# Two-body wave functions and compositeness from scattering amplitudes:
II. Application to the physical $N^{\ast}$ and $\Delta^{\ast}$ resonances

Takayasu Sekihara<EMAIL_ADDRESS>Graduate School of Life and
Environmental Science, Kyoto Prefectural University, Sakyo-ku, Kyoto 606-8522,
Japan  Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki,
Osaka, 567-0047, Japan  RIKEN Cluster for Pioneering Research, RIKEN, Wako
351-0198, Japan

###### Abstract

The meson–baryon molecular components for the $N^{\ast}$ and $\Delta^{\ast}$
resonances are investigated in terms of the compositeness, which is defined as
the norm of the two-body wave function from the meson–baryon scattering
amplitudes. The scattering amplitudes are constructed in a $\pi N$-$\eta
N$-$\sigma N$-$\rho N$-$\pi\Delta$ coupled-channels problem in a meson
exchange model together with several bare $N^{\ast}$ and $\Delta^{\ast}$
states, and parameters are fitted so as to reproduce the on-shell $\pi N$
partial wave amplitudes up to the center-of-mass energy
$1.9\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ with the orbital angular momentum
$L\leq 2$. As a result, the Roper resonance $N(1440)$ is found to be dominated
by the $\pi N$ and $\sigma N$ molecular components while the bare-state
contribution is small. The squared wave functions in coordinate space imply
that both in the $\pi N$ and $\sigma N$ channels the separation between the
meson and baryon is about more than $1\text{\,}\mathrm{f}\mathrm{m}$ for the
$N(1440)$ resonance. On the other hand, dominant meson–baryon molecular
components are not observed in any other $N^{\ast}$ and $\Delta^{\ast}$
resonances in the present model, although they have some fractions of the
meson–baryon clouds.

##  I Introduction

The spectroscopy of the nucleon resonances $N^{\ast}$ and $\Delta^{\ast}$ is a
key clue to understand nonperturbative aspects of quantum chromodynamics
(QCD), which is the fundamental theory of hadrons and strong interactions
Zyla:2020zbs . Historically, the $\Delta(1232)$ resonance opens the door to
the color degrees of freedom Han:1965pf , which is an essential idea of QCD.
The masses, widths, and other properties such as transition strength of the
nucleon resonances have been good tests to examine the behavior of constituent
quarks inside them. Today rich spectra of the nucleon resonances have been
revealed in the $\pi N$ coupled-channels scattering amplitudes by dynamical
coupled-channels models Krehl:1999km ; Chen:2007cy ; Ronchen:2012eg ;
Kamano:2013iva  as well as the $K$-matrix approaches Anisovich:2011fc ;
Workman:2012hx . In addition, results from the lattice QCD simulations have
been used to discuss the $N^{\ast}$ and $\Delta^{\ast}$ physics Lang:2016hnn ;
Liu:2015ktc ; Liu:2016uzk ; Wu:2016ixr ; Wu:2017qve . Recent studies on the
$N^{\ast}$ and $\Delta^{\ast}$ spectroscopy can be found in, e.g., Refs.
Segovia:2015hra ; Ronchen:2014cna ; Mart:2015jof ; Ronchen:2015vfa ;
Beck:2016hcy ; Kamano:2016bgm ; Chen:2017pse ; Ronchen:2018ury ; Chen:2019fzn
. Motivated by these rich spectra of the nucleon resonances, we are now in a
phase of clarifying their internal structure.

In this study I utilize the $\pi N$ scattering amplitudes for the calculations
of the meson–baryon molecular components for the $N^{\ast}$ and
$\Delta^{\ast}$ resonances. This can be done in a strategy developed in my
first paper of a series Sekihara:2016xnq , where I have shown that the two-
body wave function of the bound state, both in the stable and decaying cases,
can be extracted from the residue of the off-shell scattering amplitude at the
bound-state pole. Furthermore, the normalization of the two-body wave function
from the residue is automatically achieved. In this sense, once the $\pi N$
coupled-channels interactions are fixed so as to reproduce the empirical $\pi
N$ scattering amplitudes, one can discuss the internal structure of the
$N^{\ast}$ and $\Delta^{\ast}$ resonances with the meson–baryon wave functions
from the off-shell parts of the scattering amplitudes and its norm, which is
called compositeness Hyodo:2011qc ; Aceti:2012dd ; Hyodo:2013nka ;
Sekihara:2014kya . (For calculations of the compositeness, see, e.g., Refs.
Aceti:2014ala ; Nagahiro:2014mba ; Kamiya:2015aea ; Sekihara:2015gvw ;
Guo:2016wpy ; Guo:2015daa ; Lu:2016gev ; Kang:2016ezb ; Albaladejo:2016hae ;
Kamiya:2016oao ; Kamiya:2017pcq ; Tsuchida:2017gpb ; Oller:2017alp ;
Guo:2019kdc . See also review articles on hadronic molecules Guo:2017jvc  and
hadron–hadron scattering Oller:2019opk .) In particular, from the $\pi N$
scattering amplitudes I can unveil amounts of the meson–baryon molecular
components of $N^{\ast}$ and $\Delta^{\ast}$ resonances which are claimed to
be “dynamically generated” without bare $N^{\ast}$ and $\Delta^{\ast}$ states.
Furthermore, even for the resonances which originate from bare states, I can
evaluate how much the bare states are dressed in the meson–baryon clouds in
terms of the compositeness. These are the aim of the present paper, the second
paper of a series for the two-body wave function and compositeness in general
quantum systems following the first paper Sekihara:2016xnq .

This paper is organized as follows. First, I briefly show the formulae of the
two-body wave function and compositeness from the scattering amplitude of
constituents in Sec. II. Next, in Sec. III I construct an effective model to
describe the $\pi N$ scattering amplitudes and $N^{\ast}$ and $\Delta^{\ast}$
resonances. In Sec. IV I utilize the $\pi N$ scattering amplitudes in the
effective model for the calculations of the two-body wave functions and
compositeness for the $N^{\ast}$ and $\Delta^{\ast}$ resonances, and I discuss
their meson–baryon molecular components. Section V is devoted to the
conclusion of this study.

##  II Formulae of compositeness

First of all, I briefly summarize the formulae of the two-body wave functions
and compositeness from the scattering amplitudes, which were expressed in
detail in Refs. Sekihara:2015gvw ; Sekihara:2016xnq . Below I focus on
unstable resonance states, but formulations and equations are applicable to
stable bound states as well. After showing general formulae in Sec. II.1, I
rewrite them in terms of the complex scaling method for numerical calculations
of resonance wave functions in Sec. II.2.

###  II.1 Two-body wave functions from scattering amplitudes

In this study I consider a two-body to two-body scattering, $k(\bm{q})\to
j(\bm{q}^{\prime})$, where $k(\bm{q})$ and $j(\bm{q}^{\prime})$ denote the
channels (relative three-momenta) of the two-body initial and final states,
respectively. The scattering is governed by the two-body interaction
$V_{\alpha,jk}(E;q^{\prime},q)$ in momentum space, where $E$ is the center-of-
mass energy of the system, $q^{(\prime)}\equiv|\bm{q}^{(\prime)}|$, and
partial-wave projection to a certain quantum number $\alpha$ of the system was
performed. I allow that the interaction may intrinsically depend on the energy
$E$. Then, the scattering amplitude $T_{\alpha,jk}(E;q^{\prime},q)$ in a
coupled-channels problem is a solution of the Lippmann–Schwinger equation:

$\displaystyle T_{\alpha,jk}(E;q^{\prime},q)=V_{\alpha,jk}(E;q^{\prime},q)$
$\displaystyle+\sum_{l=1}^{N_{\rm
chan}}\int_{0}^{\infty}dk\,k^{2}\frac{V_{\alpha,jl}(E;q^{\prime},k)T_{\alpha,lk}(E;k,q)}{E-\mathcal{E}_{l}(k)+i0}.$
(1)

Here $N_{\rm chan}$ is the number of channels and $\mathcal{E}_{j}(q)$ is the
on-shell energy of the system in the $j$th channel as a function of the
relative three-momentum $q$, for which I take the semirelativistic option:

$\mathcal{E}_{j}(q)=\sqrt{m_{j}+q^{2}}+\sqrt{M_{j}+q^{2}},$ (2)

for stable particle channels, and

$\mathcal{E}_{j}(E;q)=\sqrt{m_{j}+q^{2}}+\sqrt{M_{j}+q^{2}}+\Sigma_{j}(E;q),$
(3)

for unstable particle channels with the self-energy $\Sigma_{j}$ whose
practical form is given in Sec. III.3. The masses of two particles in the
$j$th channel are $m_{j}$ and $M_{j}$.

In my model space I treat only two-body to two-body scattering. In the study
of the $N^{\ast}$ and $\Delta^{\ast}$ resonances below, I will implement one-
body states (bare $N^{\ast}$ and $\Delta^{\ast}$) into the two-body
interaction $V_{\alpha,jk}$ and three-body state ($\pi\pi N$) into the self-
energy $\Sigma_{j}$.

In the physical scattering, the initial and final states are on their mass
shell, $E=\mathcal{E}_{j}(q^{\prime})=\mathcal{E}_{k}(q)$, and hence the
scattering amplitude $T_{\alpha,jk}(E;q^{\prime},q)$ is a function only of the
energy $E$, which is called the on-shell amplitude. The on-shell amplitude is
a solution of the Lippmann–Schwinger equation (1) and can be evaluated by the
Heitler equation

$\displaystyle T_{\alpha,jk}^{\text{on-shell}}(E)=K_{\alpha,jk}^{\text{on-
shell}}(E)$ $\displaystyle\quad+\sum_{l=\text{stable}}K_{\alpha,jl}^{\text{on-
shell}}(E)\left[-i\frac{\rho_{l}(E)}{2}\right]T_{\alpha,lk}^{\text{on-
shell}}(E),$ (4)

with the on-shell $K$-matrix $K_{\alpha,jk}^{\text{on-shell}}$ and phase space
$\rho_{l}$ where $l$ represents channels of stable particles. The $K$-matrix
is a solution of the following equation

$\displaystyle K_{\alpha,jk}(E;q^{\prime},q)=V_{\alpha,jk}(E;q^{\prime},q)$
$\displaystyle+\sum_{l=1}^{N_{\rm
chan}}\mathcal{P}\int_{0}^{\infty}dk\,k^{2}\frac{V_{\alpha,jl}(E;q^{\prime},k)K_{\alpha,lk}(E;k,q)}{E-\mathcal{E}_{l}(k)},$
(5)

and its on-shell part, $K_{\alpha,jk}^{\text{on-shell}}$, is obtained by
taking the on-shell momenta $q^{\prime}$ and $q$ for the parameters of
$K_{\alpha,jk}$ as $\mathcal{E}_{j}(q^{\prime})=\mathcal{E}_{k}(q)=E$. By
means of $\mathcal{P}$ I take the Cauchy principal value for the integral over
the momentum variable in stable open channels, but it returns to the usual
integral in closed channels or unstable channels. The phase space is defined
for stable channels as

$\rho_{j}(E)=\frac{E^{4}-(m_{j}^{2}-M_{j}^{2})^{2}}{4\pi
E^{3}}k_{j}(E)\theta(E-m_{j}-M_{j}),$ (6)

with the Heaviside step function $\theta(x)$ and the on-shell relative three-
momentum

$k_{j}(E)=\frac{\lambda^{1/2}(E^{2},m_{j}^{2},M_{j}^{2})}{2E}.$ (7)

Here $\lambda(x,y,z)\equiv x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$ is the Källen
function.

Besides the on-shell amplitude, mathematically one may treat
$T_{\alpha,jk}(E;q^{\prime},q)$ as a function of the three independent
variables $E$, $q^{\prime}$, and $q$ as an off-shell amplitude. In particular,
calculation of the off-shell amplitude with the complex energy $E$ is
essential to extract the two-body wave function from the scattering amplitude,
as seen below.

The scattering amplitude may have resonance poles in the complex energy plane.
Each pole position $E_{\rm pole}$ coincides with an eigenvalue of the
Schrödinger equation for a resonance state as an eigenstate. The pole for a
certain resonance state exists at the same position in the on-shell and off-
shell amplitudes. In particular, the resonance pole in the off-shell amplitude
can be described as

$T_{\alpha,jk}(E;q^{\prime},q)=\frac{\gamma_{j}(q^{\prime})\gamma_{k}(q)}{E-E_{\rm
pole}}+(\text{regular at }E=E_{\rm pole}),$ (8)

with the residue $\gamma_{j}(q^{\prime})\gamma_{k}(q)$.

As pointed out first by Weinberg Weinberg:1965zz  and discussed in Refs.
Sekihara:2015gvw ; Sekihara:2016xnq , the residue of the off-shell amplitude
contains information on the two-body wave function of the corresponding
resonance state as

$\gamma_{j}(q)=\frac{1}{(2\pi)^{3/2}}[E_{\rm
pole}-\mathcal{E}_{j}(q)]\tilde{R}_{j}(q),$ (9)

where $\tilde{R}_{j}(q)$ is the radial part of the resonance wave function in
the $j$th channel in momentum space. An important point is that one cannot
introduce any scaling factor for this wave function $\tilde{R}_{j}(q)$ because
the Lippmann–Schwinger equation (1) is an inhomogeneous integral equation.
Therefore, the wave function in Eq. (9) should be automatically scaled.
Indeed, the normalization of the wave function in Eq. (9) is guaranteed by the
fact that the residue of the resonance propagator $1/(E-\hat{H})$, where
$\hat{H}$ is the full Hamiltonian, is chosen to be exactly unity in the
present formulation Sekihara:2016xnq .

From the residue, one can calculate the norm of the two-body wave function in
the $j$th channel. In the present notation, the expression of the norm is

$X_{j}=\int_{0}^{\infty}dq\,\tilde{\text{P}}_{j}(q),$ (10)

with the density distribution $\tilde{\text{P}}_{j}(q)$ defined as

$\tilde{\text{P}}_{j}(q)\equiv\frac{q^{2}}{(2\pi)^{3}}\tilde{R}_{j}(q)^{2}=q^{2}\left[\frac{\gamma_{j}(q)}{E_{\rm
pole}-\mathcal{E}_{j}(q)}\right]^{2}.$ (11)

I call $X_{j}$ the compositeness of the channel $j$ for the resonance state.
Here I note that in Eq. (11) one should calculate the complex number squared
rather than the absolute value squared so as to achieve the normalizability
for resonance wave functions Hernandez:1984zzb .

The wave function in coordinate space can be calculated by the Fourier
transformation, which brings the radial part of the wave function in
coordinate space $R_{j}(r)$:

$R_{j}(r)=\frac{i^{L}}{2\pi^{2}}\int_{0}^{\infty}dq\,q^{2}\tilde{R}_{j}(q)j_{L}(qr),$
(12)

where $j_{L}(x)$ is the spherical Bessel function with the orbital angular
momentum $L$ for the channel $j$. Therefore, after omitting the irrelevant
factor $i^{L}$ from the angular part, the density distribution in coordinate
space becomes

$\displaystyle\text{P}_{j}(r)=$ $\displaystyle
r^{2}\left[\frac{1}{2\pi^{2}}\int_{0}^{\infty}dq\,q^{2}\tilde{R}_{j}(q)j_{L}(qr)\right]^{2}$
$\displaystyle=$
$\displaystyle\frac{2r^{2}}{\pi}\left[\int_{0}^{\infty}dq\,q^{2}\frac{\gamma_{j}(q)}{E_{\rm
pole}-\mathcal{E}_{j}(q)}j_{L}(qr)\right]^{2},$ (13)

which is related to the compositeness as

$X_{j}=\int_{0}^{\infty}dr\,\text{P}_{j}(r).$ (14)

Because the wave function from the scattering amplitude is automatically
scaled, the compositeness (10) and (14) should be normalized correctly.
Indeed, for energy-independent interactions, the compositeness was proved to
be exactly unity in Ref. Hernandez:1984zzb  in the nonrelativistic single-
channel resonances, and the validity was extended to coupled-channels problems
in semirelativistic cases in Ref. Sekihara:2016xnq .

However, when one treats energy dependent interaction and/or unstable
constituent with its self-energy, the sum of the compositeness of all channels
deviates from unity. This can be interpreted as a missing-channel
contribution, which is not explicit degrees of freedom in the model space of
the two-body channels but is implemented into the interaction and/or into the
self-energy. I represent the missing-channel contribution as $Z$:

$Z\equiv 1-\sum_{j=1}^{N_{\rm chan}}X_{j}.$ (15)

Here I note that both the compositeness $X_{j}$ and the missing-channel
contribution $Z$ are model dependent quantities because they are not physical
observables (see discussion in Ref. Sekihara:2016xnq ).

For resonance states, the compositeness $X_{j}$ and the missing-channel
contribution $Z$ become complex in general. Therefore, in contrast to the
stable bound states, one cannot make a probabilistic interpretation for them.
Actually, for stable bound states, $X_{j}$ and $Z$ are real and bound in the
range $[0,1]$ and hence a sum rule

$\sum_{j=1}^{N_{\rm chan}}|X_{j}|+|Z|=1$ (16)

is satisfied. On the other hand, this relation is not satisfied for resonance
states because both $X_{j}$ and $Z$ are complex. To interpret such complex
values, I introduce a quantity $U$ for resonance states to measure the
deviation from the sum rule (16), according to Refs. Kamiya:2015aea ;
Kamiya:2016oao

$U\equiv\sum_{j=1}^{N_{\rm chan}}|X_{j}|+|Z|-1.$ (17)

Because of the definition of $Z$ in Eq. (15), $U$ satisfies $U\geq 0$.
Furthermore, $U$ becomes small if $\text{Im}\,X_{j}$ and $\text{Im}\,Z$ are
negligible and $\text{Re}\,X_{j}$ and $\text{Re}\,Z$ are not negatively large.
In such a case, the wave function of the resonance state considered is similar
to that of a certain stable bound state. In particular, $U$ goes to zero in a
stable bound-state limit, so $U$ is understood as an uncertainty of the
interpretation of the complex-valued $X_{j}$ and $Z$, as discussed in Refs
Kamiya:2015aea ; Kamiya:2016oao .

In this line, I employ quantities introduced in Refs. Sekihara:2015gvw ;
Sekihara:2016xnq :

$\tilde{X}_{j}\equiv\frac{|X_{j}|}{1+U},\quad\tilde{Z}\equiv\frac{|Z|}{1+U}.$
(18)

The quantities $\tilde{X}_{j}$ and $\tilde{Z}$ are real, bound in the range
$[0,1]$, and automatically satisfy the sum rule:

$\sum_{j=1}^{N_{\rm chan}}\tilde{X}_{j}+\tilde{Z}=1.$ (19)

Then, to estimate uncertainties of the probabilistic interpretation of
$\tilde{X}_{j}$ and $\tilde{Z}$, I utilize the relation (17), which means that
$U$ measures the deviation from sum rule for a bound state (16). Therefore, a
contribution from each $X_{j}$ or $Z$ to the deviation $U$ can be estimated by
$U$ divided by the number of the degrees of freedom $N_{\rm chan}+1$, to which
I refer as the reduced uncertainty $U_{\rm r}$:

$U_{\rm r}\equiv\frac{U}{N_{\rm chan}+1}.$ (20)

In this sense, if and only if $U_{\rm r}\ll 1$, one can interpret
$\tilde{X}_{j}$ ($\tilde{Z}$) as the probability of finding the composite
(missing) part, and $U_{\rm r}$ can be considered as the uncertainty of the
probabilities $\tilde{X}_{j}$ and $\tilde{Z}$.

Figure 1: Feynman diagrams for the meson–baryon interactions in the $\pi N$ coupled-channels scattering amplitudes. Table 1: Explicit channels for each quantum number $J^{P}$, $I$ ($L_{2I\,2J}$ for the elastic $\pi N$) considered in the present study. The meson–baryon states are specified by their orbital angular momentum $L$ and total spin $S$ as ($L,S$). $J^{P}$, $I$ ($L_{2I\,2J}$) | Channel 1 | Channel 2 | Channel 3 | Channel 4 | Channel 5 | Channel 6 | Channel 7 | Channel 8
---|---|---|---|---|---|---|---|---
$1/2^{-}$, $1/2$ ($S_{11}$) | $\pi N(0,1/2)$ | $\eta N(0,1/2)$ | $\sigma N(1,1/2)$ | $\rho N(0,1/2)$ | $\rho N(2,3/2)$ | $\pi\Delta(2,3/2)$ |  |
$1/2^{-}$, $3/2$ ($S_{31}$) | $\pi N(0,1/2)$ | $\rho N(0,1/2)$ | $\rho N(2,3/2)$ | $\pi\Delta(2,3/2)$ |  |  |  |
$1/2^{+}$, $1/2$ ($P_{11}$) | $\pi N(1,1/2)$ | $\eta N(1,1/2)$ | $\sigma N(0,1/2)$ | $\rho N(1,1/2)$ | $\rho N(1,3/2)$ | $\pi\Delta(1,3/2)$ |  |
$1/2^{+}$, $3/2$ ($P_{31}$) | $\pi N(1,1/2)$ | $\rho N(1,1/2)$ | $\rho N(1,3/2)$ | $\pi\Delta(1,3/2)$ |  |  |  |
$3/2^{+}$, $1/2$ ($P_{13}$) | $\pi N(1,1/2)$ | $\eta N(1,1/2)$ | $\sigma N(2,1/2)$ | $\rho N(1,1/2)$ | $\rho N(1,3/2)$ | $\pi\Delta(1,3/2)$ |  |
$3/2^{+}$, $3/2$ ($P_{33}$) | $\pi N(1,1/2)$ | $\rho N(1,1/2)$ | $\rho N(1,3/2)$ | $\pi\Delta(1,3/2)$ |  |  |  |
$3/2^{-}$, $1/2$ ($D_{13}$) | $\pi N(2,1/2)$ | $\eta N(2,1/2)$ | $\sigma N(1,1/2)$ | $\rho N(0,3/2)$ | $\rho N(2,1/2)$ | $\rho N(2,3/2)$ | $\pi\Delta(0,3/2)$ | $\pi\Delta(2,3/2)$
$3/2^{-}$, $3/2$ ($D_{33}$) | $\pi N(2,1/2)$ | $\rho N(0,3/2)$ | $\rho N(2,1/2)$ | $\rho N(2,3/2)$ | $\pi\Delta(0,3/2)$ | $\pi\Delta(2,3/2)$ |  |
$5/2^{-}$, $1/2$ ($D_{15}$) | $\pi N(2,1/2)$ | $\eta N(2,1/2)$ | $\rho N(2,1/2)$ | $\rho N(2,3/2)$ | $\pi\Delta(2,3/2)$ |  |  |
$5/2^{-}$, $3/2$ ($D_{35}$) | $\pi N(2,1/2)$ | $\rho N(2,1/2)$ | $\rho N(2,3/2)$ | $\pi\Delta(2,3/2)$ |  |  |  |

###  II.2 Complex scaling method for resonances

To calculate numerically the resonance wave function from the residue of the
scattering amplitude, one has to perform an analytic continuation to reach the
resonance pole $E_{\rm pole}$ in the complex energy plane. One way to do this
is the complex scaling method Aoyama:2006 ; Myo:2020rni , which I employ in
the present study.

In the complex scaling method, one transforms the relative coordinate $\bm{r}$
and relative momenta $\bm{q}$ for two-body states into the complex-scaled
values in the following manner:

$\bm{r}\to\bm{r}e^{i\theta},\quad\bm{q}\to\bm{q}e^{-i\theta},$ (21)

with the scaling angle $\theta$. Then, the equations relevant to my study
become:

$\displaystyle
T_{\alpha,jk}(E;q^{\prime}e^{-i\theta},qe^{-i\theta})=V_{\alpha,jk}(E;q^{\prime}e^{-i\theta},qe^{-i\theta})$
$\displaystyle+e^{-3i\theta}\sum_{l=1}^{N_{\rm
chan}}\int_{0}^{\infty}dk\,k^{2}\frac{V_{\alpha,jl}(E;q^{\prime}e^{-i\theta},ke^{-i\theta})}{E-\mathcal{E}_{l}(ke^{-i\theta})}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\times
T_{\alpha,lk}(E;ke^{-i\theta},qe^{-i\theta}),$ (22) $\displaystyle
T_{\alpha,jk}(E;q^{\prime}e^{-i\theta},qe^{-i\theta})$
$\displaystyle=\frac{\gamma_{j}(q^{\prime}e^{-i\theta})\gamma_{k}(qe^{-i\theta})}{E-E_{\rm
pole}}+(\text{regular at }E=E_{\rm pole}),$ (23)
$\gamma_{j}(qe^{-i\theta})=\frac{1}{(2\pi)^{3/2}}[E_{\rm
pole}-\mathcal{E}_{j}(qe^{-i\theta})]\tilde{R}_{j}(qe^{-i\theta}),$ (24)
$X_{j}=\int_{0}^{\infty}dq\,\tilde{\text{P}}_{j}^{(\theta)}(q)=\int_{0}^{\infty}dr\,\text{P}_{j}^{(\theta)}(r),$
(25)
$\tilde{\text{P}}_{j}^{(\theta)}(q)=e^{-3i\theta}q^{2}\left[\frac{\gamma_{j}(qe^{-i\theta})}{E_{\rm
pole}-\mathcal{E}_{j}(qe^{-i\theta})}\right]^{2},$ (26)
$\text{P}_{j}^{(\theta)}(r)=\frac{2r^{2}e^{-3i\theta}}{\pi}\left[\int_{0}^{\infty}dq\,q^{2}\frac{\gamma_{j}(qe^{-i\theta})j_{L}(qr)}{E_{\rm
pole}-\mathcal{E}_{j}(qe^{-i\theta})}\right]^{2}.$ (27)

The definitions of the missing-channel contribution $Z$ (15), uncertainties
$U$ (17) and $U_{\rm r}$ (20), and real-valued quantities $\tilde{X}$ and
$\tilde{Z}$ (18) are unchanged.

An important property of the complex scaling method is that the pole position
$E_{\rm pole}$ and compositeness $X_{j}$ do not depend on the scaling angle
$\theta$ while density distributions $\text{P}_{j}^{(\theta)}$ and
$\tilde{\text{P}}_{j}^{(\theta)}$ depend on $\theta$.

##  III The $\pi N$ coupled-channels scattering amplitudes

As described in the previous section, one can extract normalized two-body wave
functions of resonance states from the residues of the off-shell scattering
amplitudes at the resonance poles. This fact is of special important when
investigating the internal structure of the $N^{\ast}$ and $\Delta^{\ast}$
resonances, because nowadays precise $\pi N$ scattering amplitudes are
available from the partial wave analysis of the experimental data (see, e.g.,
the database of SAID SAID ), which allows us to construct sophisticated models
for the $\pi N$ coupled-channels scattering amplitudes as done in Refs.
Ronchen:2012eg ; Kamano:2013iva .

In this study I investigate the meson–baryon molecular components of the
$N^{\ast}$ and $\Delta^{\ast}$ resonances by constructing a meson–baryon
coupled-channels model for the $\pi N$ amplitudes partly based on Ref.
JuliaDiaz:2007kz . For this purpose I take into account the $\pi N$, $\eta N$,
$\sigma N$, $\rho N$, and $\pi\Delta$ channels. The interactions are
calculated according to the Feynman diagrams shown in Fig. 1. Model parameters
are fixed so as to reproduce the results of the SAID partial wave analysis for
the on-shell $\pi N$ amplitudes SAID  up to
$E=$1.9\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ with the orbital angular
momentum $L\leq 2$, i.e., $S_{11}$, $S_{31}$, $P_{11}$, $P_{31}$, $P_{13}$,
$P_{33}$, $D_{13}$, $D_{33}$, $D_{15}$, and $D_{35}$ partial waves of the
elastic $\pi N$ scattering. Here and below I use the notation $L_{2I\,2J}$
with the orbital angular momentum $L$, isospin $I$, and total angular momentum
$J$ for the $\pi N$ system.

I first summarize the notation of the $\pi N$ scattering amplitudes in Sec.
III.1 and then construct the tree-level interactions in Sec. III.2. The self-
energies for the unstable channels, i.e., $\sigma N$, $\rho N$, and
$\pi\Delta$, are introduced in Sec. III.3. In Sec. III.4 bare $N^{\ast}$ and
$\Delta^{\ast}$ states are introduced. Finally in Sec. III.5 the model
parameters are fitted to reproduce the on-shell $\pi N$ partial wave
amplitudes. Throughout the calculations isospin symmetry is assumed.

###  III.1 Notation of the scattering amplitudes

First of all, I fix the quantum number of the $\pi N$ scattering. When one
considers the elastic $\pi N$ scattering, its partial wave can be uniquely
specified by $L_{2I\,2J}$. In a general coupled-channels analysis, however,
the quantum number of the system should be specified by the spin/parity
$J^{P}$ and isospin $I$, because the orbital angular momentum $L$ may change
in different channels such as $\pi N$ and $\sigma N$. Therefore, for the
coupled-channels scattering amplitude in the Lippmann–Schwinger equation (1),
I use the notation $\alpha=(J^{P},I)$, but for the $\pi N$ partial waves I use
the notation $L_{2I\,2J}$ as well.

For each quantum number $\alpha=(J^{P},I)$, I take into account the $\pi N$,
$\eta N$, $\sigma N$, $\rho N$, and $\pi\Delta$ channels with their orbital
angular momenta $L\leq 2$. The explicit channels considered are summarized in
Table 1. From the coupled-channels amplitude, I calculate the normalized on-
shell $\pi N$ amplitude $L_{2I\,2J}(E)$ as

$L_{2I\,2J}(E)=-\frac{\rho_{\pi N}(E)}{2}T_{(J^{P},I),11}^{\text{on-
shell}}(E),$ (28)

where the phase space $\rho_{\pi N}$ of the $\pi N$ channel is calculated
according to Eq. (6). This normalized $\pi N$ amplitude satisfies the optical
theorem

$\text{Im}\,L_{2I\,2J}(E)=\left|L_{2I\,2J}(E)\right|^{2}.$ (29)

below the inelastic threshold for the $\pi N$ state.

Table 2: Effective Lagrangians for the $\pi N$ coupled-channels interaction. Vertex | $\mathcal{L}_{\rm int}$
---|---
$\pi NN$ | $\displaystyle-\frac{D+F}{2f_{\pi}}\bar{N}\gamma^{\mu}\gamma_{5}\partial_{\mu}\vec{\pi}\cdot\vec{\tau}N$
$\pi N\Delta$ | $\displaystyle-\frac{f_{\pi N\Delta}}{m_{\pi}}\bar{N}\partial_{\mu}\vec{\pi}\cdot\vec{T}\Delta^{\mu}+\text{h.c.}$
$\rho NN$ | $\displaystyle\frac{g_{\rho NN}}{2}\bar{N}\left(\gamma^{\mu}-\frac{\kappa_{\rho}}{2m_{N}}\sigma^{\mu\nu}\partial_{\nu}\right)\vec{\rho}_{\mu}\cdot\vec{\tau}N$
$\pi\pi\rho$ | $\displaystyle g_{\pi\pi\rho}(\vec{\pi}\times\partial^{\mu}\vec{\pi})\cdot\vec{\rho}_{\mu}$
$\sigma NN$ | $g_{\sigma NN}\sigma\bar{N}N$
$\pi\pi\sigma$ | $\displaystyle-\frac{g_{\pi\pi\sigma}}{2m_{\pi}}\partial_{\mu}\vec{\pi}\cdot\partial^{\mu}\vec{\pi}\sigma$
$\eta NN$ | $\displaystyle\frac{D-3F}{2\sqrt{3}f_{\eta}}\partial_{\mu}\eta\bar{N}\gamma^{\mu}\gamma_{5}N$
$\pi\rho NN$ | $\displaystyle\frac{(D+F)g_{\rho NN}}{2f_{\pi}}\bar{N}\gamma^{\mu}\gamma_{5}(\vec{\pi}\times\vec{\rho}_{\mu})\cdot\vec{\tau}N$
$\rho\rho NN$ | $\displaystyle\frac{g_{\rho NN}^{2}\kappa_{\rho}}{8m_{N}}\bar{N}\sigma^{\mu\nu}(\vec{\rho}_{\mu}\times\vec{\rho}_{\nu})\cdot\vec{\tau}N$

###  III.2 Tree-level interactions

Next I formulate the tree-level $\pi N$ coupled-channels interactions
diagrammatically shown in Fig. 1. For this purpose I employ effective
Lagrangians summarized in Table 2. Here the notations for the hadron fields
are: $N=(p,n)^{\rm t}$ and

$\vec{\pi}=(\pi^{1},\pi^{2},\pi^{3})=\left(\frac{\pi^{+}+\pi^{-}}{\sqrt{2}},-\frac{\pi^{+}-\pi^{-}}{\sqrt{2}i},\pi^{0}\right),$
(30)

hence

$\vec{\pi}\cdot\vec{\tau}=\begin{pmatrix}\pi^{0}&\sqrt{2}\pi^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}\\\ \end{pmatrix},$ (31)

where $\vec{\tau}=(\tau^{1},\tau^{2},\tau^{3})$ is the Pauli matrices acting
on the isospin states. The $\vec{\rho}$ field is expressed in the same manner
to the $\vec{\pi}$ field. The $\Delta$ field
$\Delta=(\Delta^{++},\Delta^{+},\Delta^{0},\Delta^{-})^{\rm t}$ is tied to the
isospin transition operators $\vec{T}=(T^{1},T^{2},T^{3})$ from isospin $3/2$
to $1/2$:

$T^{1}=\begin{pmatrix}-1/\sqrt{2}&0&1/\sqrt{6}&0\\\
0&-1/\sqrt{6}&0&1/\sqrt{2}\\\ \end{pmatrix},$ (32)
$T^{2}=\begin{pmatrix}-i/\sqrt{2}&0&-i/\sqrt{6}&0\\\
0&-i/\sqrt{6}&0&-i/\sqrt{2}\\\ \end{pmatrix},$ (33)
$T^{3}=\begin{pmatrix}0&\sqrt{2/3}&0&0\\\ 0&0&\sqrt{2/3}&0\\\ \end{pmatrix},$
(34)

hence

$\vec{\pi}\cdot\vec{T}=\begin{pmatrix}-\pi^{-}&\sqrt{2/3}\pi^{0}&\sqrt{1/3}\pi^{+}&0\\\
0&-\sqrt{1/3}\pi^{-}&\sqrt{2/3}\pi^{0}&\pi^{+}\\\ \end{pmatrix}.$ (35)

For the $\pi NN$ and $\eta NN$ vertices, I employ the chiral Lagrangian with
the meson decay constants, $f_{\pi}$ and $f_{\eta}$, at their physical values
Zyla:2020zbs , $f_{\pi}=$92.1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ and
$f_{\eta}=1.3f_{\pi}$, and fix the coupling constants $D+F=1.26$ and
$D-F=0.33$ by the weak decay of octet baryons. Therefore, both the $\pi NN$
and $\eta NN$ couplings do not contain free parameters, except for the cutoffs
[see Eq. (36)].

As for the $\pi N\Delta$ vertex, I treat the coupling constant $f_{\pi
N\Delta}$ as a free parameter. I use the real-valued physical $\Delta(1232)$
mass $m_{\Delta}=$1210\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ for the
propagator of the diagram (1c) in Fig. 1. I allow that the parameters for the
$\Delta$ in the diagram (1c) may differ from those for the bare $\Delta$
introduced in Sec. III.4 for better reproduction of the experimental data.

In the case of the $\rho$ and $\sigma$ exchanges, the coupling constants
$g_{\rho NN}$, $\kappa_{\rho}$, $g_{\pi\pi\rho}$, $g_{\sigma NN}$, and
$g_{\pi\pi\sigma}$ are free parameters. For these $t$-channel $\rho$ and
$\sigma$ exchanges, I use the real-valued physical mass for the $\rho$ meson,
$m_{\rho}=$775.3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$, but a real-valued
bare mass for the $\sigma$ meson,
$m_{\sigma_{0}}=$700\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ (see Sec.
III.3).

To regularize the divergences from the integrals of the Lippmann–Schwinger
equation (1), I introduce a dipole form factor

$\mathcal{F}(\Lambda,q)\equiv\left(\frac{\Lambda^{2}}{\Lambda^{2}+q^{2}}\right)^{2},$
(36)

with a cutoff $\Lambda$ for each meson–baryon–baryon vertex with $q$ being the
three-momentum of the meson. I also use the dipole form factor (36) for the
meson–meson–meson vertex with $q$ being the three-momentum of the exchanged
meson. The cutoffs $\Lambda$ are model parameters and may take different
values for different vertices.

The energies of the mesons and baryons in the initial and final states are
respectively fixed to their on-shell values [see Eq. (44) in Appendix B].
Therefore, the meson–baryon interactions of the diagrams in Fig. 1 do not
depend on the center-of-mass energy $E$ but only on the relative momenta $q$
and $q^{\prime}$. The explicit forms of the interaction terms and their
partial-wave projections are given in Appendix C.

###  III.3 Self-energies for the $\sigma N$, $\rho N$, and $\pi\Delta$
channels

Let us turn to the self-energies for the $\sigma N$, $\rho N$, and $\pi\Delta$
channels.

Because the $\sigma$, $\rho$, and $\Delta$ resonances are unstable particles,
one should take into account their self-energies as in Eq. (3). I take the
strategy to calculate the self-energy developed in Refs. Kamano:2008gr ;
Kamano:2013iva . For the $\rho$ and $\Delta$ resonances, I use the same
formulae and parameters in Ref. Kamano:2013iva . On the other hand, to
describe the $\sigma$ resonance I use the same formula but different
parameters so as to reproduce the $\pi\pi$ ($I=0$, $L=0$) phase shift in
particular near the $\pi\pi$ threshold where the $\sigma$ resonance exists.
The effective interaction of the $\pi\pi$ ($I=0$, $L=0$) scattering is

$\mathcal{V}_{\sigma}(E_{2};q^{\prime},q)=\frac{g_{0}}{E_{2}-m_{\sigma_{0}}}f_{g}(q^{\prime})f_{g}(q)+h_{0}f_{h}(q^{\prime})f_{h}(q),$
(37)

with monopole form factors

$f_{g}(q)=\frac{\lambda_{g}^{2}}{q^{2}+\lambda_{g}^{2}},\quad
f_{h}(q)=\frac{\lambda_{h}^{2}}{q^{2}+\lambda_{h}^{2}}.$ (38)

Here $E_{2}$ is the total energy of the $\pi\pi$ system and $q^{(\prime)}$ is
the relative momentum of the initial (final) $\pi\pi$ state. The bare $\sigma$
mass $m_{\sigma_{0}}$, coupling constants $g_{0}$ and $h_{0}$, and cutoffs
$\lambda_{g}$ and $\lambda_{h}$ are the model parameters for the $\sigma$
resonance. As a result of the fit to the $\pi\pi$ ($I=0$, $L=0$) phase shift,
I obtain $m_{\sigma_{0}}=$700\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$,
$g_{0}=$616\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}^{-1}$$,
$h_{0}=$1189\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}^{-2}$$,
$\lambda_{g}=$178\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$, and
$\lambda_{h}=$217\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ for the $\sigma$
resonance.

With the parameters for the $\sigma$, $\rho$, and $\Delta$ resonances, I find
resonance poles at $E_{2}=486-213i\,$\mathrm{M}\mathrm{e}\mathrm{V}$$,
$765-75i\,$\mathrm{M}\mathrm{e}\mathrm{V}$$, and
$1210-55i\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ for the $\sigma$, $\rho$, and
$\Delta$ resonances, respectively.

Figure 2: Branch cuts for the $\pi N$, $\eta N$, $\sigma N$, $\rho N$, and $\pi\Delta$ channels. The solid lines indicate the usual cuts, i.e., cuts with the complex-scaling angle $\theta=0^{\circ}$, while the dashed lines are those with $\theta=45^{\circ}$. The line for the $\eta N$ channel is shifted slightly upward for a better visualization. The boxes denote the branch points, whose positions are written in the parentheses in units of MeV. Table 3: Bare $N^{\ast}$ and $\Delta^{\ast}$ states introduced in the present model. Their fitted bare masses $M_{0}$, cutoffs $\Lambda$, and coupling constants $g$ are also shown. The channels are specified by the indices $1$, $2$, $\ldots$, $8$ in the same order as in Table 1. $J^{P}$, $I$ ($L_{2I\,2J}$) |  $M_{0}$ [MeV] |  $\Lambda$ [MeV] | $g_{1}$ | $g_{2}$ | $g_{3}$ | $g_{4}$ | $g_{5}$ | $g_{6}$ | $g_{7}$ | $g_{8}$
---|---|---|---|---|---|---|---|---|---|---
$1/2^{-}$, $1/2$ ($S_{11}$) | $1919$ | $855$ | $2.054$ | $-0.570$ | $0.415$ | $-1.817$ | $-0.031$ | $-0.043$ |  |
” | $2150$ | $575$ | $1.095$ | $-0.761$ | $-0.227$ | $-2.651$ | $-0.386$ | $0.181$ |  |
$1/2^{+}$, $1/2$ ($P_{11}$) | $1912$ | $634$ | $-1.116$ | $0.299$ | $2.221$ | $0.157$ | $0.719$ | $0.610$ |  |
$3/2^{+}$, $3/2$ ($P_{33}$) | $1314$ | $476$ | $1.089$ | $-0.310$ | $-0.288$ | $0.046$ |  |  |  |
$3/2^{-}$, $1/2$ ($D_{13}$) | $2043$ | $608$ | $0.148$ | $0.033$ | $-1.028$ | $-1.935$ | $-0.152$ | $-0.213$ | $-0.149$ | $-0.068$
$3/2^{-}$, $3/2$ ($D_{33}$) | $1824$ | $446$ | $0.168$ | $-2.488$ | $0.142$ | $0.262$ | $2.762$ | $0.150$ |  |
$5/2^{-}$, $1/2$ ($D_{15}$) | $1821$ | $494$ | $0.142$ | $0.189$ | $0.060$ | $0.096$ | $0.167$ |  |  |

The $\sigma N$, $\rho N$, and $\pi\Delta$ self-energies, $\Sigma_{\sigma N}$,
$\Sigma_{\rho N}$, and $\Sigma_{\pi\Delta}$, respectively, are calculated in
the same manner as in Ref. Kamano:2013iva , and then I obtain the kinetic
energy (3) in these channels:

$\mathcal{E}_{\sigma
N}(E;q)=\sqrt{m_{\sigma_{0}}^{2}+q^{2}}+\sqrt{m_{N}^{2}+q^{2}}+\Sigma_{\sigma
N}(E;q),$ (39) $\mathcal{E}_{\rho
N}(E;q)=\sqrt{m_{\rho_{0}}^{2}+q^{2}}+\sqrt{m_{N}^{2}+q^{2}}+\Sigma_{\rho
N}(E;q),$ (40)
$\mathcal{E}_{\pi\Delta}(E;q)=\sqrt{m_{\pi}^{2}+q^{2}}+\sqrt{m_{\Delta_{0}}^{2}+q^{2}}+\Sigma_{\pi\Delta}(E;q),$
(41)

with the bare $\rho$ mass
$m_{\rho_{0}}=$812\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ and bare $\Delta$
mass $m_{\Delta_{0}}=$1280\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$
Kamano:2013iva . I note that the kinetic energies for the unstable channels as
well as the self-energies depend on the energy $E$ because they implicitly
involve the three-body $\pi\pi N$ state. Therefore, this $E$ dependence in the
kinetic energy may give a deviation of the compositeness from unity and hence
a nonzero missing-channel contribution $Z$ (15) corresponding to the implicit
$\pi\pi N$ state, as discussed in Ref. Sekihara:2016xnq .

By calculating the energy $E$ which satisfies $E=\mathcal{E}_{\sigma N,\rho
N,\pi\Delta}(E;q)$ with the momentum $q$ from $0$ to $+\infty$, I obtain the
branch cuts for these channels, which are plotted in Fig. 2 as solid lines
together with the branch cuts for the stable channels $\pi N$ and $\eta N$. I
note that for the energy satisfying
$\text{Re}\,E>2m_{\pi}+m_{N}=$1215\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$
and $\text{Im}\,E<0$ I perform the analytic continuation to the second Riemann
sheet of the $\pi\pi N$ channel by deforming appropriately the momentum
integral paths in the formulae of the self-energies.

Here it is instructive to see how these branch cuts behave with a finite value
of the complex-scaling angle $\theta$ in the complex scaling method. For this
purpose, in Fig. 2 I also plot the branch cuts with $\theta=45^{\circ}$ in the
complex scaling method as the dashed lines. As one can see, each branch cut
rotates clockwise with a finite value of the scaling angle. In practical
calculations, one can reach the second Riemann sheet in each channel for the
complex energy $E$ in the region sandwiched between the solid and dashed lines
of Fig. 2.

Figure 3: Diagrams for the $\pi\pi N$ three-body unitary cut. (a) and (b) are
taken into account in the present model, while (c) and (d) are not included.

Finally I comment on the three-body unitarity. In the $\pi N$ scattering the
$\pi\pi N$ three-body channel opens at the $\pi\pi N$ threshold. In the
present model I implicitly include the $\pi\pi N$ discontinuities arising from
the $s$-channel propagation of the three-body states in the self-energies for
the $\sigma N$, $\rho N$, and $\pi\Delta$ channels. The three-body cut for
this process is depicted in Figs. 3(a) and 3(b) as the thick dashed lines. On
the other hand, I do not include the $\pi\pi N$ discontinuities induced by the
$t$-channel $\pi$ exchanges in the $\pi\Delta\to\sigma N$, $\rho N$
transitions and by the $u$-channel $N$ exchange in the $\pi\Delta\to\pi\Delta$
transition, which are depicted in Figs. 3(c) and 3(d), respectively. In this
sense, I partly take into account the $\pi\pi N$ three-body unitarity. To
satisfy the three-body $\pi\pi N$ unitarity fully, it is necessary to include
the so-called $Z$ diagrams, which corresponds to Figs. 3(c) and 3(d), in
addition to the usual two-body to two-body interaction terms Matsuyama:2006rp
.

###  III.4 Bare $N^{\ast}$ and $\Delta^{\ast}$ states

Now I introduce bare $N^{\ast}$ and $\Delta^{\ast}$ states, which are embedded
into the $\pi N$ coupled-channels interactions. They are described as
$s$-channel interactions

$\displaystyle V_{jk}^{\rm bare}(E;q^{\prime},q)=$
$\displaystyle\frac{g_{j}g_{k}}{2m_{\pi}(E-M_{0})}\left(\frac{q^{\prime}}{m_{\pi}}\right)^{L^{\prime}}\left(\frac{q}{m_{\pi}}\right)^{L}$
$\displaystyle\times\mathcal{F}(\Lambda,q^{\prime})\mathcal{F}(\Lambda,q).$
(42)

These interactions are added to each partial-wave component of the $\pi N$
coupled-channels interaction $V$ in Eq. (1). Here $M_{0}$ is the bare mass,
$g_{j}$ is the coupling constant of the bare state to the channel $j$, form
factor $\mathcal{F}$ is the same as in Eq. (36), and $L^{(\prime)}$ is the
orbital angular momentum of the initial (final) meson–baryon state. The bare
mass $M_{0}$, coupling constant $g_{j}$, and cutoff $\Lambda$ are free
parameters and are fixed in the fits. I use a single cutoff $\Lambda$ for each
$N^{\ast}$ or $\Delta^{\ast}$ state to reduce the number of parameters.

Because the bare-state contributions are implemented into the two-body
coupled-channels interactions as in Eq. (42), they depend on the energy $E$,
which will bring nonzero missing-channel contribution $Z$ (15) corresponding
to the implicit bare states Sekihara:2016xnq .

A problem is to specify the number of bare $N^{\ast}$ and $\Delta^{\ast}$
states in each partial wave. In the present study I take into account the bare
states only if the bare states can significantly improve the fitting and
reproduce well resonance-like behaviors of the on-shell $\pi N$ amplitudes. In
this strategy I introduce the bare $N^{\ast}$ and $\Delta^{\ast}$ states
listed in Table 3, in which the fitted values of the parameters are shown as
well.

Table 4: Fitted values of coupling constants and cutoff parameters. Cutoff parameters are in units of MeV. $f_{\pi N\Delta}$ | $-0.439$ |  | $\Lambda_{\pi NN}$ | $566$
---|---|---|---|---
$g_{\pi\pi\rho}$ | $2.876$ |  | $\Lambda_{\pi N\Delta}$ | $510$
$g_{\rho NN}$ | $8.783$ |  | $\Lambda_{\pi\pi\rho}$ | $1000$
$\kappa_{\rho}$ | $4.806$ |  | $\Lambda_{\rho NN}$ | $564$
$g_{\pi\pi\sigma}$ | $3.188$ |  | $\Lambda_{\pi\pi\sigma}$ | $1000$
$g_{\sigma NN}$ | $21.571$ |  | $\Lambda_{\sigma NN}$ | $564$
|  |  | $\Lambda_{\eta NN}$ | $843$

###  III.5 Fit to the experimental $\pi N$ amplitudes

Figure 4: Fitted $\pi N$ scattering amplitudes $L_{2I\,2J}$. Lines (points)
represent results of the present calculation (SAID analysis SAID ). Table 5:
Nucleon resonances obtained in this study. I here show their pole positions in
the present model $E_{\rm pole}$ and their locations in terms of the Riemann
sheets, pole positions listed in Particle Data Group (PDG) Zyla:2020zbs ,
compositeness $X$ (25), missing-channel contributions $Z$ (15), uncertainties
$U$ (17) and $U_{\rm r}$ (20), and real-valued quantities $\tilde{X}$ and
$\tilde{Z}$ (18). The notation of the pole locations is explained in the text.
The indices $1$, $2$, $3$ ($1$, $2$) for the $\rho N$ ($\pi\Delta$) channel
represent the same order as in Table 1.

|  $N(1535)$ $1/2^{-}$ |  $N(1650)$ $1/2^{-}$ |  $N(1440)_{1}$ $1/2^{+}$ |  $N(1440)_{2}$ $1/2^{+}$
---|---|---|---|---
$E_{\rm pole}$ [MeV] | $1527-45i$ | $1699-70i$ | $1362-106i$ | $1361-114i$
Location | (22111) | (22112) | (21112) | (21111)
$E_{\rm pole}$(PDG) [MeV] | $(1500\text{--}1520)-(55\text{--}75)i$ | $(1640\text{--}1670)-(50\text{--}85)i$ | $(1360\text{--}1380)-(80\text{--}95)i$ | $(1360\text{--}1380)-(80\text{--}95)i$
$X_{\pi N}$ | $-0.09+0.01i\phantom{-}$ | $-0.05-0.02i\phantom{-}$ | $0.47+0.35i$ | $0.55+0.28i$
$X_{\eta N}$ | $-0.42-0.19i\phantom{-}$ | $-0.02-0.01i\phantom{-}$ | $-0.00-0.01i\phantom{-}$ | $-0.00-0.01i\phantom{-}$
$X_{\sigma N}$ | $-0.00+0.03i\phantom{-}$ | $-0.00+0.09i\phantom{-}$ | $0.40+0.15i$ | $0.46+0.03i$
$X_{\rho N(1)}$ | $0.37-0.09i$ | $0.21-0.60i$ | $-0.00+0.00i\phantom{-}$ | $-0.00+0.00i\phantom{-}$
$X_{\rho N(2)}$ | $0.31-0.00i$ | $0.09-0.01i$ | $-0.00-0.03i\phantom{-}$ | $-0.01-0.03i\phantom{-}$
$X_{\pi\Delta(1)}$ | $0.06-0.01i$ | $0.07+0.12i$ | $0.03-0.24i$ | $0.02-0.06i$
$Z$ | $0.78+0.25i$ | $0.71+0.43i$ | $0.11-0.23i$ | $-0.01-0.22i\phantom{-}$
$U$ | $1.15$ | $0.87$ | $0.55$ | $0.40$
$U_{\rm r}$ | $0.16$ | $0.12$ | $0.08$ | $0.06$
$\tilde{X}_{\pi N}$ | $0.04$ | $0.03$ | $0.38$ | $0.44$
$\tilde{X}_{\eta N}$ | $0.22$ | $0.01$ | $0.00$ | $0.00$
$\tilde{X}_{\sigma N}$ | $0.01$ | $0.05$ | $0.28$ | $0.33$
$\tilde{X}_{\rho N(1)}$ | $0.18$ | $0.34$ | $0.00$ | $0.00$
$\tilde{X}_{\rho N(2)}$ | $0.14$ | $0.05$ | $0.02$ | $0.02$
$\tilde{X}_{\pi\Delta(1)}$ | $0.03$ | $0.08$ | $0.15$ | $0.05$
$\tilde{Z}$ | $0.38$ | $0.44$ | $0.17$ | $0.16$

|  $N(1520)$ $3/2^{-}$ |  $N(1675)$ $5/2^{-}$ |  $\Delta(1232)$ $3/2^{+}$ |  $\Delta(1700)$ $3/2^{-}$
---|---|---|---|---
$E_{\rm pole}$ [MeV] | $1506-53i$ | $1652-57i$ | $1216-54i$ | $1666-84i$
Location | (22112) | (22 - 12) | (2 - - 11) | (2 - - 12)
$E_{\rm pole}$(PDG) [MeV] | $(1505\text{--}1515)-(52.5\text{--}60)i$ | $(1655\text{--}1665)-(62.5\text{--}75)i$ | $(1209\text{--}1211)-(49\text{--}51)i$ | $(1640\text{--}1690)-(100\text{--}150)i$
$X_{\pi N}$ | $0.05+0.16i$ | $-0.03+0.05i\phantom{-}$ | $-0.03+0.59i\phantom{-}$ | $-0.03+0.00i\phantom{-}$
$X_{\eta N}$ | $0.01-0.00i$ | $0.06+0.24i$ |  |
$X_{\sigma N}$ | $0.15+0.11i$ |  |  |
$X_{\rho N(1)}$ | $0.01-0.06i$ | $0.01-0.00i$ | $0.00-0.00i$ | $0.48-0.09i$
$X_{\rho N(2)}$ | $0.04-0.02i$ | $0.02-0.01i$ | $0.01+0.00i$ | $0.05-0.02i$
$X_{\rho N(3)}$ | $0.08-0.04i$ |  |  | $0.09-0.02i$
$X_{\pi\Delta(1)}$ | $-0.00+0.00i\phantom{-}$ | $0.08+0.07i$ | $0.00-0.00i$ | $-0.08-0.23i\phantom{-}$
$X_{\pi\Delta(2)}$ | $0.02-0.00i$ |  |  | $-0.01+0.04i\phantom{-}$
$Z$ | $0.62-0.15i$ | $0.86-0.35i$ | $1.02-0.59i$ | $0.50+0.31i$
$U$ | $0.24$ | $0.37$ | $0.77$ | $0.54$
$U_{\rm r}$ | $0.03$ | $0.06$ | $0.15$ | $0.08$
$\tilde{X}_{\pi N}$ | $0.14$ | $0.04$ | $0.33$ | $0.02$
$\tilde{X}_{\eta N}$ | $0.01$ | $0.18$ |  |
$\tilde{X}_{\sigma N}$ | $0.15$ |  |  |
$\tilde{X}_{\rho N(1)}$ | $0.05$ | $0.01$ | $0.00$ | $0.32$
$\tilde{X}_{\rho N(2)}$ | $0.04$ | $0.02$ | $0.01$ | $0.03$
$\tilde{X}_{\rho N(3)}$ | $0.08$ |  |  | $0.06$
$\tilde{X}_{\pi\Delta(1)}$ | $0.00$ | $0.08$ | $0.00$ | $0.16$
$\tilde{X}_{\pi\Delta(2)}$ | $0.02$ |  |  | $0.03$
$\tilde{Z}$ | $0.51$ | $0.67$ | $0.66$ | $0.39$

The present model for the $\pi N$ coupled-channels amplitudes contains the
model parameters: coupling constants, cutoffs, and bare-state masses. They can
be determined by fitting the experimental data of the on-shell $\pi N$
scattering amplitudes. In the present study, while I fix the cutoffs for the
meson–meson–meson vertices as
$\Lambda_{\pi\pi\rho}=\Lambda_{\pi\pi\sigma}=$1000\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$,
I allow the other model parameters to vary freely so as to reproduce the on-
shell $\pi N$ scattering amplitudes. The fixed value
$\Lambda_{\pi\pi\rho}=\Lambda_{\pi\pi\sigma}=$1000\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$
is a typical value of the hadronic scale, and I checked that the on-shell $\pi
N$ scattering amplitudes only weakly depend on the meson–meson–meson cutoffs
$\Lambda_{\pi\pi\rho}$, $\Lambda_{\pi\pi\sigma}$.

For the on-shell $\pi N$ scattering amplitudes, I employ the database of SAID
SAID . Restricting the center-of-mass energy
$E\leq$1.9\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ and orbital angular
momentum $L\leq 2$, I obtain the model parameters listed in Tables 3 and 4 by
the fitting.

The results of the on-shell $\pi N$ amplitudes are shown in Fig. 4. As one can
see, the present model reproduces the on-shell $\pi N$ amplitudes fairly well
except for the $S_{31}$, $P_{31}$, and $P_{13}$ amplitudes. In particular, the
resonance-like behavior in $S_{11}$, $P_{11}$, $P_{33}$, $D_{13}$, $D_{33}$,
and $D_{15}$ is well reproduced. Indeed, fittings to these amplitudes are
significantly improved by introducing two bare $N^{\ast}$ states in the
$S_{11}$ and one bare $N^{\ast}$ or $\Delta^{\ast}$ state in the $P_{11}$,
$P_{33}$, $D_{13}$, $D_{33}$, and $D_{15}$, respectively.

On the other hand, I can quantitatively reproduce the $S_{31}$ and $P_{31}$
amplitudes only in the low-energy region
$E\lesssim$1250\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$. For the $P_{13}$
amplitude, I can only reproduce the smallness of the absolute value of the
amplitude ($\lesssim 0.1$). I expect that one could cure these discrepancies
in the $S_{31}$, $P_{31}$, and $P_{13}$ amplitudes by employing further
diagrams of the meson–baryon interactions or phenomenological contact
potentials as done in Ref. Kamano:2013iva  for the $S_{31}$ partial wave, and
by taking into account other meson–baryon channels such as $K\Lambda$ and
$K\Sigma$. I note that, although the experimental data imply resonances around
$1600\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ and
$1700\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ in the $S_{31}$ and $P_{13}$,
respectively, I do not include bare states in these partial waves because bare
states will not significantly improve the fittings.

Figure 5: Density distributions of the $\pi N$ and $\sigma N$ components for
the $N(1440)$ resonance in coordinate space. (a) For the $N(1440)_{1}$ pole
with the complex-scaling angle $\theta=50^{\circ}$. (b) For the $N(1440)_{2}$
pole with the complex-scaling angle $\theta=35^{\circ}$.

##  IV The meson–baryon compositeness for the $N^{\ast}$ and $\Delta^{\ast}$
resonances

In this section I calculate the compositeness of the $\pi N$, $\eta N$,
$\sigma N$, $\rho N$, and $\pi\Delta$ channels for the $N^{\ast}$ and
$\Delta^{\ast}$ resonances.

One of the most interesting features in hadron physics is the competition
between hadron degrees of freedom and quark degrees of freedom. In the present
$N^{\ast}$/$\Delta^{\ast}$ case, bare states which are expected to originate
from quark degrees of freedom are embedded into the $\pi N$ coupled channels.
As a consequence, even if a meson–baryon interaction is strongly attractive
enough to make a bound state, the bound state is in general contaminated by
bare $N^{\ast}$/$\Delta^{\ast}$ states which couple to the meson–baryon
system. Conversely, it is inevitable that a bare $N^{\ast}$/$\Delta^{\ast}$
state is dressed in meson–baryon clouds. The compositeness is applicable to
evaluating both the dominance of the meson–baryon molecular components and the
fractions of the meson–baryon clouds for physical nucleon resonances.

In the present study, I employ the interaction diagrams in Fig. 1 and several
bare $N^{\ast}$ and $\Delta^{\ast}$ states for the $\pi N$ coupled-channels
scattering, and fix the model parameters so as to reproduce the on-shell $\pi
N$ amplitude, as explained in the previous section. I perform the analytic
continuations of the scattering amplitudes to the complex energy plane in the
complex scaling method, and find resonance poles corresponding to the
$N(1535)$ and $N(1650)$ in the spin/parity $J^{P}=1/2^{-}$, $N(1440)$ in
$1/2^{+}$, $N(1520)$ in $3/2^{-}$, $N(1675)$ in $5/2^{-}$, $\Delta(1232)$ in
$3/2^{+}$, and $\Delta(1700)$ in $3/2^{-}$, where the names are taken from
Particle Data Group (PDG) Zyla:2020zbs . Their pole positions are listed in
Table 5 together with the results of the compositeness $X$ (25), missing-
channel contributions $Z$ (15), uncertainties $U$ (17) and $U_{\rm r}$ (20),
and real-valued quantities $\tilde{X}$ and $\tilde{Z}$ (18).

Below I discuss the internal structure of the $N^{\ast}$ and $\Delta^{\ast}$
resonances on their resonance poles.

###  IV.1 $N(1440)$

The $N(1440)$ resonance in $J^{P}=1/2^{+}$, also known as the Roper resonance,
is one of the most interesting states among the nucleon resonances. The Roper
resonance is lighter than the lowest negative-parity nucleon excitations,
i.e., $N(1535)$ in $J^{P}=1/2^{-}$ and $N(1520)$ in $J^{P}=3/2^{-}$, which
cannot be easily explained if one assumes that the Roper resonance is a radial
excitation of nucleon as a three-quark system. A promising physical
interpretation is that the Roper resonance is the first radial excitation of
nucleon but consists of a dressed-quark core augmented by a meson cloud
Burkert:2017djo . Importance of the contribution from the $\pi N$ coupled-
channels dynamics was also pointed out in, e.g., Refs. Krehl:1999km ;
Lang:2016hnn ; Liu:2016uzk ; Wu:2017qve ; Suzuki:2009nj ; Golli:2017nid . In
this sense, the use of the two-body wave functions and compositeness in my
approach is quite suitable for studying the internal structure of the Roper
resonance.

In the present $\pi N$ coupled-channels model, I find two poles of the
scattering amplitude corresponding to the Roper resonance at $E_{\rm
pole}=1362-106i\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ and
$1361-114i\,$\mathrm{M}\mathrm{e}\mathrm{V}$$. The pole positions deviate only
slightly from the value reported by PDG. The former pole is found with the
complex-scaling angle $\theta\geq 45^{\circ}$ in the complex scaling method.
This indicates that the former pole exists in the second Riemann sheets of the
$\pi N$ and $\pi\Delta$ channels but in the first Riemann sheets of the $\eta
N$, $\sigma N$, and $\rho N$ channels (see Fig. 2), to which I refer as
$(21112)$ in the order $\pi N$, $\eta N$, $\sigma N$, $\rho N$, and
$\pi\Delta$. On the other hand, the latter pole is found with the scaling
angle $25^{\circ}\leq\theta\leq 40^{\circ}$, and hence it exists in the sheet
$(21111)$. Because the former pole is closer to the physical region, i.e., the
real energy axis, the former is the resonance pole for the Roper resonance and
write it as $N(1440)_{1}$. On the other hand, the latter pole is the shadow
pole for the Roper resonance Suzuki:2009nj  and write it as $N(1440)_{2}$. I
note that the pole positions $E_{\rm pole}$ do not depend on the scaling angle
$\theta$ but one can switch the Riemann sheets at a certain energy by varying
$\theta$.

From the residues of the off-shell scattering amplitudes at the poles, I can
calculate the two-body wave functions and compositeness as their norms both
for the $N(1440)_{1}$ and $N(1440)_{2}$, according to the method in Sec. II.
The results of the compositeness $X_{j}$ (25) of the $j$th meson–baryon
channel and missing-channel contribution $Z$ (15) are listed in Table 5. I
checked that the compositeness does not depend on the scaling angle $\theta$.
As one can see from the Table, although the values of the compositeness are
complex due to the resonance nature, the real parts of $X_{\pi N}$ and
$X_{\sigma N}$ for the both poles are as large as $0.5$, which should be
compared with unity, and their imaginary parts are smaller than the real
parts. On the other hand, the missing-channel contribution $Z$ is close to
zero. Because the nonzero value of $Z$ comes from the bare state in the
meson–baryon interaction (42) as well as the $\pi\pi N$ state in the self-
energies of Eqs. (39)–(41), the present result strongly implies that, while
the $\pi N$ and $\sigma N$ molecular components dominate both the
$N(1440)_{1}$ and $N(1440)_{2}$ states as “thick meson clouds”, the bare-state
contribution is small.

The complex-valued compositeness, however, cannot be interpreted as
probabilities of finding meson–baryon components. To draw a more definite
conclusion, I calculate the real-valued quantities $\tilde{X}$ and $\tilde{Z}$
(18) together with $U$ (17) and $U_{\rm r}$ (20) for the $N(1440)_{1}$ and
$N(1440)_{2}$ states. The results are listed in Table 5. The value of the
reduced uncertainty $U_{\rm r}$ is smaller than $0.1$ for both poles.
Therefore, according to the discussions in the end of Sec. II.1, I can
interpret $\tilde{X}_{j}$ ($\tilde{Z}$) as the probability of finding the
composite (missing) part with small uncertainties. From the values in Table 5,
I can conclude more definitely that the $\pi N$ and $\sigma N$ molecular
components, whose contributions are almost the same as each other, dominate
the $N(1440)_{1}$ and $N(1440)_{2}$ states while the bare-state contribution
is about less than $20\%$ in the present model. These finding are consistent
with the previous studies in, e.g., Refs. Krehl:1999km ; Lang:2016hnn ;
Liu:2016uzk ; Wu:2017qve ; Suzuki:2009nj ; Golli:2017nid ; Burkert:2017djo ,
in which a significant contribution from meson–baryon coupled channels was
reported. In particular, the present study supports the scenario drawn in Ref.
Krehl:1999km  that the $\pi N\to\sigma N$ transition potential and treatment
of the $\pi\pi N$ components in the unstable $\sigma N$ channel are important
for the description of the Roper resonance.

The squared wave functions in coordinate space (27) represent the behavior of
relative motions between the mesons and baryons as the density distributions.
The density distributions $\text{P}_{j}^{(\theta)}$ for the $N(1440)_{1}$ and
$N(1440)_{2}$ states are plotted in Fig. 5 as functions of the relative
distance $r$ between the mesons and baryons. The scaling angle is fixed as
$\theta=50^{\circ}$ for the $N(1440)_{1}$ and $\theta=35^{\circ}$ for the
$N(1440)_{2}$. Although the density distributions are complex for resonances
in general and depend on $\theta$, they provide information on the typical
distance between the mesons and baryons. The results of the density
distributions imply that both in the $\pi N$ and $\sigma N$ channels the
meson–baryon separation is about more than $1\text{\,}\mathrm{f}\mathrm{m}$
for the $N(1440)_{1}$ and $N(1440)_{2}$ states.

###  IV.2 $\Delta(1232)$

The $\Delta(1232)$ resonance in $J^{P}=3/2^{+}$ is also interesting, because
there are several suggestions of its large $\pi N$ component. Historically, it
was pointed out in Ref. Chew:1955zz  that the $\Delta(1232)$ resonance can
occur by the attractive $p$-wave $\pi N$ interaction. A hint of the large
effect of the meson cloud is seen, for instance, in the $M1$ transition form
factor for $\gamma^{\ast}N\to\Delta$ at $Q^{2}=0$ Sato:2009de . Further
studies on the dynamical generation of the $\Delta(1232)$ resonance can be
seen, e.g., in Refs. Aceti:2014ala ; Sekihara:2015gvw ; Golli:2019vxw . I can
examine the picture of a large $\pi N$ component in terms of the
compositeness.

In the present model, I observe the resonance pole for the $\Delta(1232)$ at
$1216-54i\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ in the Riemann sheet
$(2\text{~{}-~{}-~{}}11)$, where hyphen represents a decoupled channel. The
compositeness calculated from the residue and pole position for the
$\Delta(1232)$ is listed in Table 5. The $\pi N$ compositeness $X_{\pi N}$ has
nonnegligible imaginary part but its real part is small. The other
meson–baryon channels, i.e., the $\rho N(L=1,S=1/2)$, $\rho N(L=1,S=3/2)$, and
$\pi\Delta$ channels give negligible contributions to the compositeness, and
hence the missing-channel contribution $Z$ is almost unity in the real part
and negatively large in the imaginary part. Because the bare $\Delta^{\ast}$
state exists near the physical pole position, I can expect that $Z$ is
dominated by the bare state. Therefore, the results imply that the bare-state
contribution is the most essential for the physical $\Delta(1232)$ resonance.
Nevertheless, I expect that the large absolute value $|X_{\pi N}|$ reflects
the importance of the $\pi N$ channel in the $\Delta(1232)$ and affects the
properties of the $\Delta(1232)$ as the meson clouds.

Besides, the reduced uncertainty $U_{\rm r}$ is as large as $0.15$ owing to
the large imaginary part in the $\pi N$ channel. Therefore, although
$\tilde{X}_{\pi N}$ takes a nonnegligible value $\sim 0.3$, I cannot
definitely interpret it as the probability of finding $\pi N$ component inside
the $\Delta(1232)$ resonance.

###  IV.3 $N(1535)$, $N(1650)$, $N(1520)$, $N(1675)$, and $\Delta(1700)$

Next I consider the other resonances: $N(1535)$ and $N(1650)$ in
$J^{P}=1/2^{-}$, $N(1520)$ in $3/2^{-}$, $N(1675)$ in $5/2^{-}$, and
$\Delta(1700)$ in $3/2^{-}$. The results of their pole positions and
compositeness are listed in Table 5.

The $N(1535)$ and $N(1650)$ resonances exist in the Riemann sheets $(22111)$
and $(22112)$, respectively. The results of the compositeness for the
$N(1535)$ imply that the bare-state contribution is dominant but the coupling
to the $\eta N$ channel, whose branch point is the closest to the $N(1535)$
pole position, would be large. However, the real part of the $\eta N$
compositeness $X_{\eta N}$ is negatively large, which is canceled with the
real parts of the $\rho N$ compositeness. As a consequence, the reduced
uncertainty $U_{\rm r}$ is as large as $\tilde{X}_{\eta N,\rho N(1),\rho
N(2)}$ for the $N(1535)$ and I cannot interpret $\tilde{X}_{\eta N,\rho
N(1),\rho N(2)}$ as the probabilities of finding the $\eta N$ and $\rho N$
components, respectively.

On the other hand, for the $N(1650)$, because $U_{\rm r}=0.12$, I can
interpret $\tilde{X}_{\rho N(1)}$ and $\tilde{Z}$ as the probabilities with
uncertainties $\sim 0.1$. The results of $\tilde{X}_{j}$ and $\tilde{Z}$
indicate that about half of the $N(1650)$ comes from missing channels, in the
present case the bare state, while it has a certain fraction of the $\rho
N(L=0,S=1/2)$ cloud.

Similarly, the poles of the $N(1520)$, $N(1675)$, and $\Delta(1700)$
resonances are found in the $(22112)$, $(22\text{~{}-~{}}12)$, and
$(2\text{~{}-~{}-~{}}12)$ Riemann sheets, respectively. The results of the
compositeness indicate that they are dominated by missing channels, i.e., the
bare states. The $\Delta(1700)$ resonance has a certain fraction of the $\rho
N(L=0,S=3/2)$ cloud, while the $N(1520)$ and $N(1675)$ resonances have only
small fractions of meson–baryon clouds.

These results indicate that, although the $N(1535)$, $N(1650)$, $N(1520)$,
$N(1675)$, and $\Delta(1700)$ resonances have some fractions of the
meson–baryon clouds, they do not have dominant meson–baryon molecular
components. The largest fractions of the meson–baryon clouds are the $\rho N$
for the $N(1650)$ and $\Delta(1700)$ resonances, which amount to $\sim 0.3$
with uncertainties $\sim 0.1$. These are because the resonance pole positions
are close to the $\rho N$ branch point and coupling constants of the
$N^{\ast}$/$\Delta^{\ast}$ bare states to the $\rho N$ channel are large, as
seen in Table 3.

##  V Conclusion

In this study I have investigated the internal structure of the nucleon
resonances $N^{\ast}$ and $\Delta^{\ast}$ in terms of the meson–baryon two-
body wave functions and compositeness. One of the most essential parts in my
approach is to extract the meson–baryon two-body wave functions by using the
pole positions for the nucleon resonances in the complex energy plane of the
$\pi N$ coupled-channels scattering amplitudes and residues for them. Here the
scattering amplitudes are solutions of the Lippmann–Schwinger equation. In
this strategy, for each resonance pole I can obtain wave functions of the $\pi
N$ and coupled channels which are automatically scaled, thanks to the
inhomogeneous property of the Lippmann–Schwinger equation. As a consequence,
by calculating the compositeness, which is defined as the norm of the two-body
wave function from the meson–baryon scattering amplitudes, and by comparing
the compositeness with unity, I can evaluate the dominance of the meson–baryon
molecular components as well as the fractions of meson–baryon clouds for
physical nucleon resonances.

For this purpose I have constructed a meson exchange model in a $\pi N$-$\eta
N$-$\sigma N$-$\rho N$-$\pi\Delta$ coupled-channels problem and involve
several bare $N^{\ast}$ and $\Delta^{\ast}$ states. The coupling constants,
cutoffs, and bare-state masses as the model parameters were fixed so as to
reproduce the experimental data of the on-shell $\pi N$ scattering amplitudes
in the center-of-mass energy
$E\leq$1.9\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ and orbital angular
momentum $L\leq 2$. The constructed model reproduced the on-shell $\pi N$
amplitudes fairly well and generated resonance poles corresponding to the
$N(1535)$ and $N(1650)$ in the spin/parity $J^{P}=1/2^{-}$, $N(1440)$ in
$1/2^{+}$, $N(1520)$ in $3/2^{-}$, $N(1675)$ in $5/2^{-}$, $\Delta(1232)$ in
$3/2^{+}$, and $\Delta(1700)$ in $3/2^{-}$ in the complex energy plane of the
scattering amplitudes. In the present model the Roper resonance $N(1440)$ is
composed of two poles in different $\pi\Delta$ sheets.

Then I have calculated the meson–baryon wave functions and compositeness from
the scattering amplitudes for these nucleon resonances. As a result, the Roper
resonance $N(1440)$, for both the two poles, was found to be dominated by the
$\pi N$ and $\sigma N$ molecular components, whose contributions are almost
the same as each other, while the bare-state contribution is about less than
$20\%$ in the present model. The squared wave functions in coordinate space
imply that both in the $\pi N$ and $\sigma N$ channels the separation between
the meson and baryon is about more than $1\text{\,}\mathrm{f}\mathrm{m}$ for
the $N(1440)$ resonance. On the other hand, dominant meson–baryon molecular
components were not observed in any other $N^{\ast}$ and $\Delta^{\ast}$
resonances in the present model, although they have some fractions of the
meson–baryon clouds.

Here I emphasize that the present strategy to calculate the compositeness is
in general applicable as long as the Lippmann–Schwinger equation is fully
solved for hadron–hadron scatterings. In this sense, more definitely
conclusion about the composite nature of nucleon resonances will be drawn with
more sophisticated models such as Refs. Ronchen:2012eg ; Kamano:2013iva ;
Ronchen:2018ury , in which they precisely reproduced experimental data of not
only the on-shell $\pi N$ amplitudes but also the pion- and photon-induced
reactions. Furthermore, excited baryons with non-zero strangeness, i.e.,
$\Lambda^{\ast}$, $\Sigma^{\ast}$, $\Xi^{\ast}$, and $\Omega^{\ast}$ states,
will be important as a next target, because they will be discovered and be
investigated extensively in near future experiments at J-PARC, JLab, etc., as
well as in the relativistic heavy ion collisions. To construct scattering
amplitudes for these resonances, approaches in, e.g., Refs. Kamano:2014zba ;
Kamano:2015hxa  will be helpful.

Finally I comment on the model dependence of the two-body wave functions and
compositeness. Because the compositeness as well as the wave functions is not
observable, the compositeness is in general a model dependent quantity. This
fact has a special meaning when one discusses a hadron–hadron molecular
component in a hadron resonance. The strong interactions between hadrons
emerge as a nonperturbative phenomenon of the underlying theory, QCD, and
hence one cannot obtain the strong interactions between hadrons by
analytically solving QCD. This is in contrast to the electromagnetic case, in
which the electromagnetic interactions can be directly obtained by the
fundamental theory, quantum electrodynamics (QED). Therefore, to pin down the
hadron–hadron interaction and calculate its off-shell part, which plays an
essential role in the two-body wave functions and compositeness, we have to
fix a scheme based on a certain principle such as meson exchanges and employ
effective Lagrangians which govern the hadron–hadron interaction. The present
article indeed suggests a strategy in this line to elucidate the internal
structure of hadron resonances in terms of the hadron–hadron molecular
components.

###### Acknowledgements.

The author acknowledges H. Kamano, S. X. Nakamura, and T. Sato for helpful
discussions on the $\pi N$ partial-wave analysis and on the $N^{\ast}$ and
$\Delta^{\ast}$ physics. He is also very grateful to D. Jido and T. Hyodo for
fruitful discussions on the compositeness. This work is partly supported by
JSPS KAKENHI Grant No. JP15K17649.

##  Appendix A Masses of hadrons

In this study I employ isospin symmetric masses for hadrons:
$m_{\pi}=$138.0\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$,
$m_{\eta}=$547.9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$, and
$m_{N}=$938.9\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$. These values are used
in the $t$-channel pion propagators and in the $s$\- and $u$-channel nucleon
propagators as well as in the initial and final states. The masses in the
unstable initial and final states are taken from the bare states in the self-
energies: $m_{\sigma_{0}}=$700\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$,
$m_{\rho_{0}}=$812\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$, and
$m_{\Delta_{0}}=$1280\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$. The value
$m_{\sigma_{0}}=$700\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ is used for the
$t$-channel $\sigma$ exchange in the $\pi N\to\pi N$ interaction, while
physical values $m_{\rho}=$775.3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ and
$m_{\Delta}=$1210\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ are used for the
$t$-channel $\rho$ exchange and $u$-channel $\Delta$ exchange in the $\pi
N\to\pi N$ interaction, respectively.

##  Appendix B Partial waves of the meson–baryon interactions

In this Appendix I summarize the notations and partial-wave projections of the
meson–baryon ($MB$) interactions.

###  B.1 Notations

The meson–baryon scatterings are denoted by
$M(k^{\mu},\lambda_{M})B(p^{\mu},\lambda_{B})\to
M^{\prime}(k^{\prime\mu},\lambda_{M^{\prime}})B^{\prime}(p^{\prime\mu},\lambda_{B^{\prime}})$,
where $k^{(\prime)\mu}$ and $p^{(\prime)\mu}$ are momenta and $\lambda_{M,B}$
are helicities of the meson and baryon, respectively. Since I consider
scatterings in the center-of-mass frame, I can write the three-momenta as
$\bm{q}\equiv\bm{k}=-\bm{p}$ and
$\bm{q}^{\prime}\equiv\bm{k}^{\prime}=-\bm{p}^{\prime}$. Without loss of
generality, I can choose the coordinates such that

$\bm{q}=(0,0,q),\quad\bm{q}^{\prime}=(q^{\prime}\sin\theta,0,q^{\prime}\cos\theta),$
(43)

with the scattering angle $\theta$. The masses of the meson $M^{(\prime)}$ and
baryon $B^{(\prime)}$ are expressed as $m_{M^{(\prime)}}$ and
$m_{B^{(\prime)}}$, respectively. Throughout this study I fix the energies of
the four-momenta $k^{\mu}$ and $p^{\mu}$ to their on-shell values as

$k^{0}=\sqrt{m_{M}^{2}+q^{2}},\quad p^{0}=\sqrt{m_{B}^{2}+q^{2}},$ (44)

and similarly for $k^{\prime 0}$ and $p^{\prime 0}$.

###  B.2 Partial-wave projections

I calculate the partial-wave matrix elements of the interaction $V_{\alpha}$
by following the Jacob–Wick formulation Jacob:1959at , where $\alpha$
specifies the quantum numbers of the system. In the $\pi N$ coupled-channels
scattering case, I take $\alpha=(J^{P},I)$ with the total angular momentum
$J$, parity $P$, and isospin $I$.

First, according to Feynman diagrams, I calculate the interactions in terms of
the helicity eigenstates

$V_{MB\to M^{\prime}B^{\prime}}=V_{MB\to
M^{\prime}B^{\prime}}(\bm{q}^{\prime},\lambda_{M^{\prime}},\lambda_{B^{\prime}},\bm{q},\lambda_{M},\lambda_{B}).$
(45)

Then, the interactions are projected to the total angular momentum $J$ as

$\displaystyle
V^{J}(q^{\prime},\lambda_{M^{\prime}},\lambda_{B^{\prime}},q,\lambda_{M},\lambda_{B})$
$\displaystyle=2\pi\int_{-1}^{1}d\cos\theta\,d^{J}_{\lambda_{M}-\lambda_{B}\,\lambda_{M^{\prime}}-\lambda_{B^{\prime}}}(\theta)$
$\displaystyle\phantom{=}\times
V(\bm{q}^{\prime},\lambda_{M^{\prime}},\lambda_{B^{\prime}},\bm{q},\lambda_{M},\lambda_{B}),$
(46)

where I omitted the subscript $MB\to M^{\prime}B^{\prime}$ of $V$, and
$d_{m^{\prime}\,m}^{j}$ is the Wigner $d$-matrix.

The interaction of the total angular momentum $J$ is projected to the states
with definite orbital angular momenta and spins for the meson–baryon channels
as

$\displaystyle V_{\alpha}(q^{\prime},q)$
$\displaystyle=\kappa(q^{\prime},q)\sum_{\lambda_{M},\lambda_{B},\lambda_{M^{\prime}},\lambda_{B^{\prime}}}\frac{\sqrt{(2L+1)(2L^{\prime}+1)}}{2J+1}$
$\displaystyle\quad\times\langle
j_{M^{\prime}}\,j_{B^{\prime}}\,\lambda_{M^{\prime}}\,-\lambda_{B^{\prime}}|S^{\prime}\,S_{z}^{\prime}\rangle\langle
L^{\prime}\,S^{\prime}\,0\,S_{z}^{\prime}|J\,S_{z}^{\prime}\rangle$
$\displaystyle\quad\times\langle
j_{M}\,j_{B}\,\lambda_{M}\,-\lambda_{B}|SS_{z}\rangle\langle
L\,S\,0\,S_{z}|J\,S_{z}\rangle$ $\displaystyle\quad\times
V^{J}(q^{\prime},\lambda_{M^{\prime}},\lambda_{B^{\prime}},q,\lambda_{M},\lambda_{B}),$
(47)

where $j_{M^{(\prime)}}$ and $j_{B^{(\prime)}}$ are the spins of the meson
$M^{(\prime)}$ and baryon $B^{(\prime)}$, respectively, $L^{(\prime)}$ and
$S^{(\prime)}$ are the orbital angular momentum and spin in the initial
(final) state, respectively, $S_{z}\equiv\lambda_{M}-\lambda_{B}$, and
$S_{z}^{\prime}\equiv\lambda_{M^{\prime}}-\lambda_{B^{\prime}}$. $\langle
j_{M}\,j_{B}\,\lambda_{M}\,-\lambda_{B}|S\,S_{z}\rangle$ is the Clebsch–Gordan
coefficient. The factor $\kappa(q^{\prime},q)$, which is defined as

$\kappa(q^{\prime},q)\equiv\frac{1}{(2\pi)^{3}}\sqrt{\frac{m_{B}m_{B^{\prime}}}{4\omega_{M}(q)E_{B}(q)\omega_{M^{\prime}}(q^{\prime})E_{B^{\prime}}(q^{\prime})}},$
(48)

with $\omega_{M^{(\prime)}}(q)\equiv\sqrt{q^{2}+m_{M^{(\prime)}}}$ and
$E_{B^{(\prime)}}(q)\equiv\sqrt{q^{2}+m_{B^{(\prime)}}^{2}}$, was introduced
so as to satisfy the optical theorem with the correct coefficients. The
interaction (47) is applicable to the Lippmann–Schwinger equation (1).

##  Appendix C Explicit forms of the meson–baryon interactions

In this Appendix I show the explicit forms of the interactions
$M(k^{\mu},\lambda_{M})B(p^{\mu},\lambda_{B})\to
M^{\prime}(k^{\prime\mu},\lambda_{M^{\prime}})B^{\prime}(p^{\prime\mu},\lambda_{B^{\prime}})$
used in the present study. The interactions are written in terms of the
helicity eigenstates, i.e., those in Eq. (45), except for the $s$-channel
contributions of the bare $N^{\ast}$ and $\Delta^{\ast}$ states in Appendix
C.16.

Table 6: Isospin factors for the interactions. | $I=1/2$ | $I=3/2$
---|---|---
$\tau^{j}\tau^{i}$ | $3$ | $0$
$\tau^{i}\tau^{j}$ | $-1$ | $2$
$\delta_{ij}$ | $1$ | $1$
$i\epsilon_{jik}\tau^{k}$ | $2$ | $-1$
$T^{i}T^{\dagger j}$ | $4/3$ | $1/3$
$\tau^{i}$ | $-\sqrt{3}$ | $0$
$\tau^{j}$ | $-\sqrt{3}$ | $0$
$T^{\dagger j}\tau^{i}$ | $\sqrt{6}$ | $0$
$T^{\dagger i}\tau^{j}$ | $\sqrt{8/3}$ | $-\sqrt{5/3}$
$T^{\dagger j}$ | $-\sqrt{2}$ | $0$
$T^{\dagger j}T^{i}$ | $2$ | $0$

For the incoming and outgoing nucleons, I express the helicity eigenstates by
the the Dirac spinors $u_{N}(-\bm{q},\lambda_{N})$ and
$\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})$, respectively. The
spinors for the incoming and outgoing $\rho$ meson are
$e_{\rho}^{\mu}(\bm{q},\lambda_{M})$ and
$e_{\rho}^{\mu\ast}(\bm{q}^{\prime},\lambda_{M^{\prime}})$, respectively. The
helicity eigenstates of the $\Delta$ baryon as the Rarita-Schwinger spinor are
$u_{\Delta}^{\mu}(-\bm{q},\lambda_{\Delta})$ for the incoming and
$\bar{u}_{\Delta}^{\mu}(-\bm{q}^{\prime},\lambda_{\Delta^{\prime}})$ for the
outgoing states. The explicit forms of the spinors are given in Ref.
Sekihara:2018tsb .

In this study I multiply a factor $i$ for the incoming $\sigma$ and $\rho$
mesons and accordingly a factor $-i$ for the outgoing $\sigma$ and $\rho$
mesons to obtain a real-valued interaction.

###  C.1 $\pi N\to\pi N$

The $\pi^{i}N\to\pi^{j}N$ interactions, where the isospin indices for mesons
$i,j=1,2,3$ correspond to those in Eq. (30), are given as

$V_{\pi N\to\pi
N}=\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{1a}}+\bar{V}_{\text{1b}}+\bar{V}_{\text{1c}}+\bar{V}_{\text{1d}}+\bar{V}_{\text{1e}})u_{N}(-\bm{q},\lambda_{N}),$
(49)

with

$\bar{V}_{\text{1a}}=(\tau^{j}\tau^{i})\left(\frac{D+F}{2f_{\pi}}\right)^{2}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma_{5}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\pi
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q),$ (50)
$\bar{V}_{\text{1b}}=(\tau^{i}\tau^{j})\left(\frac{D+F}{2f_{\pi}}\right)^{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma_{5}\mathcal{F}(\Lambda_{\pi
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q),$ (51)
$\bar{V}_{\text{1c}}=(T^{i}T^{\dagger j})\left(\frac{f_{\pi
N\Delta}}{m_{\pi}}\right)^{2}k_{\mu}\left[\frac{S_{\Delta}^{\mu\nu}(p-k^{\prime})+S_{\Delta}^{\mu\nu}(p^{\prime}-k)}{2}\right]k_{\nu}^{\prime}\mathcal{F}(\Lambda_{\pi
N\Delta},q^{\prime})\mathcal{F}(\Lambda_{\pi N\Delta},q),$ (52)
$\displaystyle\bar{V}_{\text{1d}}=$
$\displaystyle(i\epsilon_{jik}\tau^{k})\frac{g_{\pi\pi\rho}g_{\rho
NN}}{2}\left[\frac{S_{\rho}(k-k^{\prime})+S_{\rho}(p^{\prime}-p)}{2}\right]$
$\displaystyle\times\left\\{\ooalign{\hfil/\hfil\crcr$k$}+\ooalign{\hfil/\hfil\crcr$k$}^{\prime}+\frac{\kappa_{\rho}}{4m_{N}}[(\ooalign{\hfil/\hfil\crcr$k$}+\ooalign{\hfil/\hfil\crcr$k$}^{\prime})(\ooalign{\hfil/\hfil\crcr$p$}-\ooalign{\hfil/\hfil\crcr$p$}^{\prime})-(\ooalign{\hfil/\hfil\crcr$p$}-\ooalign{\hfil/\hfil\crcr$p$}^{\prime})(\ooalign{\hfil/\hfil\crcr$k$}+\ooalign{\hfil/\hfil\crcr$k$}^{\prime})]\right\\}\mathcal{F}(\Lambda_{\pi\pi\rho},|\bm{q}-\bm{q}^{\prime}|)\mathcal{F}(\Lambda_{\rho
NN},|\bm{q}-\bm{q}^{\prime}|),$ (73)
$\bar{V}_{\text{1e}}=(\delta_{ij})\left(-\frac{g_{\pi\pi\sigma}g_{\sigma
NN}}{m_{\pi}}\right)k^{\mu}k_{\mu}^{\prime}\left[\frac{S_{\sigma}(k-k^{\prime})+S_{\sigma}(p^{\prime}-p)}{2}\right]\mathcal{F}(\Lambda_{\pi\pi\sigma},|\bm{q}-\bm{q}^{\prime}|)\mathcal{F}(\Lambda_{\sigma
NN},|\bm{q}-\bm{q}^{\prime}|).$ (74)

The explicit values of the isospin factors are listed in Table 6. Propagators
$S_{N}$, $S_{\Delta}$, $S_{\rho}$, and $S_{\sigma}$ are respectively:

$\begin{split}&S_{N}(p)=\frac{\ooalign{\hfil/\hfil\crcr$p$}+m_{N}}{(p^{\mu})^{2}-m_{N}^{2}},\quad
S_{\Delta}^{\mu\nu}(p)=\frac{\ooalign{\hfil/\hfil\crcr$p$}+m_{\Delta}}{(p^{\mu})^{2}-m_{\Delta}^{2}}\left[-g^{\mu\nu}+\frac{1}{3}\gamma^{\mu}\gamma^{\nu}+\frac{2p^{\mu}p^{\nu}}{3m_{\Delta}^{2}}-\frac{p^{\mu}\gamma^{\nu}-p^{\nu}\gamma^{\mu}}{3m_{\Delta}}\right],\\\
&S_{\rho}(p)=\frac{1}{(p^{\mu})^{2}-m_{\rho}^{2}},\quad
S_{\sigma}(p)=\frac{1}{(p^{\mu})^{2}-m_{\sigma_{0}}^{2}},\end{split}$ (75)

where the physical masses $m_{N}$, $m_{\Delta}$, $m_{\rho}$ are used for the
$N$, $\Delta$, and $\rho$ exchanges, while the bare mass $m_{\sigma_{0}}$ is
used for the $\sigma$ exchange. The dipole form factor $\mathcal{F}$ was
defined in Eq. (36). I note that I use an idea of the unitary transformation
method Sato:1996gk ; Sato:2009de  to calculate denominators of the
propagators. Owing to the treatment of the energies of the four-momenta in Eq.
(44), these interaction terms are independent of the center-of-mass energy
$E$.

###  C.2 $\pi N\to\eta N$

The $\pi^{i}N\to\eta N$ interactions, where the $\eta N$ state is purely
isospin $I=1/2$, are given as

$V_{\pi N\to\eta
N}=\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{2a}}+\bar{V}_{\text{2b}})u_{N}(-\bm{q},\lambda_{N}),$
(76)

with

$\bar{V}_{\text{2a}}=(\tau^{i})\left[-\frac{(D+F)(D-3F)}{4\sqrt{3}f_{\pi}f_{\eta}}\right]\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma_{5}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\eta
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q),$ (77)
$\bar{V}_{\text{2b}}=(\tau^{i})\left[-\frac{(D+F)(D-3F)}{4\sqrt{3}f_{\pi}f_{\eta}}\right]\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma_{5}\mathcal{F}(\Lambda_{\eta
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q).$ (78)

###  C.3 $\eta N\to\eta N$

The $\eta N\to\eta N$ interactions are given as

$V_{\eta N\to\eta
N}=\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{3a}}+\bar{V}_{\text{3b}})u_{N}(-\bm{q},\lambda_{N}),$
(79)

with

$\bar{V}_{\text{3a}}=\left(\frac{D-3F}{2\sqrt{3}f_{\eta}}\right)^{2}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma_{5}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\eta
NN},q^{\prime})\mathcal{F}(\Lambda_{\eta NN},q),$ (80)
$\bar{V}_{\text{3b}}=\left(\frac{D-3F}{2\sqrt{3}f_{\eta}}\right)^{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma_{5}\mathcal{F}(\Lambda_{\eta
NN},q^{\prime})\mathcal{F}(\Lambda_{\eta NN},q).$ (81)

###  C.4 $\pi N\to\sigma N$

The $\pi^{i}N\to\sigma N$ interactions, where the $\sigma N$ state is purely
$I=1/2$, are given as

$V_{\pi N\to\sigma
N}=\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{4a}}+\bar{V}_{\text{4b}}+\bar{V}_{\text{4c}})u_{N}(-\bm{q},\lambda_{N}),$
(82)

with

$\bar{V}_{\text{4a}}=(\tau^{i})\frac{g_{\sigma
NN}(D+F)}{2f_{\pi}}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\sigma
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q),$ (83)
$\bar{V}_{\text{4b}}=(\tau^{i})\frac{g_{\sigma
NN}(D+F)}{2f_{\pi}}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\mathcal{F}(\Lambda_{\sigma
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q),$ (84)
$\bar{V}_{\text{4c}}=(\tau^{i})\left[-\frac{g_{\pi\pi\sigma}(D+F)}{2m_{\pi}f_{\pi}}\right]k_{\mu}(k-k^{\prime})^{\mu}(\ooalign{\hfil/\hfil\crcr$k$}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime})\gamma_{5}S_{\pi}(p^{\prime}-p)\mathcal{F}(\Lambda_{\pi\pi\sigma},|\bm{q}-\bm{q}^{\prime}|)\mathcal{F}(\Lambda_{\pi
NN},|\bm{q}-\bm{q}^{\prime}|).$ (85)

The $\pi$ propagator is

$S_{\pi}(p)=\frac{1}{(p^{\mu})^{2}-m_{\pi}^{2}},$ (86)

with the physical pion mass $m_{\pi}$. I do not include the pion propagator of
$S_{\pi}(k-k^{\prime})$ for $\bar{V}_{\text{4c}}$ because the $\pi\pi\sigma$
vertex interaction is “real” and hence it causes divergence. Similarly, I will
omit propagators of momenta associated with the “real” vertex interactions.

###  C.5 $\eta N\to\sigma N$

The $\eta N\to\sigma N$ interactions are given as

$V_{\eta N\to\sigma
N}=\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{5a}}+\bar{V}_{\text{5b}})u_{N}(-\bm{q},\lambda_{N}),$
(87)

with

$\bar{V}_{\text{5a}}=-\frac{g_{\sigma
NN}(D-3F)}{2\sqrt{3}f_{\eta}}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\sigma
NN},q^{\prime})\mathcal{F}(\Lambda_{\eta NN},q),$ (88)
$\bar{V}_{\text{5b}}=-\frac{g_{\sigma
NN}(D-3F)}{2\sqrt{3}f_{\eta}}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\mathcal{F}(\Lambda_{\sigma
NN},q^{\prime})\mathcal{F}(\Lambda_{\eta NN},q).$ (89)

###  C.6 $\sigma N\to\sigma N$

The $\sigma N\to\sigma N$ interactions are given as

$V_{\sigma N\to\sigma
N}=\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{6a}}+\bar{V}_{\text{6b}})u_{N}(-\bm{q},\lambda_{N}),$
(90)

with

$\bar{V}_{\text{6a}}=g_{\sigma
NN}^{2}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\mathcal{F}(\Lambda_{\sigma
NN},q^{\prime})\mathcal{F}(\Lambda_{\sigma NN},q),$ (91)
$\bar{V}_{\text{6b}}=g_{\sigma
NN}^{2}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\mathcal{F}(\Lambda_{\sigma
NN},q^{\prime})\mathcal{F}(\Lambda_{\sigma NN},q).$ (92)

###  C.7 $\pi N\to\rho N$

The $\pi^{i}N\to\rho^{j}N$ interactions are given as

$V_{\pi N\to\rho
N}=e_{\rho\mu}^{\ast}(\bm{q}^{\prime},\lambda_{M^{\prime}})\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{7a}}^{\mu}+\bar{V}_{\text{7b}}^{\mu}+\bar{V}_{\text{7c}}^{\mu}+\bar{V}_{\text{7d}}^{\mu})u_{N}(-\bm{q},\lambda_{N}),$
(93)

with

$\bar{V}_{\text{7a}}^{\mu}=(\tau^{j}\tau^{i})\frac{g_{\rho
NN}(D+F)}{4f_{\pi}}\left[\gamma^{\mu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\mu}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma^{\mu})\right]\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q),$ (94)
$\bar{V}_{\text{7b}}^{\mu}=(\tau^{i}\tau^{j})\frac{g_{\rho
NN}(D+F)}{4f_{\pi}}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\left[\gamma^{\mu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\mu}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma^{\mu})\right]\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q),$ (95)
$\bar{V}_{\text{7c}}^{\mu}=(i\epsilon_{jik}\tau^{k})\frac{g_{\pi\pi\rho}(D+F)}{2f_{\pi}}(2k^{\mu}-k^{\prime\mu})(\ooalign{\hfil/\hfil\crcr$k$}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime})\gamma_{5}S_{\pi}(p^{\prime}-p)\mathcal{F}(\Lambda_{\pi\pi\rho},|\bm{q}-\bm{q}^{\prime}|)\mathcal{F}(\Lambda_{\pi
NN},|\bm{q}-\bm{q}^{\prime}|),$ (96)
$\bar{V}_{\text{7d}}^{\mu}=(i\epsilon_{jik}\tau^{k})\left[-\frac{g_{\rho
NN}(D+F)}{2f_{\pi}}\right]\gamma^{\mu}\gamma_{5}\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q).$ (97)

###  C.8 $\eta N\to\rho N$

The $\eta N\to\rho^{j}N$ interactions are given as

$V_{\eta N\to\rho
N}=e_{\rho\mu}^{\ast}(\bm{q}^{\prime},\lambda_{M^{\prime}})\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{8a}}^{\mu}+\bar{V}_{\text{8b}}^{\mu})u_{N}(-\bm{q},\lambda_{N}),$
(98)

with

$\bar{V}_{\text{8a}}^{\mu}=(\tau^{j})\left[-\frac{g_{\rho
NN}(D-3F)}{4\sqrt{3}f_{\eta}}\right]\left[\gamma^{\mu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\mu}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma^{\mu})\right]\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\eta NN},q),$ (99)
$\bar{V}_{\text{8b}}^{\mu}=(\tau^{j})\left[-\frac{g_{\rho
NN}(D-3F)}{4\sqrt{3}f_{\eta}}\right]\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\left[\gamma^{\mu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\mu}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma^{\mu})\right]\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\eta NN},q).$ (100)

###  C.9 $\sigma N\to\rho N$

The $\sigma N\to\rho^{j}N$ interactions are given as

$V_{\sigma N\to\rho
N}=e_{\rho\mu}^{\ast}(\bm{q}^{\prime},\lambda_{M^{\prime}})\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{9a}}^{\mu}+\bar{V}_{\text{9b}}^{\mu})u_{N}(-\bm{q},\lambda_{N}),$
(101)

with

$\bar{V}_{\text{9a}}^{\mu}=(\tau^{j})\frac{g_{\rho NN}g_{\sigma
NN}}{2}\left[\gamma^{\mu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\mu}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma^{\mu})\right]\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\sigma NN},q),$ (102)
$\bar{V}_{\text{9b}}^{\mu}=(\tau^{j})\frac{g_{\rho NN}g_{\sigma
NN}}{2}\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\left[\gamma^{\mu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\mu}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma^{\mu})\right]\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\sigma NN},q).$ (103)

###  C.10 $\rho N\to\rho N$

The $\rho^{i}N\to\rho^{j}N$ interactions are given as

$V_{\rho N\to\rho
N}=e_{\rho\mu}^{\ast}(\bm{q}^{\prime},\lambda_{M^{\prime}})\bar{u}_{N}(-\bm{q}^{\prime},\lambda_{N^{\prime}})(\bar{V}_{\text{10a}}^{\mu\nu}+\bar{V}_{\text{10b}}^{\mu\nu}+\bar{V}_{\text{10c}}^{\mu\nu})e_{\rho\nu}(\bm{q},\lambda_{M})u_{N}(-\bm{q},\lambda_{N}),$
(104)

with

$\displaystyle\bar{V}_{\text{10a}}^{\mu\nu}=$
$\displaystyle(\tau^{j}\tau^{i})\frac{g_{\rho
NN}^{2}}{4}\left[\gamma^{\mu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\mu}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma^{\mu})\right]\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\left[\gamma^{\nu}-\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\nu}\ooalign{\hfil/\hfil\crcr$k$}-\ooalign{\hfil/\hfil\crcr$k$}\gamma^{\nu})\right]$
(113) $\displaystyle\times\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\rho NN},q),$ (114)
$\displaystyle\bar{V}_{\text{10b}}^{\mu\nu}=$
$\displaystyle(\tau^{i}\tau^{j})\frac{g_{\rho
NN}^{2}}{4}\left[\gamma^{\nu}-\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\nu}\ooalign{\hfil/\hfil\crcr$k$}-\ooalign{\hfil/\hfil\crcr$k$}\gamma^{\nu})\right]\frac{S_{N}(p-k^{\prime})+S_{N}(p^{\prime}-k)}{2}\left[\gamma^{\mu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\mu}\ooalign{\hfil/\hfil\crcr$k$}^{\prime}-\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma^{\mu})\right]$
(123) $\displaystyle\times\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\rho NN},q),$ (124)
$\bar{V}_{\text{10c}}^{\mu\nu}=(i\epsilon_{jik}\tau^{k})\frac{g_{\rho
NN}^{2}\kappa_{\rho}}{8m_{N}}(\gamma^{\nu}\gamma^{\mu}-\gamma^{\mu}\gamma^{\nu})\mathcal{F}(\Lambda_{\rho
NN},q^{\prime})\mathcal{F}(\Lambda_{\rho NN},q).$ (125)

###  C.11 $\pi N\to\pi\Delta$

The $\pi^{i}N\to\pi^{j}\Delta$ interaction is given as

$V_{\pi
N\to\pi\Delta}=\bar{u}_{\Delta\mu}(-\bm{q}^{\prime},\lambda_{\Delta^{\prime}})(\bar{V}_{\text{11a}}^{\mu}+\bar{V}_{\text{11b}}^{\mu})u_{N}(-\bm{q},\lambda_{N}),$
(126)

with

$\bar{V}_{\text{11a}}^{\mu}=(T^{\dagger j}\tau^{i})\frac{f_{\pi
N\Delta}(D+F)}{2m_{\pi}f_{\pi}}k^{\prime\mu}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\pi
N\Delta},q^{\prime})\mathcal{F}(\Lambda_{\pi NN},q),$ (127)
$\bar{V}_{\text{11b}}^{\mu}=(T^{\dagger i}\tau^{j})\frac{f_{\pi
N\Delta}(D+F)}{2m_{\pi}f_{\pi}}k^{\mu}S_{N}(p-k^{\prime})\ooalign{\hfil/\hfil\crcr$k$}^{\prime}\gamma_{5}\mathcal{F}(\Lambda_{\pi
NN},q^{\prime})\mathcal{F}(\Lambda_{\pi N\Delta},q).$ (128)

###  C.12 $\eta N\to\pi\Delta$

The $\eta N\to\pi^{j}\Delta$ interaction is given as

$V_{\eta
N\to\pi\Delta}=\bar{u}_{\Delta\mu}(-\bm{q}^{\prime},\lambda_{\Delta^{\prime}})\bar{V}_{\text{12}}^{\mu}u_{N}(-\bm{q},\lambda_{N}),$
(129)

with

$\bar{V}_{\text{12}}^{\mu}=(T^{\dagger j})\left[-\frac{f_{\pi
N\Delta}(D-3F)}{2\sqrt{3}m_{\pi}f_{\eta}}\right]k^{\prime\mu}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\ooalign{\hfil/\hfil\crcr$k$}\gamma_{5}\mathcal{F}(\Lambda_{\pi
N\Delta},q^{\prime})\mathcal{F}(\Lambda_{\eta NN},q).$ (130)

###  C.13 $\sigma N\to\pi\Delta$

The $\sigma N\to\pi^{j}\Delta$ interaction is given as

$V_{\sigma
N\to\pi\Delta}=\bar{u}_{\Delta\mu}(-\bm{q}^{\prime},\lambda_{\Delta^{\prime}})\bar{V}_{\text{13}}^{\mu}u_{N}(-\bm{q},\lambda_{N}),$
(131)

with

$\bar{V}_{\text{13}}^{\mu}=(T^{\dagger j})\frac{g_{\sigma NN}f_{\pi
N\Delta}}{m_{\pi}}k^{\prime\mu}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\mathcal{F}(\Lambda_{\pi
N\Delta},q^{\prime})\mathcal{F}(\Lambda_{\sigma NN},q).$ (132)

###  C.14 $\rho N\to\pi\Delta$

The $\rho^{i}N\to\pi^{j}\Delta$ interaction is given as

$V_{\rho
N\to\pi\Delta}=\bar{u}_{\Delta\mu}(-\bm{q}^{\prime},\lambda_{\Delta^{\prime}})\bar{V}_{\text{14}}^{\mu\nu}e_{\rho\nu}(\bm{q},\lambda_{M})u_{N}(-\bm{q},\lambda_{N}),$
(133)

with

$\bar{V}_{\text{14}}^{\mu\nu}=(T^{\dagger j}\tau^{i})\frac{g_{\rho NN}f_{\pi
N\Delta}}{2m_{\pi}}k^{\prime\mu}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}\left[\gamma^{\nu}-\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\nu}\ooalign{\hfil/\hfil\crcr$k$}-\ooalign{\hfil/\hfil\crcr$k$}\gamma^{\nu})\right]\mathcal{F}(\Lambda_{\pi
N\Delta},q^{\prime})\mathcal{F}(\Lambda_{\rho NN},q).$ (134)

###  C.15 $\pi\Delta\to\pi\Delta$

The $\pi^{i}\Delta\to\pi^{j}\Delta$ interaction is given as

$V_{\pi\Delta\to\pi\Delta}=\bar{u}_{\Delta\mu}(-\bm{q}^{\prime},\lambda_{\Delta^{\prime}})\bar{V}_{\text{15}}^{\mu\nu}u_{\Delta\nu}(-\bm{q},\lambda_{\Delta}),$
(135)

with

$\bar{V}_{\text{15}}^{\mu\nu}=(T^{\dagger j}T^{i})\left(\frac{f_{\pi
N\Delta}}{m_{\pi}}\right)^{2}k^{\prime\mu}\frac{S_{N}(p+k)+S_{N}(p^{\prime}+k^{\prime})}{2}k^{\nu}\mathcal{F}(\Lambda_{\pi
N\Delta},q^{\prime})\mathcal{F}(\Lambda_{\pi N\Delta},q).$ (136)

###  C.16 $s$-channel exchange of bare $N^{\ast}$ and $\Delta^{\ast}$ states

To take into account the bare $N^{\ast}$ and $\Delta^{\ast}$ states for the
meson–baryon scattering in $s$ channel, I add

$V_{jk}^{\rm
bare}(E;q^{\prime},q)=\frac{g_{j}g_{k}}{2m_{\pi}(E-M_{0})}\left(\frac{q^{\prime}}{m_{\pi}}\right)^{L^{\prime}}\left(\frac{q}{m_{\pi}}\right)^{L}\mathcal{F}(\Lambda,q^{\prime})\mathcal{F}(\Lambda,q),$
(137)

to the corresponding partial-wave components $V_{\alpha,jk}$ in Eq. (1). Here,
$M_{0}$ is the bare mass of the $N^{\ast}$ and $\Delta^{\ast}$ states, $g_{j}$
is the coupling constant for the bare state to the $j$th meson–baryon channel,
and $\Lambda$ is the cutoff. I note that only this bare-state contribution
depends on the center-of-mass energy $E$ among the meson–baryon interaction
terms.

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