††thanks: Official email<EMAIL_ADDRESS>
# Quark propagator and di-lepton production rate in a hot, dense and very
strongly magnetized rotating Quark-Gluon Plasma

Aritra Das<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni-752050, India

###### Abstract

In this paper, a theoretical calculation of thermal di-lepton production rate
is reported from a hot and dense, unbounded rotating quark gluon plasma in the
presence of very strong uniform background magnetic field typically generated
at heavy-ion collision experiments. In this extreme magnetic field, the quarks
as well as anti-quarks are approximated to be confined in the lowest Landau
level (LLL). Firstly, I have converted the expression of the quark propagator
in LLL approximation to the momentum space representation. After that, using
the derived quark propagator, the di-lepton production rate (DPR) is
calculated from the photon polarization tensor (PPT) with the help of the
framework of thermal field theory. The system is first confined in a cylinder
of radius $R_{\scriptscriptstyle{\perp}}$ and then
$R_{\scriptscriptstyle{\perp}}$ is taken to be very large in order to consider
the situation of unbounded system. Depending on the charge of participating
quarks, the role of rotation is to alter the chemical potential. The DPR is
suppressed prominently for low invariant mass with respect to the LLL-
approximated non-rotating case that was reported earlier.

QGP, rotation, electromagnetic probes, dileptons, photon polarization tensor

##  I Introduction

In the peripheral relativistic heavy ion collisions, a tremendous amount of
magnetic field is generated in the direction perpendicular to the reaction
plane due to the relative motions of the participating ions. Theoretical
calculations as well as numerical simulations indicate that the generated
magnetic field, whose strength reaches upto $B\sim 10^{18}$G at RHIC and
$B\sim 10^{19}$G at LHC [3, 1, 2], is one of the highest among all produced in
the Universe. Recent studies on the effect of magnetic field on color-
deconfined quark-gluon plasma indicates the emergence of several novel
phenomena like chiral magnetic effect [4, 5], chiral vortical effect [6],
chiral magnetic wave [7], magnetic [8, 9, 10] and inverse-magnetic catalysis
[11, 12, 13], superconductivity of QCD vacuum [14, 15] and many more.

Apart from the strong magnetic field, a high vorticity is also present in the
created medium. It happens due to the conversion of large angular momentum,
carried by the initial colliding nuclei, to vorticity as a result of the
conservation of angular momentum. The generated vorticity, calculated to be
$\sim 10^{22}\;\text{s}^{-1}$ [16], induces a spin polarization [17, 18, 20],
directed perpendicular to the reaction plane, when averaged over the whole
fluid. Recent experimental observation of $\Lambda$ and
$\overline{\Lambda}$-hyperon spin polarization [19] by STAR collaboration has
given a new direction to this already matured field of research. This
experimental observation along with some novel theoretical findings, like
rotational suppression of chiral condensate [22], further boosted interest in
the heavy-ion community to re-investigate various phenomena. For example, in
Ref. [21, 23, 24, 25] chiral phase transition is investigated in the ambit of
effective model under rotation. Next, the investigation of QCD phase
transitions under rotation gained the attention of the lattice QCD community
after the formulation of lattice Quantum Chromodynamics (LQCD) in a rotating
frame reported in [26]. Thereafter, using LQCD simulations, the confinement
deconfinement transitions [27, 28, 29], equation of state [30] in rotating QCD
medium was studied. Moreover, shear [31] and bulk [32] viscosities were
studied in a while ago.

The combined effect of magnetic field with rotation showed “rotational
magnetic inhibition” [33] where by increasing $\Omega$, the amount of
suppression of the dynamical mass with increasing the background B-field was
enhanced. This conclusion was confirmed for non-homogeneous condensate by
taking into account 1-flavor [34] NJL model and 2-flavor [35] NJL model.

Nowadays, there is growing interest in probing the properties of QGP, after
witnessing the emergence of novel phenomena in the these extreme environments.
Electromagnetic probes, like di-leptons, real photons, are considered one of
the cleanest among all the probes of QGP. Once produced, they interact very
feebly $(\alpha_{s}\gg\alpha_{\textsf{EM}})$ with the strongly interacting
plasma due to their electromagnetic nature. The investigation of thermal
dilepton production rate (DPR) in the presence of background magnetic field
was reported in Ref. [36] using Ritus eigenfunction method. Also, several
calculations were carried out by restricting the strength of the field, e.g.
in strong field [37] (the case of lowest Landau level), weak field [38]. In
Ref. [39], authors calculated DPR by considering arbitrary B-field but
restricted the component of di-lepton momentum along the direction of B-field,
i.e. for $p_{\scriptscriptstyle{\perp}}=0$ but $p_{z}\neq 0$, only. This
restriction was later lifted in Ref. [40, 41] by including
$p_{\scriptscriptstyle{\perp}}\neq 0$ in the calculation.111Authors of Ref.
[42], also calculated DPR in magnetised medium taking into account all LL and
argued that the energy momentum distribution of di-leptons can indicate to
vacuum dichroism and vacuum birefringes. Also the production rate of real
photons [43, 44] along with (photon + dilepton) emission anisotropy [45] was
also studied in recent times. In a non-magnetized medium and in the presence
of a background rotation, prompt photon spectrum [47] and thermal dilepton
rate [46] was reported of late. As for the production rate of electromagnetic
probes in hot, dense, magnetorotating QGP medium is concerned, there is a
beautiful work [48] that investigated the prompt photon spectrum.

In this work, I have calculated the DPR from a rotato-magnetic QGP medium in
LLL approximation. In Sec. II, the technique of incorporating the effect of
rotation in the background field is introduced. In Sec. III, I derive the
expression of quark propagator in a strong magnetic field under rotation.
Next, in Sec. IV, the imaginary part of Lorentz-traced photon polarization
tensor is computed under LLL approximation. The exact analytic expression of
DPR is obtained in Sec. V. In Sec. VI, I have tried to interpret the obtained
expression of the rate along with some approximations and limitations
involved. Next, I have studied the behavior of the rate by varying various
parameters of the expression in Sec. VII. In Sec. VIII, I conclude and put
forward some suggestions to further improve the work.

##  II Setup

In this section, I discuss about Dirac equation for a quark in the presence of
an uniform space-time independent background magnetic field with magnitude $B$
in a reference frame rotating with an angular velocity $\Omega$. Keeping in
mind the situation created in HIC experiments, both the magnetic field $B$ and
the angular velocity $\Omega$ are taken along the same direction. Along that
direction, I orient the $z$-axis of the coordinate system. Also, the direction
of propagation of the two participating ions is taken along the $x$-axis. In
this case, the metric appropriate to the rotating frame is written as

$\displaystyle
g_{\mu\nu}=\begin{pmatrix}1-(x^{2}+y^{2})\Omega^{2}&y\Omega&-x\Omega&0\\\
y\Omega&-1&0&0\\\ -x\Omega&0&-1&0\\\ 0&0&0&-1\end{pmatrix}.$ (1)

The Dirac equation, describing a quark of flavor $f$ in a rotating frame, is
written as

$\displaystyle\left[i\gamma^{\mu}\left(\partial_{\mu}+\Gamma_{\mu}\right)-m_{f}\right]\psi=0,$
(2)

where $m_{f}$ is the mass of the quark and $\Gamma^{\mu}$ is the affine
connection which is defined by

$\displaystyle\Gamma^{\mu}=-\frac{i}{4}\omega_{\mu\hat{i}\hat{j}}\sigma^{\hat{i}\hat{j}},$
(3)

where Latin indices with hat, i.e.,
$\hat{i},\hat{j},\ldots=\hat{t},\hat{x},\hat{y},\hat{z}$, denote the
directions of the Cartesian coordinate system in the laboratory frame and
Greek indices $\mu,\nu,\ldots=t,x,y,z$ denote the general coordinates in the
rotating frame. The quantities $\omega_{\mu\hat{i}\hat{j}}$,
$\sigma^{\hat{i}\hat{j}}$ in Eq. (3), are defined by

$\displaystyle\omega_{\mu\hat{i}\hat{j}}\equiv
g_{\lambda\rho}e^{\lambda}_{\hat{i}}\left(\partial_{\mu}e^{\rho}_{\hat{j}}+\Gamma^{\rho}_{\mu\nu}e^{\nu}_{\hat{j}}\right),\qquad\sigma^{\hat{i}\hat{j}}=\frac{i}{2}\left[\gamma^{\hat{i}},\gamma^{\hat{j}}\right],$
(4)

where

$\displaystyle\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu}\right)$
(5)

is the Christoffel connection and $e^{\mu}_{\hat{i}}$ is the vierbein that
connects the general coordinates with the Cartesian coordinate is written as
$x^{\mu}=e^{\mu}_{\hat{i}}x^{\hat{i}}$. In this case, the components of
vierbein are given by

$\displaystyle
e^{t}_{\hat{t}}=e^{x}_{\hat{x}}=e^{y}_{\hat{y}}=e^{z}_{\hat{z}}=1,\quad
e^{x}_{\hat{t}}=y\Omega,\quad e^{y}_{\hat{t}}=-x\Omega,$ (6)

with other components being zero. For the Christoffel connection, the non-zero
components are written as

$\displaystyle\Gamma^{x}_{ty}=\Gamma^{x}_{yt}=-\Omega,\quad\Gamma^{y}_{tx}=\Gamma^{y}_{xt}=\Omega,\quad\Gamma^{x}_{tt}=-x\Omega^{2},\quad\Gamma^{y}_{tt}=-y\Omega^{2}.$
(7)

With this, the only component of affine connection that survives is given as

$\displaystyle\Gamma_{t}=-\frac{i}{2}\Omega\sigma^{\hat{x}\hat{y}}.$ (8)

Now the components of the gamma matrices in the rotating frame are given by
$\gamma^{\mu}=e^{\mu}_{\hat{i}}\gamma^{\hat{i}}$. In this case, one has

$\displaystyle\gamma^{t}=\gamma^{\hat{t}},\quad\gamma^{x}=y\Omega\gamma^{\hat{i}}+\gamma^{\hat{x}},\quad\gamma^{y}=-x\Omega\gamma^{\hat{t}}+\gamma^{\hat{y}},\quad\gamma^{z}=\gamma^{\hat{z}}.$
(9)

Now, in the presence of background magnetic field, the partial derivatives
$\partial_{\mu}$ are replaced by covariant derivatives $D_{\mu}$ as
$\partial_{\mu}\rightarrow D_{\mu}=\partial_{\mu}-iq_{f}\mathcal{A}_{\mu}$,
where $\mathcal{A}_{\mu}$ is the gauge field responsible for the generation of
the background magnetic field. It can be chosen in various gauges, but for our
purpose I choose it in the symmetric gauge where
$\mathcal{A}^{\mu}=\dfrac{B}{2}(0,-y,x,0)$. Also $q_{f}$ is the charge of
quark of flavor $f$. In this paper, I consider only light $u$ and $d$ quarks
whose charges are given by $q_{u}=\dfrac{2}{3}e$ and $q_{d}=-\dfrac{1}{3}e$,
respectively. Here $e$, taken as the absolute value of electron’s charge, is
positive. Thus, the general form of the Dirac equation describing rotating
fermions in the magnetized QGP is given as

$\displaystyle\left[i\gamma^{\mu}\left(D_{\mu}+\Gamma_{\mu}\right)-m_{f}\right]\psi=0.$
(10)

In the case of uniform rotation $\bm{\Omega}=\Omega\bm{\hat{z}}$, and magnetic
field $\bm{B}=B\bm{\hat{z}}$, it becomes

$\displaystyle\left[i\gamma^{\hat{t}}\left(\partial_{t}+\Omega
y\partial_{x}-\Omega
x\partial_{y}-\frac{i}{2}\Omega\sigma^{\hat{x}\hat{y}}\right)+i\gamma^{\hat{x}}\left(\partial_{x}+i\frac{q_{f}B}{2}y\right)+i\gamma^{\hat{y}}\left(\partial_{y}-i\frac{q_{f}B}{2}x\right)+i\gamma^{\hat{z}}\partial_{z}-m_{f}\right]\psi=0.$
(11)

Note that I have retained gamma matrices in the laboratory frame but
coordinates in the rotating frame for the convenience of our calculation. From
now on, I simply write $\gamma^{\hat{t}},\gamma^{\hat{x}},\cdots$ as
$\gamma^{0},\gamma^{1},\cdots$. In this way

$\displaystyle\sigma^{\hat{x}\hat{y}}=\sigma^{12}=\begin{pmatrix}\sigma_{z}&&0\\\
0&&\sigma_{z}\end{pmatrix},$ (12)

where $\sigma_{z}$ is the $z$ component of the Pauli spin matrix.

##  III The Quark Propagator

This section deals with the expression of the quark propagator in the rotato-
magnetic background that is necessary for the calculation of DPR. I shall
quote the expression of the quark propagator in a magnetised rotating medium
in subsec. III.1 of this section. Then, in subsec. III.2, I shall represent it
in a suitable form for my calculation of DPR under LLL approximation valid for
in the case of very strong B-field.

###  III.1 The Quark Propagator in a bounded magneto-rotating system

The equation that the quark propagator of flavor $f$ satisfies is given as

$\displaystyle\bigg{[}i\gamma^{0}\left(\partial_{t}+\Omega
y\partial_{x}-\Omega
x\partial_{y}-\frac{i}{2}\Omega\sigma^{12}\right)+i\gamma^{1}\left(\partial_{x}+i\frac{q_{f}B}{2}y\right)+i\gamma^{2}\left(\partial_{y}-i\frac{q_{f}B}{2}x\right)+i\gamma^{3}\partial_{z}-m_{f}\Bigg{]}S^{\scriptscriptstyle{(f)}}_{\alpha\beta}(\mathcal{X},\mathcal{X}^{\prime})$
$\displaystyle\hskip
369.88582pt=\delta^{(4)}(\mathcal{X}-\mathcal{X}^{\prime})\delta_{\alpha\beta},$
(13)

where I have collectively denoted the space-time coordinates by
$\mathcal{X}^{\mu}=(t,x,y,z)=(t,\bm{r})$ and
${\mathcal{X}^{\prime}}^{\mu}=(t^{\prime},x^{\prime},y^{\prime},z^{\prime})=(t^{\prime},\bm{r}^{\prime})$.
Before proceeding to the solution of Eq. (13), one can note that it is
convenient to confine the system in a cylinder with radius
$R_{\scriptscriptstyle{\perp}}$ and work with the cylindrical polar
coordinates where $x=r_{\scriptscriptstyle{\perp}}\cos\phi$,
$y=r_{\scriptscriptstyle{\perp}}\sin\phi$, with $t,z$ remaining unchanged.
Here $r_{\scriptscriptstyle{\perp}}=\left|\bm{r}\cdot\bm{\hat{z}}\right|$ is
the distance of point $\bm{r}$ from the axis of rotation. The solution of Eq.
(13) was investigated thoroughly in Ref. [34]. Below, I just quote the
expression

$\displaystyle
S^{\scriptscriptstyle{(f)}}_{\alpha\beta}(\mathcal{X},\mathcal{X}^{\prime})$
$\displaystyle=i\sum_{n,\ell}\int\frac{dk_{0}dk_{z}}{(2\pi)^{2}}\mathcal{C}_{n,\ell,s_{q}}^{2}\,e^{-ik_{0}(t-t^{\prime})+ik_{z}(z-z^{\prime})}\left[\mathbbm{P}^{\scriptscriptstyle{(f)}}_{\lambda_{n},\ell}(\xi,\phi)\right]_{\alpha\rho}\left(\frac{\gamma\cdot\tilde{k}_{\lambda_{n},\ell,+}+m_{\scriptscriptstyle{f}}}{(k_{0}+\Omega
j)^{2}-\epsilon^{\scriptscriptstyle{(f)}\,2}_{\lambda_{n},k_{z}}}\right)_{\rho\lambda}\left[\mathbbm{P}^{\scriptscriptstyle{(f)}}_{\lambda_{n},\ell}(\xi^{\prime},\phi^{\prime})\right]^{\dagger}_{\lambda\beta}.$
(14)

As we can see Eq. (14) involves many terms, which are explained one after
another. Here $\alpha$, $\beta$, $\lambda$, $\rho$ denotes Lorentz indices.
The term $j$ is a positive or negative half-integer written as

$\displaystyle j\equiv\ell+\frac{1}{2},$ (15)

and $\tilde{k}_{\lambda_{n},\ell,\kappa}$ is the _Ritus momentum_ which is
given by

$\displaystyle\tilde{k}_{\lambda_{n},\ell,\kappa}=\left(k_{0}+\kappa\Omega
j,0,\kappa s_{\ell}\sqrt{2\lambda_{n}|q_{f}B|},k_{z}\right),$ (16)

with $s_{\ell}=\textsf{sgn}(\ell)$, $\kappa=\pm 1$. Here $\kappa=1$ and
$\kappa=-1$ are related to the particle and antiparticle solution of the Dirac
equation in rotato-magnetic medium, respectively. Also, for any four vector
$a^{\mu}=(a_{0},a_{x},a_{y},a_{z})$, I define

$\displaystyle\gamma\cdot
a=\not{a}=\gamma^{0}a_{0}-\gamma^{1}a^{1}-\gamma^{2}a^{2}-\gamma^{3}a^{3},\quad
a^{\mu}_{\shortparallel}=(a_{0},0,0,a_{z}),\quad
a^{\mu}_{\scriptscriptstyle{\perp}}=(0,a_{x},a_{y},0)$
$\displaystyle\gamma\cdot
a_{\shortparallel}=\not{a}_{\shortparallel}=\gamma^{0}a_{0}-\gamma^{3}a^{3},\quad\gamma\cdot
a_{\scriptscriptstyle{\perp}}=\gamma^{1}a^{1}+\gamma^{2}a^{2},$ $\displaystyle
A^{2}=a_{0}^{2}-a_{x}^{2}-a_{y}^{2}-a_{z}^{2},\quad
a_{\shortparallel}^{2}=a_{0}^{2}-a_{z}^{2},\quad
a_{\scriptscriptstyle{\perp}}^{2}=a_{x}^{2}+a_{y}^{2},\quad
a^{2}=|\bm{a}|^{2}=a_{x}^{2}+a_{y}^{2}+a_{z}^{2}.$ (17)

The factor
$\mathbbm{P}^{\scriptscriptstyle{(f)}}_{\lambda_{n},\ell}(\xi,\phi)$ is given
as

$\displaystyle\mathbbm{P}^{\scriptscriptstyle{(f)}}_{\lambda_{n},\ell}(\xi)=\mathit{P}_{+}f^{+}_{\lambda_{n},\ell,s_{q}}(\xi,\phi)+\mathit{P}_{-}f^{-}_{\lambda_{n},\ell,s_{q}}(\xi,\phi),\qquad\text{with}\quad\xi=\frac{|q_{f}B|r_{\scriptscriptstyle{\perp}}^{2}}{2},$
(18)

where $\mathit{P}_{\pm}=\dfrac{1}{2}\left(\openone\pm
i\gamma^{1}\gamma^{2}\right)$ and

$\displaystyle
f^{+}_{\lambda_{n},\ell,s_{q}}(\xi,\phi)=\mathcal{C}^{+}_{n,\ell,s_{q}}e^{i\ell\phi}\,\,\Phi^{+}_{\lambda_{n},\ell,s_{q}}(\xi),$
$\displaystyle
f^{-}_{\lambda_{n},\ell,s_{q}}(\xi,\phi)=\mathcal{C}^{-}_{n,\ell,s_{q}}e^{i(\ell+1)\phi}\,\,\Phi^{-}_{\lambda_{n},\ell,s_{q}}(\xi).$
(19)

The subscripts and superscripts appearing in $f$ bear different meanings which
are explained one after another:  (i) The indices $+$ and $-$ in the
superscript of the function $f$ are related to the spin up and spin down of a
quark, respectively.  (ii) $s_{q}=\textsf{sgn}(q_{f}B)$, with sgn denoting the
sign function. It takes the value of $\pm 1$. In this paper, I have taken the
background magnetic field to point along the $z$-direction such that
$s_{q}=+1$ for positively charged particles and $s_{q}=-1$ for negatively
charged particles.  (iii) $\lambda_{n}$, with ($n=0,1,2,\cdots$), is a
parameter analogous to Landau levels in the case of non-rotating magnetic
medium. The method to determine it’s value, for system with finite
$R_{\scriptscriptstyle{\perp}}$, is discussed later in this subsection. In the
limit of $R_{\scriptscriptstyle{\perp}}\rightarrow\infty$, it takes the value
of positive integer including zero.  (iv) $\ell$is the orbital angular
momentum quantum number whose range depends on $n$ as shown in table 1.  The
pre-factors $\mathcal{C}^{\pm}_{n,\ell,s_{q}}$ are defined by

$\displaystyle\mathcal{C}^{\pm}_{n,\ell,s_{q}}=\left(\frac{|q_{f}B|}{2\pi\int\limits^{\alpha_{b}}_{0}d\xi\left[\Phi^{\pm}_{\lambda_{n},\ell,s_{q}}(\xi)\right]^{2}}\right)^{1/2},$
(20)

where $\alpha_{b}\equiv\dfrac{|q_{f}B|}{2}R_{\scriptscriptstyle{\perp}}^{2}$.
The functions $\Phi^{\pm}_{\lambda_{n},\ell,s_{q}}$ appearing in Eq. (19) and
in Eq. (20) are defined by

$\displaystyle\Phi^{+}_{\lambda_{n},\ell,s_{q}}(\xi)=\frac{1}{|\ell|!}\left(\frac{|q_{f}B|}{2\pi}\frac{\left(\mathcal{N}^{+}_{\lambda_{n},s_{q}}+|\ell|\right)!}{\mathcal{N}^{+}_{\lambda_{n},s_{q}}!}\right)^{1/2}e^{-\xi/2}\xi^{\frac{|\ell|}{2}}\,\,{{}_{1}F_{1}\left(-\mathcal{N}^{+}_{\lambda_{n},s_{q}};|\ell|+1;\xi\right)},$
$\displaystyle\Phi^{-}_{\lambda_{n},\ell,s_{q}}(\xi)=\frac{1}{|\ell+1|!}\left(\frac{|q_{f}B|}{2\pi}\frac{\left(\mathcal{N}^{-}_{\lambda_{n},s_{q}}+|\ell+1|\right)!}{\mathcal{N}^{-}_{\lambda_{n},s_{q}}!}\right)^{1/2}e^{-\xi/2}\xi^{\frac{|\ell+1|}{2}}\,\,{{}_{1}F_{1}\left(-\mathcal{N}^{-}_{\lambda_{n},s_{q}};|\ell+1|+1;\xi\right)}.$
(21)

The function ${}_{1}F_{1}$ in Eq. (21) is the confluent hypergeometric
function. The terms $\mathcal{N}^{\pm}_{\lambda_{n},s_{q}}$ are defined as

$\displaystyle\mathcal{N}^{+}_{\lambda_{n},s_{q}}$
$\displaystyle=\lambda_{n}+\frac{s_{q}(\ell+1)-|\ell|-1}{2},$
$\displaystyle\mathcal{N}^{-}_{\lambda_{n},s_{q}}$
$\displaystyle=\lambda_{n}+\frac{s_{q}\ell-|\ell+1|-1}{2}.$ (22)

As mentioned earlier, the $\lambda_{n}$ (for $n=0,1,2,\cdots$) are Landau
level like numbers ($\lambda_{n}\in\mathbbm{R}$) when
$R_{\scriptscriptstyle{\perp}}$ is finite. It’s values depend on $\ell$ and it
is computed from the root of the following equations

${}_{1}F_{1}\left(-\lambda-\frac{s_{q}(\ell+1)-|\ell|-1}{2},|\ell|+1,\alpha_{b}\right)=0,\text{
for }\ell\geq 0,$
${}_{1}F_{1}\left(-\lambda-\frac{s_{q}\ell-|\ell+1|-1}{2},|\ell+1|+1,\alpha_{b}\right)=0,\text{
for }\ell\leq-1.$ (23)

For $\lambda_{n}$ that satisfy Eq. (23), one can show numerically that
$\mathcal{C}^{+}_{n,\ell,s_{q}}=\mathcal{C}^{-}_{n,\ell,s_{q}}\equiv\mathcal{C}_{n,\ell,s_{q}}$.

###  III.2 Quark Propagator under the Lowest Landau Level Approximation

In this section, I concentrate on the lowest Landau level (LLL) term of the
quark propagator written in Eq. (13) for the case of no boundary
($R_{\scriptscriptstyle{\perp}}\rightarrow\infty$). Now, one can immediately
point out some of the restrictions imposed on parameters $\lambda_{n}$,
$\ell$.

  1. (a)

First, for the fermion with no boundary
$R_{\scriptscriptstyle{\perp}}\rightarrow\infty$, $\lambda_{n}$ must be
positive integer including $0$. Otherwise,
$f^{\sigma}_{n,\ell,s_{q}}(\xi\rightarrow\infty,\phi)$ can not be kept finite
[33].

$\displaystyle\lambda_{n}\xrightarrow[]{R_{\scriptscriptstyle{\perp}}\rightarrow\infty}n,\quad\text{where}\quad
n\in(0,\mathbbm{Z}^{+}).$ (24)

  2. (b)

The confluent Hypergeometric function ${}_{1}F_{1}(a,b;z)$ yields finite
number of terms only when

$\displaystyle(a<0)\quad\textsf{and}\quad(b<a\quad\textsf{or}\quad b>0).$ (25)

Using conditions given in (a) and (b), the allowed values of $\ell$ are
determined for each values of $n$ and are as shown in Tab 1.

HLL ($n\geq 1$)
---
$s_{q}=1$ | $\ell=-n,-n+1,\cdots,0,1,\cdots$
$s_{q}=-1$ | $\ell=n-1,n-2,\cdots,0,-1,\cdots$

LLL ($n=0$)
---
| $\sigma=1$ | $\sigma=-1$
$s_{q}=1$ | $\ell=0,1,\cdots$ | States not allowed
$s_{q}=-1$ | States not allowed | $\ell=-1,-2,\cdots$

Table 1: Table depicting the dependencies of $\ell$ on spin, $s_{q}$ for
states in LLL and higher Landau levels (HLL). In HLL both spins can propagate
in contrast to LLL.

Before proceeding to the discussion of LLL scenario, let us note that after
determining the constant $\mathcal{C}^{\pm}_{n,\ell,s_{q}}$ defined in Eq.
(20), the function $f^{\pm}_{n,\ell,s_{q}}$ can be written down for the
unbounded case ($R_{\scriptscriptstyle{\perp}}\rightarrow\infty$) in a compact
manner. For this purpose, I introduce

$\displaystyle\ell_{+}\equiv\ell\quad\text{and}\quad\ell_{-}\equiv\ell+1,$
(26)

As a result, the functional form of $f^{\pm}_{n,\ell,s_{q}}$ can be written as

$\displaystyle
f^{\sigma}_{n,\ell,s_{q}}(\xi,\phi)=\left(\frac{|q_{f}B|}{2\pi}\frac{\mathcal{N}^{\sigma}_{n,s_{q}}!}{\left(\mathcal{N}^{\sigma}_{n,s_{q}}+\left|\ell_{\sigma}\right|\right)!}\right)^{1/2}e^{i\ell_{\sigma}\phi}\;_{1}F_{1}\left(-\mathcal{N}^{\sigma}_{n,s_{q}},|\ell_{\sigma}|+1,\xi\right),\quad\text{with}\quad\sigma=\pm
1,$ (27)

where

$\displaystyle\mathcal{N}^{\sigma}_{n,s_{q}}=n+\frac{s_{q}\ell_{-\sigma}-|\ell_{\sigma}|-1}{2},\qquad\ell_{\sigma}=\ell+\delta_{\sigma,-1}.$
(28)

The $\ell_{\pm}$ are related to $j$ via the relation
$j=\ell_{+}+\dfrac{1}{2}=\ell_{-}-\dfrac{1}{2}$.

I again emphasize that (i) $\sigma=\pm 1$ is the spin of fermion, (ii) $n$ is
the landau level index, (iii) $\ell$ is the orbital angular momentum and (iv)
$s_{q}=\textsf{sgn}(q_{f}B)$. For LLL, the allowed values $\ell$ are given by

$\displaystyle\ell=\begin{cases}0,1,2,\cdots,&\quad\text{when}\quad
s_{q}=+1,\\\ -1,-2,-3,\cdots,&\quad\text{when}\quad
s_{q}=-1,\end{cases}\quad\quad\text{for}\quad n=0.$ (29)

Moreover, using the fact that only $f^{+}_{0,\ell,+1}$ and $f^{-}_{0,\ell,-1}$
survives among all other terms and
$[\mathit{P}_{\pm},\gamma_{\shortparallel}]=0$, one can write the quark
propagator in the LLL, from Eq. (13) and Eq. (27), as

$\displaystyle
S^{\textsf{LLL}}_{f,\Omega,s_{q}}(t-t^{\prime},r_{\scriptscriptstyle{\perp}},\phi-\phi^{\prime},z-z^{\prime}\,;\,0,r^{\prime}_{\scriptscriptstyle{\perp}},0,0)=i\frac{|q_{f}B|}{2\pi}e^{-\frac{\xi+\xi^{\prime}}{2}}\int\\!\frac{dk_{0}dk_{z}}{(2\pi)^{2}}e^{-i\left[k_{0}(t-t^{\prime})-k_{z}(z-z^{\prime})\right]}\mathcal{I}^{(s_{q})}(k_{0},k_{z};\xi,\xi^{\prime}),$
(30)

where

$\displaystyle\mathcal{I}^{(s_{q})}(k_{0},k_{z};\xi,\xi^{\prime})=\displaystyle{\begin{cases}\sum\limits_{\ell=0}^{\infty}\frac{\left(\xi\xi^{\prime}\right)^{\frac{\ell}{2}}}{\ell!}e^{i\ell(\phi-\phi^{\prime})}\frac{(k_{0}+j\Omega)\gamma^{0}-k_{z}\gamma^{3}+m_{f}}{\left(k_{0}+j\Omega\right)^{2}-k_{z}^{2}-m_{f}^{2}}\mathit{P}_{+},&\text{for}\quad
s_{q}=1,\\\
\sum\limits_{\ell=-1}^{-\infty}\frac{\left(\xi\xi^{\prime}\right)^{-\frac{\ell+1}{2}}}{(-\ell-1)!}e^{i(\ell+1)(\phi-\phi^{\prime})}\frac{(k_{0}+j\Omega)\gamma^{0}-k_{z}\gamma^{3}+m_{f}}{\left(k_{0}+j\Omega\right)^{2}-k_{z}^{2}-m_{f}^{2}}\mathit{P}_{-},&\text{for}\quad
s_{q}=-1,\end{cases}}$ (31)

where is $j$ defined by Eq. (15). Next changing the summation index
$\ell\rightarrow-\ell-1$ for $s_{q}=-1$ case in Eq. (31), I can write the
quark propagator displayed in Eq. (30) in a compact form as

$\displaystyle
S^{\textsf{LLL}}_{f,\Omega,s_{q}}(t-t^{\prime},r_{\scriptscriptstyle{\perp}},\phi-\phi^{\prime},z-z^{\prime}\,;\,0,r^{\prime}_{\scriptscriptstyle{\perp}},0,0)=i\frac{|q_{f}B|}{2\pi}e^{-\frac{\xi+\xi^{\prime}}{2}}$
$\displaystyle\times\int\\!\frac{dk_{0}dk_{z}}{(2\pi)^{2}}e^{-i\left[k_{0}(t-t^{\prime})-k_{z}(z-z^{\prime})\right]}\sum_{\ell=0}^{\infty}\frac{\left(\xi\xi^{\prime}\right)^{\frac{\ell}{2}}}{\ell!}e^{is_{q}\ell(\phi-\phi^{\prime})}\frac{\left[k_{0}+s_{q}\Omega\left(\ell+\frac{1}{2}\right)\right]\gamma^{0}-k_{z}\gamma^{3}+m_{f}}{\left[k_{0}+s_{q}\Omega\left(\ell+\frac{1}{2}\right)\right]^{2}-k_{z}^{2}-m_{f}^{2}}\mathit{P}_{s_{q}},$
(32)

where

$\displaystyle\mathit{P_{s_{q}}}=\frac{1}{2}\left(\openone\pm
s_{q}\,i\gamma^{1}\gamma^{2}\right).$ (33)

Note that the purpose of the variable change $\ell\rightarrow-\ell-1$ for the
summation in case of $s_{q}=-1$ is to write the sum over $\ell$ to run from
$0$ to $\infty$ for both $s_{q}=\pm 1$. Now, I make a change of variable
$\tilde{k}_{0}=k_{0}+s_{q}\left(\ell+\frac{1}{2}\right)\Omega$ after
exploiting the translational invariance by defining $\bar{t}=t-t^{\prime}$,
$\bar{z}=z-z^{\prime}$ and $\bar{\phi}=\phi-\phi^{\prime}$ coordinate to get

$\displaystyle
S^{\textsf{LLL}}_{f,\Omega,s_{q}}(\bar{t},r_{\scriptscriptstyle{\perp}},\bar{\phi},\bar{z};r^{\prime}_{\scriptscriptstyle{\perp}})=$
$\displaystyle
i\frac{|q_{f}B|}{2\pi}\sum_{\ell=0}^{\infty}e^{-\frac{\xi+\xi^{\prime}}{2}}\frac{\left(\xi\xi^{\prime}\right)^{\frac{\ell}{2}}}{\ell!}e^{is_{q}\ell(\bar{\phi}+\Omega\bar{t})}e^{is_{q}\frac{\Omega}{2}\bar{t}}\int\\!\frac{d\tilde{k}_{0}dk_{z}}{(2\pi)^{2}}\frac{\tilde{k}_{0}\gamma^{0}-k_{z}\gamma^{3}+m_{f}}{\tilde{k}^{2}_{0}-k_{z}^{2}-m_{f}^{2}}e^{-i(\tilde{k}_{0}\bar{t}-k_{z}\bar{z})}\mathit{P}_{s_{q}}.$
(34)

Now, the sum over $\ell$ can be performed by using the identity

$\displaystyle\sum_{\ell=0}^{\infty}(\xi\xi^{\prime})^{\ell/2}e^{is_{q}(\phi+\Omega
t)\ell}\frac{1}{\ell!}=\exp\left(\sqrt{\xi\xi^{\prime}}e^{is_{q}(\phi+\Omega
t)}\right),$ (35)

which leads to the simplified exact expression of the propagator in LLL as

$\displaystyle
S^{\textsf{LLL}}_{f,\Omega,s_{q}}\left(\bar{t},r_{\scriptscriptstyle{\perp}},\bar{\phi},\bar{z};r^{\prime}_{\scriptscriptstyle{\perp}}\right)=$
$\displaystyle
i\frac{|q_{f}B|}{2\pi}e^{-\frac{\xi+\xi^{\prime}}{2}}\exp\left(\sqrt{\xi\xi^{\prime}}e^{is_{q}(\bar{\phi}+\Omega\bar{t})}\right)e^{is_{q}\frac{\Omega}{2}\bar{t}}\int\\!\frac{d\tilde{k}_{0}dk_{z}}{(2\pi)^{2}}\frac{\tilde{k}_{0}\gamma^{0}-k_{z}\gamma^{3}+m_{f}}{\tilde{k}^{2}_{0}-k_{z}^{2}-m_{f}^{2}}e^{-i(\tilde{k}_{0}\bar{t}-k_{z}\bar{z})}\mathit{P}_{s_{q}}.$
(36)

At this point, I make the approximation $\bar{\phi}+\Omega\bar{t}=0$. This is
a good approximation in the early stage of heavy ion collision where the
fermion is totally dragged by the vortical motion [53]. The reason being the
absence of radial flow in such an early time.

Employing this approximation and using

$\displaystyle\exp\left(-\frac{\xi+\xi^{\prime}}{2}\right)e^{\sqrt{\xi\xi^{\prime}}}=\exp\left[-\frac{\left(\sqrt{\xi}-\sqrt{\xi^{\prime}}\right)^{2}}{2}\right]$
(37)

and finally making a change of variable
$\tilde{k}_{0}=k_{0}+s_{q}\frac{\Omega}{2}$, the propagator is obtained as

$\displaystyle
S^{\textsf{LLL}}_{f,\Omega,s_{q}}\left(\bar{t},\bar{r}_{\scriptscriptstyle{\perp}},\bar{\phi},\bar{z}\right)=i\frac{|q_{f}B|}{2\pi}\exp\left(-\frac{|q_{f}B|}{4}\bar{r}_{\scriptscriptstyle{\perp}}^{2}\right)\int\\!\frac{d^{2}k_{\shortparallel}}{(2\pi)^{2}}\frac{\left(k_{0}+s_{q}\frac{\Omega}{2}\right)\gamma^{0}-k_{z}\gamma^{3}+m_{f}}{\left(k_{0}+s_{q}\frac{\Omega}{2}\right)^{2}-k_{z}^{2}-m_{f}^{2}}e^{-i(k_{0}\bar{t}-k_{z}\bar{z})}\mathit{P}_{s_{q}},$
(38)

where
$\bar{r}_{\scriptscriptstyle{\perp}}=r_{\scriptscriptstyle{\perp}}-r^{\prime}_{\scriptscriptstyle{\perp}}$.
Now the Fourier transformation can be performed as the propagator depends on
the relative coordinates

$\displaystyle
S_{f,\Omega,s_{q}}^{\textsf{LLL}}(P)=\int\\!d^{4}\bar{\mathcal{X}}\;e^{ip\cdot\bar{\mathcal{X}}}\;S^{\textsf{LLL}}_{f,\Omega,s_{q}}(\bar{t},\bar{r}_{\scriptscriptstyle{\perp}},\bar{\phi},\bar{z}),$
(39)

where $\bar{\mathcal{X}}=\mathcal{X}-\mathcal{X}^{\prime}$. Writing out the
position measure in cylindrical polar coordinates with
$\bar{x}=\bar{r}_{\scriptscriptstyle{\perp}}\cos\bar{\phi}$,
$\bar{y}=\bar{r}_{\scriptscriptstyle{\perp}}\sin\bar{\phi}$ and perpendicular
momentum components $p_{x}=p_{\scriptscriptstyle{\perp}}\cos\alpha$,
$p_{y}=p_{\perp}\sin\alpha$, one gets

$\displaystyle
S_{f,\Omega,s_{q}}^{\textsf{LLL}}(P)=i\exp\left(-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{|q_{f}B|}\right)\frac{\left(k_{0}+s_{q}\frac{\Omega}{2}\right)\gamma^{0}-k_{z}\gamma^{3}+m_{f}}{\left(k_{0}+s_{q}\frac{\Omega}{2}\right)^{2}-k_{z}^{2}-m_{f}^{2}}(\openone+is_{q}\gamma^{1}\gamma^{2}).$
(40)

To arrive at the last expression, I have written
$e^{-i\bm{p}_{\perp}.\bar{\bm{r}}_{\scriptscriptstyle{\perp}}}=e^{-ip_{\perp}\bar{r}_{\scriptscriptstyle{\perp}}\cos(\bar{\phi}-\alpha)}$
and made use of integration

$\displaystyle\int\limits_{0}^{2\pi}\\!d\bar{\phi}\;\exp\left[ip_{\perp}\bar{r}_{\scriptscriptstyle{\perp}}\cos(\bar{\phi}-\alpha)\right]=2\pi
J_{0}(p_{\perp}\bar{r}_{\scriptscriptstyle{\perp}}),$ (41)
$\displaystyle\int\limits_{0}^{\infty}\\!d\bar{r}_{\scriptscriptstyle{\perp}}\;\bar{r}_{\scriptscriptstyle{\perp}}\exp\left(-\frac{|q_{f}B|\bar{r}_{\scriptscriptstyle{\perp}}^{2}}{4}\right)J_{0}\left(p_{\perp}\bar{r}_{\scriptscriptstyle{\perp}}\right)=\frac{2}{|q_{f}B|}\exp\left(-\frac{p_{\perp}^{2}}{|q_{f}B|}\right).$
(42)

Now, noting the appearance of $\Omega/2$ term, one can conclude that the
effect of rotation is the same as that of altering the density of the medium
by introducing an effective chemical potential $\mu_{\text{eff}}=\pm\Omega/2$
when $s_{q}=\pm 1$.

##  IV Photon Polarization tensor

Figure 1: Feynman Diagram to compute the one loop photon polarization tensor.
The red dashed line indicates the only possible cut allowed to get the quark
antiquark annihilation process. The gray background indicates the fact that
the quark propagators are affected by the background magnetic field and
rotation.

In this section, I compute the photon polarization tensor (PPT) in the
presence of magnetized and rotating QGP. The expression of PPT, reading from
the Feynman diagram shown in Fig. 1, is written as

$\displaystyle-i\Pi^{\mu\nu}_{f,B,\Omega,s_{q}}(p_{0},p_{z},p_{\scriptscriptstyle{\perp}})=N_{c}q_{f}^{2}\,iT\\!\sum_{k_{0}}\int\frac{d^{3}k}{(2\pi)^{3}}\textsf{Tr}\left[\gamma^{\mu}S^{\textsf{LLL}}_{f,\Omega,s_{q}}(K)\gamma^{\nu}S^{\textsf{LLL}}_{f,\Omega,s_{q}}(K-P)\right]$
$\displaystyle=-N_{c}q_{f}^{2}\int\frac{d^{2}\bm{k}_{\bm{\scriptscriptstyle{\perp}}}}{(2\pi)^{2}}\exp\left(-\frac{\bm{k}_{\bm{\scriptscriptstyle{\perp}}}^{2}+(\bm{k}-\bm{p})_{\mathbbm{\scriptscriptstyle{\perp}}}^{2}}{|q_{f}B|}\right)\,iT\\!\sum_{k_{0}}\int\limits_{-\infty}^{\infty}\frac{dk_{z}}{2\pi}\textsf{Tr}\left[\gamma^{\mu}\frac{1}{\not{\widetilde{k}}_{\shortparallel}+m_{f}}2\mathit{P}_{s_{q}}\gamma^{\nu}\frac{1}{\not{\widetilde{k}}_{\shortparallel}-\widetilde{p}_{\shortparallel}+m_{f}}2\mathit{P}_{s_{q}}\right],$
(43)

where I have defined

$\displaystyle\widetilde{k}_{\shortparallel}^{\mu}=\left(k_{0}+s_{q}\frac{\Omega}{2},0,0,k_{z}\right).$
(44)

The last expression should not be confused with the parallel component of
Ritus four-momentum as the latter involves various labels on its subscript.
Note that, a replacement

$\displaystyle\int\limits_{-\infty}^{\infty}\frac{dk_{0}}{2\pi}\Rightarrow
iT\sum_{k_{0}},\quad\text{where}\quad k_{0}=\mu+i(2m+1)\pi T,$ (45)

for fermions in the loop, according to the prescription of imaginary time
formalism (ITF) of thermal field theory, is made [55]. The sum is over
discrete Matsubara modes characterized by an integer $m(=0,\pm 1,\pm
2,\ldots)$. Here $T=\beta^{-1}$ is the temperature $\mu$ is the chemical
potential of the plasma. The three-momentum integral measure is split into
longitudinal part $d^{2}\bm{k}_{\scriptscriptstyle{\perp}}$ and the transverse
part $dk_{z}$. The Dirac trace is performed numerous times in the literature
[37, 49, 50] before for non-rotating case. In this calculation, the only
difference from the previous one is the replacement $k_{0}\rightarrow
k_{0}+s_{q}\Omega/2$. The trace is calculated as

$\displaystyle\mathcal{T}_{\shortparallel}^{\mu\nu}=\textsf{Tr}\left[\gamma^{\mu}\frac{1}{\not{\widetilde{k}}_{\shortparallel}+m_{f}}2\mathit{P}_{s_{q}}\gamma^{\nu}\frac{1}{\not{\widetilde{k}}_{\shortparallel}-\widetilde{p}_{\shortparallel}+m_{f}}2\mathit{P}_{s_{q}}\right]$
$\displaystyle=4\left[\widetilde{k}^{\mu}_{\shortparallel}\left(\widetilde{k}^{\nu}_{\shortparallel}-p^{\nu}_{\shortparallel}\right)+\widetilde{k}^{\nu}_{\shortparallel}\left(\widetilde{k}^{\mu}_{\shortparallel}-p^{\mu}_{\shortparallel}\right)-\eta_{\shortparallel}^{\mu\nu}\left\\{\left({\eta_{\shortparallel}}\right)_{\lambda\rho}\widetilde{k}^{\lambda}_{\shortparallel}\left(\widetilde{k}^{\rho}_{\shortparallel}-p_{\shortparallel}^{\rho}\right)-m_{f}^{2}\right\\}\right].$
(46)

where
$\eta_{\mu\nu}=\left(\eta_{\shortparallel}\right)_{\mu\nu}-\left(\eta_{\scriptscriptstyle{\perp}}\right)_{\mu\nu}=\textsf{diag}(1,-1,-1,-1)$
is the flat-metric tensor which is broken into parallel and perpendicular part
as $\left(\eta_{\shortparallel}\right)_{\mu\nu}=\textsf{diag}(1,0,0,-1)$ and
$\left(\eta_{\scriptscriptstyle{\perp}}\right)_{\mu\nu}=\textsf{diag}(0,1,1,0)$,
respectively.

The perpendicular integration is performed by using the following integration

$\displaystyle\int\limits_{0}^{2\pi}\\!d\phi\;\exp\left(\frac{2k_{\scriptscriptstyle{\perp}}p_{\scriptscriptstyle{\perp}}}{|q_{f}B|}\cos\phi\right)=2\pi
I_{0}\left(\frac{2k_{\scriptscriptstyle{\perp}}p_{\scriptscriptstyle{\perp}}}{|q_{f}B|}\right),$
(47)
$\displaystyle\int\limits_{0}^{\infty}\\!dk_{\scriptscriptstyle{\perp}}\;k_{\scriptscriptstyle{\perp}}I_{0}\left(\frac{2k_{\scriptscriptstyle{\perp}}p_{\scriptscriptstyle{\perp}}}{|q_{f}B|}\right)\exp\left(-\frac{k^{2}_{\scriptscriptstyle{\perp}}}{|q_{f}B|}\right)=\frac{|q_{f}B|}{4}\exp\left(\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}\right),$
(48)

and the result is given as

$\displaystyle\int\frac{d^{2}\bm{k}_{\bm{\scriptscriptstyle{\perp}}}}{(2\pi)^{2}}\exp\left(-\frac{\bm{k}_{\bm{\scriptscriptstyle{\perp}}}^{2}+(\bm{k}-\bm{p})_{\mathbbm{\scriptscriptstyle{\perp}}}^{2}}{|q_{f}B|}\right)=\frac{|q_{f}B|}{8\pi}\exp\left(-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}\right).$
(49)

The $I_{0}$ in Eq. (47) and (48) is the bessel-$I$ function of the order $0$.
It is very evident that the perpendicular momentum is disentangled from
parallel momentum in PPT.

Therefore, substituting Eq. (49) and Eq. (46) in Eq. (43), I get

$\displaystyle-i\Pi^{\mu\nu}_{f,B,\Omega,s_{q}}(p_{0},p_{z},p_{\scriptscriptstyle{\perp}})=$
$\displaystyle-
N_{c}q_{f}^{2}\frac{|q_{f}B|}{8\pi}e^{-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}}\,iT\sum_{k_{0}}\int\limits_{-\infty}^{\infty}\frac{dk_{z}}{2\pi}\frac{\mathcal{T}^{\mu\nu}_{\shortparallel}}{\left[\left(k_{0}+s_{q}\frac{\Omega}{2}\right)^{2}-k_{z}^{2}-m_{f}^{2}\right]\left[\left(k_{0}+s_{q}\frac{\Omega}{2}-p_{0}\right)^{2}-(k_{z}-p_{z})^{2}-m_{f}^{2}\right]}.$
(50)

##  V Dilepton Production Rate

Lepton pairs are generated as Landau level states in the rotato-magnetic
environment inside the fireball. However, they get converted into the plane
wave states $\ket{Q}$, $\ket{Q^{\prime}}$ as soon as they leave the medium. It
happens due to the fact that the non-trivial backgrounds under consideration
only exist in the formed QGP medium. Now, the probability of re-scattering of
lepton pairs inside the plasma is very small. So, one can make an
approximation by which all the lepton pairs, produced in the magnetized
medium, are converted into plane waves. In this scenario, the matrix amplitude
of initial quark and antiquark states
$\ket{q(n,\ell,p_{z}),\overline{q}(n^{\prime},\ell^{\prime},p^{\prime}_{z})}$
to get converted into di-leptons is given as
$\braket{Q_{1},Q_{2}}{\mathcal{M}}{q(n,\ell,p_{z}),\overline{q}(n^{\prime},\ell^{\prime},p^{\prime}_{z})}$.
In this way, the representation of final lepton-antilepton pairs by plane
waves is a very good approximation [41].
Also, the lepton antilepton pairs are assumed to be massless $M_{l}=0$ because
they are the smallest among all the scales involved. Hence, in the scenario in
which the quarks move in the strongly magnetized and rotating medium but not
the final lepton pairs, the dilepton production per unit space-time and four
momentum volume is written as

$\displaystyle\frac{dN}{d^{4}\mathcal{X}d^{4}P}=\frac{\alpha_{\textsf{EM}}}{12\pi^{4}}\frac{1}{p_{0}^{2}-\bm{p}_{\scriptscriptstyle{\perp}}^{2}-p_{z}^{2}}\frac{\eta_{\mu\nu}}{e^{p_{0}/T}-1}\sum_{f=u,d}\textsf{Im}\;\Pi^{\mu\nu}_{f,B,\Omega}(p_{0},p_{z},p_{\scriptscriptstyle{\perp}}).$
(51)

Note that, $s_{q}$ is $+1$ and $-1$ for $q_{u}$ and $q_{d}$, respectively.
Also, I have
$\mathcal{T}_{\shortparallel\;\mu}^{\mu}=\eta_{\mu\nu}\mathcal{T}_{\shortparallel}^{\mu\nu}=8m_{f}^{2}$.
The frequency sum is performed in Appendix A. Setting $a=\bar{a}=\Omega/2$,
$E_{1}=\sqrt{k_{z}^{2}+m_{f}^{2}}$, $E_{2}=\sqrt{(k_{z}-p_{z})^{2}+m_{f}^{2}}$
in Eq. (88) of Appendix A, I get

$\displaystyle\eta_{\mu\nu}\Pi^{\mu\nu}_{f,B,\Omega,s_{q}}(p_{0},p_{z},p_{\scriptscriptstyle{\perp}})=-N_{c}q_{f}^{2}m_{f}^{2}\frac{|q_{f}B|}{8\pi^{2}}e^{-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}}$
$\displaystyle\times\int\limits_{-\infty}^{\infty}dk_{z}\sum_{r_{1}=\pm
1}\sum_{r_{2}=\pm
1}\frac{r_{1}r_{2}}{\epsilon_{0,k_{z}}^{\scriptscriptstyle{(f)}}\epsilon_{0,k_{z}-p_{z}}^{\scriptscriptstyle{(f)}}}\frac{1-\widetilde{n}^{(+)}\left(r_{1}\epsilon_{0,k_{z}}^{\scriptscriptstyle{(f)}}-s_{q}\frac{\Omega}{2}\right)-\widetilde{n}^{(-)}\left(r_{2}\epsilon_{0,k_{z}-p_{z}}^{\scriptscriptstyle{(f)}}+s_{q}\frac{\Omega}{2}\right)}{p_{0}-r_{1}\epsilon_{0,k_{z}}^{\scriptscriptstyle{(f)}}-r_{2}\epsilon_{0,k_{z}-p_{z}}^{\scriptscriptstyle{(f)}}+i\varepsilon}.$
(52)

Introducing $\tilde{n}(x)=1/[\exp(x)+1]$, Eq. (52) is re-cast as

$\displaystyle\eta_{\mu\nu}\Pi^{\mu\nu}_{f,B,\Omega,s_{q}}(p_{0},p_{z},p_{\scriptscriptstyle{\perp}})=-N_{c}q_{f}^{2}m_{f}^{2}\frac{|q_{f}B|}{8\pi^{2}}e^{-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}}$
$\displaystyle\times\int\limits_{-\infty}^{\infty}dk_{z}\sum_{r_{1}=\pm
1}\sum_{r_{2}=\pm
1}\frac{r_{1}r_{2}}{\epsilon_{0,k_{z}}^{\scriptscriptstyle{(f)}}\epsilon_{0,k_{z}-p_{z}}^{\scriptscriptstyle{(f)}}}\frac{1-\widetilde{n}\left(r_{1}\epsilon_{0,k_{z}}^{\scriptscriptstyle{(f)}}-\mu_{s_{q}}^{\ast}\right)-\widetilde{n}\left(r_{2}\epsilon_{0,k_{z}-p_{z}}^{\scriptscriptstyle{(f)}}+\mu_{s_{q}}^{\ast}\right)}{p_{0}-r_{1}\epsilon_{0,k_{z}}^{\scriptscriptstyle{(f)}}-r_{2}\epsilon_{0,k_{z}-p_{z}}^{\scriptscriptstyle{(f)}}+i\varepsilon},$
(53)

where an _effective chemical_ potential is introduced via

$\displaystyle\mu^{\ast}_{s_{q}}=\mu+s_{q}\frac{\Omega}{2}.$ (54)

For simplicity, I consider the mass of up and down quark same $m_{q}\equiv
m_{u}=m_{d}=5\;\text{MeV}$. Therefore, I drop the subscript and/or superscript
$f$ in the expression of energy. As I am working in LLL approximation the
subscript $0$ denoting LLL is also dropped from now on.

The sums over $r_{1},r_{2}$ are performed as

$\displaystyle\eta_{\mu\nu}\Pi^{\mu\nu}_{f,B,\Omega,s_{q}}(p_{0},p_{z},p_{\scriptscriptstyle{\perp}})=-N_{c}q_{f}^{2}m_{q}^{2}\frac{|q_{f}B|}{8\pi^{2}}e^{-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}}$
$\displaystyle\times\int\limits_{-\infty}^{\infty}\frac{dk_{z}}{\epsilon_{k_{z}}\epsilon_{k_{z}-p_{z}}}\Bigg{[}\frac{1-\widetilde{n}\left(\epsilon_{k_{z}}-\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}+\mu^{\ast}_{s_{q}}\right)}{p_{0}-\epsilon_{k_{z}}-\epsilon_{k_{z}-p_{z}}+i\varepsilon}+\frac{\widetilde{n}\left(\epsilon_{k_{z}}-\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}-\mu^{\ast}_{s_{q}}\right)}{p_{0}-\epsilon_{k_{z}}+\epsilon_{k_{z}-p_{z}}+i\varepsilon}$
$\displaystyle\hskip
56.9055pt-\frac{\widetilde{n}\left(\epsilon_{k_{z}}+\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}+\mu^{\ast}_{s_{q}}\right)}{p_{0}+\epsilon_{k_{z}}-\epsilon_{k_{z}-p_{z}}+i\varepsilon}-\frac{1-\widetilde{n}\left(\epsilon_{k_{z}}+\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}-\mu^{\ast}_{s_{q}}\right)}{p_{0}+\epsilon_{k_{z}}+\epsilon_{k_{z}-p_{z}}+i\varepsilon}\Bigg{]}.$
(55)

The imaginary part is calculated by employing the formula

$\displaystyle\textsf{Im}\frac{1}{u\pm i\varepsilon}=\mp\pi\delta(u).$ (56)

From here, $B$, $\Omega$ are also dropped from the expression of polarization
tensor which is evident from the context. Also, the the presence of $s_{q}$ is
automatically understood if not explicitly specified. So, one can write

$\displaystyle\textsf{Im}\Pi^{\mu}_{f\;\mu}=N_{c}q_{f}^{2}m_{q}^{2}\frac{|q_{f}B|}{8\pi^{2}}e^{-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}}\int\limits_{-\infty}^{\infty}\frac{dk_{z}}{\epsilon_{k_{z}}\epsilon_{k_{z}-p_{z}}}\Bigg{\\{}\Big{[}1-\widetilde{n}\left(\epsilon_{k_{z}}-\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}+\mu^{\ast}_{s_{q}}\right)\Big{]}\delta\bigg{(}p_{0}-\epsilon_{k_{z}}$
$\displaystyle-\epsilon_{k_{z}-p_{z}}\bigg{)}+\Big{[}\widetilde{n}\left(\epsilon_{k_{z}}-\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}-\mu^{\ast}_{s_{q}}\right)\Big{]}\delta\bigg{(}p_{0}-\epsilon_{k_{z}}+\epsilon_{k_{z}-p_{z}}\bigg{)}-\Big{[}\widetilde{n}\left(\epsilon_{k_{z}}+\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}\right.$
$\displaystyle\left.+\mu^{\ast}_{s_{q}}\right)\Big{]}\delta\bigg{(}p_{0}+\epsilon_{k_{z}}-\epsilon_{k_{z}-p_{z}}\bigg{)}-\Big{[}1-\widetilde{n}\left(\epsilon_{k_{z}}+\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}-\mu^{\ast}_{s_{q}}\right)\Big{]}\delta\Big{(}p_{0}+\epsilon_{k_{z}}+\epsilon_{k_{z}-p_{z}}\Big{)}\Bigg{\\}}.$
(57)

From the Appendix B, it is evident that there are two kinematic regions that
give rise to imaginary part of the potential and leads to various processes.
They are

  1. (I)

Unitary cut:
$\left(\infty>p_{0}\geqslant+\sqrt{p_{z}^{2}+4m_{q}^{2}}\right)\quad\textsf{and}\quad\left(-\infty<p_{0}\leqslant-\sqrt{p_{z}^{2}+4m_{q}^{2}}\right)$,

  2. (II)

Landau cut:  $-|p_{z}|\leqslant p_{0}\leqslant|p_{z}|$.

Figure 2: [color online] Kinematic region in $\angle p_{0}$ plane where
$\Pi^{\mu}_{\mu}$ gives non-zero imaginary part. Here only the
$\textsf{Re}(p_{0})>0$ portion is shown. The red line denotes Landau cut part
which does not contribute to the DPR when both quark and antiquark resides at
LLL. However, the green line denotes unitary cut part which do contribute to
DPR in the LLL scenario.

However, the imaginary part arising from Landau cut region does not contribute
to any processes when both the quark and/or antiquark reside in LLL.222Upon
lifting the condition of LLL approximation, the Landau cut do contributes to
the DPR [40] The reason is the unavailability of phase space for producing
lepton anti-lepton pairs which is evident from the following argument —
Let $Q=(q_{0},\bm{q})=\left(\sqrt{|\bm{q}|^{2}+M_{l}^{2}},\bm{q}\right)$ and
$Q^{\prime}=(q_{0}^{\prime},\bm{q}^{\prime})=\left(\sqrt{|\bm{q}^{\prime}|^{2}+M_{l}^{2}},\bm{q}^{\prime}\right)$
be the four-momentum of the produced lepton and anti-lepton pair for on-shell
final di-leptons, respectively. Then, following the conservation of energy
momentum, one has $P=Q+Q^{\prime}$ and $Q^{2}={Q^{\prime}}^{2}=M_{l}^{2}$,
respectively. Squaring both sides of energy-momentum conservation relation
yields

$\displaystyle P^{2}=Q^{2}+{Q^{\prime}}^{2}+2Q\cdot
Q^{\prime}=2M_{l}^{2}+2\left(\sqrt{|\bm{q}|^{2}+M_{l}^{2}}\sqrt{|\bm{q}^{\prime}|^{2}+M_{l}^{2}}-\bm{q}\cdot\bm{q}^{\prime}\right).$
(58)

Lastly, the Schwartz inequality
$\sqrt{|\bm{q}|^{2}+M_{l}^{2}}\sqrt{|\bm{q}^{\prime}|^{2}+M_{l}^{2}}-\bm{q}\cdot\bm{q}^{\prime}\geqslant
M_{l}^{2}$ leads to $P^{2}=p_{0}^{2}-p_{\perp}^{2}-p_{z}^{2}\geqslant
4M_{l}^{2}$, where $M_{l}$ is the mass of leptons (anti-leptons). In the
Landau cut region, one has
$P^{2}=p^{2}_{0}-p_{z}^{2}-p_{\perp}^{2}=-|p_{z}^{2}-p_{0}^{2}|-p_{z}^{2}\leqslant
0$ leading to the non-availability of phase space as claimed. Moreover, as the
condition $p_{0}>0$ and $p_{0}<0$ corresponds to the creation and decay of
virtual photons, respectively, I concentrate only on $p_{0}>0$ case. As a
result, I only concentrate on $p_{0}\geqslant\sqrt{p_{z}^{2}+4m_{q}^{2}}$
region.

Among the four terms appearing in the R.H.S of Eq. (57), the last one inside
$k_{z}$ integration with
$\delta\left(p_{0}+\epsilon_{k_{z}}+\epsilon_{k_{z}-p_{z}}\right)$ does not
contribute to the production of dileptons. So, I concentrate on the first
three terms inside the $k_{z}$ integral in Eq. (57). To perform the $k_{z}$
integration, I make use of the identity of Dirac delta function

$\displaystyle\delta(f(u))=\sum_{i}\left|\dfrac{\partial f(u)}{\partial
u}\right|_{u=u_{i}}^{-1}\delta(u-u_{i}),\quad\text{with }u_{i}\text{ being the
}i\text{-th root of }f(u)=0.$ (59)

The equation $p_{0}-\epsilon_{k_{z}}-\epsilon_{k_{z}-p_{z}}=0$ provides two
solution for $k_{z}$

$\displaystyle
k_{z}^{(\pm)}=\frac{1}{2}\left(p_{z}\pm\frac{\sqrt{p_{0}^{2}\,(p_{0}^{2}-p_{z}^{2})(p_{0}^{2}-p_{z}^{2}-4m_{q}^{2})}}{p_{0}^{2}-p_{z}^{2}}\right),$
(60)

for $r_{1},r_{2}=\pm 1$. The condition
$0\leqslant\sqrt{p_{z}^{2}+4m_{q}^{2}}\leqslant p_{0}$ further simplifies the
expression of $k_{z}^{(\pm)}$ in Eq.(60) to

$\displaystyle k^{(\pm)}_{z}=\frac{1}{2}\left(p_{z}\pm
p_{0}\sqrt{1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}}\right).$ (61)

It leads to

$\displaystyle k^{(\pm)}_{z}-p_{z}=-\frac{1}{2}\left(p_{z}\mp
p_{0}\sqrt{1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}}\right),$ (62)

with energies

$\displaystyle\epsilon_{k^{(\pm)}_{z}}=\epsilon^{(\pm)},\quad\epsilon_{k^{(\pm)}_{z}-p_{z}}=\epsilon^{(\mp)},$
(63)

where $\epsilon^{(\pm)}$ are defined as

$\displaystyle\epsilon^{(\pm)}\equiv\frac{1}{2}\left(p_{0}\pm
p_{z}\sqrt{1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}}\right).$ (64)

###  V.1 Process - I

In this case, I concentrate on quark and anti-quark annihilation process
$q\overline{q}\rightarrow\gamma^{*}$ where the relevant term is written as

$\displaystyle\bigg{(}\textsf{Im}\Pi^{\mu}_{f\;\mu}\bigg{)}_{\text{I}}=N_{c}q_{f}^{2}m_{q}^{2}\frac{|q_{f}B|}{8\pi}\exp\left(-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}\right)\int\limits_{-\infty}^{\infty}\frac{dk_{z}}{\epsilon_{k_{z}}\epsilon_{k_{z}-p_{z}}}\Big{[}1-\widetilde{n}\left(\epsilon_{k_{z}}-\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}+\mu^{\ast}_{s_{q}}\right)\Big{]}$
$\displaystyle\hskip
284.52756pt\times\delta\bigg{(}p_{0}-\epsilon_{k_{z}}-\epsilon_{k_{z}-p_{z}}\bigg{)}.$
(65)

Applying the identity in Eq. (59) along with $\epsilon^{(\pm)}>0$333For the
proof of the positivity condition of $\epsilon^{(\pm)}$, see Appendix B and
$p_{\shortparallel}^{2}>0$, the delta function in Eq. (65) becomes after few
steps of algebra,

$\displaystyle\delta\left(p_{0}-\epsilon_{k_{z}}-\epsilon_{k_{z}-p_{z}}\right)=\frac{2\epsilon^{(+)}\epsilon^{(-)}}{(p_{0}^{2}-p_{z}^{2})}\left(1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}\right)^{-\frac{1}{2}}\left[\delta\left(k_{z}-k_{z}^{(+)}\right)+\delta\left(k_{z}-k_{z}^{(-)}\right)\right],$
(66)

Here, I have used

$\displaystyle\left[\frac{\partial}{\partial
k_{z}}\delta\left(p_{0}-\epsilon_{k_{z}}-\epsilon_{k_{z}-p_{z}}\right)\right]_{k_{z}=k_{z}^{(\pm)}}=\pm\frac{p_{0}^{2}-p_{z}^{2}}{2\epsilon^{(\pm)}\epsilon^{(\mp)}}\sqrt{1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}}.$
(67)

Therefore, an analytic expression of first term is obtained as

$\displaystyle\bigg{(}\textsf{Im}\Pi^{\mu}_{f\;\mu}\bigg{)}_{\text{I}}=N_{c}q_{f}^{2}m_{q}^{2}\frac{|q_{f}B|}{8\pi}\exp\left(-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}\right)\frac{2}{(p_{0}^{2}-p_{z}^{2})}\left(1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}\right)^{-\frac{1}{2}}\Bigg{[}2-\widetilde{n}\left(\epsilon^{(-)}-\mu_{s_{q}}^{\ast}\right)$
$\displaystyle\hskip
142.26378pt-\widetilde{n}\left(\epsilon^{(+)}-\mu_{s_{q}}^{\ast}\right)-\widetilde{n}\left(\epsilon^{(+)}+\mu_{s_{q}}^{\ast}\right)-\widetilde{n}\left(\epsilon^{(-)}+\mu_{s_{q}}^{\ast}\right)\Bigg{]}$
(68)

###  V.2 Process - II

In this case, the corresponding processes are decay processes
$q(\overline{q})\rightarrow q(\overline{q})\gamma^{*}$. The expression
relating to this written as

$\displaystyle\bigg{(}\textsf{Im}\Pi^{\mu}_{f\;\mu}\bigg{)}_{\text{II}_{\pm}}=\pm
N_{c}q_{f}^{2}m_{q}^{2}\frac{|q_{f}B|}{8\pi}\exp\left(-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}\right)$
$\displaystyle\times\int\limits_{-\infty}^{\infty}\frac{dk_{z}}{\epsilon_{k_{z}}\epsilon_{k_{z}-p_{z}}}\Big{\\{}\left[\widetilde{n}\left(\epsilon_{k_{z}}\mp\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon_{k_{z}-p_{z}}\mp\mu^{\ast}_{s_{q}}\right)\right]\delta\bigg{(}p_{0}\mp\epsilon_{k_{z}}\pm\epsilon_{k_{z}-p_{z}}\bigg{)}\Bigg{\\}}.$
(69)

Here the subscript $\text{II}_{+}$ refers to the term with
$\delta\left(p_{0}-\epsilon_{k_{z}}+\epsilon_{k_{z}-p_{z}}\right)$ and
$\text{II}_{-}$ refers to that with
$\delta\left(p_{0}+\epsilon_{k_{z}}-\epsilon_{k_{z}-p_{z}}\right)$. In this
particular kinematic region, I have

$\displaystyle\left[\frac{\partial}{\partial
k_{z}}\delta\left(p_{0}\mp\epsilon_{k_{z}}\pm\epsilon_{k_{z}-p_{z}}\right)\right]_{k_{z}=k_{z}^{(\pm)}}=\frac{8m_{q}^{2}p_{0}p_{z}}{(p_{0}^{2}-p_{z}^{2})\epsilon^{(+)}\epsilon^{(-)}},$
(70)

The delta function is simplified by Eq. (59) as

$\displaystyle\delta\left(p_{0}\mp\epsilon_{k_{z}}\pm\epsilon_{k_{z}-p_{z}}\right)=\frac{\epsilon^{(+)}\epsilon^{(-)}(p_{0}^{2}-p_{z}^{2})}{8m_{q}^{2}p_{0}|p_{z}|}\left[\delta\left(k_{z}-k^{(+)}_{z}\right)+\delta\left(k_{z}-k^{(-)}_{z}\right)\right].$
(71)

After performing delta function integration, I arrive at

$\displaystyle\bigg{(}\textsf{Im}\Pi^{\mu}_{f\;\mu}\bigg{)}_{\text{II}_{\pm}}=\pm
N_{c}q_{f}^{2}m_{q}^{2}\frac{|q_{f}B|}{8\pi}\exp\left(-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{f}B|}\right)\frac{\epsilon^{(+)}\epsilon^{(-)}(p_{0}^{2}-p_{z}^{2})}{8m_{q}^{2}p_{0}|p_{z}|}$
$\displaystyle\times\Bigg{[}\widetilde{n}\left(\epsilon^{(+)}\mp\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon^{(-)}\mp\mu^{\ast}_{s_{q}}\right)+\widetilde{n}\left(\epsilon^{(+)}\mp\mu^{\ast}_{s_{q}}\right)-\widetilde{n}\left(\epsilon^{(-)}\mp\mu^{\ast}_{s_{q}}\right)\Bigg{]}$
$\displaystyle=0.$ (72)

Hence, when both of the quark and antiquark resides on LLL, the calculation
showed that decay processes are kinematically prohibited in the region where
$p_{0}\geqslant\sqrt{p_{z}^{2}+4m_{q}^{2}}$. In general, decay processes can
never arise from the unitary cut region of the complex $p_{0}$ plane [59].
Moreover, when both of the quark-antiquark resides on LLL, the Landau cut part
does not support the processes where $P^{2}<0$.

###  V.3 The Di-lepton Production Rate

The dilepton production per unit four-spacetime volume and per unit four-
momentum volume in a hot rotating magnetized medium under LLL approximation is
given, upon substituting Eq. (68) in Eq. (51), as

$\displaystyle\frac{d\mathcal{N}}{d^{4}Pd^{4}\mathcal{X}}\Bigg{|}_{q\bar{q}\rightarrow\gamma^{*}}=N_{c}\frac{\alpha^{2}_{\textsf{EM}}}{12\pi^{4}}\frac{m_{q}^{2}}{p_{\shortparallel}^{2}(p_{\shortparallel}^{2}-p_{\scriptscriptstyle{\perp}}^{2})}{\frac{1}{e^{p_{0}/T}-1}}\left(1-\frac{4m_{q}^{2}}{p_{\shortparallel}^{2}}\right)^{-\frac{1}{2}}\Bigg{\\{}\left(q_{u}/e\right)^{2}|q_{u}B|\exp\left(-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{u}B|}\right)\Bigg{[}2$
$\displaystyle-\sum_{I=H,L}\Bigg{(}\widetilde{n}\left(\epsilon_{I}-\mu^{\ast}_{+}\right)+\widetilde{n}\left(\epsilon_{I}+\mu^{\ast}_{+}\right)\Bigg{)}\Bigg{]}+\left(q_{d}/e\right)^{2}|q_{d}B|\exp\left(-\frac{p_{\scriptscriptstyle{\perp}}^{2}}{2|q_{d}B|}\right)\Bigg{[}2-\sum_{I=H,L}\Bigg{(}\widetilde{n}\left(\epsilon_{I}-\mu^{\ast}_{-}\right)$
$\displaystyle\hskip
312.9803pt+\widetilde{n}\left(\epsilon_{I}+\mu^{\ast}_{-}\right)\Bigg{)}\Bigg{]}\Bigg{\\}}.$
(73)

In the last line I have introduced two modes, independent of the sign of
$p_{z}$, as

$\displaystyle\epsilon_{H,L}=\frac{1}{2}\left(p_{0}\pm|p_{z}|\sqrt{1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}}\right),$
(74)

with $H$ and $L$ corresponds to the upper and lower signs before $|p_{z}|$,
respectively. From the expression of Eq. (73), one immediately notices that in
the massless limit $m_{q}\rightarrow 0$, the rate vanishes. The reason for
such a consequence is a mechanism [42] similar to the “helicity suppression”
present in the lepton decay of charged pions.

##  VI Discussions

To arrive at the simplified result of differential DPR, I have made some
assumptions, working best in the early stage of heavy ion collision, under
which the validity of our calculation can be justified.444The typical scale
hierarchy that works best in my calculation is $\sqrt{|eB|}>T>\mu>\Omega$.
These are listed below —

  1. (I)

At the initial stage of HIC, the generated magnetic field is very high. It is
the highest among all the scales present in the problem. In this scenario, one
can assume that the energy gap between the zero-th and first Landau level is
so high that the quark antiquark pairs are confined in the LLL [51, 52]. Thus,
the LLL approximation is justified only in the initial stage.

  2. (II)

Next, during the early stages of HIC, the radial expansion is not yet
developed by particle interactions. In this case one can assume that the
quarks are totally dragged by the vortical motion of the fluid. Therefore, the
approximation $\bar{\phi}+\Omega\bar{t}=0$ is a good one [53] in this
scenario.

  3. (III)

In my calculation, I have assumed no radial boundary that confines the quarks
from escaping the rotating medium. However, to preserve causality, one must
have $g_{00}\leqslant 1$. This is ensured by the condition $\Omega
R_{\scriptscriptstyle{\perp}}\leqslant 1$ [21]. Thus, to secure the validity
of our calculation, I have considered the case of slow rotation.

  4. (IV)

When tackling the problem of rotating medium, one should keep in mind that the
temperature of the plasma depends on $r_{\scriptscriptstyle{\perp}}$, the
distance of the point in concern from the axis of rotation. Specifically, it
changes with distance as
$T\left(r_{\scriptscriptstyle{\perp}}\right)=T\left(r_{\scriptscriptstyle{\perp}}=0\right)/\sqrt{g^{00}}$,
where $g^{00}$ is the zero-th component of metric tensor in curved space time
[60]. For the rigid rotation studied here
$g^{00}=1-\Omega^{2}r_{\scriptscriptstyle{\perp}}^{2}$. Following [28], I have
neglected the variation of $T$ with $r_{\scriptscriptstyle{\perp}}$ by
considering it the same with all over the space that at the rotation axis.

After shedding some light on the assumptions made in our calculation, I wish
to comment on the translational symmetry in various backgrounds.

  1. 1.

$\bm{eB\neq 0}$ and $\bm{\Omega=0:}$
In the case of non-zero background magnetic field in a non-rotating scenario,
the rotational symmetry is explicitly broken in the direction perpendicular to
the reaction plane. Due to this explicit breaking, the translational symmetry
in one of the coordinate among four space-time coordinates is lost
irrespective of the choice vector potential $\mathcal{A}^{\mu}$. The exact
coordinate, in which the symmetry is broken, depends on the choice of the
background gauge $\mathcal{A}^{\mu}$ used to represent the B-field as shown in
the table 2.

Gauge choice ($\mathcal{A}^{\mu}$) | Broken Coordinates | Unbroken Coordinates
---|---|---
$B(0,-y,0,0)$ |  $y$-coordinate |  $t$, $x$, $z$
$B(0,0,x,0)$ |  $x$-coordinate |  $t$, $y$, $z$
$\dfrac{B}{2}(0,-y,x,0)$ |  $r_{\scriptscriptstyle{\perp}}(=\sqrt{x^{2}+y^{2}})$-coordinate |  $t$, $\phi$, $z$
Table 2: The interdependency between the gauge choice and coordinate in which
the translational symmetry is broken.

This loss is reflected by the appearance of a phase factor in the mathematical
expression of the propagator

$\displaystyle
S_{f,B}\left(\mathcal{X},\mathcal{X}^{\prime}\right)=\exp\left[i\Phi\left(\mathcal{X},\mathcal{X}^{\prime}\right)\right]S_{f,B}\left(\mathcal{X}-\mathcal{X}^{\prime}\right),$
(75)

where $\Phi$ is called the Schwinger Phase factor. The total phase factor
becomes $1$ for the diagrams where the fermions appears only in the form of a
closed loop [54] (for example in 1-PI diagrams such as photon polarization
tensor diagram, triangle diagrams etc.).

  2. 2.

$\bm{eB=0}$ and $\bm{\Omega\neq 0:}$
In the absence of background $B$ field in a rotating medium, the angular
velocity $\Omega$, generated perpendicular to the reaction plane, breaks
rotational symmetry. However, the system possesses invariance under rotations
in the $x-y$ plane. Therefore, in this case, the most natural choice of the
coordinate system to work with is the cylindrical polar coordinate due to the
geometry of the system. The translational symmetry remains intact along
$t,\phi$ and $z$ direction but it is broken in radial directions. This is very
evident from the following argument— consider two observers sitting on a
rotating plate at the same radial coordinate but with different angles will
agree on some measurement. But in the opposite scenario, i.e. with same
azimuthal but different radial coordinate, they will differ. From the
intuition of classical mechanics, it is clear that they will have different
radial velocity.

  3. 3.

$\bm{eB\neq 0}$ and $\bm{\Omega\neq 0:}$
For a rotating plasma in the presence of B-field, one can expect the non-
breaking of translational symmetry only in two coordinates. However, in our
case the translational symmetry is lost only in one direction, thanks to the
fact that both of the $B$-field and $\Omega$ are pointing along the same
direction. Now, few things can be noted from the factor in the propagator that
breaks translational invariance.

     * •

In the presence of background B-field and rotations, the factor that breaks
translational invariance, denoted by $\aleph$, is of the following form

$\displaystyle\aleph\left(\bar{t},\xi,\bar{\phi};\xi^{\prime}\right)=\exp\left[-\Xi\left(\bar{t},\xi,\bar{\phi};\xi^{\prime}\right)\right],$
(76)

where

$\displaystyle\Xi\left(\bar{t},\xi,\bar{\phi};\xi^{\prime}\right)=\frac{\xi+\xi^{\prime}}{2}-\sqrt{\xi\xi^{\prime}}e^{i(\bar{\phi}+\Omega\bar{t})}\quad\text{with}\quad\bar{t}=t-t^{\prime},\,\bar{\phi}=\phi-\phi^{\prime}.$
(77)

     * •

Due to the factor of
$S(\mathcal{X},\mathcal{X}^{\prime})S(\mathcal{X}^{\prime},\mathcal{X})$ in
one loop calculations, the translationaly non-invariant factor, after few
steps of algebra, becomes

$\displaystyle\exp\left[-\Xi\left(\bar{t},\xi,\bar{\phi};\xi^{\prime}\right)-\Xi\left(-\bar{t},\xi^{\prime},-\bar{\phi};\xi\right)\right]=\exp\left[-\left\\{\xi+\xi^{\prime}-2\sqrt{\xi\xi^{\prime}}\cos\left(\bar{\phi}+\Omega\bar{t}\right)\right\\}\right]$
$\displaystyle=\exp\left[-\frac{|q_{f}B|}{2}\left\\{r_{\scriptscriptstyle{\perp}}^{2}+{r_{\scriptscriptstyle{\perp}}^{\prime}}^{2}-2r_{\scriptscriptstyle{\perp}}r_{\scriptscriptstyle{\perp}}^{\prime}\cos\left(\bar{\phi}+\Omega\bar{t}\right)\right\\}\right].$
(78)

     * •

For $\Omega=0$, expanding the Cosine and using
$x=r_{\scriptscriptstyle{\perp}}\cos\phi,y=r_{\scriptscriptstyle{\perp}}\sin\phi,x^{\prime}=r_{\scriptscriptstyle{\perp}}^{\prime}\cos\phi^{\prime},y^{\prime}=r_{\scriptscriptstyle{\perp}}^{\prime}\sin\phi^{\prime}$,
one gets back the translational invariance in the radial coordinates.
Mathematically, it turns out that

$\displaystyle\lim_{\Omega\rightarrow
0}\exp\left[-\frac{|q_{f}B|}{2}\left\\{r^{2}_{\scriptscriptstyle{\perp}}+{r^{\prime}_{\scriptscriptstyle{\perp}}}^{2}-2r_{\scriptscriptstyle{\perp}}r_{\scriptscriptstyle{\perp}}^{\prime}\cos\left(\bar{\phi}+\Omega\bar{t}\right)\right\\}\right]=\exp\left[-\frac{|q_{f}B|}{2}\left\\{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}\right\\}\right].$
(79)

This result is in accordance with the non-rotating case where the phase factor
disappears in the diagrams with closed fermion loops.

The expression of DPR can be interpreted as follows: One can rearrange the
Fermi-Dirac distribution part of the result in Eq. (73) as

$\displaystyle
2-\widetilde{n}\left(\epsilon_{H}-\mu_{s_{q}}^{\ast}\right)-\widetilde{n}\left(\epsilon_{L}-\mu_{s_{q}}^{\ast}\right)-\widetilde{n}\left(\epsilon_{H}+\mu_{s_{q}}^{\ast}\right)-\widetilde{n}\left(\epsilon_{L}+\mu_{s_{q}}^{\ast}\right)$
$\displaystyle=\left[1-\widetilde{n}\left(\epsilon_{H}-\mu_{s_{q}}^{\ast}\right)\right]\left[1-\widetilde{n}\left(\epsilon_{L}+\mu_{s_{q}}^{\ast}\right)\right]-\widetilde{n}\left(\epsilon_{H}-\mu_{s_{q}}^{\ast}\right)\widetilde{n}\left(\epsilon_{L}+\mu_{s_{q}}^{\ast}\right)$
$\displaystyle\hskip
113.81102pt+\left[1-\widetilde{n}\left(\epsilon_{L}-\mu_{s_{q}}^{\ast}\right)\right]\left[1-\widetilde{n}\left(\epsilon_{H}+\mu_{s_{q}}^{\ast}\right)\right]-\widetilde{n}\left(\epsilon_{L}-\mu_{s_{q}}^{\ast}\right)\widetilde{n}\left(\epsilon_{H}+\mu_{s_{q}}^{\ast}\right).$
(80)

The total DPR in the original system can be interpreted as equal to the sum of
that in the two fictitious systems – one with chemical potential of
$\mu^{\ast}_{+}=\mu+\Omega/2$ and another with that of
$\mu^{\ast}_{-}=\mu-\Omega/2$ (see Fig. 3). The quark flavors can be
identified by noting that particles with $s_{z}=\frac{1}{2}$ possesses lower
energy than that of spin $s_{z}=-\frac{1}{2}$ particles. In both of the
systems, a $u$($\bar{d}$) with energy $\epsilon_{L}$ and a $d$($\bar{u}$) with
energy $\epsilon_{H}$ annihilates with each other to produce a virtual photon.
Note that the statistical weight factors $1-\widetilde{n}(E-\mu)$ and
$\widetilde{n}(E-\mu)$ corresponds to a particle in the final state and
initial state [56], respectively.555To get the weight factors involving
antipaticles, the chemical potential $\mu$ is replaced by $-\mu$ in the Fermi-
Dirac distribution function. These weight factors are called Pauli suppression
factor.

Figure 3: [color online] Schematic figures of the two fictitious systems with
chemical potential $\mu^{\ast}_{+}$ (left side) and $\mu^{\ast}_{-}$ (right
side). The small arrows over the particle symbols represents the
$z$-components of spin.

##  VII Results

For the illustration purpose, I have plotted the dilepton production rate,
denoted by DR, with invariant mass squared666From the conservation of energy
momentum valid outside the medium, the di-lepton’s invariant mass squared
$M^{2}=(Q_{1}+Q_{2})^{2}$ is the same as the four momentum square of the
virtual photon, i.e., $P^{2}=M^{2}$. $M^{2}$ at non-vanishing $\Omega$ scaled
with DR at $\Omega=0$ for various $\Omega=0$, $0.01$, $0.05$, $0.1$ and $0.15$
and $0.2$ GeV in the left panel of Fig. 4. The magnetic field is fixed at
$|eB|=15m_{\pi}^{2}$, which is reasonably strong in the context of non-central
HIC. From the nature of the plot, it is clear that the effect of rotation is
to suppress the rate compared to that of non-rotating case. The suppression
gets stronger as the invariant mass $M^{2}$ is lowered. With increasing
invariant mass the DPR eventually merge up with the non-rotating strong field
case. As $\Omega$ is increased, the DPR increases more rapidly with invariant
mass which is clearly seen from the left panel of Fig. 4.

Figure 4: [Left Panel] The ratio of DPR at non-zero $\Omega$ and at $\Omega=0$
(along the $y$-axis) for various values of $\Omega$ as a function of the
invariant mass of di-leptons $M^{2}$ (along the $x$-axis). [Right Panel] The
DPR as a function of $\Omega$ for $|eB|$ values starting from $5m_{\pi}^{2}$
to $20m_{\pi}^{2}$ in steps of $5m_{\pi}^{2}$.

Next, the behaviour of DPR with $\Omega$ is illustrated in the right panel of
Fig. 4 for the values of the magnetic field $|eB|$ starting from
$5m_{\pi}^{2}$ to $20m_{\pi}^{2}$ in steps of $5m_{\pi}^{2}$. The rate
decreases almost steadily with $\Omega$. However, as the magnetic field is
increased, the curve falls more sharply with angular velocity. For example the
change of DR with angular velocity is more at $|eB|=20m_{\pi}^{2}$ than that
at $|eB|=5m_{\pi}^{2}$ in the range $\Omega\in[0,0.2]\;\text{GeV}$. It is
evident that the rate increase with magnetic field. The dependence on $|eB|$
on DPR at LLL approximation comes from the term
$|eB|\exp(-p_{\scriptscriptstyle{\perp}}^{2}/2|eB|)$. This behaviour indicates
that with increasing magnetic field the influence of rotation on di-lepton
production rate is reduced.
The result of LLL-approximated case can be recovered by taking
$\Omega\rightarrow 0$ limit for $\Omega\neq 0$. However, the $|eB|\rightarrow
0$ limit can not be taken since this calculation is valid only for high
$|eB|$.

Figure 5: The $x$-axis and $y$-axis shows the angular velocity $\Omega$ and
DPR, respectively.

Finally, the plot of dilepton production rate as a function of $\Omega$ for
different temperature of the plasma starting from $T=0.17$ GeV to $0.25$ GeV
is displayed in Fig. 5. Expectedly, the rate increases with temperature. At
this point, one can observe that the rate decreases monotonically with
$\Omega$ for various temperatures in a similar fashion.

The choice of the parameter value $\Omega$ deserve some comments. The
vorticity created in the medium depends on the C.M energy
($\sqrt{S_{\textsf{NN}}}$) and the impact parameter (usually denoted $b$) of
the two colliding nuclei. The STAR collaboration measured the vorticity $\sim
6$ MeV from $\sqrt{S_{\textsf{NN}}}$ averaged polarization. Furthermore, the
AMPT [57] and HIJING [58] predicted it $\sim 27.6$ MeV and $\sim 20$ MeV,
respectively. However, in Fig. 4 and 5, I have kept $\Omega$ flexible so that
its influence on the rate can be compared. This was done purely from
theoretical perspective.

##  VIII Conclusion

In this work, I have calculated an analytical expression of dilepton
production rate from a rotating color deconfined QGP medium in the presence of
strong background magnetic field. I have considered the case of
$\bm{B}\parallel\bm{\Omega}$, which is suitable in the context of heavy ion
collision experiments. To achieve this goal, I have reformulated the
expression of quark propagator in expressed in terms of four-position basis to
four momentum basis under the approximation that fermions are totally dragged
by the vortical motion of the created medium. In this reformulation of
propagator, I only worked with lowest Landau level approximation which is
valid when the generated magnetic field is stronger than all the scales
present in the theory. In LLL, only spin up $u$, $\bar{d}$ quark and spin down
$\bar{u}$, $d$ can propagate. The effect of rotation can be interpreted as an
alteration of the quark chemical potential of the system which depends on the
charge of the quark under consideration. The dilepton production rate is
suppressed compared to the non-rotating due to the presence of rotation. When
the invariant mass is reduced, the rate of suppression is found out to be more
rapid for higher values of $\Omega$. With increasing $|eB|$, the rate gets
enhanced in this calculations.

As an outlook one can take a more pragmatic approach to study the DPR by
lifting the first three approximations mentioned in Sec. VI. One such approach
is to relax the condition that both the quark and antiquark is in LLL. In this
circumstance, it is reasonable to expect that there will be contributions to
the rate due to the transition of quarks (anti-quarks) between various Landau
levels like $\Omega=0$ case [40, 41]. Also, one can take magneto-rotating
system bounded in a cylindrical box whose radius is of the order of the size
of the system ($R\sim 10$ fm). Then, one must take into a boundary condition
(e.g, MIT boundary condition [21], global boundary condition [34]) to strictly
ensure that causality in the calculation. Lastly, one can reformulate the
rate-formula of DPR from the very beginning by employing the tools of basic
thermal quantum field theory when the initial quarks are not in momentum
eigenstates. In this direction, one can utilize the skill of QFT in non-
trivial backgrounds and consult the classic paper of Weldon [59].

###### Acknowledgements.

I would like to acknowledge School of Physical Science, NISER, Bhuwaneshwar.

##  Appendix A Fermionic Frequency Sum

The frequency sum of interest is written as

$\displaystyle\mathcal{F}\equiv
T\\!\sum_{k_{0}}\frac{1}{\left[(k_{0}+a)^{2}-E_{1}^{2}\right]}\frac{1}{(p_{0}-k_{0}-\bar{a})^{2}-E_{2}^{2}}.$
(81)

In the ITF formalism of thermal field theory, the fermionic momentum at finite
density is replaced by $k_{0}\rightarrow i(2l+1)\pi
T+\mu=\widetilde{\omega}_{l}+\mu$ and $p_{0}\rightarrow i2\pi mT=\omega_{m}$.
Thus, the frequency sum can be rewritten as

$\displaystyle\mathcal{F}$
$\displaystyle=T\\!\sum_{l=-\infty}^{\infty}\frac{1}{\left[(i\widetilde{\omega}_{l}+\mu+a)^{2}-E_{1}^{2}\right]}\frac{1}{(i\omega_{m}-i\widetilde{\omega}_{l}-\mu-\bar{a})^{2}-E_{2}^{2}}$
$\displaystyle=\sum_{r_{1}=\pm 1}\sum_{r_{2}=\pm
1}\frac{r_{1}r_{2}}{4E_{1}E_{2}}T\\!\sum_{l=-\infty}^{\infty}\frac{1}{i\widetilde{\omega}_{l}+\mu+a-r_{1}E_{1}}\frac{1}{i(\omega_{m}-\widetilde{\omega}_{l})-\mu-\bar{a}-r_{2}E_{2}}.$
(82)

The fraction in the right hand side of the above equation can be written as

$\displaystyle\frac{1}{i\widetilde{\omega}_{l}+\mu+a-r_{1}E_{1}}$
$\displaystyle=\tilde{n}^{\scriptscriptstyle{(+)}}\left(r_{1}E_{1}-a\right)\int\limits_{0}^{\beta}d\tau_{1}\,e^{-\tau_{1}\left(i\widetilde{\omega}_{l}+\mu+a-r_{1}E_{1}\right)}$
$\displaystyle\frac{1}{i(\omega_{m}-\widetilde{\omega}_{l})-\mu-\bar{a}-r_{2}E_{2}}$
$\displaystyle=\tilde{n}^{\scriptscriptstyle{(-)}}\left(r_{2}E_{2}+\bar{a}\right)\int\limits_{0}^{\beta}d\tau_{2}\,e^{-\tau_{2}\left[i(\omega_{m}-\widetilde{\omega}_{l})-\mu-\bar{a}-r_{2}E_{2}\right]}.$
(83)

where $\beta\equiv T^{-1}$,
$\tilde{n}^{\scriptscriptstyle{\pm}}(E)\equiv\left[e^{\beta(E\mp\mu)}+1\right]^{-1}$.
Thus, substituting Eq. (83) in Eq. (82), the sum is written as

$\displaystyle\mathcal{F}$ $\displaystyle=\sum_{r_{1},r_{2}=\pm
1}\frac{r_{1}r_{2}}{4E_{1}E_{2}}\,\tilde{n}^{\scriptscriptstyle{(+)}}\left(r_{1}E_{1}-a\right)\tilde{n}^{\scriptscriptstyle{(-)}}\left(r_{2}E_{2}+\bar{a}\right)\int\limits_{0}^{\beta}d\tau_{1}\,d\tau_{2}e^{-i\tau_{2}\omega_{m}}e^{-\mu(\tau_{1}-\tau_{2})}e^{-\tau_{1}a+\tau_{2}\bar{a}}$
$\displaystyle\hskip 227.62204pt\times
e^{\tau_{1}r_{1}E_{1}+\tau_{2}r_{2}E_{2}}\,T\\!\\!\\!\sum_{l=-\infty}^{\infty}e^{-i\widetilde{\omega}_{l}(\tau_{1}-\tau_{2})},$
(84)

Next, I employ the identity

$\displaystyle
T\\!\\!\\!\sum_{l=-\infty}^{\infty}e^{-i\widetilde{\omega}_{l}(\tau_{1}-\tau_{2})}=\delta(\tau_{1}-\tau_{2})$
(85)

and perform the integration over delta function to get

$\displaystyle\mathcal{F}=\sum_{r_{1},r_{2}=\pm
1}\frac{r_{1}r_{2}}{4E_{1}E_{2}}\,\tilde{n}^{\scriptscriptstyle{(+)}}\left(r_{1}E_{1}-a\right)\tilde{n}^{\scriptscriptstyle{(-)}}\left(r_{2}E_{2}+\bar{a}\right)\int\limits_{0}^{\beta}d\tau_{1}e^{-\tau(i\omega_{m}+a-\bar{a}-r_{1}E_{1}-r_{2}E_{2})}.$
(86)

Performing the $\tau_{1}$ integral and using $e^{i\beta\omega_{m}}=1$, I get
after simplifying

$\displaystyle\mathcal{F}$ $\displaystyle=\sum_{r_{1},r_{2}=\pm
1}-\frac{r_{1}r_{2}}{4E_{1}E_{2}}\,\tilde{n}^{\scriptscriptstyle{(+)}}\left(r_{1}E_{1}-a\right)\tilde{n}^{\scriptscriptstyle{(-)}}\left(r_{2}E_{2}+\bar{a}\right)\frac{e^{\beta(r_{1}E_{1}-a-\mu)}e^{\beta(r_{2}E_{2}+\bar{a}+\mu)}-1}{i\omega_{m}-r_{1}E_{1}-r_{2}E_{2}+a-\bar{a}},$
$\displaystyle=\sum_{r_{1},r_{2}=\pm
1}-\frac{r_{1}r_{2}}{4E_{1}E_{2}}\frac{1-\tilde{n}^{\scriptscriptstyle{(+)}}\left(r_{1}E_{1}-a\right)-\tilde{n}^{\scriptscriptstyle{(-)}}\left(r_{2}E_{2}+\bar{a}\right)}{i\omega_{m}-r_{1}E_{1}-r_{2}E_{2}+a-\bar{a}}.$
(87)

At last, continuing analytically to the real value of energy,
$i\omega_{m}\rightarrow p_{0}+i\varepsilon$, I get

$\displaystyle\mathcal{F}$ $\displaystyle=\sum_{r_{1},r_{2}=\pm
1}-\frac{r_{1}r_{2}}{4E_{1}E_{2}}\frac{1-\tilde{n}^{\scriptscriptstyle{(+)}}\left(r_{1}E_{1}-a\right)-\tilde{n}^{\scriptscriptstyle{(-)}}\left(r_{2}E_{2}+\bar{a}\right)}{p_{0}-r_{1}E_{1}-r_{2}E_{2}+a-\bar{a}+i\varepsilon}.$
(88)

##  Appendix B Kinematics at lowest Landau levels

Here I investigate kinematics of various processes in LLL. I start with the
expression of $k_{z}^{(\pm)}$ and quote it again

$\displaystyle
k_{z}^{(\pm)}=\frac{1}{2}\left(p_{z}\pm\frac{\sqrt{p_{0}^{2}\,(p_{0}^{2}-p_{z}^{2})(p_{0}^{2}-p_{z}^{2}-4m_{q}^{2})}}{p_{0}^{2}-p_{z}^{2}}\right).$
(89)

  1. 1.

For the production of di-leptons, one must have

$p_{0}>0,$ (90)

giving

$\displaystyle
k_{z}^{(\pm)}=\frac{1}{2}\left(p_{z}\pm\frac{p_{0}}{p_{0}^{2}-p_{z}^{2}}\sqrt{(p_{0}^{2}-p_{z}^{2})(p_{0}^{2}-p_{z}^{2}-4m_{q}^{2})}\right),$
(91) $\displaystyle
k_{z}^{(\pm)}-p_{z}=-\frac{1}{2}\left(p_{z}\mp\frac{p_{0}}{p_{0}^{2}-p_{z}^{2}}\sqrt{(p_{0}^{2}-p_{z}^{2})(p_{0}^{2}-p_{z}^{2}-4m_{q}^{2})}\right)=-k_{z}^{(\mp)},$
(92)
$\displaystyle\epsilon_{k_{z}^{(\pm)}}=\frac{1}{2}\left(p_{0}\pm\frac{p_{z}}{p_{0}^{2}-p_{z}^{2}}\sqrt{(p_{0}^{2}-p_{z}^{2})(p_{0}^{2}-p_{z}^{2}-4m_{q}^{2})}\right),$
(93)
$\displaystyle\epsilon_{k_{z}^{(\pm)}-p_{z}}=\frac{1}{2}\left(p_{0}\mp\frac{p_{z}}{p_{0}^{2}-p_{z}^{2}}\sqrt{(p_{0}^{2}-p_{z}^{2})(p_{0}^{2}-p_{z}^{2}-4m_{q}^{2})}\right).$
(94)

  2. 2.

Now, for the energy value to be real,

$\displaystyle\left[\left(p_{0}^{2}-p_{z}^{2}\geqslant
0\right)\land\left(p_{0}^{2}-p_{z}^{2}\geqslant
4m_{q}^{2}\right)\right]\lor\left[\left(p_{0}^{2}-p_{z}^{2}\leqslant
0\right)\land\left(p_{0}^{2}-p_{z}^{2}\leqslant 4m_{q}^{2}\right)\right].$
(95)

    1. (I)

Now, if the condition in the left side of or is satisfied, the and forces one
to choose $p_{0}^{2}-p_{z}^{2}\geqslant 4m_{q}^{2}$. Hence, $p_{0}>0$ implies

$\displaystyle
p_{0}\geqslant\sqrt{p_{z}^{2}+4m_{q}^{2}}\quad\quad\text{Kinematic Region-I}$
(96)

    2. (II)

Similarly, if the condition in the right side of or is satisfied, the and
forces one to choose $p_{0}^{2}-p_{z}^{2}\leqslant 0$. This will imply
$(p_{0}\geqslant-p_{z})\land(p_{0}\leqslant p_{z})$ for $p_{z}>0$ case. For
$p_{z}<0$, I have $(p_{0}\leqslant-p_{z})\land(p_{0}\geqslant p_{z})$. It is
because for both sign of $p_{z}$ the other condition cannot be met
simultaneously. Combining both conditions with $p_{0}>0$ gives

$\displaystyle 0\leqslant p_{0}<|p_{z}|\quad\quad\quad\text{Kinematic Region-
II}$ (97)

    3. (III)

Lastly, $p_{0}\geqslant\sqrt{p_{z}^{2}+4m_{q}^{2}}$ implies

$\displaystyle\sqrt{1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}}\leqslant 1,$
(98) $\displaystyle p_{0}>|p_{z}|.$ (99)

From the conditions in Eq. (98) and ref. (99), one can write

$\displaystyle
p_{0}>|p_{z}|\geqslant|p_{z}|\sqrt{1-\frac{4m_{q}^{2}}{p_{0}^{2}-p_{z}^{2}}}\;.$
(100)

In the region $p_{0}\geqslant\sqrt{p_{z}^{2}+4m_{q}^{2}}$, it implies that
$\epsilon_{H}>0$ and $\epsilon_{L}>0$.

## References

  * [1] V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod. Phys. A 24, 5925-5932 (2009), [arXiv:0907.1396 [nucl-th]].
  * [2] K. Tuchin, Phys. Rev. C 87, no.2, 024912 (2013), [arXiv:1206.0485 [hep-ph]].
  * [3] L. McLerran and V. Skokov, Nucl. Phys. A 929, 184-190 (2014), [arXiv:1305.0774 [hep-ph]].
  * [4] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 78, 074033 (2008), [arXiv:0808.3382 [hep-ph]].
  * [5] D. E. Kharzeev, Prog. Part. Nucl. Phys. 75, 133-151 (2014) [arXiv:1312.3348 [hep-ph]].
  * [6] D. E. Kharzeev, J. Liao, S. A. Voloshin and G. Wang, Prog. Part. Nucl. Phys. 88, 1-28 (2016) [arXiv:1511.04050 [hep-ph]].
  * [7] D. E. Kharzeev and H. U. Yee, Phys. Rev. D 83, 085007 (2011), [arXiv:1012.6026 [hep-th]].
  * [8] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Phys. Rev. Lett. 73, 3499-3502 (1994), [erratum: Phys. Rev. Lett. 76, 1005 (1996)], [arXiv:hep-ph/9405262 [hep-ph]].
  * [9] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B 462, 249-290 (1996), [arXiv:hep-ph/9509320 [hep-ph]].
  * [10] I. A. Shovkovy, Lect. Notes Phys. 871, 13-49 (2013), [arXiv:1207.5081 [hep-ph]].
  * [11] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz and A. Schafer, Phys. Rev. D 86, 071502 (2012), [arXiv:1206.4205 [hep-lat]].
  * [12] F. Bruckmann, G. Endrodi and T. G. Kovacs, JHEP 04, 112 (2013), [arXiv:1303.3972 [hep-lat]].
  * [13] J. O. Andersen, W. R. Naylor and A. Tranberg, Rev. Mod. Phys. 88, 025001 (2016), [arXiv:1411.7176 [hep-ph]].
  * [14] M. N. Chernodub, Phys. Rev. D 82, 085011 (2010), [arXiv:1008.1055 [hep-ph]].
  * [15] M. N. Chernodub, Phys. Rev. Lett. 106, 142003 (2011), [arXiv:1101.0117 [hep-ph]].
  * [16] F. Becattini, G. Inghirami, V. Rolando, A. Beraudo, L. Del Zanna, A. De Pace, M. Nardi, G. Pagliara and V. Chandra, Eur. Phys. J. C 75, no.9, 406 (2015), [erratum: Eur. Phys. J. C 78, no.5, 354 (2018)], [arXiv:1501.04468 [nucl-th]].
  * [17] F. Becattini, F. Piccinini and J. Rizzo, Phys. Rev. C 77, 024906 (2008), [arXiv:0711.1253 [nucl-th]].
  * [18] F. Becattini, V. Chandra, L. Del Zanna and E. Grossi, Annals Phys. 338, 32-49 (2013), [arXiv:1303.3431 [nucl-th]].
  * [19] L. Adamczyk et al. [STAR], Nature 548, 62-65 (2017), [arXiv:1701.06657 [nucl-ex]].
  * [20] Z. T. Liang and X. N. Wang, Phys. Lett. B 629, 20-26 (2005), [arXiv:nucl-th/0411101 [nucl-th]].
  * [21] M. N. Chernodub and S. Gongyo, JHEP 01, 136 (2017), [arXiv:1611.02598 [hep-th]].
  * [22] Y. Jiang and J. Liao, Phys. Rev. Lett. 117, no.19, 192302 (2016), [arXiv:1606.03808 [hep-ph]].
  * [23] M. N. Chernodub and S. Gongyo, Phys. Rev. D 95, no.9, 096006 (2017), [arXiv:1702.08266 [hep-th]].
  * [24] X. Wang, M. Wei, Z. Li and M. Huang, Phys. Rev. D 99, no.1, 016018 (2019) [arXiv:1808.01931 [hep-ph]].
  * [25] Z. Zhang, C. Shi, X. T. He, X. Luo and H. S. Zong, Phys. Rev. D 102, no.11, 114023 (2020), [arXiv:2012.01017 [hep-ph]].
  * [26] A. Yamamoto and Y. Hirono, Phys. Rev. Lett. 111, 081601 (2013), [arXiv:1303.6292 [hep-lat]].
  * [27] V. V. Braguta, A. Y. Kotov, D. D. Kuznedelev and A. A. Roenko, Pisma Zh. Eksp. Teor. Fiz. 112, no.1, 9-16 (2020)
  * [28] V. V. Braguta, A. Y. Kotov, D. D. Kuznedelev and A. A. Roenko, Phys. Rev. D 103, no.9, 094515 (2021), [arXiv:2102.05084 [hep-lat]].
  * [29] J. C. Yang and X. G. Huang, [arXiv:2307.05755 [hep-lat]].
  * [30] V. V. Braguta, I. E. Kudrov, A. A. Roenko, D. A. Sychev and M. N. Chernodub, JETP Lett. 117, no.9, 639-644 (2023),
  * [31] S. Satapathy, R. Singh, P. Panday, S. A. Khan and D. Dey, [arXiv:2309.05284 [hep-ph]].
  * [32] S. Satapathy, [arXiv:2307.09953 [nucl-th]].
  * [33] H. L. Chen, K. Fukushima, X. G. Huang and K. Mameda, Phys. Rev. D 93, no.10, 104052 (2016), [arXiv:1512.08974 [hep-ph]].
  * [34] N. Sadooghi, S. M. A. Tabatabaee Mehr and F. Taghinavaz, Phys. Rev. D 104, no.11, 116022 (2021), [arXiv:2108.12760 [hep-ph]].
  * [35] S. M. A. T. Mehr and F. Taghinavaz, Annals Phys. 454, 169357 (2023), [arXiv:2201.05398 [hep-ph]].
  * [36] N. Sadooghi and F. Taghinavaz, Annals Phys. 376, 218-253 (2017), [arXiv:1601.04887 [hep-ph]].
  * [37] A. Bandyopadhyay, C. A. Islam and M. G. Mustafa, Phys. Rev. D 94, no.11, 114034 (2016) [arXiv:1602.06769 [hep-ph]].
  * [38] A. Bandyopadhyay and S. Mallik, Phys. Rev. D 95, no.7, 074019 (2017), [arXiv:1704.01364 [hep-ph]].
  * [39] S. Ghosh and V. Chandra, Phys. Rev. D 98, no.7, 076006 (2018), [arXiv:1808.05176 [hep-ph]].
  * [40] A. Das, A. Bandyopadhyay and C. A. Islam, Phys. Rev. D 106, no.5, 056021 (2022), [arXiv:2109.00019 [hep-ph]].
  * [41] X. Wang and I. A. Shovkovy, Phys. Rev. D 106, no.3, 036014 (2022), [arXiv:2205.00276 [nucl-th]].
  * [42] K. Hattori, H. Taya and S. Yoshida, JHEP 01, 093 (2021), [arXiv:2010.13492 [hep-ph]].
  * [43] X. Wang, I. A. Shovkovy, L. Yu and M. Huang, Phys. Rev. D 102, no.7, 076010 (2020), [arXiv:2006.16254 [hep-ph]].
  * [44] X. Wang and I. Shovkovy, Phys. Rev. D 104, no.5, 056017 (2021), [arXiv:2103.01967 [nucl-th]].
  * [45] X. Wang and I. A. Shovkovy, [arXiv:2307.07557 [hep-ph]].
  * [46] M. Wei, C. A. Islam and M. Huang, Phys. Rev. D 105, no.5, 054014 (2022), [arXiv:2111.05192 [hep-ph]].
  * [47] M. Buzzegoli and K. Tuchin, [arXiv:2308.10349 [hep-ph]].
  * [48] M. Buzzegoli, J. D. Kroth, K. Tuchin and N. Vijayakumar, [arXiv:2306.03863 [hep-ph]].
  * [49] K. Hattori and K. Itakura, Annals Phys. 446, 169114 (2022), [arXiv:2205.04312 [hep-ph]].
  * [50] K. Hattori and K. Itakura, Annals Phys. 446, 169115 (2022), [arXiv:2205.06411 [hep-ph]].
  * [51] K. Hattori, K. Itakura and S. Ozaki, Prog. Part. Nucl. Phys. 133, 104068 (2023), [arXiv:2305.03865 [hep-ph]].
  * [52] K. Fukushima, Prog. Part. Nucl. Phys. 107, 167-199 (2019), [arXiv:1812.08886 [hep-ph]].
  * [53] A. Ayala, L. A. Hernández, K. Raya and R. Zamora, Phys. Rev. D 103, no.7, 076021 (2021), [erratum: Phys. Rev. D 104, no.3, 039901 (2021)], [arXiv:2102.03476 [hep-ph]].
  * [54] T. K. Chyi, C. W. Hwang, W. F. Kao, G. L. Lin, K. W. Ng and J. J. Tseng, Phys. Rev. D 62, 105014 (2000), [arXiv:hep-th/9912134 [hep-th]].
  * [55] M. G. Mustafa, Eur. Phys. J. ST 232, no.9, 1369-1457 (2023), [arXiv:2207.00534 [hep-ph]].
  * [56] H. A. Weldon, Phys. Rev. D 28, 2007 (1983)
  * [57] Y. Jiang, Z. W. Lin and J. Liao, Phys. Rev. C 94, no.4, 044910 (2016), [erratum: Phys. Rev. C 95, no.4, 049904 (2017)], [arXiv:1602.06580 [hep-ph]].
  * [58] W. T. Deng and X. G. Huang, Phys. Rev. C 93, no.6, 064907 (2016), [arXiv:1603.06117 [nucl-th]].
  * [59] H. A. Weldon, Phys. Rev. D 42, 2384-2387 (1990)
  * [60] R. C. Tolman, Phys. Rev. 35, 904-924 (1930)