# Pion couplings with low-lying nucleon resonances

Janardan Prasad Singh111Retired from the services of The Maharaja Sayajirao
University of Baroda
Physics Department, Faculty of Science, The Maharaja Sayajirao University of
Baroda, Vadodara-390002, Gujarat, India

###### Abstract

We have calculated coupling constants of a neutral pion with the lowest two
nucleon resonances. This includes both the diagonal as well as non-diagonal
coupling constants involving a nucleon resonance and a nucleon. For this, we
first calculate vacuum-to-pion correlation function of the interpolating
fields of two nucleons and then take its matrix elements with respect to a
nucleon spinor and/or a nucleon resonance spinor(s). Using different QCD sum
rules obtained from different matrix elements we eliminate unwanted coupling
constants and solve for the desired ones. This is a simple extension of the
projected correlation function approach used in the literature where we use
multiple of states involving those of a nucleon and its resonances. We have
also checked the stability of our results with respect to variation of
different QCD and phenomenological input parameters.

Keywords : QCD sum rules, Nucleon resonances, non-local operators, Projected
correlation function

##  1 Introduction

Strong interaction is responsible for generating baryonic resonances as well
as their couplings with mesons. While the masses and the widths of the
baryonic resonances are known fairly well, their coupling constants with
mesons are not known so well. Determination of these couplings will be useful
for construction of NN potential and in other phenomenological analyses such
as photoproduction of mesons off a nucleon target. A broad experimental effort
has been underway for the past two decades, with measurements of exclusive
meson photoproduction and electroproduction reactions, including many
polarization observables [1]. Theoretical estimates of these coupling
constants have been made using phenomenological Lagrangians [2, 3, 4] or using
QCD based methods. While the phenomenological Lagrangians are good for a broad
understanding of hadronic interactions, QCD based approaches give a
microscopic view of hadronic interactions and provide a test of QCD at low
energies. QCD-based methods have been applied, in past, to determine the
masses and widths of nucleon resonances [5, 6, 7] and other light baryon
resonances [8]. In recent past, these methods have also been applied to
calculate coupling constant of a pion with a nucleon resonance [9, 10, 11, 12,
13]. QCD sum rule based on three-point correlation function has been used for
calculating $N^{(*)}\pi N^{(*)}$ coupling constant in [12]. It is known that
an approach based on three-point function gives results that may contain non-
negligible contribution from higher resonances $\pi$(1300) and $\pi$(1800)
[13]. $N^{*}N^{*}\pi$ coupling constant has also been derived in [13] within
the light-cone QCD sum rule method by considering vacuum-to-pion correlation
function of the interpolating fields of two nucleons. To eliminate the
undesired coupling constants involving nucleon and the Roper resonance
N(1440), which invariably appears on the phenomenological side, the authors
use all the independent Lorentz structures as well as their derivatives
including a second derivative for one of them. It is known that the sum rules
obtained from derivatives of the correlation functions give bad sum rules[14].
Also, though the tensor Dirac structure has nice features for calculating
$\pi$NN coupling constant, other Dirac structures have been found not to be
reliable for the same as they contain large contributions from the continuum
[15, 16].

In this work we use a projected correlation function approach as outlined in
Ref. [15]. From the vacuum-to pion correlation function of the interpolating
fields of two nucleons and by taking its matrix elements with respect to
nucleon and/ or nucleon resonance spinor(s), we get different algebraic
equations involving desired coupling constants of a pion with a nucleon and
nucleon resonances. From these equations undesired coupling constants can be
eliminated algebraically and desired ones can be extracted.

In Sec. 2, we first derive a correlation function of two interpolating fields
of a nucleon between vacuum and one-pion state; next we have derived projected
correlation functions from which we get the desired sum rules. In Sec. 3, we
first fix on a form of interpolating field for nucleon and its resonances
which is suitable for our purpose. The sum rules constructed from this
interpolating field have been solved numerically for different parameter
values to get the coupling constants of pions with the two low-lying nucleon
resonances. In Sec. 4, we discuss our approach and compare our results with
those available in the literature. Some of the intermediate expressions are
given in the Appendix.

##  2 The Sum Rules

We start with the following correlation function :

$\Pi(q,p)=i\int\mathrm{d^{4}}x\,e^{iqx}\langle
0|T\\{J_{p}(x)\bar{J_{p}}(0)\\}|\pi^{0}(p)\rangle$ (1)

where $J_{p}(x)$ is the most general form of the interpolating field of a
proton [14, 16] and its resonances [17, 12, 13] made up of three quark fields
without any derivative :

$J_{p}=2\epsilon^{abc}[(u^{aT}Cd^{b})\gamma_{5}u^{c}+t(u^{aT}C\gamma_{5}d^{b})u^{c}]$
(2)

where a, b, c are color indices and t is an arbitrary real parameter.
Generally Ioffe current $(t=-1)$[14] has been used for cases where only proton
is involved. But different values of the parameter t have also been used [16],
particularly in cases where nucleon resonances are involved [17, 6, 12, 13].
Stability of the results with respect to variation in t should be taken as a
required condition for checking the suitability of the interpolating field. We
parameterize the coupling of $J_{p}$ with the nucleon and its resonance states
as

$\displaystyle\langle 0|J_{p}(0)|N(k,s)\rangle=\lambda\,u_{N}(k,s)$ (3)
$\displaystyle\langle
0|J_{p}(0)|N_{+}(k,s)\rangle=\lambda_{1}\,u_{N_{+}}(k,s)$ (4)
$\displaystyle\langle
0|J_{p}(0)|N_{-}(k,s)\rangle=\lambda_{2}\,i\gamma_{5}\,u_{N_{-}}(k,s)$ (5)

where N, $N_{+}$ and $N_{-}$ stand for the nucleon, N(1440) and N(1535) states
and + and - denote the parity of the state. The correlation function
$\Pi$(q,p) can be evaluated phenomenologically by saturating it with the
nucleon- and its resonance-states. We can also evaluate it using operator
product expansion (OPE) technique exploiting the construction of $J_{p}$ as
given in Eq. (2). For the phenomenological evaluation we use the following
Lagrangian [9, 11] :

$\displaystyle\mathcal{L}=g_{1}\bar{N}i\gamma_{5}\overrightarrow{\tau}.\overrightarrow{\pi}N+g_{2}\bar{N_{+}}i\gamma_{5}\overrightarrow{\tau}.\overrightarrow{\pi}N_{+}+g_{3}\bar{N_{-}}i\gamma_{5}\overrightarrow{\tau}.\overrightarrow{\pi}N_{-}+g_{4}(\bar{N}i\gamma_{5}\overrightarrow{\tau}.\overrightarrow{\pi}N_{+}$
(6)
$\displaystyle+\bar{N_{+}}i\gamma_{5}\overrightarrow{\tau}.\overrightarrow{\pi}N)+g_{5}(\bar{N}\overrightarrow{\tau}.\overrightarrow{\pi}N_{-}+\bar{N_{-}}\overrightarrow{\tau}.\overrightarrow{\pi}N)+g_{6}(\bar{N_{+}}\overrightarrow{\tau}.\overrightarrow{\pi}N_{-}+\bar{N_{-}}\overrightarrow{\tau}.\overrightarrow{\pi}N_{+})$

The phenomenological expression for the correlation function $\Pi$(q,p) is
given in the Appendix. The Wilson coefficients of the OPE are calculated in
two steps : the light-cone expansion of the correlation function is performed
first. The short-distance expansion of the vacuum-to-pion matrix elements of
light-cone operators is performed next. In the Appendix, we have given the
result of parametrization of the vacuum-to pion matrix elements of the light-
cone operators.

We define the projected correlation function as [15]

$\Pi_{+}^{ij}(q,p)=\bar{u}_{i}(q,r)\gamma_{0}\Pi(q,p)\gamma_{0}u_{j}(q-p,s)$
(7)

where each of (i,j) stand for N, $N_{+}$ or $N_{-}$. Projected correlation
functions have been successfully used for calculating not only diagonal
coupling constants, such as $\pi$NN [15] and $\eta$NN, $\eta^{\prime}$NN [18]
coupling constants, but also non-diagonal ones, such as $\pi\Lambda\Sigma$
coupling constant [16]. $\Pi_{+}^{ij}$(q,p) can be regarded as a function of
$q_{0}$ in the reference frame in which q=0 . The odd and even parts of
$\Pi_{+}^{ij}(q_{0})$ satisfy dispersion relation

$\displaystyle\Pi_{+o}^{ij}(q_{0}^{2})=-\frac{1}{\pi}\int
dq^{\prime}_{0}\frac{1}{q_{0}^{2}-q^{\prime
2}_{0}}\textit{Im}\Pi_{+}^{ij}(q^{\prime}_{0})$ (8)
$\displaystyle\Pi_{+e}^{ij}(q_{0}^{2})=-\frac{1}{\pi}\int
dq^{\prime}_{0}\frac{q^{\prime}_{0}}{q_{0}^{2}-q^{\prime
2}_{0}}\textit{Im}\Pi_{+}^{ij}(q^{\prime}_{0})$ (9)

On taking Borel transform [14, 15, 16, 18] with respect to $q_{0}^{2}$ they
take the form

$\displaystyle B[\Pi_{+o}^{ij}(q_{0}^{2})]=\frac{1}{\pi}\int
dq^{\prime}_{0}\,\ e^{-q^{\prime 2}_{0}/M^{2}}\textit{
Im}\Pi_{+}^{ij}(q^{\prime}_{0})$ (10) $\displaystyle
B[\Pi_{+e}^{ij}(q_{0}^{2})]=\frac{1}{\pi}\int dq^{\prime}_{0}q^{\prime}_{0}\,\
e^{-q^{\prime 2}_{0}/M^{2}}\textit{Im}\Pi_{+}^{ij}(q^{\prime}_{0})$ (11)

where M is the Borel mass parameter.

For calculating the correlation function using OPE, we have used vacuum-to-
pion matrix elements of non-local operators as given by Eqs. (A2-A5).
Following is the result of our OPE evaluation of the correlation function
$\Pi(q,p)$:

$\displaystyle\Pi(q,p)=-\frac{t_{3}}{16\pi^{2}}i\gamma_{5}\left[\frac{\langle\bar{q}q\rangle}{f_{\pi}}\Big{\\{}(q^{2}-p.q)ln(-q^{2})+\frac{2}{3}\frac{(p.q)^{2}}{q^{2}}\Big{\\}}-18.91\sqrt{2}f_{3\pi}\frac{(p.q)^{2}}{q^{2}}\right]$
(12)
$\displaystyle-\frac{t_{6}}{288}i\gamma_{5}\frac{\langle\bar{q}q\rangle}{f_{\pi}}\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle\frac{1}{q^{2}}\left(1+\frac{q.p}{q^{2}}+\frac{4}{3}\frac{(q.p)^{2}}{q^{4}}\right)-i\gamma_{5}f_{3\pi}\frac{(q.p)^{2}}{q^{2}}\times$
$\displaystyle\left(\frac{t_{11}}{8\sqrt{2}\pi^{2}}-t_{6}\frac{18.91}{72\sqrt{2}}\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle\frac{1}{q^{4}}\right)$
$\displaystyle+i\gamma_{5}\not{q}\bigg{[}\frac{t_{1}}{72\pi^{2}}f_{\pi}\delta^{2}\frac{q.p}{q^{2}}(1+\frac{q.p}{q^{2}})+\frac{t_{2}}{24\pi^{2}}f_{\pi}\Big{\\{}-q.pln(-q^{2})+\frac{(q.p)^{2}}{q^{2}}$
$\displaystyle-\delta^{2}\frac{q.p}{q^{2}}(1+\frac{q.p}{q^{2}})+2a^{\prime}\frac{(q.p)^{3}}{q^{4}}\Big{\\}}-\frac{t_{4}}{9}\frac{(\langle\bar{q}q\rangle)^{2}}{f_{\pi}}\frac{q.p}{q^{4}}(1+\frac{2q.p}{q^{2}})$
$\displaystyle-\frac{f_{\pi}}{288}\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle\Big{\\{}t_{2}\left[\frac{2q.p}{q^{4}}(1+\frac{2q.p}{q^{2}})+\frac{4}{3}\delta^{2}\frac{q.p}{q^{6}}+24a^{\prime}\frac{(q.p)^{3}}{q^{8}}\right]$
$\displaystyle-\frac{4}{9}t_{5}\delta^{2}\frac{q.p}{q^{6}}\Big{\\}}-\frac{t_{9}}{108}\langle\bar{q}g\sigma.Gq\rangle\frac{\langle\bar{q}q\rangle}{f_{\pi}}\frac{q.p}{q^{6}}\bigg{]}$
$\displaystyle+i\gamma_{5}\not{p}\bigg{[}\frac{t_{1}}{16\pi^{2}}f_{\pi}\Big{\\{}(q^{2}-q.p)ln(-q^{2})+\delta^{2}(-ln(-q^{2})+\frac{8}{9}\frac{q.p}{q^{2}})+2a^{\prime}\frac{(q.p)^{2}}{q^{2}}\Big{\\}}$
$\displaystyle+\frac{t_{2}}{48\pi^{2}}f_{\pi}\Big{\\{}(2q.p-q^{2})ln(-q^{2})+\delta^{2}ln(-q^{2})-6a^{\prime}\frac{(q.p)^{2}}{q^{2}}\Big{\\}}$
$\displaystyle-\frac{t_{4}}{18}\frac{(\langle\bar{q}q\rangle)^{2}}{f_{\pi}}\frac{1}{q^{2}}+\frac{1}{288}\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle\Big{\\{}t_{5}\frac{f_{\pi}}{q^{2}}\left(1+\frac{q.p}{q^{2}}-\frac{\delta^{2}}{9q^{2}}+4a^{\prime}\frac{(q.p)^{2}}{q^{4}}\right)$
$\displaystyle+t_{2}\frac{f_{\pi}}{q^{4}}\left(q^{2}+2q.p+\frac{\delta^{2}}{3}+12a^{\prime}\frac{(q.p)^{2}}{q^{2}}\right)\Big{\\}}+\frac{t_{9}}{432}\langle\bar{q}g\sigma.Gq\rangle\frac{\langle\bar{q}q\rangle}{f_{\pi}}\frac{1}{q^{4}}\bigg{]}$
$\displaystyle+\gamma_{5}\sigma^{\mu\nu}q_{\mu}p_{\nu}\bigg{[}\frac{-t_{3}}{48\pi^{2}}\frac{\langle\bar{q}q\rangle}{f_{\pi}}\Big{\\{}ln(-q^{2})-\frac{q.p}{q^{2}}-2c^{\prime}\frac{(p.q)^{2}}{q^{4}}\Big{\\}}-\frac{t_{4}}{3}\langle\bar{q}q\rangle
f_{\pi}\frac{1}{q^{2}}\times$
$\displaystyle\Big{\\{}1+\frac{q.p}{q^{2}}+\frac{5}{9}\frac{\delta^{2}}{q^{2}}(1+\frac{2q.p}{q^{2}})+4a^{\prime}\frac{(q.p)^{2}}{q^{4}}\Big{\\}}+\frac{t_{7}}{864}\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle\frac{\langle\bar{q}q\rangle}{f_{\pi}}\times$
$\displaystyle\frac{1}{q^{4}}(1+\frac{2q.p}{q^{2}})+\frac{t_{10}}{18}\langle\bar{q}q\rangle
f_{\pi}\delta^{2}\frac{1}{q^{4}}(1+\frac{2q.p}{q^{2}})-\frac{t_{8}}{288}\langle\bar{q}g\sigma.Gq\rangle\times$
$\displaystyle\frac{f_{\pi}}{q^{4}}(1+\frac{2q.p}{q^{2}})\bigg{]}$

where $t_{1}=7+6t+7t^{2}$, $t_{2}=(1+t)^{2}$, $t_{3}=5+2t-7t^{2}$,
$t_{4}=1+2t-3t^{2}$, $t_{5}=19+14t+19t^{2}$, $t_{6}=21-6t-15t^{2}$,
$t_{7}=17+2t-19t^{2}$, $t_{8}=20+24t-44t^{2}$, $t_{9}=5+6t-11t^{2}$,
$t_{10}=3-2t-t^{2}$, $t_{11}=(1-t)^{2}$. The constants $a^{\prime}$ and
$c^{\prime}$ are defined as :
$a^{\prime}=3/10+3a_{2}/35,c^{\prime}=3/10+3C_{2}/35$. In the above equation,
terms such as $\textit{p.q}/q^{2}$ will be important for $g_{4}$ and $g_{5}$
coupling constants. Most of the terms in OPE as given in Eq. (12) reduce to
the corresponding terms given in Ref.[15] for $t=-1$. However, there are
significant differences in form of sign differences of some of the terms, and
also on account of higher powers of p.x considered. The sign of $\delta^{2}$
used in this work is in conformity with that used in Ref. [16] and is opposite
to that used in Refs. [15] and [19] resulting in a sign difference. We can
calculate the projected correlation function from the OPE expression as given
in Eq. (12) and separate its odd and even parts. It is usual to model the
continuum part of $\Pi(q,p)_{ph}$ by $\Pi(q,p)_{OPE}$ with
$q_{0}^{2}>s_{\pi}$, a threshold parameter and only terms with postive powers
of $q_{0}$ are taken into account in doing so [5, 8, 14, 18, 15, 16]. The
parameters $\delta^{2}$ and $C_{2}$ are defined as [20, 21, 22]

$\langle
0|\bar{q}g\tilde{G}_{\mu\nu}\gamma^{\nu}q|\pi^{0}(p)\rangle=ip_{\mu}f_{\pi}\delta^{2}$
$C_{2}=-\frac{f_{\pi}f_{3\pi}}{\langle\bar{q}q\rangle}\frac{6.44}{\sqrt{2}}$

In Eq. (12), $a_{2}$ and $C_{2}$ are the coefficients of the Gegenbauer
polynomials in the leading twist-2 pion wave function and two-particle pion
wave function of twist-3 respectively[21]. $f_{3\pi}$ appears in two-particle
wave function of twist-3 as well as in three-particle pion wave function [21].
It has been estimated that $a_{2}\simeq 0.4$ and $f_{3\pi}\simeq
0.0045GeV^{2}$. The coefficient of $\bar{u}_{N}i\gamma_{5}u_{N}$ in the
phenomenological as well as in the OPE expressions of the projected
correlation function have been Borel transformed and are given below :

$\displaystyle
B[\Pi_{+o}^{NN}]_{ph}=\lambda^{2}g_{1}(m)e^{-\frac{m^{2}}{M^{2}}}(\frac{2m}{M^{2}}-\frac{g^{\prime}_{1}}{g_{1}}+\frac{1}{2E_{k1}}+\frac{1}{2m})+\lambda_{1}^{2}g_{2}(m_{1})e^{-\frac{m_{1}^{2}}{M^{2}}}\times$
(13)
$\displaystyle\frac{(m+m_{1})(3m-m_{1}-2\omega_{p1})}{4m_{1}E_{k2}}(\frac{2m_{1}}{M^{2}}-\frac{g^{\prime}_{2}}{g_{2}}+\frac{1}{2E_{k2}}+\frac{1}{2m_{1}})$
$\displaystyle+\lambda_{2}^{2}g_{3}(m_{2})e^{-\frac{m_{2}^{2}}{M^{2}}}\frac{(m_{2}-m)(3m+m_{2}-2\omega_{p1})}{4m_{2}E_{k3}}(\frac{2m_{2}}{M^{2}}-\frac{g^{\prime}_{3}}{g_{3}}+\frac{1}{2E_{k3}}+\frac{1}{2m_{2}})$
$\displaystyle+\lambda\lambda_{1}g_{4}(m_{1})e^{-\frac{m_{1}^{2}}{M^{2}}}\frac{(m+m_{1})(m-\omega_{p1})}{2m_{1}E_{k4}}(\frac{2m_{1}}{M^{2}}-\frac{g^{\prime}_{4}}{g_{4}}+\frac{1}{2E_{k4}}+\frac{1}{2m_{1}})$
$\displaystyle+\lambda\lambda_{1}g_{4}(m)e^{-\frac{m^{2}}{M^{2}}}\frac{(3m-m_{1}-2\omega_{p1})}{2E_{k^{\prime}4}}(\frac{2m}{M^{2}}-\frac{g^{\prime}_{4}}{g_{4}}+\frac{1}{2E_{k^{\prime}4}}+\frac{1}{2m})$
$\displaystyle+\lambda\lambda_{2}g_{5}(m_{2})e^{-\frac{m_{2}^{2}}{M^{2}}}\frac{(m_{2}-m)(m-\omega_{p1})}{2m_{2}E_{k5}}(\frac{2m_{2}}{M^{2}}-\frac{g^{\prime}_{5}}{g_{5}}+\frac{1}{2E_{k5}}+\frac{1}{2m_{2}})$
$\displaystyle-\lambda\lambda_{2}g_{5}(m)e^{-\frac{m^{2}}{M^{2}}}\frac{(3m+m_{2}-\omega_{p1})}{2E_{k^{\prime}5}}(\frac{2m}{M^{2}}-\frac{g^{\prime}_{5}}{g_{5}}+\frac{1}{2E_{k^{\prime}5}}+\frac{1}{2m})$
$\displaystyle+\lambda_{1}\lambda_{2}g_{6}(m_{2})e^{-\frac{m_{2}^{2}}{M^{2}}}\frac{(m_{2}-m)(3m-m_{1}-2\omega_{p1})}{4m_{2}E_{k6}}(\frac{2m_{2}}{M^{2}}-\frac{g^{\prime}_{6}}{g_{6}}+\frac{1}{2E_{k6}}+\frac{1}{2m_{2}})$
$\displaystyle-\lambda_{1}\lambda_{2}g_{6}(m_{1})e^{-\frac{m_{1}^{2}}{M^{2}}}\frac{(m_{1}+m)(3m+m_{2}-2\omega_{p1})}{4m_{1}E_{k^{\prime}6}}(\frac{2m_{1}}{M^{2}}-\frac{g^{\prime}_{6}}{g_{6}}+\frac{1}{2E_{k^{\prime}6}}+\frac{1}{2m_{1}})$

$\displaystyle
B[\Pi_{+o}^{NN}]_{OPE}=M^{2}\Big{[}-t_{1}\frac{f_{\pi}}{8\pi^{2}}L^{-4/9}E_{k1}\omega_{p1}+t_{2}\frac{f_{\pi}}{12\pi^{2}}L^{-4/9}E_{k1}\omega_{p1}$
(14)
$\displaystyle+\frac{t_{3}}{48\pi^{2}}\frac{\langle\bar{q}q\rangle}{f_{\pi}}(m+E_{k1}-3\omega_{p1})\Big{]}E_{0}\left(\frac{s_{\pi}}{M^{2}}\right)$
$\displaystyle+\frac{t_{1}}{9\pi^{2}}f_{\pi}\delta^{2}E_{k1}\omega_{p1}+\frac{t_{4}}{3}\langle\bar{q}q\rangle
f_{\pi}(m+E_{k1})$
$\displaystyle\frac{1}{M^{2}}\Big{[}(-t_{4}\frac{5}{27}+\frac{t_{10}}{18})\langle\bar{q}q\rangle
f_{\pi}\delta^{2}(m+E_{k1})L^{32/75}-\frac{1}{288}\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle$
$\displaystyle\times\Big{\\{}2t_{5}f_{\pi}L^{-4/9}E_{k1}\omega_{p1}+4t_{2}f_{\pi}L^{-4/9}E_{k1}\omega_{p1}+t_{6}\frac{\langle\bar{q}q\rangle}{f_{\pi}}\omega_{p1}$
$\displaystyle-\frac{t_{7}}{3}\frac{\langle\bar{q}q\rangle}{f_{\pi}}(m+E_{k1})\Big{\\}}-\frac{t_{9}}{72}f_{\pi}\langle\bar{q}g\sigma.Gq\rangle
L^{-14/27}(m+E_{k1})\Big{]}$

Figure 1: Feynman diagrams for the phenomenological form of the correlation
function

In above equations m, $m_{1}$ and $m_{2}$ are masses of N, $N_{+}$ and $N_{-}$
states. The continuum contribution of B[$\Pi_{+o}^{NN}]_{ph}$, modelled as
explained above, has been transferred and combined with
B[$\Pi_{+o}^{NN}]_{OPE}$ in form of $E_{0}$, where $E_{0}(x)=1-e^{-x}$.
$g^{\prime}_{i}$ denotes derivative of the respective coupling constant.
Though $\omega_{p1}\simeq$0.01 GeV is small, it has been retained for the sake
of demonstration. We have done a similar exercise with
$\Pi_{+o}^{N_{+}N_{+}}$, $\Pi_{+o}^{N_{-}N_{-}}$, $\Pi_{+o}^{N_{+}N}$ and
$\Pi_{+o}^{N_{-}N}$ also getting five equations in all. L=
$\alpha_{s}(\mu^{2})/\alpha_{s}(M^{2})$ and it appears in the OPE expressions
of the correlation function for renormalization group improvement; the
anomalous dimensions of various operators have been taken from Refs. [14, 22].
The emitted pion has significant energies, $\omega_{p4}\simeq$0.42 GeV for
$N_{+}\to N$ and $\omega_{p5}\simeq$0.49 GeV for $N_{-}\to N$. A
characteristic feature of $(\Pi^{N_{-}j}_{+o})_{ph}$ is that due to appearance
of $\gamma_{5}$ in the matrix element $\langle 0|J_{p}(0)|N_{-}(k,s)\rangle$
the pole at $q^{2}=m^{2}_{2}$ gets removed. We are giving below the expression
for B[$\Pi_{+o}^{N_{-}N}]_{OPE}$ with higher powers of $\omega_{p5}$ to show
its characteristic difference.

$\displaystyle
B[\Pi_{+o}^{N_{-}N}]_{OPE}=M^{2}\Big{[}\left(\frac{t_{2}}{24\pi^{2}}-\frac{t_{1}}{16\pi^{2}}\right)f_{\pi}L^{-4/9}(E_{k5}+m-\omega_{p5})\omega_{p5}+\frac{t_{3}}{48\pi^{2}}\frac{\langle\bar{q}q\rangle}{f_{\pi}}$
(15)
$\displaystyle\times(E_{k5}+m-3\omega_{p5})\Big{]}E_{0}\left(\frac{s_{\pi}}{M^{2}}\right)$
$\displaystyle+\frac{t_{1}}{72\pi^{2}}f_{\pi}\delta^{2}[4(E_{k5}+m)\omega_{p5}-5\omega_{p5}^{2}]+\frac{t_{2}}{24\pi^{2}}f_{\pi}\delta^{2}\omega_{p5}^{2}-\frac{t_{3}}{24\pi^{2}}\frac{\langle\bar{q}q\rangle}{f_{\pi}}c^{\prime}$
$\displaystyle\times(E_{k5}+m)\omega_{p5}^{2}+\frac{t_{4}}{3}f_{\pi}\langle\bar{q}q\rangle(E_{k5}+m)$
$\displaystyle+\frac{1}{M^{2}}\Big{[}\frac{t_{4}}{3}\langle\bar{q}q\rangle\Big{\\{}-f_{\pi}(E_{k5}+m)(\frac{5}{9}\delta^{2}L^{32/75}+4a^{\prime}\omega_{p5}^{2})-\frac{2}{3}\frac{\langle\bar{q}q\rangle}{f_{\pi}}L^{4/9}\omega_{p5}^{2}\Big{\\}}$
$\displaystyle+\frac{t_{10}}{18}\langle\bar{q}q\rangle
f_{\pi}\delta^{2}L^{32/75}(E_{k5}+m)-\frac{1}{288}\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle\Big{\\{}t_{5}f_{\pi}L^{-4/9}$
$\displaystyle\times(E_{k5}+m-\omega_{p5})\omega_{p5}+2t_{2}f_{\pi}L^{-4/9}(E_{k5}+m+\omega_{p5})\omega_{p5}$
$\displaystyle+t_{6}\frac{\langle\bar{q}q\rangle}{f_{\pi}}\omega_{p5}-\frac{t_{7}}{3}\frac{\langle\bar{q}q\rangle}{f_{\pi}}(E_{k5}+m)\Big{\\}}-\frac{t_{9}}{72}f_{\pi}\langle\bar{q}g\sigma.Gq\rangle
L^{-14/27}(E_{k5}+m)\Big{]}$

Figure 2: Plots of $B[\Pi_{+o}^{NN}]$, $B[\Pi_{+o}^{N_{+}N_{+}}]$,
$B[\Pi_{+o}^{N_{-}N_{-}}]$, $B[\Pi_{+o}^{N_{+}N}]$ and $B[\Pi_{+o}^{N_{-}N}]$,
obtained from OPE, as a function of t for a typical case with $M^{2}$= 3.0
$GeV^{2}$, $s_{\pi}$=2.7 $GeV^{2}$, $\langle\bar{q}q\rangle=-0.0117$ $GeV^{3}$
and other QCD parameters as given in Table 2 have been shown in Fig. (a).
Plots of the same functions as functions of $cos\theta$ ($t=tan\theta$) have
been shown in Fig. (b).

We have shown plots of $B[\Pi_{+o}^{NN}]$, $B[\Pi_{+o}^{N_{+}N_{+}}]$,
$B[\Pi_{+o}^{N_{-}N_{-}}]$, $B[\Pi_{+o}^{N_{+}N}]$ and $B[\Pi_{+o}^{N_{-}N}]$,
obtained using OPE, as a function of $t$ in Fig. 2a and as a function of
$cos\theta$ in Fig. 2b, where $t=tan\theta$, for a typical case with $M^{2}$=
3.0 $GeV^{2}$, $s_{\pi}$=2.7 $GeV^{2}$, $\langle\bar{q}q\rangle=-0.0117$
$GeV^{3}$ and other QCD parameters as given in Table 2. Fig. 2a shows behavior
of the function close to the origin whereas Fig. 2b shows the behavior for
large $t$. We observe stability of the diagonal projected correlation
functions $B[\Pi_{+o}^{NN}]$, $B[\Pi_{+o}^{N_{+}N_{+}}]$ and
$B[\Pi_{+o}^{N_{-}N_{-}}]$ at $t\simeq 0.2$, similar to an observation made in
[16]. Qualitatively similar behavior is obtained for $M^{2}$= 2.5 $GeV^{2}$,
3.5 $GeV^{2}$ and $\langle\bar{q}q\rangle$=$-0.0145$ $GeV^{3}$. In the next
section we will consider a combination of all the five projected correlation
functions to extract the desired coupling constants of pions with the nucleon
and its resonances; there we will test the stability of the combination of
projected correlation functions with respect to variation in t in a more
comprehensive way.

##  3 Numerical Estimates

We have nine coupling constants of a neutral pion with N, $N_{+}$ and $N_{-}$
states including the three non-diagonal ones with different arguments. The
$\pi$NN coupling constant $g_{1}$ can be determined by retaining only the
first term in Eq. (13) and a lower continuum threshold $s^{\prime}_{\pi}\simeq
2.0GeV^{2}$ on the OPE side as is done in a standard way [19, 15, 16]. Using
four equations obtained from the four projected correlation functions we can
eliminate four coupling constants, namely, $g_{4}$(m), $g_{5}$(m),
$g_{6}(m_{1})$ and $g_{6}(m_{2})$ getting an equation involving coupling
constants $g_{2}(m_{1})$, $g_{3}(m_{2})$, $g_{4}(m_{1})$ and $g_{5}(m_{2})$
(hereafter referred to simply as $g_{2}$, $g_{3}$, $g_{4}$ and $g_{5}$). As is
clear from Eq. (13), we also get $g^{\prime}_{i}$ as an unknown constant along
with each $g_{i}$ on the phenomenological side of the sum rule. We have
checked the value of $g^{\prime}_{1}$ from the sum rule for $g_{1}$ only for
different values of $\langle\bar{q}q\rangle$ and for different values of
$M^{2}$ in the range where the sum rule is applicable. We found that $-1.0$
$GeV^{-1}<g^{\prime}_{1}/g_{1}<0.55$ $GeV^{-1}$. We assume that
$g^{\prime}_{i}/g_{i}$ (i=2,3,4,5) also lies in approximately the same range.
Calling $g^{\prime}_{i}/g_{i}=\alpha$ for i=2,3 and
$g^{\prime}_{i}/g_{i}=\beta$ for i=4,5, we will vary $\alpha$ and $\beta$
independently between -1.0 Ge$V^{-1}$ and 0.55 Ge$V^{-1}$ and check the
resulting variations in $g_{i}$’s. The equation obtained from combining
different projected correlation functions, as stated above, can be solved for
$g_{2}$, $g_{3}$, $g_{4}$ and $g_{5}$ in such a way that the phenomenological
and the OPE sides of the combination of the projected correlation functions
match over a range of Borel mass parameter which covers the continuum
threshold.

Before we proceed further, we will check the stability of our results with
respect to variation of the parameter t. In Figs. (3a, 3b) we have shown plots
of a combination of projected correlation functions containing coupling
constants $g_{2}$, $g_{3}$, $g_{4}$ and $g_{5}$. While Fig. (3a) displays the
behavior of the function close to the origin and shows

Figure 3: Plots of a combination of projected correlation functions as a
function of $t$ for $s_{\pi}=2.7$ $GeV^{2}$, $\langle\bar{q}q\rangle=-0.0117$
$GeV^{3}$, other QCD parameters as given in Table 2 and $M^{2}$=2.7$GeV^{2}$,
3.0 $GeV^{2}$ and 3.3 $GeV^{2}$ have been shown in Fig. (a). Plots of the same
combination of projected correlation functions as functions of $cos\theta$
($t=tan\theta$) have been shown in Fig. (b).

a maxima at $t\simeq-1.0$, in Fig (3b) the same function has been plotted with
respect to $cos\theta$ ($t=tan\theta$) for displaying its behavior for large
t. We will also determine the coupling constants $\lambda(t),\lambda_{1}(t)$
and $\lambda_{2}(t)$ as given in Eqs (3-5) from a simultaneous fit of mass sum
rule for N, $N_{+}$ and $N_{-}$ ignoring the mass widths of $N_{+}$ and
$N_{-}$; an explicit demonstration will be given later. The Wilson
coefficients with t-dependence and including radiative corrections have been
taken from [7] and consistancy with the results given in [23] for $t=-1$ was
checked. For the evaluation we have used $\alpha_{s}$ to two-loop order
including $\Lambda_{QCD}$=0.355 GeV from [24]. We have plotted chirality
conserving part of the

Figure 4: Plots of the chirality conserving part of the polarisation operator
constructed from the interpolating field given in Eq. (2) as functions of $t$
for $s_{0}=2.7$ $GeV^{2}$, $\langle\bar{q}q\rangle=-0.0117$ $GeV^{3}$, other
QCD parameters as given in Table 2 and $M^{2}$=2.5 $GeV^{2}$, 3.0 $GeV^{2}$
and 3.5 $GeV^{2}$ have been shown in Fig. (a). Plots of the same functions as
functions of $cos\theta$ ($t=tan\theta$) have been shown in Fig. (b). The
three lowest mass states N, $N_{+}$ and $N_{-}$ are explicitly included in the
physical spectrum.

polarization operator, constructed from the interpolating field $J_{p}$ given
in Eq. (2), as a function of t in Fig. (4a); the same function has been
plotted as a function of cos$\theta$ in Fig. (4b) to display its behavior for
large t. The plot shows a minimum at $t\simeq-0.2$ and is qualitatively
similar to the one given in Ref. [25] where a plot of chirality non-conserving
part of polarization operator without radiative correction is given. It is
found that the ratio of the combination of projected correlation functions to
the chirality conserving part of the polarization operator has a plateau for
$t\lesssim-2.0$. Based on these observations, we have determined the coupling
constants $g_{i}$’s in three regions of t: close to the extremums of the two
curves, namely around $t=-0.2$ and around $t=-1.0$, and around $t=-2.5$ where
the ratio of the two functions has a plateau. At each value of t, we first
determine $\lambda(t),\lambda_{1}(t)$ and $\lambda_{2}(t)$ which were, in
turn, used to determine $g_{i}(t)$ for i=2, 3, 4. In Table 1, we have
displayed our results of $g_{i}(t)$’s assuming $g_{5}$ is known from [26, 4].
The numerical values of $g_{i}(t)$ close to $t=-0.2$ are unstable and
unreasonably large. The numerical values of $g_{i}(t)$ at $t=-2.5\pm 0.5$ are
relatively stable but values of $g_{4}$ are unreasonably large, and hence we
discard both these results. In contrast, the numerical values of $g_{i}(t)$’s
around $t=-1$ are reasonable and the results are stable over a range
$t=-1.0\pm 0.1$. In view of these observations, from here onwards we will take
$t=-1.0$ and analyze the results in more detail.

Table 1: Our results on coupling constants of a neutral pion with low-lying nucleon resonances for various parameter values, $\langle\bar{q}q\rangle$ =$-0.0117$ Ge$V^{3}$ and other QCD parameters as given in Table 2 $t$ | $s_{\pi}$ | $\alpha$ | $\beta$ | range of $M^{2}$ where phen. and OPE sides match | $g_{2}$ | $g_{3}$ | $g_{4}$ | $g_{5}$
---|---|---|---|---|---|---|---|---
-0.1 | 2.7 | -1.0 | -1.0 | 2.8 - 3.2 | 12.2 | -9.62 | 350.0 | -1.1
-0.2 | 2.7 | -1.0 | -1.0 | 2.8 - 3.35 | 213.1 | -6.34 | 188.3 | -1.1
-0.3 | 2.7 | -1.0 | -1.0 | 2.8 - 3.25 | 162.0 | -4.5 | 180.0 | -1.1
-0.8 | 2.7 | -1.0 | -1.0 | 2.25 - 2.75 | 8.0 | 7.0 | 19.5 | -1.1
-0.9 | 2.7 | -1.0 | 0.0 | 2.25 - 2.7 | 6.0 | 14.1 | -7.5 | -1.1
-0.95 | 3.0 | 0.0 | 0.0 | 2.4 - 3.0 | 7.65 | 12.6 | -9.2 | -1.1
-1.0 | 2.7 | -1.0 | -1.0 | 2.6- 3.15 | 6.0 | 13.1 | -10.4 | -1.1
-1.0 | 3.0 | 0.0 | -1.0 | 2.65 -3.25 | 7.0 | 14.5 | -8.8 | -1.1
-1.05 | 2.7 | -1.0 | -1.0 | 2.7 -3.35 | 7.4 | 11.5 | -17.3 | -1.1
-1.1 | 2.7 | 0.0 | -1.0 | 2.8- 3.5 | 5.0 | 6.8 | -18.0 | -1.1
-1.3 | 2.7 | -1.0 | -1.0 | 2.65 - 3.2 | -28.6 | 11.4 | -28.0 | -1.1
-2.0 | 2.7 | -1.0 | -1.0 | 2.6 - 3.0 | 15.4 | 10.6 | -95.0 | -1.1
-2.5 | 2.7 | -1.0 | -1.0 | 2.7 - 3.15 | 13.0 | 10.9 | -89.0 | -1.1
-3.0 | 2.7 | -1.0 | -1.0 | 2.75 - 3.4 | 10.9 | 14.3 | -80.0 | -1.1

In Fig. 5, we have shown a fit for the chirality conserving part of the
polarization operator constructed from the interpolating field given in Eq.
(2) $(t=-1)$ when the physical spectrum is saturated with the three lowest
mass states over a range of $M^{2}=(1.7-3.3)GeV^{2}$. This is a demonstration
of a typical case for $\langle\bar{q}q\rangle=-0.0117$ Ge$V^{3}$, where the
three coupling constants appearing in Eqs. (3), (4) and (5) are uniquely fixed
as $\lambda^{2}=7.256\times 10^{-4}$ $GeV^{6}$, $\lambda_{1}^{2}=6.027\times
10^{-4}$ $GeV^{6}$ and $\lambda_{2}^{2}=1.369\times 10^{-4}$ $GeV^{6}$. A
similar fit for $\langle\bar{q}q\rangle=-0.0145$ Ge$V^{3}$ yields
$\lambda^{2}=7.758\times 10^{-4}$ $GeV^{6}$, $\lambda_{1}^{2}=6.242\times
10^{-4}$ $GeV^{6}$ and $\lambda_{2}^{2}=1.210\times 10^{-4}$ $GeV^{6}$.

Figure 5: Plots of the chirality conserving part of the polarisation operator
constructed from the interpolating field given in Eq. (2) $(t=-1)$ as a
function of Borel mass squared $M^{2}$. The three lowest mass states N,
$N_{+}$ and $N_{-}$ are explicitly included in the physical spectrum.

It may be emphasized that in the current approach separating the contributions
of $g_{1}$ and $g_{2}$ is not difficult even though N and $N_{+}$ states have
the same parities. As is clear from Eq. (13), the dominant behaviors of the
coefficients of $g_{1},g_{2}$ and $g_{3}$ in the considered range of $M^{2}$
are determined by the exponential factors. The parities of the states in the
projected correlation function approach have a minor role in the form of an
overall multiplicative factor.

Figure 6: Plots of the coefficients of $g_{1},g_{2}$ and $g_{3}$ as a function
of Borel mass squared $M^{2}$ as they appear in Eq. (13) have been shown in
Fig.(a). Plots of the coefficients of $g_{4}$ and $g_{5}$ as a function of
Borel mass squared $M^{2}$ as they appear in Eq. (13) have been shown in
Fig.(b).

We have shown plots of coefficients of $g_{1},g_{2}$ and $g_{3}$ appearing in
Eq. (13) in Fig. (6a) and of coefficients of $g_{4}$ and $g_{5}$ in Fig. (6b).
It is clear that in addition to different exponential behaviors there is a
wide gap between the coefficients of $g_{1}$ and $g_{2}$ which is the result
of large gap between the two masses m and $m_{1}$. Similar behavior is
observed for other B[$\Pi_{+o}^{ij}]_{ph}$. The gap between the coefficients
of $g_{2}$ and $g_{3}$ is not so large, though they have opposite parities,
largely because the masses of $N_{+}$ and $N_{-}$ states are so close.

For determining $g_{2}$, $g_{3}$ and $g_{4}$ from our sum rules, we have also
assumed $-1.1\leq g_{5}\leq-0.6$ based on estimates given in Ref.[26, 4] and
references therein. We have varied the continuum threshold $s_{\pi}$ between
(2.7 - 3.0) Ge$V^{2}$ which covers the mass of the next nucleon resonance
N(1650) along with its width. For matching the two sides of the combination of
the projected correlation functions we have chosen the range of $M^{2}$ to lie
between (2.7 - 3.3) Ge$V^{2}$. The reason for this choice is as follows : As
stated earlier, the Wilson coefficients of OPE are based on light-cone
expansion as well as on short-distance expansion. For light-cone expansion it
has been argued [13]that the upper limit is obtained by requiring that higher
states and continuum contributions constitute, say, 40% of the perturbative
contribution. The lower limit of $M^{2}$ is determined from the condition that
higher twist contributions are less than the leading twist contributions. Our
chosen range of $M^{2}$ is consistent with this criterion. We will comment
later on this choice from the point of view of the short distance expansion.
In the literature, [14, 8, 18, 15, 16] a range for the values of the quark
condensate $\langle\bar{q}q\rangle$ has been suggested, hence we have varied
it between $-(0.0117-0.0145)$ Ge$V^{3}$ and solved the equation. In Table 3,
we have listed a sample of our results obtained from variation of various
parameters. In Figs. (7- 10) we have shown plots of some of our results for
the sake of illustration. It is observed that the largest changes in the
values of the coupling constants occur due to changes in
$\langle\bar{q}q\rangle$ which appears as a factor in most of the terms of of
the OPE expression for $\Pi$(q,p). Furthermore, we note that it is $g_{4}$
which undergoes maximum change. Changes in the coupling constants due to
variations in $\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle$, $m_{0}^{2}$ and
$\delta^{2}$ within the range suggested in the literatureare, is small and
hence we have ignored them. The mass sum rule of nucleon and its resonances
gives only a magnitude of $\lambda$, $\lambda_{1}$ and $\lambda_{2}$; we have
chosen signs of $\lambda,\lambda_{1}$ and $\lambda_{2}$ to be +ve. If we
reverse the signs of $\lambda$ or $\lambda_{1}$ and $\lambda_{2}$, then the
signs of $g_{4}$ and $g_{5}$ will get reversed.

We could have done a similar analysis with $\Pi_{+e}^{ij}$. However, this sum
rule has a larger dependence on the continuum threshold $s_{\pi}$ [15, 16]
resulting in a larger uncertainty on our end reults of $g_{i}$’s. Hence, we
have not tried it.

We have replaced the first term containing $g_{1}$ in B$[\Pi_{+o}^{ij}]_{ph}$
with the OPE expression with lower continuum threshold $s^{\prime}_{\pi}$ as
mentioned earlier[19, 15, 16]. Based on the current literature [14, 15, 16,
18, 19, 21], in Table 2 we have listed the numerical values of the QCD and
phenomenological parameters used in this work. For $\langle\bar{q}q\rangle$
and $s_{\pi}$ a range of their values is given.

Table 2: Numerical values of the QCD and phenomenological parameters
used in this work ($\langle\bar{q}g\sigma.Gq\rangle$= $m_{0}^{2}\langle\bar{q}q\rangle$)  $\Lambda_{QCD}$ | $\langle\bar{q}q\rangle$ | $\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle$ | $m_{0}^{2}$ | $\delta^{2}$ | $f_{\pi}$
---|---|---|---|---|---
(GeV) | ($GeV^{3}$) | ($GeV^{4}$) | ($GeV^{2}$) | ($GeV^{2}$) | (GeV)
0.355 | -0.0117 | 0.012 | 0.8 | 0.2 | 0.093
| -0.0145 |  |  |  |
$f_{3\pi}$ | $s_{\pi}$ | $s^{\prime}_{\pi}$ | $m_{1}$ | $m_{2}$ | $a_{2}$
($GeV^{2}$) | ($GeV^{2}$) | ($GeV^{2}$) | (GeV) | (GeV) |
0.0045 | 2.7 | 2.0 | 1.44 | 1.54 | 0.4
| 3.0 |  |  |  |

We are giving below a typical equation involving the coupling constants we
want to determine for typical parameter values
$\langle\bar{q}q\rangle$=$-0.0117$ Ge$V^{3}$, $s_{\pi}$= 2.7 Ge$V^{2}$,
$\alpha$=$\beta$ =$-1.0$ and the remaining parameters as given in Table 2 at
$M^{2}$=2.8 Ge$V^{2}$:

$0.917g_{2}+6.263g_{3}+1.238g_{4}+3.641g_{5}=69.977$ (16)

Had we used the phenomenological expression for the first term containing
$g_{1}$ in B$[\Pi_{+o}^{ij}]_{ph}$ with the experimental value $g_{1}$=13.0 in
place of the OPE expression as done above, we would have got 70.553 on the rhs
in Eq. (16) making little difference on the end results. We have displayed in
Table 3, a sample of results for the coupling constants obtained along with
the parameters used and the range of $M^{2}$ over which the phenomenological
and the OPE sides of the expressions match. We have also shown some of these
results graphically.

Table 3: Our results on coupling constants of a neutral pion with low-lying nucleon resonances for various parameter values $\langle\bar{q}q\rangle$ | $s_{\pi}$ | $\alpha$ | $\beta$ | range of $M^{2}$ where phen. and OPE sides match | $g_{2}$ | $g_{3}$ | $g_{4}$ | $g_{5}$
---|---|---|---|---|---|---|---|---
-0.0117 | 2.7 | -1.0 | -1.0 | 2.6 - 3.15 | 6.0 | 13.1 | -10.4 | -1.1
-0.0117 | 3.0 | -1.0 | -1.0 | 2.6 - 3.2 | 6.9 | 14.1 | -10.5 | -0.6
-0.0117 | 3.0 | 0.0 | 0.0 | 2.7 - 3.25 | 7.8 | 13.2 | -10.4 | -1.1
-0.0117 | 2.7 | 0.0 | 0.0 | 2.7 - 3.25 | 7.9 | 11.48 | -10.4 | -0.6
-0.0117 | 2.7 | -1.0 | 0.0 | 2.7 - 3.25 | 7.0 | 11.25 | -11.2 | -0.6
-0.0117 | 3.0 | -1.0 | 0.0 | 2.7 - 3.25 | 7.0 | 12.95 | -11.2 | -1.1
-0.0117 | 3.0 | 0.0 | -1.0 | 2.65 - 3.25 | 7.0 | 14.5 | -8.8 | -1.1
-0.0117 | 2.7 | 0.0 | -1.0 | 2.7 - 3.25 | 7.0 | 12.68 | -8.8 | -0.6
-0.0145 | 2.7 | 0.55 | 0.55 | 2.7 - 3.3 | 5.2 | 13.02 | -23.2 | -0.6
-0.0145 | 3.0 | 0.55 | 0.55 | 2.7 - 3.35 | 6.2 | 14.85 | -22.8 | -1.1
-0.0145 | 3.0 | 0.0 | 0.0 | 2.7 - 3.3 | 6.9 | 15.58 | -22.8 | -0.6
-0.0145 | 2.7 | 0.0 | 0.0 | 2.7 - 3.3 | 6.0 | 14.0 | -22.8 | -0.6
-0.0145 | 2.7 | 0.0 | 0.55 | 2.7 - 3.3 | 4.2 | 13.22 | -22.9 | -1.1
-0.0145 | 3.0 | 0.0 | 0.55 | 2.65 - 3.3 | 4.2 | 14.48 | -22.6 | -0.6
-0.0145 | 3.0 | 0.55 | 0.0 | 2.7 - 3.35 | 6.0 | 15.95 | -20.8 | -1.1
-0.0145 | 2.7 | 0.55 | 0.0 | 2.7 - 3.3 | 5.0 | 14.18 | -22.0 | -0.6

The maximum percentage changes in the numerical values of $g_{2}$, $g_{3}$ and
$g_{4}$ with respect to their central values, due to variation of a single
parameter, as given in Table 3, were found to be as follows: due to variation
of $\langle\bar{q}q\rangle$ alone $g_{2}$, $g_{3}$ and $g_{4}$ change by 32%,
18% and 78% respectively; changes in $\alpha$ and $\beta$ lead to 47%, 12% and
20% changes in the values of $g_{2}$, $g_{3}$ and $g_{4}$ respectively;
finally variation of $s_{\pi}$ changes $g_{2}$, $g_{3}$ and $g_{4}$ by 18%,
13% and 10% respectively. Looking at the contributing components to
B$[\Pi_{+o}^{ij}]_{OPE}$, we find that the continuum contribution for the
diagonal cases with (ij)=(NN), $(N_{+}N_{+})$ and $(N_{-}N_{-})$ are upto 46%
of the total sum of the respective case and the last term in the OPE
contribution is upto 5% of the same. For the non-diagonal cases the situation
is not so favorable. Continuum contribution to $B[\Pi_{+o}^{N_{-}N}]_{OPE}$ is
(1.5 - 2.0) times larger than the sum, considered as a limiting case for the
validity of the sum rule [14], and the last term included in the OPE is upto
1/3 of the sum. For the $B[\Pi_{+o}^{N_{+}N}]_{OPE}$, the situation is worse
than this. However, we may look at this problem from a different perspective.
Allowing for the mixing of the states with different energies but with the
same quantum numbers, calling $u_{+}$=$(u_{N_{+}}+u_{N})/2$,
$u_{-}$=$(u_{N_{+}}-u_{N})/2$ and
$\Pi_{+o}^{(1)}$=$\bar{u}_{+}(q)\gamma_{0}\Pi(q,p)\gamma_{0}u_{N}(q-p)$,
$\Pi_{+o}^{(2)}$=$\bar{u}_{-}(q)\gamma_{0}\Pi(q,p)\gamma_{0}u_{N}(q-p)$, we
find that the largest continuum contribution in the considered range of
$M^{2}$ is for B[$\Pi_{+o}^{(2)}$] and is 82% of the total sum whereas the
largest contribution for the last term in the OPE is for $\Pi_{+o}^{(1)}$ and
is 13% of the total sum. It is in this sense that we consider all the five sum
rules as acceptable within the range of $M^{2}$ considered. We also tried
$\Pi_{+0}^{N_{-}N_{+}}$ sum rule and found that this utterly fails to do the
job.

It was observed that the renormalization group improvement of
$(\Pi_{+o}^{ij})_{OPE}$ through the introduction of L has a significant
bearing on the final results of the coupling constants $g_{i}$’s.

Based on the analysis of our results, we find the following values of the
coupling constants of the pion with the nucleon and its resonances :

$\displaystyle g_{2}\equiv g_{N_{+}\pi^{0}N_{+}}=5.95\pm 1.95,g_{3}\equiv
g_{N_{-}\pi^{0}N_{-}}=13.6\pm 2.35,$ (17) $\displaystyle g_{4}\equiv
g_{N_{+}\pi^{0}N}=-16.0\pm 7.2,g_{5}\equiv g_{N_{-}\pi^{0}N}=-0.85\pm 0.25,$

where we have introduced the conventioinal notation for the sake of clarity
and the result for $g_{5}$ is not our independent result as stated earlier. To
the best of our knowledge, the results on $g_{N_{+}\pi^{0}N_{+}}$ and
$g_{N_{+}\pi^{0}N}$ are new in this work.

##  4 Discussions

Though in this approach of projected correlation function one takes into
account all the Lorentz structures simultaneously, due to appearance of
$E_{0}(s_{\pi}/M^{2})$ in Eqs.(14) and (15), the model continuum contribution
is kept low for a given sum rule. By first taking different matrix elements
between a pair of states from among nucleon and nucleon resonances and then
applying dispersion relation, we change the residues of the poles while the
arguments of the coupling constants $g_{i}(q_{0})$ are decided by the pole
position only. We exploited this fact to get different algebraic equations
which were solved numerically for the desired coupling constants. We have
applied this approach of projected correlation function in a case where
multiple of states are simultaneously and explicitly taken into account. The
appearance of derivative of the coupling constant, $g^{\prime}_{i}$, is
intrinsic to this approach. We have made a reasonable assumption that
$g^{\prime}_{i}/g_{i}$ for i=2, 3, 4, 5 is approximately the same as
$g^{\prime}_{1}/g_{1}$, though they all may not be the same at a time and the
latter can be determined separately.

In Ref.[13] the authors have applied light-cone QCD sum rule with all possible
Lorentz structures in the correlation function and their derivatives. In
addition, they have used $s_{\pi}$= (4.0, 4.5) Ge$V^{2}$ and
$1.5GeV^{2}<M^{2}<2.5GeV^{2}$ and they get $g_{N_{-}\pi^{0}N_{-}}$=$10\pm 2$.
In Ref. [12] authors have used QCD sum rule based on three-point function and
have obtained $g_{N_{-}\pi^{0}N_{-}}$=$6.56\pm 1.96$,
$g_{N_{-}\pi^{0}N}$=$0.882\pm 0.264$. An and Saghai [26] have applied chiral
constituent quark approach considering $N_{-}$-resonance as a mixture of a
three-quark state with exact spin-flavor and orbital $SU(6)\times O(3)$
symmetry, and a five quark state; the authors were able to get satisfactory
results with respect to all known partial decay widths of $N_{-}$-resonace
with $g_{N_{-}\pi^{0}N}\simeq-0.6$ in particle basis and
$g_{N_{-}\pi^{0}N}\simeq-1.1$ in the isospin basis. Treating baryons in a
mirror assignment where $N_{-}$ is considered predominantly as the chiral
partner of the nucleon, Olbrich et al. [4] have got $g_{N_{-}\pi
N}\simeq-0.7$. Jido et al [11] have pointed out that in the soft pion limit,
$g_{N_{-}\pi N}$ coupling constant vanishes due to chiral symmetry while
$g_{N_{-}\eta N}$ coupling constant remains finite.

Our spectral function does not include N(1650)-state which has the same
quantum numbers as N(1535)-state and is in so close proximity to it that it
may have a significant bearing on the determination of the coupling constants
$g_{i}$’s. The widths of both of these states are $\sim$100 MeV and our guess
is that taking N(1650)-state into account will lead to decrease in the
numerical values of the coupling constants $g_{i}$’s.

Figure 7: A combination of projected correlation functions plotted as a
function of Borel mass squared, $M^{2}$. The values of parameters used for
Fig. (a) are : $\alpha=-1.0$, $\beta=-1.0$, $s_{\pi}$=2.7 Ge$V^{2}$,
$\langle\bar{q}q\rangle$=$-0.0117GeV^{3}$, and $g_{2}$=6.0, $g_{3}$=13.0,
$g_{4}=-10.5$, $g_{5}=-1.1$; for Fig. (b) the values of the parameters are :
$\alpha$=0.0, $\beta$=0.0, $s_{\pi}$=2.7 Ge$V^{2}$,
$\langle\bar{q}q\rangle$=$-0.0117GeV^{3}$, and $g_{2}$=7.9, $g_{3}$=11.48,
$g_{4}=-10.4$, $g_{5}=-0.6$

Figure 8: A combination of projected correlation functions plotted as a
function of Borel mass squared, $M^{2}$. The values of parameters used for
Fig. (a) are : $\alpha=-1.0$, $\beta$=0.0, $s_{\pi}$=3.0 Ge$V^{2}$,
$\langle\bar{q}q\rangle$=$-0.0117GeV^{3}$, and $g_{2}$=7.0, $g_{3}$=12.66,
$g_{4}=-11.2$, $g_{5}=-0.6$; for Fig. (b) the values of the parameters are :
$\alpha$=0.0, $\beta=-1.0$, $s_{\pi}$=3.0 Ge$V^{2}$,
$\langle\bar{q}q\rangle$=$-0.0117GeV^{3}$, and $g_{2}$=7.0, $g_{3}$=14.5,
$g_{4}=-8.8$,$g_{5}=-1.1$

Figure 9: A combination of projected correlation functions plotted as a
function of Borel mass squared, $M^{2}$. The values of parameters used for
Fig. (a) are : $\alpha$=0.55, $\beta$=0.55, $s_{\pi}$=3.0 Ge$V^{2}$,
$\langle\bar{q}q\rangle$=$-0.0145GeV^{3}$, and $g_{2}$=6.2, $g_{3}$=14.85,
$g_{4}=-22.8$, $g_{5}=-1.1$; for Fig. (b) the values of the parameters are :
$\alpha$=0.0, $\beta$=0.0, $s_{\pi}$=2.7 Ge$V^{2}$,
$\langle\bar{q}q\rangle$=$-0.0145GeV^{3}$, and $g_{2}$=6.0, $g_{3}$=14.0,
$g_{4}=-22.8$, ,$g_{5}=-0.6$

Figure 10: A combination of projected correlation functions plotted as a
function of Borel mass squared, $M^{2}$. The values of parameters used for
Fig. (a) are : $\alpha$=0.0, $\beta$=0.55, $s_{\pi}$=3.0 Ge$V^{2}$,
$\langle\bar{q}q\rangle$=$-0.0145GeV^{3}$, and $g_{2}$=4.2, $g_{3}$=14.48,
$g_{4}=-22.6$, $g_{5}=-0.6$; for Fig. (b) the values of the parameters are :
$\alpha$=0.55, $\beta$=0.0, $s_{\pi}$=3.0 Ge$V^{2}$,
$\langle\bar{q}q\rangle$=$-0.0145GeV^{3}$, and $g_{2}$=6.0, $g_{3}$=15.95,
$g_{4}=-20.8$, $g_{5}=-1.1$

## APPENDIX

Using the phenomenological Lagrangian $\mathcal{L}$ as given in Eq. (6), we
can calculate the correlation function and this gives

$\displaystyle\Pi(q,p)_{ph}=g_{1}\lambda^{2}\frac{(\not{q}+m)i\gamma_{5}(\not{q}-\not{p}+m)}{(q^{2}-m^{2})[(q-p)^{2}-m^{2}]}+g_{2}\lambda_{1}^{2}\frac{(\not{q}+m_{1})i\gamma_{5}(\not{q}-\not{p}+m_{1})}{(q^{2}-m_{1}^{2})[(q-p)^{2}-m_{1}^{2}]}$
(A-1)
$\displaystyle+g_{3}\lambda_{2}^{2}\frac{\gamma_{5}(\not{q}+m_{2})i\gamma_{5}(\not{q}-\not{p}+m_{2})(-\gamma_{5})}{(q^{2}-m_{2}^{2})[(q-p)^{2}-m_{2}^{2}]}$
$\displaystyle+g_{4}\lambda\lambda_{1}\Big{[}\frac{(\not{q}+m_{1})i\gamma_{5}(\not{q}-\not{p}+m)}{(q^{2}-m_{1}^{2})[(q-p)^{2}-m^{2}]}+\frac{(\not{q}+m)i\gamma_{5}(\not{q}-\not{p}+m_{1})}{(q^{2}-m^{2})[(q-p)^{2}-m_{1}^{2}]}\Big{]}$
$\displaystyle+g_{5}\lambda\lambda_{2}\Big{[}\frac{i\gamma_{5}(\not{q}+m_{2})(\not{q}-\not{p}+m)}{(q^{2}-m_{2}^{2})[(q-p)^{2}-m^{2}]}+\frac{(\not{q}+m)(\not{q}-\not{p}+m_{2})i\gamma_{5}}{(q^{2}-m^{2})[(q-p)^{2}-m_{2}^{2}]}\Big{]}$
$\displaystyle+g_{6}\lambda_{1}\lambda_{2}\Big{[}\frac{i\gamma_{5}(\not{q}+m_{2})(\not{q}-\not{p}+m_{1})}{(q^{2}-m_{2}^{2})[(q-p)^{2}-m_{1}^{2}]}+\frac{(\not{q}+m_{1})(\not{q}-\not{p}+m_{2})i\gamma_{5}}{(q^{2}-m_{1}^{2})[(q-p)^{2}-m_{2}^{2}]}\Big{]}$
$\displaystyle+\text{contribution from continuum and higher states}$

Below we list vacuum-to-pion matrix elements of non-local operators used in
the text based on Refs.[21, 16]:

$\langle
0|\bar{u}(0)i\gamma_{5}u(x)|\pi^{0}(p)\rangle=-\frac{\langle\bar{u}u\rangle}{f_{\pi}}+\textit{ip.x}\frac{\langle\bar{u}u\rangle}{2f_{\pi}}+(\textit{p.x})^{2}\Big{[}\frac{\langle\bar{u}u\rangle}{6f_{\pi}}-\frac{18.91}{2\sqrt{2}}f_{3\pi}\Big{]}$
(A-2) $\displaystyle\langle
0|\bar{u}(0)\gamma_{\mu}\gamma_{5}u(x)|\pi^{0}(p)\rangle=if_{\pi}p_{\mu}(1-\frac{i}{2}\textit{p.x}-\frac{(\textit{p.x})^{2}}{2}a^{\prime})-\frac{i}{18}f_{\pi}\delta^{2}(\textit{p.x}x_{\mu}$
(A-3) $\displaystyle-\frac{5}{2}x^{2}p_{\mu})(1-\frac{i}{2}\textit{p.x})$
$\langle
0|\bar{u}(0)\gamma_{5}\sigma^{\mu\nu}u(x)|\pi^{0}(p)\rangle=-i(p_{\mu}x_{\nu}-p_{\nu}x_{\mu})\frac{\langle\bar{u}u\rangle}{6f_{\pi}}\Big{[}1-\frac{i}{2}\textit{p.x}-\frac{c^{\prime}}{2}(p.x)^{2}\Big{]}$
(A-4) $\displaystyle\langle
0|u^{a}(x)g_{s}G^{n}_{\mu\nu}(x/2)\bar{u}^{b}(0)|\pi^{0}(p)\rangle=i\frac{f_{3\pi}}{16\sqrt{2}}t^{n}_{ab}\gamma_{5}(\sigma_{\lambda\mu}p_{\nu}-\sigma_{\lambda\nu}p_{\mu})p^{\lambda}$
(A-5)
$\displaystyle+\frac{if_{\pi}\delta^{2}}{252.16}t^{n}_{ab}\gamma_{5}[10(\gamma_{\mu}p_{\nu}-\gamma_{\nu}p_{\mu})\textit{p.x}+16\not{p}(p_{\mu}x_{\nu}-p_{\nu}x_{\mu})]$
$\displaystyle-\frac{i}{48}t^{n}_{ab}f_{\pi}\delta^{2}\epsilon_{\mu\nu}^{\quad\alpha\beta}\gamma_{\alpha}p_{\beta}(1-\frac{i}{2}\textit{p.x})$

Following is the Borel transform of a function used in the text [14, 15]:

$B[f(p_{0}^{2})]=\lim\limits_{\begin{subarray}{c}n\to\infty\\\
-p_{0}^{2}\to\infty\\\
-p_{0}^{2}/n=M^{2}\end{subarray}}\frac{(-p_{0}^{2})^{n+1}}{n!}\Big{(}\frac{d}{dp_{0}^{2}}\Big{)}^{n}f(p_{0}^{2})$
(A-6)

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