# Astrophysical observables for regular black holes with sub-Planckian
curvature

Wei Zeng1<EMAIL_ADDRESS>Yi Ling 2,3,1<EMAIL_ADDRESS>Qing-Quan
Jiang1<EMAIL_ADDRESS>1 School of Physics and Astronomy, China West
Normal University, Nanchong 637002, China
2 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, China
3 School of Physics, University of Chinese Academy of Sciences, Beijing
100049, China

###### Abstract

We investigate the photon sphere and the extremal stable circular orbit (ESCO)
for massive particles over a recently proposed regular black holes with sub-
Planckian curvature and Minkowskian core. We derive the effective potential
for geodesic orbits and determine the radius of circular photon orbits, with
an analysis on the stability of these orbits. We extend our analysis to the
background of compact massive object (CMO) without horizon, whose mass is
below the lowest bound for the formation of a black hole. For massive
particles, the ESCOs become double-valued in CMO phase and we calculate the
innermost stable circular orbits (ISCO) and the outermost stable circular
orbits (OSCO). By comparing wih Bardeen black hole and Hayward black hole, it
is also found that the locations of photon sphere and ESCO in CMO phase with
Minkowskian core are evidently different from the ones in CMO phase with dS
core, which potentially provides a way to distinguish these two sorts of black
holes by astronomical observation.

##  I Introduction

It had been a challenging problem to identify a compact massive object to be a
black hole by astronomical observation until recently remarkable progress was
made by the Event Horizon Telescope (EHT) collaboration team. The successful
detection of black holes at the center of M87 and Sgr A* has announced the
coming of a new age for the study of black holes as genuine astrophysical
objectsEventHorizonTelescope:2019dse EventHorizonTelescope:2022xnr . In this
direction, detecting the photon sphere closely surrounding the black hole
silhouette plays a key role in measuring various parameters of black holes,
such as the size, the mass as well as the spin of black holes.

It is well known that the existence of black hole solutions in general
relativity also brings some fundamental problems which are notoriously hard to
solve in theoretical physics. Among of them the most famous one is the
singularity problem, which tells us that the scalar curvature becomes
divergent at the center of black holesHawking:1966sx ; Penrose:1964wq ;
Joshi:2011rlc ; Goswami:2005fu ; Janis:1968zz . This trouble becomes severe
after the Hawking radiation was discovered, which leads to the evaporation of
black holes and finally the information loss paradoxHawking:1976ra ;
Giddings:1992hh ; Hawking:1975vcx ; Preskill:1992tc ; Xiang:2013sza ;
Chen:2014jwq ; Casadio:2000py ; Unruh:1976db ; Page:1993wv . Theoretically,
people believe that the infinity of the Kritchmann scalar curvature just
implies the classical general relativity would not be applicable to the space
time with extremal environment, and the singularity at the center of a
classical black hole might be removed or avoided by the quantum effects of
gravityGaray:1994en ; tHooft:1984kcu ; Han:2004wt ; Donoghue:1993eb ;
Callan:1992rs ; DeWitt:1967yk ; DeWitt:1967ub ; DeWitt:1967uc ; Calmet:2017qqa
; Ali:2015tva . Before a complete theory of quantum gravity could be
established, it is very desirable to construct some black hole solutions which
are singularity free. In literature, such regular black holes were constructed
at the phenomenological level indeed, which now may be classified by their
asymptotic behavior near the center of black holes. One sort of regular black
holes has asymptotically de-Sitter core, including the famous Bardeen black
hole, Hayward black hole as well as the Frolov black holeBardeen:1968 ;
Hayward:2005gi ; Frolov:2014jva ; while the other sort of regular black holes
has asymptotically Minkowskian core, which is characterized by an
exponentially suppressed Newton potentialXiang:2013sza ; Ling:2021olm ;
Li:2016yfd ; Ben-Amots:2011lbe ; Simpson:2019mud ; Culetu:2013fsa ;
Martinis:2010zk . In particular, a new class of regular black holes with
asymptotically Minkowskian core was constructed recently in Ling:2021olm ,
whose scalar curvature is not only finite, but also maintains to be sub-
Planckian during the course of evaporation, irrespective of the mass of the
black hole. Such a scenario is quite compatible with the expectation of
quantum gravity, which asserts that the energy scale of any object should be
bounded by the Planck energy in the context of quantum theory of gravity. In
addition, the connection between regular black holes with asymptotically dS
core and those with asymptotically Minkowskian core has been disclosed in
Ling:2021olm . Recently, more features of this sort of black holes have been
investigated in Ling:2021abn ; Ling:2022vrv .

Currently, although it is still far to provide a complete understanding on
these regular black holes, it is very intriguing to link these regular black
holes to astrophysical observation and explore if there exists any evidence or
signature implying that the detected black holes in the sky would be regular
black hole rather than the traditional black hole with singularity. Obviously,
at the current stage it is impractical to distinguish singular black holes and
regular black holes by diving into the horizon or detecting any signal coming
out from the interior of the horizon. Nevertheless, the difference inside
black holes may leave signatures on the featured phenomena outside black
holes, such as the photon sphere or the trajectories of massive
particlesClaudel:2000yi ; Boonserm:2018orb ; Berry:2020ntz ; Decanini:2009mu ;
Stefanov:2010xz ; Wei:2011zw ; Wei:2013mda ; Cunha:2018acu . With this
purpose, in this paper we intend to provide detailed analysis on the photon
sphere and the extremal stable circular orbit (ESCO) for massive particles
over recently proposed regular black holes with sub-Planckian curvature, and
expect our theoretical investigation will provide basis and useful information
for distinguishing different kinds of black holes by astrophysical observation
in near future.

The paper is organized as follows. In next section we will present a short
review on regular black holes with sub-Planckian curvature which are
characterized by two factors $(x,n)$. We will focus on the structure of its
horizon, which usually contains two branches, namely, the inner horizon and
the outer horizon. When the mass is below some value, the horizon will
disappear and the black hole becomes a horizonless compact massive object. In
section three and four, we consider two regular black holes with $(x=2/3,n=2)$
and $(x=1,n=3)$, respectively, and determine the radius of circular photon
orbits, with an analysis on the stability of these orbits. For massive
particles, the curve of ESCO becomes double-valued in CMO phase and we
calculate the innermost stable circular orbits (ISCO) and the outermost stable
circular orbits(OSCO). By comparing with those in Bardeen black hole and
Hayward black holes, we find that the locations of photon sphere and ESCO over
the horizonless CMO with Minkowskian core are evidently different from the
ones over the CMO with dS core, which potentially provides a way to
distinguish these two sorts of black holes by astronomical observation.

##  II The regular black hole with sub-Planckian Kretschmann scalar curvature

In Ref.Ling:2021olm , a new sort of regular black holes with sub-Planckian
Kretschmann scalar curvature was proposed with the following metric

$ds^{2}=-(1+2\psi(r))dt^{2}+\frac{1}{(1+2\psi(r))}dr^{2}+r^{2}d\Omega^{2},$
(1)

where $\psi(r)$ is understood as the modified Newton potential which is
specified as

$\psi(r)=-\frac{M}{r}e^{\frac{-\alpha_{0}M^{x}}{r^{n}}}.$ (2)

The modified Newton potential is characterized by an exponentially suppressing
form with a deviation parameter $\alpha_{0}$. Obviously, when $\alpha=0$, it
goes back to the standard Schwarzschild black hole. The behavior of
Kretschmann scalar curvature at the core depends on both factors $x$ and $n$.
In Ref.Ling:2021olm , it has been pointed out that to guarantee the existence
of the horizon and the scalar curvature to be sub-Planckian, two factors must
satisfy the conditions $n\geq x\geq n/3$ and $n\geq 2$. Furthermore, as
$r\rightarrow 0$, the space time is characterized by a Minkowskian core, in
contrast to the well-known regular black holes such as Bardeen black hole,
Hayward black hole as well as Frolov black hole, which are characterized by a
de-Sitter core. Nevertheless, a one-to-one correspondence between such regular
black holes with Minkowskian core and the ones with de-Sitter core has been
constructed in Ling:2021olm . Explicitly, given a regular black hole with
Minkowskian core which is described by Eq.(2), then correspondingly there
exists a regular black hole with de-Sitter core whose Newton potential
$\psi(r)$ is given by

$\psi(r)=-\frac{Mr^{\frac{n}{x}-1}}{(r^{n}+x\alpha_{0}M^{x})^{1/x}}.$ (3)

Specially, Bardeen black hole is just given by specifying $x=2/3,n=2$, while
Hayward black hole is given by $x=1,n=3$. These two sorts of regular black
holes exhibit identical asymptotic behavior at large scales but possess
different cores at the center. Therefore, it is quite intriguing to explore
practical ways to distinguish them by possible astronomical observation in
future. Here as the starting point we will investigate this issue from the
theoretical point of view by comparing the distinct behavior of photon spheres
and ESCO of massive particles over such two sorts of backgrounds.

Now we present some basic properties of the regular black holes with the
metric given by Eq.(1) and Eq.(2). In general, the location of the horizon is
determined by $1+2\psi(r_{h})=0$, namely

$2M=r_{h}e^{{\alpha_{0}M^{x}}/{r_{h}^{n}}}.$ (4)

From this equation, one may obtain the radius of the horizon $r_{h}$ as the
function of the mass

$r_{H}=\left[-\frac{M^{x}n\alpha_{0}}{W(-2^{-n}M^{x-n}n\alpha_{0})}\right]^{\frac{1}{n}},$
(5)

where $W$ is the Lambert-W function which allows two possible branches of
solutions with appropriate values of $\alpha_{0}$, corresponding to the outer
horizon and the inner horizon, respectively.

Next we consider the geodesics of photons and massive particles over such a
background. Firstly, since there are two Killing vectors, namely
$(\partial/\partial t)^{a}$ and $(\partial/\partial\phi)^{a}$ for the space
time, correspondingly there are two conservative quantities for freely moving
particles, namely the energy $E$ and the angular momentum $L$. Now it is
straightforward to derive the effective potential for geodesic orbits as

$V_{\epsilon}(r)=\left(1+2\psi\right)\left(\frac{L^{2}}{r^{2}}-\epsilon\right),$
(6)

where $\epsilon=0$ for photons and $\epsilon=-1$ for massive particles. As a
result, the location of the photon sphere $r_{c}$ is determined by the
vanishing of the derivative of the effective potential with respect to the
radial coordinate, giving rise to

$-3Mr_{c}^{n}+e^{M^{x}r_{c}^{-n}\alpha_{0}}r_{c}^{1+n}+M^{1+x}n\alpha_{0}=0.$
(7)

In parallel, for massive particles the location of ESCO is denoted as $r_{e}$
which is determined by the equation

$\displaystyle
r_{e}^{2n}\left(-6M+e^{M^{x}r_{e}^{-n}\alpha_{0}}r_{e}\right)-M^{x}nr_{e}^{n}\left[2M(-4+n)-e^{M^{x}r_{e}^{-n}\alpha_{0}}nr_{e}\right]\alpha_{0}$
(8)
$\displaystyle-M^{2x}n^{2}\left(2M+e^{M^{x}r_{e}^{-n}\alpha_{0}}r_{e}\right)\alpha_{0}^{2}=0.$

The derivation of this equation can be found in next sections for special x
and n.

Following the strategy presented in Berry:2020ntz , we may define two
dimensionless quantities as

$\omega=\frac{r}{\alpha_{0}^{1/(n-x)}};\quad\quad\quad
z=\frac{M}{\alpha_{0}^{1/(n-x)}},$ (9)

then Eq.(5), Eq.(7), Eq.(8) can be rewritten as

$\omega_{H}=2z\left[\frac{\theta}{W(\theta)}\right]^{\frac{1}{n}}\quad;\quad\theta=-\frac{n}{2^{n}z^{n-x}},$
(10)

and

$-3z\omega_{c}^{n}+e^{z^{x}\omega_{c}^{-n}}\omega_{c}^{1+n}+z^{1+x}n=0,$ (11)

and

$\displaystyle\omega_{e}^{2n}\left(-6z+e^{z^{x}\omega_{e}^{-n}}\omega_{e}\right)-z^{x}n\omega_{e}^{n}\left[2z(-4+n)-e^{z^{x}\omega_{e}^{-n}}n\omega_{e}\right]$
(12)
$\displaystyle-z^{2x}n^{2}\left(2z+e^{z^{x}\omega_{e}^{-n}}\omega_{e}\right)=0.$

It means that under the condition $x\neq{n}$, we may plot the locations of
horizon, the photon sphere and the ESCO as the functions of $z$, which exhibit
a universal behavior independent of the deviation parameter $\alpha_{0}$.
Nevertheless, one can easily derive the effects of the deviation parameter
with the use of parameters $M=z\alpha_{0}^{\frac{1}{n-x}}$ and
$r=\omega\alpha_{0}^{\frac{1}{n-x}}$.

##  III The regular black hole with $x=2/3$ and $n=2$

In this section we investigate the photon sphere and ESCO over the regular
black hole with $x=2/3$ and $n=2$, which corresponds to Bardeen black hole at
large scales. The thermodynamic behavior of this black hole as well as the
feature of Kretschmann scalar curvature has been investigated in detail in
Ling:2021olm .

Figure 1:  The outer horizon (black solid line) and the inner horizon (black
dashed line) of the black hole with $x=2/3$ and $n=2$. The vertical dashed
line in orange is the borderline between the region in black hole phase
(right) and the region in horizonless CMO phase (left).

Firstly, we present the basic properties of the regular black hole with
$x=2/3$ and $n=2$. The location of the horizon is given by Eq.(5). Remind that
Lambert-W function has two branches in the real regionCorless:1996zz ;
Valluri:2000zz ; Valluri:2009zz , which give rise to the inner horizon and the
outer horizon separately for this regular black hole as

$\omega_{H^{+}}=2z\left[\frac{\theta}{W_{0}(\theta)}\right]^{1/2};\quad\quad\omega_{H^{-}}=2z\left[\frac{\theta}{W_{-1}(\theta)}\right]^{1/2};\quad\quad\theta=-\frac{1}{2z^{4/3}}.$
(13)

We plot the inner horizon and outer horizon as the function of the mass in
Fig.(1). From this figure one notices that with the decrease of the mass, the
inner horizon and the outer horizon become closer and finally merge at some
critical point. As a matter of fact, the location of such a critical point can
be evaluated by the feature of Lambert-W function as follows.

Two branches of Lambert-W function merge at $W_{-1}(\theta)=W_{0}(\theta)$,
implying that $W(\theta)=-1$. Using the identity of Lambert-W function

$\theta=W(\theta)e^{W(\theta)},$ (14)

one finds the inner horizon and outer horizon merge at $\theta=-1/e$, namely

$z=\frac{M}{\alpha_{0}^{3/4}}=\left(\frac{e}{2}\right)^{3/4}=1.258...;\quad\omega=\frac{r}{\alpha_{0}^{3/4}}=1.526....$
(15)

As a result, the condition for the existence of the horizon is
$\theta\geq-1/e$, namely

$z\geq\left(\frac{e}{2}\right)^{3/4},$ (16)

below of which the horizon disappears such that the black hole becomes a CMO.
It is interesting to notice that this situation is in contrast to the standard
Schwarzschild black hole, where the horizon does not disappear as the mass
decreases, but shrinks to zero with its mass.

###  III.1 Photon Spheres

Next we consider the property of photon spheres over this background. The
effective potential is given by

$V_{\epsilon}(\omega)=\left(1-\frac{2ze^{-\frac{z^{2/3}}{\omega^{2}}}}{\omega}\right)\left(\frac{L^{2}}{\omega^{2}}-\epsilon\right).$
(17)

For null trajectories, the effective potential is

$V_{0}(\omega)=\left(1-\frac{2ze^{-\frac{z^{2/3}}{\omega^{2}}}}{\omega}\right)\left(\frac{L^{2}}{\omega^{2}}\right).$
(18)

Thus, the location of the circular photon orbit can be determined by
$V_{0}^{\prime}(\omega_{c})=0$, leading to

$-3z\omega_{c}^{2}+e^{\frac{z^{2/3}}{\omega_{c}^{2}}}\omega_{c}^{3}+2z^{5/3}=0.$
(19)

We numerically plot the radius of the photon sphere as the function of the
mass in Fig.(2). First of all, we notice without surprise that the radius of
the photon sphere is always larger than the outer horizon. In particular, if
we rewrite the equation for the photon sphere in terms of $M$ and $r$, then it
becomes

Figure 2:  The location of photon spheres in the spacetime with $x=2/3$ and
$n=2$. The blue line represents the location of photon spheres in black hole
phase, while the red solid line is the location of unstable photon spheres in
CMO phase, and the red dashed line is the location of stable photon spheres in
CMO phase. The turning point of each curve is marked by a black dot as well.

$-3Mr_{c}^{2}+e^{\frac{M^{2/3}\alpha_{0}}{r_{c}^{2}}}r_{c}^{3}+2M^{5/3}\alpha_{0}=0.$
(20)

Obviously, when $\alpha_{0}$ is vanishing, the location of the photon sphere
is $r_{c}=3m$, a well-known result for Schwarzschild black hole. Secondly, we
find in the horizonless CMO phase, the radius of the photon sphere becomes
double-valued, as illustrated in Fig.(2). This phenomenon is consistent with
the results presented in Cunha:2017qtt ; Guo:2020qwk , where it is found that
photon spheres always appear in pairs outside a horizonless compact object.
The location of the turning point is given by $dz/d\omega_{c}=0$, namely

$\frac{dz}{d{\omega_{c}}}=\frac{2e^{\frac{z^{2/3}}{\omega_{c}^{2}}}z^{2/3}+6z\omega_{c}-3e^{\frac{z^{2/3}}{\omega_{c}^{2}}}\omega_{c}^{2}}{10z+2e^{\frac{z^{2/3}}{\omega_{c}^{2}}}\omega_{c}-9z^{1/3}\omega_{c}^{2}}=0.$
(21)

Numerically one finds the coordinates for the turning point are
$(z=1.037...,\omega=1.929...)$.

As a summary of this subsection, we argue that the photon spheres exhibit
abundant structure in this background. In the region of black hole phase
$(z\geq 1.258)$, the radius of the photon sphere is single-valued and
approaches $3M$ as $\alpha_{0}$ goes to zero. In the region of CMO phase with
$(1.258\geq z\geq 1.037)$, the radius of the photon sphere has two solutions
with the same mass; while in the region with $(z<1.037)$, we find no photon
sphere exists.

###  III.2 The stability of photon spheres

Next we are concerned with the stability of photon spheres, which is related
to the second order of the derivative of the effective potential. The stable
photon orbit locates at a place where $V^{\prime\prime}_{0}(\omega_{c})\geq
0$,

$V^{\prime\prime}_{0}(\omega_{c})=\frac{2e^{-\frac{z^{2/3}}{\omega_{c}^{2}}}L^{2}\left(-12z\omega_{c}^{4}+3e^{\frac{z^{2/3}}{\omega_{c}^{2}}}+18z^{5/3}\omega_{c}^{2}-4z^{7/3}\right)}{\omega_{c}^{9}}\geq
0.$ (22)

So, we have

$-12z\omega_{c}^{4}+3e^{\frac{z^{2/3}}{\omega_{c}^{2}}}\omega_{c}^{5}+18z^{5/3}\omega_{c}^{2}-4z^{7/3}\geq
0.$ (23)

One can figure out the stable region of photon orbits by numerically solving
this equation. Combining Eq. (19) and Eq. (23), we can get the demarcation
point of whether the photon sphere orbit is stable or unstable, as illustrated
in Fig.(2). We find that the photon sphere is not stable in the region of
black hole phase. While in the region of CMO phase, the upper branch is
unstable, but the lower branch is stable.

###  III.3 Circular orbits for massive particles

For massive particles, we set $\epsilon=-1$ in the effective potential

$V_{-1}(\omega)=\left(1-\frac{2ze^{-\frac{z^{2/3}}{\omega^{2}}}}{\omega}\right)\left(\frac{L^{2}}{\omega^{2}}+1\right).$
(24)

So, the radius of circular orbits can be determined by
$V^{\prime}_{-1}(\omega_{e})=0$, leading to

$z\omega_{e}^{4}-2\omega_{e}^{5/3}z^{2}-L^{2}\left(-3z\omega_{e}^{2}+e^{\frac{z^{2/3}}{\omega_{e}^{2}}}\omega_{e}^{3}+2z^{5/3}\right)=0.$
(25)

With the above equation the angular momentum can be written as

$L^{2}=\frac{z\omega_{e}^{4}-2\omega_{e}^{5/3}z^{2}}{-3z\omega_{e}^{2}+e^{\frac{z^{2/3}}{\omega^{2}}}\omega_{e}^{3}+2z^{5/3}}.$
(26)

Figure 3:  The location of ESCO for massive particles in the spacetime with
$x=2/3$ and $n=2$. The purple solid line represents the location of ESCO in
black hole phase, while the purple dashed line is the ESCO in CMO phase, where
the upper branch is ISCO and the lower branch is OSCO. In the region shaded in
orange, all the circular orbits are unstable, while in the region shaded in
pink, all the circular orbit are stable. The region shaded in green denotes
the place where the circular orbit does not exist. The turning point of each
curve is marked by a black dot as well.

The positivity of the angular momentum $(0\leq L^{2}<\infty)$ constrains the
conditions for the existence of the circular orbit to be

$z\omega_{e}^{4}-2\omega_{e}^{5/3}z^{2}\geq
0;\quad-3z\omega_{e}^{2}+e^{\frac{z^{2/3}}{\omega_{e}^{2}}}\omega_{e}^{3}+2z^{5/3}>0.$
(27)

Furthermore, the stability of these circular orbits is determined by the
second derivatives of the effective potential, namely
$V_{-1}^{\prime\prime}(\omega_{e})\geq 0$, which requires

$-\omega_{e}\left(\omega_{e}^{4}+4z^{2/3}\omega_{e}^{2}-4z^{4/3}\right)+e^{\frac{-z^{2/3}}{\omega_{e}^{2}}}\left(6z\omega_{e}^{4}-8z^{5/3}\omega_{e}^{2}+8z^{7/3}\right)\geq
0.$ (28)

The equal sign in the above equation gives rise to the location of ESCO which
is plotted as a curve in $(z,\omega)$ plane, as illustrated in Fig.(3). In
this figure we notice that in black hole phase, the ESCO is single-valued and
it is an ISCO; while in the region of CMO phase, there are two possible ESCO
locations for a fixed value of $z$. In this case the upper branch is an ISCO
and the lower branch is an OSCO. Moreover, the curve of ESCO does not
terminate at the horizon, but at the photon sphere.

The turning point of the curve of ESCO can be obtained by solving
$dz/d\omega_{e}=0$, which gives rise to

$16z^{3}-16z^{3/7}\omega_{e}^{2}+4e^{\frac{z^{2/3}}{\omega_{e}^{2}}}z^{4/3}\omega_{e}^{3}-4z^{5/3}\omega_{e}^{4}-12e^{\frac{z^{2/3}}{\omega_{e}^{2}}}z^{2/3}\omega_{e}^{5}+24z\omega_{e}^{6}-5e^{\frac{z^{2/3}}{\omega_{e}^{2}}}\omega_{e}^{7}=0$
(29)

The numerical analysis indicates that the turning point is located at

$z=\frac{M}{\alpha_{0}^{3/4}}=0.850...;\quad\omega=\frac{r}{\alpha_{0}^{3/4}}=2.660....$
(30)

At the turning point, the ESCO turns from an ISCO into an OSCO.

As the summary of this subsection, we remark that the circular orbits for
massive particles exist only in the region outside photon pheres. In
particular, those between the curve of ESCO and the curve of photon sphere are
unstable, while those outside the curve of ESCO are stable, as illustrated in
Fig.(3).

Figure 4:  The comparison of photon spheres and ESCO in the spacetime with
Minkowskian core $(x=2/3,n=2)$ and those in Bardeen spacetime with dS core.

###  III.4 In comparison with Bardeen black hole

It is quite instructive to compare the photon sphere as well as ESCO over the
spacetime with $(x=2/3,n=2)$ and those over Bardeen spacetime, since these two
spacetimes have the same asymptotic behavior at large scales in radial
direction.

Bardeen black hole has a dS core. The previous investigation on photon orbits
over Bardeen black hole can be found in Stuchlik:2014qja . With the same value
of the deviation parameter $\alpha_{0}$, the effective potential is given by

$V_{\epsilon}(\omega)=\left[1-\frac{2z\omega^{2}}{\left(\omega^{2}+\frac{2}{3}z^{2/3}\right)^{3/2}}\right]\left(\frac{L^{2}}{\omega^{2}}-\epsilon\right).$
(31)

Thus the radius of the photon circular orbits is determined by
$V^{\prime}_{0}(\omega_{c_{2}})=0$, namely

$-\frac{2}{\omega_{c_{2}}^{3}}+\frac{54\sqrt{3}z\omega_{c_{2}}}{\left(3\omega_{c_{2}}^{2}+2z^{2/3}\right)^{5/2}}=0.$
(32)

For massive particles, the ESCO can be obtained in a parallel way as we
demonstrate for the black hole with $x=2/3$ and $n=2$, which is given by

$\displaystyle
64z^{2}\left({2z^{2/3}+3\omega_{e_{2}}^{2}}\right)^{1/2}-162z^{2/3}\omega_{e_{2}}^{4}\left(2z^{2/3}+3\omega_{e_{2}}^{2}\right)^{1/2}$
(33)
$\displaystyle-27\omega_{e_{2}}^{6}\left[\left(2z^{2/3}+3\omega_{e_{2}}^{2}\right)^{1/2}-6\left(3\right)^{1/2}z\right]=0.$

We compare the radius of photon spheres as well as ESCO in these two
spacetimes with the same deviation parameter in Fig.(4). In black hole phase,
it is hard to distinguish the regular black hole with $(x=2/3,n=2)$ and
Bardeen black hole by detecting either the photon sphere or the ESCO, since
these curves are almost overlapped. Nevertheless, one may theoretically
analyze their differences by expanding the radius of the photon sphere in
series of the deviation parameter $\alpha_{0}$ for these two black holes. For
Bardeen black holes, one finds

$r_{c_{2}}=3M+\frac{5}{9M^{1/3}}\alpha_{0}+\mathcal{O}[\alpha_{0}]^{2},$ (34)

while for the black hole with $(x=2/3,n=2)$, one has

$r_{c}=3M-\frac{5}{9M^{1/3}}\alpha_{0}+\mathcal{O}[\alpha_{0}]^{2}.$ (35)

Firstly, in comparison with the standard result in Schwarzschild black hole,
which is $3M$, we find the radius of the photon sphere in Bardeen black hole
becomes larger, while the radius of the photon sphere in black hole with
$(x=2/3,n=2)$ becomes smaller. In particular, with the decrease of the mass,
we find the discrepancy becomes larger. We remark that the above result
coincides with the result in Ling:2022vrv , where the shadow of these regular
black holes are compared and it is found that the shadow of the black hole
with $(x=2/3,n=2)$ is more deformed and the size is smaller. We may also
expand the radius of ESCO in two spacetimes near $\alpha\rightarrow 0$. For
Bardeen black holes, one finds:

$r_{e_{2}}=6M+\frac{19}{18M^{1/3}}\alpha_{0}+\mathcal{O}[\alpha_{0}]^{2}.$
(36)

For the black hole with $(x=2/3,n=2)$, one has

$r_{e}=6M+\frac{19}{3M^{1/3}}\alpha_{0}+\mathcal{O}[\alpha_{0}]^{2}.$ (37)

We find both radii are larger than $6M$, but the radius of ESCO in the black
hole with $(x=2/3,n=2)$ is a little bit larger than that in Bardeen black
hole. In the region of CMO phase, which could be viewed as the remnants of the
evaporation of these two black holes, the behavior of these curves are
different and in principle can be distinguished by the observation, as
demonstrated in Fig.(4). From Eq.(32) and Eq.(33), one can obtain the turning
points of the curves of photon sphere and ESCO by $dz/d\omega_{e}=0$ for
Bardeen spacetime. We collect the locations of these turning points in two
different space times in table (1).

##  IV The regular black hole with $x=1$ and $n=3$

In this section we investigate photon spheres and ESCO for massive particles
in the spacetime with $x=1$ and $n=3$, which corresponds to Hayward black hole
at large scales. Its thermodynamic behavior as well as the feature of
Kretschmann scalar curvature can be found in Ling:2021olm .

In this background, the outer horizon and the inner horizon are separately
given by

$\omega_{H^{+}}=2z\left[\frac{\theta}{W_{0}(\theta)}\right]^{1/3};\quad\omega_{H^{-}}=2z\left[\frac{\theta}{W_{-1}(\theta)}\right]^{1/3};\quad\theta=-\frac{3}{8z^{2}}.$
(38)

Figure 5:  The outer horizon (black solid line) and the inner horizon (black
dashed line) of the black hole with $x=1$ and $n=3$. The vertical dashed line
in orange is the borderline between the region in black hole phase (right) and
the region in CMO phase (left).

We plot the locations of the outer horizon and the inner horizon in Fig.(5).
The point where the inner horizon and the outer horizon meet is given by

$z=\frac{M}{\alpha_{0}^{1/2}}=1.010...;\quad\omega=\frac{r}{\alpha_{0}^{1/2}}=1.465....$
(39)

###  IV.1 Photon Spheres

The effective potential to determine the geodesic trajectory of photons is
given by

$V_{0}(\omega)=\left(1-\frac{2ze^{\frac{-z}{\omega^{3}}}}{\omega}\right)\left(\frac{L^{2}}{\omega^{2}}\right).$
(40)

Setting $V_{0}^{\prime}(\omega_{c})=0$, one finds the radius of circular
photon orbits is determined by

$-3z\omega_{c}^{3}+e^{\frac{z}{\omega_{c}^{3}}}\omega_{c}^{4}+3z^{2}=0.$ (41)

Similarly, the stability of photon spheres requires
$V_{0}^{\prime\prime}(\omega_{c})\geq 0$, leading to

$-4z\omega_{c}^{6}+e^{\frac{z}{\omega_{c}^{3}}}\omega_{c}^{7}+10z^{2}\omega_{c}^{3}-3z^{3}\geq
0.$ (42)

We obtain the turning point by $dz/d\omega_{c}=0$, which gives rise to

$z=\frac{M}{\alpha_{0}^{1/2}}=0.794...;\quad\omega=\frac{r}{\alpha_{0}^{1/2}}=1.731....$
(43)

We plot the radius of photon spheres as the function of the mass in Fig.(6).

Figure 6:  The location of photon spheres in the spacetime with x = 1 and n =
3. The blue line represents the location of photon spheres in black hole
phase, while the red solid line is the location of unstable photon spheres in
CMO phase, and the red dashed line is the location of stable photon spheres in
CMO phase. The turning point of each curve is marked by a black dot as well.

###  IV.2 Circular orbits for massive particles

For massive particles, the effective potential is given by

$V_{-1}(\omega)=\left(1-\frac{2ze^{\frac{-z}{\omega^{3}}}}{\omega}\right)\left(\frac{L^{2}}{\omega^{2}}+1\right).$
(44)

Figure 7:  The location of ESCO for massive particles in the spacetime with
$x=1$ and $n=3$. The purple solid line represents the location of ESCO in
black hole phase, while the purple dashed line is the ESCO in CMO phase, where
the upper branch is ISCO and the lower branch is OSCO. In the region shaded in
orange, all the circular orbits are unstable, while in the region shaded in
pink, all the circular orbit are stable. The region shaded in green denotes
the place where the circular orbit does not exist. The turning point of each
curve is marked by a black dot as well.

Then the radius of circular orbits $\omega_{e}$ is determined by
$V_{-1}^{\prime}(\omega_{e})=0$, which gives

$z\omega_{e}^{2}\left(\omega_{e}^{3}-3z\right)-L^{2}\left(-3z\omega_{e}^{3}+e^{\frac{z}{\omega_{e}^{3}}}\omega_{e}^{4}+3z^{2}\right)=0.$
(45)

Furthermore, the angular momentum for all circular orbits must satisfy

$L^{2}=\frac{z\omega_{e}^{2}\left(\omega_{e}^{3}-3z\right)}{-3z\omega_{e}^{3}+e^{\frac{z}{\omega_{e}^{3}}}\omega_{e}^{4}+3z^{2}}\geq
0,$ (46)

which gives the region where the circular orbit for massive objects is
allowed. The location of ESCO is given by

$e^{\frac{z}{\omega_{e}^{3}}}\omega_{e}^{7}-18z^{3}+3z^{2}\omega_{e}\left(2\omega_{e}^{2}-3e^{\frac{z}{\omega_{e}^{3}}}\right)+z\left(-6\omega_{e}^{6}+9e^{\frac{z}{\omega_{e}^{3}}}\omega_{e}^{4}\right)=0,$
(47)

which gives a curve in $(z,\omega)$ plane. We plot the location of ESCO as the
function of mass in Fig.(7). Keep going further, the turning point of the
curve of ESCO is calculated by $dz/d\omega_{e}=0$, which turns out to be

$z=\frac{M}{\alpha_{0}^{1/2}}=0.595...;\quad\omega=\frac{r}{\alpha_{0}^{1/2}}=2.403....$
(48)

###  IV.3 In comparison with Hayward black hole

The effective potential of Hayward black hole with de-Sitter core is

$V_{\epsilon}(\omega)=\left(1-\frac{2z\omega^{2}}{\omega^{3}+z}\right)\left(\frac{L^{2}}{\omega^{2}}-\epsilon\right).$
(49)

Figure 8:  The comparison of photon spheres and ESCO in the spacetime with
Minkowskian core $(x=1,n=3)$ and those in Hayward spacetime with dS core.

For null trajectories, the radius of photon spheres is given by
$V_{0}^{\prime}(\omega_{c_{2}})=0$, leading to

$\omega_{c_{2}}^{5}\left(-3z+\omega_{c_{2}}\right)+2z\omega_{c_{2}}^{3}+z^{2}=0.$
(50)

For massive particles, the ESCO is given by

$-8z^{2}+11z\omega_{e_{2}}^{3}+\omega_{e_{2}}^{5}\left(-6z+\omega_{e_{2}}\right)=0.$
(51)

Previously, the similar discussion on circular orbits in Hayward spacetime can
be found in Chiba:2017nml . Now we compare photon spheres as well as the ESCO
in these two different spacetimes in Fig.(8). In black hole phase, the radius
of circular orbits for photons and massive particles can be expanded as the
series of the deviation parameter $\alpha$ as follows (The left equation is
for Hayward spacetime and the right equation is for the spacetime with
$(x=1,n=3)$).
Photon sphere:

$r_{c_{2}}=3M+\frac{2}{9M}\alpha_{0}+\mathcal{O}[\alpha_{0}]^{2};\quad
r_{c}=3M-\frac{4}{3M}\alpha_{0}+\mathcal{O}[\alpha_{0}]^{2}.$ (52)

ESCO:

$r_{e_{2}}=6M+\frac{11}{36M}\alpha_{0}+\mathcal{O}[\alpha_{0}]^{2};\quad
r_{e}=6M+\frac{19}{M}\alpha_{0}+\mathcal{O}[\alpha_{0}]^{2}.$ (53)

It is noticed that the radius of the photon sphere in Hayward black hole is
larger than the standard result $3M$ in Schwarzschild black hole, while that
in spacetime with $(x=1,n=3)$ is smaller than $3M$. The radius of ESCO in both
spacetimes is larger than standard result $6M$ in Schwarzschild black hole. In
CMO phase, both photon spheres and ESCO exhibit distinct behavior in these two
spacetimes. In particular, setting $dz/d\omega_{e}=0$ and $dz/d\omega_{c}=0$,
we can get the turning point of the curves for photon sphere and ESCO in two
different spacetimes, we collect these results in table (1).

Table 1: The turning points of the curves representing the location of horizon, photon sphere as well as ESCO in $(z,\omega)$ plane for various regular black holes. | Horizon |  Photon sphere |  ESCO
---|---|---|---
$n=2,x=1$ |  ($1.359$, $1.649$) |  ($1.050$, $1.953$) |  ($0.806$, $2.660$)
$n=2,x=2/3$ |  ($1.259$, $1.527$) |  ($1.037$, $1.929$) |  ($0.850$, $2.807$)
$Bardeen$ |  ($1.092,1.189$) |  ($0.927,1.592$) |  ($0.788,2.424$)
$n=3,x=1$ |  ($1.010$, $1.465$) |  ($0.794$, $1.731$) |  ($0.595$, $2.403$)
$Hayward$ |  ($0.918,1.224$) |  ($0.743,1.549$) |  ($0.576,2.258$)

##  V CONCLUSION AND DISCUSSION

In this paper we have investigated the circular orbits for photons and massive
particles in regular black holes with sub-Planckian curvature and Minkowskian
core, which exhibit abundant structure and are contrasted with the results in
standard Schwarzschild black hole. It is found that in black hole phase, the
location of photon sphere as well as ESCO is singe-valued, while in CMO phase
it is double-valued. We have also studied the stability of these circular
orbits. In black hole phase, the photon sphere is unstable, while in CMO phase
the upper branch is unstable but the lower branch is stable. The ESCO has two
branches in CMO phase as well. The upper one is ISCO while the lower one is
OSCO. In the region between the curve of photon sphere and the curve of ESCO,
all the circular orbits for massive particles are unstable, while outside the
curve of ESCO, all the circular orbits are stable. We have also compared the
locations of photon sphere and ESCO in spacetime with Minkowskian core and
those in spacetime with dS core. In black hole phase, we have found the radius
of the photon sphere in spacetime with Minkowskian core is smaller than that
in spacetime with dS core. In CMO phase, we have calculated the turning points
of the curves of photon sphere as well as ESCO, and found these points are
distinct in these two different sorts of spacetimes. The work in this paper
provides the theoretical basis for the possible detection of regular black
holes in astrophysical observation.

As the first step, we have only studied the astrophysical observables in
spherically symmetric spacetime. Definitely one could keep going on and extend
the analysis to rotating Kerr-like black holes with sub-Planckian curvature.
In this circumstance, the null geodesics with circular orbit near the horizon
of the black hole has been derived and the shadow of black holes has been
plotted in Ling:2022vrv . Nevertheless, the detailed analysis on the circular
orbits for massive particles and the CMO phase as the remnant of the
evaporation of black holes are still absent. In addition, to conjecture that
the remnant of the evaporation could be a candidate of dark matter, the
stability of horizionless CMO should be considered seriously, thus it is very
worthy of studying the quasinormal modes of this sort of spacetime in CMO
phase.

## Acknowledgments

We are very grateful to Meng-He Wu for helpful discussions. This work is
supported in part by the Natural Science Foundation of China under Grant No.
11875053 and 12035016. It is also supported by Beijing Natural Science
Foundation under Grant No. 1222031, and by Sichuan Youth Science and
Technology Innovation Research Team with Grant No. 21CXTD0038, by Central
Government Funds of Guiding Local Scientific and Technological Development for
Sichuan Province with Grant No. 2021ZYD0032.

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#  Threshold photoproduction of neutral pions off protons in nuclear model
with explicit mesons

D.V<EMAIL_ADDRESS>and M. Mikkelsen
Aarhus University, Aarhus, Denmark

###### Abstract

We apply the nuclear model with explicit mesons to photoproduction of neutral
pions off protons at the threshold. In this model the nucleons do not interact
with each other via a potential but rather emit and absorb mesons that are
treated explicitly on equal footing with the nucleons. We calculate the total
cross section of the reaction for energies close to threshold and compare the
calculations with available experimental data. We show that the model is able
to reproduce the experimental data and determine the range of the parameters
where the model is compatible with the experiment.

##  1 Introduction

In [1] a nuclear model with explicit mesons has been proposed where the
nucleons do not interact with each other via a potential but rather emit and
absorb mesons. The mesons are treated explicitly on equal footing with the
nucleons. It has been shown—using the sigma-meson in a proof of concept
calculation—that the model is able to reproduce the bound state of a neutron
and a proton, the deuteron.

One possible advantage of this model over the conventional interaction
models—where the particles interact directly via a phenomenological
potential—is the reduced number of parameters. There are basically only two
parameters per meson: the range and the strength of the meson-nucleon
coupling. For example, in this model the two parameters for the pion-nucleon
coupling determine all the forces mitigated by the pions: the central, tensor,
and the three-body force (as the two-pion exchange force). Moreover these two
parameters also determine the low-energy physics with pions, in particular the
photoproduction of pions off nucleons. Should the model turn out to actually
work in practice it must be able—with the two parameters for pions plus two
parameters for sigma-mesons—to describe the deuteron, the low-energy nucleon-
nucleon scattering, the pion-nucleon scattering, the pion photoproduction,
and—with the resulting tree-body force—also the triton.

In this investigation we are going to apply the model to neutral pion
photoproduction off protons where there exist precise experimental data [2, 3,
4]. We shall calculate the total cross section for this reaction and then
compare it with the experimental data with a view to determine the model’s
pion parameters that are compatible with the experimental data, if there are
any.

We first introduce the model with one possible pion-nucleon coupling operator
that conserves isospin, angular momentum, and parity; we then derive the
equations describing the dressing of bare protons with pions in one-pion
approximation; derive the formulae for the neutral pion photoproduction cross
section off the dressed proton; calculate the total cross-section, compare it
with the experimental data, and determine the range of the parameters of the
model where it is compatible with the experiment.

##  2 Nuclear interaction model with explicit pions

In the model with explicit pions the nucleons do not interact directly via a
potential but instead emit and absorb pions via a suitable pion-nucleon
coupling operator. In the absence of any extra energy the pion emitted by a
nucleon is virtual: it cannot leave the nucleon as it finds itself in a
classically forbidden region under a potential barrier about the size of the
mass of the pion. A bare nucleon surrounded by virtual pions is called a
physical or dressed nucleon. A dressed nucleon exists in a superposition of
states with different number of pions. The corresponding multi-component wave-
function of the dressed nucleon, $\Psi_{N}$, can be written as

$\Psi_{N}=\left(\begin{array}[]{c}\psi_{\tilde{N}}\\\ \psi_{{\tilde{N}}\pi}\\\
\psi_{{\tilde{N}}\pi\pi}\\\ \vdots\\\ \end{array}\right)\,,$ (1)

where $\psi_{\tilde{N}}$ is the wave-function of the bare nucleon,
$\psi_{{\tilde{N}}\pi}$ is the wave-function of the bare nucleon plus one
pion, $\psi_{{\tilde{N}}\pi\pi}$ is the wave-function for the bare nucleon
plus two pions and so on. The Hamiltonian that acts on this multi-component
wave-function is given as

$H=\left(\begin{array}[]{cccc}{\check{m}}_{\tilde{N}}+K_{\tilde{N}}&W^{\dagger}&0&\dots\\\
W&{\check{m}}_{\tilde{N}}+K_{\tilde{N}}+{\check{m}}_{\pi}+K_{\pi}+V_{C}&W^{\dagger}&\dots\\\
0&W&{\check{m}}_{\tilde{N}}+K_{\tilde{N}}+2{\check{m}}_{\pi}+K_{\pi_{1}}+K_{\pi_{2}}+V_{C}&\dots\\\
\vdots&\vdots&\vdots&\ddots\end{array}\right)\,,$ (2)

where $K_{\tilde{N}}$ is the kinetic energy of the bare nucleon, $K_{\pi}$ is
the kinetic energy of the pion, ${\check{m}}_{\tilde{N}}\equiv
m_{\tilde{N}}c^{2}$ and ${\check{m}}_{\pi}\equiv m_{\pi}c^{2}$ are the masses
of the bare nucleon and the pion in units of energy, $V_{C}$ is the Coulomb
interaction between the charged particles, if any, and $W$ ($W^{\dagger}$) is
the operator that generates (annihilates) a pion. Assuming that nuclear
interaction conserves isospin, angular momentum, and parity, one possible
operator that generates a pion—a scalar isovector with negative parity—can be
written as (cf. [5])

$W(r)=(\vec{\tau}\vec{\pi})(\vec{\sigma}\vec{r})f(r)\;,$ (3)

where $\vec{\tau}$ is the isovector of Pauli matrices that act in the isospin
space of the nucleon, $\vec{\pi}$ is the isovector of pions, $\vec{\sigma}$ is
the vector of Pauli matrices that act in the spin space of the nucleon,
$\vec{r}$ is the relative coordinate between the nucleon and the pion, and
$f(r)$ is a phenomenological (short-range) form-factor. The isospin factor
$\vec{\tau}\vec{\pi}$ is given as

$\vec{\tau}\vec{\pi}=\tau_{0}\pi^{0}+\sqrt{2}\tau_{-}\pi^{+}+\sqrt{2}\tau_{+}\pi^{-}\;,$
(4)

where $\pi^{0}$, $\pi^{+}$, and $\pi^{-}$ are the physical pions and where the
$\tau$-matrices are given as

$\tau_{0}=\left(\begin{tabular}[]{cc}1&0\\\
0&-1\end{tabular}\right)\,,\;\tau_{-}=\left(\begin{tabular}[]{cc}0&0\\\
1&0\end{tabular}\right)\,,\;\tau_{+}=\left(\begin{tabular}[]{cc}0&1\\\
0&0\end{tabular}\right)\,.$ (5)

The $W$ operator adds an extra pion to a state of nucleons and pions and the
$W^{\dagger}$ operator annihilates a pion222The action of $W^{\dagger}$
includes integration $\int_{V}d^{3}r$ to get rid of the coordinate of the
annihilated pion..

The ground state of the corresponding Schrödinger equation,

$H\Psi_{N}=E\Psi_{N}\,,$ (6)

is the physical (dressed) nucleon – a bare nucleon surrounded by a cloud of
under-the-barrier (virtual) pions. The ground state energy in the rest frame
of the nucleon gives the mass of the physical nucleon.

##  3 Dressing of the proton in one-pion approximation

Generation of a pion requires more than ${\check{m}}_{\pi}\approx$140 MeV of
extra energy. In the absence of the extra energy the pion generated by a bare
proton finds itself in a classically forbidden region under the potential
barrier of those 140 MeV. Its wave-function falls off asymptotically as
$e^{-r/\lambda}$ where $\lambda\approx\hbar c/{\check{m}}_{\pi}\approx$1.4 fm.
The pion is virtual as it cannot leave the proton.

Given the amount of energy in excess of the pion mass (plus recoil)—from an
incoming photon, for example—the virtual pion can be knocked off the proton
and become a physical pion. That would be what is called a pion
photoproduction reaction.

Each additional virtual pion adds yet another 140 MeV to the potential barrier
under which the corresponding system of particles resides. Therefore one might
assume that the first virtual pion—which is under the smallest barrier—should
account for the largest contribution to the dressing of the proton. That is
the one-pion approximation to which we are going to restrict ourselves in this
investigation. This assumption must be of course checked by direct
calculations. However adding a second pion is a much bigger task which will
hopefully be dealt with in a separate investigation.

In the one-pion approximation the physical proton is a superposition of two
coupled components: the bare-proton component and the bare-nucleon plus one
pion component. The Hamiltonian for the physical proton is then the given as

$H=\left(\begin{array}[]{cc}K_{\tilde{p}}+{\check{m}}_{\tilde{N}}&W^{\dagger}\\\
W&K_{\tilde{N}}+{\check{m}}_{\tilde{N}}+K_{\pi}+{\check{m}}_{\pi}\end{array}\right)\,.$
(7)

The Hamiltonian acts on a two-component wave-function of the physical proton,

$\Psi_{p}=\left(\begin{array}[]{l}\psi_{\tilde{p}}(\vec{R})\\\
\psi_{{\tilde{N}}\pi}(\vec{R},\vec{r})\end{array}\right)\,,$ (8)

where $\vec{R}$ is the center-of-mass coordinate and $\vec{r}$ the relative
coordinate between the pion and the nucleon. The kinetic energy operators are
given as

$\displaystyle K_{\tilde{p}}$ $\displaystyle=$
$\displaystyle\frac{-\hbar^{2}}{2m_{\tilde{N}}}\frac{\partial^{2}}{\partial\vec{R}^{2}}\,,$
(9) $\displaystyle K_{\tilde{N}}+K_{\pi}$ $\displaystyle=$
$\displaystyle\frac{-\hbar^{2}}{2M_{{\tilde{N}}\pi}}\frac{\partial^{2}}{\partial\vec{R}^{2}}+\frac{-\hbar^{2}}{2m_{{\tilde{N}}\pi}}\frac{\partial^{2}}{\partial\vec{r}^{2}}\,,$
(10)

where $m_{\tilde{N}}$, $m_{\pi}$ are the masses of the bare nucleon and the
pion, $M_{{\tilde{N}}\pi}=m_{\tilde{N}}+m_{\pi}$ is the total mass, and
$m_{{\tilde{N}}\pi}=m_{\tilde{N}}m_{\pi}/(m_{\tilde{N}}+m_{\pi})$ is the
reduced mass of the bare nucleon and the pion.

In the rest frame of the proton the dependence on $\vec{R}$ disappears and
$\psi_{\tilde{p}}$ can be chosen as

$\psi_{\tilde{p}}=\frac{p\uparrow}{\sqrt{V}}\,,$ (11)

where $p$ is the isospin state of the proton,

$p=\left(\begin{array}[]{c}1\\\ 0\end{array}\right)\,,$ (12)

($\tau_{0}p=p$), and $\uparrow$ is the spin state of the proton,

$\uparrow=\left(\begin{array}[]{c}1\\\ 0\end{array}\right)\,,$ (13)

($\sigma_{0}\uparrow=\uparrow$). The function $\psi_{\tilde{p}}$ is normalized
to one proton in the volume $V$.

The corresponding two-component Schrödinger equation is given as

$\displaystyle\int_{V}d^{3}r(\vec{\tau}\vec{\pi})^{\dagger}(\vec{\sigma}\vec{r})^{\dagger}f(r)\psi_{{\tilde{N}}\pi}$
$\displaystyle=$ $\displaystyle\tilde{E}\frac{p\uparrow}{\sqrt{V}}\,,$
$\displaystyle(\vec{\tau}\vec{\pi})(\vec{\sigma}\vec{r})f(r)\frac{p\uparrow}{\sqrt{V}}+\left(\frac{-\hbar^{2}}{2m_{{\tilde{N}}\pi}}\frac{\partial^{2}}{\partial\vec{r}^{2}}+{\check{m}}_{\pi}\right)\psi_{{\tilde{N}}\pi}$
$\displaystyle=$ $\displaystyle\tilde{E}\psi_{{\tilde{N}}\pi}\,,$ (14)

where $\tilde{E}\equiv E-{\check{m}}_{\tilde{N}}$. The second row of the
equation indicates that $\psi_{{\tilde{N}}\pi}$ must have the same spin-
isospin structure as

$(\vec{\tau}\vec{\pi})(\vec{\sigma}\vec{r})p\uparrow\,,$ (15)

suggesting that one can search for $\psi_{{\tilde{N}}\pi}$ in the form

$\psi_{{\tilde{N}}\pi}=(\vec{\tau}\vec{\pi})(\vec{\sigma}\vec{r})\frac{p\uparrow}{\sqrt{V}}\varphi(r)\;,$
(16)

where $\varphi(r)$ is a scalar radial function with the dimension length-5/2
such that the integral

$\int_{V}d^{3}R\int_{V}d^{3}r|\psi_{{\tilde{N}}\pi}|^{2}$ (17)

is dimensionless.

Using the formulae

$\displaystyle(\vec{\tau}\vec{\pi})^{\dagger}(\vec{\tau}\vec{\pi})$
$\displaystyle=$ $\displaystyle 3\,,$ (18)
$\displaystyle(\vec{\sigma}\vec{r})(\vec{\sigma}\vec{r})$ $\displaystyle=$
$\displaystyle r^{2}\,,$ (19) $\displaystyle\nabla^{2}(\vec{r}\varphi(r))$
$\displaystyle=$
$\displaystyle\vec{r}(\varphi^{\prime\prime}+\frac{4}{r}\varphi^{\prime})\;,$
(20)

the Schrödinger equation (3) can be rewritten as

$\displaystyle 3\int_{V}d^{3}r~{}r^{2}f(r)\varphi(r)$ $\displaystyle=$
$\displaystyle\tilde{E}$
$\displaystyle-\frac{\hbar^{2}}{2m_{{\tilde{N}}\pi}}(\varphi^{\prime\prime}+\frac{4}{r}\varphi^{\prime})+f(r)$
$\displaystyle=$ $\displaystyle(\tilde{E}-{\check{m}}_{\pi})\varphi\,,$ (21)

where the boundary conditions for $\varphi(r)$ are (assuming that the
formfactor $f(r)$ is finite and short-ranged)

$\left\\{\begin{array}[]{lcl}\varphi(r\to 0)&\to&\mathrm{const}\;,\\\
\varphi(r\to\infty)&\to&0\;.\end{array}\right.$ (22)

The constant and the ground state energy, $\tilde{E}_{0}$, must be found from
the self-consistent solution to the equations where the boundary conditions
are satisfied and where the mass of the bare proton is chosen such that the
ground state energy gives the experimental mass of the proton,
${\check{m}}_{N}={\check{m}}_{\tilde{N}}+\tilde{E}_{0}$.

Notice that

$(\vec{\tau}\vec{\pi})p=p\pi^{0}+\sqrt{2}n\pi^{+}\,,$ (23)

where

$n=\left(\begin{array}[]{c}0\\\ 1\end{array}\right)\,,$ (24)

($\tau_{0}n=-n$) is the isostate of the neutron. That is, the physical proton
can be found in a superposition of a bare-proton plus $\pi^{0}$ state and a
bare-neutron plus $\pi^{+}$ state.

The Schrödinger equation (3) can be solved by the gaussian method [6, 7, 8]
where the integro-differential boundary condition problem is reformulated as a
generalized eigenvalue problem. The radial function $\varphi(r)$ is
represented as a linear combination of gaussians (which satisfy the boundary
conditions (22)),

$\varphi(r)=\sum_{n=1}^{K}c_{n}e^{-\alpha_{n}r^{2}}\,,$ (25)

where $K$ is the number of gaussians in the expansion333For this one-
dimensional problem the ground state wave-function needs around four to five
gaussians to converge.. The $\psi_{{\tilde{N}}\pi}$ is then given as

$\psi_{{\tilde{N}}\pi}=\sum_{n=1}^{K}c_{n}\psi_{{\tilde{N}}\pi}^{\alpha_{n}}\,.$
(26)

where

$\psi_{{\tilde{N}}\pi}^{\alpha}=(\vec{\tau}\vec{\pi})(\vec{\sigma}\vec{r})\frac{p\uparrow}{\sqrt{V}}e^{-\alpha
r^{2}}\,.$ (27)

The function $\psi_{\tilde{p}}$ in the center-of-mass frame of the proton is
simply a constant,

$\psi_{\tilde{p}}=c_{0}\frac{p\uparrow}{\sqrt{V}}\,.$ (28)

The Schrödinger equation (3) with the boundary conditions (22) now turns into
a generalized matrix eigenvalue problem,

$\mathcal{H}c=\tilde{E}\mathcal{N}c\,,$ (29)

where

$c=\left(\begin{array}[]{c}c_{0}\\\ c_{1}\\\ \vdots\\\
c_{K}\end{array}\right)$ (30)

is the vector of coefficients to be found by solving the eigensystem, and
where the matrix elements of the (symmetric positive definite) overlap matrix
${\mathcal{N}}$ are given as (cf. [8])

$\displaystyle{\mathcal{N}}_{00}$ $\displaystyle=$
$\displaystyle\langle\psi_{\tilde{p}}|\psi_{\tilde{p}}\rangle=1\;,$ (31)
$\displaystyle{\mathcal{N}}_{0i}$ $\displaystyle=$ $\displaystyle 0\;,$ (32)
$\displaystyle{\mathcal{N}}_{ij}$ $\displaystyle=$
$\displaystyle\langle\psi_{{\tilde{N}}\pi}^{\alpha_{i}}|\varphi_{{\tilde{N}}\pi}^{\alpha_{j}}\rangle=3\frac{3}{2}\frac{1}{\alpha_{i}+\alpha_{j}}\left(\frac{\pi}{\alpha_{i}+\alpha_{j}}\right)^{3/2}\;,$
(33)

where $i,j=1,\dots,K$.

Assuming for simplicity that the formfactor $f(r)$ is a gaussian,

$f(r)=\mathcal{A}e^{-\kappa r^{2}}\,,$ (34)

the matrix elements of the (symmetric) Hamiltonian matrix ${\mathcal{H}}$ are
given as

$\displaystyle{\mathcal{H}}_{00}$ $\displaystyle=$
$\displaystyle\langle\psi_{\tilde{p}}|K_{\tilde{p}}|\psi_{\tilde{p}}\rangle=0\;,$
(35) $\displaystyle{\mathcal{H}}_{0i}$ $\displaystyle=$
$\displaystyle\langle\varphi_{{\tilde{N}}\pi}^{\alpha_{i}}|W|\varphi_{p}\rangle=3\mathcal{A}\frac{3}{2}\frac{1}{\alpha_{i}+\kappa}\left(\frac{\pi}{\alpha_{i}+\kappa}\right)^{3/2}\;,$
(36) $\displaystyle{\mathcal{H}}_{ij}$ $\displaystyle=$
$\displaystyle\langle\varphi_{{\tilde{N}}\pi}^{\alpha_{i}}|K_{\tilde{N}}+K_{\pi}+m_{\pi}|\varphi_{{\tilde{N}}\pi}^{\alpha_{j}}\rangle=3\frac{\hbar^{2}}{2m_{{\tilde{N}}\pi}}15\frac{\alpha_{i}\alpha_{j}}{(\alpha_{i}+\alpha_{j})^{2}}\left(\frac{\pi}{\alpha^{\prime}+\alpha}\right)^{3/2}+m_{\pi}{\mathcal{N}}_{ij}\;,$
(37)

where $i,j=1,\dots,K$.

The generalized eigenvalue problem can be easily solved by standard methods
[9].

The range parameters $\alpha_{k}$ of the gaussians are often chosen
stochastically using random [6] or quasi-random [10] sequences. However for
the current one-dimensional problem we simply perform global optimization in
the space of $\alpha_{k}$ using the Melder-Need downhill simplex method [9,
11].

In the following we shall parametrize $\kappa$ and $\mathcal{A}$ as

$\kappa\doteq\frac{1}{b_{w}^{2}}\;,\,\mathcal{A}\doteq\frac{S_{w}}{b_{w}}\,,$
(38)

where $b_{w}$ [fm] and $S_{w}$ [MeV] are the parameters of the model.

##  4 Pion photoproduction off proton

A photon with the energy in excess of the pion mass (plus recoil) can knock
off the virtual pion from the dressed proton in what is called the pion
photoproduction reaction. Since the wavelength of a photon with the energy
$\hbar\omega~{}{\scriptstyle{}^{>}_{\sim}}~{}{\check{m}}_{\pi}$ is on the
order of the size of the dressed proton we have to calculate the cross-section
without the use of the usual multipole expansion of the electromagnetic
operator (cf. [12]).

The interaction between the electromagnetic field with potential $\vec{A}$ in
the radiation gauge and a particle with charge $e$, mass $m_{e}$, position
$\vec{r}_{e}$ and momentum $\vec{p}_{e}$ is given as (neglecting the term of
the second order in the field and the contribution from the magnetic moment of
the particle)

$V_{\vec{A}}=\frac{-e}{m_{e}c}\vec{A}(\vec{r}_{e},t)\vec{p}_{e}\;.$ (39)

The free electromagnetic field can be represented as a superposition of normal
modes. With plane waves as normal modes the electromagnetic potential
$\vec{A}(\vec{r},t)$ of the free field is given (in the radiation gauge) as

$\vec{A}(\vec{r},t)=\sum_{\vec{k},\lambda}\sqrt{\frac{2\pi\hbar
c^{2}}{\omega_{k}V}}\left(a_{\vec{k}\lambda}\vec{e}_{\vec{k}\lambda}e^{+i(\vec{k}\vec{r}-\omega_{k}t)}+a_{\vec{k}\lambda}^{\dagger}\vec{e}_{\vec{k}\lambda}^{~{}*}e^{-i(\vec{k}\vec{r}-\omega_{k}t)}\right)\;,$
(40)

where $\vec{k}$, $\omega_{k}=kc$, $\lambda$, $\vec{e}_{\vec{k}\lambda}$ are
the wavenumber, frequency, polarization index, and polarization vector of the
normal mode; $a_{\vec{k}\lambda}$, $a_{\vec{k}\lambda}^{\dagger}$ are the
(bosonic) annihilation/generation operators of a photon with the corresponding
quantum numbers; and where the normal modes are normalized to one photon in
the volume $V$.

In the case of the neutral pion photoproduction off a proton it is the proton
that interacts with the electromagnetic field. The corresponding interaction
operator is given as

$V_{\vec{A}}=\frac{-e}{m_{p}c}\vec{A}(\vec{r}_{p},t)\vec{p}_{p}=\frac{-e}{m_{p}c}\vec{A}\left(\vec{R}-\frac{m_{\pi}}{m_{p}+m_{\pi}}\vec{r},t\right)\left(\frac{m_{p}}{m_{p}+m_{\pi}}\vec{P}-\vec{p}\right)\,,$
(41)

where $e$ is the charge of the proton, $\vec{r}_{p}$ is the coordinate of the
proton, $\vec{p}_{p}$ is the proton momentum operator,
$\vec{r}=\vec{r}_{\pi}-\vec{r}_{p}$ is the relative pion-proton coordinate,
$\vec{R}=(m_{p}\vec{r}_{p}+m_{\pi}\vec{r}_{\pi})/(m_{p}+m_{\pi})$ is the
coordinate of the center of mass of the pion-proton system, $\vec{p}$ is the
relative pion-proton momentum operator, and $\vec{P}$ is the total pion-proton
momentum operator.

We assume that when driven away from its virtual neutral pion by the
electromagnetic field the bare proton gets dressed immediately by a new
virtual pion. We shall therefore use the physical mass, $m_{p}$, of the proton
in the corresponding formulae.

The initial state of the pion photoproduction reaction consists of a dressed
proton and a plane-wave photon with wavenumber $\vec{k}$ and polarisation
$\lambda$ in the state $a_{\vec{k}\lambda}^{\dagger}|0\rangle$, where
$|0\rangle$ is the electromagnetic vacuum. In the final state there is a
physical proton plus a neutral pion (in a relative plane-wave motion) and no
more photons, that is, the electromagnetic vacuum $\langle 0|$. The
electromagnetic part of the matrix element of this transition is given as

$\left\langle
0\left|\frac{-e}{m_{p}c}\vec{A}\left(\vec{R}-\frac{m_{\pi}}{m_{p}+m_{\pi}}\vec{r},t\right)a_{\mathbf{k}\lambda}^{\dagger}\right|0\right\rangle=\frac{-e}{m_{p}}\sqrt{\frac{2\pi\hbar}{\omega_{k}V}}\vec{e}_{\vec{k}\lambda}e^{i\vec{k}(\vec{R}-\frac{m_{\pi}}{m_{p}+m_{\pi}}\vec{r})-i\omega_{k}t}\;.$
(42)

The probability per unit time of the transition from the initial to the final
state is then given by the Fermi’s golden rule,

$dw_{fi}=\frac{2\pi}{\hbar}|M_{fi}|^{2}d\rho_{f}\;,$ (43)

where $d\rho_{f}$ is the density of the final states and where the transition
matrix element is given as (in the lab frame where $\vec{P}=0$)

$M_{fi}=\frac{+e}{m_{p}}\sqrt{\frac{2\pi\hbar}{\omega_{k}V}}\left\langle
p\xi\pi^{0}\frac{e^{i\vec{q}\vec{r}}}{\sqrt{V}}\frac{e^{i\vec{Q}\vec{R}}}{\sqrt{V}}\right|e^{i\vec{k}(\vec{R}-\frac{m_{\pi}}{m_{p}+m_{\pi}}\vec{r})}(\vec{e}_{\vec{k}\lambda}\vec{p})\left|(\vec{\tau}\vec{\pi})(\vec{\sigma}\vec{r})\varphi(r)\frac{p\uparrow}{\sqrt{V}}\right\rangle\;,$
(44)

where $\xi$ is the spin-state of the final proton (can be either $\uparrow$ or
$\downarrow$), $\hbar\vec{q}$ is the pion-proton relative momentum in the
final state, and $\vec{Q}=\vec{k}$ is the recoil.

Integration over $d^{3}R$ gives $V$, the isospin factor gives

$\langle p\pi^{0}|\vec{\tau}\vec{\pi}|p\rangle=1\;,$ (45)

and, recalling that $\vec{p}=-i\hbar\frac{\partial}{\partial\vec{r}}$, the
matrix element becomes

$M_{fi}=\frac{-ie\hbar}{m_{p}V}\sqrt{\frac{2\pi\hbar}{\omega_{k}}}\left\langle\xi\left|\big{\langle}e^{i\vec{s}\vec{r}}|\left(\vec{e}_{\vec{k}\lambda}\frac{\partial}{\partial\vec{r}}\right)(\vec{\sigma}\vec{r})|\varphi(r)\big{\rangle}\right|\uparrow\right\rangle\;,$
(46)

where $\vec{s}\doteq\vec{q}+\frac{m_{\pi}}{m_{p}+m_{\pi}}\vec{k}$.

In the inner matrix element the integral over angles can be evaluated
analytically: integrating by parts once and then using the following formulae,

$\displaystyle e^{-i\vec{s}\vec{r}}$ $\displaystyle=$
$\displaystyle\sum_{l}(2l+1)(-i)^{l}j_{l}(sr)P_{l}\left(\frac{\vec{s}\vec{r}}{sr}\right)\,,$
(47) $\displaystyle\int d\Omega~{}r_{i}r_{j}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{3}r^{2}\delta_{ij}\,,$ (48)

(where $P_{l}$ is the Legendre polynomial and $j_{l}$ is the spherical Bessel
function) gives

$\displaystyle\left\langle
e^{i\vec{s}\vec{r}}\left|\left(\vec{e}\frac{\partial}{\partial\vec{r}}\right)(\vec{\sigma}\vec{r})\right|\varphi(r)\right\rangle$
$\displaystyle=$ $\displaystyle+i(\vec{e}\vec{s})\int
d^{3}re^{-i\vec{s}\vec{r}}(\vec{\sigma}\vec{r})\varphi(r)$ (49)
$\displaystyle=$ $\displaystyle-i(\vec{e}\vec{s})\int
d^{3}r3ij_{1}(sr)\frac{(\vec{s}\vec{r})}{sr}(\vec{\sigma}\vec{r})\varphi(r)$
(50) $\displaystyle=$
$\displaystyle(\vec{e}\vec{s})(\vec{\sigma}\vec{s})\frac{4\pi}{s}\int_{0}^{\infty}r^{3}drj_{1}(sr)\varphi(r)\,.$
(51)

Thus the inner matrix element is given as

$\big{\langle}e^{i\vec{s}\vec{r}}\big{|}\left(\vec{e}\frac{\partial}{\partial\vec{r}}\right)(\vec{\sigma}\vec{r})\big{|}\varphi(r)\big{\rangle}=(\vec{e}\vec{s})(\vec{\sigma}\vec{s})F(s)\,.$
(52)

where the factor $F(s)$ is determined by the spherical Bessel transform of
$\varphi(r)$,

$F(s)=\frac{4\pi}{s}\int_{0}^{\infty}r^{3}drj_{1}(sr)\varphi(r)\,.$ (53)

If $\varphi(r)$ is represented as a linear combination of gaussians,

$\varphi(r)=\sum_{n}c_{n}e^{-\alpha_{n}r^{2}}\,,$ (54)

the radial integral in $F(s)$ can be calculated analytically either using the
formula

$\frac{4\pi}{s}\int_{0}^{\infty}r^{3}drj_{1}(sr)e^{-\alpha
r^{2}}=\frac{1}{2\alpha}e^{-\frac{1}{4\alpha}s^{2}}\left(\frac{\pi}{\alpha}\right)^{3/2}\,,$
(55)

which gives

$F(s)=\sum_{n}c_{n}\frac{4\pi}{s}\int_{0}^{\infty}r^{3}drj_{1}(sr)e^{-\alpha_{n}r^{2}}=\sum_{n}c_{n}\frac{1}{2\alpha_{n}}e^{-\frac{1}{4\alpha_{n}}s^{2}}\left(\frac{\pi}{\alpha_{n}}\right)^{3/2}\,,$
(56)

or like this444 First the matrix element with the operator $r_{i}r_{j}$ is
calculated as  $\displaystyle\langle e^{i\vec{s}\vec{r}}|r_{i}r_{j}|e^{-\alpha
r^{2}}\rangle=\frac{-\partial^{2}}{\partial s_{i}\partial s_{j}}\langle
e^{i\vec{s}\vec{r}}|e^{-\alpha r^{2}}\rangle=\frac{-\partial^{2}}{\partial
s_{i}\partial
s_{j}}e^{-\frac{1}{4\alpha}s^{2}}\left(\frac{\pi}{\alpha}\right)^{3/2}=\left(\frac{1}{2\alpha}\delta_{ij}-\frac{1}{4\alpha^{2}}s_{i}s_{j}\right)e^{-\frac{1}{4\alpha}s^{2}}\left(\frac{\pi}{\alpha}\right)^{3/2}\,.$
(57) And now the sought matrix element evaluates as  $\displaystyle\langle
e^{i\vec{s}\vec{r}}|(\vec{e}\frac{\partial}{\partial\vec{r}})(\vec{\sigma}\vec{r})|e^{-\alpha
r^{2}}\rangle=(\vec{e}\vec{\sigma})\langle e^{i\vec{s}\vec{r}}|e^{-\alpha
r^{2}}\rangle-2\alpha\langle
e^{i\vec{s}\vec{r}}|(\vec{e}\vec{r})(\vec{\sigma}\vec{r})|e^{-\alpha
r^{2}}\rangle$
$\displaystyle=(\vec{e}\vec{\sigma})e^{-\frac{1}{4\alpha}s^{2}}\left(\frac{\pi}{\alpha}\right)^{3/2}-2\alpha\left(\frac{1}{2\alpha}(\vec{e}\vec{\sigma})-\frac{1}{4\alpha^{2}}(\vec{e}\vec{s})(\vec{\sigma}\vec{s})\right)e^{-\frac{1}{4\alpha}s^{2}}\left(\frac{\pi}{\alpha}\right)^{3/2}$
$\displaystyle=\frac{1}{2\alpha}(\vec{e}\vec{s})(\vec{\sigma}\vec{s})e^{-\frac{1}{4\alpha}s^{2}}\left(\frac{\pi}{\alpha}\right)^{3/2}$
(58) .

The total matrix element is now given as

$M_{fi}=\frac{-ie\hbar}{m_{p}V}\sqrt{\frac{2\pi\hbar}{\omega_{k}}}(\vec{e}_{\vec{k}\lambda}\vec{s})\langle\xi|\vec{\sigma}\vec{s}|\uparrow\rangle
F(s)\;.$ (59)

The transition probability is determined by the absolute square of the matrix
element,

$|M_{fi}|^{2}=\frac{2\pi}{V^{2}}\frac{e^{2}\hbar^{3}}{m_{p}^{2}\omega_{k}}\big{|}(\vec{e}_{\vec{k}\lambda}\vec{s})\big{|}^{2}\big{|}\langle\xi|\vec{\sigma}\vec{s}|\uparrow\rangle\big{|}^{2}F^{2}(s)\;.$
(60)

Assuming that the beam and the target are unpolarized we need to average the
transition probability over the polarizations of the photon (and the target)
in the initial state and sum over the spin states of the proton in the final
state. Summation over the photon polarizations gives

$\sum_{\lambda}(\vec{e}_{\vec{k}\lambda}^{~{}\star}\vec{s})(\vec{e}_{\vec{k}\lambda}\vec{s})=s^{2}-\frac{(\vec{k}\vec{s})^{2}}{k^{2}}=q^{2}-\frac{(\vec{k}\vec{q})^{2}}{k^{2}}=q^{2}\sin^{2}(\theta_{q})\,,$
(61)

where $\cos(\theta_{q})=(\vec{q}~{}\vec{k})/(qk)$.

Summation over the spin states of the final proton gives,555
$\langle\uparrow|\vec{\sigma}\vec{s}|\uparrow\rangle=s_{z}\,,\,\langle\downarrow|\vec{\sigma}\vec{s}|\uparrow\rangle=s_{x}+is_{y}\,,\,s_{z}^{2}+(s_{x}-is_{y})(s_{x}+is_{y})=s^{2}\,.$
(62)

$\sum_{\xi}\left|\langle\xi|~{}\vec{\sigma}\vec{s}~{}|\uparrow\rangle\right|^{2}=s^{2}\,.$
(63)

We do not really need to average over the spin states of the final proton as
the two states give identical contribution. The averaged matrix element is
given as

$\frac{1}{2}\sum_{\lambda}\sum_{\xi}|M_{fi}|^{2}=\frac{\pi}{V^{2}}\frac{e^{2}\hbar^{3}}{m_{p}^{2}\omega_{k}}q^{2}\sin^{2}(\theta_{q})s^{2}F^{2}(s)\;.$
(64)

The number of final states around $\vec{q}$ is given as

$dn_{\vec{q}}=\frac{Vd^{3}q}{(2\pi)^{3}}=\frac{Vd\Omega_{q}}{(2\pi)^{3}}q^{2}dq=\frac{Vd\Omega_{q}}{(2\pi)^{3}}\frac{1}{2}qdq^{2}\;,$
(65)

which gives the density of the final states as

$d\rho_{f}=\frac{dn_{\vec{q}}}{dE_{q}}=\frac{Vd\Omega_{q}}{(2\pi)^{3}}q\frac{1}{2}\frac{dq^{2}}{dE_{q}}\,,$
(66)

where $E_{q}$ is the energy of the relative motion of the particles in the
final state. The averaged transition probability is then given as

$dw_{\mathrm{f}i}=\frac{2\pi}{\hbar}\frac{1}{2}\sum_{\lambda}\sum_{\xi}|M_{\mathrm{f}i}|^{2}d\rho_{\mathrm{f}}=\frac{2\pi}{\hbar}\frac{\pi}{V^{2}}\frac{e^{2}\hbar^{3}}{m_{p}^{2}\omega}q^{2}\sin^{2}(\theta_{q})s^{2}F^{2}(s)\frac{Vd\Omega_{q}}{(2\pi)^{3}}q\frac{1}{2}\frac{dq^{2}}{dE_{q}}\,.$
(67)

Finally, division by the flux density of the incoming photons, $c/V$, gives
the differential cross-section for the photoproduction of neutral pion off
protons,

$\frac{d\sigma(E_{q},\theta_{q})}{d\Omega_{q}}=\frac{e^{2}}{8\pi}\frac{1}{{\check{m}}_{p}^{2}}\frac{q^{3}}{k}\frac{d(\hbar
qc)^{2}}{dE_{q}}\sin^{2}(\theta_{q})s^{2}F^{2}(s)\,.$ (68)

The energy $E_{q}$ of the relative pion-proton motion in the final states is
given by the (relativistic, for generality) energy conservation relation,

$E_{q}=\hbar\omega+{\check{m}}_{p}-\sqrt{({\check{m}}_{p}+{\check{m}}_{\pi})^{2}+\hbar^{2}\omega^{2}}\,.$
(69)

The absolute value of the relative wave-number $\vec{q}$ can then be
calculated using the energy equation in the final state,

$E_{q}=\sqrt{{\check{m}}_{p}^{2}+(\hbar
qc)^{2}}+\sqrt{{\check{m}}_{\pi}^{2}+(\hbar
qc)^{2}}-{\check{m}}_{p}-{\check{m}}_{\pi}\,,$ (70)

which gives

$(\hbar
qc)^{2}=\frac{E_{q}(E_{q}+2{\check{m}}_{p})(E_{q}+2{\check{m}}_{\pi})(E_{q}+2{\check{m}}_{p}+2{\check{m}}_{\pi})}{4(E_{q}+{\check{m}}_{p}+{\check{m}}_{\pi})^{2}}\,,$
(71)

and

$\frac{d(\hbar
qc)^{2}}{dE_{q}}=\frac{(E_{q}^{2}+2E_{q}{\check{m}}_{p}+2{\check{m}}_{p}^{2}+2E_{q}{\check{m}}_{\pi}+2{\check{m}}_{p}{\check{m}}_{\pi})(E_{q}^{2}+2E_{q}{\check{m}}_{p}+2{\check{m}}_{\pi}^{2}+2E_{q}{\check{m}}_{\pi}+2{\check{m}}_{p}{\check{m}}_{\pi})}{2(E_{q}+{\check{m}}_{p}+{\check{m}}_{\pi})^{3}}\,.$
(72)

The total cross-section $\sigma$ is given as the integral over $\theta_{q}$,

$\sigma=\int_{0}^{\pi}2\pi\sin(\theta_{q})d\theta_{q}\frac{d\sigma}{d\Omega_{q}}\,.$
(73)

##  5 Results

Figure 1: Total cross section, $\sigma$, for the neutral pion photoproduction
off a proton as function of photon energy, $E_{\gamma}$, calculated with
several sets of model parameters and shown together with the experimental data
from [2, 3, 4]. The errorbars are about the size of the symbols.

The calculated total cross section for neutral pion photoproduction off a
proton as function of the photon energy is shown on Figure 1 together with the
recent experimental data. The model is apparently able to quantitatively
describe the experiment. However the range of the parameters, where the model
agrees with the experiment, is relatively broad: basically for any $b_{w}$ in
the range from 3.7fm to 4.0fm one can find the corresponding strength $S_{w}$
that would fit the data (perhaps with $b_{w}=$3.9fm being the favourite).

For the set of parameters $b_{w}$=3.9fm, $S_{w}$=41.5MeV the relative weight
of the $\pi^{0}$ component in the wave-function of the dressed proton is about
25% and the virtual pions contribution to the mass of the dressed proton is
about -586MeV. All these numbers might seem somewhat excessive and might
indicate that the two-pion effects are not negligible or that our
phenomenological coupling operator is not good enough. Calculations of the
deuteron with the found parameters might provide a better insight into the
validity of this model.

##  6 Conclusion

We have applied the nuclear model with explicit mesons to pion photoproduction
off protons. In this model the nucleons do not interact directly with each
other but rather emit and absorb mesons that are treated explicitly on equal
footing with the nucleons. The mesons emitted by a bare nucleon find
themselves in a classically forbidden region under a potential barrier about
the size of the mass of the mesons. The mesons cannot leave the nucleon and
can thus be referred to as virtual. A bare nucleon surrounded by virtual
mesons is the physical or dressed nucleon. We have considered the dressing of
a proton with pions in the one-pion approximation and derived the equations
for the wave-function of the virtual pion.

A photon with the energy in excess of the pion mass (plus recoil) can knock
off a virtual pion from the dressed proton. That would be the pion
photoproduction reaction. We have calculated the cross section of the neutral
pion photoproduction off protons near threshold and have compared it with the
experimental data. We have shown that the model is able to quantitatively
describe the experiment for a certain range of parameters.

In conclusion we have shown that the nuclear model with explicit mesons is
able to reproduce quantitatively the experimental cross section for neutral
pion photoproduction off protons close to threshold, and have determined the
range of the parameters where the model is compatible with the experimental
data.

##  7 Acknowledgments

Numerous discussions with Hans Fynbo and Karsten Riisager are gratefully
acknowledged.

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