General structure of fermion two-point function and its spectral representation in a hot magnetised medium

We have systematically constructed the general structure of the fermion self-energy and the effective quark propagator in presence of a nontrivial background like hot magnetised medium. This is applicable to both QED and QCD. The hard thermal loop approximation has been used for the heat bath. We have also examined transformation properties of the effective fermion propagator under some of the discrete symmetries of the system. Using the effective fermion propagator we have analysed the fermion dispersion spectra in a hot magnetised medium along with the spinor for each fermion mode obtained by solving the modified Dirac equation. The fermion spectra is found to reflect the discrete symmetries of the two-point functions. We note that for a chirally symmetric theory the degenerate left and right handed chiral modes in vacuum or in a heat bath get separated and become asymmetric in presence of magnetic field without disturbing the chiral invariance. The obtained general structure of the two-point functions is verified by computing the three-point function, which agrees with the existing results in one-loop order. Finally, we have computed explicitly the spectral representation of the two-point functions which would be very important to study the spectral properties of the hot magnetised medium corresponding to QED and QCD with background magnetic field.


I. Introduction
In non-central heavy ion collision (HIC) experiments in LHC at CERN and in RHIC at BNL, it is believed that a very strong magnetic field is created in the direction perpendicular to the reaction plane due to the spectator particles that are not participating in the collisions. The experiments conducted by PHENIX Collaboration [1] showed direct-photon anisotropy which has posed a serious challenge to the present theoretical models. It is conjectured that this excess elliptic flow may be due to the excess photons produced by the decay → ( ) and the branching ratio of which increases in presence of the magnetic field near the critical value where the condensate of is found. The estimated strength of this magnetic field depends on collision energy and impact parameter between the colliding nuclei and is about several times the pion mass squared, i.e., ∼ 15 2 at LHC in CERN [2]. Also, a class of neutron star called magnetar exhibits [3][4][5] a magnetic field of 10 18 − 10 20 Gauss at the inner core and 10 12 − 10 13 Gauss at the surface. These observations motivate to study the properties of hot magnetised medium using both phenomenology and quantum field theory.
The magnetic field created in HIC lasts for very short time ( ∼ a few fm). The strength of the field decays rapidly with time after ∼ 1 − 2 ∕ . However, the medium effects like electric conductivity can delay the decay and by the time deconfined quarks and gluons equilibrate with QGP medium, the magnetic field strength gets sufficiently weak. At that time the relevant energy scales of the system can be put in this way: < 2 ≪ 2 . In this low field limit the properties of the deconfined medium are also affected. So, it becomes important to treat the weak field limit separately. Fermion propagator in presence of a uniform background magnetic field has been derived first by Schwinger [38]. Using this, one loop fermion self-energy and the vacuum polarisation was calculated in double parameter integral in [39] and [40], respectively. The weak field expansion of this propagator was calculated order by order in powers of in [41]. Recently, the pion self-energy and its dispersion property have been studied at zero temperature [42] in weak field approximation and using full propagator at finite temperature [43]. Also a detailed study of the spectral properties of mesons has been performed in presence of magnetic field both at zero [44,45] and at non-zero temperature [46].
For hot and dense medium (e.g., QED and QCD plasma), it is well know that a bare perturbation theory breaks down due to infrared divergences. A reorganisation of the perturbation theory has been done by performing the expansion around a system of massive quasiparticles [47], where mass is generated through thermal fluctuations. This requires a resummation of certain class of diagrams, known as hard thermal loop (HTL) resummation [48], when the loop momenta are of the order of the temperature. This reorganised perturbation theory, known as HTL perturbation theory (HTLpt), leads to gauge independent results for various physical quantities [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. Within this one-loop HTLpt, the thermomagnetic correction to the quark self-energy [66], quarkgluon three point [66] function at zero chemical potential and four point [67] function at finite chemical potential in weak field limit have been computed. The fermion self-energy has also been extended to the case of non-zero chemical potential and the pressure of a weakly magnetised QCD plasma [68] has also been obtained.
In recent years a huge amount of activity is underway to explore the properties of a hot medium with a background magnetic field using phenomenology as well as using thermal field theory. In a thermal medium the bulk and dynamical properties [48,69,70] are characterised by the collective excitations in a time like region and the Landau damping in a space-like domain. The basic quantity associated with these medium properties is the two point correlation function. In this work we construct the general structure of the fermionic two point functions (e.g., self-energy and the effective propagator) in a nontrivial background like a hot magnetised medium. We then analyse its property under the transformation of some discrete symmetries of the system, the collective fermionic spectra, QED like three-point functions and the spectral representation of the two point function and its consequences in a hot magnetised medium. The formulation is applicable equally well to both QED and QCD.
The paper is organised as follows; In section II, the notation and set up are briefly discussed through a fermion propagator in a constant background field using Schwinger formalism. Section III has number of parts in which we obtain the general structure of the self-energy (subsec.III A), the effective fermion propagator (subsec.III B ), the transformation properties and discrete symmetries of the effective propagator (subsec.III C), the modified Dirac equations in general and for lowest Landau level (subsec.III D) and the dispersion properties of the various collective modes (subsec.III E) in time-like region. In section IV the general structure of the self-energy and the propagator has been verified from one-loop direct calculation. The spectral representation of the effective propagator in space-like domain has been obtained in section V. We have presented some detailed calculations for various sections and subsections in Appendix A-E. Finally, we conclude in section VI.

II. Charged Fermion Propagator in Background Magnetic Field within Schwinger Formalism
In this section we set the notation and briefly outline the fermionic propagator in presence of a background magnetic field following Schwinger formalism [38] . Without any loss of generality, the background magnetic field is chosen along the direction, ⃗ = ̂ , and the vector potential in a symmetric gauge reads as Below we also outline the notation we shall be using throughout: where ∥ and ⟂ are, respectively, the parallel and perpendicular components, which would be separated out due to the presence of the background magnetic field. Now, the fermionic two-point function is written as where the parameter is called Schwinger proper time variable [38]. We note that and are mass and absolute charge of the fermion of flavour , respectively. The phase factor, ( , ′ ), is independent of but is responsible for breaking of both gauge and translational invariance. Remaining part, denoted as ( − ′ ), is translationally invariant. However, as shown below, ( , ′ ) drops out for a gauge invariant calculation. Now ( , ′ ) reads as where is just a number. The integral in the exponential is independent of the path taken between and ′ and choosing it as a straight line one can write Using the gauge transformation ( ) → ( )+ Λ( ), and choosing symmetric gauge as given in (1), the phase factor Φ( , ′ ) becomes 1, if we take [66] From equation (3), the momentum space propagator can be obtained as where = 1 2 = 1 ̃ . (12) and the four velocity in the rest frame of the heat bath and the direction of the magnetic field , respectively, given as = (0, 0, 0, 1).
One can notice that in a hot magnetised medium both and are correlated as given in (12) and the contribution due to magnetic field in (9) in presence of heat bath becomes a thermo-magnetic contribution. We also further note that in absence of heat bath, (11) reduces to (9), which is not obvious by inspection but we would see later.

III. General Structure of Fermion Two-point Function in a Hot Magnetised Medium
In previous section the modification of a free propagator has been discussed briefly in presence of a background magnetic field. In this section we would like to obtain the most general structure of a fermion self-energy, the effective fermion propagator and some of its properties in a nontrivial background like hot magnetised medium. We would also discuss the modified Dirac equation and the fermion dispersion spectrum in a hot magnetised medium. For thermal bath we would use HTL approximation and any other approximation required for the purpose will be stated therein.

A. General Structure of the Fermion Self-Energy
The fermionic self-energy is a matrix as well as a Lorentz scalar. However, in presence of nontrivial background, e.g., heat bath and magnetic field, the boost and rotational symmetries of the system are broken. The general structure of fermion self-energy for hot magnetised medium can be written by the following arguments. The self-energy Σ( ) is a 4 × 4 matrix which depends, in present case, on the four momentum of the fermion , the velocity of the heat bath and the direction of the magnetic field . Now, any 4 × 4 matrix can be expanded in terms of 16 basis matrices: {1, 5 , , 5 , }, which are the unit matrix, the four -matrices, the six matrices, the four 5 matrices and finally 5 . So, the general structure can be written as where various coefficients are known as structure functions. We note that the combinations involving do not appear due to antisymmetric nature of it in any loop order of self-energy. Also in a chirally invariant theory, the terms 1 and 5 will not appear as they would break the chiral symmetry. The term 5 ∕ would appear in the self-energy if fermions interact with an axial vector 1 . By dropping those in (14) for chirally symmetric theory, one can now write Now we point out that some important information is encoded into the fermion propagator in (7) through (11) for a hot magnetised medium. This suggests that ∕ should not appear in the fermion self-energy 2 and the most general form of the fermion self-energy for a hot magnetised medium becomes When a fermion propagates in a vacuum, then = ′ = ′ = 0 and Σ( ) = − ∕ . But when it propagates in a background of pure magnetic field without any heat bath, then ≠ 0, = 0 and the structure functions, ′ and ′ , will depend only on the background magnetic field as we will see later. When a fermion propagates in a heat bath, then ≠ 0, ≠ 0 but both ′ and ′ vanish because there would not be any thermo-magnetic corrections as can also be seen later. We now write down the right chiral projection operator,  + and the left chiral projection operator  − , respectively, defined as: which satisfy the usual properties of projection operator: Using the chirality projection operators, the general structure of the self-energy in (16) can be casted in the following form where ∕ and ∕ are defined as From (16) one obtains the general form of the various structure functions as which are also Lorentz scalars . Beside and , they would also depend on three Lorentz scalars defined by we may interpret , ⟂ , as Lorentz invariant energy, transverse momentum, longitudinal momentum respectively. All these structure functions for 1-loop order in a weak field and HTL approximations have been computed in Appendix A and quoted here 3 as We note that the respective vacuum contributions in , ′ and ′ have been dropped by the choice of the renormalisation prescription, and the general structure of the self-energy, as found in appendix A, agrees with that in (16).

B. Effective Fermion Propagator
The effective fermion propagator is given by Dyson-Schwinger equation (see Fig. 1) which reads as Using (19) the inverse fermion propagator can be written as * −1 ( ) where ∕ and ∕ can be obtained from two four vectors given by 3 In weak field approximation the domain of applicability becomes 2 ℎ (∼ 2 2 ) < < 2 instead of 2 < < 2 as discussed in Appendix A. with Using (26) in (24), the propagator can now be written as where we have used the properties of the projection operators  ± =  ∓ ,  2 ± =  ± , and  +  − =  −  + = 0. It can be checked that * ( ) * −1 ( ) =  + +  − = 1 . Also we have where we have used 2 = 1, 2 = −1, ⋅ = 0, ⋅ = 0 , and ⋅ = − . Note that we have suppressed the functional dependencies of , , ,  ± and ′ and would bring them back whenever necessary. For the lowest Landau Level (LLL), = 0 ⇒ ⟂ = 0, and these relations reduce to The poles of the effective propagator, 2 = 0 and 2 = 0, give rise to quasi-particle dispersion relations in a hot magnetised medium. There will be four collective modes with positive energies: two from 2 = 0 and two from 2 = 0. Nevertheless, we will discuss dispersion properties later.

C. Transformation Properties of Structure Functions and Propagator
First, we outline some transformation properties of the various structure functions as obtained in (23a), (23b), (23c) and (23d).
Now based on the above we also note down the transformation properties of those quantities appearing in the propagator: .
1. For : 2. For  ± : Using the above transformation properties, it can be shown that ∕ , ∕ , 2 and 2 , respectively given in (27a), (27b), (30a) and (30b) transform as and Now we are in a position to check the transformation properties of the effective propagator under some of the discrete symmetries:

Reflection
Under reflection the fermion propagator transforms [71] as The effective propagator, * ( 0 , ⟂ , ), in (29) transforms under reflection as However, now considering the rest frame of the heat bath, = (1, 0, 0, 0), and the background magnetic field along -direction, As seen in both cases the reflection symmetry is violated as we will see later while discussing the dispersion property of a fermion.

Parity
Under parity a fermion propagator transforms [71] as The effective propagator, * ( 0 , ⟂ , ), in (29) under parity transforms as which does not obey (44), indicating that the effective propagator in general frame of reference is not parity invariant due to the background medium. However, now considering the rest frame of the heat bath, = (1, 0, 0, 0), and the background magnetic field along -direction, = (0, 0, 0, 1), one can write (45) by using (37a), (37b) and 0 = − 0 as which indicates that the propagator is parity invariant in the rest frame of the magnetised heat bath. We note that other discrete symmetries can also be checked but leave them on the readers.

For General Case
The effective propagator that satisfy the modified Dirac equation with spinor is given by Using the chiral basis one can write (47) as where and are two component Dirac spinors with ≡ (1, ⃗ ) and̄ ≡ (1, −⃗ ), respectively. One can obtain nontrivial solutions with the condition We note that for a given 0 (= ), either 2 = 0, or 2 = 0, but not both of them are simultaneously zero. This implies that i) when 2 = 0, = 0 ; ii) when 2 = 0, = 0. These dispersion conditions are same as obtained from the poles of the effective propagator in (29) as obtained in subsec. III B.
2. For LLL, 2 = 0 in (31a) indicates that 0 = ± , = = 0. The two solutions obtained, respectively, in (B7) and (B8) in Appendix B are given as The spin operator along the direction is given by where with single index denotes Pauli spin matrices whereas that with double indices denote generator of Lorentz group in spinor representation. Now, So, the modes (−) and (+) have spins along the direction of magnetic field whereas (+) and (−) have spins opposite to the direction of magnetic field. Now we discuss the helicity eigenstates of the various modes in LLL. The helicity operator is defined as When a particle moves along + direction,̂ =̂ and when it moves along − direction,̂ = −̂ . Thus Thus, and In presence of magnetic field, the component of momentum transverse to the magnetic field is Landau quantised and takes discrete values given by 2 ⟂ = 2 | |, where is a given Landau levels. In presence of pure background magnetic field and no heat bath ( = 0), the Dirac equation gives rise a dispersion relation where one can define Now we discuss the dispersion properties of a fermions in a hot magnetised medium. For general case (for higher LLs, ≠ 0) the dispersion curves obtained by solving, 2 = 0 and 2 = 0 given in (30a) and (30b), numerically. We note that the roots of . The corresponding eigenstates are obtained in (54a), (54b), (52a) and (52b) in subsection III D 1. We have chosen = 0.2 GeV, = 0.3 and = 0.5 2 , where is the pion mass. In Fig. 2 the dispersion curves for higher Landau levels are shown where all four modes can propagate for a given choice of . This is because the corresponding states for these modes are neither spin nor helicity eigenstates as shown in subsec. III D 1. We also note that there will be negative energy modes which are not displayed here but would be discussed in the analysis of the spectral representation of the effective propagator section V.
At LLL = 0 → ⟂ = 0 and the roots of 0 = ± give rise to two right handed modes (±) with energy (±) whereas those for 0 = ± produce 4 two left handed modes (±) with energy (±) . In Appendix D the analytic solutions for the dispersion relations in LLL are presented which show four different modes and the corresponding eigenstates are obtained in subsec. III D 2. Now at LLL we discuss two possibilities below: (i) for positively charged fermion = 1, = 1 implies = 0 and = −1 implies = −1. Now we note that can never be negative. This implies that the modes with = 1 and = −1 (spin down) cannot propagate in LLL. Now, the right handed mode (+) and the left handed mode (−) have spin up as shown in subsec. III D 2, will propagate in LLL for > 0. The (+) mode has helicity to chirality ratio +1 is a quasiparticle whereas the mode (−) left handed has that of −1 known as plasmino (hole). However, for < 0, the right handed mode flips to plasmino (hole) as its chirality to helicity ratio becomes -1 whereas the left handed mode becomes particle as its chirality to helicity ratio becomes +1. The dispersion behaviour of the two modes are shown in the left panel of Fig. 3

which begins at mass
as given in (D13).
(ii) for negatively charged fermion = −1, = 1 implies = −1 and = −1 implies = 0. Thus, the modes with = −1 and = +1 (spin up) cannot propagate in LLL. However, the modes (+) and (−) have spin down as found in subsec. III D 2 will propagate in LLL. Their dispersion are shown in the right panel of Fig. 3 which begin at mass * + as given in (D13). For > 0 the mode (+) has helicity to chirality ratio +1 whereas (−) has that of −1 and vice-versa for < 0. In the absence of the background magnetic field ( = 0), the two modes, the left handed (+) and the right handed (+) fermions, merge together whereas the other two modes, the left handed (−) and the right handed (−) fermions, also merge together. This leads to degenerate (chirally symmetric) modes for which the dispersion plots start at ℎ and one gets back the usual HTL result [49] with quasiparticle and plasmino modes in presence of heat bath as shown in Fig. 4.
As evident from the dispersion plots (Figs. 2 and 3) both left and right handed modes are also degenerate at = 0 in presence of magnetic field but at non-zero | | both left and right handed modes get separated from each others, causing a chiral asymmetry without disturbing the chiral invariance (subsec. III C 1) in the system. Also in subsec. III C 2 it was shown that the fermion propagator does not obey the reflection symmetry in presence of medium, which is now clearly evident from all dispersion plots as displayed above.

IV. Three Point Function
The ( + 1)-point functions are related to the -point functions through Ward-Takahashi (WT) identity. The 3-point function is related to the 2-point function as where = − . We note that recently the general form of the thermo-magnetic corrections for 3-point [66,67] and 4-point [67] functions have been given in terms of the involved angular integrals, which satisfy WT identies. Nevertheless, to validate the general structure of the self-energy in (16) vis-a-vis the inverse propagator in (25), we obtain below the temporal component of the 3-point function at ⃗ = 0; ⃗ = ⃗ and = .
Using (23a), (23b), (23c) and (23d), we can obtain ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Thermo-magnetic correction where are the Legendre functions of the second kind given in (A7a) and (A7b). Important to note that the thermo-magnetic (TM) correction Γ 0 TM matches exactly with that from direct calculation in (C5) in Appendix C. The result also agrees with the HTL 3-point function [66,67] in absence of background magnetic field by setting = 0 ⇒ ′ = 0 as We now note that this is the three-point function with pure background magnetic field but no heat bath. The gauge boson is oriented along the field direction and there is no polarisation in the transverse direction. Thus, only the longitudinal components (i.e, (0,3)-components) of the 3-point function would be relevant for pure background magnetic field in contrast to that of (70) for pure thermal background.

V. Spectral Representation of the Effective Propagator
In this section we obtain the spectral representation of the effective propagator in a hot magnetised medium. This quantity is of immense interest for studying the various spectral properties, real and virtual photon production, damping rates and various transport coefficients etc. of the hot magnetised medium, in particular, for hot magnetised QCD medium.

A. General Case
The effective propagator as obtained in (29) is given by where ∕ and ∕ can be written in the rest frame of the heat bath and the magnetic field in the -direction following (27a) and (27b), respectively, as wherê = ∕| |, = | | and, and ⟂ are given, respectively, in (22b) and (22c). We also note that though 2 = 2 ; 3 = 3 , but they are treated separately for the sake of notations that we would be using, for convenience, as and . One can decompose the effective propagator into six parts by separating out the matrices as In subsection III E we have discussed that 2 = 0 yields four poles, leading to four modes with both positive and negative energy as ± (+) ( ⟂ , ) and ± (−) ( ⟂ , ). Similarly, 2 = 0 also yields four poles, namely ± (+) ( ⟂ , ) and ± (−) ( ⟂ , ).
With this information one can obtain the spectral representation [49,[72][73][74] of the effective propagator in (76) as where the spectral function corresponding to each of the term can be written as where = 1, 2, 3. We note that the delta-functions are associated with pole parts originating from the time like domain ( 2 0 > 2 ) whereas the cut parts ( ) are associated with the Landau damping arises from the space-like domain, 2 0 < 2 , of the propagator. The residues ( ) are determined at the various poles as As a demonstration, we present analytical expressions of three residues corresponding to the pole 0 = + (+) as . The other poles of 2 = 0 can trivially be found out by replacing (+) in the above expressions. The expressions for the residues for parts can similarly be expressed as the parts, but we do not show them. Below in Fig. 5 we present the residues corresponding to the first Landau level where all the terms are present. We take the value of the magnetic field as 2 ∕2 and temperature to be 200 MeV. where

B. LLL Case
For LLL, as ⟂ = 0, so 2 ( ) and 3 ( ) in (74) and (75)  The spectral function corresponding to LLL reads as where one needs to determine which can again be represented in terms of different residues corresponding to different poles of 2 ( 2 ) = 0 as in Eq.(79). In Fig. 6, the variation of the residues for the lowest Landau level are shown.
In appendix E we have demonstrated how one gets back the HTL spectral functions when magnetic field is withdrawn from the thermal medium.

VI. Conclusions
In this article the general structure of fermionic self-energy for a chirally invariant theory has been formulated for a hot and magnetised medium. Using this we have obtained a closed form of the general structure of the effective fermion propagator. The collective excitations in such a nontrivial background has been obtained for a time-like momenta in the weak field and HTL approximation in the domain 2 ℎ (∼ 2 2 < | | < 2 . We found that the left and right handed modes get separated and become asymmetric in presence of magnetic field which were degenerate and symmetric otherwise. The transformation of the effective propagator in a hot magnetised medium under some of the discrete symmetries have been studied and its consequences are also reflected in the collective fermion modes in the Landau levels. We have also obtained the Dirac spinors of the various collective modes by solving the Dirac equation with the effective two-point function. Further, we checked the general structure of the two-point function by obtaining the three-point function using the Ward-Takahashi identity, which agrees with the direct calculation of one-loop order in weak field approximation. We also found that only the longitudinal component of the vertex would be relevant when there is only background magnetic field. The spectral function corresponding to the effective propagator is explicitly obtained for a hot magnetised medium which will be extremely useful for studying the spectral properties, e.g., photon/dilepton production, damping rate, transport coefficients for a hot magnetised medium. This has pole contribution due to the various collective modes originating from the time-like domain and a Landau cut contribution appearing from the space-like domain. It has explicitly been shown that the spectral function reduces to that obtained for thermal medium in absence of the magnetic field. Our formulation is in general applicable to both QED and QCD with nontrivial background like hot magnetised medium

VII. Acknowledgement
The authors would like to acknowledge useful discussions with Palash B Pal, Najmul Haque, Chowdhury Aminul Islam, Arghya Mukherjee and Bithika Karmakar. AB and MGM were funded by the Department of Atomic Energy (DAE), India via the project TPAES whereas AD and PKR were funded by the project DAE/ALICE/SINP. AB was also partially supported by the National Post Doctoral Program CAPES (PNPD/CAPES), Govt. of Brazil.

A. Computations of structure functions in one-loop in a weak field approximation for hot magnetised QCD medium:
Here, we present the computations of the various structure functions in (21a) to (21d) in 1-loop order (Fig.7) in a weak field and HTL approximations following the imaginary time formalism. In Fig.7 the modified quark propagator (bold line) due to background magnetic field is given in (A3). Since glouns are chargeless, their propagators do not change in presence of magnetic field. The gluon propagator in Feynman gauge, is given as [41] ( ) = − 2 . (A1) We note that we would like to explore the fermion spectrum in a hot magnetised background in the limit 2 < < 2 . In this domain the fermion propagator is obtained by expanding the sum over all Landau levels in powers of in (7) where the first term is the free propagator and the second one is [ ] correction to it. Now combining (A2) and (11) the fermion propagator in background magnetic field reads as where the fermion mass in the numerator has been neglected in the weak field domain, 2 < ( ) < 2 . The one-loop quark self-energy upto (| |) can be written as where is the QCD coupling constant, = 4∕3 is the Casimir invariant of (3) group, is the temperature of the system. The first term is the thermal bath contribution in absence of magnetic field ( = 0) whereas the second one is from the magnetised thermal bath.
Using (A4) in (21a) and (21b), the structure functions and , respectively, become where the contributions coming from Σ vanish due to the trace of odd number of -matrices. Following the well known results in Ref. [69], one can write where the Legendre functions of the second kind read as and the thermal mass [69,72] of the quark is given as The thermal part of the self-energy in (A4) becomes Again using (A4) in (21c) and (21d), the structure functions ′ and ′ , respectively, become where the contributions coming from Σ 0 vanish due to the trace of odd number of -matrices. For computing the above thermomagnetic structure functions, one needs to use the following two traces: With this one can obtain where the boson propagator in Saclay representation is given by where the sum is over 0 = 2 and 2 = 2 + 2 . Also the fermion propagator in Saclay representation reads where the sum above is over 0 = (2 + 1) . Now following HTL approximation in presence of magnetic field [66,68] the (A14) and (A15) are simplified as Using the results of the HTL angular integrations [67] ∫ Ω 4 with the magnetic mass is obtained as 2 ( , , ) = 16 2 ln(2) − 2 . (A20) We note here that for → 0, the magnetic mass diverges but it can be regulated by the the thermal mass ℎ in (A8) as is done in Refs. [66,67]. Then the domain of applicability becomes 2 ℎ (∼ 2 2 ) < < 2 instead of 2 < < 2 . The thermo-magnetic part of the self-energy in (A4) becomes Now combining (A9), (A21) and (A4), the general structure of quark self-energy in hot magnetised QCD becomes which agrees quite well with the general structure as discussed in (16) and also with results directly calculated in Refs. [66][67][68].

B. Solution of the Modified Dirac equation at Lowest Landau Level (LLL)
At LLL, → 0 ⇒ ⟂ = 0 and the effective Dirac equation becomes where = with ( ) are 2 × 1 blocks. Now, the condition for the non-trivial solution to exist is given as (1) which leads to the following conditions: • Case-II: For 0 = , one gets whereas for 0 = − , one finds (B8)

C. Verification of the Three Point Function from Direct Calculation
In this appendix we would verify the general structure of the temporal 3-point function as obtained in sec. IV using the general structure of the self-energy.
We begin with the one-loop level 3-point function in a hot magnetised medium in [67] within HTL approximation [48,75] as where the external four-momentum = − . The HTL correction part [49,74,75] is given as where = 0 − 0 . We note that this expression matches exactly with the expression obtained in (72) from the general structure of fermion self-energy.