1 Introduction

The propagation of vector gauge bosons in a material medium in presence of a magnetic field produces many interesting observational effects. As for example the photons with different polarizations have different dispersion properties which lead to the Faraday rotation. This has also been observed for various astrophysical objects [1,2,3,4] and in the millisecond pulsations of solar radio emission [5]. In view of the theoretical perspective the general feature is associated with the propagation of a photon in an externally magnetized material medium. The subject of the propagation of photons in magnetized plasmas has been studied in large extent and also covered in standard electromagnetic theory [6, 7] and plasma physics [8, 9] books. However, in most cases it was assumed that the medium consists of non-relativistic and non-degenerate electrons and nucleons. This suggests a modification of theoretical tools in which a general formalism based on quantum field theory proves to be helpful [10]. A quantum field theoretical formalism to calculate Faraday rotation in different kinds of media (hot magnetized one) have been done in Refs. [11, 12]. Also high-intensity laser fields are used to create ultrarelativistic electron-positron plasmas which play an important role in various astrophysical situations. Some properties of such plasma are studied using QED at finite temperature [13, 14].

In the regime of Quantum Chromo Dynamics (QCD), nuclear matter dissolves into a thermalized color deconfined state Quark Gluon Plasma (QGP) under extreme conditions such as very high temperature and/or density. To probe different characteristics of this novel state, various high energy Heavy-Ion-Collisions (HIC) experiments are under way, e.g., RHIC@BNL, LHC@CERN and upcoming FAIR@GSI. Depending on the impact parameter of the collision, a relativistic HIC can be central or non-central. In recent years the non-central HIC is getting more and more attention in the heavy-ion community because of some distinct features which appear due to the non-centrality of the collision. One of those is the prospect of producing a very strong magnetic field in the direction perpendicular to the reaction plane due to the relatively higher rapidity of the spectator particles that are not participating in the collisions. Presently immense activities are in progress to study the properties of strongly interacting matter in presence of an external magnetic field, resulting in the emergence of several novel phenomena [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. This suggests that there is clearly an increasing demand to study the effects of intense background magnetic fields on various aspects and observables of non-central heavy-ion collisions. Also experimental evidences of photon anisotropy, provided by the PHENIX Collaboration [33], have posed a challenge for existing theoretical models. This kind of current experimental evidences have prompted that a modification of the present theoretical tools are much needed by considering the effects of intense background magnetic field on various aspects and observables of non-central HIC. In a field theoretic calculation n-point functions are the basic quantities to compute the various observables of a system. With this perspective in very recent works, based on various symmetries of the system for a nontrivial medium like a hot magnetized one, the general structure of fermionic 2- and 3-point function [34], and 4-point function [35] were computed. Also the spectral representation of two point function [34] were obtained for such system. In this paper we consider gluon that propagates in a hot magnetized QCD plasma for which we aim at the general structure of the gauge boson self-energy, the effective propagator and its dispersion property. This formalism is also applicable to QED system. The general propagators for fermion obtained in Ref. [34] and for the gauge boson obtained here have already been used to compute the quark-gluon free energy for a hot magnetized deconfined QCD system in Ref. [36].

This paper has been organized as follows: in Sect. 2 the general structure of a gauge boson self-energy in a hot magnetized medium is discussed progressively. It includes two parts: a brief review of the general structure in presence of only thermal medium in Sect. 2.1 and then a generalization of it to a hot magnetized medium in Sect. 2.2. In Sect. 3 we discuss the general structure for the gauge boson propagator using the results of Sect. 2. Section 4 begins with the domain of applicability depending upon the scales (mass, temperature and the magnetic field strength) associated with the system. In Sects. 4.1 and 4.2 we elaborately compute the various form factors, Debye screening mass, dispersion relations within strong and weak field approximation, respectively. Finally, we conclude in Sect. 5.

2 General structure of a gauge boson self-energy

In this section we first briefly review the formalism of the general structure for a gauge boson self-energy by considering only thermal bath without the presence of any magnetic field in Sect. 2.1 and it will then be followed by a formalism for a magnetized hot medium in Sect. 2.2.

2.1 Finite temperature and zero magnetic field case

We begin with the general structure of the gauge boson self-energy in vacuum, given as

$$\begin{aligned} \Pi ^{\mu \nu }(P) = V^{\mu \nu }\Pi (P^2), \end{aligned}$$
(1)

where the form factor \(\Pi (P^2)\) is Lorentz invariant and depends only on the four scalar \(P^2\). The vacuum projection operator is

$$\begin{aligned} V^{\mu \nu } = g^{\mu \nu }-\frac{P^\mu P^\nu }{P^2}, \end{aligned}$$
(2)

with the metric \(g^{\mu \nu }\equiv (1,-1,-1,-1)\) and \(P^\mu \equiv (p_0, \varvec{p})=(p^0,p^1,p^2,p^3)\). The self-energy satisfies the gauge invariance through the transversality condition

$$\begin{aligned} P_\mu \Pi ^{\mu \nu } (P)= 0 , \end{aligned}$$
(3)

and it is also symmetric

$$\begin{aligned} \Pi ^{\mu \nu } (P)= \Pi ^{\nu \mu } (P). \end{aligned}$$
(4)

The conditions in Eqs. (3) and (4) are sufficient to obtain ten components of \(\Pi ^{\mu \nu }\).

The presence of finite temperature (\(\beta =1/T\)) or heat bath breaks the Lorentz (boost) invariance of the system. In finite temperature one accumulates four-vectors and tensors to form a general covariant structure of the gauge boson self-energy. Those are \(P^\mu , \, g^{\mu \nu }\) from vacuum and the four-velocity \(u^\mu \) of the heat bath, discreetly introduced because of the medium. With these one can form four symmetric basis tensors, namely \(P^\mu P^\nu , P^\mu u^\nu + u^\mu P^\nu , u^\mu u^\nu \) and \(g^{\mu \nu }\). These four tensors can be reduced to two independent mutually orthogonal projection tensors by virtue of the constraints provided by the transversality condition in Eq. (3). One uses them to construct manifestly Lorentz-invariant structure of the gauge boson self-energy and propagator at finite temperature which have been discussed in the literature in details [37,38,39]. Nevertheless, we briefly discuss some of the essential points that would be very useful in constructing those general structures for a magnetized hot medium.

We now begin by defining Lorentz scalars, vectors and tensors that characterize the heat bath or hot medium in a local rest frame:

$$\begin{aligned} u^\mu&=(1,0,0,0), \nonumber \\ P^\mu u_\mu&=P\cdot u=p_0 , \end{aligned}$$
(5a)
$$\begin{aligned} {\tilde{P}}^\mu&= P^\mu - (P\cdot u)u^\mu = P^\mu - p_0 u^\mu , \end{aligned}$$
(5b)
$$\begin{aligned} {\tilde{g}}^{\mu \nu }&= g^{\mu \nu } - u^\mu u^\nu \end{aligned}$$
(5c)
$$\begin{aligned} {\tilde{P}}^2&= {\tilde{P}}^\mu {\tilde{P}}_\mu = P^2-p_0^2= -p^2, \end{aligned}$$
(5d)

where \(p=|\varvec{p}|\). We note here that one can only construct two independent Lorentz scalars as given in Eqs. (5a) and (5d). One can further redefine four vector \(u^\mu \) by projecting the vacuum projection tensor upon it as

$$\begin{aligned} {\bar{u}}^\mu= & {} V^{\mu \nu }u_\nu = u^\mu - \frac{(P\cdot u)P^\mu }{P^2} = u^\mu - \frac{p_0 P^\mu }{P^2}. \end{aligned}$$
(6)

which is orthogonal to \(P^\mu \). Now one can construct two independent and mutually transverse second rank projection tensors in terms of those redefined set of four-vectors and tensor as

$$\begin{aligned} A^{\mu \nu }&= {\tilde{g}}^{\mu \nu } - \frac{{\tilde{P}}^\mu {\tilde{P}}^\nu }{{\tilde{P}}^2}, \end{aligned}$$
(7a)
$$\begin{aligned} B^{\mu \nu }&= \frac{1}{{\bar{u}}^2} {\bar{u}}^\mu {\bar{u}}^\nu . \end{aligned}$$
(7b)

Moreover, sum of these two projection operators lead to the well known vacuum projection tensor \(V^{\mu \nu }\) as

$$\begin{aligned} A^{\mu \nu } + B^{\mu \nu }= & {} g^{\mu \nu } - \frac{P^\mu P^\nu }{P^2} = V^{\mu \nu }. \end{aligned}$$
(8)

So, the general (manifestly) covariant form of the self-energy tensor can be written as

$$\begin{aligned} \Pi ^{\mu \nu } = \Pi _T A^{\mu \nu } + \Pi _L B^{\mu \nu }, \end{aligned}$$
(9)

where \(\Pi _{L}\) and \(\Pi _T\) are, respectively, the longitudinal and transverse form factors. Eventually one can obtain these two form factors as

$$\begin{aligned} \Pi _L&= -\frac{P^2}{p^2} \Pi _{00}, \end{aligned}$$
(10a)
$$\begin{aligned} \Pi _T&= \frac{1}{D-2} (\Pi ^\mu _\mu - \Pi _L ), \end{aligned}$$
(10b)

where D is the space-time dimension of a given theory. The above Lorentz-invariant form factors would depend on the two independent Lorentz scalars \(p_0\) and \( p=\sqrt{p_0^2-P^2}\) as defined, respectively, in Eqs. (5a) and (5d) besides the temperature \(T=1/\beta \).

2.2 Finite temperature and finite magnetic field case

The finite temperature breaks the Lorentz (boost) symmetry whereas the presence of magnetic field breaks the rotational symmetry in the system. In presence of both finite temperature (\(\beta =1/T\)) and finite magnetic field B, the four-vectors and tensors available to form the general covariant structure of the gauge boson self-energy are \(P^\mu \), \(g^{\mu \nu }\), the electromagnetic field tensor \(F^{\mu \nu }\) and it’s dual \({{\tilde{F}}}^{\mu \nu }\), and the four velocity of the heat bath, \(u^\mu \). As seen in Sect. 2.1 at finite T the heat bath introduces a preferred direction that breaks the boost invariance. On the other hand, the presence of the magnetic field breaks the rotational symmetry in the system because it introduces an anisotropy in space. For hot magnetized system, one can define a new four vector \(n^\mu \) which is associated with the electromagnetic field tensor \(F^{\mu \nu }\). We define the electromagnetic field tensor as

$$\begin{aligned} F^{\mu \nu }= \left( {\begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -B &{} 0 \\ 0 &{} B &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{array} } \right) . \end{aligned}$$
(11)

In the rest frame of the heat bath, i.e., \(u^\mu =(1,0,0,0)\), \(n^\mu \) can be defined uniquely as projection of \(F^{\mu \nu }\) along \(u^\mu \),

$$\begin{aligned} n_\mu \equiv \frac{1}{2B} \epsilon _{\mu \nu \rho \lambda }\, u^\nu F^{\rho \lambda } = \frac{1}{B}u^\nu {{\tilde{F}}}_{\mu \nu } = (0,0,0,1), \end{aligned}$$
(12)

which is in the z-direction. This also establishes a connection between the heat bath and the magnetic field.

Now for a hot magnetized case one has Lorentz vectors, \(P^\mu \), \(u^\mu \) and \(n^\mu \) along with metric tensor \(g^{\mu \nu }\), from which one can form seven symmetric basis tensors, namely \(P^\mu P^\nu , P^\mu n^\nu + n^\mu P^\nu , n^\mu n^\nu \), \(P^\mu u^\nu + u^\mu P^\nu \), \(u^\mu u^\nu \), \(u^\mu n^\nu + n^\mu u^\nu \) and \(g^{\mu \nu }\). These seven tensors reduce to four because of constraints provided by the gauge invariance condition in Eq. (3). Below we obtain the four basis tensors.Footnote 1

We first form the transverse four momentum and the transverse metric tensor as

$$\begin{aligned} P_\perp ^\mu&=P^\mu - (P\cdot u)u^\mu + (P\cdot n)n^\mu \nonumber \\&= P^\mu - p_0 u^\mu + p^3 n^\mu = P^\mu -P_\parallel ^\mu , \end{aligned}$$
(13a)
$$\begin{aligned} g_\perp ^{\mu \nu }&= g^{\mu \nu } - u^\mu u^\nu + n^\mu n^\nu = g^{\mu \nu } - g_\parallel ^{\mu \nu }, \end{aligned}$$
(13b)

where

$$\begin{aligned} P_\parallel ^\mu&= p_0 u^\mu -p^3 n^\mu , \end{aligned}$$
(14a)
$$\begin{aligned} P_\parallel ^2&= P_\parallel ^\mu P^\parallel _\mu = p_0^2-p_3^2, \end{aligned}$$
(14b)
$$\begin{aligned} g_\parallel ^{\mu \nu }&= u^\mu u^\nu - n^\mu n^\nu , \end{aligned}$$
(14c)
$$\begin{aligned} P_\perp ^\mu P^\perp _{\mu }&=P_\perp ^2= P^2-p_0^2+p_3^2=P^2-P_\parallel ^2=-p_\perp ^2,\nonumber \\ \end{aligned}$$
(14d)

where \(P^2=P_\parallel ^2+P_\perp ^2=P_\parallel ^2-p_\perp ^2\), \(P_\parallel ^2=p_0^2-p_3^2\) and \(p_\perp ^2=p_1^2+p_2^2\). We further note that the three independent Lorentz scalars are \(p_0\), \(p^3=P\cdot n\) and \(P_\perp ^2\).

We take \(B^{\mu \nu }\) in Eq. (7b) as one of projection tensors in hot magnetized system. Now \(A^{\mu \nu }A_{\mu \nu }=2\) indicates that it is a combination of two mutually orthogonal projection tensors, which yields two degenerate transverse modes for gauge boson in heat bath. Projection of \(A^{\mu \nu }\) along magnetic field direction \(n^{\mu }\) is \({{\bar{n}}}^{\mu }=A^{\mu \nu }n_{\nu }\). So we can construct another second rank tensor orthogonal to both \(P^\mu \) and \(B^{\mu \nu }\) as,

$$\begin{aligned} Q^{\mu \nu }= & {} \frac{{{{\bar{n}}}}^\mu {{\bar{n}}}^\nu }{{\bar{n}}^2} . \end{aligned}$$
(15)

We, now, construct the third projection tensor \(R^{\mu \nu }\), with a constraint such that the sum of \(R^{\mu \nu }\), \(B^{\mu \nu }\) and \(Q^{\mu \nu }\) gives the vacuum projection operator \(V^{\mu \nu }\) as

$$\begin{aligned} R^{\mu \nu }= & {} V^{\mu \nu } - B^{\mu \nu } - Q^{\mu \nu } =A^{\mu \nu }-Q^{\mu \nu } \nonumber \\= & {} g_{\perp }^{\mu \nu }-\frac{P^{\mu }_{\perp }P^{\nu }_{\perp }}{P_{\perp }^2}. \end{aligned}$$
(16)

It can be checked easily that all the projection tensors satisfy the following properties,

$$\begin{aligned} P_\mu Z^{\mu \nu }&=0, \end{aligned}$$
(17a)
$$\begin{aligned} Z^{\mu \lambda }Z_{\lambda }^{\nu }&= Z^{\mu \nu }, \end{aligned}$$
(17b)
$$\begin{aligned} Z^{\mu \nu }Z_{\mu \nu }&= 1 . \end{aligned}$$
(17c)

where \(Z=B,R,Q\). The three projection tensors are orthogonal to each other:

$$\begin{aligned} Z^{\mu \nu } Y_{\mu \nu }&= 0, \end{aligned}$$
(18a)

where \(Z\ne Y\) and \(Y=B,R,Q\).

Now we construct the fourth tensor as

$$\begin{aligned} N^{\mu \nu }= & {} \frac{{\bar{u}}^{\mu }{\bar{n}}^{\nu }+{\bar{u}}^{\nu } {\bar{n}}^{\mu }}{\sqrt{{\bar{u}}^2}\sqrt{{\bar{n}}^2}}, \end{aligned}$$
(19)

which satisfies the following properties

$$\begin{aligned} N^{\mu \rho }N_{\rho \nu }= & {} B^{\mu }_{\nu }+Q^{\mu }_{\nu }, \end{aligned}$$
(20)
$$\begin{aligned} B^{\mu \rho }N_{\rho \nu }+N^{\mu \rho }B_{\rho \nu }= & {} N^{\mu }_{\nu }\,\,, \nonumber \\ Q^{\mu \rho }N_{\rho \nu }+N^{\mu \rho }Q_{\rho \nu }= & {} N^{\mu }_{\nu }, \end{aligned}$$
(21)
$$\begin{aligned} R^{\mu \rho }N_{\rho \nu }= & {} N^{\mu \rho }R_{\rho \nu }=0. \end{aligned}$$
(22)

Now, one can write a general covariant structure of gauge boson self-energy as

$$\begin{aligned} \Pi ^{\mu \nu } = b B^{\mu \nu } + c R^{\mu \nu } + d Q^{\mu \nu }+ a N^{\mu \nu }, \end{aligned}$$
(23)

where b, c, d and a are four Lorentz-invariant form factors associated with the four basis tensors. Note that Eq. (23) can also be expressed as

$$\begin{aligned} \Pi ^{\mu \nu } = b B^{\mu \nu } + c A^{\mu \nu } + (d-c) Q^{\mu \nu }+ a N^{\mu \nu } \end{aligned}$$
(24)

This particular decomposition of the self-energy in terms of four tensor basis is exactly same that has been used in Refs. [43, 44] which, however were then applied for different perspectives.

The (00) components of the constituent tensors are given by

$$\begin{aligned} B_{00}&= {\bar{u}}^2, \end{aligned}$$
(25a)
$$\begin{aligned} R_{00}&= 0, \end{aligned}$$
(25b)
$$\begin{aligned} Q_{00}&= 0, \end{aligned}$$
(25c)
$$\begin{aligned} N_{00}&=0, \end{aligned}$$
(25d)
$$\begin{aligned} \Pi _{00}&= b B_{00} = b{\bar{u}}^2 . \end{aligned}$$
(25e)

Using these information, we obtain the form factors as

$$\begin{aligned} b&= B^{\mu \nu }\Pi _{\mu \nu }, \end{aligned}$$
(26a)
$$\begin{aligned} c&= R^{\mu \nu }\Pi _{\mu \nu }, \end{aligned}$$
(26b)
$$\begin{aligned} d&= Q^{\mu \nu }\Pi _{\mu \nu }, \end{aligned}$$
(26c)
$$\begin{aligned} a&= \frac{1}{2}N^{\mu \nu }\Pi _{\mu \nu } . \end{aligned}$$
(26d)

In absence of the magnetic field by comparing with the known general form of finite temperature self-energy in Eq. (9), as

$$\begin{aligned} \Pi _TA_{\mu \nu }+\Pi _LB_{\mu \nu } = b_0B_{\mu \nu }+c_0R_{\mu \nu }+d_0Q_{\mu \nu }+a_0 N_{\mu \nu },\nonumber \\ \end{aligned}$$
(27)

one can write

$$\begin{aligned} b_0&= \Pi _{L}, \end{aligned}$$
(28a)
$$\begin{aligned} c_0&= d_0= \Pi _{T}, \end{aligned}$$
(28b)
$$\begin{aligned} a_0&=0 \end{aligned}$$
(28c)

where we used the fact that \(R_{\mu \nu }+Q_{\mu \nu }=A_{\mu \nu }\).

3 General form of gauge boson propagator in a hot magnetized medium

In covariant gauge the inverse of the gauge boson propagator in vacuum reads as

$$\begin{aligned} ({\mathcal {D}}^0 )^{-1}_{uv} = P^2g_{\mu \nu } - \frac{\xi -1}{\xi }P_\mu P_\nu , \end{aligned}$$
(29)

where \(\xi \) is the gauge parameter. From Eq. (16) one can write

$$\begin{aligned} P_\mu P_\nu= & {} P^2 [ g_{\mu \nu } -(B_{\mu \nu } +R_{\mu \nu } +Q_{\mu \nu }) ]. \end{aligned}$$
(30)

and using in Eq. (29), we get

$$\begin{aligned} ({\mathcal {D}}^0 )^{-1}_{uv} = \frac{P^2}{\xi }g_{\mu \nu } + P^2\frac{\xi -1}{\xi } (B_{\mu \nu } +R_{\mu \nu } +Q_{\mu \nu } ). \end{aligned}$$
(31)

The inverse of the general gauge boson propagator following Dyson–Schwinger equation reads as

$$\begin{aligned} {\mathcal {D}}_{uv}^{-1}= ({\mathcal {D}}^0 )_{uv}^{-1} - \Pi _{\mu \nu }. \end{aligned}$$
(32)

From Eqs. (31) and (23) we can now readily get

$$\begin{aligned} {\mathcal {D}} _{uv}^{-1}= & {} \frac{P^2}{\xi }g_{\mu \nu } + (P_m^2 - b )B_{\mu \nu } + (P_m^2 - c )R_{\mu \nu } \nonumber \\&+ (P_m^2 - d )Q_{\mu \nu }-a N_{\mu \nu }, \end{aligned}$$
(33)

where

$$\begin{aligned} P_m^2 = P^2\frac{\xi -1}{\xi }. \end{aligned}$$
(34)

The inverse of Eq. (33) can be written as

$$\begin{aligned} {\mathcal {D}}_{\mu \rho } = \alpha P_\mu P_\rho + \beta B_{\mu \rho } + \gamma R_{\mu \rho } + \delta Q_{\mu \rho }+ \sigma N_{\mu \rho }. \end{aligned}$$
(35)

along with

$$\begin{aligned} g_\mu ^\nu= & {} {\mathcal {D}}_{\mu \rho } \left( {\mathcal {D}}^{\rho \nu }\right) ^{-1} \nonumber \\= & {} \alpha \frac{P^2}{\xi } P_\mu P^\nu + \left[ \frac{\beta P^2}{\xi } +\beta (P_m^2-b)-\sigma a\right] B_\mu ^\nu \nonumber \\&+ \left[ \frac{\delta P^2}{\xi }+\delta (P_m^2-d)-\sigma a\right] Q_\mu ^\nu \nonumber \\&+\left[ \frac{\gamma P^2}{\xi }+\gamma (P_m^2-c) \right] R_\mu ^\nu \nonumber \\&+\left[ -\beta a+\sigma (P_m^2-d)+\frac{\sigma P^2}{\xi }\right] \frac{{\bar{u}}_\mu {\bar{n}}^\nu }{\sqrt{{\bar{u}}^2} \sqrt{{\bar{n}}^2}} \nonumber \\&+\left[ -\delta a +\sigma (P_m^2-b)+\frac{\sigma P^2}{\xi }\right] \nonumber \\&\times \frac{{\bar{n}}_\mu {\bar{u}}^\nu }{\sqrt{{\bar{u}}^2}\sqrt{{\bar{n}}^2}}. \end{aligned}$$
(36)

Now using the explicit forms of \(B_\mu ^\nu , R_\mu ^\nu \), \(Q_\mu ^\nu \) and \(N_\mu ^\nu \) and equating different coefficients from both sides yield the following conditions:

$$\begin{aligned}&\alpha = \frac{\xi }{P^4},\nonumber \\&\frac{\beta P^2}{\xi }+\beta (P_m^2-b)-\sigma a =1,\nonumber \\&\frac{\delta P^2}{\xi }+\delta (P_m^2-d)-\sigma a =1,\nonumber \\&\frac{\gamma P^2}{\xi }+\gamma (P_m^2-c) = 1,\nonumber \\&-\beta a+\sigma (P_m^2-d)+\frac{\sigma P^2}{\xi } = 0,\nonumber \\&-\delta a +\sigma (P_m^2-b)+\frac{\sigma P^2}{\xi } = 0. \end{aligned}$$
(37)

Solving this we get

$$\begin{aligned} \alpha= & {} \frac{\xi }{P^4},\nonumber \\ \beta= & {} \frac{P^2-d}{(P^2-b)(P^2-d)-a^2},\nonumber \\ \gamma= & {} \frac{1}{P^2-c},\nonumber \\ \delta= & {} \frac{P^2-b}{(P^2-b)(P^2-d)-a^2},\nonumber \\ \sigma= & {} \frac{a}{(P^2-b)(P^2-d)-a^2}. \end{aligned}$$
(38)

Now the general covariant structure of the gauge boson propagator in covariant gauge can finally be obtained as

$$\begin{aligned} {\mathcal {D}}_{\mu \nu }= & {} \frac{\xi P_{\mu }P_{\nu }}{P^4}+\frac{(P^2-d) B_{\mu \nu }}{(P^2-b)(P^2-d)-a^2}+\frac{R_{\mu \nu }}{P^2-c} \nonumber \\&+\frac{(P^2-b) Q_{\mu \nu }}{(P^2-b)(P^2-d)-a^2}\nonumber \\&+\frac{a N_{\mu \nu }}{(P^2-b)(P^2-d)-a^2}. \end{aligned}$$
(39)

We recall that the breaking of boost invariance due to finite temperature leads to two modes (degenerate transverse mode and plasmino). Now, the breaking of the rotational invariance in presence of magnetic field lifts the degeneracy of the transverse modes which introduces an additional mode in the hot medium. These three dispersive modes of gauge boson can be seen from the poles of Eq. (39). The poles \((P^2-b)(P^2-d)-a^2=0\), lead to two dispersive modes. We call one mode \(n^+\) with energy \(\omega _{n^+}\) and the other one \(n^-\) with energy \(\omega _{n^-}\). The pole \(P^2-c=0\) leads to the third dispersive mode c with energy \(\omega _c\). We will discuss about these dispersive modes in details later for both strong and weak field approximation.

When we turn off the magnetic field, the general structure of the propagator in a non-magnetized thermal bath can be obtained by putting \(b_0=\Pi _L,\ c_0=d_0=\Pi _T\) and \(a_0=0\) as

$$\begin{aligned} {\mathcal {D}}_{\mu \rho }= & {} \frac{\xi P_\mu P_\rho }{P^4} + \frac{B_{\mu \rho }}{P^2-\Pi _L} + \frac{A_{\mu \rho }}{P^2-\Pi _T} \end{aligned}$$
(40)

which agrees with the known result [37,38,39, 45].

4 Form factors

Before computing the various form factors associated with the general structure we note the following points:

  1. 1.

    The magnetic field generated during the non-central HIC is time dependent but is believed to decrease rapidly with time [20, 46]. It would be extremely complicated to work with a time dependent magnetic field. Instead we work by considering a constant background magnetic field along with some limiting conditions so that the effect of magnetic field can be incorporated analytically. We note here that incorporation of magnetic field to the heat bath introduces another scale in the system. Beside the fermion mass \(m_f\) and the temperature T, the additional scale is the strength of magnetic field B. Below we would discuss the different domains of scales:

    a) Strong Field Approximation: At the time of the collision, the value of the magnetic field B is estimated upto the order of \(|eB| \sim 15 m_\pi ^2\) (where e is the electronic charge, \(m_\pi \) is the mass of a pion), which is very high compared to the temperature T and \(m_f\) in the LHC at CERN [47]. Also in the dense sector, neutron stars (NS), or more specifically magnetars are known to possess strong enough magnetic field [48,49,50]. The effect of this strong enough magnetic field can be incorporated via a simplified Lowest Landau Level (LLL) approximation in which fermions are basically confined within the LLL. In the Sect. 4.1 we will work on strong field approximation with a scale hierarchy, \(m_f< T < \sqrt{|eB|}\), where the loop momentum \(K\sim T\) within HTL approximation.

    b) Weak Field Approximation: Furthermore, it is believed that the magnetic field generated in heavy-ion collisions decreases rapidly with time. This provides us a simplified situation where one can work in weak field approximation with a scale hierarchy, \(\sqrt{|eB|}< m_f < T \) which will be discussed in details in Sect. 4.2.

  2. 2.

    We would consider \(m_f=5 \) MeV for two light quark flavors u and d.

  3. 3.

    We choose a frame of reference as shown in Fig. 1 in which one considers the external momentum of the vector boson in xz planeFootnote 2 with \(0<\theta _p< \pi /2\). So one can write

    $$\begin{aligned} P^\mu =(p_0,p\sin {\theta _p},0,p\cos {\theta _p}), \end{aligned}$$
    (41)

    and then loop momenta

    $$\begin{aligned} K^\mu =(k_0,k \sin {\theta } \cos {\phi },k \sin {\theta } \sin {\phi },k \cos {\theta }). \end{aligned}$$
    (42)
Fig. 1
figure 1

Choice of reference frame for computing the various form factors associated with the general structure of gauge boson 2-point functions. The magnetic field is along z-direction

Full size image
Fig. 2
figure 2

Gluon polarization tensor in the limit of strong field approximation

Full size image

4.1 Gauge boson in strongly magnetized medium

4.1.1 One-loop gluon self-energy

When the external magnetic field is very strong, \( eB\rightarrow \infty \), it pushes all the Landau levels (\(n\ge 1\)) to infinity compared to the Lowest Landau Level (LLL) with \(n=0\). For LLL approximation in the strong field limit the fermion propagator reduces to a simplified form as

(43)

where K is the fermionic four momentum and we have used the properties of generalized Laguerre polynomial, \(L_n\equiv L_n^0\) and \(L_{-1}^\alpha = 0\). In strong field approximation or in LLL, \(eB \gg k_\perp ^2\), an effective dimensional reduction from \((3+1)\) to \((1+1)\) takes place.

Now in the strong field limit the self-energy (Fig. 2) can be computed as

(44)

where ‘s’ indicates that the quantities are to be calculated in the strong field approximation and \(\textsf {Tr}\) represents only the Dirac trace. We have suppressed the color indices for convenience. Now one can notice that the longitudinal and transverse parts are completely separated and the Gaussian integration over the transverse momenta can be done trivially, which leads to

$$\begin{aligned} \Pi _{\mu \nu }^s(P)= & {} \sum _f i ~e^{{-p_\perp ^2}/{2|q_fB|}}~\frac{g^2 |q_fB|}{2\pi } \nonumber \\&\times \int \frac{d^2K_\parallel }{(2\pi )^2} \frac{{{{\mathcal {S}}}}_{\mu \nu }^s}{(K_\parallel ^2-m_f^2)(Q_\parallel ^2-m_f^2)} \nonumber \\= & {} -\sum _fe^{{-p_\perp ^2}/{2|q_fB|}}~\frac{g^2 |q_fB|}{2\pi }~T\sum \limits _{k_0} \nonumber \\&\times \int \frac{dk_3}{2\pi } \frac{\mathcal{S}_{\mu \nu }^s}{(K_\parallel ^2-m_f^2)(Q_\parallel ^2-m_f^2)}, \end{aligned}$$
(45)

with the tensor structure \({{\mathcal {S}}}_{\mu \nu }^s\) that originates from the Dirac trace is

$$\begin{aligned} {{\mathcal {S}}}_{\mu \nu }^s = K_\mu ^\parallel Q_\nu ^\parallel + Q_\mu ^\parallel K_\nu ^\parallel - g_{\mu \nu }^\parallel \left( (K\cdot Q)_\parallel -m_f^2\right) , \end{aligned}$$
(46)

where the Lorentz indices \(\mu \) and \(\nu \) are restricted to longitudinal values because of dimensional reduction to (1+1) dimension and forbids to take any transverse values. Now we use Eqs. (14a) and (14c) to rewrite \(S_{\mu \nu }\) as

$$\begin{aligned} {{\mathcal {S}}}_{\mu \nu }^s= & {} (k_0 u_\mu - k_3 n_\mu ) (q_0 u_\nu - q_3 n_\nu ) \nonumber \\&+ (q_0 u_\mu - q_3 n_\mu ) (k_0 u_\nu - k_3 n_\nu ) \nonumber \\&- (u_\mu u_\nu - n_\mu n_\nu )\left( (k\cdot q)_\parallel -m_f^2\right) \nonumber \\= & {} u_\mu u_\nu \left( k_0 q_0 + k_3 q_3 +m_f^2\right) \nonumber \\&+ n_\mu n_\nu \left( k_0 q_0 + k_3 q_3 -m_f^2\right) \nonumber \\&- \left( u_\mu n_\nu + n_\mu u_\nu \right) \left( k_0 q_3 + k_3 q_0 \right) . \end{aligned}$$
(47)

4.1.2 Form factors and Debye mass

First we evaluate the form factors in Eqs. (26a), (26b), (26c) and (26d) in strong field approximation as

$$\begin{aligned} c&= R^{\mu \nu }(\Pi _{\mu \nu }^{g}+\Pi _{\mu \nu }^s) =c_{YM}+c_s \nonumber \\&= \frac{C_Ag^2T^2}{3}\frac{1}{2} \left[ \frac{p_0^2}{p^2}-\frac{P^2}{p^2}{\mathcal {T}}_P(p_0,p)\right] \quad \text{ where }\,\, c_s=0, \end{aligned}$$
(48a)
$$\begin{aligned} b&= B^{\mu \nu }(\Pi _{\mu \nu }^{g}+\Pi _{\mu \nu }^s) = b_{YM}+\frac{u^\mu u^\nu }{{\bar{u}}^2} \Pi _{\mu \nu }^s \nonumber \\&=b_{YM}+b_s =\frac{C_Ag^2T^2}{3{{\bar{u}}}^2}\left[ 1-{\mathcal {T}}_P(p_0,p)\right] \nonumber \\&\quad - \sum _fe^{{-p_\perp ^2}/{2|q_fB|}}~~\frac{g^2 |q_fB|}{2\pi {\bar{u}}^2}~T \nonumber \\&\quad \times \sum \limits _{k_0}\int \frac{dk_3}{2\pi } \frac{k_0 q_0 + k_3q_3 +m_f^2}{(K_\parallel ^2-m_f^2) (Q_\parallel ^2-m_f^2)}, \end{aligned}$$
(48b)
$$\begin{aligned} d&= d_{YM}+Q^{\mu \nu }\Pi _{\mu \nu }^s = d_{YM}+d_s \end{aligned}$$
(48c)
$$\begin{aligned}&=\frac{C_Ag^2T^2}{3}\frac{1}{2}\left[ \frac{p_0^2}{p^2}- \frac{P^2}{p^2}{\mathcal {T}}_P(p_0,p)\right] \nonumber \\&\quad +\sum _f e^{{-p_\perp ^2}/{2 |q_fB|}}~\frac{g^2|q_f B|}{2\pi }\frac{p_{\perp }^2}{p^2}~ T \nonumber \\&\quad \times \sum _{k_0}\int \frac{dk_3}{2\pi } \frac{k_0 q_0+k_3q_3-m_f^2}{(K_\parallel ^2-m_f^2)(Q_\parallel ^2-m_f^2)}, \end{aligned}$$
(48d)
$$\begin{aligned} a&=\frac{1}{2}N^{\mu \nu }(\Pi _{\mu \nu }^{g}+\Pi ^s_{\mu \nu }) \nonumber \\&=\frac{1}{2}N^{\mu \nu } \Pi ^s_{\mu \nu }=a_s,\,\text{ where }\,\,a_{YM}=0, \end{aligned}$$
(48e)

where \(\Pi _{\mu \nu }^{g}\) is the Yang–Mills (YM) contribution from ghost and gluon loop which remain unaffected in presence of magnetic field and can be written as

$$\begin{aligned} \Pi _{\mu \nu }^{g}(P)=-\frac{N_cg^2T^2}{3} \int \frac{d \Omega }{2 \pi }\left( \frac{p_{0} {\hat{K}}_{\mu } {\hat{K}}_{\nu }}{{\hat{K}} \cdot P}-g_{\mu 0} g_{\nu 0}\right) . \end{aligned}$$
(49)

Now, combining Eq. (48b) and the Hard Thermal Loop (HTL) approximation [51] one can have

$$\begin{aligned} b_s\approx & {} -\sum _fe^{{-p_\perp ^2}/{2|q_fB|}}~\frac{g^2|q_fB|}{2\pi {\bar{u}}^2}~T \sum \limits _{k_0}\int \frac{dk_3}{2\pi } \nonumber \\&\times \left[ \frac{1}{(K_\parallel ^2-m_f^2)}+\frac{2\left( k_3^2+m_f^2\right) }{(K_\parallel ^2-m_f^2)(Q_\parallel ^2-m_f^2)}\right] \nonumber \\= & {} \sum _fe^{{-p_\perp ^2}/{2|q_fB|}}~\frac{g^2|q_fB|}{2\pi {\bar{u}}^2}~\int \frac{dk_3}{2\pi } \nonumber \\&\times \left[ -\frac{n_F(E_{k_3})}{E_{k_3}} +\left\{ \frac{n_F(E_{k_3})}{E_{k_3}}+\frac{p_3 k_3}{E_{k_3}} \frac{\partial n_F(E_{k_3})}{\partial k_3} \right. \right. \nonumber \\&\times \left. \left. \left( \frac{p_3 k_3/E_{k_3}}{p_0^2-p_3^2(k_3/E_{k_3})^2}\right) \right\} \right] \nonumber \\= & {} \sum _fe^{{-p_\perp ^2}/{2|q_fB|}}~\frac{g^2|q_fB|}{2\pi {\bar{u}}^2}\int \frac{dk_3}{2\pi }\,\frac{p_3 k_3}{E_{k_3}} \frac{\partial n_F(E_{k_3})}{\partial E_{k_3}} \nonumber \\&\times \left( \frac{p_3 k_3/E_{k_3}}{p_0^2-p_3^2(k_3/E_{k_3})^2}\right) . \end{aligned}$$
(50)

Using Eqs. (48b), (50) in Eq. (25e) one also can directly calculate the Debye screening mass in QCD as

$$\begin{aligned} (m_D^2)_s= & {} \left. {{\bar{u}}}^2 b \right| _{p_0=0,\ \varvec{p} \rightarrow 0} =m_D^2+\sum _f (\delta m_{D,f}^2)_s \nonumber \\= & {} m_D^2 - \sum _f\frac{g^2|q_fB|}{2\pi }~\int \frac{dk_3}{2\pi } \frac{\partial n_F(E_{k_3})}{\partial E_{k_3}} \nonumber \\= & {} \frac{g^2N_c T^2}{3}+ \sum _f\frac{g^2|q_fB|}{2\pi T} \nonumber \\&\times \int _{-\infty }^\infty \frac{dk_3}{2\pi } n_F(E_{k_3}) (1-n_F(E_{k_3}) ). \end{aligned}$$
(51)

which reduces to the expression of QED Debye mass calculated in Refs. [52, 53] without QCD factors where three distinct scales (\(m_f^2\), \(T^2\) and eB) were clearly evident for massive quarks.

Now using Eq. (51) in Eq. (50) along with \(E_{k_3}\sim k_3\), the form factor b can be expressed in terms of \(m_D\) as

$$\begin{aligned} b= & {} \frac{C_Ag^2T^2}{3{\bar{u}}^2} [1-{\mathcal {T}}_P(p_0,p) ] \nonumber \\&-\sum _f e^{{-p_\perp ^2}/{2 |q_fB|}}~\left( \frac{\delta m_{D,f}}{{\bar{u}}}\right) ^2\frac{p_3^2}{p_0^2-p_3^2} .\ \end{aligned}$$
(52)

The form factor d then becomes

$$\begin{aligned} d\approx & {} \frac{C_Ag^2T^2}{3}\frac{1}{2} \left[ \frac{p_0^2}{p^2}- \frac{P^2}{p^2}{\mathcal {T}}_P(p_0,p)\right] \nonumber \\&+\sum _f e^{{-p_\perp ^2}/{2 |q_fB|}}~\left( \frac{\delta m_{D,f}}{{\bar{u}}} \right) ^2\frac{p_3^2}{p_0^2-p_3^2} . \end{aligned}$$
(53)

where the expression for \((\Pi _\mu ^\mu )^s\) is given in Eq. (B2) in Appendix B.

The form factor \(d_s\) can be calculated as

$$\begin{aligned} d_s= & {} Q^{\mu \nu } \Pi _{\mu \nu }^s,\nonumber \\\approx & {} -\sum _f i~e^{{-p_\perp ^2}/{2 |q_fB|}}~ \frac{g^2|q_f B|}{2\pi }\frac{p_{\perp }^2}{p^2} \nonumber \\&\times \int \frac{d^2K_\parallel }{(2\pi )^2}\bigg [\frac{\big (k_0^2+k_3^2- m_f^2\big )}{(K_\parallel ^2-m_f^2)(Q_\parallel ^2-m_f^2)}\bigg ],nn \end{aligned}$$
(54)
$$\begin{aligned}\approx & {} \sum _f e^{{-p_\perp ^2}/{2 |q_fB|}}~\delta m_{D,f}^2~\frac{p_{\perp }^2}{p^2}\frac{p_3^2}{p_0^2-p_3^2} \end{aligned}$$
(55)

for \(k_3 \sim E_{k_3}\). Now using (55) in (48d), the form factor d can be written as

$$\begin{aligned} d\approx & {} \frac{C_Ag^2T^2}{3}\frac{1}{2}\left[ \frac{p_0^2}{p^2}- \frac{P^2}{p^2}{\mathcal {T}}_P(p_0,p)\right] \nonumber \\&+\sum _f e^{{-p_\perp ^2}/{2 |q_fB|}}~\delta m_{D,f}^2~\frac{p_{\perp }^2}{p^2}\frac{p_3^2}{p_0^2-p_3^2} , \end{aligned}$$
(56)

where \(p_3=p\cos \theta _p\) and \(p_\perp =p\sin \theta _p\) as given in Eq. (41).

Also

$$\begin{aligned} 2a= & {} N^{\mu \nu } \Pi _{\mu \nu }^s = \sum _fi~e^{{-p_\perp ^2}/{2 |q_fB|}}~\frac{g^2|q_f B|}{2\pi \sqrt{{\bar{u}}^2} \sqrt{{\bar{n}}^2}} \nonumber \\&\times \int \frac{d^2K_\parallel }{(2\pi )^2} \bigg [\frac{-2\frac{{\bar{u}} \cdot n}{{\bar{u}}^2}\big (k_0^2+k_3^2+m_f^2\big )+4k_0k_3}{(K_\parallel ^2-m_f^2)(Q_\parallel ^2-m_f^2)}\bigg ]\nonumber \\= & {} \sum _f e^{{-p_\perp ^2}/{2 |q_fB|}}~\frac{g^2|q_f B|}{2\pi \sqrt{{\bar{u}}^2}\sqrt{{\bar{n}}^2}} \nonumber \\&\times \int \frac{dk_3}{2\pi }\left[ -2\frac{{\bar{u}} \cdot n}{{\bar{u}}^2} \frac{\partial n_F(E_{k_3})}{\partial E_{k_3}}\frac{p_3^2 k_3^2/E_{k_3}^2}{\big (p_0^2-p_3^2 k_3^2/E_{k_3}^2\big )}\right. \nonumber \\&\left. +\frac{2\partial n_F(E_{k_3})}{\partial E_{k_3}}\frac{p_0p_3 k_3^2/E_{k_3}^2}{\big (p_0^2-p_3^2 k_3^2/E_{k_3}^2\big )}\right] \nonumber \\\approx & {} \sum _f 2 ~e^{{-p_\perp ^2}/{2 |q_fB|}} \frac{\sqrt{{\bar{n}}^2}}{\sqrt{{\bar{u}}^2}}\delta m_{D,f}^2~ \frac{p_0p_3}{p_0^2-p_3^2}, \end{aligned}$$
(57)

where \({\bar{n}}^2=-p_{\perp }^2/p^2=-\sin ^2{\theta _p}\) and \({\bar{u}}^2=-p^2/P^2\).

Also in the strong field approximation, \( |eB|> T^2 > m_f^2\), one can neglect the quark mass \(m_f\), to get an analytic expression of Debye mass as

$$\begin{aligned} (m_D^2)_s= & {} \frac{g^2N_c T^2}{3} \nonumber \\&+ \sum _f\frac{g^2|q_fB|}{2\pi T}~\int _{-\infty }^\infty \frac{dk_3}{2\pi }~ n_F(k_3)\left( 1-n_F(k_3)\right) \nonumber \\= & {} \frac{g^2N_c T^2}{3}+\sum _f \frac{g^2 |q_fB|}{4\pi ^2}\nonumber \\= & {} m_D^2+\sum _f (\delta m_{D,f}^2)_s\nonumber \\= & {} m_D^2+(\delta m_D^2)_s, \end{aligned}$$
(58)

which agrees with that obtained in Ref. [53].

4.1.3 Dispersion

As discussed after Eq. (39), the dispersion relations for gluon in strong field approximation with LLL read as

$$\begin{aligned} P^2-c&= 0, \end{aligned}$$
(59a)
$$\begin{aligned} (P^2-b)(P^2-d)-a^2&=(P^2-\omega _n^+)(P^2-\omega _n^-)=0, \end{aligned}$$
(59b)

with

$$\begin{aligned} \omega _{n^+}&=\frac{b+d+\sqrt{\left( b-d\right) ^2+4a^2}}{2}, \end{aligned}$$
(60a)
$$\begin{aligned} \omega _{n^-}&=\frac{b+d-\sqrt{\left( b-d\right) ^2+4a^2}}{2}, \end{aligned}$$
(60b)

where the form factors are given, respectively, in Eqs. (48a), (52), (53) and (57).

Fig. 3
figure 3

The plot of dispersion of the three modes (\(n^-\), c and \(n^+\) modes) of a gauge boson in strong field approximation for propagation angles \(\theta _p=0,\,\, \pi /4\) and \(\pi /2\) at \(eB=20m_{\pi }^2\) and \(T=0.2\) GeV. \(\omega =p\) represents the light cone

Full size image

The solutions of above three dispersion relations are named as c-mode, \(n^+\)-mode and \(n^-\)-mode with energies \(\omega _c\), \(\omega _{n^+}\) and \(\omega _{n^-}\), respectively. The dispersion plot for the three modes of gluon in strong field approximation is shown in Fig. 3 for \(|eB|=20m_{\pi }^2\), \(T=0.2 \) GeV and for three propagation angles \(\theta _p=0,\,\, \pi /4\) and \(\pi /2\). We have used both magnetic field and temperature dependent coupling constant [36] for the purpose. As found \(c_s=0\) in Eq. (48a) which implies that the c-mode is unaffected by the magnetic field and propagates like HTL transverse mode irrespective of the propagation angle as shown in Fig. 3. The reason for which could be understood in the following way: in strong field approximation there is an effective dimensional reduction from (3+1) to \((1+1)\) dimension in LLL. Fermions at LLL can move only along the direction of external magnetic field. The electric field corresponding to the c mode is always transverse to the external magnetic field irrespective of the propagation angle of gluon. Thus, the fermions are not affected by the gluon excitation [40] and the quark loop contribution (\(c_s\)) becomes zero.

Now we note that at \(\theta _p=0\) the form factor \(a=0\) as it is proportional to \(\sin \theta _p\cos \theta _p\). In this case both \(n^-\) and c modes are degenerate as the form factors coincide with the HTL \(\Pi _T\) without the quark loop contribution. This is because quark loop contribution in the form factor d in Eq. (56) is proportional to \( \sin ^2\theta _p\cos ^2\theta _p\). This makes \(n^-\) and c mode to coincide with the HTL transverse dispersive mode. This can be seen from the left panel of Fig. 3. It could also be understood in the following way: when gluon propagates along the direction of external magnetic field, i.e., \(\theta _p=0\), the two transverse modes become rotationally symmetric about the external magnetic field and become degenerate which is shown in the left panel of Fig. 3. The electric fields corresponding to the \(n^-\) and c modes are perpendicular to the external magnetic field. Thus two transverse electric fields can not excite the fermions whose movement are restricted to the direction of external magnetic field in LLL [40]. This makes the quark loop contribution zero as noted earlier. In addition to the two transverse modes \(n^-\) and c, there is also a longitudinal excitation \(n^+\) at \(\theta _p=0\). At any intermediate angle of propagation, e.g, \(\theta _p=\pi /4\), the degeneracy of the transverse modes is lifted as shown in the middle panel of Fig. 3. Here both the transverse and longitudinal modes can excite the fermions as the corresponding electric fields are not orthogonal to the external magnetic field. As the propagation angle increases, the pole position corresponding to the \(n^-\) mode shifts from transverse channel and approaches the longitudinal channel [40]. At \(\theta _p=\pi /2\), the form factor a in Eq. (57) and the quark contribution of the form factor d in Eq. (53) also vanish because of their \(\theta _p\) dependence. Thus, the \(n^-\) mode merges with HTL longitudinal mode whereas the \(n^+\) mode merges with c mode. This is reflected in the right panel of Fig. 3.

4.2 Gauge boson in weakly magnetized hot medium

4.2.1 One-loop gluon self-energy

The fermion propagator in a weak magnetic field, i.e., \(\sqrt{|eB|} <(K\sim T)\) and \(m_f\), can be written up to \({\mathcal {O}}[(eB)^2]\) as

(61)

where \(S_0\) is the continuum free field propagator in absence of B whereas \(S_1\) and \(S_2\) are, respectively, \({\mathcal {O}}[(eB)]\) and \({\mathcal {O}}[(eB)^2]\) correction terms in presence of B. The contribution to the gluon self-energy due to the quark loop can be written from the Feynman diagram Fig. 4 as

$$\begin{aligned} \Pi ^{w,q}_{\mu \nu }(P)= & {} \sum _f \frac{ig^2}{2}\int \frac{d^4K}{(2\pi )^4}\textsf {Tr} [\gamma _\mu S^w_m(K)\gamma _\nu S^w_m(Q) ]. \end{aligned}$$
(62)
Fig. 4
figure 4

The order of \((eB)^2\) correction to the gluon polarization tensor \((\delta \Pi _{\mu \nu }^{a})\) in weak field approximation

Full size image
Fig. 5
figure 5

The order of \((eB)^2\) correction to the gluon polarization tensor \((\delta \Pi _{\mu \nu }^{b})\) in weak field approximation

Full size image

We have suppressed the color indices here also for convenience. Using Eq. (61) the self-energy in weak field approximation upto an \({\mathcal {O}}[(eB)^2]\) and also adding pure YM contribution, total gluon self-energy in weak field approximation can be decomposed as

$$\begin{aligned} \Pi ^w_{\mu \nu }(P)= & {} \Pi _{\mu \nu }^{g}(P) + \Pi _{\mu \nu }^{0}(P) +\delta \Pi _{\mu \nu }^{a}(P) \nonumber \\&+2\delta \Pi _{\mu \nu }^{b}(P) + {\mathcal {O}}[(eB)^3], \end{aligned}$$
(63)

where the first term \(\Pi _{\mu \nu }^{g}\) is the YM contribution which is given in Eq. (49). The last three terms in Eq. (63) appear from the expansion of quark loop contribution to the gluon self-energy. The term \(\Pi _{\mu \nu }^{0}\), containing two \(S_0\) , is the leading order perturbative term in absence of B whereas the remaining two terms are \((eB)^2\) order corrections as shown in Figs. 4 and 5. However, we note that \({\mathcal {O}}[(eB)]\) vanishes according to Furry’s theorem since the expectation value of any odd number of electromagnetic currents must vanish due to the charge conjugation symmetry.

Now the second and third terms in Eq. (63) can be written as

$$\begin{aligned} \Pi _{\mu \nu }^{0}(P)= & {} \sum _f\frac{ig^2}{2}\int \frac{d^4K}{(2\pi )^4} \textsf {Tr}\left[ \gamma _\mu S_0(K)\gamma _\nu S_0(Q)\right] \nonumber \\= & {} \sum _f\frac{ig^2}{2}\int \frac{d^4K}{(2\pi )^4}\left[ 8K_\mu K_\nu -4K^2g_{\mu \nu }\right] \nonumber \\&\times \frac{1}{{(K^2-m_f^2)(Q^2-m_f^2)}}, \end{aligned}$$
(64)
$$\begin{aligned} \delta \Pi _{\mu \nu }^{a}(P)= & {} \sum _f\frac{ig^2}{2}\int \frac{d^4K}{(2\pi )^4} \textsf {Tr}\left[ \gamma _\mu S_1(K)\gamma _\nu S_1(Q)\right] ,\nonumber \\= & {} \sum _f\frac{ig^2}{2}(q_fB)^2 \nonumber \\&\times \int \frac{d^4K}{(2\pi )^4} \frac{U_{\mu \nu }}{(K^2-m_f^2)^2(Q^2-m_f^2)^2}, \end{aligned}$$
(65)

where in the numerator we have neglected the mass of the quark and the external momentum P due to HTL approximation. The tensor structure of the self-energy correction in weak field approximation comes out to be

$$\begin{aligned} U_{\mu \nu }= & {} 4(K \cdot u) (Q \cdot u) (2n_\mu n_\nu + g_{\mu \nu } ) \nonumber \\&+4(K \cdot n) (Q \cdot n) (2u_\mu u_\nu - g_{\mu \nu } ) \nonumber \\&- 4 [(K \cdot u)(Q \cdot n)+(K \cdot n)(Q \cdot u) ] \nonumber \\&\times (u_\mu n_\nu + u_\nu n_\mu ) +4m_f^2 g_{\mu \nu } \nonumber \\&+ 8m_f^2 (g_{1\mu }g_{1\nu } +g_{2\mu }g_{2\nu } ) . \end{aligned}$$
(66)

The third term in Eq. (63) can be written as

$$\begin{aligned}&\delta \Pi _{\mu \nu }^{b}(P) \nonumber \\&\quad = \sum _f\frac{ig^2}{2}\int \frac{d^4K}{(2\pi )^4}\textsf {Tr} [\gamma _\mu S_2(K)\gamma _\nu S_0(Q) ] \nonumber \\&\quad =\sum _f ig^2(q_fB)^2\int \frac{d^4K}{(2\pi )^4} \nonumber \\&\qquad \times \left[ \frac{X_{\mu \nu }}{(K^2-m_f^2)^3(Q^2-m_f^2)}- \frac{(K_\parallel ^2-m_f^2) W_{\mu \nu }}{(K^2-m_f^2)^4(Q^2-m_f^2)}\right] \nonumber \\ \end{aligned}$$
(67)

where

$$\begin{aligned} X_{\mu \nu }&= 4 [(K\cdot u) (u_\mu Q_\nu + u_\nu Q_\mu ) \nonumber \\&\quad - (K\cdot n) (n_\mu Q_\nu + n_\nu Q_\mu ) \nonumber \\&\qquad + \{ (K\cdot n)(Q\cdot n) - (K\cdot u)(Q\cdot u) +m_f^2 \} g_{\mu \nu } ], \end{aligned}$$
(68a)
$$\begin{aligned} W_{\mu \nu }&= 4 (K_\mu Q_\nu + Q_\mu K_\mu ) - 4 (K\cdot Q-m_f^2 ) g_{\mu \nu }. \end{aligned}$$
(68b)

4.2.2 Computation of form factors and Debye mass of \({\mathcal {O}}\left[ (eB)^0\right] \) term

In this subsection, we calculate the \({\mathcal {O}}\left[ (eB)^0\right] \) terms in the form factors b, c, d in the weak magnetic field limit which are denoted by \(b_0,c_0,d_0\), respectively.

The form factor \(b_0\) in absence of magnetic field can be written from Eq. (25e) as

$$\begin{aligned} b_0(p_0,p)= & {} \frac{1}{{\bar{u}}^2} [\Pi ^{g}_{00}(P) +\Pi ^{0}_{00}(P) ] . \end{aligned}$$
(69)

where

$$\begin{aligned} \Pi ^{0}_{00}(P)= & {} \sum _f\frac{ig^2}{2}\int \frac{d^4K}{(2\pi )^4} \left[ 8k_0^2-4K^2\right] \nonumber \\&\times \frac{1}{{(K^2-m_f^2)(Q^2-m_f^2)}} . \end{aligned}$$
(70)

Using the hard thermal loop (HTL) approximation [38] and performing the frequency sum, one can write

$$\begin{aligned} \Pi ^{0}_{00}(P)= & {} -2g^2N_f\int \frac{k^2dk}{2\pi ^2}~\frac{dn_F(k)}{dk}\nonumber \\&\times \int \frac{d\Omega }{4\pi }\left( 1-\frac{p_0}{P\cdot {\hat{K}}}\right) , \end{aligned}$$
(71)

for \( m_f=0\) .

Now the QCD Debye mass in the absence of the magnetic field can directly be obtained using Eq. (25e) as

$$\begin{aligned} m_D^2= & {} \Pi ^{0}_{00}\Big \vert _{\begin{array}{c} p_0=0 \\ \mathbf{p} \rightarrow 0 \end{array}} = {{\bar{u}}^2} b_0\Big \vert _{\begin{array}{c} p_0=0 \\ \mathbf{p}\rightarrow 0 \end{array}} = \frac{N_cg^2T^2}{3} \nonumber \\&-2g^2\int \frac{k^2dk}{2\pi ^2}~\frac{dn_F(k)}{dk} =\frac{g^2T^2 }{3}\left( N_c+\frac{N_f}{2}\right) . \nonumber \\ \end{aligned}$$
(72)

Using Eq. (72) in Eq. (71), we get

$$\begin{aligned} \Pi ^{0}_{00}(P)= & {} \frac{N_fg^2T^2}{6}\int \frac{d\Omega }{4\pi } \left( 1-\frac{p_0}{p_0-\varvec{p}\cdot \hat{\varvec{k}}}\right) \nonumber \\= & {} \frac{N_fg^2T^2}{6}\left( 1-\frac{p_0}{2p}\log \frac{p_0+p}{p_0-p}\right) , \end{aligned}$$
(73)

where we use \(p=\sqrt{p_1^2+p_3^2}\) as p lies in xz plane as shown Fig. 1. The form factor in Eq. (69) becomes

$$\begin{aligned} b_0(p_0,p)= & {} \frac{m_D^2}{{\bar{u}}^2}\left( 1-\frac{p_0}{2p}\log \frac{p_0+p}{p_0-p}\right) , \end{aligned}$$
(74)

which agrees with the HTL longitudinal form factor \(\Pi _L(p_0,p)\) [38]. Similarly, we will calculate here the coefficients \(c_0\) and \(d_0\) explicitly.

$$\begin{aligned} c_0(p_0,p)= & {} R^{\mu \nu } [\Pi _{\mu \nu }^g(P)+\Pi ^{0}_{\mu \nu }(P) ]\nonumber \\= & {} (\Pi ^{g})^{\mu }_\mu (P) +(\Pi ^{0})^{\mu }_\mu (P) \nonumber \\&+ \frac{1}{p_\perp ^2} [(p_0^2-p_\perp ^2) \{\Pi _{00}^{g}(P)+\Pi _{00}^{0}(P)\}\nonumber \\&+\,p^2\{\Pi _{33}^{g}(P)+\Pi _{33}^{0}(P)\} \nonumber \\&- 2p_0p_3\{\Pi _{03}^{g}(P)+\Pi _{03}^{0}(P)\}], \end{aligned}$$
(75)

and

$$\begin{aligned} d_0(p_0,p)= & {} Q^{\mu \nu }[\Pi _{\mu \nu }^g(P)+\Pi ^{0}_{\mu \nu } (P)]\nonumber \\= & {} -\frac{p^2}{p_\perp ^2} \Biggl [\{\Pi _{33}^{g}(P) +\Pi _{33}^{0}(P)\} \nonumber \\&\qquad \quad -\frac{2p_0p_3}{p^2} \{\Pi _{03}^{g}(P)+\Pi _{03}^{0}(P)\} \nonumber \\&\quad \qquad + \frac{p_0^2p_3^2}{p^4} \{\Pi _{00}^{g}(P)+\Pi _{00}^{0}(P)\}\Biggr ]. \end{aligned}$$
(76)

Now from Eq. (49), we can write

$$\begin{aligned} \Pi ^{g}_{00}(P)= & {} \frac{N_c\,g^2T^2}{3}\left( 1-\frac{p_0}{2p} \log \frac{p_0+p}{p_0-p}\right) , \end{aligned}$$
(77)
$$\begin{aligned} \Pi ^{g}_{03}(P)= & {} \frac{N_c\,g^2T^2}{3}\frac{p_0p_3}{p^2} \left( 1-\frac{p_0}{2p}\log \frac{p_0+p}{p_0-p}\right) , \end{aligned}$$
(78)
$$\begin{aligned} \Pi ^{g}_{33}(P)= & {} \frac{N_c\,g^2T^2}{3}\frac{3p_3^2-p^2}{p^2} \frac{p_0^2}{2p^2}\left( 1-\frac{p_0}{2p} \log \frac{p_0+p}{p_0-p}\right) \nonumber \\&+ \frac{N_c\,g^2T^2}{3} \frac{p_3^2-p^2}{2p^2} \frac{p_0}{2p}\log \frac{p_0+p}{p_0-p}. \end{aligned}$$
(79)

We note that 00 component from the quark contribution \(\Pi _{00}^{0}\) is already calculated in Eq. (73) and one needs to calculate the remaining two components of \(\Pi _{\mu \nu }^0(P)\) which are as follows:

$$\begin{aligned} \Pi ^{0}_{03}(P)= & {} \sum _f\frac{ig^2}{2}\int \frac{d^4K}{(2\pi )^4} \frac{8k_0k_3}{{K^2Q^2}} \nonumber \\= & {} -\frac{N_fg^2T^2}{6}\int \frac{d\Omega }{4\pi } \frac{p_0{\hat{k}}_3}{P\cdot {\hat{K}}}\nonumber \\= & {} \frac{N_fg^2T^2}{6}\frac{p_0p_3}{p^2} \left( 1-\frac{p_0}{2p}\log \frac{p_0+p}{p_0-p}\right) , \end{aligned}$$
(80)

and

$$\begin{aligned} \Pi ^{0}_{33}(P)= & {} \sum _f\frac{ig^2}{2}\int \frac{d^4K}{(2\pi )^4} \frac{8k_3^2+4K^2}{{(K^2-m_f^2)(Q^2-m_f^2)}} \nonumber \\= & {} -\frac{N_fg^2T^2}{6}\int \frac{d\Omega }{4\pi }\frac{p_0{\hat{k}}_3^2}{P\cdot {\hat{K}}}\nonumber \\= & {} \frac{N_fg^2T^2}{6}\frac{3p_3^2-p^2}{p^2}\frac{p_0^2}{2p^2} \left( 1-\frac{p_0}{2p}\log \frac{p_0+p}{p_0-p}\right) \nonumber \\&+ \frac{N_fg^2T^2}{6}\frac{p_3^2-p^2}{2p^2}\frac{p_0}{2p} \log \frac{p_0+p}{p_0-p}. \end{aligned}$$
(81)

Using the results from Eqs. (73), (77)–(81), \(c_0(p_0,p)\) and \(d_0(p_0,p)\) become

$$\begin{aligned} c_0(p_0,p)= & {} d_0(p_0,p) \nonumber \\= & {} \frac{m_D^2}{2p^2}\left[ p_0^2- \left( p_0^2-p^2\right) \frac{p_0}{2p}\log \frac{p_0+p}{p_0-p}\right] ,\nonumber \\ \end{aligned}$$
(82)

which agrees with the HTL transverse form factor \(\Pi _T(p_0,p)\) [38].

This implies that the zero magnetic field contribution of the fourth form factor a should vanish. Below we obtain the same from Eqs. (26d) and (64) as,

$$\begin{aligned} a_0= & {} \frac{1}{2}N^{\mu \nu }\Big [\Pi _{\mu \nu }^{g}+\Pi _{\mu \nu }^{0}\Big ]\nonumber \\= & {} \frac{1}{2\sqrt{{\bar{u}}^2}\sqrt{{\bar{n}}^2}}\bigg [u^\mu n^\nu +n^\mu u^\nu -2 \frac{{\bar{u}} \cdot n}{{\bar{u}}^2}{\bar{u}}^\mu {\bar{u}}^\nu \bigg ] \nonumber \\&\times \Big [\Pi _{\mu \nu }^{g} +\Pi _{\mu \nu }^{0}\Big ]\nonumber \\= & {} \frac{1}{2\sqrt{{\bar{u}}^2}\sqrt{{\bar{n}}^2}}\bigg [-2\frac{{\bar{u}} \cdot n}{{\bar{u}}^2}\Big [\Pi _{00}^{g}+\Pi _{00}^{0}\Big ] + 2\Big [\Pi _{03}^{g}+\Pi _{03}^{0}\Big ]\bigg ]\nonumber \\= & {} 0 \end{aligned}$$
(83)

4.2.3 Computation of form factors and Debye mass of \({\mathcal {O}}\left[ (eB)^2\right] \) terms

In this subsection, we calculate the \({\mathcal {O}}\left[ (eB)^2\right] \) coefficients of b, c, d, a which are denoted by \(b_2,c_2,d_2,a_2\), respectively. The form factor \(b_2\), i.e., \({\mathcal {O}}(eB)^2\) term of the coefficient b, has been computed in Eq. (E13) of appendix E1 as

$$\begin{aligned} b_2= & {} \frac{1}{{\bar{u}}^2} [\delta \Pi _{00}^{a}(P)+2\delta \Pi _{00}^{b}(P)]\nonumber \\= & {} \frac{\delta m_D^2}{{\bar{u}}^2}+ \sum _f\frac{g^2(q_fB)^2}{{\bar{u}}^2 \pi ^2} \nonumber \\&\times \left[ \left( g_k+\frac{\pi m_f-4T}{32m_f^2T}\right) (A_0-A_2)\right. \nonumber \\&\left. +\left( f_k+\frac{8T-\pi m_f}{128 m_f^2 T}\right) \left( \frac{5A_0}{3}-A_2\right) \right] . \end{aligned}$$
(84)

and also the Debye screening mass of \({\mathcal {O}}(eB)^2\) as obtained in Eq. (E8) of appendix E1 as

$$\begin{aligned} \delta m_D^2= & {} -\sum _f\frac{g^2}{3\pi ^2}(q_fB)^2 \left[ \left( \frac{\partial }{\partial (m_f^2)}\right) ^2 + m_f^2\left( \frac{\partial }{\partial (m_f^2)}\right) ^3\right] \nonumber \\&\times \ m_f^2 \sum \limits _{l=1}^\infty (-1)^{l+1} \left[ K_2\left( \frac{m_fl}{T}\right) -K_0 \left( \frac{m_fl}{T}\right) \right] \nonumber \\= & {} \frac{g^2}{12\pi ^2T^2}\sum _f(q_fB)^2\sum \limits _{l=1}^\infty (-1)^{l+1} l^2K_0\left( \frac{m_fl}{T}\right) .\nonumber \\ \end{aligned}$$
(85)

We obtain \({\mathcal {O}}(eB)^2\) term of the coefficient c in Eq. (E15) of appendix E2 as

$$\begin{aligned} c_2= & {} R^{\mu \nu }(\delta \Pi _{\mu \nu }^{a} + 2\delta \Pi _{\mu \nu }^{b})\nonumber \\= & {} -\sum _f\frac{4g^2(q_fB)^2}{3\pi ^2}g_k \nonumber \\&+ \sum _f\frac{g^2(q_fB)^2}{2\pi ^2} \left( g_k + \frac{\pi m_f - 4T}{32m_f^2T}\right) \nonumber \\&\times \left[ -\frac{7}{3} \frac{p_0^2}{p_{\perp }^2} + \left( 2+\frac{3}{2} \frac{p_0^2}{p_{\perp }^2}\right) A_0 \right. \nonumber \\&+\left( \frac{3}{2}+\frac{5}{2}\frac{p_0^2}{p_{\perp }^2}+\frac{3}{2} \frac{p_3^2}{p_{\perp }^2} \right) A_2 - \frac{3p_0p_3}{p_{\perp }^2}A_1 \nonumber \\&\left. - \frac{5}{2}\left( 1-\frac{p_3^2}{p_{\perp }^2}\right) A_4- \frac{5p_0p_3}{p_{\perp }^2}A_3 \right] . \end{aligned}$$
(86)

We calculate the \({\mathcal {O}}(eB)^2\) term of the coefficient d in appendix E3 as

$$\begin{aligned} d_2= & {} Q^{\mu \nu }(\delta \Pi _{\mu \nu }^a + 2\delta \Pi _{\mu \nu }^b) = F_1+F_2 , \end{aligned}$$
(87)

where expressions for \(F_1\) and \(F_2\) can be found in Eqs. (E18) and (E19), respectively.

The \({\mathcal {O}}(eB)^2\) term of the coefficient a is calculated in appendix E4 as

$$\begin{aligned} a_2= & {} N^{\mu \nu }(\delta \Pi _{\mu \nu }^a + 2\delta \Pi _{\mu \nu }^b) = G_1+G_2 , \end{aligned}$$
(88)

where \(G_1\) and \(G_2\) are given in Eqs. (E21) and (E22) respectively.

Fig. 6
figure 6

Gluon dispersion curves for \(\theta _p=\pi /3\) but with varying magnetic field strength \(eB=m_\pi ^2/2, \ \ m_\pi ^2/10\ \ \mathrm{and} \, \ m_\pi ^2/800 (\sim 0)\) for \(N_f=2\)

Full size image

4.2.4 Dispersion

In weak field approximation the dispersion relation can now be written as

$$\begin{aligned}&P^2-c=P^2-\Pi _T-c_2 =0, \end{aligned}$$
(89a)
$$\begin{aligned}&(P^2-b)(P^2-d)-a^2 \nonumber \\&\quad = (P^2-\Pi _L-b_2)(P^2-\Pi _T-d_2)-a_2^2\nonumber \\&\quad =\left( P^2-\frac{b_0+b_2+d_0+d_2+\sqrt{\left( b_0+b_2-d_0-d_2\right) ^2 +4a_2^2}}{2}\right) \nonumber \\&\qquad \times \left( P^2-\frac{b_0+b_2+d_0+d_2-\sqrt{\left( b_0+b_2-d_0-d_2\right) ^2 +4a_2^2}}{2}\right) \nonumber \\&\quad =0 \end{aligned}$$
(89b)

which give rise to c,  \(n^+\) and \(n^-\) dispersive modes with energies \(\omega _c\), \(\omega _{n^+}\) and \(\omega _{n^-}\) respectively.

In this section, we consider that the magnetic field is the smallest scale and calculate all the quantities up to \({\mathcal {O}}[\left( eB\right) ^2]\). Within this approximation, Eq. (89b) can be approximated as

$$\begin{aligned}&(P^2-b_0-b_2 ) (P^2-d_0-d_2 )=0, \end{aligned}$$
(90)

as there is no contribution of \({\mathcal {O}}[\left( eB\right) ^2]\) from \(a_2\) and it only starts contributing \(\mathcal O[\left( eB\right) ^4]\) onwards. Thus \(a_2\) can safely be neglected in the weak field approximation. Now one can write the dispersion relation in weak field approximation as

$$\begin{aligned} P^2-b= & {} 0,\nonumber \\ P^2-c= & {} 0,\nonumber \\ P^2-d= & {} 0, \end{aligned}$$
(91)

where the respective dispersive modes are denoted by b-,c-, d-mode.

We note that the dispersion relations are scaled by plasma frequency of non-magnetized medium, \(\omega _p=m_D/\sqrt{3}\) where \(m_D^2\) is given in Eq. (72). As seen that there are three distinct modes when a gluon propagates in hot magnetized material medium. The magnetized plasmon mode with energy \(\omega _b\) appears due to the form factor b whereas two transverse modes with energy \(\omega _c\) and \(\omega _d\) are, respectively, due to the form factors c and d. The presence of magnetic field lifts the degeneracy of the transverse mode found only in a thermal medium.

Fig. 7
figure 7

Gluon dispersion curves for \(\theta _p=\pi /6\) but with varying magnetic field strength \(eB=m_\pi ^2/4, \ \ m_\pi ^2/10\ \ \mathrm{and} \, \ m_\pi ^2/800 (\sim 0)\) for \(N_f=2\)

Full size image

Now, the dispersion curves for gluon are displayed in Fig. 6 when it propagates at an angle \(\theta _p=\pi /3\) with the direction of the magnetic field. We have chosen three different values of magnetic field \(|eB|=m_\pi ^2/2, \ \ m_\pi ^2/10\ \ \mathrm{and} \, \ m_\pi ^2/800 (\sim 0) \); \(m_\pi \) is the pion mass. For a given magnetic field strength, say \(|eB|=m_\pi ^2/2\), one finds two modes (viz., b and d mode) with vanishing plasma frequency and one mode (viz., c mode) with finite plasma frequency. The zero plasma frequency for b and d modes could be the artefact of the weak field approximation used in the series expanded version of the Schwinger propagator, i.e. Eq. (61) where the propagator is expanded in a series of eB by considering eB as the lowest scale. This expansion constrains the arbitrariness of the value of p as it is valid only when \(p\gtrsim \sqrt{eB}\). Hence in the limit \(p\rightarrow 0\) with finite value of eB (however small), as p then becomes the lowest scale and Eq. (61) is not valid. For d mode with a very small magnetic field, the dispersion curve for d at \(p=0\) jumps to zero abruptly. This is because, taking \(p\rightarrow 0\) limit before taking \(eB \rightarrow 0\) again violates the condition \(p\gtrsim \sqrt{eB}\) and leaves behind a zero frequency mode. However, the situation is different while taking \(eB\rightarrow 0\) limit first though, as in that case considering \(eB=0\), one gets back two HTL dispersive modes for gluon propagation. In Fig. 7 we have also displayed the dispersion of gluon when it propagates at an angle \(\theta _p=\pi /6\).

5 Conclusion

In this article, we have constructed the general structure of two point functions (self-energy and propagator) of a gauge boson when it travels through a magnetized thermal medium. The Lorentz (boost) invariance is broken due to the presence of heat bath whereas rotational invariance is broken due to the presence of a background magnetic field. Based on gauge invariance and symmetry properties of the gauge boson self-energy, the general Lorentz structure of gauge boson two point functions is obtained by using four linearly independent basis tensors. We used the effective two point functions to study the dispersion spectra of a gluon in hot magnetized medium. In strong field approximation, one finds three modes which in limiting cases (propagation angle \(\pi /2\)) merge with the thermal modes. On the other hand in weak field approximation one also finds three distinct modes, viz., one magnetized plasmon, two transverse mode. The calculation for photon can trivially be obtained from this calculation. We further note that the effective propagator obtained here can conveniently be used to study various quantities in QED and QCD plasma. We, finally, note that in a following calculation [36], various thermodynamic quantities are computed using the general structure of the gauge boson here and fermions in Ref. [34] of a magnetized hot QCD plasma.