General structure of gauge boson propagator and its spectra in a hot magnetized medium

We have systematically constructed the general structure of the gauge boson self-energy and the effective propagator in presence of a nontrivial background like hot magnetized material medium using \emph{three mutually orthogonal projection tensors}. The hard thermal loop approximation has been used for the heat bath to obtain various form factors associated with the two point function. We have analyzed the dispersion spectra of the gauge boson using the effective propagator both in weak and strong magnetic field approximation. The formalism is applicable to both QED and QCD. The presence of a heat bath usually breaks the Lorentz (boost) symmetry whereas that of magnetic field breaks the rotational symmetry in the system. The presence of only thermal background leads to a longitudinal (plasmon) mode and a two-fold degenerate transverse mode. In presence of a hot but weakly magnetized background the degeneracy of the two transverse mode is lifted at low momenta where one remains massive and the other one becomes massless. We also note that the plasmon mode also becomes massless. On the other hand, in strong field approximation the three modes behave almost like that of a free massless gauge boson in Lowest Landau Level due to the dimensional reduction.


Introduction
The propagation of vector gauge bosons in a material medium in the presence of a magnetic field produces many interesting observational effects. As for example the photons with different polarizations have different dispersion properties 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 in one case and a plane wave electromagnetic field on the other. 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] book. 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 lin 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 in high-intensity laser fields are used to ultrarelativistic electron-positron plasmas which play an important role in various astrophysical situations. Various 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 appears 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 the 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], has 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. In this perspective in very recent works where three of the present authors were involved, the general structure of fermionic 2-and 3-point function [34], and 4-point function [35], based on various symmetries of the system for a nontrivial medium like a hot magnetized one. Also the spectral representation of two point function [34] were obtained for such system. In this paper we consider photon that propagates in a hot magnetized QED 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 QCD system. This paper, we call it as paper-I, has been organized as follows: in section 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 subsection 2.1 and then a generalization of it to a hot magnetized medium in subsection 2.2. In section 3 we discuss the general structure for the gauge boson propagator using the results of section 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 subsections 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 section 5 along with a statement that the general propagators for fermion in Ref. [34] and the gauge boson obtained here will be used to compute the quark-gluon free energy for a hot magnetized deconfined QCD system in paper-II.

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 subsection 2.1 and it will then be followed by a formalism for a magnetized hot medium in subsection 2.2.

Finite temperature and zero magnetic field case
We begin with the general structure of the gauge boson self-energy in vacuum, given as where the form factor Π(P 2 ) is Lorentz invariant and depends only on the four scalar P 2 . The vacuum projection operator is with the metric g µν ≡ (1, −1, −1, −1) and P µ ≡ (p 0 , p). The self-energy satisfies the gauge invariance through the transversality condition P µ Π µν (P 2 ) = 0, (2.3) and it is also symmetric The conditions in Eqs. (2.3) and (2.4) are sufficient to obtain ten components of Π µν . The presence of finite temperature (β = 1/T ) or heat bath breaks the Lorentz (boost) invariance of the system. In finite temperature one accumulates all the necessary four vectors and tensors present to form a general covariant structure of the gauge boson self-energy. Those are P µ , g µν from vacuum and the four-velocity u µ of the heat bath, discreetly introduced because of the medium. With these one can form four types of tensors, namely P µ P ν , P µ u ν + u µ P ν , u µ u ν and g µν . These four tensors can be reduced to two independent mutually orthogonal projection tensors by virtue of the constraints provided by the transversality condition given above in Eq. (2.3). 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 [36][37][38]. 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: where p = |p|. We note here that one can only construct two independent Lorentz scalars as given in Eqs. (2.5a) and (2.5d). One can further redefine four vector u µ by projecting the vacuum projection tensor upon it as which is orthogonal to P µ . 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 Moreover, sum of these two projection operators lead to the well known vacuum projection tensor V µν as So, the general (manifestly) covariant form of the self-energy tensor can be written as where Π L and Π T are, respectively, the longitudinal and transverse form factors. Eventually one can obtain these two form factors as 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 = p 2 0 − P 2 as defined, respectively, in Eqs. (2.5a) and (2.5d) besides the temperature T = 1/β.

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 (β = 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 µ , g µν , the electro-magnetic field tensor F µν and it's dualF µν , and the four velocity of the heat bath, u µ . As seen in section 2.1 at finite T the heat bath introduces a preferred direction that breaks the boost invariance. On the other hand, the presence of magnetic field breaks the rotational symmetry in the system because it introduces an anisotropy in space. For hot magnetised system, one can define a new four vector n µ which is associated with the electromagnetic field tensor F µν . In the rest frame of the heat bath, i.e., u µ = (1, 0, 0, 0), n µ can be defined uniquely as projection of F µν along u µ , 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 P µ , g µν , u µ and n µ from which one can form seven types of tensors, namely P µ P ν , P µ n ν + n µ P ν , n µ n ν , P µ u ν + u µ P ν , u µ u ν , u µ n ν + n µ u ν and g µν . These seven tensors reduce to four because of constraints provided by the gauge invariance condition in Eq. (2.3). However, these four tensors are not mutually transverse. So, all of them are not independent as well as not projection tensors. This will result in coupling of form factors with each other. If one uses these four tensors 1 [39], to construct the general structure of the gauge boson self-energy, it leads to a complicated structure of the effective propagator which also looks a bit unusual. Instead we will show that, for a hot magnetized medium, three mutually orthogonal projection tensors are sufficient for restoring the manifestly Lorentz-invariant structure of self-energy. As we will see below that it also leads to a very simplified general structure of the gauge boson effective propagator.
We first form the transverse four momentum and the transverse metric tensor as The newly defined momentum and metric tensor orthogonal to both u and b, satisfy the following We further note that the three independent Lorentz scalars are p 0 , p 3 = P · n and P 2 ⊥ . Now following the footsteps of the two previous sections, we can construct a second rank tensor orthogonal to P µ , u µ , and n µ by using those redefined set of four vectors, e.g., which satisfies the transversality condition and projection operator property The second projection tensor is which was used in finite temperature and zero magnetic field case. We, now, construct the third projection tensor Q µν , such that the sum of R µν , B µν and Q µν gives the vacuum projection operator V µν as It can be checked easily that the projection tensor Q µν satisfies all the necessary properties, i.e., The three projection tensors are orthogonal to each other: Now, one can write a general covariant structure of gauge boson self-energy as where b, c and d are three Lorentz-invariant form factors associated with the three projection tensors. Below we define some essential relations of the constituent tensors for obtaining those form factors: contracting with the general metric tensor yields along with (00) components of the constituent tensors as Using these information, we obtain the form factors as In absence of the magnetic field by comparing with the known general form of finite temperature self-energy in Eq. (2.9), as where we used the fact that R µν + Q µν = A µν .

General form of gauge boson propagator in a hot magnetized medium
The inverse of the gauge boson propagator in vacuum is written as where ξ is the gauge parameter. From Eq. (2.19) one can write and using in Eq. (3.1), we get The inverse of the general gauge boson propagator following Dyson-Schwinger equation reads as The inverse of Eq. (3.5) can be written as along with Now using the explicit forms of B ν µ , R ν µ and Q ν µ and equating different coefficients from both sides yield the following conditions: Coefficients of n µ n ν : Now the general covariant structure of the gauge boson propagator can finally be written as We note that there are three different modes in presence of both temperature and magnetic field as evident from the poles of the above propagator. 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 mode which introduces an additional mode in the hot medium. When we turn off the magnetic field, the general structure of the propagator in a non-magnetised thermal bath becomes which agrees with the known result [36][37][38].

Form Factors
Before computing the various form factors associated with the general structure we note following points: 1. Now we note that the magnetic field generated during the non-central HIC is time dependent which is believed to decrease rapidly with time [20,41]. It would be extremely complicated to work with a time dependent magnetic field. Instead one can work in two limiting conditions to analytically incorporate the effects of magnetic field into the system. At the time of the collision, the value of the magnetic field B is very high compared to the temperature T (T 2 < eB where e is the electronic charge) as estimated upto the order of eB ∼ 15m 2 π in the LHC at CERN [42]. Also in the dense sector, neutron stars (NS), or more specifically magnetars are known to possess strong enough magnetic field [43][44][45]. The effect of the strong enough magnetic field can be incorporated via a simplified Lowest Landau Level (LLL) approximation which considers that in such strong magnetic fields, fermions are basically confined within the LLL. Again, a rapidly decaying magnetic field provides us another simplified situation in the regime where it becomes sufficiently weaker than the associated temperature scale (T 2 > eB). In this weakly magnetized hot medium one can write down the corrections due to the magnetic field as the higher order perturbations to the unmagnetized part. In the following two 2. Since m f is the lowest scale in weak field approximation, we would restrict it by the thermal mass m th ∼ eT (gT ), where e 2 (g 2 ) = 4πα (α s ) in QED (QCD) coupling and T is the temperature of the system.
3. We choose the frame of reference as shown in Fig. 1 in which one considers the external momentum of the vector boson in xz plane with 0 < θ p < π/2. So one can write P µ = (p 0 , p 1 , 0, p 3 ) = (p 0 , p sin θ p , 0, p cos θ p ) (4.1) and then loop momenta K µ = (k 0 , k sin θ cos φ, k sin θ sin φ, k cos θ). 4. For simplicity we consider photon (QED case) and extending it to gluon (QCD case) will be discussed. We consider strong field approximation: m 2 f T 2 eB, where m f is the fermion mass and T is the temperature. When the external magnetic field is very strong, eB → ∞, it pushes all the Landau levels (n ≥ 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 where K is the fermionic four momentum and we have used the properties of generalized Laguerre polynomial, L n ≡ L 0 n and L α −1 = 0. In strong field approximation or in LLL, eB k 2 ⊥ , an effective dimensional reduction from (3+1) to (1+1) takes place. Now in the strong field limit the self-energy can be computed as where 's' indicates that the quantities are to be calculated in the strong field approximation and Tr represents only the Dirac trace. 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 with the tensor structure S s µν that originates from the Dirac trace is where the Lorentz indices µ and ν are restricted to longitudinal values because of dimensional reduction to (1+1) dimension and forbids to take any transverse values. Now we use Eq. (2.13b) and Eq. (2.13c) to rewrite S µν as

Form factors and Debye mass
First we evaluate the form factors in Eqs. (2.26a), (2.26b) and (2.26c) in strong field approximation as Now, combining Eq. (4.8b) and the Hard Thermal Loop (HTL) approximation [46] one can have Using Eq. (4.9) in Eq. (2.26d) one also can directly calculate the Debye screening mass in QED as which matches with the expression of QED Debye mass calculated in Refs. [47,48] where three distinct scales (m 2 f , T 2 and eB) were clearly evident for massive quarks. Now using Eq. (4.10) in Eq. (4.9), the form factor b can be expressed in terms The form factor d then becomes where the expression for (Π µ µ ) s is given in Eq. (A.3) in Appendix A. Also in the strong field approximation, |eB| T 2 m 2 f , one can neglect the quark mass m f , as to get an analytic expression of Debye mass as which agrees with that obtained in Ref. [48].

Dispersion
As discussed in Eq. (3.10), the dispersion relations for photon in strong field approximation with LLL read where the form factors are given, respectively, in Eqs. (4.8a), (4.11) and (4.12) . As found c = 0 and both b and d are non-zero but have negligibly small contributions (∼ 0). So, in strong field approximation with LLL, all three dispersion dispersion relations resemble the free dispersion of photon. The reason for which could be understood in the following way: the projection operators, (1 ± iγ 1 γ 2 ), that appears in the (anti)fermion propagator in Eq. (4.3) aligns the spin of the fermions in LLL along the magnetic field direction. This means that the two fermions from LLL make the photon spin equals to zero in the field direction and there is no polarization in the transverse direction as evident from Eq. (4.8a). This is because in strong field approximation there is an effective dimensional reduction from (3+1) to (1 + 1) dimension in LLL, which causes three dispersion relations to merge to a single one.

Gauge boson in weakly magnetised hot medium
As discussed earlier we recall that, in weak field approximation, one is restricted in the domain m 2 f |eB| T 2 . Since fermion mass is very small, we use the thermal mass of fermion m f = m th f = eT / √ 8 for QED system [37,38].

One-loop photon self-energy
The fermion propagator in a weak magnetic field can be written up to an O[(eB) 2 ] as (4.15) where S 0 is the continuum free field propagator in absence of B whereas S 1 and S 2 are, respectively, O[(eB)] and O[(eB) 2 ] correction terms in presence of B. The photon self-energy from the Feynman diagram in Fig. 3 can be written as where the first term Π 0 µν , containing two S 0 , is the leading order perturbative term in absence of B whereas the remaining two terms are (eB) 2 order corrections. However, P P Q = K − P K γ µ γ ν Figure 3: Photon polarization tensor in weak field approximation.
we note that 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 first and second terms in Eq. (4.17) can be written as 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 U µν = 4(K · u)(Q · u) (2n µ n ν + g µν ) + 4(K · n)(Q · n) (2u µ u ν − g µν ) −4 [(K · u)(Q · n) + (K · n)(Q · u)] (u µ n ν + u ν n µ ) + 4m 2 f g µν +8m 2 f (g 1µ g 1ν + g 2µ g 2ν ) . (4.20) The third term in Eq. (4.17) can be written as where Π 0 00 (P ) = ie 2 .

(4.24)
Using the hard thermal loop (HTL) approximation [37] and performing the frequency sum, one can write for m f = 0 . Now the QED Debye mass in the absence of the magnetic field can directly be obtained using Eq. (2.26d) as Using Eq. (4.26) in Eq. (4.25), we get where we use p = p 2 1 + p 2 3 as p lies in xz plane as shown Fig. 1. The form factor in Eq. (4.23) becomes which agrees with the HTL longitudinal form factor Π L (p 0 , p) [37]. Similarly, we will calculate here the coefficients c 0 and d 0 explicitly. and We note that Π 0 00 is already calculated in Eq. (4.27) and one needs to calculate the remaining two components of Π 0 µν (P ) which are as follows: and Π 0 33 (P ) = ie 2 Using the results from Eqs. (4.27), (4.31), and (4.32), c 0 (p 0 , p) and d 0 (p 0 , p) become which agrees with the HTL transverse form factor Π T (p 0 , p) [37].
We obtain O(eB) 2 term of the coefficient c in Eq. (E.2) of appendix E as We calculate the O(eB) 2 term of the coefficient d in appendix F as where F 1 and F 2 can be found in Eqs. (F.2) and (F.3), respectively.

Dispersion
In weak field approximation the dispersion relation can now be written as We note that the dispersion relations are scaled by m D / √ 3. As seen that there are three distinct modes when a photon propagates in hot magnetised material medium. The magnetised plasmon mode with energy ω b appears due to the form factor b whereas two transverse modes with energy ω c and ω 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. 5 when it propagates at an angle θ p = π/3 with the direction of the magnetic field. We have chosen three different values of magnetic field |eB| = m 2 π /2, m 2 π /10 and m 2 π /800(∼ 0); m π is the pion mass. For a given magnetic field strength, say |eB| = m 2 π /2, one finds two zero modes and one massive mode. The two zero modes comprise a plasmon and a transverse mode. We note that there is a cancellation between thermal and magnetic mass that causes massless mode to appear in a hot magnetized medium. As the field strength decreases, the dispersion curves of photon approach to those of thermal medium which can be seen for the field strength, |eB| = m 2 π /800(∼ 0). In Fig. 6 we have also displayed the dispersion of photon when it propagates at an angle θ p = π/6.

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 heat bath whereas rotational invariance is broken due to presence of a background magnetic field. Based on gauge invariance and symmetry property of the gauge boson self-energy, the general Lorentz structure is obtained by using three mutually orthogonal projection tensors in contrary to four tensors all of which are not projection tensor and also not orthogonal. We emphasize that because of the three independent projection tensors we could able to obtain a very simplified structure of the effective gauge boson propagator in a magnetized hot medium akin to that of a thermal medium.
We used this effective propagator to study the dispersion spectra of a photon in hot magnetized medium. In strong field approximation, one finds three degenerate massless modes of a gauge boson akin in LLL due to dimensional reduction. On the other hand in weak field approximation one finds there distinct modes, viz., one magnetized massless plasmon, one massless transverse mode and one massive transverse mode. Interestingly, the massless modes or zero modes appear due to the cancellation of thermal and magnetic mass. The calculation for photon can trivially be extended to gluon. We further note that the effective propagator obtained here can conveniently be used to study the various quantities in QED and QCD plasma. We, further, note that a calculation is in progress to obtain various thermodynamic quantities using the general structure of the gauge boson here and fermions in Ref. [34] of a magnetized hot QCD plasma.

Acknowledgement
BK and MGM would like to acknowledge very useful discussion with Palash B Pal. BK would also like to thank Avik Banerjee for helpful discussion. AB is supported by the National Post Doctoral Program CAPES (PNPD/CAPES), Govt. of Brazil. NH is funded by the Alexander von Humboldt Foundation, Germany as an Alexander von Humboldt postdoctoral fellow. BK and MGM were funded by the Department of Atomic Energy (DAE), India via the project TPAES.
Let us take m f = yT and k = xT .
The integrals can be represented by the well-known functions as, f n+1 (y) = 1 Γ(n + 1) In the regime of HTL perturbation theory and weak magnetic field, one can use high temperature expansion for f 1 as,