The Coupling Constant of the Pseudovector Pion-nucleon Interaction

for the Magnetic Moment of Nucleon

Susumu Kinpara

Institute for Quantum Medical Science (QST)
Chiba 263-8555, Japan

###### Abstract

The role of the suppression of the coupling constant in the pseudovector pion-
nucleon interaction is examined to account for the calculation of the magnetic
moment of nucleon. Among three kinds of the higher-order corrections the
vacuum polarization of the pion propagator contributes to the numerical value
considerably. On the other hand the lowest-order vertex correction for the
pion-nucleon-nucleon vertex does not give large effect in the approximation of
the on-shell momenta for nucleons.

## 1 Introduction

Nucleon is the fundamental particle interacting with other particles by
exchanging a lot of bosons. Especially pion generates the strong force and the
virtual processes around the nucleon modify the dynamical properties of the
propagation. Supposing that pion plays a significant role to describe the
nuclear system the field theoretical method is necessary to examine the
reaction mechanism. Including the effects of the interaction the propagators
are required to extract the constants of the structure inherent to the
phenomena.
One of the subjects which attracts our interest is the electromagnetic
interaction and the magnetic moment of nucleon. The cloud of pion surrounding
the bare nucleon gives rise to an additional effect which is to be included by
the perturbative calculation to construct the anomalous part in conjunction
with the self-energy since the strength of the nuclear force is stronger than
the electromagnetic interaction related with photons.
The virtual pions also interact with photon and the anomalous part consists of
the pion and the nucleon current parts. Both of the results of the lowest-
order corrections are found to be convergent about the integral related to the
diagrams with the pseudovector coupling $\pi$-N interaction. In addition to
the result of the calculation which is convergent the numerical values explain
the experimental data well provided the coupling constant of the $\pi$-N
interaction $f$ is reduced about 20$\%$ from the standard value [1]. The
present study examines the source of the shift of the coupling constant on the
basis of the next-order perturbative calculation.

## 2 The vertex correction in the pseudovector $\pi$-N interaction

The pion-nucleon-nucleon three-point vertex is corrected same as the case of
the photon-nucleon-nucleon vertex by using the pion propagator. In momentum
space the Feynman rule makes us write the vertex explicitly as

$\displaystyle\qquad\qquad\Gamma_{0}(p^{\prime},p)=-\frac{f}{m}\tau_{a}\gamma_{5}\gamma\cdot
q\qquad\qquad(q\equiv p^{\prime}-p)$ (1)

for the lowest-order in the pseudovector coupling interaction with the
coupling constant $f$ (=1) and the mass of pion $m$. Through the calculation
of the higher-order the incoming and the outgoing momenta $p$ and $p^{\prime}$
are restricted to the on-shell state and the vertex function is put between
the Dirac spinors so that it is dependent solely on the momentum transfer
$q^{2}$.
Using the rule of the diagram in momentum space the vertex function of the
second-order is given as

$\displaystyle\Gamma_{2}(p^{\prime},p)=(\frac{f}{m})^{3}\int\frac{d^{4}k}{(2\pi)^{4}}\,i\Delta_{0}(k)\,\tau_{j}\gamma_{5}\gamma\cdot(-k)\,iG_{0}(p^{\prime}-k)$
$\displaystyle\times\,\tau_{a}\gamma_{5}\gamma\cdot(-q)\,iG_{0}(p-k)\,\tau_{j}\gamma_{5}\gamma\cdot
k$ (2)

where the $\Delta_{0}$ and $G_{0}$ are the free pion and nucleon propagators.
The non-perturbative term has not been added in the present study following
the fact that the calculation of the anomalous interaction also does not use
the term. The integrand in Eq. (2) has the order $k\sim-2$ and it is not
apparent that the renormalization is possible or not different from the
pseudoscalar interaction which is $k\sim-4$ and divergent only
logarithmically.
The $k$-integral is performed by the dimensional reguralization method using
the related formulas and which makes the original form convert to the integral
on the parameters $s$ and $t$ as

$\displaystyle\Gamma_{2}(p^{\prime},p)=-\frac{f}{m}\tau_{a}\gamma_{5}\gamma\cdot
q\,\rho\,(a_{0}+a_{1}\frac{q^{2}}{M^{2}}+a_{2}(\frac{q^{2}}{M^{2}})^{2}+O((q^{2})^{3}))$
(3) $\displaystyle\rho\equiv\frac{M^{2}}{(4\pi)^{2}}(\frac{f}{m})^{2}$ (4)
$\displaystyle
a_{i}\equiv\frac{1}{2}\int_{0}^{1}ds\int_{-s}^{s}dt\;c_{i}(s,t)$ (5)
$\displaystyle c_{0}(s,t)=2(\eta-1-{\rm log}\frac{M^{2}}{m^{2}}-2\,{\rm
log}s)(1+6s^{2})-4+4s+6s^{2}$ (6) $\displaystyle
c_{1}(s,t)=-\frac{1}{2}-s+t+\frac{3}{2}(s^{2}-t^{2})+\frac{t^{2}}{2s^{2}}\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad+(\eta-1-{\rm log}\frac{M^{2}}{m^{2}}-2\,{\rm
log}s)\\{1-3(s^{2}-t^{2})\\}$ (7) $\displaystyle
c_{2}(s,t)=\frac{1}{4s^{4}}\\{s^{4}-t^{4}+\frac{1}{4}(1-6s^{2})(s^{2}-t^{2})^{2}\\}$
(8) $\displaystyle\eta\equiv\frac{2}{\epsilon}-\gamma+1-{\rm
log}\frac{m^{2}}{4\pi\mu^{2}}$ (9)

up to the $O((q^{2})^{2})$ order in the series of $q^{2}/M^{2}$ where $M$ is
the nucleon mass. The definitions of the quantities $\epsilon$, $\gamma$ and
$\mu$ in Eq. (9) are seen in Ref. [1]. It is noted that the expansion in
$q^{2}/M^{2}$ is correct within the radius of the convergence. The coefficient
$a_{1}$ is convergent and we do not need to subtract the divergent term beyond
the constant term. The numerical values are $a_{1}=0$ and $a_{2}=1/15$ in the
approximation neglecting the mass of pion $m^{2}\approx 0$. Although the
$a_{0}$ is divergent it is not essential to derive the vertex function since
the renormalization enables to eliminate the term legitimately.
The dependence of $a_{1}$ on the pion mass $m$ is thought to be small as shown
below. The correction to $c_{1}(s,t)$ is denoted by $c_{1}^{\prime}(s,t)$ and
which is given as

$\displaystyle
c_{1}^{\prime}(s,t)=\frac{\lambda(1-s)}{s^{2}+\lambda(1-s)}\cdot\frac{s^{2}-t^{2}}{4}\\{\,\frac{2(1-3s^{2})}{s^{2}}-\frac{\lambda(1-s)}{s^{2}+\lambda(1-s)}\,\\}$
$\displaystyle-\\{1-3(s^{2}-t^{2})\\}\,{\rm log}[1+\lambda s^{-2}(1-s)]\qquad$
(10)

While $\lambda\equiv m^{2}/M^{2}$ is small ($\lambda\sim 0.02$) the expansion
in $\lambda$ is not possible at $s\sim 0$ and the $s,t$-integral is done
numerically and it yields $a_{1}^{\prime}=-0.023$. The $m^{2}\neq 0$ effect on
$a_{2}$ has not been calculated for the $q^{2}-m^{2}$ term arising from the
$\sim(q^{2})^{2}$ in Eq. (3) is suppressed by the factor $\lambda$.
Taking into account the vertex correction the coupling constant $f$ is
modified as $f\rightarrow ff_{1}(q^{2})$ with

$\displaystyle f_{1}(q^{2})=1+\rho\,[\,(a_{1}+2\lambda
a_{2})\frac{q^{2}-m^{2}}{M^{2}}+a_{2}(\frac{q^{2}-m^{2}}{M^{2}})^{2}+O((\frac{q^{2}-m^{2}}{M^{2}})^{3})\,]$
(11)

effectively. The subtraction of the constant term is determined so as to
satisfy the renormalization condition $f_{1}(m^{2})=1$ about the external line
of pion. At the region $q^{2}\sim M^{2}$ typical for the use of the pion
propagator the factor does not shift so much as $f_{1}(M^{2})=1.013$. It does
not necessarily mean that the the vertex corrections have an influence on the
momentum dependence of the effective coupling constant only weakly since the
result is obtained under the on-shell condition for both nucleons. In fact the
argument $q^{2}$ generally exceeds the radius of the convergence and the
series is not convergent.

## 3 The higher-order effects to the pion and nucleon propagators

In the calculation of the anomalous part of the magnetic moment the internal
line of the pion propagator is changed from the non-interacting one as
$\Delta_{0}(q)\rightarrow\Delta(q)$. It is expressed by the Dyson equation and
the solution is given as $\Delta(q)=(\Delta(q)^{-1}-\Pi(q))^{-1}$ using the
polarization function $\Pi(q)$. So our interest is to determine the form of
$\Pi(q)$ by the perturbative calculation of the vacuum polarization. It has
been verified that the non-perturbative term in the pseudovector coupling does
not contribute to $\Pi(q)$. In the present study the form of the lowest-order
approximation $\Pi(q)\approx\Pi_{0}(q)$ is applied to give the correction.
The factor $f_{3}(q^{2})$ of the modification $f\rightarrow ff_{3}(q^{2})$ due
to the vacuum polarization is given as

$\displaystyle f_{3}(q^{2})=(1-\frac{\Pi_{0}(q)}{q^{2}-m^{2}})^{-\frac{1}{2}}$
(12)

applying the renormalized form of $\Pi_{0}(q)$. One of the examples of the
effective coupling constant is the anomalous interaction in the
electromagnetic process. The diagram consists of two parts that is the part of
the pion current and the nucleon current and their higher-order corrections
are represented by the common factor $f_{3}(q^{2})^{2}$ approximately when the
momentum of the virtual photon approaches to zero.
The renormalized form of $\Pi_{0}(q)$ has been calculated and it is given as
follows

$\displaystyle\Pi_{0}(q)=\frac{1}{\pi^{2}}(\frac{f}{m})^{2}M^{2}\,[\,q^{2}(I(q)-I_{0})-m^{2}\,I_{1}(q^{2}-m^{2})\,]$
(13) $\displaystyle I(q)\equiv\int_{0}^{1}dz\,{\rm
log}\,[1-\frac{z(1-z)q^{2}}{M^{2}}]$ (14)

where $I_{0}$ and $I_{1}$ are the coefficients of the expansion in the
vicinity of $q^{2}=m^{2}$ such as
$I(q)=I_{0}+I_{1}(q^{2}-m^{2})+O((q^{2}-m^{2})^{2})$. It corresponds to the
polarization function in the medium of the nuclear matter at the limit of the
zero density of nucleon with $m\rightarrow 0$ about the pion mass [2]. At
$q^{2}=M^{2}$ typical of the momentum for the pion propagator the
$f_{3}(q^{2})$ gives $f_{3}(M^{2})=0.74$ suitable to examine the effect of the
decrease of the coupling constant $f$.
An interesting property of the pseudovector coupling interaction is the non-
perturbative term which is ascribed to the $\theta$-function in the time-
ordered product. It is generated from the non-perturbative relation by using
the equation of motion for the $\pi$-N system besides the cancellation between
the normal-dependent term and the contact interaction in the calculation of
the perturbative expansion. Imposing the on-shell condition the non-
perturbative term in the vertex vanishes and then the vertex correction and
the vacuum polarization mentioned above are applicable to the higher-order
corrections by the renormalized functions.
The calculation of the self-energy for the nucleon propagator encounters the
difficulty preparing the finite result under the usual manner of the
renormalization by the counter terms of the mass and the field because of the
derivative coupling. In such a case the non-perturbative term plays a decisive
role to construct the self-energy and enables to obtain the convergent result.
The divergent parts occurring in the dimensional reguralization integral are
cancelled in the fraction along with the usual procedure of the counter terms.
Determining the self-energy $\Sigma(p)$ the renormalized propagator of nucleon
$G(p)$ is expressed

$\displaystyle G(p)=\frac{1}{\gamma\cdot
p-M-\Sigma(p)}=\frac{\alpha(p^{2})\gamma\cdot p+\beta(p^{2})M}{p^{2}-M^{2}}$
(15) $\displaystyle\Sigma(p)=Mc_{1}(p^{2})-\gamma\cdot p\,c_{2}(p^{2})$ (16)

in two kinds of the expressions. These four quantities $c_{i}(p^{2})$
($i$=1,2), $\alpha(p^{2})$ and $\beta(p^{2})$ are not independent and
connected by Eq. (15) in conjunction with the renormalization conditions. It
is given by the relations between the coefficients of the expansion in the
series of $p^{2}-M^{2}$. For example in the case of $c_{i}^{(n)}$ such that
$c_{i}(p^{2})=\sum_{n=0}^{\infty}n!^{-1}c_{i}^{(n)}(p^{2}-M^{2})^{n}$ the
renormalization condition imposes the relations $c\equiv
c_{1}^{(0)}=c_{2}^{(0)}$ and $c_{1}^{(1)}-c_{2}^{(1)}=c_{2}^{(0)}/2M^{2}$ and
$\alpha(M^{2})=\beta(M^{2})=1$ is satisfied to construct the rationalized form
of $G(p)$.
It is useful to express the coefficients of the expansion $\alpha^{(n)}$ and
$\beta^{(n)}$ in terms of $c_{i}^{(n)}$ at the region around the on-shell
point $p^{2}=M^{2}$. The coefficients of the rationalized form of $G(p)$ has
been used to carry out the perturbative calculation along with the method of
the inversion of the matrix for processes related to the external pions. The
relation is presented up to $n=2$ order as

$\displaystyle\alpha^{(1)}=\frac{c^{2}}{4M^{2}(1+c)}-c_{2}^{(1)}+M^{2}(c_{1}^{(2)}-c_{2}^{(2)})\qquad\quad$
(17)
$\displaystyle\beta^{(1)}=\alpha^{(1)}+\frac{c}{2M^{2}(1+c)}\qquad\qquad\qquad\quad\qquad\quad$
(18)
$\displaystyle\alpha^{(2)}=2\alpha^{(1)\,2}-\frac{c^{2}\,c_{2}^{(1)}}{2M^{2}(1+c)^{2}}+\frac{c\,(c_{1}^{(2)}-c_{2}^{(2)})}{1+c}-c_{2}^{(2)}\quad\qquad$
$\displaystyle+\frac{2M^{2}}{3}(c_{1}^{(3)}-c_{2}^{(3)})$ (19)
$\displaystyle\quad\beta^{(2)}=\alpha^{(2)}+M^{-2}\alpha^{(1)}+\frac{1}{M^{4}(1+c)^{2}}(-\frac{c^{2}}{4}+M^{2}c_{2}^{(1)})$
(20)

In our previous study $\alpha^{(1)}$ and $\beta^{(1)}$ has been applied to
calculate the cross section of the photoproduction of $\pi^{+}$ from the
threshold to the resonance energy region. The approximate value of the self-
energy is determined so as to reproduce the low-energy pion-nucleon elastic
scattering. Meanwhile the $n=2$ model is appropriate since the magnetic moment
of nucleon is calculated by using the values ($c=2$, $\,c_{1}^{(1)}=M^{-2}$,
$c_{2}^{(1)}=c_{i}^{(n\geq 2)}=0$). It yields
$\beta^{(1)}=2\alpha^{(1)}=2/3M^{2}$ and $\beta^{(2)}=2\alpha^{(2)}=4/9M^{4}$.
The expansion of $\alpha(p^{2})$ and $\beta(p^{2})$ at the on-shell point does
not have difficulty taking into account the complex poles. The prescription
follows the way of the perturbative calculation in which the poles out of the
real axis are neglected [3].
Different from above two corrections the nucleon propagator does not supply
the factor multiplied by the free propagator. When the approximation
$\alpha(p^{2})\approx\beta(p^{2})$ is suitable also for the off-shell regions
it is possible to create the multiplicative factor $f_{2}(p^{2})$ such as
$\sqrt{\alpha(p^{2})},\,\sqrt{\beta(p^{2})}$ and
$\sqrt{(\alpha(p^{2})+\beta(p^{2}))/2}$ according to the definition of it.
While three possible factors are separated from each other except the on-shell
point they all increase monotonously as $p^{2}$ goes to infinity. It indicates
that the factor plays a role to restore too strong effect of the vacuum
polarization in the region of the nucleon momentum $p^{2}<M^{2}$.
The value of the pion momentum $q^{2}=(p^{\prime}-p)^{2}\sim M^{2}$ has been
introduced tentatively. The choice is not only an assumption and expected by
the consideration of the $k$-integral to obtain the self-energy of the nucleon
in the lowest-order calculation. The denominator of the integrand has the zero
at the specific values $\bar{k}^{\prime}=\pm\,pz$ of the variable $k^{\prime}$
transformed from the momentum $k$ of the internal nucleon by $k^{\prime}\equiv
k+pz$ where $p$ is the momentum of the on-shell nucleon. The parameter $z$ of
the Feynman formula of the integral is replaced with the average
$\bar{z}=1/2$. According to the sign of $\bar{k}^{\prime}$ the $\bar{k}^{2}$
has the values 0 or $M^{2}$ and the former is chosen as $\bar{k}^{2}$ for
nucleon.
The same procedure is applied to $k^{2}$ of the pion propagator for the
internal line by changing as $M\leftrightarrow m$ and $p\rightarrow-p$ and it
results in $M^{2}$ or 0 according to $\bar{k}^{\prime}=\pm\,pz$. To derive
these relations the term of the pion mass $m^{2}$ is dropped approximately.
The numerical values of these factors are $f_{3}(M^{2})=0.74$ and
$f_{2}(0)=0.88$ for $\sqrt{\alpha(0)}$. The use of the upper sign for
$\bar{k}^{\prime}$ ($\bar{k}^{\prime}=+\,pz$) is consistent with the integral
of the nucleon current part in the calculation of the anomalous magnetic
moment as described below.
It is important to search for the factor $f_{3}$ in another way to understand
the suppression of the effective coupling constant. Regarding to the nucleon
current part the shift of the variable $k\rightarrow k^{\prime}=k-\alpha$ is
same as that of the integral in Eq. (2). The variable $k^{\prime}$ may be
represented by the parameter $\alpha\equiv px+p^{\prime}y$ with the average
values $x=y=1/3$ and $\bar{k}^{\prime}=+\alpha$ is chosen since the integrand
is transformed to $\sim({k^{\prime}}^{2}-\alpha^{2})^{-1}$. Using it
$\bar{k}^{2}\sim 4\alpha^{2}$ is deduced. Substituting $q^{2}=k^{2}$ the
average value of $\alpha^{2}$ is $\alpha^{2}\sim 4M^{2}/9-{\bar{k}}^{2}/9$.
The joint use of these relations yields the expected value $\bar{k}^{2}$ of
the pion momentum $k$ as $\bar{k}^{2}\sim 1.23M^{2}$ and the factor results in
$f_{3}(\bar{k}^{2})=0.69$.
The numerical value of the factor is expected to be close to that extracted
from the experimental value. The coefficients of the anomalous magnetic
moments of proton ($\kappa_{p}$) and neutron ($\kappa_{n}$) are connected with
the self-energy stemmed from the non-perturbative term and the perturbative
terms of the vertex corrections by virtue of the Ward-Takahashi identity for
the case of the photon-nucleon-nucleon vertex as

$\displaystyle\kappa_{p}\tau_{p}+\kappa_{n}\tau_{n}=z\tau_{p}+a\tau_{3}+b(1+\tau_{n})$
(21)

in terms of the third component of the isospin matrix $\tau_{3}$ and
$\tau_{p/n}\equiv(1\pm\tau_{3})/2$. Then the vertex corrections of the part of
the pion current ($a$) and the nucleon current ($b$) are given as

$\displaystyle a=(-2z+2\kappa_{p}-\kappa_{n})/3$ (22) $\displaystyle
b=(-z+\kappa_{p}+\kappa_{n})/3$ (23)

Using the parameter $z=2(1+\lambda)^{-1}$ by the $n=2$ model (the lowest-order
in $\gamma\cdot p-M$) for the self-energy and the experimental values of the
anomalous magnetic moment the $a$ and $b$ are required to be $a=0.528$ and
$b=-0.692$. On the other hand the perturbative calculations of the vertex
corrections become $a^{\prime}=0.831$ and $b^{\prime}=-1.086$ and the expected
values of the factor $f^{\prime}$ ($f\rightarrow ff^{\prime}$) result in 0.797
and 0.798 to reproduce $a$ and $b$ respectively.

## 4 Summary and remarks

Three corrections have been investigated to explain the role of the
suppression of the coupling constant for the anomalous part of the magnetic
moment. The perturbative lowest-order calculation of the vertex correction is
convergent after the renormalization of the coupling constant however the
amount of the vertex function is not enough to observe the considerable shift
of the factor. The exact treatment is necessary to obtain the true value since
it is not realistic to think that the result is unchanged when the on-shell
condition is not acceptable for the vertex connected to the internal fermion
line. As for the pion propagator the polarization function is found to possess
an interesting property at the large momentum transfer regions. Because of the
radius of the convergence the expansion is not available to examine the
contribution of the lowest-order alone. The factor is necessarily defined by
the series in the inverse of the coupling constant when the degree of the off-
shell is large.

## References

  * [1] S. Kinpara, arXiv:nucl-th/1809.00475.
  * [2] B. D. Serot and J. D. Walecka, Advances in Nuclear Physics, Vol. 16, Plenum Press, New York (1986).
  * [3] G. Feldman, Proc. Roy. Soc. A223, 112 (1954).