{2\epsilon }\\ & \left[ \left( 1+2\ln 2\right) \left( \frac{1}{\epsilon }+2\gamma _E\right) -\frac{4}{3}+\frac{22}{3}\ln 2 +2\ln ^2 2 +\mathcal {O}[\epsilon ] \right] \, . \end{aligned}$$

(A.6f)

1.3 A.3 Braaten–Pisarski–Yuan (BPY) prescription

Lets consider a complex function f(z) having branch cut

$$\begin{aligned} f(z) =\frac{1}{2\pi i} \oint \frac{f(\xi )\ \text{d}\xi }{\xi -z} \, . \end{aligned}$$

(A.7)

considering $\xi =x+i\epsilon$, one can write

$$\begin{aligned} f(z) =\frac{1}{2\pi i} \int \limits _{-\infty }^{+\infty } \frac{f(x+i\epsilon ) -f(x-i\epsilon )}{x-z} \, dx \\ = \frac{1}{2\pi i} \int \limits _{-\infty }^{+\infty } \frac{\text {Disc} f(x+i\epsilon )}{x-z} dx \,. \end{aligned}$$

(A.8)

where the discontinuity is related to the imaginary part of a complex function as

$$\begin{aligned} \text {Disc} f(x+i\epsilon ) = f(x+i\epsilon )-f(x-i\epsilon ) = 2i \, \text { Im} f(x+i\epsilon ) \, . \end{aligned}$$

(A.9)

Combining (A.8) and (A.9) one can write

$$\begin{aligned} f(z) = \frac{1}{\pi } \int \limits _{-\infty }^{+\infty } \frac{\text { Im} \, f(x+i\epsilon )}{x-z} dx = \int \limits _{-\infty }^{+\infty } \frac{ \rho (x)}{x-z} dx \end{aligned}$$

(A.10)

where the spectral density $\rho$ is defined as

$$\begin{aligned} \rho (x) = \frac{1}{\pi } \text {Im} \, f(x+i\epsilon ) \, . \end{aligned}$$

(A.11)

The spectral density $\rho _1 (\omega _1)$ is related to the any complex function $F_1(k_0)$ as given in (A.10)

$$\begin{aligned} F_1(k_0) =\int \limits _{-\infty }^{+\infty } \frac{\rho _1(\omega _1) \text{d}\omega _1}{\omega _1-k_0-i\epsilon _1} \, . \end{aligned}$$

(A.12)

We note that $K\equiv (k_0,\vec{\varvec{k}})$ is the fermionic momentum with $k_0 =(2m+1) i\pi T$.

Lets have,

$$\begin{aligned} \int \limits _{0}^{1/T} \text{d}\tau _1 \ e^{x\tau _1}= &\, {} \frac{ e^{x/T}-1}{x} \, \, \Rightarrow \,\, \frac{1}{x} = \frac{1}{e^{x/T}-1} \int \limits _{0}^{1/T} \text{d}\tau _1 \ e^{x\tau _1} \,, \end{aligned}$$

(A.13)

where T is the temperature. Now, considering $x=(\omega _1-k_0-i\epsilon _1)$ one can write (A.13) as

$$\begin{aligned} \frac{1}{\omega _1-k_0-i\epsilon _1} = \frac{1}{e^{\frac{(\omega _1-k_0)}{T}}-1} \int \limits _{0}^{1/T} \text{d}\tau _1 \ e^{(\omega _1-k_0-i\epsilon _1)\tau _1} \, . \end{aligned}$$

(A.14)

Combining (A.14) with (A.12), one gets

$$\begin{aligned} F_1(k_0) = \int \limits _{-\infty }^{+\infty } \frac{\rho _1(\omega _1) \text{d}\omega _1}{e^{\frac{(\omega _1-k_0)}{T}}-1} \int \limits _{0}^{1/T} \text{d}\tau _1 \ e^{(\omega _1-k_0-i\epsilon _1)\tau _1} \, . \end{aligned}$$

(A.15)

Now, using $e^{k_0/T} = e^{(2m+1)i\pi } = -1$, one can write as

$$\begin{aligned} F_1(k_0)= &\, {} - \int \limits _{-\infty }^{+\infty } \frac{\rho _1(\omega _1) \text{d}\omega _1}{e^{\frac{\omega _1}{T}}-1} \int \limits _{0}^{1/T} \text{d}\tau _1 \ e^{(\omega _1-k_0-i\epsilon _1)\tau _1} \nonumber \\= &\, {} - \int \limits _{-\infty }^{+\infty } n_{\mathrm{F}}(\omega _1) \ \rho _1(\omega _1) \, \text{d}\omega _1 \int \limits _{0}^{1/T} \text{d}\tau _1 \ e^{(\omega _1-k_0-i\epsilon _1)\tau _1} \,. \end{aligned}$$

(A.16)

Similarly, one can write another complex function $F_2(q_0)$ as

$$\begin{aligned} F_2(q_0)= &\, {} - \int \limits _{-\infty }^{+\infty } n_{\mathrm{F}}(\omega _2) \ \rho _2(\omega _2) \, \text{d}\omega _2 \int \limits _{0}^{1/T} \text{d}\tau _2 \ e^{(\omega _2-q_0-i\epsilon _2)\tau _2} \,, \end{aligned}$$

(A.17)

where $q_0=(p_0-k_0)$ and P is the bosonic momentum with $p_0=2mi\pi T$.

We would like to compute the imaginary part of the product of two complex functions $T\sum _{k_0} F_1(k_0)F_2(q_0)$:

$$\begin{aligned} &\text {Im} \, \, \, T\sum _{k_0} F_1(k_0)F_2(q_0)\\ & \quad = \, {} \text {Im} \, \, \, T\sum _{k_0} \int \limits _{-\infty }^{+\infty } \text{d}\omega _1 \int \limits _{-\infty }^{+\infty } \text{d}\omega _2\, n_{\mathrm{F}}(\omega _1) n_{\mathrm{F}}(\omega _2) \rho _1(\omega _1) \rho _2(\omega _2) \nonumber \\{} & \qquad {} \times \int \limits _{0}^{1/T} \text{d}\tau _1 \int \limits _{0}^{1/T} \text{d}\tau _2 \, \, e^{(\omega _1-k_0-i\epsilon _1)\tau _1}\,\, e^{(\omega _2-q_0-i\epsilon _2)\tau _2} \nonumber \\& \quad = \, {} \text {Im} \, \int \limits _{-\infty }^{+\infty } \text{d}\omega _1 \int \limits _{-\infty }^{+\infty } \text{d}\omega _2\, n_{\mathrm{F}}(\omega _1) n_{\mathrm{F}}(\omega _2) \rho _1(\omega _1) \rho _2(\omega _2) \nonumber \\{} &\qquad {} \times \int \limits _{0}^{1/T} \text{d}\tau _1 \int \limits _{0}^{1/T} \text{d}\tau _2 \, \, e^{(\omega _1-i\epsilon _1)\tau _1}\,\, e^{(\omega _2-p_0-i\epsilon _2)\tau _2} \, \underbrace{T\sum _{k_0} e^{-k_0(\tau _1-\tau _2)}}_{\delta (\tau _2-\tau _1)}. \end{aligned}$$

(A.18)

Performing $\tau _2$-integration using $\delta$-function, one can write

$$\begin{aligned} &\text {Im} \, \, \, T\sum _{k_0} F_1(k_0)F_2(q_0)\\ &\quad = \, {} \text {Im} \, \int \limits _{-\infty }^{+\infty } \text{d}\omega _1 \int \limits _{-\infty }^{+\infty } \text{d}\omega _2\, n_{\mathrm{F}}(\omega _1) n_{\mathrm{F}}(\omega _2) \rho _1(\omega _1) \rho _2(\omega _2) \nonumber \\{} & \qquad {} \times \int \limits _{0}^{1/T} \text{d}\tau _1 \, e^{(\omega _1+\omega _2-p_0-i\epsilon )\tau _1} \,, \end{aligned}$$

(A.19)

where $\epsilon =\epsilon _1+\epsilon _2$. Now performing the $\tau _1$-integration, one gets

$$\begin{aligned} & \text {Im} \, \, \, T\sum _{k_0} F_1(k_0)F_2(q_0)\\ & \quad = \, {} \text {Im} \, \int \limits _{-\infty }^{+\infty } \text{d}\omega _1 \int \limits _{-\infty }^{+\infty } \text{d}\omega _2\, n_{\mathrm{F}}(\omega _1) n_{\mathrm{F}}(\omega _2) \rho _1(\omega _1) \rho _2(\omega _2) \nonumber \\{} &\qquad {} \times \frac{e^{(\omega _1+\omega _2-p_0)/T} -1}{ \omega _1+\omega _2 -p_0 -i\epsilon }\, \nonumber \\& \quad = \, {} \, \int \limits _{-\infty }^{+\infty } \text{d}\omega _1 \int \limits _{-\infty }^{+\infty } \text{d}\omega _2\, n_{\mathrm{F}}(\omega _1) n_{\mathrm{F}}(\omega _2) \rho _1(\omega _1) \rho _2(\omega _2) \nonumber \\{} & \qquad {} \times \left( e^{(\omega _1+\omega _2-p_0)/T} -1\right) \, \text {Im} \left( \frac{1}{ \omega _1+\omega _2 -p_0 -i\epsilon }\right) \,. \end{aligned}$$

(A.20)

Now using

$$\begin{aligned} e^{-{p_0}/{T}}&= e^{-2m\pi i}= 1 \, , \end{aligned}$$

(A.21a)

$$\begin{aligned} p_0&=\omega +i\epsilon ' \, , \end{aligned}$$

(A.21b)

$$\begin{aligned} \frac{1}{ \omega _1+\omega _2 -p_0 -i\epsilon }&= \frac{1}{ \omega _1+\omega _2 -\omega -i\epsilon ' -i\epsilon } \\ &=\frac{1}{ \omega _1+\omega _2 -\omega -i\epsilon ''} \, , \end{aligned}$$

(A.21c)

$$\begin{aligned} \text {Im} \left( \frac{1}{\omega _1+\omega _2-\omega -i\epsilon ''}\right)&= -\pi \delta \left( \omega _1+\omega _2-\omega \right) \, , \end{aligned}$$

(A.21d)

one gets

$$\begin{aligned} & \text {Im} \, \, \, T\sum _{k_0} F_1(k_0)F_2(q_0)\\ = &\, {} - \, \pi \, \int \limits _{-\infty }^{+\infty } \text{d}\omega _1 \int \limits _{-\infty }^{+\infty } \text{d}\omega _2\, n_{\mathrm{F}}(\omega _1) n_{\mathrm{F}}(\omega _2) \rho _1(\omega _1) \rho _2(\omega _2) \nonumber \\{} & {} \times \left( e^{(\omega _1+\omega _2)/T} -1\right) \, \delta \left( \omega _1+\omega _2-\omega \right) \nonumber \\= &\, {} \pi \left( 1-e^{\beta \omega } \right) \int \limits _{-\infty }^{+\infty } \text{d}\omega _1 \int \limits _{-\infty }^{+\infty } \text{d}\omega _2\, \, \, n_{\mathrm{F}}(\omega _1) n_{\mathrm{F}}(\omega _2) \nonumber \\{} & {} \times \rho _1(\omega _1) \rho _2(\omega _2) \, \delta \left( \omega _1+\omega _2-\omega \right) \,, \end{aligned}$$

(A.22)

where $\beta =1/T$.

We finally obtain following (A.9) and (A.22), the discontinuity or the imaginary part of a product of two complex functions [81] as

$$\begin{aligned}& \text {Disc} \, \, T\sum _{k_0} F_1(k_0)F_2(q_0)\\ = &\, {} 2i \,\, \text {Im} \, \, \, T\sum _{k_0} F_1(k_0)F_2(q_0) \nonumber \\= &\, {} 2 \pi i \left( 1-e^{\beta \omega } \right) \int \limits _{-\infty }^{+\infty } \text{d}\omega _1 \int \limits _{-\infty }^{+\infty } \text{d}\omega _2\, \, \, n_{\mathrm{F}}(\omega _1) n_{\mathrm{F}}(\omega _2) \nonumber \\{} & {} \times \rho _1(\omega _1) \rho _2(\omega _2) \, \delta \left( \omega _1+\omega _2-\omega \right) \,. \end{aligned}$$

(A.23)