# Gamma-ray burst Jet Progenitor:
long vs short bursts by angular momentum and mass ratio

She-Sheng Xue

###### Abstract

In this study, we investigate the homologous collapse dynamics of a slowly
rotating stellar nuclear core. We incorporate the virial theorem of hadron
collisional relaxations to analyze the conversion of gravitational potential
energy into kinetic energy of hadrons. Additionally, we consider the
production of photons and electron-positron pairs resulting from hadron
collisions. Our analysis reveals a gravo-thermal process wherein gravitational
energy transforms into energy associated with photons and pairs. The presence
of axial symmetric centrifugal potential reduces the gravitational potential
energy gain. The presence of axial symmetric centrifugal potential mitigates
the gain in gravitational potential energy. Consequently, we observe the
formation of highly energetic and opaque photon-pair jets, whose size, peak
spectra, energy and number densities are consistent with observed energetic
features and empirical relationships of Gamma-Ray Burst (GRB) progenitors.
Furthermore, we find that the angular momentum and mass ratio ($J/M$) of the
collapsing core and binary coalescence play crucial roles in determining the
jet angle and the differentiation between short and long bursts. Our findings
contribute to a deeper understanding of the mechanisms governing GRBs’
progenitor systems.

##  1 Introduction

Gamma-ray bursts (GRBs) are the most energetic and complex events in the
Universe. Within mere seconds, their progenitors release high-energy photons
of tremendous energy ranging from $10^{49}$ to $10^{54}$ ergs. Despite
significant advancements in our comprehension of GRBs, achieved through
extensive research efforts [1, 2, 3, 4, 5, 6, 7], a fundamental question
remains unanswered: what are the dominant dynamics of GRB progenitors that
ultimately yield the formation of an energetic and optically thick plasma
composed of photons and electron-positron pairs, i.e., a fireball or jet ?
Massive stellar gravitational potentials are indeed vast reservoirs of energy,
and during stellar collapse and coalescence, significant amounts of
gravitational energy are released. The critical question is how this immense
gravitational energy is converted into photon and pair energies within just a
few seconds during GRBs.

In a self-gravitating system, Lynden-Bell and Wood [8, 9] pioneered the study
of gravo-thermal catastrophe using the framework of violent hadron
collisionless relaxation and the virial theorem. Their work demonstrated the
relaxation process in which hadrons gain their mean kinetic heat energy from
gravitational potential energy.

On the other hand, the production of photons by hadron collisions has been
extensively studied in the context of heavy-ion collisions [10, 11, 12, 13,
14, 15]. Based on the homologous core collapse scenario [16], we show that
hadrons gain gravitational energy through collisional relaxation. Hadron-
hadron collisions subsequently produce photons, leading to the conversion of
gravitational energy into the energies of opaque photon-pair spheres. This
process accounts for the origins of GRBs’ energies and their empirical
correlations [17].

Furthermore, we investigate the formation of photon-pair jets in scenarios
involving rotating core collapse and binary coalescence. We find that the
angular momenta and mass ratios of these systems determine the jet angle and
the distinct characteristics of long versus short GRBs.

We organize this article as follows. Based on the adiabatic assumption,
Sections 2, 3, and 4 present discussions on gravo-thermal catastrophe, the
virial theorem, and photon production from hadron relaxations and collisions.
In Section 5, we discuss the homologous collapse of a slowly rotating core.
Sections 6 and 7 study the properties and correlations of photon-pair spheres
and jets. The characteristics of long versus short GRBs, as determined by
progenitor angular momenta and mass ratios, are described in Sections 7 and 8.

The stellar core density is characterised by the nuclear saturation density
$n_{0}\approx 0.16\,{\rm fm}^{-3}=1.6\times 10^{38}/{\rm cm}^{3}$ or
$\rho_{0}=mn_{0}\approx 2.4\times 10^{35}{\rm ergs}/{\rm cm}^{3}$, and
$m\approx 1$ GeV is the typical baryon (neutron) mass. The light speed $c=1$,
Planck constant $\hbar=1$ and Boltzmann constant $k=1$ are used unless
otherwise specified.

##  2 Adiabatic approximation in gravitational collapse

Gravitational collapse is a dynamic avalanche involving complex microscopic
and macroscopic processes occurring at vastly different scales. It is
challenging for both analytical and numerical approaches to analyze all these
dynamics without any approximations. To make appropriate approximations, we
need to understand the dominant physical processes and their variations across
the relevant length and time scales.

We consider a rotating stellar core of mass $M$, radius $R$, and angular
velocity $\Omega$ undergoing a gravitational collapse process. The most
relevant timescales $\tau_{\rm grav}$ are collapsing $|R|/|\dot{R}|$, rotating
$1/\Omega$ and hydrodynamics timescale $\sim R/v_{s}$, where $v_{s}$ is the
sound velocity. These macroscopic scales are much larger than the microscopic
scales of baryon collisional relaxation, photon production, and
thermalization.

This disparity in scales implies that:

  * •

these microscopic processes occur not only “instantaneously” but also
“locally” compared with gravitational collapsing and hydrodynamic processes.

  * •

There is no causal correlation among the concurrent microscopic processes at
different space-time points on macroscopic scales.

In other words, the vast difference between microscopic and macroscopic
timescales suggests that gravitational collapse can be considered a very slow,
adiabatic process compared to these rapid and local microscopic processes.
Therefore, these microscopic processes can be approximately analyzed as if the
self-gravitating cores were static.

Based on this adiabatic approximation, we discuss isothermal configurations
during gravitational collapses.

##  3 Virial theorem and gravo-thermal configuration

We consider a stellar core composed of $N$ baryons of mass $m$111By the term
baryons, we indicate nucleons, nuclei, hadrons, quark-gluon plasma that carry
baryon numbers.. In gravitational collapse, baryons gain their mean kinetic
energy $F$ from the gravitational energy $U$ via collisionless relaxation.
This gravo-thermal catastrophe phenomenon of stellar cores is studied by
considering violent relaxation [8] and equipartition theorem [9]. In these
studies, the stellar core is an isothermal core of baryon temperature $T\ll m$
and internal “heat” energy $F=\frac{3}{2}TN$. It is an equilibrium or
equipartition system, where the Clausius virial theorem applies,

$\displaystyle 2F+U+U_{c}=3PV,$ (3.1)

where $V$ is the core volume, and $P$ is the external pressure acting on the
core surface. Here, $U_{c}\ll U$ represents the centrifugal potential energy
for a slowly rotating core, which we will further justify.

These studies indicate that the baryon collisionless relaxation process in
gravitational potential and under external pressure leads to two key
consequences:

  * •

The gravitational energy is converted to the internal heat energy of baryons.

  * •

The balance between gravitational dynamics and thermodynamics is established,
making the virial theorem (3.1) applicable.

The second point holds, provided the time scales of microscopic relaxation
processes are much smaller than those of macroscopic gravitational and
hydrodynamical processes, justifying adiabatic approximation. Following the
discussions in the pioneer works [18, 19], it is noted that the gravitational
evolution time scale due to stellar evaporation and consumption is longer than
the core relaxation time scale, the core will maintain an approximate
isothermal core profile (3.1) and evolve homologously.

We generalise these discussions to a small fluid element of volume $dV$ inside
the stellar core. Given that the baryon-baryon collision timescale at nuclear
density is small, equipartition or equilibrium can be locally achieved through
the relaxation processes of baryon-baryon collisions. The local virial theorem
(3.1) is given by [17]

$\displaystyle 2dF+dU+dU_{c}=3pdV,$ (3.2)

and $dF=(3/2)T(\rho/m)dV$, $dU=(1/2)\rho\phi dV$ and $dU_{c}=\phi_{c}\rho dV$.
Here baryon mass density $\rho=mn$ and number density $n$. The terms $\phi$
and $\phi_{c}$ are gravitational and centrifugal potential in a rotating
frame, respectively. For each fluid element of mass energy $\rho dV$, the
baryon temperature $T$ characterises the mean kinetic energy $dF$ of baryon
motions and collisions, $dF\ll\rho dV$ for $T\ll m$. The fluid element’s
rotating energy is much smaller than its gravitational energy, i.e.,
$dU_{c}\ll|dU|$. For an isothermal core, integrating Eq. (3.2) over the core
volume $\int dV$, one obtains the virial theorem (3.1), using $\int pdV=PV$.
This shows that the local equilibrium properties extend to the entire core
under the adiabatic approximation.

Moreover, we approximately describe the internal pressure $p$ by the
polytropic equation of state (EoS)

$\displaystyle p=\kappa\rho^{\gamma},$ (3.3)

with two parameters: mean thermal index $\gamma$ and coefficient $\kappa$. As
a result, combining the local virial theorem (3.2) and the EoS (3.3), we
obtain the baryon temperature:

$\displaystyle 3T/m$ $\displaystyle\approx$ $\displaystyle
3v_{s}^{2}(1/\gamma)-(1/2)\phi-\phi_{c},$ (3.4)

where $v_{s}$ is the sound velocity given by,

$\displaystyle v_{s}^{2}$ $\displaystyle=$ $\displaystyle\partial
p/\partial\rho=\gamma p/\rho.$ (3.5)

This relationship shows that via collisional relaxations, baryons gain heat
energy (temperature) from the gravitational potential, which is effectively
reduced by the centrifugal potential.

##  4 Photon production from hadron collisions

The baryon-baryon collisions produce photons and pairs of light-charged
leptons and quarks. Consequently, the mean kinetic heat energy of baryons is
converted to the energy of photons, and pairs of other light-charged leptons
and quarks. From the studies of heavy-ion collisions [10, 11, 12, 13, 14, 15,
11, 12], we learn about photon production in these collisions and can obtain
the photon number and energy densities as follow [17],

$\displaystyle n_{\gamma}$ $\displaystyle\approx$
$\displaystyle\frac{4}{3}\frac{\alpha\alpha_{s}}{\pi}T^{3}\ln\left(1+\frac{2.9}{4\pi\alpha_{s}}\right),$
(4.1) $\displaystyle\rho_{\gamma}$ $\displaystyle\approx$
$\displaystyle\frac{8}{3}\frac{\alpha\alpha_{s}}{\pi}T^{4}\ln\left(1+\frac{2.9}{4\pi\alpha_{s}}\right),$
(4.2)

where $\alpha=1/137$. The QCD coupling $\alpha_{s}\approx 0.5$ indicates that
the photon production described by equation (4.2) is dominant over other QED
processes producing photons.

The gravo-thermal catastrophe produces not only heat energy for massive
baryons, given by $(3/2)T(\rho/m)dV$ (as in equation (3.2)), but also energy
for relativistic particles, described by $\rho_{\gamma}dV$ (as in equation
(4.2)). For a baryon temperature $T$ on the order of ${\mathcal{O}}(10^{2})$
MeV, the photon density $n_{\gamma}$ (as in equation (4.2)) is so large that
relativistic particles become opaque and thermalized by collisions among
themselves. The mean temperature $T_{\gamma}$ of thermalized photons can be
estimated by equating photon thermal energy density to that given in equation
(4.2)

$\displaystyle\frac{\pi^{2}}{15}T_{\gamma}^{4}\approx\rho_{\gamma},\quad\Rightarrow\quad
T_{\gamma}\approx 0.21T.$ (4.3)

Moreover, these photons couple to pairs of charged leptons (quarks) and anti-
leptons (-quarks). Among these charged particle pairs, we consider pairs of
electrons and positrons since their masses are smaller than the photon
characteristic energy given in equation (4.3). Therefore, the fourth and back
processes involving the production and annihilation of electron-positron
pairs, $\gamma+\gamma\leftrightarrow e^{+}+e^{-}$, are kinematically possible.
The cross-section $\sigma_{\gamma\gamma\leftrightarrow e^{+}e^{-}}$ is of the
order of the Thomson cross section $\sigma_{\gamma}$, and the corresponding
mean-free length is given by $\sim(\sigma_{\gamma\gamma\leftrightarrow
e^{+}e^{-}}n_{\gamma})^{-1}$, which is very small. This indicates that via
processes $\gamma+\gamma\leftrightarrow e^{+}+e^{-}$, photons and electron-
positron pairs participate in thermalisation or thermal equipartition in
particle energy and number. As a result, these processes lead to an
electrically neutral, deeply opaque, and thermalised plasma (fluid) of photons
and electron-positron pairs.

As a consequence, the isothermal core profile is approximately described by
baryon temperature $T$ (equation (3.4)) and photon temperature $T_{\gamma}$
(equation (4.3)), as well as the photon production number and energy densities
(equations (4.1) and (4.2)). The core will maintain this isothermal profile
and evolve homologously.

##  5 Gravitational collapse of rotating stellar cores

Observations of neutron stars, supernovas, and GRBs suggest that rotation is
crucial in the core collapse scenario. However, there is still a lack of
realistic models for the final evolutionary state of a massive collapsing star
that consistently includes rotation. We proceed along the lines of previous
Newtonian and rotating core collapse studies, which construct a set of
differentially rotating polytropes in equilibrium as pre-collapse iron core
models [20, 21].

We consider a slowly rotating stellar core undergoing a gravitational collapse
process, whose rotation energy is much smaller than gravitational energy. In
the Newtonian approximation, we describe the process by the continuity
equation for the baryon number conservation, Euler’s equation for energy-
momentum conservation, Poisson’s equation for gravitational potential $\phi$
are,

$\displaystyle\frac{\partial\rho}{\partial t}+{\bf\nabla}\cdot(\rho\bf{u})$
$\displaystyle=$ $\displaystyle 0,$ (5.1)
$\displaystyle\Big{(}\frac{\partial{\bf u}}{\partial
t}\Big{)}+{\bf\nabla}\Big{(}\frac{1}{2}|{\bf u}|^{2}\Big{)}+(\nabla\times{\bf
u})\times{\bf u}+\nabla h+{\bf\nabla}\phi+{\bf\nabla}\phi_{c}$
$\displaystyle=$ $\displaystyle 0,$ (5.2) $\displaystyle\nabla^{2}\phi-4\pi
G\rho$ $\displaystyle=$ $\displaystyle 0,$ (5.3)

where baryon fluid is described by density $\rho$ and velocity $\bf{u}$.
Equation of State (3.3) yields the baryon heat function $h=H/\rho$ ($H$ for
Enthalpy)

$\displaystyle\nabla h=\nabla p/\rho;\quad h=\int\nabla
p/\rho=\frac{\kappa\gamma}{\gamma-1}\rho^{\gamma-1}.$ (5.4)

We formally add the gradient of centrifugal potential $\phi_{c}$ given by
${\bf\nabla}\phi_{c}=j{\bf\nabla}\Omega$, where $j$ and $\Omega$ are angular
momentum and velocity respectively. These equations reduce to their
counterparts in the hydrodynamic equilibrium case, describing the core’s state
when the dynamic effects of collapse are not present.

###  5.1 Static equilibrium configuration

To develop a simple analytical model for a rigidly rotating stellar core
undergoing gravitational homologous collapse, we start with the established
equilibrium conditions and rotation laws. In hydrodynamic equilibrium, the
integrability condition of Euler’s equation (5.2) necessitates that the
angular momentum $j$ depends only on angular velocity $\Omega$. We adopt the
rotation law from Refs. [22, 23, 24],

$\displaystyle j=A^{2}(\Omega_{c}-\Omega),$ (5.5)

where $\Omega_{c}$ is the value at the centre of the coordinate. The rotation
parameter $A$ is a positive constant (length scale), which implicitly depends
on $\Omega_{c}$.

In the Newtonian approximation, the rotation law simplifies to

$\displaystyle\Omega$ $\displaystyle\approx$
$\displaystyle\Omega_{c}/(1+x^{2}/A^{2}),\quad j\approx x^{2}\Omega,$
$\displaystyle\phi_{c}$ $\displaystyle=$ $\displaystyle\int jd\Omega\approx
A^{2}x^{2}\Omega/2+{\rm const},$ (5.6)

where $x=R\sin\theta$ and $z=R\cos\theta$ are the cylindrical coordinates, $R$
is the radial coordinate, and $\theta$ is the zenith (polar) angle. The
rotating axis is along the $\hat{z}$ direction.

We focus on the limit of constant angular velocity, namely a rigid rotating
core. In this limit,

$\displaystyle\Omega\rightarrow\Omega_{c},\quad x^{2}/A^{2}\ll 1,$ (5.7)

and up to the leading order $\mathcal{O}(x^{2}/A^{2})$,

$\displaystyle j\approx
x^{2}\Omega_{c},\quad\phi_{c}\approx\frac{A^{2}x^{2}\Omega_{c}}{2}.$ (5.8)

We set the centrifugal potential to zero at the rotating axis $x=0$, thus
ensuring that $\phi_{c}=0$ at $x=0$.

In the static equilibrium case, integrating Euler’s equation (5.2) yields

$h+\phi+\phi_{c}={\rm constant},$ (5.9)

which is uniform across the space.

To generalize this to a model of a rigidly and slowly rotating stellar core
undergoing homologous collapse, we need to consider the density, velocity, and
potential changes during the collapse. Given the homologous nature of the
collapse, the density profile remains similar while scaling with time.

###  5.2 Homologous collapse of rigid rotating core

To analyze the homologous collapse of a nearly spherically symmetric, rigidly
and slowly rotating stellar baryon core, we follow the approach of Goldreich
and Weber [16] in the rotating frame. We assume the baryon fluid flow is
vorticity-free, meaning $(\nabla\times\mathbf{u})=0$. The velocity ${\bf
u}=\nabla v$ can thus be expressed as a gradient of a scalar function $v$:

$\displaystyle{\bf u}=\nabla v,\quad v=(1/2)(\dot{a}/a)R^{2},\quad{\bf
u}=(\dot{a}/a){\bf R}=\dot{a}{\bf r}.$ (5.10)

Here, $a(t)$ is a length scale function, and ${\bf R}$ is the radial vector in
the physical coordinates. The core density $\rho$ and sound velocity $v_{s}$
(3.5) in the core can be expressed in terms of a homologous function $f(r)$,

$\rho=\rho_{c}f^{3},\quad v_{s}^{2}=\gamma
p/\rho=(v_{s}^{c})^{2}f^{3(\gamma-1)},$ (5.11)

where $r$ is the dimensionless radius

$r=R/a(t),\quad{\bf r}={\bf R}/a(t).$ (5.12)

Here $\rho_{c}$ is the central density, and $v_{s}^{c}$ is the central sound
velocity, given by $(v_{s}^{c})^{2}\equiv\gamma p_{c}/\rho_{c}$. The length
scale function $a(t)$ relates to the central density and sound velocity by:

$a(t)=v_{s}^{c}/(\gamma\pi G\rho_{c})^{1/2}.$ (5.13)

Given the functional forms of $\rho$ and ${\bf u}$, the continuity equation
(5.1) reduces to the trivial relation $\dot{f}=0$, indicating that the
homologous profile $f(r)$ is time-independent. The integrated Euler equation
(5.2) yields

$\frac{\partial\upsilon}{\partial
t}+\frac{1}{2}|{\bf\nabla}\upsilon|^{2}+h+\phi+\phi_{c}={\rm constant},$
(5.14)

which reduces to the equation (5.9) in the static equilibrium case $\dot{a}=0$
and $v=0$.

We are now in the position of postulating the rotation law for a rigidly and
slowly rotating stellar core, which undergoes homologous gravitational
collapse. Since the stellar rotating scale $\Omega^{-1}$ and collapsing
timescales $\tau_{\rm grav}=R/\dot{R}$ are much larger than microscopic
interaction and thermalization timescales, we approximately consider the
gravitational collapse of rotating stellar core is an adiabatic process.
Namely, at each collapse step at time $t$, the stellar core is approximately
in an equilibrium state as if the collapsing stellar core adiabatically
evolves in a time sequence of equilibrium states during gravitational
collapse. The rigid core angular velocity $\Omega(t)$ increases in time.
Therefore, we generalize the equilibrium rotation law (5.6) to

$j(t)=r^{2}\sin^{2}\theta
a^{2}(t)\Omega(t);\quad\phi_{c}(t)=\frac{A^{2}(t)}{2}r^{2}\sin^{2}\theta
a^{2}(t)\Omega(t),$ (5.15)

where $A(t)$ is the time-dependent rotation parameter.

To further describe the centrifugal potential $\phi_{c}$ of a slowly and
rigidly rotating, homologously collapsing stellar core, we assume a time-
dependent rotation parameter $A(t)$ in (5.15) depends on the centre sound
velocity

$\displaystyle A^{2}(t)={\mathcal{A}}^{2}[v^{c}_{s}(t)]^{2},$ (5.16)

and ${\mathcal{A}}$ is a constant that does not depend on time. Thus, the
centrifugal potential of a slowly and rigidly rotating and homologous
collapsing core can be expressed by

$\displaystyle\phi_{c}=\frac{[v^{c}_{s}(t)]^{2}}{2}r^{2}\sin^{2}\theta{\mathcal{A}}^{2}a^{2}\Omega,\quad{\mathcal{A}}^{2}a^{2}\Omega={\mathcal{A}}^{2}a^{2}(t)\Omega(t),$
(5.17)

in the rotating frame. We will show soon that
${\mathcal{A}}a^{2}\Omega={\mathcal{A}}a^{2}(t)\Omega(t)$ is time-independent,
based on total core mass and angular momentum conservation.

By integrating these equations, the evolution of the collapsing core can be
numerically simulated or approximately analyzed, incorporating the effects of
rotation and centrifugal potential. This approach can yield insights into the
final state of the collapsing core, and help understand phenomena such as
supernovae and gamma-ray bursts.

###  5.3 Spherical and homologous solution in space

Using the ansatz $v=(1/2)(\dot{a}/a)R^{2}$, Euler’s equation (5.14) is
separable in spatial and temporal variables,

$\displaystyle h+\phi=\frac{(v_{s}^{c})^{2}}{\gamma-1}\frac{\lambda}{6}r^{2},$
(5.18) $\displaystyle
a^{\frac{\gamma}{2-\gamma}}\ddot{a}=-\frac{\lambda}{6}\frac{2\gamma}{\gamma-1}\left(\frac{\kappa^{\frac{1}{\gamma-1}}}{\pi
G}\right)^{\frac{\gamma-1}{2-\gamma}}.$ (5.19)

where the positive eigenvalue $\lambda$ relates to the material binding energy
of baryon fluid per energy density $\rho$ [16] or external pressure acting on
the core surface. The core outer radius $r_{s}$ decreases as the $\lambda$
value increases. Equation (5.19) describes the time evolution of the length
scale function $a(t)$ in the collapsing core.

The spatial solution to the Euler’s equation (5.18) is

$\displaystyle
h=\frac{(v_{s}^{c})^{2}}{\gamma-1}f^{3(\gamma-1)},\quad\phi=\frac{(v_{s}^{c})^{2}}{\gamma-1}\left(\frac{\lambda}{6}~{}r^{2}-f^{3(\gamma-1)}\right).$
(5.20)

Poisson’s equation (5.3) yields a second-order differential equation,

$\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{df^{3(\gamma-1)}}{dr}\right)+\frac{4(\gamma-1)}{\gamma}f^{3}=\lambda,$
(5.21)

the boundary conditions are $f^{\prime}(0)=0$ and $f(0)=1$. Following [16] for
the $\gamma=4/3$ solution, we obtain general solutions for $1<\gamma<4/3$
[17]. As shown in Fig. 1, three solutions of homologous density profile
$f^{3}$ correspond to the selected values $(\gamma,\lambda)$:

solutions | $\gamma$ | $\lambda$ | $r_{s}$
---|---|---|---
orange | $1.24$ | $1.0\times 10^{-4}$ | $23$
blue | $1.23$ | $8.0\times 10^{-5}$ | $36$
green | $1.225$ | $8.0\times 10^{-6}$ | $34$

(5.22)

The homologous profile $f(r)$ becomes tangent to $f=0$ at the core outer
radius $r_{s}$, namely $f^{\prime}(r_{s})\approx 0$. The core homologous
profile $f(r)$ and outer radius $r_{s}$ do not depend on time.

In our analysis, we approximately decouple the centrifugal potential
$\phi_{c}$ from the Euler equation (5.14), because the rotational potential
energy is much smaller than the heat and gravitational potential energies.
Namely, we neglect the back-reaction dynamics of the rotation potential
$\phi_{c}$ in homologous gravitational collapse. The gravitational potential
$\phi$ and heat function $h$ are balanced with the $\lambda$ term in Eq. 5.18.
The homologously collapsing cores remain spherically symmetric without high
moments.

The hydrodynamics of slow-rotating core collapse up until the core bounce can
be understood in this relatively simple analytic model. This model provides a
framework for understanding the centrifugal potential of a slowly and rigidly
rotating core collapse until the core bounces, laying the groundwork for more
detailed numerical simulations or further analytical refinements.

Figure 1: Colours on line for different eigenvalues (5.22). We show numerical
solutions to the eigenvalue problem of Eqs. (5.18) and (5.19). The homologous
density profile $f^{3}$ is plotted as a function of dimensionless radius
$r=R/a$. The density profile $f^{3}$ does not change in time and vanishes at
the core outer radius $r_{s}$ (5.22).  Figure 2: Corresponding to the density
profile $f^{3}$ in Fig. 1, the scale length $a(t)/a_{\rm min}$ is plotted as
in terms of $t/\tau_{\rm min}$, using units $a_{\rm min}$ (5.24) and
$\tau_{\rm min}$ (5.25). Note that the initial time $t_{\rm initial}\sim
10^{7}\tau_{\rm min}\sim{\mathcal{O}}(1)$ second, given by the initial
conditions (5.27). The shadow in between these colour lines indicates other
possible physical solutions.

###  5.4 Initial central density and length scale function in time

The temporal equation (5.19) governs how the scale length $a(t)$ changes in
time, and the solution is

$\displaystyle\frac{a(t)}{a_{\rm min}}$ $\displaystyle=$
$\displaystyle\left\\{1+\Big{[}\frac{\lambda\gamma}{3(2-\gamma)(\gamma-1)^{2}}\Big{]}^{\frac{1}{2}}\Big{(}\frac{t}{\tau_{\rm
min}}\Big{)}\right\\}^{2-\gamma}.$ (5.23)

In a collapsing process from the initial time $t_{\rm intial}$ to the final
time $t_{\rm final}=\tau_{\rm min}$, the scale function $a(t)$ decreases
significantly, approaching the minimum,

$\displaystyle a_{\rm min}$ $\displaystyle=$
$\displaystyle(3\gamma)^{\frac{2-\gamma}{2(\gamma-1)}}\Big{(}\frac{\kappa^{\frac{1}{\gamma-1}}}{\pi
G}\Big{)}^{1/2}=3.06\times 10^{5}(3\gamma)^{-1/2}(\frac{\rho_{c}^{\rm
max}}{\rho_{0}})^{-1/2}({\rm cm}).$ (5.24) $\displaystyle\tau_{\rm min}$
$\displaystyle\equiv$ $\displaystyle\Big{(}\frac{1}{\pi G\rho_{c}^{\rm
max}}\Big{)}^{1/2}=1.02\times 10^{-5}(\frac{\rho_{c}^{\rm
max}}{\rho_{0}})^{-1/2}({\rm sec}),$ (5.25)

and $\tau_{\rm min}=(3\gamma)^{1/2}a_{\rm min}$. It gives the basic
macroscopic length scale of the homologous collapse. The centre density
$\rho_{c}$ and sound velocity $v_{s}^{c}$ increase in time,

$\rho_{c}=\rho_{c}^{\rm max}\Big{(}\frac{a}{a_{\rm
min}}\Big{)}^{-\frac{2}{2-\gamma}};\quad(v_{s}^{c})^{2}=\frac{1}{3}\Big{(}\frac{a}{a_{\rm
min}}\Big{)}^{-2\frac{\gamma-1}{2-\gamma}}=\frac{1}{3}\Big{(}\frac{\rho_{c}}{\rho_{c}^{\rm
max}}\Big{)}^{\gamma-1},$ (5.26)

approaching their maximal values $\rho^{\rm
max}_{c}=(3\gamma\kappa)^{-\frac{1}{\gamma-1}}$ and $1/\sqrt{3}$. The maximal
centre density $\rho^{\rm max}_{c}=(3\gamma\kappa)^{-\frac{1}{\gamma-1}}$ is
about $10$ times the nuclear saturation energy density $\rho_{0}\approx
2.4\times 10^{35}{\rm ergs}/{\rm cm}^{3}$, corresponding to the number density
$n_{0}\approx 1.6\times 10^{38}/{\rm cm}^{3}$.

Instead of using collapsing time $t$, we choose the central density
$\rho_{c}(t)$ as the primary variable to describe the homologous gravitational
collapse of a stellar core. This choice is physically intuitive and aligns
with the key stages of the collapse, characterized by changes in the core
density. The initial value $\rho_{c}^{\rm in}=\rho_{c}(t_{\rm initial})$
depends on the homologous core mass $M$ and radius $r_{s}$. From Eqs.
(5.23-5.26), we obtain

$\displaystyle\rho^{\rm in}_{c}\sim 10^{-6}\rho^{\rm max}_{c}$ (5.27)

for $t_{\rm initial}\sim 10^{6}~{}\tau_{\rm min}\sim{\mathcal{O}}(1)$ seconds.
The final value $\rho_{c}^{\rm fi}=\rho_{c}(t_{\rm final})$ is the maximal
value $\rho_{c}^{\rm max}$. The core undergoes a homologous collapsing
process: starting from the initial time $t_{\rm initial}\sim{\mathcal{O}}(1)$
seconds and initial core centre density $\rho^{\rm in}_{c}$, ending at the
final time $t_{\rm final}=\tau_{\rm min}\sim 10^{-5}$ seconds and final core
centre density is $\rho^{\rm max}_{c}\approx 10\rho_{0}$. The duration is
about ${\mathcal{O}}(1)$ seconds.

###  5.5 Core mass and angular momentum conservations

Using Eq. (5.21) and outer boundary condition $f^{\prime}(r_{s})\approx 0$ of
the homologously collapsing core, we calculate its total core mass

$\displaystyle M=\int d^{3}R\rho$ $\displaystyle=$ $\displaystyle
a^{3}(t)\rho_{c}(t)4\pi\int_{0}^{r_{s}}r^{2}drf^{3}$ $\displaystyle=$
$\displaystyle
a^{3}(t)\rho_{c}(t)\frac{\lambda\gamma}{4(\gamma-1)}\frac{4\pi}{3}r^{3}_{s},$

and total core angular momentum

$\displaystyle J=\int d^{3}R\rho j$ $\displaystyle=$ $\displaystyle
a^{5}(t)\rho_{c}(t)\Omega(t)\frac{8\pi}{3}\int_{0}^{r_{s}}r^{4}drf^{3}$
$\displaystyle\approx$ $\displaystyle
a^{5}(t)\rho_{c}(t)\Omega(t)\frac{\lambda\gamma}{4(\gamma-1)}\frac{8\pi}{15}r^{5}_{s}.$

The total core mass and angular momentum conservation lead to the constants
222$a^{3}(t)\rho_{c}(t)=(\kappa/\pi G)^{3/2}$ for $\gamma=4/3$

$\displaystyle a^{3}(t)\rho_{c}(t)={\rm const.},\quad a^{2}(t)\Omega(t)={\rm
const.},$ (5.30)

in the homologous collapsing process from the initial core density
$\rho_{c}^{\rm in}$ to the final core density $\rho_{c}^{\rm fi}=\rho_{c}^{\rm
max}$.

The ratio of total core mass and angular momentum

$\displaystyle J/M\approx(2/5)a^{2}(t)\Omega(t)r^{2}_{s},$ (5.31)

is constant in time, and proportional to the rotation parameter
${\mathcal{A}}^{2}a^{2}(t)\Omega(t)$ (5.17) for
${\mathcal{A}}^{2}=2r_{s}^{2}/5$. Therefore, we will use the ratio $J/M$ to
characterize system rotation and mass energies. In our case, $J/M\ll 1$ for a
slow-rotating system of the rotational energy being much smaller than mass-
energy.

##  6 Photon-pair sphere/jet energetic properties

###  6.1 Photon-pair sphere/jet temperature profiles

Based on the adiabatic approximation discussed in Sec. 5, we can use the
“local” virial theorem (3.4) to obtain the baryon heat energy $dF$ (3.2) and
temperature $T$ (3.4). As discussed in Sec. 4, the baryon heat energy is
transformed into the thermal energy of a photon-pair plasma through hadronic
photon production, resulting in a photon temperature $T_{\gamma}$ given by Eq.
(4.3). Using these relations, the centrifugal potential (5.17) and homologous
solution (5.20), we derive

$\displaystyle\frac{T_{\gamma}}{m}$ $\displaystyle\approx$ $\displaystyle
0.21\frac{(v^{c}_{s})^{2}}{6}\Big{\\{}\frac{1}{\gamma(\gamma-1)}\left[(7\gamma-6)f^{3(\gamma-1)}-\frac{\lambda}{6}\gamma
r^{2}\right]$ (6.1) $\displaystyle-$
$\displaystyle\frac{1}{2}{\mathcal{A}}^{2}a^{2}\Omega
r^{2}\sin^{2}\theta\Big{\\}}.$

The first term inside the braces corresponds to the gravitational potential
energy contribution to the temperature. The second term accounts for the
rotational energy’s influence, which decreases the photon-pair plasma
temperature due to the repulsive nature of the centrifugal potential
$\phi_{c}$ acting against gravitational collapse. For $\theta\not=0$, the
photon-pair temperature $T_{\gamma}$ becomes smaller, compared with non-
rotating cases $\Omega=0$.

The expression for $T_{\gamma}$ highlights the interplay between the
gravitational and centrifugal potentials in determining the thermal energy
distribution of the photon-pair plasma during the homologous collapse of the
rotating stellar core.

At the end of the homologous collapse $t=t_{\rm final}$, the core’s central
density reaches its maximum value $\rho_{c}=\rho_{c}^{\rm max},$ and the sound
speed squared is $(v^{c}_{s})^{2}=1/3$, the produced photon-pair plasma forms
an axial symmetric jet, described in the cylindrical coordinates. For
illustration, we use the parameters, we use parameters
$(\gamma,\lambda_{m})=(1.23,8\times 10^{-5})$ (5.22) (the blue case) and
${\mathcal{A}}^{2}a^{2}\Omega=0.05$, indicating a small centrifugal potential
$\phi_{c}$.

In Fig. 3, we plot the photon-pair plasma temperature $T_{\gamma}$ for the
non-rotating case $\Omega=0$ and rotating cases $\Omega(t)\not=0$. The maximal
temperature is located at the centre $r=0$. The vanishing temperature
$T_{\gamma}(r,\theta)=0$ determines the photon-pair sphere radius
$r_{\gamma}<r_{s}$ or the jet boundary

$\displaystyle r^{j}_{\gamma}(\theta)\leq r_{\gamma},\quad
R^{j}_{\gamma}(\theta)=a(t)r^{j}_{\gamma}(\theta).$ (6.2)

The photon-pair sphere volume $V_{\gamma}$ is larger than the photon-pair jet
volume $V^{j}_{\gamma}$, see Fig. 4. In the rotating case, we plot the jet
boundary $r^{j}_{\gamma}(\theta)$ in Fig. 5. As a self-consistency check, we
find that the baryon temperature $T$ and photon temperature $T_{\gamma}$ are
much smaller than baryon mass ($T_{\gamma}\approx 0.21T\ll m$). It fills the
condition for applying the virial theorem (3.2).

Figure 3: Colours online. Using results (6.1) for maximal sound velocity
$(v_{s}^{c})^{2}=1/3$, we plot the temperature $T_{\gamma}/m$ in the
cylindrical coordinates $y=R\sin\theta$ and $z=R\cos\theta$, where the
$\theta$ is the zenith angle. The temperature unit is of typical baryon mass
$m\approx 1$ GeV. The parameters are the $Blue$ case parameters (5.22), and
the initial conditions are (5.27). The upper panel is a non-rotating case
${\mathcal{A}}^{2}a^{2}\Omega=0$. The lower panel is a rotating case of
${\mathcal{A}}^{2}a^{2}\Omega=0.05$. The temperature vanishes at the photon-
pair sphere radius $r_{\gamma}$ or the jet boundary $r^{j}_{\gamma}(\theta)$.
Both are smaller than the core outer radius $r_{s}$ (5.22), see Fig. 1.

###  6.2 Photon-pair sphere/jet density and size profiles

We use photon productions (4.1) and (4.2) by baryon collisions to
approximately obtain the photon-pair jet energy and number densities

$\displaystyle\frac{\rho_{\gamma}}{\rho_{0}}\propto\Big{(}\frac{T_{\gamma}}{m}\Big{)}^{4},\quad\frac{n_{\gamma}}{n_{0}}\propto\Big{(}\frac{T_{\gamma}}{m}\Big{)}^{3},$
(6.3)

in terms of photon-pair thermal temperature $T_{\gamma}$. The results show
that the photon-pair energy density $\rho_{\gamma}$ is in the range of
$(10^{25}\sim 10^{30})~{}{\rm ergs}/{\rm cm}^{3}$, and the number density
$n_{\gamma}$ is in the range of $(10^{37}\sim 10^{30})/{\rm cm}^{3}$.

Using Eqs. (5.23), (5.24) and (5.26), we obtain the photon-pair sphere radius

$\displaystyle R_{\gamma}$ $\displaystyle=$ $\displaystyle 3.06\times
10^{5}(3\gamma)^{-1/2}r_{\gamma}\left(\frac{\rho_{0}}{\rho_{c}^{\rm
max}}\right)^{\frac{\gamma-1}{2}}\left(\frac{\rho_{0}}{\rho^{\rm
in}_{c}}\right)^{\frac{2-\gamma}{2}}({\rm cm}),$ (6.4)

and the photon-pair jet boundary

$\displaystyle R^{j}_{\gamma}(\theta)$ $\displaystyle=$ $\displaystyle
3.06\times
10^{5}(3\gamma)^{-1/2}r_{\gamma}^{j}(\theta)\left(\frac{\rho_{0}}{\rho_{c}^{\rm
max}}\right)^{\frac{\gamma-1}{2}}\left(\frac{\rho_{0}}{\rho^{\rm
in}_{c}}\right)^{\frac{2-\gamma}{2}}({\rm cm}).$ (6.5)

They are expressed as a function of initial core centre density $\rho^{\rm
in}_{c}$ when a homologous collapse initiates. The photon-pair sphere size
$R_{\gamma}\sim(10^{7}\sim 10^{10})~{}{\rm cm}$ and the photon-pair
cylindrical jet width $R^{j}_{\gamma}(\pi/2)\sim(10^{6}\sim 10^{9})~{}{\rm
cm}$.

Both sizes $R_{\gamma}$ and $R^{j}_{\gamma}$ are much larger than the photon
mean-free path $\lambda_{\gamma}=(\sigma_{\gamma}n_{\gamma})^{-1}$, namely
$\sigma_{\gamma}n_{\gamma}R_{\gamma}\gg 1$ or
$\sigma_{\gamma}n_{\gamma}R^{j}_{\gamma}\gg 1$. The photon-pair sphere/jet is
deeply opaque. On the other hand, we find in Fig. 3 that the photon-pair
sphere/jet temperature $T_{\gamma}$ can be well above the critical energy
threshold $2m_{e}=1.02$ MeV of electron-positron ($e^{+}e^{-}$) pair
production. The photon-pair sphere/jet energy density $\rho_{\gamma}$ can be
well above the critical energy density $\rho_{e}=m_{e}^{4}=5.93\times
10^{-11}\rho_{0}$ of electron-positron pairs. This means the photons are
quickly thermalised to form an electron-position-photon plasma, as briefly
discussed in Sec. 3. The photon-pair sphere/jet undergoes ultra-relativistic
hydrodynamic expansion, which has been well-studied in the literature, and
will not be studied in this article.

We compare the photon-pair sphere and jet boundaries in Fig. 4 and define the
cylindrical jet angle $\theta_{\rm jet}$ as the ratio

$\theta_{\rm
jet}\approx\frac{R^{j}_{\gamma}(\theta=\pi/2)}{R^{j}_{\gamma}(\theta=0)},\quad
R^{j}_{\gamma}(\theta=0)=R_{\gamma},$ (6.6)

that characterizes the axial symmetric jet along the $\hat{z}$ direction.

Figure 4: We compare the photon-pair sphere and jet boundaries of Fig. 3. The
photon-pair sphere radius $r_{\gamma}\approx 30$ and $R_{\gamma}=a_{\rm
min}r_{\gamma}\sim(10^{7}-10^{10})$ cm. The photon-pair jet boundary
$r^{j}_{\gamma}(\theta)$ is given in Fig. 5.  Figure 5: The photon-pair jet
boundary $r^{j}_{\gamma}(\theta)$ shows that (i) $\theta\rightarrow 0$ and
$r^{j}_{\gamma}(\theta)$ approaches to the radius $r_{\gamma}\approx 30$ of
the photon-pair sphere; (ii) $\theta\rightarrow\pi/2$ and $r_{\gamma}(\theta)$
approaches to $r^{j}_{\gamma}(\pi/2)\approx 1$ of the photon-pair jet. The
cylindrical jet width $R^{j}_{\gamma}=a_{\rm
min}r^{j}_{\gamma}(\pi/2)\sim(10^{6}-10^{8})$ cm. The jet angle (6.6)
$\theta_{\rm jet}\approx 1/30\approx 1^{\circ}$.

###  6.3 Photon-pair sphere/jet energy and spectrum peak

The total relativistic particle energy $E_{\gamma}$ and number $N_{\gamma}$ of
the photon-pair sphere/jet can be obtained by numerically integrating the
energy and number densities (6.3) over the photon-pair jet volume

$\displaystyle
V^{j}_{\gamma}=\int_{V^{j}_{\gamma}}d^{3}R=a^{3}\int_{V^{j}_{\gamma}}2\pi
r^{2}dr\sin\theta d\theta.$ (6.7)

We also compute the total baryon core mass

$\displaystyle M=\int_{\rm core}\rho d^{3}R=\rho_{c}a^{3}F_{0},$ (6.8)

and number $N=M/m$, where $F_{0}\equiv\int_{0}^{r_{s}}4\pi r^{2}drf^{3}$.

At the end of homologous collapse when the central sound velocity
$v^{c}_{s}=\sqrt{1/3}$, we define integration

$\displaystyle
Y_{n}=2\pi\int_{V^{j}_{\gamma}}\left(\frac{T_{\gamma}}{m}\right)^{n}r^{2}dr\sin\theta
d\theta$ (6.9)

over photon-pair jet volume $V^{j}_{\gamma}$. From Eq. (6.3), we obtain the
total energy and photon number of photon-pair jet systems

$\displaystyle\frac{E_{\gamma}}{M}$ $\displaystyle\propto$
$\displaystyle\frac{Y_{4}}{F_{0}}\big{(}\frac{\rho_{c}^{\rm
max}}{\rho_{0}}\big{)}^{4(1-\gamma)}\big{(}\frac{\rho_{0}}{\rho^{\rm
in}_{c}}\big{)}^{5-4\gamma},$ (6.10) $\displaystyle\frac{N_{\gamma}}{N}$
$\displaystyle\propto$
$\displaystyle\frac{Y_{3}}{F_{0}}\big{(}\frac{\rho_{c}^{\rm
max}}{\rho_{0}}\big{)}^{3(1-\gamma)}\big{(}\frac{\rho_{0}}{\rho^{\rm
in}_{c}}\big{)}^{4-3\gamma}.$ (6.11)

In addition, we define the mean temperature of the photon-pair sphere/jet as
averaged value of $T_{\gamma}$ (6.1) over photon-pair jet volume
$V^{j}_{\gamma}$,

$\displaystyle\frac{\langle T_{\gamma}\rangle}{m}$ $\displaystyle\propto$
$\displaystyle\frac{Y_{1}}{Y_{0}}\left(\frac{\rho_{0}}{\rho_{c}^{\rm
max}}\right)^{\gamma-1}\left(\frac{\rho^{\rm
in}_{c}}{\rho_{0}}\right)^{\gamma-1},$ (6.12)

which represents the characteristic temperature of the photon-pair sphere/jet.
The photon-pair spectrum has a maximum at the peak energy $E_{p}\sim\langle
T_{\gamma}\rangle$.

In Eqs. (6.10) and (6.12), we express total photon-pair energy $E_{\gamma}$
and spectral peak $E_{p}$ in terms of the initial core centre density
$\rho^{\rm in}_{c}=\rho_{c}(t_{\rm initial})$ at the time $t_{\rm initial}$
when the core initiates homologous collapse. If the core mass $M$ and baryon
number $N$ are fixed and conserved, the smaller initial core centre density
$\rho^{\rm in}_{c}$ is, the more extended core mass density profile
$\rho(r,t_{\rm initial})$ is, i.e., the larger $r_{s}$ is. As a consequence,
more gravitational energy can be gained in collapse and converted to the
photon-pair sphere/jet energy $E_{\gamma}$ and number $N_{\gamma}$. In
contrast, these quantities are also functions of the material binding energy
parameter $\lambda$, the averaged thermal index $\gamma$ and rotation
parameter $J/M\propto{\mathcal{A}}^{2}a^{2}\Omega$. The rotation represents
the repulsive reactions against gravitational collapse and energy gain.

For gravitational collapses from the initial core density $\rho_{c}^{\rm
in}\sim 10^{-6}\rho_{0}$ to the final one $\rho_{c}^{\rm max}\sim 10\rho_{0}$,
we obtain

$\displaystyle\frac{E_{\gamma}}{M}\sim(2\sim 5)\times
10^{-2},\quad\frac{E_{p}}{m}\sim(8\sim 10)\times 10^{-5}$
$\displaystyle\frac{N_{\gamma}}{N},\sim(1\sim 3)\times 10,\quad
R_{\gamma}\sim(5\sim 9)\times 10^{8}{\rm cm},$ (6.13)

where the variations are due to the ranges of the parameters $\lambda$,
$\gamma$ and $J/M\propto{\mathcal{A}}^{2}a^{2}\Omega$. These quantities are
conserved in the subsequent processes of the ultra-relativistic hydrodynamic
expansion and internal/external collisions up to the transparent point when
pairs annihilate and produce photons stream to detectors.

We have to point out that the photon-pair sphere/jet energetic properties
(6.4,6.5) and (6.10,6.12) explicitly depend on the initial and final core
central densities $\rho_{c}^{\rm in}$ and $\rho_{c}^{\rm max}$. They
characterize the gravitation gain and conversion to baryon kinetic energy and
photon-pair thermal energy in gravitational collapses. These characteristics
should also be held for gravitational collapses of non-spherical cores or
coalescence of binary systems by introducing effective initial and final
central densities $(\rho_{c}^{\rm in})_{\rm eff}$ and $(\rho_{c}^{\rm
max})_{\rm eff}$ of the system.

By simplifying out $(\rho_{c}^{\rm in})$ and $(\rho_{c}^{\rm max})$ among Eqs.
(6.4,6.5) and (6.10,6.12), we obtained theoretical correlations among the
spectral peak $E_{p}$, total photon-pair energy $E_{\gamma}$ and peak
luminosity $L_{\gamma}\propto cR_{\gamma}^{2}\langle
T_{\gamma}\rangle^{4}\propto cE_{\gamma}/R_{\gamma}$,

$\displaystyle E_{p}\propto E_{\gamma}^{\chi},\quad E_{p}\propto
L_{\gamma}^{\frac{\chi}{\chi+1}},\quad L_{\gamma}\propto E_{\gamma}^{1+\chi},$
(6.14)

with the unique positive parameter
$\chi=\frac{\gamma-1}{4\gamma-5}\sim{\mathcal{O}}(1)$ 333It is an inverse of
the original $\chi-$definition [17]. Although many intermediate processes have
not been considered in this simplified model, the scaling laws (6.14) of GRBs’
progenitors give an insight into the Amati and Yonetoku empirical relations,
as well as the $L_{\gamma}\propto E_{\gamma}^{1+\chi}$ relation in a unified
framework.

$J/M$ | $\theta_{\rm jet}$ | $E^{j}_{p}/E_{p}$ | $E^{j}_{\gamma}/E_{\gamma}$ | $L^{j}_{\gamma}/L_{\gamma}$ | $R^{j}_{\gamma}/R_{\gamma}$ | $N^{j}_{\gamma}/N_{\gamma}$ | $V^{j}_{\gamma}/V_{\gamma}$
---|---|---|---|---|---|---|---
$0.001$ | $13.2^{\circ}$ | $1.01$ | $0.998$ | $4.29$ | $6.9/30$ | $0.99$ | $0.92$
$0.005$ | $7.8^{\circ}$ | $1.25$ | $0.99$ | $7.24$ | $4.1/30$ | $0.97$ | $0.63$
$0.01$ | $6.3^{\circ}$ | $1.38$ | $0.98$ | $8.90$ | $3.3/30$ | $0.96$ | $0.52$
$0.02$ | $4.8^{\circ}$ | $1.52$ | $0.97$ | $11.6$ | $2.5/30$ | $0.93$ | $0.42$
$0.03$ | $4.2^{\circ}$ | $1.60$ | $0.95$ | $12.9$ | $2.2/30$ | $0.91$ | $0.37$
$0.04$ | $3.8^{\circ}$ | $1.66$ | $0.94$ | $14.1$ | $2.0/30$ | $0.89$ | $0.33$
$0.05$ | $3.4^{\circ}$ | $1.71$ | $0.93$ | $15.5$ | $1.8/30$ | $0.88$ | $0.31$
$0.06$ | $3.2^{\circ}$ | $1.74$ | $0.92$ | $16.2$ | $1.7/30$ | $0.86$ | $0.29$
$0.07$ | $3.1^{\circ}$ | $1.78$ | $0.91$ | $17.1$ | $1.6/30$ | $0.85$ | $0.27$
$0.08$ | $2.9^{\circ}$ | $1.81$ | $0.90$ | $18.0$ | $1.5/30$ | $0.84$ | $0.26$
$0.09$ | $2.7^{\circ}$ | $1.84$ | $0.89$ | $19.1$ | $1.4/30$ | $0.83$ | $0.25$
$0.10$ | $2.5^{\circ}$ | $1.86$ | $0.88$ | $20.3$ | $1.3/30$ | $0.82$ | $0.24$
$0.15$ | $2.1^{\circ}$ | $1.95$ | $0.85$ | $23.2$ | $1.1/30$ | $0.78$ | $0.20$
Table 1: For selected values of the rotation parameter
${\mathcal{A}}^{2}a^{2}\Omega$, i.e., the ratio $J/M$ (5.31), we tabulate the
jet angle $\theta_{\rm jet}$ (6.6) and the ratios of the spectral peak
$E^{j}_{p}/E_{p}$, total energy $E^{j}_{\gamma}/E_{\gamma}$, maximal
luminosity
$L^{j}_{\gamma}/L_{\gamma}\propto(E^{j}_{\gamma}/E_{\gamma})(R_{\gamma}/R^{j}_{\gamma})$,
minimal size $R^{j}_{\gamma}/R_{\gamma}$, total particle number
$N^{j}_{\gamma}/N_{\gamma}$ and volume $V^{j}_{\gamma}/V_{\gamma}$ between the
photon-pair sphere and jet cases for comparisons. The photon-pair jet (with
superscript $j$) and sphere quantities are given by Eqs. (6.4,6.11,6.12).
Their ratios do not explicitly depend on the initial (final) core centre
density $\rho_{c}^{\rm in}$ ($\rho_{c}^{\rm fi}$) of gravitational collapses.

##  7 Long vs short GRBs by angular momentum and mass ratio

###  7.1 Trends of jet features varying with $J/M$

Based on the numerical results of Eqs. (6.9-6.12) and Fig. 4 for the
parameters $\it Blue$ case (5.22), using the initial condition (5.27), we
calculate the photon-pair jet’s angle $\theta_{\rm jet}$, spectral peak
$E^{j}_{p}$ total energy $E^{j}_{\gamma}$ and luminosity $L^{j}_{\gamma}$ in
comparison with the sphere counterparts. We select the rotation parameter
${\mathcal{A}}^{2}a^{2}\Omega\propto J/M$ values to gain an insight into the
trends of how jet geometric and energetic features vary with the rotation
angular momentum in Table 1.

###  7.2 A separatrix in angular momentum $J/M$

The photon-pair (approximate) sphere case represents one distinct class of the
GRBs progenitors with small angular momenta ($J/M\gtrsim 0$), which are more
likely for the cases of massive core collapses, relating to long GRBs. The
photon-pair jet case with nontrivial angular momenta represents another
distinct class of the GRBs progenitors, which are more likely for the cases of
binary coalescence, relating to short GRBs. We will come back to this point
later. In the cases of binary coalescence, systems have larger angular momenta
$J/M$, where the core mass $M$ should be an effective value of total system
mass, the central density $\rho_{c}$ (5.11) and its initial value (5.27)
should be effective one. Since the separation between long and short GRBs
classes is distinct, there must be a separatrix in angular momentum $J/M$. It
occurs when the centrifugal potential $\phi_{c}$ is too large to be balanced
by the gravitational potential $\phi$ and the material binding energy
$\lambda$. Using the equilibrium equation (5.9) and gravitational potential
$\phi$ balanced by the heat function $h$, we give a separatrix,

$\displaystyle\phi_{c}\approx\frac{(v_{s}^{c})^{2}}{\gamma-1}\frac{\lambda}{6}r^{2}\quad\Rightarrow\quad
J/M\propto({\mathcal{A}}^{2}a^{2}\Omega)_{\rm
sepa}\sim\frac{\lambda}{3(\gamma-1)}\approx\lambda$ (7.1)

in this simplified model. When the repulsive rotation energy is larger than
the material binding energy, namely $J/M>\lambda$, they are for the cases of
binary coalescence. On the contrary, $J/M<\lambda$ are for the cases of
massive core collapses. The real situation is much more complex.

###  7.3 Long vs short GRBs

The following qualitative observations and comments are in order from the
trends of increasing or decreasing quantities as increasing angular momentum
$J/M$ in Table 1.

  1. (i)

Due to the angular momentum and repulsive centrifugal potential increase, the
gravitational energy gain becomes smaller. The ratios
$E_{\gamma}^{j}/E_{\gamma}$ and $N_{\gamma}^{j}/N_{\gamma}$ are smaller than
one but still in the order of unity, because most photons and pairs are
produced at the centre ($z=0,y=0$) of the sphere and jet, see Fig. 3.

  2. (ii)

The jet spectral peak $E_{p}^{j}/E_{p}$ significantly increases up to twice
larger than the spherical one, because the photon-pair energy density
increases as the jet volume $V^{j}$ decreases. As a result, the spectral peak
$E^{j}_{p}$ increases and total energy $E^{j}_{\gamma}$ decreases as angular
momentum increases. It gives a possible explanation for two distinct Amati
relations $E_{p}\propto E_{\gamma}^{\chi}$ (6.14) respectively for long and
short bursts’ data scattering on the $E_{p}$-$E_{\gamma}$ plane, see Fig 7
(right) of the reference [25], where two distinct Amati relations are observed
for long and short bursts.

  3. (iii)

The spectral peak $E^{j}_{p}$ and the peak luminosity $L^{j}_{\gamma}$
increase as angular momentum increases. It implies the Yonetoku relation
$E_{p}\propto L_{\gamma}^{\chi/(\chi+1)}$ (6.14) is not the same for long and
short bursts’ data scattering on the $E_{p}$-$L_{\gamma}$ plane, as shown in
Figure 7 (left) of the reference [25].

  4. (iv)

The jet and sphere size ratio
$R_{\gamma}^{j}/R_{\gamma}\sim{\mathcal{O}}(10^{-1})$. It implies that long
bursts duration $T_{90}\sim R_{\gamma}/(2c\Gamma^{2})$ are about 10 times
lager than short bursts $T^{j}_{90}\sim R^{j}_{\gamma}/(2c\Gamma_{j}^{2})$. We
assume hydrodynamic expansion Lorentz factors
$\Gamma/\Gamma_{j}\sim{\mathcal{O}}(1)$, considering the obtained result
$E^{j}_{\gamma}/E_{\gamma}\sim{\mathcal{O}}(1)$ in Table 1. It is
qualitatively consistent with two classes of long (soft) and short (hard)
bursts’ data scattering on the $E_{p}$-$T_{90}$ plane, see Figures 1 and 6
(up-right) of the reference [25]. However, the separatrix (7.1) of this
simplified model cannot infer the separatrix $T_{90}\approx 1\sim 2$ seconds
between long GRBs (massive core collapses) and short GRBs (binary
coalescence).

  5. (v)

The trends of numerical results in Table 1 imply possibly positive or negative
correlations between the jet angle $\theta_{\rm jet}$ and burst duration
$T_{90}^{j}$, spectral peak $E_{p}^{j}$, total energy $E_{\gamma}^{j}$,
luminosity $L_{\gamma}^{j}$. Short GRBs should be more jetting than long GRBs.

These insights underscore the importance of considering rotational effects and
jet dynamics in studying GRB spectra and their correlations.

##  8 Conclusion and remarks

To understand GRBs’ jet progenitors, we adopt an adiabatic approximation and a
simplified model to study photon-pair jets generated by the gravitational
collapse of slowly rotating cores and binary systems. Our results
qualitatively show photon-pair jets features: (i) the jet angle $\theta_{\rm
jet}$ relates to progenitor angular momenta and masses ratio $J/M$; (ii)
variation trends of GRBs’ peak spectra, total energy and luminosity in terms
of jet angle, positive and negative correlations among them; (iii) the
distinction between short and long GRBs’ properties is due to the different
angular momentum of systems that undergo gravitational processes of binary
coalescence or massive core collapse.

More theoretical studies and numerical simulations of massive rotating core
collapse and binary coalescence are necessarily required to better understand
the dynamics of massive rotating core collapse and binary coalescence. These
studies will help refine our understanding of the photon-pair jet formation
process and the resultant GRB properties.

The data analyses on possible correlations of GRBs’ jet energetic, geometric
and time-duration features are inviting. Systematic analyses of GRB data are
required to explore possible correlations between jet energetics, geometric
features, and time-duration characteristics. Understanding these correlations
will provide deeper insights into the jet progenitor systems and the physical
mechanisms driving GRBs.

In summary, our simplified model provides a qualitative framework to
understand the generation of photon-pair jets from gravitationally collapsing
systems. It highlights the significant role of progenitor angular momentum in
determining the jet characteristics and, consequently, the observable
properties of GRBs. Further theoretical and observational studies are crucial
to validate and expand upon these findings, paving the way for a more
comprehensive understanding of GRB jet progenitors and their evolution.

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# Remote Preparation of the Two-Particle State

YAN Feng-Li, ZHANG Guo-Hua  College of Physics Science and Information
Engineering, Hebei Normal University, Shijiazhuang 050016, China

###### Abstract

We present a scheme of remote preparation of the two-particle state by using
two Einstein-Podolsky-Rosen pairs or two partial entangled two-particle states
as the quantum channel. The probability of the successful remote state
preparation is obtained.

###### pacs:

03.65.Ta, 03.67.Hk, 03.67.Lx

Quantum entanglement is a valuable resource for the implementation of quantum
computation and quantum communication protocols, like quantum teleportation
[1,2], quantum key distribution [3,4], quantum secure direct communication
[5], dense coding [6-8], quantum computation [9], remote state preparation
[10-15] and so on. Quantum teleportation is regarded as one of the most
profound results of quantum information theory. In the original quantum
teleportation protocol of Bennett et al. [1], it was showed that an unknown
state of a qubit can be perfectly transported from a sender Alice to a remote
receiver Bob with the aid of long-range Einstein-Podolsky-Rosen correlation
and transmission of two bits of classical information without transmission of
the carrier of the quantum state. They have also generalized the protocol for
an unknown qubit by using a maximally entangled state in $d\times d$
dimensional Hilbert space and sending $2{\rm log}_{2}d$ bits of classical
information.

Another important application of quantum entanglement is remote state
preparation, where two spatially distant people Alice and Bob can prepare an
quantum state known to Alice but unknown to Bob, with the aid of previously
shared quantum entanglement and the classical communication. Recently, Pati
[10] presented a protocol of remotely preparing a special ensemble of states.
Lo [11] showed that remote state preparation requires less classical
communication than teleportation for the special ensembles of states, but for
general states the classical communication cost of teleportation would be
equal to that of remote state preparation. Bennett et al. [12] studied the
trade off between entanglement cost and classical communication cost in remote
state preparation. Since then, some people have investigated various
theoretical protocols about generalization of remote state preparation. Zhan
[13] gave a scheme for preparing remotely a three-particle GHZ state. Huang et
al. [14] put forward a protocol for preparing remotely the multipartite pure
state. A scheme for preparing remotely a two-particle entangled state via two
pairs of entangled particles was presented by Liu et al. [15].

In this paper, we propose a scheme of preparing remotely the two-particle
state. Two Einstein-Podolsky-Rosen pairs and two partial entangled two-
particle states as the quantum channel are considered, respectively. The
probability of the successful remote state preparation is calculated.

Let us first begin our remote state preparation with two Einstein-Podolsky-
Rosen pairs as the quantum channel. We suppose that a sender Alice wants to
help a remote receiver Bob to prepare a two-particle state in the following
formation

$|\phi\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle,$
(1)

where $\alpha$ and $\gamma$ are real numbers, $\beta$ and $\delta$ are complex
numbers and $\beta^{*}\delta$ is real,
$|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}+|\delta|^{2}=1$. We suppose that Alice
knows $\alpha$, $\beta$, $\gamma$ and $\delta$ completely, but Bob does not
know them at all. We also assume that the quantum channel shared by Alice and
Bob is composed of two Einstein-Podolsky-Rosen pairs

$|\psi\rangle_{12}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)_{12},$ (2)
$|\psi\rangle_{34}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)_{34},$ (3)

where particles 1 and 3 belong to Alice while Bob has particles 2 and 4. In
order to help Bob to remotely prepare a two-particle state stated in Eq.(1) on
the particles 2 and 4, Alice must make a measurement on her two particles 1
and 3. The measurement basis chosen by Alice is a set of mutually orthogonal
basis vectors
$\\{|\varphi\rangle_{13},|\varphi_{\perp}\rangle_{13},|\psi\rangle_{13},|\psi_{\perp}\rangle_{13}\\}$,
where

$\displaystyle|\varphi\rangle_{13}=(\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle)_{13},$
$\displaystyle|\varphi_{\perp}\rangle_{13}=(-\delta^{*}|00\rangle+\gamma|01\rangle-\beta^{*}|10\rangle+\alpha|11\rangle)_{13},$
$\displaystyle|\psi\rangle_{13}=(\gamma|00\rangle+\delta|01\rangle-\alpha|10\rangle-\beta|11\rangle)_{13},$
$\displaystyle|\psi_{\perp}\rangle_{13}=(\beta^{*}|00\rangle-\alpha|01\rangle-\delta^{*}|10\rangle+\gamma|11\rangle)_{13}.$

Here $\\{|00\rangle_{13},|01\rangle_{13},|10\rangle_{13},|11\rangle_{13}\\}$
is the computation basis. Then we have

$\displaystyle|\psi\rangle_{12}\otimes|\psi\rangle_{34}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[|\varphi\rangle_{13}(\alpha|00\rangle+\beta^{*}|01\rangle+\gamma|10\rangle+\delta^{*}|11\rangle)_{24}$
$\displaystyle+|\varphi_{\perp}\rangle_{13}(-\delta|00\rangle+\gamma|01\rangle-\beta|10\rangle+\alpha|11\rangle)_{24}$
$\displaystyle+|\psi\rangle_{13}(\gamma|00\rangle+\delta^{*}|01\rangle-\alpha|10\rangle-\beta^{*}|11\rangle)_{24}$
$\displaystyle+|\psi_{\perp}\rangle_{13}(\beta|00\rangle-\alpha|01\rangle-\delta|10\rangle+\gamma|11\rangle)_{24}].$

Thus if Alice performs a measurement in the basis
$\\{|\varphi\rangle_{13},|\varphi_{\perp}\rangle_{13},|\psi\rangle_{13},|\psi_{\perp}\rangle_{13}\\}$
on her two particles 1 and 3, then each outcome will occur with the equal
probability $\frac{1}{4}$. If Alice’s measurement result is
$|\varphi_{\perp}\rangle$, then particles 2 and 4 will collapse into the state

$(-\delta|00\rangle+\gamma|01\rangle-\beta|10\rangle+\alpha|11\rangle)_{24}.$
(6)

After that if Alice communicates to Bob of her actual measurement outcome via
a classical channel, then Bob will be able to apply the following unitary
transformation

$U=(|0\rangle\langle 1|+|1\rangle\langle 0|)_{2}\otimes(|0\rangle\langle
1|-|1\rangle\langle 0|)_{4}$ (7)

on his particles 2 and 4. The resulting state of Bob’s particles will be the
original state $|\phi\rangle$.

Likewise if the actual result of Alice’s measurement is
$|\psi_{\perp}\rangle_{13}$, then Bob gets the state

$(\beta|00\rangle-\alpha|01\rangle-\delta|10\rangle+\gamma|11\rangle)_{24}.$
(8)

When Bob received the classical information of the actual measurement result
sent by Alice, he can perform an appropriate operation

$U^{\prime}=(|0\rangle\langle 0|-|1\rangle\langle
1|)_{2}\otimes(|0\rangle\langle 1|-|1\rangle\langle 0|)_{4}$ (9)

on his particles 2 and 4 to obtain the state $|\phi\rangle$. So when these two
measurement outcomes happen, Alice can help Bob to remotely prepare the two-
particle state $|\phi\rangle$.

However, when the measurement outcome is $|\varphi\rangle_{13}$
($|\psi\rangle_{13}$), the remote state preparation can not be successful, as
the state of the particles 2 and 4 will be
$(\alpha|00\rangle+\beta^{*}|01\rangle+\gamma|10\rangle+\delta^{*}|11\rangle)_{24}$
(
$(\gamma|00\rangle+\delta^{*}|01\rangle-\alpha|10\rangle-\beta^{*}|11\rangle)_{24}$).
Because Bob does not know the coefficients $\alpha$, $\beta$, $\gamma$ and
$\delta$ at all, he cannot transform either the state
$(\alpha|00\rangle+\beta^{*}|01\rangle+\gamma|10\rangle+\delta^{*}|11\rangle)_{24}$
or
$(\gamma|00\rangle+\delta^{*}|01\rangle-\alpha|10\rangle-\beta^{*}|11\rangle)_{24}$
into the state $|\phi\rangle$. But, if $\alpha$, $\beta$, $\gamma$ and
$\delta$ are real numbers, the situation would be changed. When Alice’s
measurement result $|\varphi\rangle_{13}$ or $|\psi\rangle_{13}$ occurs, it is
not difficult for Bob to prepare the two-particle state $|\phi\rangle$ by
performing the suitable unitary operation determined by the outcome of Alice’s
measurement. Here we omit the concrete steps.

Now, we present a scheme for preparing remotely a two-particle state via two
non-maximally entangled states. Suppose that Alice still wishes to help Bob to
prepare remotely the state $|\phi\rangle$ in Eq.(1), but the two entangled
states shared by Alice and Bob are two non-maximally entangled states

$|\psi\rangle_{12}=(a|00\rangle+b|11\rangle)_{12},$ (10)
$|\psi\rangle_{34}=(c|00\rangle+d|11\rangle)_{34},$ (11)

where the parameters $a$, $b$, $c$ and $d$ are real numbers,
$|a|^{2}+|b|^{2}=1$, $|c|^{2}+|d|^{2}=1$, and $|a|\leq|b|,$ $|c|\leq|d|$. We
also assume that the particles 1 and 3 belong to Alice while Bob has particles
2 and 4. Since Alice knows the parameters $\alpha$, $\beta$, $\gamma$ and
$\delta$ exactly, she can perform a measurement on particles 1 and 3 in the
basis
$\\{|\varphi\rangle_{13},|\varphi_{\perp}\rangle_{13},|\psi\rangle_{13},|\psi_{\perp}\rangle_{13}\\}$.
A simple algebraic rearrangement of the expression
$|\psi\rangle_{12}\otimes|\psi\rangle_{34}$ in terms of the states
$|\varphi\rangle_{13},$ $|\varphi_{\perp}\rangle_{13},$ $|\psi\rangle_{13},$
$|\psi_{\perp}\rangle_{13}$ yields

$\displaystyle|\psi\rangle_{12}\otimes|\psi\rangle_{34}$ $\displaystyle=$
$\displaystyle|\varphi\rangle_{13}(ac\alpha|00\rangle+ad\beta^{*}|01\rangle+bc\gamma|10\rangle+bd\delta^{*}|11\rangle)_{24}$
$\displaystyle+|\varphi_{\perp}\rangle_{13}(-ac\delta|00\rangle+ad\gamma|01\rangle-
bc\beta|10\rangle+bd\alpha|11\rangle)_{24}$
$\displaystyle+|\psi\rangle_{13}(ac\gamma|00\rangle+ad\delta^{*}|01\rangle-
bc\alpha|10\rangle-bd\beta^{*}|11\rangle)_{24}$
$\displaystyle+|\psi_{\perp}\rangle_{13}(ac\beta|00\rangle-ad\alpha|01\rangle-
bc\delta|10\rangle+bd\gamma|11\rangle)_{24}.$

Therefore, if the actual result of Alice’s measurement on the two particles 1
and 3 is $|\varphi_{\perp}\rangle_{13}$ with the probability
$|bd\alpha|^{2}+|bc\beta|^{2}+|ad\gamma|^{2}+|ac\delta|^{2}$ then the state of
particles 2 and 4 will be

$|\varphi\rangle_{24}=\frac{(-ac\delta|00\rangle+ad\gamma|01\rangle-
bc\beta|10\rangle+bd\alpha|11\rangle)_{24}}{\sqrt{|bd\alpha|^{2}+|bc\beta|^{2}+|ad\gamma|^{2}+|ac\delta|^{2}}}.$
(13)

When Bob is informed the actual measurement outcome
$|\varphi_{\perp}\rangle_{13}$ by Alice via a classical channel, he can get
the original state described in Eq.(1) with certain probability.

Firstly, Bob operates a unitary operation

$U_{1}=(|0\rangle\langle 1|+|1\rangle\langle 0|)_{2}\otimes(|0\rangle\langle
1|-|1\rangle\langle 0|)_{4}$ (14)

on particles 2 and 4. Obviously $U_{1}$ will transform the state
$|\varphi\rangle_{24}$ into

$|\varphi^{\prime}\rangle_{24}=\frac{(bd\alpha|00\rangle+bc\beta|01\rangle+ad\gamma|10\rangle+ac\delta|11\rangle)_{24}}{\sqrt{|bd\alpha|^{2}+|bc\beta|^{2}+|ad\gamma|^{2}+|ac\delta|^{2}}}.$
(15)

Secondly, Bob introduces an auxiliary two-level particle $a$ with the initial
state $|0\rangle_{a}$ and performs a collective unitary transformation

$\displaystyle
U_{2}=\left(\begin{array}[]{cccccccc}\frac{ac}{bd}&A&0&0&0&0&0&0\\\
A&-\frac{ac}{bd}&0&0&0&0&0&0\\\ 0&0&\frac{a}{b}&B&0&0&0&0\\\
0&0&B&-\frac{a}{b}&0&0&0&0\\\ 0&0&0&0&\frac{c}{d}&C&0&0\\\
0&0&0&0&C&-\frac{c}{d}&0&0\\\ 0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&-1\\\
\end{array}\right)$ (24)

on particles 2, 4 and $a$ under the basis
$\\{|000\rangle_{24a},|001\rangle_{24a},|010\rangle_{24a},|011\rangle_{24a},|100\rangle_{24a},|101\rangle_{24a},$
$|110\rangle_{24a},|111\rangle_{24a}\\},$ where

$\begin{array}[]{ccc}A=\sqrt{1-(\frac{ac}{bd})^{2}},&B=\sqrt{1-(\frac{a}{b})^{2}},&C=\sqrt{1-(\frac{c}{d})^{2}}.\end{array}$

Since it has been assumed that $|a|\leq|b|$ and $|c|\leq|d|$, so one has
$|ac|^{2}\leq|bd|^{2}$. The unitary transformation $U_{2}$ will transform
$|\varphi^{\prime}\rangle_{24}|0\rangle_{a}$ into

$\displaystyle|\varphi^{\prime\prime}\rangle_{24a}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{|bd\alpha|^{2}+|bc\beta|^{2}+|ad\gamma|^{2}+|ac\delta|^{2}}}$
$\displaystyle{[ac(\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle)_{24}}|0\rangle_{a}$
$\displaystyle+(\alpha\sqrt{(bd)^{2}-(ac)^{2}}|00\rangle+c\beta\sqrt{b^{2}-a^{2}}|01\rangle$
$\displaystyle+a\gamma\sqrt{d^{2}-c^{2}}|10\rangle)_{24}|1\rangle_{a}].$

Finally, Bob performs a measurement on auxiliary particle $a$ in the basis
$\\{|0\rangle_{a},|1\rangle_{a}\\}$. If the result of his measurement is
$|1\rangle_{a}$, then the remote preparation of the original state fails. If
the measurement outcome $|0\rangle_{a}$ occurs with probability
$\frac{|ac|^{2}}{|bd\alpha|^{2}+|bc\beta|^{2}+|ad\gamma|^{2}+|ac\delta|^{2}}$,
then the remote preparation of a two-particle state $|\phi\rangle$ is
successfully realized. Evidently, when actual measurement outcome
$|\varphi_{\perp}\rangle_{13}$ is obtained, the probability of successfully
remote state preparation is
$(|bd\alpha|^{2}+|bc\beta|^{2}+|ad\gamma|^{2}+|ac\delta|^{2})\frac{|ac|^{2}}{|bd\alpha|^{2}+|bc\beta|^{2}+|ad\gamma|^{2}+|ac\delta|^{2}}=|ac|^{2}$.

Similarly, by Eq.(12), if Alice’s measurement result on particles 1 and 3 is
$|\psi_{\perp}\rangle_{13}$ with the probability
$|ad\alpha|^{2}+|ac\beta|^{2}+|bd\gamma|^{2}+|bc\delta|^{2}$, the state of
particles 2 and 4 will become

$|\psi\rangle_{24}=\frac{(ac\beta|00\rangle-ad\alpha|01\rangle-
bc\delta|10\rangle+bd\gamma|11\rangle)_{24}}{\sqrt{|ad\alpha|^{2}+|ac\beta|^{2}+|bd\gamma|^{2}+|bc\delta|^{2}}}.$
(26)

Now Bob operates the following unitary transformation

$U^{\prime}_{1}=(|0\rangle\langle 0|-|1\rangle\langle
1|)_{2}\otimes(-|0\rangle\langle 1|+|1\rangle\langle 0|)_{4}$ (27)

on particles 2 and 4. Hence the state shown in Eq.(18) was transformed into

$|\psi^{\prime}\rangle_{24}=\frac{(ad\alpha|00\rangle+ac\beta|01\rangle+bd\gamma|10\rangle+bc\delta|11\rangle)_{24}}{\sqrt{|ad\alpha|^{2}+|ac\beta|^{2}+|bd\gamma|^{2}+|bc\delta|^{2}}}.$
(28)

Next, Bob introduces an auxiliary particle $a$ with the initial state
$|0\rangle_{a}$ and performs a unitary transformation

$\displaystyle
U^{\prime}_{2}=\left(\begin{array}[]{cccccccc}\frac{c}{d}&C&0&0&&0&0&0\\\
C&-\frac{c}{d}&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&-1&0&0&0&0\\\
0&0&0&0&\frac{ac}{bd}&A&0&0\\\ 0&0&0&0&A&-\frac{ac}{bd}&0&0\\\
0&0&0&0&0&0&\frac{a}{b}&B\\\ 0&0&0&0&0&0&B&-\frac{a}{b}\\\ \end{array}\right)$
(37)

on particles 2, 4 and $a$ under the basis $\\{|000\rangle_{24a},$
$|001\rangle_{24a},$ $|010\rangle_{24a},|011\rangle_{24a},$
$|100\rangle_{24a},$ $|101\rangle_{24a},$ $|110\rangle_{24a},$
$|111\rangle_{24a}\\}$. It is straightforward to verify that the resulting
state of of particles 2, 4 and $a$ is

$\displaystyle|\psi^{\prime\prime}\rangle_{24a}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{|ad\alpha|^{2}+|ac\beta|^{2}+|bd\gamma|^{2}+|bc\delta|^{2}}}$
$\displaystyle[ac(\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle)_{24}|0\rangle_{a}$
$\displaystyle+(a\alpha\sqrt{d^{2}-c^{2}}|00\rangle+\gamma\sqrt{(bd)^{2}-(ac)^{2}}|10\rangle$
$\displaystyle+c\delta\sqrt{b^{2}-a^{2}}|11\rangle)_{24}|1\rangle_{a}].$

The above equation shows that Bob can construct a two-particle state, which
Alice wishes to prepare remotely, with certain probability by performing a
measurement on auxiliary particle $a$ in the basis
$\\{|0\rangle_{a},|1\rangle_{a}\\}$. If Bob’s actual measurement result is
$|0\rangle_{a}$, then remote state preparation is successful; otherwise remote
state preparation fails. It is easy to prove that the successful probability
of remote state preparation in this case is
$(|ad\alpha|^{2}+|ac\beta|^{2}+|bd\gamma|^{2}+|bc\delta|^{2})\frac{|ac|^{2}}{|ad\alpha|^{2}+|ac\beta|^{2}+|bd\gamma|^{2}+|bc\delta|^{2}}=|ac|^{2}$.

However, if the Alice’s actual measurement outcome on particles 1 and 3 is
$|\varphi\rangle_{13}$ ($|\psi\rangle_{13}$), Bob will obtain the state
$(ac\alpha|00\rangle+ad\beta^{*}|01\rangle+bc\gamma|10\rangle+bd\delta^{*}|11\rangle)_{24}$
($(ac\gamma|00\rangle+ad\delta^{*}|01\rangle-bc\alpha|10\rangle-
bd\beta^{*}|11\rangle)_{24}$). Since Bob has no knowledge of these states, he
can not unitary convert each of them into the original state, so remote state
preparation fails. But, when $\alpha$, $\beta$, $\gamma$ and $\delta$ are real
numbers, the situation is not the same. In this case Bob can prepare the state
in every Alice’s measurement outcome with certain probability. For saving
space we omit the concrete steps for preparing.

Synthesizing two cases (the Alice’s the actual measurement outcome is either
$|\varphi_{\perp}\rangle_{13}$ or $|\psi_{\perp}\rangle_{13}$) the probability
of the successful remote state preparation is $2|ac|^{2}$. If
$|a|=|b|=|c|=|d|=\frac{1}{\sqrt{2}},$ namely, the quantum channel consists of
two Einstein-Podolsky-Rosen pairs, the probability equals to 50%.

In summary, we have presented a scheme of the remote preparation of a two-
particle state via two Einstein-Podolsky-Rosen pairs and two partial entangled
two-particle state, respectively. The two-particle state can be prepared
probabilistically if the sender performs a measurement and the receiver
introduces an appropriate unitary transformation with the aid of long-range
Einstein-Podolsky-Rosen correlation and transmission of two bits of classical
information. Furthermore we obtained the probability of the successful remote
state preparation of the two-particle state.

Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grant No: 10671054 and Hebei Natural Science Foundation of China under
Grant Nos: A2005000140;07M006, and the Key Project of Science and Technology
Research of Education Ministry of China under Grant No:207011.

## References

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  * (2) Gao T, Wang Z X and Yan F L 2003 Chin. Phys. Lett. 20 2094
  * (3) Ekert A K 1991 Phys. Rev. Lett. 67 661
  * (4) Wang X B, Hiroshima T, Tomita A and Hayashi M 2007 Phys. Rep. 448 1
  * (5) Long G L, Deng F G, Wang C, Li X H, Wen K and Wang W Y 2007 Front. Phys. China,  2 251
  * (6) Bennett C H and Wiesner S J 1992 Phys. Rev. Lett. 69 2881
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  * (11) Lo H K 2000 Phys. Rev. A 62 012313
  * (12) Bennett C H, DiVincenzo D P, Shor P W, Smolin J A, Trehal B M and Wootters W K 2001 Phys. Rev. Lett. 87 077902
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# Taxpayer deductions and the endogenous probability of tax penalisation

Alex A.T. Rathke NECCT/FEA-RP/USP, University of São Paulo. Avenida dos
Bandeirantes 3900, 14040-905 Ribeirão Preto, SP, Brazil. E-mail:
<EMAIL_ADDRESS>
###### Abstract

We propose a parametric specification of the probability of tax penalisation
faced by a taxpayer, based on the amount of deduction chosen by her to reduce
total taxation. Comparative analyses lead to a closed-form solution for the
optimum tax deduction, and provide the maximising conditions with respect to
the probability parameters.

Keywords: probability of tax penalisation; tax deductions; expected tax
penalty.
JEL Classification: C02; D01; K34.

##  1 Introduction

Theoretical literature on tax evasion and optimal tax decisions predominantly
follows the studies of [1] and [6]. In the standard model, the taxpayer aims
to obtain a gain from under-reporting the true amount of taxable income,
however she may be penalised with legal fines if the evasion is caught by tax
authorities. Overall results show that, if the expected penalty cost is convex
increasing, then there may be an equilibrium amount of non-reported income
which maximises her expected utility.

Most studies assume that the probability of penalisation is either convex
increasing or completely exogenous, which leads to the existence of the
equilibrium. These assumptions imply analytical limitations. On the one hand,
a fully-convex probability applies only to an upper-closed support, e.g. if
the penalisation probability is a function of the total income, then it is not
differentiable on the full support. On the other hand, an exogenous
probability function limits the analysis of the taxpayer’s behaviour [2], as
her perception of risk will not change if the amount of under-reported income
changes.

This study derives a specification of the probability of penalisation, as a
function of the amount of tax deduction chosen by the taxpayer to reduce her
total taxation. We apply the properties of the traditional hazard function [3]
to obtain an endogenous probability function with generalised parameters. For
any probability function defined on the support $[0,\infty)$, we find that the
optimal tax deduction exists if the mode of the probability distribution is
not zero. Comparative analyses show how the probability parameters reflect the
changes in tax enforcement policies. Our model applies directly to the
standard tax evasion studies, where the penalisation probability is a function
of the total income or the reported income.

##  2 The model

###  2.1 Taxable income and optimal tax deduction

Let a taxpayer obtain income $\pi(w)$ based on her own effort $w$, which must
be taxed at a tax rate $t\in[0,1]$. Tax rules allow the taxpayer to deduce
some amount $m\geq 0$ before paying taxes. Tax rules also require the taxpayer
to report her income minus deductions to the tax authorities, for mandatory
inspection. Assume the taxpayer cannot under-report $\pi$, and the after-tax
income enjoyed by the taxpayer is $U(w)=\pi-t(\pi-m)$.

The taxpayer analyses the tax rules and chooses the amount $m$ by self-
assessment, however she has no information about how the tax authorities will
evaluate $m$. If tax authorities interpret the deduction $m$ as abusive, they
reclaim the payment of taxes over some excess deduction $\beta m$,
$\beta\in[0,1]$, plus a penalty rate $z\geq 0$ over this amount to compensate
the abuse [1, 6].

Larger deductions are more likely to be scrutinised by tax authorities. For
the event of penalisation by tax authorities $h\in\Omega$, define a non-
negative random variable $X(h):\Omega\rightarrow\mathbb{R}_{+}$. The
probability of $X$ to be located within some interval $[0,m]$ is equal to the
cumulative probability of $X$ at $m$, $P(X\leq m)=F_{X}(m)=F$, which is the
same as the probability of the event of tax penalisation $h$ given the
deduction $m$. The usual properties apply: $F(0)=0$, $\partial F/\partial
m=f>0$, $\lim_{m\to\infty}F=1$.

The expected after-tax income of the taxpayer is

$E(U)=\pi-t(\pi-m)-F\cdot[\beta mt(1+z)].\\\ $ (1)

Simplify the expression $\beta(1+z)=B$. Maximisation with respect to $m$ leads
to the first and second-order conditions respectively equal to

$1=(F+fm)B;\\\ $ (2)

$\dfrac{\partial f}{\partial m}=f^{\prime}>-\dfrac{2f}{m}.\\\ $ (3)

Eq. 2 derives a positive equilibrium $\text{arg max}_{m}E(U)=m_{*}>0$, since
we have $1>[F(0)+f(0)\cdot 0]=0$ for any probability function $F$. The
convexity condition in Eq. 3 shows that $m_{*}$ is a maximum point if the
probability distribution $f$ is increasing at the left of the support
$[0,\infty)$ such that for some amount $m$, we have
$m_{*}\in(0,m]\subset[0,\infty)\rightarrow
f^{\prime}(m_{*})>0>-2f(m_{*})/m_{*}$.

###  2.2 The probability of tax penalisation and comparative statics

We derive a specification of $F$ with respect to the endogenous variable $m$
based on a survival process.

The taxpayer chooses the amount of tax deduction $m>0$, producing a
probability of tax penalisation equal to $F_{X}(m)>0$. Assume now that the
taxpayer may increase the deduction to some amount $m+\delta$, $\delta>0$,
which reduces the total taxes by $\delta t$, however it increases the
probability of penalisation. Given an initial deduction $m$, the increase in
the probability of penalisation produced by $\delta$ is equal to

$P(m\leq X\leq m+\delta|X\geq
m)=\dfrac{F_{X}(m+\delta)-F_{X}(m)}{1-F_{X}(m)}.\\\ $ (4)

Dividing Eq. 4 by $\delta$ and taking the limit $\delta\rightarrow 0$, we
obtain

$\lim_{\delta\to
0}\dfrac{F_{X}(m+\delta)-F_{X}(m)}{\delta\cdot[1-F_{X}(m)]}=\dfrac{f_{X}(m)}{1-F_{X}(m)}=\lambda,\\\
$ (5)

which defines the traditional hazard function $\lambda(m)$ [3] satisfying
$m\geq 0\rightarrow\lambda\geq 0$, $\int_{0}^{\infty}\lambda dm=\infty$.
Integrating both sides of Eq. 5 on the interval $[0,m]$ and solving for
$F_{X}(m)=F$, we obtain the probability of tax penalisation equal to

$F=1-e^{-\int_{0}^{m}\lambda dy}.\\\ $ (6)

We assume that the hazard of penalisation $\lambda$ is a monotone increasing
function of $m$, since tax authorities are stricter against larger tax
deductions [2]. Therefore, conventional Laplace transform allows a parametric
representation; i.e. we may parametrise $\lambda=Am^{k-1}$, where $A>0$ and
$k>0$ are general scale and shape parameters respectively111Formally, any
completely monotone function $g:(0,\infty)\rightarrow\mathbb{R}_{+}$ may be
represented as a mixture of exponentials, see [5]., for the probability of tax
penalisation becomes

$F=1-e^{-\frac{A}{k}m^{k}}.\\\ $ (7)

Substituting Eq. 7 in Eq. 2 and Eq. 3, we obtain the optimal tax deduction
$m_{*}$ and the maximising conditions in parametric form, for we have

$\begin{array}[]{rl}0<m_{*}&=\left(\dfrac{1-Wk}{A}\right)^{1/k}<\left(\dfrac{1+k}{A}\right)^{1/k};\\\
\\\
-1<W&=W_{0}\left[\left(1-\dfrac{1}{B}\right)\cdot\dfrac{e^{1/k}}{k}\right]<\dfrac{1}{k};\\\
\\\ -\dfrac{1}{e}&<\left(1-\dfrac{1}{B}\right)\cdot\dfrac{e^{1/k}}{k};\\\ \\\
0<B<1&\rightarrow 1<W_{0}\left(\dfrac{B}{e(1-B)}\right)^{-1}<k,\\\
\end{array}$ (8)

where $W_{0}:\mathbb{R}\rightarrow[-1,\infty)$ is the principal real-valued
branch of the Lambert $W$-function [4], which provides an analytical solution
for $m_{*}$. The closed-form expression for $m_{*}$ is straightforward. The
inequalities in Eq. 8 with respect to $W$, $k$, $B$ derive from the second-
order condition for $m_{*}$ to be a maximum point222$k>1$ means that the
probability function $F$ is S-shaped with a convex region at the left of the
support $[0,\infty)$, therefore the optimal point $m_{*}$ exists for some
interval $m_{*}\in(0,m]$, $f^{\prime}(m_{*})>0$. If $k\leq 1$, the probability
$F$ is concave on the full support, for the second-order condition in Eq. 3 is
not satisfied..

Assume the conditions in Eq. 8 are satisfied. Comparative analyses regarding
$m_{*}$ provide the following:

$\dfrac{\partial m_{*}}{\partial A}=-\dfrac{m_{*}}{Ak}<0;\\\ $ (9)
$\dfrac{\partial m_{*}}{\partial
B}=-\dfrac{m_{*}^{1-k}}{AB^{2}\cdot(1-1/B)}\cdot\dfrac{W}{W+1}<0;\\\ $ (10)
$\dfrac{\partial m_{*}}{\partial
k}=\dfrac{m_{*}}{k}\cdot\left(\dfrac{W-(1+1/k)\frac{W}{W+1}}{Wk-1}-log(m_{*})\right).\\\
$ (11)

The negative effects from variables $A$ and $B$ are intuitive, since $A$ is an
inverse scale of $m_{*}$, while $B$ increases after increase of either the
reclaim value $\beta$ or the penalty rate $z$ [1, 6]. The impact of the shape
parameter $k$ in Eq. 11 is more intricate. Close inspection shows that the
derivative $\partial m_{*}/\partial k$ depends on the variation of
$W\in(-1,1/k)$ within its range, such that $W\geq 0\rightarrow\partial
m_{*}/\partial k>0$, $\lim_{W\to-1}\partial m_{*}/\partial k=-\infty$,
$\lim_{W\to 1/k}\partial m_{*}/\partial k=0_{+}$. Since Eq. 11 is continuous,
it means that $\partial m_{*}/\partial k$ is inverse-U shaped with a positive
maximum, and there is a value $W$ for which Eq. 11 is zero.

##  3 Discussion

In our analysis, the probability $F$ is fully specified by the parameters $A$
and $k$333Eq. 7 becomes a version of the Weibull distribution., which models
the expected behaviour of tax authorities given the amount of tax deduction
$m$. Higher tax enforcement implies a higher probability of penalisation, so
the curve $F$ moves to the left of the support, which implies a larger $A$ or
lower $k$. Lower tax enforcement implies the opposite effect on parameters.
Eq. 7 applies directly to the usual enforcement policies including audit rules
and fines [2]. Variations on the hazard function $\lambda$ may reflect
different assumptions or more detailed parametrisations, which derive the
probability function $F$ by direct manipulation within Eq. 6.

Some extensions are immediate. For the standard model in tax literature, $m$
represents the amount of evaded income, and $F$ may be a function of the
reported income $\pi-m\geq 0$ [1], with analogous specification. If $m$ is
limited to a closed interval, say $m\in[0,\pi]$, $F(\pi)=1$, then Eq. 7
simplifies to

$F(m\in[0,\pi])=1-\left(1-\dfrac{m^{k}}{\pi}\right)^{\frac{A\pi}{k}},\\\ $
(12)

therefore the probability $F$ becomes a function of the tax deduction $m$ and
the total income $\pi$ [6]. For analytical simulations or estimation of the
optimal point $m_{*}$, the Lambert W-function in Eq. 8 adopts several
approximations, see [4].

## References

  * [1] Allingham, M. G., & Sandmo, A. (1972). Income tax evasion: A theoretical analysis. _Journal of Public Economics, 1_ , 323-338.
  * [2] Alm, J., & Malézieux, A. (2021). 40 years of tax evasion games: a meta-analysis. _Experimental Economics, 24_(3), 699-750.
  * [3] Cox, D. R. (1972). Regression models and life tables. _Journal of the Royal Statistical Society: Series B (Methodological), 34_(2), 187-202.
  * [4] Iacono, R., & Boyd, J. (2017). New approximations to the principal real-valued branch of the Lambert W-function. _Advances in Computational Mathematics, 43_(6), 1403-1436.
  * [5] McHugh, J. A. M. (1975). A proof of Bernstein’s theorem on regularly monotonic functions. _Proceedings of the American Mathematical Society, 47_(2), 358-360.
  * [6] Srinivasan, T. N. (1973). Tax evasion: A model. _Journal of Public Economics, 2_(4), 339-346.