#  The Yukawa potential under weak magnetic field

Fábio L. Braghin1, Marcelo Loewe2,3, Cristian Villavicencio4
1 Instituto de Física, Federal University of Goiás, Av. Esperança, s/n,
74690-900, Goiânia, GO, Brazil
2 Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián,
Santiago, Chile
3 Centre for Theoretical and Mathematical Physics, and Department of Physics,
University of Cape Town,Rondebosch 7700, South Africa
4 Centro de Ciencias Exactas $\&$ Departamento de Ciencias Básicas, Facultad
de Ciencias,
Universidad del Bí0-Bío, Casilla 447, Chillán, Chile

###### Abstract

Weak magnetic field induced corrections for the Yukawa potential due to one
pion exchange between two constituent quarks (nucleons) are presented. For
that, the constant magnetic field effect on the pion propagator and on the
pion form factor are taken into account. An effective gluon propagator
parameterized with an effective gluon mass ($M_{g}\sim 0.5$ GeV) is
considered. In the limit of magnetic field weak with respect to the
constituent quark mass and pion mass, analytical and semi-analytical
expressions can be obtained. Different types of contributions are found,
isotropic or anisotropic, dependent on the pion mass and also on the
constituent quark and effective gluon masses. Overall the corrections are of
the order of $2\%$ to $5\%$ of the Yukawa potential at distances close to
$2$fm, and they decrease slower than the Yukawa potential. The anistropic
corrections are considerably smaller than the isotropic components. A sizable
splitting between results due to magnetic field dependent neutral or charged
pion mass is found.

##  1  Introduction

The Yukawa potential, due to one pion exchange, [1] is a cornerstone of
Nuclear and Particle Physics being responsible, for example, for the long
range nucleon potential. Besides its real importance for Particle and Nuclear
Physics, the Yukawa potential appears in other areas of Physics under
different names, such static screened coulomb potential in solid state and
plasma physics [2]. More recently, bound states of dark matter components have
been envisaged by considering this type of interaction [3]. Therefore, it is
interesting to understand how it behaves under different external conditions
such as magnetic fields.

In the last years many effects in hadron degrees of freedom due to strong
magnetic fields have been investigated [4, 5]. Initially, indirect effects
were searched in relativistic heavy ion collisions [6, 7, 8], in dense
stars/magnetars, including in low density outer crust regions, [9, 10, 11, 12]
and in the early Universe [13, 14]. Expectations of strong magnetic fields in
peripheral relativistic heavy ion collisions (r.h.i.c.) were somewhat
diminished in the last years [15]. Modification in the hadrons dynamics can
occur both at the more fundamental level, for quarks and gluons, and for the
hadron level. A magnetic field leads to the magnetic catalysis, due to the
high degeneracy of the lowest Landau levels [16, 4, 17], and correspondingly
it increases the quark effective masses. Less understood is the role of
magnetic fields on hadron masses/structure and dynamics in general. In this
respect effective models can be very useful to provide a framework to perform
feasible calculations, usually making possible to reach reasonably correct
results when compared to Lattice QCD [18, 19, 23, 20]. Usually these models
are compatible with the framework of the constituent quark model. Both the
Nambu-Jona-Lasinio (NJL) model and the sigma model are suitable frameworks to
the investigation of the quark-antiquark mesons providing usually very good
results for meson dynamics and other global properties of low energy hadrons,
the list of works is extense, to quote few examples [21, 22, 24, 25, 26, 27].
Even in such simplified effective models, there are difficulties involved in
this calculation of properties under strong magnetic fields. Different
calculations indicate that neutral (charged) pions have their masses decreased
(increased) with the magnetic field strength [22, 24, 25, 18, 19]. Effects of
magnetic fields on hadron couplings can also be computed although exploited
less extensively in the literature. Some derivations of the contribution of
magnetic fields to hadron or quark couplings are found in in [28, 29, 30, 31].
In spite of considering strong magnetic fields, they will be taken as
relatively weak with respect to hadron mass scales, i.e. $eB\ll M_{q}^{2}$ (as
a constituent quark mass) and/or $eB<m_{\pi}^{2}$ (pion mass). Therefore we
shall fix $eB\sim 0.1M_{q}^{2}$. In this limit, it is possible to perform
expansions in the quark and pion propagators and to present analytical or
semi-analytical calculations for the resulting form factors and overall
contributions for the Yukawa potential. The only contribution we do not take
into account is the magnetic field effect on the gluon (effective) propagator
and quark running coupling constant. The constituent quark model (CQM), in a
wide variety of different versions, has provided a sound framework for the
description of global properties of hadron structure and interactions, for
example in [32, 33]. It is based in the idea that dressed valence quarks give
rise to the hadron observables such as masses and coupling constants.

Pion constituent quark couplings and form factors have been derived at one
loop level by considering standard techniques [34]. Full form factors can be
obtained as Lagrangian interactions by considering an important term of the
QCD effective action for quarks, that is a quark-antiquark interaction
mediated by a (non perturbative) one gluon exchange. This one gluon exchange
is included by means of an (external) effective gluon propagator that accounts
for non perturbative effects of the gluon non-Abelian dynamics and it will be
parameterized by a gluon mass. The pion pseudoscalar coupling can be obtained
by standard analytical methods, being represented as Fig (1). The coupling
under magnetic field arises by considering the same method [28].

Figure 1:  One loop structure of the pion form factor: solid line for quark
with incoming and outgoing momenta $K$ and $K+Q$, wiggly line for the gluon
and dashed line for the pion with momentum $Q$.

In this work, we investigate weak constant magnetic field corrections to the
Yukawa potential. Different types of effects will be considered at this order
of correction: the magnetic field correction to the pion propagator [35], the
weak magnetic field contributions for the pion coupling to constituent quarks.
These first two effects show to be of the order of $(eB)^{2}$. We also
consider that magnetic fields lead to corrections of quark and pion masses
[16, 21, 22]. The work is organized as follows. In the next section, the
corresponding corrections to the pion propagator and to the pion form factor,
due to a constant weak magnetic field, are presented. In Section (3) the
Fourier transformation of the momentum dependent potential is calculated
mostly analytically. In Section (4), numerical estimation are displayed and
discussed. In the final part there is a Summary.

##  2  The Yukawa potential and corrections

The standard Yukawa potential can then be written as [1, 36]:

$\displaystyle V(R)=-\frac{g_{ps}^{2}}{4\pi R}e^{-m_{\pi}R},$ (1)

where $g_{ps}$ is the pseudoscalar coupling constant.

In the presence of a constant background magnetic field, the pion propagator
will receive standard corrections from the magnetic field. Besides that, the
relatively weak magnetic field will induce corrections to the pion
pseudoscalar coupling constant. However, instead of considering a punctual
coupling we shall consider the whole form factor. The following resulting
potential can be calculated as:

$\displaystyle
V(\vec{R})=\int\frac{d^{3}Q}{(2\pi)^{3}}\;(G_{ps}^{B}(\vec{Q}^{2},Q_{z}^{2}))^{2}\;e^{-i\vec{Q}\cdot\vec{R}}D_{\pi}^{B}(Q)$
(2)

where the leading contributions from the magnetic field can be written as:

$\displaystyle G_{ps}^{B}(\vec{Q}^{2},Q_{z}^{2})$ $\displaystyle\simeq$
$\displaystyle
g_{ps}+\left(\frac{eB}{M^{2}}\right)^{2}F_{ps}^{B}(\vec{Q},Q_{z}),$
$\displaystyle D_{\pi}^{B}(Q)$ $\displaystyle\simeq$ $\displaystyle
D_{\pi}(Q)+\left(eB\right)^{2}D_{1}(Q).$ (3)

The pseudoscalar pion coupling constant will be taken from the whole form
factor with $Q^{2}=0$, as shown in the Appendix. Therefore, we will calculate
quantities in the momentum space for the one pion exchange and then perform
the Fourier transform.

###  2.1  The pion propagator under weak magnetic field

The scalar field propagator under strong magnetic fields has been derived in
Ref. [35]. We’ll consider the same propagator for the pion by neglecting
therefore the particular quark-antiquark structure. In the limit of very weak
magnetic field, $eB\ll m^{2}$, the scalar field propagator can be written as:

$\displaystyle
iD^{B}(Q)\simeq\frac{i}{Q^{2}-m^{2}}\left[1-(eB)^{2}\left(\frac{1}{(Q^{2}-m^{2})^{2}}+\frac{2Q_{\perp}^{2}}{(Q^{2}-m^{2})^{3}}\right)\right]\equiv
D_{0}(Q)+(eB)^{2}D_{1}^{B}(Q,Q_{\perp})$ (4)

Note that the leading term is of the order of $(eB)^{2}$ and there is one
B-dependent correction that is isotropic and one that is anisotropic.

###  2.2  Constituent quark-pion Coupling under weak magnetic field

By starting with a quark -antiquark interaction mediated by a (non-
perturbative) one gluon exchange, the pion psedoscalar form factor in the
vacuum and under weak magnetic field were derived respectively in [34, 28].
The mediation of the (dressed) gluon, by means of an effective gluon
propagator, gives rise to a dressed quark current, being that it can be
parameterized in terms of a gluon effective mass of the order of magnitude of
a constituent quark mass. The quark-antiquark interaction can be Fierz
transformed leading to different Dirac and flavor types of quark currents,
dressed by components of the gluon propagator, that can couple to any meson
field. The resulting effective Lagrangian terms for the magnetic field induced
contribution for the pseudoscalar field ($P_{1},P_{2},P_{3}$) coupling to the
pseudoscalar quark current $j_{ps}^{i}$ can be written as:

$\displaystyle{\cal L}_{\pi-Q(B)}$ $\displaystyle=$ $\displaystyle
c_{i}\;F_{ps}^{B}(Q,K)\;P_{i}(Q)(j_{ps}^{i})^{\dagger},\;\;\;\;\;\;\;\mbox{{\it
i} = 1,2,3}$ (5)

$j_{ps}^{i}=\bar{\psi}i\gamma_{5}\lambda_{i}\psi$, and $c_{1}=c_{2}=-4/9$ and
$c_{3}=5/9$ are obtained from the trace in flavor indices. $c_{1},_{2}$ and
$c_{3}$ give rise respectively to the charged and neutral pion couplings. The
weak magnetic field couples to the internal quark lines of the diagram (1).
Although it may seem that the leading term would be a magnetic field to a
single internal quark line (which would yield a correction of the order of
$(eB)$) it turns out that the two possible diagrams of this type lead to a
tiny contribution that was not taken into account. The leading contribution
for the pseudoscalar pion coupling is that of one magnetic field insertion for
each of the internal quark lines shown in Fig. (4), that is of the order of
$(eB)^{2}$, differently from the axial coupling that goes with $(eB)$ [28].
Another very small contribution is due to two magnetic field insertions on a
single quark line, shown in Figs. (4) and (4), that is also or the order of
$(eB)^{2}$, but that is negligible when compared to the leading contribution.
The leading Lagrangian term can be written in terms of two parts, an isotropic
and an anisotropic ones, being respectively given by

$\displaystyle
F^{B}_{ps}(Q,K)=\left(\frac{eB_{0}}{M^{2}}\right)^{2}\left[F^{B,iso}_{ps}(Q,K)+F^{B,ani}_{ps}(Q,K)\right]$
(6)

where $Q,K$ are the pion and constituent quark momenta and the relative weak
strength of the magnetic field was factorized in the dimensionless factor.
These form factors can be written as:

$\displaystyle F_{ps}^{B,iso}(Q,K)$ $\displaystyle=$ $\displaystyle
iC_{PS}^{B}{M^{*}}^{4}\;\int\frac{d^{4}k}{(2\pi)^{4}}\frac{-k\cdot(k+Q)+M^{2}}{[k^{2}-M^{2}]^{2}[(k+Q)^{2}-M^{2}]^{2}}\frac{R(-k-K)}{K_{g}}$
$\displaystyle F_{ps}^{B,ani}(Q,K)$ $\displaystyle=$ $\displaystyle
iC_{PS}^{B}{M^{*}}^{4}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{-k_{\perp}\cdot(k_{\perp}+Q_{\perp})}{(k^{2}-M^{2})^{2}((k+Q)^{2}-M^{2})^{2}}\frac{R(-k-K)}{K_{g}}$
(7)

where $R(-k)$ is the effective gluon propagator, with a normalization $K_{g}$
discussed below, and the following constant was defined

$C_{PS}^{B}=8N_{c}\alpha K_{g}C_{i}.$

In the present work we consider an effective gluon propagator inspired in [37]
that is confining and leads to Dynamical Chiral Symmetry Breaking. It will be
given by:

$\displaystyle R(k)$ $\displaystyle=$
$\displaystyle\frac{K_{g}}{(k^{2}-M_{G}^{2})^{2}}$ (8)

where $K_{g},M_{G}$ are respectively the normalization constant and an
effective (constant) mass. With this effective gluon propagator it is possible
to carry the overall calculation analytically and free of UV divergences, in
spite of the need of a renormalization of their equations to settle the scale
of $K_{g}$. Besides that, this effective gluon propagator provides a string
tension and it provides numerical results for mesons-constituent quark form
factors that are basically the same as results from another gluon propagator
extracted from extensive calculations with Schwinger Dyson equations for
hadron structure [33]. This comparison was shown in, for example, Refs. [28,
34]. This normalization of the gluon propagator was fixed by the pseudoscalar
pion coupling constant in the vacuum, as a renormalization condition, such
that it reproduces the phenomenological value $G_{ps}\simeq 13$ [36, 38]. The
calculation of this coupling constant, by considering the same method employed
to calculate eqs. (2.2), is shown in the Appendix. It is interesting to note
that in the magnetic field contribution for the form factor above, the
Schwinger phase from the quark propagators does not contribute since it can be
gauged away for this type of diagrams [39].

Figure 2: Leading contribution: one magnetic field insertion for each internal
quark line.

Figure 3:  Two magnetic field insertions on a single quark internal line.

Figure 4: Two magnetic field insertions on a single quark internal line.

By considering the charged and neutral pion fields with pseudoscalar currents
associated, the pion-constituent quark coupling above can be written as:

$\displaystyle{\cal
L}_{\pi-Q(B)}=F_{ps}^{B}(Q,K)\left[\sqrt{2}c_{1}\left(\pi^{+}\bar{d}u+\pi^{-}\bar{u}d\right)+c_{3}\pi^{0}(\bar{u}u-\bar{d}d)\right],$
(9)

To calculate these form factors we considered the Feynman trick in terms of
Feynman parameters although it is not possible to perform all integrals
analytically. So, with the Feynman trick we perform the integral in momentum
which makes possible to perform the Fourier transform to calculate the Yukawa
potential and corrections at the cost of introducing integrals in the Feynman
parameters that is numerically simpler to be achieved.

####  2.2.1  Form factor: isotropic part $F_{ps}^{B,iso}$

The isotropic part $F_{ps}^{B,iso}$ (2.2) has the following integral:

$\displaystyle I_{4}(Q^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\int\frac{dk_{0}}{(2\pi)}\frac{-(k+K)\cdot(k+Q+K)+M^{2}}{[(k+K)^{2}-M^{2}]^{2}[(k+K+Q)^{2}-M^{2}]^{2}}R(-k)$
(10)

For the gluon propagator of eq. (8), this integral have 6 double poles, four
of them from quark propagators and 2 from the gluon propagator. For an off
shell pion, $Q_{0}\to 0$, and on shell constituent quark in its rest frame,
$K^{2}=M^{2}$, and by using the Feynman parameterization, this integral can be
written as:

$\displaystyle\frac{I_{4}(Q^{2})}{K_{g}}$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi^{2}}\int_{0}^{1}dz\int_{0}^{z-1}dy\left(\frac{-2(1-y-z)yz}{3A^{3}}+\frac{D_{iso}(1-y-z)yz}{A^{4}}\right),$
(11) where $\displaystyle A=-[M^{2}(1-z)^{2}+{\vec{Q}}^{2}y(1-y)+M^{2}_{g}z],$
(12) $\displaystyle D_{iso}=[M^{2}z^{2}-\vec{Q}^{2}[y^{2}-y]+M^{2}](1-y-z)yz.$

####  2.2.2 Form factor: Anisotropic part $F_{ps}^{B,ani}$

The anisotropic part of the magnetic field correction to the form factor (7)
is given by

$\displaystyle\frac{I_{5}(Q_{\perp}^{2},Q^{2})}{K_{g}}=\int_{k}\frac{-(k_{\perp}+K_{\perp})\cdot(k_{\perp}+K_{\perp}+Q_{\perp})}{((k+K)^{2}-M^{2})^{2}((k+Q+K)^{2}-M^{2})^{2}}\frac{1}{(k^{2}-M^{2}_{g})^{2}}.$
(13)

By employing the same Feynman trick, the momentum integration of the
anisotropic part of the form factor , for $\vec{K}=0$ and $K_{0}=M$ and
$Q_{0}=0$, this form factor can be written as:

$\displaystyle\frac{I_{5}(Q_{\perp}^{2},Q^{2})}{K_{g}}$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi^{2}}\int_{0}^{1}dz\int_{0}^{z-1}dy\left(-\frac{xyzQ_{\perp}^{2}[y^{2}-y]}{A^{4}}+\frac{1}{3A^{3}}\right).$
(14)

##  3  The magnetic field corrections to the Yukawa potential

By replacing the magnetic field dependent quantities in eq. (2) the potential
in the momentum space with corrections due to weak magnetic fields is:

$\displaystyle\tilde{V}(Q)$ $\displaystyle=$ $\displaystyle
G_{ps}^{B}(K,Q)D_{\pi}^{B}(Q^{2})G_{ps}^{B}(K,Q)$ (15) $\displaystyle=$
$\displaystyle\left(g_{ps}+F_{ps}^{B}(K,Q)\frac{(eB)^{2}}{M^{4}}\right)\left(D_{0}(Q)+(eB)^{2}D_{1}^{B}(Q,Q_{\perp})\right)\left(g_{ps}+F_{ps}^{B}(K,Q)\frac{(eB)^{2}}{M^{4}}\right)+...$
$\displaystyle\simeq$ $\displaystyle
V_{0}(Q^{2})+V_{\pi}^{B}(Q)+V_{FF}^{B}(Q^{2})+...$

where, the first term yields the usual Yukawa potential, and the two types of
corrections are due, respectively, to the pion propagator ($V_{\pi}^{B}(Q)$)
and to the pion form factor ($V_{FF}^{B}(Q^{2})$).

Note that both the B-dependent corrections to the pion propagator and to the
pion coupling are of the order of $(eB)^{2}$. They will be computed separately
as suggested above. For the corresponding potential in position space we take
$Q_{0}=0$ and $K^{2}=M^{2}$. The following Fourier transformation for each of
the contributions shown above will be computed:

$\displaystyle
V(\vec{R})=\int\tilde{V}(\vec{Q}^{2})e^{-i\vec{Q}\cdot\vec{R}}\frac{d^{3}Q}{(2\pi)^{3}}.$
(16)

By defining $E_{Q}=\sqrt{\vec{Q}^{2}+m^{2}}$ we have the following magnetic
field corrections:

$\displaystyle V_{FF}^{B}(\vec{R})$ $\displaystyle=$
$\displaystyle-i\left(\frac{eB}{M^{2}}\right)^{2}g_{ps}\int
F_{ps}^{B}(K,Q)\frac{1}{E_{Q}^{2}}e^{-i\vec{Q}\cdot\vec{R}}\frac{d\vec{Q}}{(2\pi)^{3}},$
(17) $\displaystyle V_{\pi}^{B}(\vec{R})\equiv
I_{1}(R)+I_{2}(R_{z},R_{\perp})$ $\displaystyle=$
$\displaystyle-i(eB)^{2}g_{ps}^{2}\int\frac{1}{E_{Q}^{4}}\left(\frac{1}{E_{Q}^{2}}+\frac{2Q_{\perp}^{2}}{E_{Q}^{4}}\right)e^{-i\vec{Q}\cdot\vec{R}}\frac{d\vec{Q}}{(2\pi)^{3}},$
(18)

where the both parts, $V_{\pi}^{B}(\vec{R})$ and $V_{FF}^{B}(\vec{R})$, have
two parts each one, as calculated below - one isotropic and one anisotropic.

###  3.1  Contribution of the B-dependent meson propagator

By denoting $Q=|\vec{Q}|$ and $R=|\vec{R}|$, the first integral of Eq. (18)

$\displaystyle I_{1}(R)$ $\displaystyle=$
$\displaystyle-\frac{g_{ps}^{2}}{(2\pi)^{2}}(eB)^{2}\int_{0}^{\infty}dQ\;Q^{2}\frac{e^{iQ|\vec{R}|}-e^{-iQ|\vec{R}|}}{(iQ|\vec{R}|)\;(Q^{2}+m^{2})^{3}}$
(19)

There are two triple poles at $Q=\pm im$, so that, by calculating the
residues, it yields

$\displaystyle I_{1}(R)$ $\displaystyle=$
$\displaystyle-\frac{g_{ps}^{2}}{32\pi}\frac{(eB)^{2}}{m^{4}}(m^{2}|\vec{R}|+m)e^{-m|\vec{R}|},$
(20)

The next correction is anisotropic with an integral denoted by $I_{2}$, it can
be written as:

$\displaystyle I_{2}(R_{z},R_{\perp})$ $\displaystyle=$
$\displaystyle-(eB)^{2}\frac{g_{ps}^{2}}{(2\pi)^{3}}\int
dQ_{z}d^{2}Q_{\perp}\;2Q_{\perp}^{2}\frac{e^{i(Q_{z}R_{z}+Q_{\perp}\cdot
R_{\perp})}}{(Q^{2}+m^{2})^{4}}$ (21)

where

$R_{z}=|z_{1}-z_{2}|,\;\;\;\;R_{\perp}=|R_{1}^{\perp}-R_{2}^{\perp}|.$

The first integration in Eq. (21) is the angular one, for which one obtains
the Bessel function since:

$\displaystyle 2\int_{0}^{\pi}d\theta e^{\beta\cos(\theta)}=2\pi J_{0}(\beta)$
(22)

where $\beta=R_{\perp}Q_{\perp}$. The second integration that can be done
analytically ($dQ_{z}$) can be solved by recurring to the residue theorem by
considering the 4th order poles. With an integration in the upper semiplane
the following pole is taken into account
$Q_{z}=i\sqrt{Q_{\perp}^{2}+m^{2}}\equiv iE_{p}$. It yields:

$\displaystyle I_{2}(R_{z},R_{\perp})=-\frac{g_{ps}^{2}}{\pi
2^{4}}(eB)^{2}{\cal J}_{2}(R,R_{z})\equiv-\frac{g_{ps}^{2}}{\pi
2^{4}}(eB)^{2}\int
dQ_{\perp}Q_{\perp}^{3}e^{-R_{z}E_{p}}J_{0}(R_{\perp}k_{\perp})\times\left(\frac{R_{z}^{3}}{6E_{p}^{4}}+\frac{R_{z}^{2}}{E_{p}^{5}}+\frac{5R_{z}}{2E_{p}^{6}}+\frac{5}{2E_{p}^{7}}\right)$
(23)

being the remaining integral solved numerically. This result is quite
anisotropic.

The resulting magnetic field dependent correction due to the pion propagator
can be written as:

$\displaystyle V_{\pi}^{B}(R,R_{z})$ $\displaystyle=$
$\displaystyle-\frac{g_{ps}^{2}(eB)^{2}}{32}\left[\left(\frac{|\vec{R}|}{m^{2}}+\frac{1}{m^{3}}\right)e^{-m|\vec{R}|}+2{\cal
J}_{2}(R,R_{z})\right],$ (24)

###  3.2  Contribution of the pion form factor

The pion form factor contribution for the magnetic field dependent Yukawa
potential (3.2) has two terms and, accordingly, the corresponding
contributions for the Yukawa potential splits into two terms:

$\displaystyle V_{FF}^{B}(R)=V_{iso}(R)+V_{ani}(R_{z},R_{\perp})$
$\displaystyle=$ $\displaystyle
2i\left(\frac{eB_{0}}{M^{2}}\right)^{2}\;\;g_{ps}C_{PS}^{B}C_{i}\int\frac{d^{3}Q}{(2\pi)^{3}}e^{-i\vec{Q}\cdot\vec{R}}\frac{-iI_{4}(\vec{Q}^{2})}{\vec{Q}^{2}+m^{2}}$
$\displaystyle+$ $\displaystyle
2i\left(\frac{eB_{0}}{M^{2}}\right)^{2}\;\;g_{ps}C_{PS}^{B}C_{i}\int\frac{d^{2}Q_{\perp}dQ_{z}}{(2\pi)^{3}}e^{-i(Q_{z}R_{z}+Q_{\perp}\cdot
R_{\perp})}\frac{-iI_{5}(Q_{\perp}^{2},Q^{2})}{\vec{Q}^{2}+m^{2}},$

being $\vec{Q}^{2}=Q_{z}^{2}+Q_{\perp}^{2}$.

####  3.2.1  Isotropic term

For the Fourier transform of $V_{iso}$ first the angular integration is done
and it leads to the following integral in 3- momentum spherical coordinates:

$\displaystyle V_{iso}$ $\displaystyle=$ $\displaystyle
C_{iso}\int\frac{Q^{2}\;dQ}{(2\pi)^{2}}\frac{(e^{-iQR}-e^{iQR})}{iQR}\;\frac{I_{4}(Q^{2})}{\vec{Q}^{2}+m_{\pi}^{2}}\equiv
C_{iso}[{\cal F}_{4a}(R)+{\cal F}_{4b}(R)]$ (26)

where
$C_{iso}=2i\left(\frac{eB_{0}}{M^{2}}\right)^{2}\;\;g_{ps}C_{PS}^{B}C_{i}K_{g}$.
In this integral there are the two simple poles in the complex plane from the
pion propagator and two double or triple poles (for each of the parts of
integral $I_{4}$) from the form factor eq. (11). These poles can be written as

$\displaystyle|\vec{Q}|=\pm i\phi\equiv\pm
i\sqrt{\frac{M^{2}(1-z)^{2}+M^{2}_{g}z}{y(1-y)}}$ (27)

By performing the integrals with the residue theorem, by taking into account
all the poles above and below the real axis, the following equations are
obtained

$\displaystyle{\cal F}_{4a}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{3}}\int_{y,z}[(1-y-z)yz]\left[\frac{2(e^{-\phi
R}-e^{-m_{\pi}R})}{R(\phi^{2}-m_{\pi}^{2})^{3}}+\frac{Re^{-\phi
R}}{4\phi^{2}(\phi^{2}-m_{\pi}^{2})}+\frac{e^{-\phi
R}}{4\phi^{3}(\phi^{2}-m_{\pi}^{2})}+\frac{e^{-\phi
R}}{\phi(\phi^{2}-m_{\pi}^{2})^{2}}\right]$ (28) $\displaystyle{\cal F}_{4b}$
$\displaystyle=$
$\displaystyle\frac{-1}{32\pi^{3}}\int_{y}\int_{z}\;\frac{(1-y-z)yz}{[(y(1-y)]^{4}}F_{4b}$
(29)

where

$\displaystyle F_{4b}$ $\displaystyle=$ $\displaystyle\frac{R^{2}e^{-\phi
R}}{8\phi^{3}(m_{\pi}^{2}-\phi^{2})}+\frac{3Re^{-\phi
R}}{8\phi^{2}(m_{\pi}^{2}-\phi^{2})}+\frac{3e^{-\phi
R}}{8\phi^{5}(m_{\pi}^{2}-\phi^{2})}+\frac{3Re^{-\phi
R}}{4\phi^{2}\left(m_{\pi}^{2}-\phi^{2}\right)^{2}}-\frac{3e^{-\phi
R}}{4\phi^{3}(m_{\pi}^{2}-\phi^{2})^{2}}$ (30) $\displaystyle+$
$\displaystyle\frac{6e^{-\phi
R}}{2\phi(m_{\pi}^{2}-\phi^{2})^{3}}+\frac{6\left(-e^{-\phi
R}+e^{-m_{\pi}R}\right)}{(m_{\pi}^{2}-\phi^{2})^{4}}$

There is only one term (the last one) in each of these equations (${\cal
F}_{4a}$ and ${\cal F}_{4b}$) due to the poles from the pion propagator, being
all the others due to the structure of the form factor.

####  3.2.2  Anisotropic term

The anisotropic contribution of the form factor is written in terms of the
momentum integral eq. (13) that, for $\vec{K}=0$ and $K_{0}=M$ and $Q_{0}=0$,
is given by:

$\displaystyle I_{5}(Q^{2},Q_{\perp})$ $\displaystyle=$ $\displaystyle-
K_{g}\frac{6i}{12\pi^{2}}\int_{0}^{1}dy\int_{0}^{1-y}dz\left(\frac{-i}{4}\frac{D(1-y-z)yz}{(y(1-y))^{4}\tilde{A}^{4}}+\frac{ix(1-x-z)z}{(y(1-y))^{3}\tilde{A}^{3}\times
12}\right)$ (31) $\displaystyle D$ $\displaystyle=$
$\displaystyle\vec{Q}_{\perp}^{2}y(1-y)$ $\displaystyle\tilde{A}(x,y)$
$\displaystyle=$ $\displaystyle-[{\vec{Q}}^{2}+\phi]$

The Fourier Transform of the potential can be written as:

$\displaystyle V_{ani}(R_{z},R_{\perp})\equiv{\cal F}(V^{B,ani}_{ps}(Q))$
$\displaystyle=$ $\displaystyle
C_{iso}\int\frac{d^{2}Q_{\perp}dQ_{z}}{(2\pi)^{3}}e^{-i(Q_{z}\cdot
R_{z}+Q_{\perp}\cdot
R_{\perp})}\frac{I_{5}(\vec{Q}^{2},\vec{Q}_{\perp}^{2})}{\vec{Q}^{2}+M_{\pi}^{2}}$
(32)

By considering the cylindrical coordinate system, the angular integration
leads to a Bessel function of the first kind, $J_{0}(Q_{\perp}R_{\perp})$.
After that, The integration in $dQ_{z}$ can be done with the residue theorem
for the following imaginary poles:

$E^{\perp}\equiv\sqrt{\phi+\vec{Q}_{\perp}^{2}},\qquad
E_{\pi}^{\perp}\equiv\sqrt{m_{\pi}^{2}+\vec{Q}_{\perp}^{2}}.$ (33)

The following equation is obtained

$\displaystyle V_{ani}(R)=\frac{C_{iso}}{(2\pi)}\int
dQ_{\perp}\;Q_{\perp}\;J_{0}(Q_{\perp}R_{\perp})\frac{[y(1-y)]^{3}}{i(1-y-z)yz}\;I_{p}(Q_{\perp}^{2}),$
(34)

Where the following equation for $I_{p}(Q_{\perp}^{2})$ was defined:

$\displaystyle I_{p}(Q_{\perp}^{2})$ $\displaystyle=\frac{\pi
Q_{\perp}^{2}R_{z}e^{-(E^{\perp})R_{z}}}{2(E^{\perp})^{2}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{3}}-\frac{\pi
Q_{\perp}^{2}e^{-(E^{\perp})R_{z}}}{(E^{\perp})\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{4}}+\frac{\pi
Q_{\perp}^{2}e^{-E_{\pi}^{\perp}R_{z}}}{(E_{\pi}^{\perp})\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{4}}+\frac{\pi
R_{z}e^{-(E^{\perp})R_{z}}}{6(E^{\perp})^{2}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{2}}$
$\displaystyle-\frac{\pi
e^{-(E^{\perp})R_{z}}}{3e\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{3}}+\frac{\pi
e^{-(E_{\pi}^{\perp})R_{z}}}{3(E_{\pi}^{\perp})\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{3}}+\frac{5\pi
Q_{\perp}^{2}e^{-(E^{\perp})R_{z}}}{16(E^{\perp})^{7}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)}$
$\displaystyle+\frac{5\pi
Q_{\perp}^{2}R_{z}e^{-(E^{\perp})R_{z}}}{16(E^{\perp})^{6}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)}+\frac{\pi
Q_{\perp}^{2}R_{z}^{2}e^{-(E^{\perp})R_{z}}}{8(E^{\perp})^{5}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)}-\frac{3\pi
Q_{\perp}^{2}e^{-(E^{\perp})R_{z}}}{8(E^{\perp})^{5}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{2}}-\frac{\pi
e^{-(E^{\perp})R_{z}}}{8(E^{\perp})^{5}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)}$
$\displaystyle+\frac{\pi
Q_{\perp}^{2}R_{z}^{3}e^{-(E^{\perp})R_{z}}}{48(E^{\perp})^{4}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)}-\frac{3\pi
Q_{\perp}^{2}R_{z}e^{-(E^{\perp})R_{z}}}{8(E^{\perp})^{4}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{2}}-\frac{\pi
R_{z}e^{-(E^{\perp})R_{z}}}{8(E^{\perp})^{4}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)}-\frac{\pi
Q_{\perp}^{2}R_{z}^{2}e^{-(E^{\perp})R_{z}}}{8(E^{\perp})^{3}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{2}}$
$\displaystyle+\frac{\pi
Q_{\perp}^{2}e^{-eR_{z}}}{2(E^{\perp})^{3}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{3}}-\frac{\pi
R_{z}^{2}e^{-(E^{\perp})R_{z}}}{24(E^{\perp})^{3}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)}+\frac{\pi
e^{-(E^{\perp})R_{z}}}{6(E^{\perp})^{3}\left((E_{\pi}^{\perp})^{2}-(E^{\perp})^{2}\right)^{2}}$
(35)

The poles of the pion propagator are responsible for the third and fifth terms
and all the others come from the pion form factor structure.

Note that in all the contributions from the pion form factor
($V_{FF}^{B}(R_{z},R)\sim V_{iso}(R)+V_{ani}(R_{z},R)$) there is a strong
dependence on the constituent quark and gluon effective masses that are not
negligible.

##  4  Numerical results

The complete leading weak magnetic field correction to the Yukawa potential
can be written in terms of three terms:

$\displaystyle V^{B}(R_{z},R)$ $\displaystyle=$ $\displaystyle
V_{\pi}^{B}(R_{z},R)+V_{FF}^{B}(R_{z},R),$ $\displaystyle V_{FF}^{B}(R_{z},R)$
$\displaystyle=$ $\displaystyle V_{iso}^{B}(R)+V_{ani}^{B}(R_{z},R),$ (36)

These terms were given in eqs. (24), (26) and (34) and they’ll be shown below
compared to the Yukawa potential ($V_{Yuk}(R)$) by means of the ratios:

$\displaystyle\frac{V^{B}(R_{z},R)}{V_{Yuk}}$ $\displaystyle=$
$\displaystyle\frac{V_{FF}^{B}(R_{z},R)+V_{\pi}^{B}(R_{z},R)}{V_{Yuk}(R)},$
(37) $\displaystyle\frac{V_{4,5}(Rz,R)}{V_{Yuk}(R)}$ $\displaystyle=$
$\displaystyle\frac{V_{FF}^{B}(R_{z},R)}{V_{Yuk}(R)},$ (38)
$\displaystyle\frac{I_{12}(R_{z},R)}{V_{Yuk}(R)}$ $\displaystyle=$
$\displaystyle\frac{V_{\pi}^{B}(R_{z},R)}{V_{Yuk}(R)},$ (39) $\displaystyle
V_{Yuk}(R)$ $\displaystyle=$ $\displaystyle-
g_{ps}^{2}\frac{e^{-m_{\pi}R}}{4\pi R}.$ (40)

The following phenomenological or experimental numerical values for the
parameters will be considered for the pseudoscalar pion coupling [36, 38],
quark effective mass, gluon effective mass and pion mass (whenever
degenerate):

$\displaystyle
g_{ps}=13,\;\;\;\;\;M_{q}=0.35\;GeV,\;\;\;\;\;M_{g}=0.5\;GeV,\;\;\;\;\;m_{\pi}=0.137GeV$
(41)

This effective gluon mass is slightly larger than most recent results in
lattice QCD and Schwinger Dyson equations calculations suggest [40]. For
values $M_{g}\sim 0.35$ GeV, numerical results from the form factor
contributions are increased nearly by a factor $(0.5/0.35)^{4}\sim 4$.
However, the large quark mass expansion performed to obtain the form factors
above [28, 34] also have to be compatible with a reasonable large gluon
effective mass, since the gluon effective propagator dresses the quark
currents. Therefore, such smaller gluon mass may invalidate the expansion and
it would lead to too large magnetic field corrections as it was verified
numerically. Besides that, there are indications, from Lattice QCD for exotic
mesons, of constituent gluons with a mass close to $1$ GeV [41]. Therefore, we
keep the value $M_{g}=0.5$ GeV to provide a first calculation for which the
gluon is attached to a (constituent) quark.

Besides the effects of the magnetic field on the shape of the Yukawa potential
discussed along this work, we also present the effect of the magnetic field
dependence of the hadron masses: pions and constituent quarks. Quark effective
masses increase with increasing magnetic fields [16] and neutral and charged
pions were found to have decreasing and increasing masses, respectively, as
found from different calculations with Lattice QCD (LQCD) and with the NJL
model [18, 19, 21, 22, 24, 25]. To present estimations of these effects on the
overall corrections to the Yukawa potential, we consider the numerical results
obtained in these just quoted for the range of weak magnetic field considered
in the present work ($eB\sim 0.01$ GeV2). The following numerical values for
will be adopted:

$\displaystyle M_{q}(B)$ $\displaystyle\sim$ $\displaystyle
M_{q}+0.020\;GeV,\;\;\;(MqB)$ $\displaystyle m_{\pi^{0}}(B)$
$\displaystyle\sim$ $\displaystyle m_{\pi^{0}}-0.020\;GeV,\;\;\;(pneB)$
$\displaystyle m_{\pi^{\pm}}(B)$ $\displaystyle\sim$ $\displaystyle
m_{\pi^{\pm}}+0.020\;GeV,\;\;\;(pchB),$ (42)

where the electromagnetic parts of the pion masses are considered in these
cases specifically: $m_{\pi^{\pm}}=0.139$GeV (pch) and
$m_{\pi^{0}}=m_{\pi}=0.135$ GeV (pne). However, it is important to point out
that the difference in the results of using $m_{\pi^{\pm}}$ or $m_{\pi^{0}}$
in the curves for magnetic field independent pion mass is not noted in the
figures below.

In figure (5) the total ratio $V^{B}(R_{z},R))/V_{Yuk}(R)$ is presented by
fixing $R_{z}=1fm$ for an unique constant value of the (degenerate, deg) pion
mass ($m_{\pi}=0.137$GeV) different situations for the quark effective mass:

$\displaystyle M=Mq\equiv
0.35\;GeV,\;\;\;\;\;\;\;M=3Mq,\;\;\;\;\;\;\;Mmix=\mbox{Mixed}\;Mq\;\mbox{
and}\;3Mq.$ (43)

The first case, $Mq$, is a typical constituent (up or down) constituent quark
mass and the second one, $3Mq$, a large quark effective mass, of the order of
magnitude of the nucleon mass, to test the resulting behavior. This may be
seen as a partial account of the whole nucleon, since the constituent quark is
considered to carry the whole nucleon mass tightly but not necessarily the
entire nucleon size. In this case, the normalization of the effective gluon
propagator ($K_{g}$) was found to increase considerably, although the form
factor contribution is reduced. The overall effect is to increase the strength
of the magnetic field correction to the Yukawa potential because of the
increase in $K_{g}$. To make clearer the effect of a larger quark mass in the
form factor and consequently in the Yukawa potential, the $Mmix$ case was done
as follows. The value of the effective gluon propagator normalization $K_{g}$
was fixed to reproduce $g_{ps}$, as shown in the Appendix, with usual quark
mass $Mq=0.35$ GeV and the whole estimation of the effects of the magnetic
field for the Yukawa potential was done by considering a large mass $3Mq$
(possibly as if one had a full nucleon mass). In this case, the contribution
of the form factor should be trivial (toward punctual particles and coupling)
and the magnetic field correction to the Yukawa potential reduces basically to
the simpler (punctual) pion exchange. Indeed, it is seen in the figure that
the strength of the curves $Mmix$ (the mixed case) is reduced and the term due
to the pion propagator is dominant.

The role of the magnetic field corrections to the charged and neutral pion
(pch and pne) form factors, by considering the different couplings according
to Eqs. (5,9), is presented in Fig. (6) by keeping the unique constant pion
mass $m_{\pi}=0.137$GeV. These $c_{1}=c_{2}$ and $c_{3}$ factors are given
respectively by:

$\displaystyle\frac{-4\sqrt{2}}{9}\times
G_{ps}^{B}\;(pch),\;\;\;\;\;\;\frac{5}{9}\times G_{ps}^{B}\;(pne).$ (44)

Whereas the neutral pion form factor provides a positive correction (i.e. more
attractive potential), the charged pion form factor leads to a reduction of
the total magnetic field correction (slightly less attractive potential). In
fact, there is a cancellation of contributions from the charged pion
propagator and the charged pion form factor that leads to a much smaller
overall modulus than the neutral pion. For the large quark mass limit, $3Mq$,
the overall correction to the (charged pion exchange) Yukawa potential becomes
repulsive. However, most of the other magnetic field corrections are basically
attractive, and they make the Yukawa potential more and more negative with
increasing distances - this helps to increase the range of the interaction.
Note that by the distance typical of the stability of the deuteron, $R\sim 2$
fm, the neutral and charged pion lead to different strengths of the order of
$2\%-3\%$ without taking into account further effects as discussed below.

Different values for $R_{z}$, 1 fm and 2 fm, were used to test the
contributions of the anisotropic terms $V_{5}$ and $I_{2}$. It turns out that
the resulting anisotropies were found to be very small, of the order of few
percent of the isotropic terms. Therefore, by fixing another value, for
example, $R_{z}=2$ fm for $R>2$fm, it leads to very tiny negligible
differences in the Figures.

Figure 5:  Total ratio $V^{B}(R_{z},R))/V_{yuk}(R)$ for $R_{z}=1fm$ for the
following cases: $M=Mq=0.35$GeV, $M=3Mq$ and the mixed calculation $Mmix$, for
degenerate $m_{\pi}$ by reducing $G_{ps}^{B}c_{i}$ with unique coupling
$c^{i}=1$ (deg).  Figure 6:  Total ratio $V^{B}(R_{z},R))/V_{yuk}(R)$ for
$R_{z}=1fm$ for the following cases: $M=Mq=0.35$GeV, $M=3Mq$ and the mixed
calculation $Mmix$, for unique $m_{\pi}$ and $G_{ps}^{B}c_{i}$ for charged and
neutral pions ($c^{1}=-4\sqrt{2}/9$ and $c^{3}=5/9$ respectively) denoted by
pch and pne.

Figs. (7) and (8) present the same cases of the previous two figures by
considering constituent quark mass under magnetic field according to Eq. (4).
The strengths of the magnetic field corrections increase slightly as compared
to the previous Figs. (5) and (6). As it can be seen, also from the previous
Figures, the difference between the mixed case $Mmix$ (or $MmixB$ in Figs.
(7)) and Fig. (8)), respectively for degenerate pion mass (deg) and neutral
pion (via the form factor) (pne), is not large.

It is important to make clear the difference between the cases with larger
quark masses, $3Mq$ and $Mmix$. In the first case, the quark mass is large in
all calculation, i.e. it leads to a more trivial punctual interaction but it
makes the normalization of the effective gluon propagator to be larger. This
effect in $K_{g}$ manifests in all the resulting curves for $V^{B}(R_{z},R)$.
The second case, $Mmix$, helps to identify the role of particular corrections
to the Yukawa potential. For large quark mass, the form factor tends to be
trivial and the most important contribution for the Yukawa potential is the
one from the pion propagator. Having shown this point, $Mmix$ will not be
exploited much further below.

Figure 7:  Total ratio $V^{B}(R_{z},R))/V_{yuk}(R)$, at $R_{z}=1fm$, for
magnetic field dependent quark effective mass (MqB), for the following cases:
$M=Mq=0.35$GeV, $M=3Mq$ and the mixed calculation $Mmix$, for unique
$m_{\pi}$, and $G_{ps}^{B}c_{i}$ by reducing $c^{i}=1$ (deg).  Figure 8:
Total ratio $V^{B}(R_{z},R))/V_{yuk}(R)$, at $R_{z}=1fm$, for magnetic field
dependent quark effective mass (MqB), for the following cases: $M=Mq=0.35$GeV,
$M=3Mq$ and the mixed calculation $Mmix$, by considering an unique constant
$m_{\pi}$ and $G_{ps}^{B}c_{i}$ for charged and neutral pions
($c^{1}=-4\sqrt{2}/9$ and $c^{3}=5/9$ respectively) denoted by pch and pne.

Fig. (9) presents similar curves to Fig. (6), by considering the neutral and
charged pion masses to be non-degenerate and with magnetic field correction
according to (4) in all the terms - by keeping the quark effective mass in the
vacuum. Constituent quark mass was kept as $Mq$ or $3Mq$. The difference in
the behavior of these magnetic field corrections for neutral or charged pions
becomes slightly larger than in the cases presented in Fig. (6) for which the
pion masses were taken for $B=0$.

Figure 9:  Total ratio $V^{B}(R_{z},R))/V_{yuk}(R)$ for $R_{z}=1fm$ for the
following cases: $M=Mq=0.35$GeV, $M=3Mq$, and $G_{ps}^{B}c_{i}$ for charged
and neutral pions ($c^{1}=-4\sqrt{2}/9$ and $c^{3}=5/9$ respectively) denoted
by pch and pne with the magnetic field dependent pion mass.

Fig. (10) presents the same cases of the previous Fig. (9) by considering the
magnetic field dependent quark effective mass - eq. (4). Again we see that, by
taking into account the magnetic field dependent masses, the strength of the
corrections to the Yukawa potential increases for both neutral and charged
pions.

Figure 10:  Total ratio $V^{B}(R_{z},R))/V_{yuk}(R)$ for $R_{z}=1fm$ for the
following cases: quark effective mass corrected by the magnetic field,
$M=MqB$, $M=3MqB$, and $G_{ps}^{B}c_{i}$ for charged and neutral pions
($c^{1}=-4\sqrt{2}/9$ and $c^{3}=5/9$ respectively) (pch and pne) with the
magnetic field dependent pion mass.

In the next figure, (11), the separate contributions $V^{B}_{FF}$ (from
$V_{4}$ and $V_{5}$) and $V_{\pi}^{B}$ (from $I_{1}+I_{2}$), eqs. (38) and
(39), are shown for the degenerate numerical factors in the form factors and
degenerate pion mass. Three cases of quark masses, $Mq$, $3Mq$ and $Mmix$ were
considered. In Fig. (12) similar curves are presented for two cases for the
magnetic field dependent quark mass $MqB$ and $3MqB$ by considering the
degenerate numerical factors of the form factors $c_{i}$. The effect due to
non degenerate pion mass is stronger in the contribution from the pion
exchange. The non degenerate pion mass effect in the form factor is small
although it is amplified if the non degenerate coupling to magnetic field, eq.
(9), is taken into account. As seen before, these two effects increase the
strength of the magnetic field corrections to the Yukawa potential and they
make the range of the potential to change differently for neutral or charged
pions. For neutral pion exchange the range of the Yukawa potential increases
whereas for the charged pion exchange it is nearly unchanged dependending on
the range of the parameters such as constituent quark mass.

Figure 11:  Separate contributions for $V^{B}_{FF}(R_{z},R)$, from isotropic
and anisotropic corrections $V_{4}$ and $V_{5}$, and for
$V_{\pi}^{B}(R_{z},R)$, from $I_{1}+I_{2}$. $R_{z}=1fm$ for Mq, 3Mq, $Mmix$,
$m_{\pi}$, and $G_{ps}^{B}$ without the isospin factor $c_{i}$.  Figure 12:
Similar curves to the previous figure: Separate contributions for
$V^{B}_{FF}(R_{z},R)$, from isotropic and anisotropic corrections $V_{4}$ and
$V_{5}$, and for $V_{\pi}^{B}(R_{z},R)$, from $I_{1}+I_{2}$. $R_{z}=1fm$ for
quark masses MqB and 3MqB, non-degenerate $m_{\pi}^{0,\pm}$, and $G_{ps}^{B}$
without the isospin factor $c_{i}$,

##  5  Summary

Magnetic field induced corrections to the Yukawa potential were presented for
$(eB)\sim 0.1M_{q}^{2}\sim 50\times 10^{12}$T. Most of the calculations are
basically analytical, with remaining numerical integrals mostly in Feynman
parameters and, in some cases, one integration in a component of the pion
three-momentum. Three types of effects were investigated, two of them
concerning the shape of the potential, besides the role of masses of pion and
of constituent quarks. Firstly, the (relatively) weak magnetic field
contribution for the pion propagator. Secondly, the contribution of the weak
magnetic field for the pseudoscalar pion form factor, $G_{ps}^{B}(Q^{2})$ in a
one loop calculation. This second contribution leads to a dependence on the
constituent quark effective mass and on a gluon effective mass which
parameterizes the gluon propagator. Finally, the effects of the magnetic field
on the pion and constituent quark masses were also considered. The first two
effects are of the order of magnitude of $(eB)^{2}$ and the third ones go with
$(eB)$. Considering the values of the (phenomenological) parameters of masses,
three different situations were analyzed. Firstly, the constituent quark mass
was taken to be $Mq=0.35$ GeV for which results are well inside the
perturbative regime for $M_{g}=0.5$ GeV. Secondly, a larger quark mass, $3Mq$,
corresponding to nearly the nucleon mass was considered. However, in this
calculation the normalization of the gluon propagator was modified accordingly
to keep $g_{ps}=13$, the physical value. Results from the form factor
contributions increase accordingly by nearly three times. Finally a mixed
calculation ($Mmix$) to keep the quark mass $Mq=0.35$GeV to fit $g_{ps}$ was
done, but increasing the quark mass of the form factors to $3Mq$. This yielded
a strong reduction of the contribution of the form factors leading to a
pointlike interaction.

The overall modification of the potential for the weak magnetic field limit is
not large in the range of distances $R\sim 1-3$ fm, the long range component
of the nucleon interactions. These magnetic field contributions become larger
at larger distances, mainly for the neutral pion exchange. Anisotropic
components are, in general, quite smaller than the isotropic ones in the range
of distances exploited in this work. However, for larger distances, the
overall Yukawa potential becomes tiny. Perturbative corrections due to the
magnetic field are, therefore, of the order of around $5\%$. Charged and
neutral pion exchanges receive different contributions from the coupling to
the magnetic field, and therefore nucleon (and more generally baryons)
interactions should manifest this splitting associated to the neutral or
charged pion exchange.

In most of the cases, the Yukawa potential becomes more attractive with
slightly longer range, given that the magnetic field correction increases the
strength of the Yukawa potential. The exception to this increase was found for
the charged pion exchange that may lead to a less attractive Yukawa potential
and a shorter range interaction. The magnetic field contributions to
constituent quark and pion masses lead overall to amplification of the effects
due to the form factor and to the pion propagator. The inclusion of the form
factor in the calculation of the potential has brought several issues into
discussion. For the limit of very large constituent quark mass and effective
gluon mass these contributions from the form factor tend to zero.
Modifications due to magnetic fields can be expected to occur in the
neighborhood (low density outer-crust) of dense stars when magnetic fields may
reach such values we consider [12]. Our results suggest non-trivial
contributions to the deuteron formed in such environment, or alternatively,
for the equation of state and other related observables at low baryon density,
may be expected. In the outer-crust of dense stars, the density is estimated
to be less than the nuclear saturation density. Note, however, that the long
range pion exchange term is suppressed in the high density regime. Besides
that, currently, the expected values of magnetic fields in heavy ions
collisions are quite reduced [15]. However, if magnetic fields with such
strengths are reached by the hadronization time scales or in the spectator
region of the collisions, in peripheral heavy ions collisions [8] or in
experiments where low density matter is obtained, Yukawa- long range nucleon
interaction may also be probed. Accordingly, deuterons formed in such magnetic
fields possibly undergo to excited states when passing to a region of zero
magnetic field. Along the same lines, other meson exchange must be verified.
This will be developed elsewhere. Magnetic field corrections to the effective
gluon propagator and quark-gluon running coupling constant were not
considered. This kind of calculation may possibly help to identify the
validity of the CQM in what concerns the hadron couplings by means of such
internal degrees of freedom. A detailed investigation of the role of the form
factor for the potential in vacuum will be performed elsewhere by one of the
authors. The role of different choices of the gluon effective propagator, and
the limit of very strong magnetic fields, are intended to be scrutinized
further in a different work.

## Acknowledgements

F.L.B. is a member of the INCT-FNA Proc. 464898/2014-5. The authors would like
to thank ICTP-SAIFR (FAPESP grant 2021/14335-0) where this work was planned.
F.L.B. acknowledges partial financial support from CNPq-312750/2021-8 and
CNPq-407162/2023-2; M.L. and C. V. acknowledge support from ANID/CONICYT
FONDECYT Regular (Chile) under Grants No. 1190192, and 1220035.

##  6  Appendix: Pseudoscalar pion coupling to constituent quarks

The one loop pseudoscalar pion form factor, for the gluon effective propagator
(8), can be written as:

$\displaystyle G_{PS}$ $\displaystyle=$ $\displaystyle
C_{ps0}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k\cdot(k+Q)+M^{2}}{(k^{2}-M^{2})((k+Q)^{2}-M^{2})((k-K)^{2}-M_{g}^{2})^{2}}$
(45)

where $C_{ps0}=8N_{c}(\alpha K_{g})$. By employing the Feynman trick to carry
the momentum integral, it can be written as:

$\displaystyle G_{PS}$ $\displaystyle=$ $\displaystyle
C_{ps0}\frac{i}{6(4\pi)^{2}}\int_{0}^{1}dz\;\int_{0}^{1-z}dy\;z\left(\frac{-2}{E}+\frac{F}{E^{2}}\right)$
(46) $\displaystyle E$ $\displaystyle=$ $\displaystyle
Q^{2}y(y-1)+K^{2}z(z-1)-2K\cdot Qyz+M^{2}(1-y-z)+M^{2}y+M_{g}^{2}z$
$\displaystyle F$ $\displaystyle=$ $\displaystyle
Q^{2}(y-1)y+K^{2}z^{2}+K\cdot Qz(1-2y)+M^{2}$ (47)

For the numerical calculation of this coupling constant, we considered off
shell pions and quarks: $Q^{2}=0$ and $K^{2}=0$.

The value of $K_{g}$ is fixed with

$K_{g}=\frac{13}{\frac{G_{PS}}{K_{g}}}.$

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11institutetext: Bose Institute, EN 80, Sector V, Bidhan Nagar, Kolkata -
700091, West Bengal, India
11email<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>
# Inclusive photon production at forward rapidities using PMD in p–Pb
collisions at $\sqrt{\it{s}_{\rm{NN}}}=8.16$ TeV with ALICE

Mohammad Asif Bhat
$($for the ALICE collaboration)

###### Abstract

We report on the performance studies of two correction methods, namely the
efficiency-purity method and the Bayesian unfolding method applied to the
pseudorapidity distribution of photons. The pseudorapidity distribution of
inclusive photons at forward rapidities in the range $(2.3<\eta<3.9)$ in p–Pb
collisions at $\sqrt{\it{s}_{\rm{NN}}}=8.16$ TeV is obtained with the HIJING
Monte Carlo event generator. The simulated data samples were obtained from the
Photon Multiplicity Detector (PMD) in ALICE.

###### keywords:

Photon Multiplicity Detector, photon multiplicity, pseudorapidity
distribution, p–Pb collisions, efficiency-purity method, Bayesian unfolding
method

##  1 Introduction

Relativistic heavy-ion collisions are believed to produce a Quark Gluon Plasma
(QGP) [1] which hadronizes dominantly into pions, of which approximately one
third are neutral pions that decay into photons. Therefore, the measurement of
photon multiplicity can provide important information about bulk physics [2]
from initial scatterings to the final state QGP effects. The measurement of
photon multiplicity in p–Pb collisions [3] is very important as it is an
intermediate step going from hadronic to heavy-ion collisions.

In this analysis we generated the pseudorapidity distribution of inclusive
photons at forward rapidities ($2.3<\eta<3.9$) in p–Pb collisions at
$\sqrt{\it{s}_{\rm{NN}}}=8.16$ TeV using the HIJING [4] Monte Carlo event
generator. The photons are detected by the Photon Multiplicity Detector (PMD)
[5, 6]. The raw pseudorapidity distribution of inclusive photons was corrected
by following two methods: (i) Efficiency-purity method (ii) Bayesian unfolding
method. We used the minimium bias events, which produce at least one hit in
V0A and V0C trigger detectors. The events with the collision vertex coordinate
along the beam axis z lying between -10 to +10 cm from the nominal interaction
point are taken for physics analysis.

##  2 Pseudorapidity distribution of photons

Photons are identified using two different conditions imposed on all detected
clusters on the PMD. These are ADC $>$ 6MPV,$N_{cell}>2$ and ADC $>$
9MPV,$N_{cell}>2$, where ADC is the sum of the ADC channel numbers of the
cells within the cluster, MPV = 72 ADC channels is the most probable value of
the pion ADC distribution and $N_{cell}$ is the average number of cells
affected by 3 GeV pion from the test beam result [7]. The pseudorapidity
distribution of photons is shown on the left side of Fig 1 . This distribution
is affected by the detector finite resolution, limited detector efficiency and
acceptance and secondary contamination such as hadron clusters. Therefore
correction is needed to be applied to get the true distribution. We used two
methods for correction as detailed in the following section.

Figure 1: Left: Uncorrected pseudorapidity distribution of photons in p–Pb
collisions at $\sqrt{\it{s}_{\rm{NN}}}=8.16$ TeV using MC event generator
HIJING. Right: Efficiency and purity values as a function of pseudorapidity.

##  3 Correction methods

###  3.1  Efficiency-purity method

Efficiency ($\epsilon_{\gamma}$) is defined as the ratio of the number of
detected photons to the number of incident photons within the same acceptance
coverage

$\epsilon_{\gamma}\;\;=\;\;\frac{N_{\gamma-detected}}{N_{\gamma-
incident}}\;\;\;,$ (1)

where $\gamma-detected$ are those $\gamma-like$ clusters whose cluster
identity matches with the incident photon identity, $\gamma-incident$ are the
incident photons produced from HIJING.
Purity $(\it{p})$ is defined as the ratio of the number of detected photons to
the number of gamma-like clusters within the same acceptance coverage

$p\;\;=\;\;\frac{N_{\gamma-detected}}{N_{\gamma-like}}\;\;\;,$ (2)

where $\gamma-like$ or $\gamma-measured$ are those clusters that satisfy the
photon-hadron discrimination conditions.
The right panel of Fig 1 shows the efficiency and purity values as a function
of pseudorapidity for two different photon-hadron discrimination conditions.
To obtain the corrected photon distribution we used the following correction
factor:

$N_{\gamma-true}=\frac{p}{\epsilon_{\gamma}}\times N_{\gamma-measured}$ (3)

###  3.2 Bayesian unfolding method

Unfolding is a technique to get the true distribution from the measured
distribution. Bayesian unfolding [8] is based on the Bayes’s theorem

$P(A|B)=\frac{P(B|A)\times P(A)}{P(B)}\;\;\;\;.$ (4)

In our notation

$\bar{A_{ij}}=\frac{A_{ji}f_{i}}{\varSigma_{k}A_{ik}f_{k}}$ (5)

$f_{i}=\varSigma_{j}\bar{A_{ij}}g_{j}$ (6)

where $f_{i}$ is the unfolded distribution, $\bar{A_{ij}}$ is the inverse of
the response matrix $A_{ij}$ and $g_{j}$ is the measured distribution. The
fluctuation in the error is controlled by the regularization parameters such
as the number of iterations and smoothing. The unfolding was carried out in
each pseudorapidity bin and for each pseudorapidity bin we have obtained the
response matrix, measured ($\gamma-like$) distribution and the unfolded
distribution for different set of regularization parameters. The sets of
parameters which gives the minimium possible fluctuations in the error were
selected. The average value of the unfolded distribution divided by the bin
width was used to get the pseudorapidity distribution of photons bin by bin.

Figure 2: Left: Response matrix for interval $(2.3<\eta<2.5)$. Right:
Efficiency-purity corrected and Bayesian-unfolded pseudorapidity distribution
of photons compared with the pseudorapidity distribution of true incident
photons from HIJING.

##  4 Results

The pseudorapidity distribution of photons obtained after applying the two
correction methods is compared with the pseudorapidity distribution of true
incident photons from HIJING as shown on right side of Fig 2 . The ratios of
the true pseudorapidity distribution with the one obtained after applying each
of the two correction methods are consistent with unity with an error of 0.5%.
This shows that while analyzing the data any one of the two correction methods
can be applied and the difference arising from using the other can be included
in the systematic uncertainty.

##  5 Summary

We reported on the corrected pseudorapidity distribution of photons obtained
by two correction methods: the efficiency-purity method and the Bayesian
unfolding method. The corrected pseudorapidity distributions of photons
obtained by these two correction methods are in good agreement with the
pseudorapidity distribution of true incident photons from HIJING.

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