for $\theta_{1},\theta_{2}\in\Theta$. the corresponding norm for
$\theta\in\Theta$ is

$\left\|\theta-\theta^{*}\right\|^{2}\coloneqq-\left.\frac{\partial^{2}}{\partial\tau^{2}}E\left[\ell\left(W,\theta^{*}+\tau\left(\theta-\theta^{*}\right)\right)\right]\right|_{\tau=0}.$
(15)

######  Assumption E8.

There is $C_{1}>0$ such that for all small $\varepsilon>0$

$\sup_{\left\\{\theta\in\Theta_{n}:\left\|\theta-\theta^{*}\right\|\leqslant\varepsilon\right\\}}\operatorname{Var}\left(\ell\left(W,\theta\right)-\ell\left(W,\theta^{*}\right)\right)\leqslant
C_{1}\varepsilon^{2}$

######  Assumption E9.

For any $\delta>0$, there exists a constant $s\in(0,2)$ such that

$\sup_{\left\\{\theta\in\Theta_{n}:\left\|\theta-\theta^{*}\right\|\leqslant\delta\right\\}}\left|\ell\left(W,\theta\right)-\ell\left(W,\theta^{*}\right)\right|\leqslant\delta^{s}U\left(W\right)$

with $E\left(\left[U\left(W\right)\right]^{\gamma}\right)\leqslant C_{2}$ for
some $\gamma\geqslant 2$.

The following theorem is now a consequence of Theorem 3.2 in Chen (2007) or
Theorem 1 in Shen and Wong (1994).

######  Theorem 5.

Let $(Y_{i,t},D_{i,t},X_{i,t}\colon t=1,\ldots,{T})_{i=1}^{n}$ be i.i.d. data
where $T\geq{2p}+1$ and Assumptions KL1-KL5 and Assumptions E1-E9 hold. Then
$\|\hat{\theta}-\theta^{*}\|=o_{p}(n^{-1/4})$.

Given the preceding result, we focus on a a shrinking neighborhood of
$\theta^{*}$. Let

$\mathcal{N}_{0}\coloneqq\left\\{\theta\in\Theta\colon\|\theta-\theta^{*}\|=o(n^{-1/4}),\leavevmode\nobreak\
d(\theta,\theta^{*})=o(1)\right\\},$

and $\mathcal{N}_{n}\coloneqq\mathcal{N}_{0}\cap\Theta_{n}$. Define
$\theta_{n}^{*}={\rm
argmin}_{\theta\in\mathcal{N}_{n}}{\|\theta-\theta^{*}\|}$. Let $\mathcal{V}$
denote the closed (under $\|\cdot\|$) linear span of $\mathcal{N}_{0}$
centered at $\theta^{*}$, and define $\mathcal{V}_{n}$ as the analogous
closure of $\mathcal{N}_{n}$.

Then we define a linear approximation to $\ell(W,\theta)-\ell(W,\theta^{*})$
as the directional derivative of $\ell$ at $(W,\theta^{*})$ in the direction
$(\theta-\theta^{*})$:

$\frac{\partial\ell\left(W,\theta^{*}\right)}{\partial\theta}[\theta-\theta^{*}]\coloneqq\left.\frac{\partial\ell\left(W,\theta^{*}+\tau(\theta-\theta^{*})\right)}{\partial\tau}\right|_{\tau=0}.$

Likewise, let $\frac{\partial
f\left(\theta^{*}\right)}{\partial\theta}[v]=\left.\frac{\partial
f\left(\theta^{*}+\tau v\right)}{\partial\tau}\right|_{\tau=0}$ for any
$v\in\mathcal{V}$.

######  Assumption E10.

Let $\mathcal{T}$ be an epsilon ball about $0\in\mathbb{R}$. (i) For all
$\theta\in\mathcal{N}_{0}$ and $W$, the derivative
$\partial\ell\left(W,\theta^{*}+\tau(\theta-\theta^{*})\right)/\partial\tau$
exists for all $\tau\in\mathcal{T}$; (ii) for all $\theta\in\mathcal{N}_{0}$,
$\text{E}\left[\ell\left(W,\theta^{*}+\tau\left(\theta-\theta^{*}\right)\right)\right]$
is finite for each $\tau\in\mathcal{T}$; (iii) for all
$\theta\in\mathcal{N}_{0}$,
$\text{E}\left[\sup_{\tau\in\mathcal{T}}\left|\frac{\partial}{\partial\tau}\ell\left(W,\theta^{*}+\tau\left[\theta-\theta^{*}\right]\right)\right|\right]$
$<\infty$.

Assumption E10 provides sufficient conditions for the set $\mathcal{V}$ to be
a Hilbert space under $\langle\cdot,\cdot\rangle$.181818See Chen et al. (2014,
p. 642). Define $v_{n}^{*}$ to be the Riesz representer of
$\frac{\partial{f}(\theta^{*})}{\partial\theta}[\cdot]$ on $\mathcal{V}_{n}$,
which exists under Assumption E11.

######  Assumption E11.

(i) $v\mapsto\frac{\partial{f}(\theta^{*})}{\partial\theta}[v]$ is a linear
functional. (ii) If $\lim_{n\rightarrow\infty}\|v_{n}^{*}\|$ is finite then
$\|v_{n}^{*}-v^{*}\|\times\|\theta_{n}^{*}-\theta^{*}\|=o(n^{-1/2})$ where
$v^{*}$ is the limit of $v_{n}^{*}$. Otherwise
${\left|\frac{\partial{f}(\theta^{*})}{\partial\theta}[\theta_{n}^{*}-\theta^{*}]\right|}/{\|v_{n}^{*}\|}=o(n^{-1/2})$.
(iii)
$\sup_{\theta\in\mathcal{N}_{0}}\frac{\left|f(\theta)-f(\theta^{*})-\frac{\partial{f}(\theta^{*})}{\partial\theta}[\theta-\theta^{*}]\right|}{\|v_{n}^{*}\|}=o(n^{-1/2})$.

Assumption E11 imposes some restrictions on the functional of interest
$\theta\mapsto f(\theta)$. Part (i) imposes that the directional derivative is
a linear functional, a mild condition that is satisfied by our examples in
Section 4. Part (ii) is a restriction on the growth rate of the dimension of
the sieve space. Part (iii) restricts the linear approximation error of
$f(\cdot)$ in a neighborhood of $\theta^{*}$, for which sufficient conditions
could be stated in terms of the smoothness of $f(\cdot)$ and the growth rate
of the dimension of the sieve space. See Chen et al. (2014) for further
discussion.

Let $u_{n}^{*}\coloneqq\frac{v_{n}^{*}}{\left\|v_{n}^{*}\right\|}$,
$\varepsilon_{n}=o\left(n^{-1/2}\right)$ and $\mu_{n}\\{g(\bm{W})\\}\coloneqq
n^{-1}\sum_{i=1}^{n}\left[g\left(W_{i}\right)-\mathrm{E}[g\left(W_{i}\right)]\right]$
denote the centered empirical process indexed by the function $g$.

######  Assumption E12.

$\mu_{n}\\{\frac{\partial\ell\left(\bm{W},\theta^{*}\right)}{\partial\theta}[v]\\}$
is linear in $v\in\mathcal{V}$.

$\sup_{\theta\in\mathcal{N}_{n}}\mu_{n}\left\\{\ell\left(\bm{W},\theta\pm\varepsilon_{n}u_{n}^{*}\right)-\ell(\bm{W},\theta)-\frac{\partial\ell\left(\bm{W},\theta^{*}\right)}{\partial\theta}\left[\pm\varepsilon_{n}u_{n}^{*}\right]\right\\}=O_{p}\left(\varepsilon_{n}^{2}\right).$

For some positive sequence $\eta_{n}\rightarrow 0$,

$\sup_{\theta\in\mathcal{N}_{n}}\left|E\left[\ell(W,\theta)-\ell\left(W,\theta\pm\varepsilon_{n}u_{n}^{*}\right)\right]-\frac{\left\|\theta\pm\varepsilon_{n}u_{n}^{*}-\theta^{*}\right\|^{2}-\left\|\theta-\theta^{*}\right\|^{2}}{2}\left(1+O\left(\eta_{n}\right)\right)\right|=O\left(\varepsilon_{n}^{2}\right).$

######  Assumption E13.

$\sqrt{n}\mu_{n}\left\\{\frac{\partial\ell\left(\bm{W},\theta^{*}\right)}{\partial\theta}\left[u_{n}^{*}\right]\right\\}\rightarrow_{d}N(0,1)$

Theorem 4 is a direct application of Lemma 2.1 in Chen and Liao (2014) so its
proof is omitted.

###  B.4 Appendix to implementation and Monte Carlo simulations section

####  B.4.1 Implicit differentiation

For implementing the estimator, it can be useful to input the gradient of the
objective function. In this section, we show how our profiling approach and
choice of sieve space simplify this task. Recall that in Section 5.1, the
profile log likelihood function with our proposed sieve space for
$F_{X_{k}^{*}}$ is

$\ell^{p}(\theta^{c})\coloneqq\sum_{i=1}^{n}\log\sum_{s=1}^{q_{n}}\omega_{s}(\theta^{c})\leavevmode\nobreak\
\ell^{c}(w_{i},\bar{x}_{n,s}^{*};\theta^{c}),$

where
$\omega(\theta^{c})=\operatorname*{arg\,max}_{\omega\in\Delta(q_{n})}\sum_{i=1}^{n}\log\sum_{s=1}^{q_{n}}\omega_{s}\leavevmode\nobreak\
\ell^{c}(w_{i},\bar{x}_{n,s}^{*};\theta^{c})$ is the solution to the inner
problem for a fixed $\theta^{c}$. Given an analytical expression for
$\ell^{c}(w_{i},x_{k}^{*};\theta^{c})$191919Given the analytical expression
for $\ell^{c}$, we use the software Google JAX to compute the derivative via
autodifferention., the challenge of computing the gradient of
$\ell^{p}(\theta^{c})$ reduces to finding the Jacobian of $\omega(\theta^{c})$
(i.e., $\frac{\partial}{\partial(\theta^{c})^{\intercal}}\omega(\theta^{c})$),
which is defined implicitly by the Karush-Kuhn-Tucker (KKT) conditions of the
inner optimization problem. In the following, we derive an analytical
expression for
$\frac{\partial}{\partial(\theta^{c})^{\intercal}}\omega(\theta^{c})$ in terms
of $\ell^{c}(w_{i},x_{k}^{*};\theta^{c})$,
$\frac{\partial}{\partial\theta^{c}}\ell^{c}(w_{i},x_{k}^{*};\theta^{c})$, and
$\omega(\theta^{c})$.

Proposition 3.3 in Kim et al. (2020) shows that $\omega(\theta^{c})$ can be
equivalently expressed as $\operatorname*{arg\,max}_{\omega\geq
0}\\{\sum_{i=1}^{n}\log\sum_{s=1}^{q_{n}}\omega_{s}\leavevmode\nobreak\
\ell^{c}(w_{i},\bar{x}_{n,s}^{*};\theta^{c})+\sum_{s=1}^{q_{n}}\omega_{s}\\}$,
where $\omega\geq 0$ means $\omega_{s}\geq 0$ for all $s=1,\dots,q_{n}$.
Letting $\lambda\in\mathbb{R}^{q_{n}}$ be the dual parameter corresponding to
the constraint $\omega\geq 0$, and
$\ell_{i}^{c}(\theta^{c})\coloneqq(\ell^{c}(w_{i},\bar{x}_{n,s}^{*};\theta^{c})\colon
s=1,\ldots,q_{n})$, the equality constraints in the KKT conditions of this
problem are,

$0_{2q_{n}\times
1}=\begin{pmatrix}\sum_{i=1}^{n}\frac{1}{\omega^{\intercal}\ell_{i}^{c}(\theta^{c})}\ell_{i}^{c}(\theta^{c})+1_{q_{n}}+\lambda\\\
\leavevmode\nobreak\ \lambda\circ\omega\end{pmatrix},$

where $\circ$ is the Hadamard product. By definition, these constraints are
identically zero for all $\theta^{c}$, so under an implicit function theorem,
$\frac{d}{d(\theta^{c})^{\intercal}}\omega(\theta^{c})=-G_{1}(\theta^{c})^{-1}G_{2}(\theta^{c})$,202020$G_{1}$
and $G_{2}$ are the partial derivatives of right hand side of the previous
equation with respect to $(\omega,\lambda)$ and $\theta^{c}$ respectively,
evaluated at $\omega(\theta^{c})$ and $\lambda(\theta^{c})$. where

$G_{1}(\theta^{c})=\begin{pmatrix}\sum_{i=1}^{n}\frac{1}{(\omega(\theta^{c})^{\intercal}\ell_{i}^{c}(\theta^{c}))^{2}}\ell_{i}^{c}(\theta^{c})(\ell_{i}^{c}(\theta^{c})^{\intercal}&I_{q_{n}\times
q_{n}}\\\
\text{diag}(\lambda(\theta^{c}))&\text{diag}(\omega(\theta^{c}))\end{pmatrix},$

and

$\displaystyle G_{2}(\theta^{c})=$
$\displaystyle\begin{pmatrix}\sum_{i=1}^{n}\left(\frac{\frac{\partial}{\partial(\theta^{c})^{\intercal}}\ell_{i}^{c}(\theta^{c})}{\omega(\theta^{c})^{\intercal}\ell_{i}^{c}(\theta^{c})}-\frac{\ell_{i}^{c}(\theta^{c})\omega(\theta^{c})^{\intercal}\frac{\partial}{\partial(\theta^{c})^{\intercal}}\ell_{i}^{c}(\theta^{c})}{(\omega(\theta^{c})^{\intercal}\ell_{i}^{c}(\theta^{c}))^{2}}\right)\\\
\leavevmode\nobreak\ 0_{q_{n}\times{\rm dim}(\theta^{c})}\end{pmatrix}$

Finally, note that the KKT conditions imply that
$\lambda(\theta^{c})=-1_{q_{n}}-\sum_{i=1}^{n}\frac{\ell_{i}^{c}(\theta^{c})}{\omega(\theta^{c})^{\intercal}\ell_{i}^{c}(\theta^{c})}$.

####  B.4.2 Details on DGP

This section gives further details on the DGP used for Monte Carlo simulations
discussed in Section 5.2. The values of the finite parameters used in the DGP
are given in the table below.

$\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}\alpha_{1,1}=0$ | $\gamma^{(1)}_{1,1}=-0.5$ | $\gamma^{(2)}_{1,1}=-0.58$ | $\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}\lambda^{u}_{1,1}=1$ | $\lambda^{k}_{1,1}=0.3$
---|---|---|---|---
$\alpha_{2,1}=0.1$ | $\gamma^{(1)}_{2,1}=-0.8$ | $\gamma^{(2)}_{2,1}=-0.83$ | $\lambda^{u}_{2,1}=1.05$ | $\lambda^{k}_{2,1}=0.35$
$\alpha_{3,1}=0.2$ | $\gamma^{(1)}_{3,1}=0.12$ | $\gamma^{(2)}_{3,1}=-0.83$ | $\lambda^{u}_{3,1}=1.01$ | $\lambda^{k}_{3,1}=0.33$
$\sigma^{2}_{1}=0.5$ |  |  |  |
$\alpha_{1,2}=-0.1$ | $\gamma^{(1)}_{1,2}=0.13$ | $\gamma^{(2)}_{1,2}=0.71$ | $\lambda^{u}_{1,2}=0.4$ | $\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}\lambda^{k}_{1,2}=1$
$\alpha_{2,2}=-0.22$ | $\gamma^{(1)}_{2,2}=0.89$ | $\gamma^{(2)}_{2,2}=-0.36$ | $\lambda^{u}_{2,2}=0.36$ | $\lambda^{k}_{2,2}=1.05$
$\alpha_{3,2}=-0.33$ | $\gamma^{(1)}_{3,2}=0.32$ | $\gamma^{(2)}_{3,2}=-0.36$ | $\lambda^{u}_{3,2}=0.44$ | $\lambda^{k}_{3,2}=1.02$
$\sigma^{2}_{2}=0.7$ |  |  |  |
$\sigma^{2}_{u}=1.5$ | $\rho=2.0$ | $\kappa=0.5$ |  |
Table 4: Finite parameter values

####  B.4.3 DGP with risk aversion

In this section, we present results from an alternative DGP in which agents
maximize their expected utility in each period which incorporates risk
aversion, through constant relative risk aversion (CRRA) preferences, and
subjective (possibly biased) beliefs. The expected utility that individual $i$
derives from choice $d$ in period $t$ is given by:

$\displaystyle
v_{i,t}(d):=\mathcal{E}_{i,t}\left(\frac{Y_{i,t}(d)^{1-\chi}}{1-\chi}\right)+\eta_{i,t}(d)$

where $\mathcal{E}_{i,t}$ denotes the expectation under individual $i$’s
subjective beliefs over $X^{*}_{u,i}$, given the information up to period $t$.
$\eta_{i,t}(d)$ are independent preference shocks, which are supposed to
follow an Extreme Value Type 1 distribution.

We assume that individuals’ subjective beliefs over $X^{*}_{u,i}$ in time $t$
are distributed $N(\mu_{i,t}+\delta X^{*}_{k,i},\Sigma_{i,t})$ where
$\mu_{i,t},\Sigma_{i,t}$ are the correct posterior mean and variance of
$X^{*}_{u,i}$ given the information up to period $t-1$. This subjective belief
process allows agents to have biased beliefs that can be correlated with the
known part of their unobserved heterogeneity, $X^{*}_{k,i}$.

Under this specification, the expected utility has the following analytical
form,

$v_{i,t}(d)=\frac{\exp\bigg{(}\mu_{i,t}(d)(1-\chi)+\frac{1}{2}\sigma_{i,t}(d)(1-\chi)^{2}\bigg{)}}{1-\chi}+\eta_{i,t}(d)$
(16)

where $\mu_{i,t}(d)$ ($\sigma_{i,t}(d)$) denote the subjective mean (variance)
of $\log(Y_{i,t}(d))$.

A naive approach to estimating ${v}_{i,t}(d)$ nonparametrically would be to
use a tensor product of polynomials $(X^{*}_{k},X,Y^{t-1},D^{t-1})$ as the
sieve space. That is, for a univariate random variable $X$, let
$\mathcal{P}_{q}(X)=\text{sp}(\\{1,X,\ldots,X^{q}\\})$. Assume $D_{t}$ is
binary, and let $\delta_{t}=1(D_{t}=1)$, then the sieve space is,

$\mathcal{P}_{q}(X^{*}_{k})\otimes\mathcal{P}_{q}(X_{1})\otimes\cdots\otimes\mathcal{P}_{q}(Y_{1})\otimes\mathcal{P}_{q}(\delta_{1})\otimes\cdots\otimes\mathcal{P}_{q}(Y_{t-1})\otimes\mathcal{P}_{q}(\delta_{t-1}).$

For an $q$-order polynomial, the number of terms would be
$(q+1)^{3}+(q+1)^{5}+(q+1)^{7}$, which grows very quickly in practical terms.

The alternative approach we consider here is to use the following
approximation

${v}_{i,t}(d)=\varphi\left(\sum_{h\in\mathcal{D}^{t-1}}1(D^{t-1}=h)(\pi_{t,h,d,0}+\pi_{t,h,d,1}^{\intercal}X+\pi_{t,h,d,2}X^{*}_{k}+\pi_{t,h,d,3}^{\intercal}Y_{i}^{t-1})\right)$

for some unknown function $\varphi$. Since the argument of $\varphi$ is
scalar-valued, this means that the nonparametric estimation problem is greatly
simplified to estimating a scalar-valued function. For this we use the sieve
space of polynomials, with the order growing at the rate of $n^{1/3}$ with 3
terms with $n=500$ and $6$ terms for $n=4,000$. Our choice of approximation is
motivated by the fact that under Lemma 1 and Equation 16, there is a set of
$\pi$ parameters such that this equality holds, with
$\varphi(\cdot)=\frac{1}{1-\chi}\exp(\cdot)$.

The finite parameters are the same as in our baseline simulations considered
in Section 5.2, with the added risk aversion parameter $\chi$, which we set to
$1.5$. $X^{*}$ and $X$ are generated from the same distributions as in the DGP
considered in Section 5.2.

With the additional $\pi$ parameters to estimate, the $\theta^{c}$ has a total
of 103 parameters. Given this large number of parameters to estimate, we
expect $n=250$ to be too small a sample size to perform well, and begin the
Monte Carlo simulations with a sample size of $n=500$. The large number of
parameters to estimate in $\theta^{c}$ results in longer but still manageable
computational times, which are reported in Table 5.

| $n=500$ | $n=1{,}000$ | $n=2{,}000$ | $n=4{,}000$
---|---|---|---|---
Time (minutes) | 3 | 7.5 | 19.5 | 56
Table 5: Time to compute the estimator: DGP with risk aversion. Computational
times were obtained using an Intel Core i9-12900K CPU, and are computed as the
average over 200 simulations.

The results of the Monte Carlo simulations are presented in Table 6 and Figure
2. Despite the increased complexity of the model, our estimation procedure
exhibits similar finite sample performance to the DGP considered in Section
5.2.

| n = 500 | n = 1,000 | n = 2,000 | n = 4,000
---|---|---|---|---
| Bias2 | Var | Bias2 | Var | Bias2 | Var | Bias2 | Var
$\alpha_{1,2}$ | 66.15 | 38.25 | 18.40 | 20.20 | 3.97 | 12.19 | 0.05 | 7.69
$\alpha_{2,1}$ | 0.17 | 28.07 | 0.05 | 12.99 | 0.08 | 5.50 | 0.05 | 2.10
$\alpha_{2,2}$ | 69.24 | 42.16 | 18.40 | 23.49 | 3.25 | 14.33 | 0.00 | 9.20
$\alpha_{3,1}$ | 1.29 | 24.63 | 0.07 | 9.98 | 0.00 | 4.73 | 0.00 | 1.83
$\alpha_{3,2}$ | 68.62 | 42.86 | 23.69 | 21.80 | 3.41 | 13.62 | 0.01 | 8.28
$\gamma_{1,1}^{(1)}$ | 0.08 | 6.61 | 0.05 | 3.30 | 0.01 | 1.72 | 0.02 | 0.95
$\gamma_{1,2}^{(1)}$ | 0.12 | 8.29 | 0.09 | 3.55 | 0.02 | 1.64 | 0.01 | 0.78
$\gamma_{2,1}^{(1)}$ | 0.03 | 7.69 | 0.08 | 3.81 | 0.04 | 2.11 | 0.02 | 1.08
$\gamma_{2,2}^{(1)}$ | 0.21 | 9.49 | 0.25 | 4.13 | 0.06 | 2.18 | 0.03 | 0.79
$\gamma_{3,1}^{(1)}$ | 0.14 | 5.52 | 0.03 | 2.52 | 0.01 | 1.38 | 0.02 | 0.72
$\gamma_{3,2}^{(1)}$ | 0.08 | 9.43 | 0.11 | 4.03 | 0.03 | 1.84 | 0.02 | 0.83
$\gamma_{1,1}^{(2)}$ | 1.65 | 35.50 | 0.00 | 12.36 | 0.22 | 5.58 | 0.01 | 2.75
$\gamma_{1,2}^{(2)}$ | 0.09 | 28.70 | 0.09 | 11.52 | 0.16 | 6.99 | 0.06 | 3.19
$\gamma_{2,1}^{(2)}$ | 1.47 | 31.77 | 0.00 | 12.37 | 0.06 | 5.50 | 0.03 | 2.79
$\gamma_{2,2}^{(2)}$ | 0.08 | 28.45 | 0.11 | 13.67 | 0.23 | 7.50 | 0.11 | 3.25
$\gamma_{3,1}^{(2)}$ | 0.73 | 25.40 | 0.02 | 11.07 | 0.13 | 4.71 | 0.01 | 2.65
$\gamma_{3,2}^{(2)}$ | 0.17 | 29.53 | 0.00 | 14.60 | 0.16 | 7.89 | 0.09 | 3.35
$\lambda_{1,1}^{k}$ | 0.34 | 20.38 | 1.18 | 6.84 | 0.02 | 4.11 | 0.01 | 1.71
$\lambda_{2,1}^{k}$ | 0.18 | 21.01 | 2.41 | 9.54 | 0.42 | 5.21 | 0.09 | 1.91
$\lambda_{2,2}^{k}$ | 0.18 | 9.49 | 0.00 | 3.31 | 0.01 | 1.60 | 0.01 | 0.80
$\lambda_{3,1}^{k}$ | 0.45 | 17.32 | 1.53 | 8.13 | 0.15 | 4.25 | 0.01 | 1.53
$\lambda_{3,2}^{k}$ | 0.03 | 10.43 | 0.21 | 3.97 | 0.01 | 2.22 | 0.01 | 1.10
$\lambda_{1,2}^{u}$ | 0.11 | 6.31 | 0.03 | 2.65 | 0.00 | 1.23 | 0.00 | 0.52
$\lambda_{2,1}^{u}$ | 0.05 | 3.54 | 0.04 | 1.41 | 0.01 | 0.78 | 0.01 | 0.43
$\lambda_{2,2}^{u}$ | 0.09 | 8.36 | 0.01 | 3.61 | 0.00 | 1.65 | 0.01 | 0.69
$\lambda_{3,1}^{u}$ | 0.06 | 3.89 | 0.02 | 1.44 | 0.01 | 0.60 | 0.00 | 0.33
$\lambda_{3,2}^{u}$ | 0.35 | 9.16 | 0.15 | 4.34 | 0.00 | 1.90 | 0.01 | 0.87
$\sigma^{2}(1)$ | 0.15 | 0.68 | 0.01 | 0.36 | 0.01 | 0.17 | 0.00 | 0.07
$\sigma^{2}(2)$ | 0.06 | 0.24 | 0.00 | 0.15 | 0.00 | 0.07 | 0.00 | 0.03
$\sigma^{2}_{u}$ | 1.38 | 19.53 | 0.02 | 6.64 | 0.01 | 3.74 | 0.00 | 1.83
Table 6: Simulation results for estimation of finite dimensional parameters.
‘Bias2’ and ‘Var’ refer to the average empirical squared bias and variance
scaled by $1{,}000$, respectively, computed over 200 simulations.
-2-101230.000.250.500.751.00$\alpha$$q_{\alpha}[X^{*}_{k}]$Sample
Size5001,0002,0004,000 Figure 2: Quantiles of Estimator of
$q_{\alpha}[X^{*}_{k}]$ under DGP with risk aversion. The red line shows the
true distribution of $X^{*}_{k}$. The blue lines show the mean, and the $5$th
and $95$th percentiles of the simulated distribution of the estimator of
$q_{\alpha}[X^{*}_{k}]$ for each sample size.


# An $N/D$ study of the $S_{11}$ channel $\pi N$ scattering amplitude

###### Abstract

Extensive dynamical $N/D$ calculations are made in the study of $S_{11}$
channel low energy $\pi$N scatterings, based on various phenomenological model
inputs of left cuts at tree level. The subtleties of the singular behavior of
the partial wave amplitude at the origin of the complex $s$ plane are
carefully analysed. Furthermore, it is found that the dispersion
representation for the phase shift, $\delta$, has to be modified in the case
of $\pi$N scatterings. An additional contribution from the dispersion integral
exists, which is, however, almost exactly cancelled the contribution from two
virtual poles located near the end points of the segment cut induced by $u$
channel nucleon exchanges. Relying very little on the details of the dynamical
inputs, the subthreshold resonance $N^{*}(890)$ survives.

Qu-Zhi Li,${}^{\dagger\,}$ Yao Ma,${}^{\dagger\,}$ Wen-Qi Niu,${}^{\dagger\,}$
Yu-Fei Wang,‡ Han-Qing Zheng${}^{\heartsuit\,,\star\,,}$111Corresponding
author.

† Department of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing 100871, P. R. China
‡ Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center
for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany
♡ College of Physics, Sichuan University, Chengdu, Sichuan 610065, P. R. China
⋆ Collaborative Innovation Center of Quantum Matter, Beijing, Peoples Republic
of China

##  1 Introduction

In a series of recent publications [1][2][3], it is suggested that there
exists a sub-threshold $1/2^{-}$ nucleon resonance hidden in $S_{11}$ channel
of $\pi N$ scatterings, with a pole mass $\sqrt{s}=(0.895\pm 0.081)-(0.164\pm
0.023)i$ GeV. The result is obtained by using the production representation
(PKU representation) for partial wave amplitudes [4][5][6][7][8]. It is found
later that the $N^{*}(890)$ pole may also be generated from a conventional and
simple $K$-matrix fit, though the latter suffers from the existence of
spurious poles on 1st Riemann sheet of complex $s$ plane [9]. Properties of
$N^{*}(890)$ are also investigated, such as its coupling to $N\gamma$ and
$N\pi$ [10, 11]. It is found that its coupling to $N\pi$ is considerably
larger than that of the $N^{*}(1535)$, while its coupling to $N\gamma$ is
comparable to that of the $N^{*}(1535)$. These results on couplings look
reasonable and are within expectations, hence providing further evidence on
the existence of $N^{*}(890)$.

However, to firmly establish the existence of such a subthreshold resonance is
still a difficult task. Besides dispersion relations, the most frequently used
tools at hand are perturbation chiral amplitudes and their unitarizations (For
a recent review, see Ref. [12]), or (unitarized) resonance models. However,
these unitarization techniques are far from being perfect when used in the
study of low energy strong interaction physics. Especially, when applying to
partial wave amplitudes with unequal mass scatterings, extra difficulties will
occur, as will be discussed at some lengths in this paper. The major
difficulties arose at $s=0$ point in $s$ plane, where chiral expansions break
down since chiral expansions and partial wave projections do not commute when
$s\to 0$. The expected decoupling property of heavy resonances when their
masses sent to infinity is also violated in partial wave amplitudes at $s=0$,
for a purely kinematical reason in partial wave projections with unequal mass
scatterings. The major task of this paper is to show how the subthreshold
resonance persist, irrespective of various difficulties and uncertainties left
in the input quantity – the left part of the scattering amplitude.

This paper will provide further evidences on the existence of $N^{*}(890)$, by
directly finding a pole in the $S$ matrix element calculated from the $N/D$
method. Early studies on low energy $\pi N$ scatterings via $N/D$ method may
be found in Ref. [13] and references therein. Nevertheless, no report on the
possible existence of a subthreshold resonance is known in the literature of
$N/D$ studies, to the best of our knowledge. In our practice of $N/D$
calculations no spurious poles on first Riemann sheet are found to emerge.
Also an $N/D$ calculation faithfully reproduces all input dynamical as well as
kinematical branch point singularities. We therefore think the $N/D$ method is
rather reliable. However, the calculations in $N/D$ studies do generate
spurious branch cuts and spurious poles on the second sheet, due to the
truncation of numerical integrations. Nevertheless, their effects can be
evaluated to see that the sum of hazardous contributions be negligible in many
cases.

This paper is organized as follows: section 1 is the introduction. In section
2 a brief introduction to the $N/D$ method is given with a solvable toy model
calculation. Also in section 2 we afford a review on the production
representation which is found very illuminating in understanding the
complicated $N/D$ calculations. A subtlety occurs when using the production
representation in dealing with $\pi$N scatterings: the dispersion
representation for the background contribution to the phase shift has to be
modified, an additional contribution emerges which is however cancelled almost
exactly by contributions from two virtual poles located near the end points of
the cut caused by $u$ channel nucleon pole exchanges. We put all the related
discussions with respect to the subtlety in the appendix. Section 3 focuses on
the singularity structure of partial wave amplitudes at $s=0$, including the
discussion on why chiral expansions break down here, and on how high energy
contributions enter through an analysis on regge asymptotic behavior of $T(s)$
when $s\to 0$. Section 4 devotes to numerical analyses on how can a
subthreshold resonance emerge under various phenomenological inputs.

##  2 The $N/D$ recipe, a prelude

###  2.1 A brief introduction to $N/D$ method

The partial wave $T$ matrix element is expressed as

$T={N}/{D}\ ,$ (1)

where $D$ contains only the $s$-channel unitarity cut or the right hand cut
$R$, whereas $N$ only contains the left hand cut ($l.h.c.$) or $L$ . See Fig.
1, $R=[s_{R},+\infty)$ whereas $L$ represents all branch cuts except $R$ in
Fig. 1. In section 3.2 we will briefly review on how to determine the cut
structure in Fig. 1 [14].

Figure 1: Branch cuts (thick blue lines) of partial wave $\pi$N elastic
scattering amplitudes, where
$c_{L}={(m_{N}^{2}-m_{\pi}^{2})^{2}}/{m_{N}^{2}}$,
$c_{R}=m_{N}^{2}+2m_{\pi}^{2}$.

Partial wave unitarity in single channel approximation

$\displaystyle\mathrm{Im}_{R}T(s)=\rho(s)|T(s)|^{2}\ ,$ (2)

where

$\rho=\sqrt{(s-s_{L})(s-s_{R})}/s$ (3)

with $s_{L}=(m_{N}-m_{\pi})^{2}$, $s_{R}=(m_{N}+m_{\pi})^{2}$, leads to the
following relations

$\displaystyle\mathrm{Im}_{R}[D(s)]$ $\displaystyle=$
$\displaystyle-\rho(s)N(s)\ ,$ $\displaystyle\mathrm{Im}_{L}[N(s)]$
$\displaystyle=$ $\displaystyle D(s)\mathrm{Im}_{L}[T(s)]\ ,$ (4)

and subsequential $N/D$ equations:

$\begin{split}&D(s)=1-\dfrac{s-s_{0}}{\pi}\int_{R}\dfrac{\rho(s^{\prime})N(s^{\prime})}{(s^{\prime}-s)(s^{\prime}-s_{0})}ds^{\prime}\
,\\\
&N(s)=N(s_{0})+\dfrac{s-s_{0}}{\pi}\int_{L}\dfrac{D(s^{\prime})\mathrm{Im}_{L}[T(s^{\prime})]}{(s^{\prime}-s)(s^{\prime}-s_{0})}ds^{\prime}\
.\end{split}$ (5)

Notice that when there appears circular cut in $T$ on $s$ plane as shown in
Fig. 1, the second equation of Eq. (2.1) should be read as
$\mathrm{disc}[N(s)]=D(s)\mathrm{disc}[T(s)]$. Also in the second equation of
Eq. (5), when the integration is performed on the circular cut which belongs
to a subset of $L$, $\mathrm{Im}_{L}[T(s)]$ should be replaced by
$\frac{1}{2i}\mathrm{disc}[T(s)]$. After getting a numerical solution of Eq.
(5), analytical continuation to the complex plane is straightforward: taking
$s$ to be complex while evaluating the integration in Eq. (5) when $s$ is in
the first sheet, and taking

$D^{\mathrm{II}}(s)=D(s)+2i\rho N(s)\ ,\,\,\,N^{\mathrm{II}}(s)=N(s)\ ,$ (6)

when $s$ lies on the second sheet. In Eq. (5), the left cut of the partial
wave $T$ matrix element, $\mathrm{Im}_{L}T$ is an input quantity. Throughout
this paper, we only discuss $\mathrm{Im}_{L}T$ extracted from tree level
amplitudes. Hence, except in the case with $t$-channel $\rho$ exchange as
discussed in section 4.2, where an arc cut in $s$ plane will be met, $L$ is
always on the real axis. For example, for pure tree level chiral amplitudes,
$L=(-\infty,0]\cup[c_{L},c_{R}]$. We will make a rather detailed discussion on
how to determine $l.h.c.$s in section 3.2.

To solve the integral equations one may substitute $D$ into the second
equation of Eq. (5) to get

$N(s)=N(s_{0})+\tilde{B}(s,s_{0})+\dfrac{s-s_{0}}{\pi}\int_{R}\dfrac{\tilde{B}(s^{\prime},s)\rho(s^{\prime})N(s^{\prime})}{(s^{\prime}-s_{0})(s^{\prime}-s)}ds^{\prime}\
,$ (7)

with

$\tilde{B}(s^{\prime},s)=\dfrac{s^{\prime}-s}{2\pi
i}\int_{L}\dfrac{\mathrm{disc}T(\tilde{s})}{(\tilde{s}-s)(\tilde{s}-s^{\prime})}d\tilde{s}\
,$ (8)

and use the inverse matrix method to obtain a numerical solution. Throughout
this paper, we set $s_{0}=1$ GeV2, a value a little bit below the elastic
threshold $s_{R}$.

As we will see later in this paper, there exists a subtlety when using Eq. (5)
to discuss un-equal mass scatterings. The problem comes from a singularity at
$s=0$ in the partial wave amplitude and in its left cut, which stems from high
energy region $t\to+\infty$ through partial wave projections, and from
relativistic spin kinematics as well. But before dealing with this problem, we
are more eager in finding a solution of Eq. (5) in a simplified toy model, in
the following subsection.

###  2.2 A toy model calculation

In $N/D$ recipe the input quantity is $\mathrm{Im}_{L}\,T$. Let us begin with
a simple version of $N/D$ study by assuming $\mathrm{Im}_{L}\,T$ being
simulated by a set of Dirac $\delta$ functions, or equivalently

$\,N=\sum_{i}\frac{\gamma_{i}}{s-s_{i}}\ ,$ (9)

which is to be used in the first equation of Eq. (5). I.e., there is no need
for a subtraction in the second equation of Eq. (5). The $T$ matrix in such a
situation is analytically solvable. We (arbitrarily) choose $Case$ I: one pole
at $s_{1}=0$, and $Case$ II: one pole at $s_{1}=-m_{N}^{2}$, and fit to the
“data” obtained from the solutions of Roy Steiner equations [15] by tuning the
parameter $\gamma_{1}$, and search for poles on the $s$-plane. Both cases give
a good fit to the data, and a sub-threshold pole emerges in each case with a
location listed in Table 1.

|  $Case$ I |  $Case$ II
---|---|---
$s_{1}$ | 0 | $-m_{N}^{2}$
$\gamma_{1}$ (GeV2) | 0.79 | 1.34
$\sqrt{s_{pole}}$(GeV) | 0.810 - 0.125i | 0.788 - 0.185i

Table 1: Subthreshold pole locations using input Eq. (9).

.

The phase shift and its PKU decomposition [4][5] are plotted in Fig. 2. On the
left of Fig. 2, we see the familiar picture that the background cut
contribution to the phase shift is concave and negative while the subthreshold
resonance pole provides a positive and convex phase shift above threshold to
counterbalance the former contribution, and the sum of the two reproduces the
steadily rising phase shift data. In order to have a better understanding of
this phenomenon, a brief introduction to the production representation of
partial wave elastic scattering $S$ matrix element is needed.

Figure 2: left: fit to the $S_{11}$ channel $\pi$N scattering phase shift
data, taking $Case$ II as an example (data from Ref. [15]); right: the
spectral function $\mathrm{Im}_{L}f(s)/s$ of $Case$ I and II. Notice that the
singularity at $s=0$ in $Case$ II is due to the kinematical singularity in
$\rho(s)$ defined in Eq. (3) rather than dynamical. For the definition of
different contributions to the phase shift, one is referred to Sec. 2.3.

###  2.3 The PKU representation

The “spectral” function drawn on the $r.h.s.$ of Fig. 2 is defined as
following:

$\displaystyle
f(s)=\dfrac{s}{{2i}\pi}\int_{L}ds^{\prime}\dfrac{\mathrm{disc}_{L}f(s^{\prime})/s^{\prime}}{(s^{\prime}-s)}+\dfrac{s}{{2i}\pi}\int_{R}ds^{\prime}\dfrac{\mathrm{disc}_{R}f(s^{\prime})/s^{\prime}}{(s^{\prime}-s)}\
,$ (10)
$\displaystyle\mathrm{disc}_{L,R}f(s)=\mathrm{disc}_{L,R}\left(\dfrac{\ln
S(s)}{2i\rho(s)}\right)\ ,$ (11)

where the partial wave $S$ matrix element $S=1+2i\rho T$ and $\rho$ is the
kinematic factor. In Eq. (10), the integration domain is depicted in Fig. 1,
it is noticed that $R=[(m_{N}+m_{\pi})^{2},+\infty)$ but actually the
integrand vanishes in the elastic region, or in other words, $R$ starts from
first inelastic threshold. Elastic partial wave $S$ matrix elements satisfy a
production representation like follows:

$S=\prod_{i}S_{i}\times e^{2i\rho(s)f(s)}\ .$ (12)

Detailed discussions on how to obtain Eq. (12) can be found in Refs. [4][5].
The production representation exhibits some nice features which are very
useful in our analyses. One advantage is that phase shifts from different
sources are additive, i.e.,

$\delta(s)=\sum_{i}\delta_{i}+\delta_{cut}\ ,$ (13)

where $\delta_{i}$ comes from the $i$-th pole contribution described by
$S_{i}$ in Eq. (12). For a resonance located at $z_{0}$ ($z_{0}^{*}$) it is

$\delta_{res}=\arctan\left[\frac{\rho(s)sG[z_{0}]}{M^{2}(z_{0})-s}\right]\,\,\,(S_{R}=e^{2i\delta_{R}})\
,$ (14)

with $G(z_{0})=\mathrm{Im}[z_{0}]/\mathrm{Re}[z_{0}\rho(z_{0})]$ and
$M^{2}(z_{0})=\mathrm{Re}[z_{0}]+\mathrm{Im}[z_{0}]\mathrm{Im}[z_{0}\rho(z_{0})]/\mathrm{Re}[z_{0}\rho(z_{0})]$.
For a virtual state located at $s_{0}$ ($s_{L}<s_{0}<s_{R}$), its contribution
is

$\delta_{v}=\arctan\left[{\frac{\rho(s)s}{s-s_{L}}\sqrt{\frac{s_{0}-s_{L}}{s_{R}s_{0}}}}\right]\,\,\,(S_{v}=e^{2i\delta_{v}})\
.$ (15)

For a bound state, the above expression changes sign, i.e.,
$\delta_{b}=-\delta_{v}$ ($S_{b}=e^{2i\delta_{b}}$). The last contribution in
Eq. (13), i.e., the background contribution to the phase shift, is

$\delta_{cut}=\rho(s)f(s)\ ,$ (16)

with function $f(s)$ defined in Eq. (10).

The above description of the production representation only repeats what have
been established in the literature. In our study of $\pi$N scatterings,
however, an unexpected new phenomenon occurs. Simply speaking, the cut
structure of the background phase shift, $\delta_{b.g.}$, has to be modified,
due to the presence of the $u$–channel nucleon exchange. 222The cut structure
of $S_{b.g.}=\exp{2i\rho f(s)}$ remains unchanged, as it should. As a result,
an additional contribution exists in the expression of $\delta_{b.g.}$, which
is, however, almost exactly cancelled by the contributions from two newly
found virtual poles located near the end points (i.e., $c_{L}$ and $c_{R}$
defined in Fig. 1) of the segment cut induced by the $u$ channel nucleon
exchange. Since the explanation is rather lengthy, we postpone the discussions
in the appendix.

The property that the phase shift contributions are additive is vital in
tracing the physical origin of the phase shift. As having been stressed
repeatedly, the positive value of $\mathrm{Im}_{L}f(s)/s$ guarantees that the
background contribution to the phase shift be negative and concave, and hence
an isolated singularity on the second sheet is needed to take charge of the
steady rise of near threshold phase shift. The circular cut caused by
$t$-channel exchanges and the $u$-channel cut (combined with effects of two
virtual poles) are not considered yet here, nevertheless they are numerically
small as comparing with the cut lying on $(-\infty,s_{L}]$.333Previous
examples can be found in Refs. [4][1][9]. Hence a large and positive
$\mathrm{Im}_{L}f(s)/s$ at $s=0$, looks important as suggested in Fig. 2. The
strong enhancement of $\mathrm{Im}_{L}f(s)/s$ at $s=0$ is due to two reasons:
one is from the kinematic singularity at $s=0$ from Eq. (3), the other is from
the possible singularity in $T(s)$ when $s\to 0$. An example for the former is
fit $Case$ II in section 2.2 where $T(0)\sim\mathrm{constant}$, while the
latter example is provided by fit $Case$ I where $T(s)\sim O(s^{-1})$ when
$s\to 0$. In general, making use of the property of real analyticity for $S$
matrix elements, one recasts $\mathrm{Im}_{L}f(s)$, when $s$ lies on the real
axis, as
$-\ln{|S(s)|}/2\rho(s)=-\ln{[1-4\rho(s)\mathrm{Im}_{L}T(s)+4\rho(s)^{2}|T|^{2}]}/4\rho(s)$.
Hence if $T(s)$ does not vanish when $s\to 0$, then $\ln{|S(s)|}$ diverges
logarithmically, since $\rho(s)\propto s^{-1}$ at origin. It may be worth
stressing that the singularities caused by relativistic kinematics truly exist
and bring physical consequences, since they enter the physical equations such
as Eq. (2). A good example to realize this comes from figure 5 and Eq. (55) of
Ref. [7]: without the kinematical singularity the data curve can simply not be
explained.

On the other side, the inelastic right hand cut contribution to the phase
shift in Eq. (10) should always be positive [3], since in that region
$|S(s)|=\eta<1$, with $\eta$ the inelasticity parameter. In $N/D$
calculations, a cutoff to the integral interval has to be adopted, i.e.,
[$s_{R},+\infty)$ to be replaced by $[s_{R},\Lambda_{R}^{2}]$. In this
situation, it is not difficult to understand that the truncated $N/D$
integration actually violates unitarity by introducing a spurious branch cut
starting from $s=\Lambda_{R}^{2}$, in the sense that the effective $\eta$
parameter exceeds unity:
${|S(s)|^{2}}={[1-4\rho(s)\mathrm{Im}_{R}T(s)+4\rho(s)^{2}|T|^{2}]}=[1+4\rho(s)^{2}|T|^{2}]>1$,
when $s>\Lambda_{R}^{2}$, since a truncation implies actually
$\mathrm{Im}_{R}T=0$. So one has to be cautious when performing the $N/D$
calculations by monitoring to what extent unitarity is violated. This may be
fulfilled at quantitative level by, for example, calculating the contribution
from the region above $\Lambda^{2}_{R}$ to the phase shift, through Eq. (10).
It is found that, when doing calculations in this paper, the violation can
either be large or small, depending on whether or not the input quantity
$T_{L}(s)$ at $s=\Lambda_{R}^{2}$ is too large or small. For the former, an
example is the $\chi$PT input which is not valid anymore at
$\Lambda_{R}^{2}=1.48$ GeV2, i.e., the value we choose in most of our
calculations. However, the $N^{*}(890)$ exists with a rather stable location,
irrespective to the pollution of truncation of integration.

Encouraged by the discussions made in section 2.2, we plan to make a more
realistic $N/D$ calculation at next steps. The first thing needs to be settled
down is of course choosing an input $\mathrm{Im}_{L}T$ as much realistic as
possible. One may choose the $\chi$PT outputs as an input here, as what is
adopted in Refs. [1][2][3]. A careful analysis reveals, however, that the
partial wave projection of $\chi$PT amplitudes encounters a rather severe
problem at $s=0$. In the following section we dedicate to the study of such a
problem.

##  3 The singularity structure of $T(s)$ at $s=0$

As suggested in the end of the above section, to make a more realistic
calculation, one may use $\chi$PT amplitudes to extract $\mathrm{Im}_{L}T$
[3]. However, the results may not be directly applicable to Eq. (5), and
should be treated with great care. The integration interval for $N(s)$ is
$L=(-\infty,0]$ for tree level estimation here. The $\mathcal{O}(p^{3})$ level
$\mathrm{Im}_{L}T$ behaves as $\mathcal{O}(s^{-7/2})$ when $s\to 0$, and at
$\mathcal{O}(p^{2})$ level it behaves as $\mathcal{O}(s^{-5/2})$, while from a
rather general argument to be discussed in section 3.2 these strong
singularities at $s=0$ are not physical. Hence one has to find a way to get
rid of these artifacts caused by partial wave projections of $\chi$PT
amplitudes.

In general, singularities of the partial wave $T$ matrix element at $s=0$ come
from two aspects:

  * •

the $\mathcal{O}(p^{n})$ ($n\geq 2$) level $\chi$PT amplitudes behave as
$s^{-n-1/2}$ when $s\to 0$ for unequal mass scatterings, after partial wave
projections, and the integer $n$ increases when the chiral order increases.

  * •

the left-hand cut around $s=0$ receives singular contribution from high energy
region of crossed channels, i.e., $t,u\to\infty$, through partial wave
projections;

We will address these problems in some details below. It is worth pointing out
beforehand, that though in $N/D$ approach one has to deal with these problems
cautiously, the calculations made in Refs. [3] and [1] are luckily insensitive
to the problem. Though the background contribution to the phase shift will be
enhanced incorrectly by the contribution in the vicinity of $s=0$, it will be
largely compensated by tuning the cutoff parameter $\Lambda_{L}^{2}$ when
evaluating the $l.h.c.$ integral. More interestingly even though the integral
of $f(s)$ is enhanced near $s=0$, its derivatives behave very differently. For
example,

$\frac{df(s)}{ds}=-\dfrac{1}{\pi}\int
ds^{\prime}\dfrac{\ln{|S(s^{\prime})|}/2\rho(s^{\prime})}{(s^{\prime}-s)^{2}}\
.$ (17)

Now the integrand near $s^{\prime}=0$ behaves like $\sim s^{\prime}\ln
s^{\prime}$ and the unwanted singularity is killed. This fact may be
translated into a more transparent language: in Refs. [3][1], except the
scattering length which anyway needs to be fitted by tuning $\Lambda^{2}_{L}$,
the other quantities such as the curvature of the phase shift curve are quite
immune to the disease infected from $s=0$.

###  3.1 Artificial singularities in partial wave $\chi$PT amplitudes

We notate the process as $N(p,\sigma)+\pi(q)\to
N(p^{\prime},\sigma^{\prime})+\pi(q^{\prime})$, where $p,\ q,\ p^{\prime},\
q^{\prime}$ are the momenta and $\sigma,\ \sigma^{\prime}$ are the spins. The
Mandelstam variables are

$s=(p+q)^{2}=(p^{\prime}+q^{\prime})^{2}\ ,\quad
t=(p-p^{\prime})^{2}=(q-q^{\prime})^{2}\ ,\quad
u=(p-q^{\prime})^{2}=(p^{\prime}-q)^{2}\ .$ (18)

The full amplitude, ${\cal T}$, can be decomposed as (the following
discussions are all for isospin $I=1/2$ only),

${\cal
T}=\bar{u}(p^{\prime},\sigma^{\prime})\left[A(s,t)+\frac{\not{q}+\not{q}^{\prime}}{2}B(s,t)\right]u(p,\sigma)\
.$ (19)

The results of the scalar functions $A,B$ are for example listed in Refs. [3,
1] (further references are found in Refs. [16][17][18][19][20][21]). From
$\mathcal{O}(p^{2})$ on, the $\chi$PT lagrangian contains $4$-point $\pi\pi
NN$ contact terms, which contributes to the scalar functions as ($C$ refers to
constants),

$A\big{[}\mathcal{O}(p^{2})\big{]}\supset C(s-u)^{2}\ ,\quad
A\big{[}\mathcal{O}(p^{3})\big{]}\supset C(s-u)^{3}\ .$ (20)

More explicitly,

  * •

at $\mathcal{O}(p^{1})$ (Born and contact diagrams):

$\begin{split}&A_{1}=\frac{g^{2}m_{N}}{F^{2}}\ \mbox{, }\\\
&B_{1}=\frac{1-g^{2}}{F^{2}}-\frac{3m_{N}^{2}g^{2}}{F^{2}(s-m_{N}^{2})}-\frac{m_{N}^{2}g^{2}}{F^{2}}\frac{1}{u-m_{N}^{2}}\
\mbox{; }\end{split}$ (21)

  * •

at $\mathcal{O}(p^{2})$ (only contact diagram):

$\begin{split}&A_{2}=-\frac{4c_{1}m_{\pi}^{2}}{F^{2}}+\frac{c_{2}(s-u)^{2}}{8m_{N}^{2}F^{2}}+\frac{c_{3}}{F^{2}}(2m_{\pi}^{2}-t)-\frac{c_{4}(s-u)}{F^{2}}\
\mbox{, }\\\ &B_{2}=\frac{4m_{N}c_{4}}{F^{2}}\ \mbox{; }\end{split}$ (22)

  * •

at $\mathcal{O}(p^{3})$ (Born diagram):

$\begin{split}&A_{3B}=-\frac{m_{N}g}{F^{2}}\times
4m_{\pi}^{2}(d_{18}-2d_{16})\ \mbox{, }\\\
&B_{3B}=\frac{4m_{\pi}^{2}g(d_{18}-2d_{16})}{F^{2}}\times\frac{su+m_{N}^{2}(2u-3m_{N}^{2})}{(s-m_{N}^{2})(u-m_{N}^{2})}\
\mbox{; }\end{split}$ (23)

  * •

at $\mathcal{O}(p^{3})$ (contact diagram):

$\begin{split}&A_{3C}=-\frac{(d_{14}-d_{15})(s-u)^{2}}{4m_{N}F^{2}}+\frac{(d_{1}+d_{2})}{m_{N}F^{2}}(s-u)(2m_{\pi}^{2}-t)\\\
&\,\,\,\,\,\,\,\,\,\,\,+\frac{d_{3}}{8m_{N}^{3}F^{2}}(s-u)^{3}+\frac{4m_{\pi}^{2}d_{5}}{m_{N}F^{2}}(s-u)\
\mbox{, }\\\ &B_{3C}=\frac{(d_{14}-d_{15})(s-u)}{F^{2}}\ \mbox{. }\end{split}$
(24)

In the expressions above, $g$ is the axial-vector coupling constant, $F$ is
the pion decay constant, and $c_{i}$, $d_{i}$ are low-energy constants.

The partial wave projection is done on helicity amplitudes:

$\begin{split}&{\cal
T}_{++}=\sqrt{\frac{1+z_{s}}{2}}\big{[}2m_{N}A(s,t)+(s-m_{\pi}^{2}-m_{N}^{2})B(s,t)\big{]}\
\mbox{, }\\\ &{\cal
T}_{+-}=\sqrt{\frac{1-z_{s}}{2s}}\big{[}(s-m_{\pi}^{2}+m_{N}^{2})A(s,t)+m_{N}(s+m_{\pi}^{2}-m_{N}^{2})B(s,t)\big{]}\
\mbox{, }\end{split}$ (25)

where $z_{s}=\cos\theta$, and $\theta$ is the scattering angle; $T_{++}$
stands for that the helicities of the initial and final nucleon are both
$+1/2$, while $T_{+-}$ means the final nucleon has helicity $-1/2$ . The
relations between the Mandelstam variables and the scattering angle
($z_{s}=\cos\theta$) are:

$\displaystyle
t(s,z_{s})=2m_{\pi}^{2}-\frac{(s+m_{\pi}^{2}-m_{N}^{2})^{2}}{2s}+\frac{[s-(m_{\pi}+m_{N})^{2}][s-(m_{\pi}-m_{N})^{2}]}{2s}z_{s}\
,$ (26) $\displaystyle
u(s,z_{s})=m_{\pi}^{2}+m_{N}^{2}-\frac{s^{2}-(m_{\pi}^{2}-m_{N}^{2})^{2}}{2s}-\frac{[s-(m_{\pi}+m_{N})^{2}][s-(m_{\pi}-m_{N})^{2}]}{2s}z_{s}\
.$ (27)

The $S_{11}$ amplitude is from $J=1/2$ partial wave helicity amplitudes:

$\begin{split}&T_{++}^{J}=\frac{1}{32\pi}\int_{-1}^{1}dz_{s}{\cal
T}_{++}(s,t(s,z_{s}))d^{J}_{1/2,1/2}(z_{s})\ \mbox{, }\\\
&T_{+-}^{J}=\frac{1}{32\pi}\int_{-1}^{1}dz_{s}{\cal
T}_{+-}(s,t(s,z_{s}))d^{J}_{-1/2,1/2}(z_{s})\ \mbox{, }\end{split}$ (28)

where $d$ stands for Wigner small-$d$ matrix. For $S_{11}$ channel,

$T(S_{11})=T_{++}^{J=1/2}+T_{+-}^{J=1/2}\ .$ (29)

From this formula the singularity at $s=0$ is obvious. On the one hand, in Eq.
(25) the kinematic effects give an $s^{-1/2}$ factor, which makes $s=0$ a
branch point. On the other hand, see for example in Eq. (24), contact terms
from $\chi$PT expansions lead to higher and higher order polynomials of $s-u$:

${\cal T}\big{[}\mathcal{O}(p^{n})\big{]}\supset C(s-u)^{n}\ .$ (30)

According to Eq. (27), $u(s\to 0)\to s^{-1}$, so when $n\geq 2$

${\cal T}\big{[}\mathcal{O}(p^{n})\big{]}(s\to 0)\sim Cs^{-n-1/2}\ ,$ (31)

where the extra factor $-1/2$ in the power comes from kinematic effects of
helicity basis. Eq. (31) indicates that the higher order of $\chi$PT
calculation is employed, the stronger singularity will occur near $s=0$ and
the chiral series breaks down. Apparently this is only an artificial problem
caused by chiral expansions since it contradicts the genuine singularity
structure expected when $s\to 0$, as discussed in the end of section 3.2: the
Froissart bound forbids a power behavior like Eq. (31) when $s\to 0$. In
principle one would not hope Eq. (31) to appear in the expression of
$\mathrm{Im}_{L}T$ when using $N/D$.

###  3.2 High energy contributions from crossed channels

One writes a spectral representation of the partial wave amplitude [14], for
$t$-channel:

$T\sim\int_{\sigma_{t}}^{+\infty}dt^{\prime}\Sigma(s,t^{\prime})\int_{-1}^{1}d\cos\theta\frac{R(\cos\theta)}{t-t^{\prime}}\
,$ (32)

where $\Sigma$ is the Mandelstam spectral function, $\sigma_{t}=4m_{\pi}^{2}$
is the threshold of $t$-channel process, and $R$ is the basis function of the
partial wave projection (usually the linear combinations of Wigner-$d$
matrices). The function $R$ can be expanded at $t=t^{\prime}$:
$R=R_{0}+(t-t^{\prime})R_{1}+\cdots$, and only the leading order causes
singularities:

$T\propto\int_{\sigma_{t}}^{+\infty}dt^{\prime}\Sigma(s,t^{\prime})\beta^{-1}\ln\left[\frac{\alpha+\beta}{\alpha-\beta}\right]\
,$ (33)

with

$\alpha=t^{\prime}-2m_{\pi}^{2}+\frac{(s+m_{\pi}^{2}-m_{N}^{2})^{2}}{2s}\
,\quad\beta=\frac{[s-(m_{\pi}+m_{N})^{2}][s-(m_{\pi}-m_{N})^{2}]}{2s}\ .$ (34)

Therefore the left-hand cut is described by the equation

$\alpha^{2}=\beta^{2}\ ;$ (35)

which gives

$s_{\pm}(t^{\prime})=m_{\pi}^{2}+m_{N}^{2}-\frac{t^{\prime}}{2}\pm\frac{1}{2}\sqrt{(t^{\prime}-4m_{\pi}^{2})(t^{\prime}-4m_{N}^{2})}\
.$ (36)

When $t^{\prime}$ takes the value from $\sigma_{t}$ to $+\infty$, the
trajectory of this solution traces out the left-hand cut. The following
conclusions can be obtained:

  * •

when $t^{\prime}\in[4m_{\pi}^{2},4m_{N}^{2}]$, the cut appears as a circle
$\text{Re}s^{2}+\text{Im}s^{2}=(m_{N}^{2}-m_{\pi}^{2})^{2}$, and the endpoint
to the right $s=m_{N}^{2}-m_{\pi}^{2}$ corresponds to
$t^{\prime}=\sigma_{t}=4m_{\pi}^{2}$;

  * •

when $t^{\prime}\in(4m_{N}^{2},+\infty)$, $s_{-}$ generates the cut
$(-\infty,m_{\pi}^{2}-m_{N}^{2})$, and $-\infty$ corresponds to
$t^{\prime}\to+\infty$;

  * •

when $t^{\prime}\in(4m_{N}^{2},+\infty)$, $s_{+}$ generates the cut
$(m_{\pi}^{2}-m_{N}^{2},0)$, and actually $0$ corresponds to
$t^{\prime}\to+\infty$;

One performs a similar analysis in $u$-channel and the solution is

$s_{1}(u^{\prime})=\frac{(m_{\pi}^{2}-m_{N}^{2})^{2}}{u^{\prime}}\ ,\
s_{2}(u^{\prime})=2(m_{\pi}^{2}+m_{N}^{2})-u^{\prime}\ .$ (37)

There is a nucleon pole $u^{\prime}=m_{N}^{2}$, giving a segment cut
$((m_{\pi}^{2}-m_{N}^{2})^{2}/m_{N}^{2},2m_{\pi}^{2}+m_{N}^{2})$. When
$u^{\prime}>\sigma_{u}=(m_{\pi}+m_{N})^{2}$, $s_{1}$ gives
$(0,(m_{N}-m_{\pi})^{2})$ and $s_{1}\to 0$ just when $u^{\prime}\to+\infty$;
while $s_{2}$ generates $(-\infty,(m_{N}-m_{\pi})^{2})$ with $s_{2}\to-\infty$
when $u^{\prime}\to+\infty$.

It was pointed out in Ref. [5] that, for meson – meson scatterings, if
$T(s,t)\sim O(t^{n})$ when $t\to\infty$ for fixed $s$, then the partial wave
amplitude behaves as $T(s)\sim O(s^{-n})$ when $s\to 0$. Considering the
Froissart bound in the crossed channel $|T(t,\cos\theta_{s}=1)|<t\ln^{2}t$, it
is expected that the proper singularity behavior for $s$-channel partial wave
amplitude near $s=0$ is no more singular than $T(s)\sim O(s^{-1})$ (up to some
logarithmic corrections). This estimation can even further be improved. It is
seen from the above discussions that as $s\to 0_{-}$, there is a high energy
contribution from the $t\to+\infty$ region. In this region the full amplitude
is governed by $t$-channel ($\pi\pi\to N\bar{N}$) reggeon exchanges. The
leading Regge trajectory is $\Delta(1232)$ with the intercept parameter
$\alpha_{\Delta}(0)\simeq 0.19$ [22], which leads to a weak singularity

$T\sim s^{-\alpha_{\Delta}(0)}\ ,$ (38)

for the partial wave amplitude, when $s\to 0_{-}$. The $s\to 0_{+}$ limit is
the same according to for example Ref. [22].

All the discussions given above in this section are on how to determine
$l.h.c.$s generated dynamically, or in other words, cuts originated from
physical absorptive singularities from crossed channels. Besides these
dynamical $l.h.c.$s there exists an additional cut $(-\infty,0]$ for pure
kinematical reasons: the nucleon spinor wave function provides a $\sqrt{s}$
branch cut which already showed up in the second equality in Eq. (28). The
effect of branch cut singularity from relativistic kinematics truly exists, as
has already been addressed in section 2.3.

##  4 Numerical analyses of $N/D$ method

It has been made clear in the above section, that using $\chi$PT inputs of
$\mathrm{Im}_{L}T$ encounters the problem that the partial wave projections of
$\chi$PT amplitudes lead to a strong but incorrect singularity at $s=0$, by
violating what is expected from rather general constraints of quantum field
theory. Nevertheless it is not clear yet, to what extent the use of $\chi$PT
results may distort the physical output. In the following we devote to the
study of this problem by invoking $\mathcal{O}(p^{2})$ and
$\mathcal{O}(p^{3})$ (tree level amplitude only) $\chi$PT results, since in
$\mathcal{O}(p^{1})$ case no free parameter is available for a data fit.
Nevertheless, the $\mathcal{O}(p^{1})$ $N/D$ unitarization can still be made
and compared with data, which ends up with a pole location
$\sqrt{s}=1.08-i0.23$ GeV and a steadily rise phase shift larger than data by
roughly 5 degrees at $\sqrt{s}=1.16$ GeV.

###  4.1 $N/D$ calculations using pure $\chi$PT inputs

The singularity of $\mathcal{O}(p^{2})$ $\mathrm{Im}_{L}T$ when $s\to 0$
behaves as $\mathcal{O}(s^{-2-1/2})$. Which, as discussed previously, is not
physical. We nevertheless still perform the $N/D$ calculation to see what
happens. In doing such a calculation it is noticed that the sick singularity
behavior destroys the validity of Eq. (8). To overcome the problem, the
auxiliary function $\tilde{B}$ in Eqs. (7) and (8) can be formally written as
$\tilde{B}(s,s_{0})=T_{L}(s)-T_{L}(s_{0})$, where $T_{L}$ is taken as the
$O(p^{1})$ partial wave amplitude (Eq. (51)) plus the $O(p^{2})$ part of
$T^{J}_{+-}$, since at $O(p^{2})$ level $T^{J}_{++}$ does not contribute to
the imaginary part on the left. In this way we avoid the discussions on
possible subtractions encountered at two endpoints of the integral defined in
Eq. (8).444One may redefine $T(s)=\bar{T}(s)/s^{2}$ and
$\bar{T}(s)={N(s)}/{D(s)}$ to avoid the singularity in the integral in Eq. (8)
at $s=0$. The results are similar to analyses presented in this manuscript.
The subtraction constant $N(s_{0})$ appeared in Eq. (7), or $T(s_{0})$, serves
as a free fit parameter.

At $\mathcal{O}(p^{2})$ level there are four low energy constants (LECs)
$c_{i}$ with $i=1,\cdots,4$. There are different sets of $c_{i}$ parameters
found in the literature (e.g., Refs. [23],[13],[24],[25],[1]). For these LECs,
certain bounds, i.e., the positivity constraints [26] are obeyed.

A good fit is obtained with $c_{1}=-0.40,\ c_{2}=3.50,\ c_{3}=-3.90,\
c_{4}=2.17$, $N(s_{0})=0.47$, and the pole locates at

$\sqrt{s}=1.01\pm 0.19\ \mathrm{GeV}.$ (39)

In addition, we have also employed different sets of $c_{i}$, and the results
change very little. For instance when we take the central values of $c_{i}$’s
from Ref. [23] (NLO): $c_{1}=-0.74\pm 0.02,\ c_{2}=1.81\pm 0.03,\
c_{3}=-3.61\pm 0.05,\ c_{4}=2.17\pm 0.03$ (in units of GeV-1), the pole
position is $0.99-0.16i$ GeV though the phase shift does not fit data very
well. It is noticed that, the spurious branch cut becomes a bit annoying here,
contributing to the phase shift at $\sqrt{s}=1.16$ GeV roughly -6∘.
Nevertheless, this spurious effect is far from being dominant as can be seen
from Fig. 3, where one finds the total background contribution exceeds -30∘,
and hence is not worrisome. It is not totally clear to us yet why and under
what situation the spurious branch cut becomes numerically visible. One
conjecture is that the $\mathcal{O}(p^{2})$ $\chi$PT input itself becomes sick
at $\Lambda_{R}^{2}=1.48$ GeV2, hence amplifies the contribution of the
spurious branch cut . Finally, since the effect generated from the cut at
$s=\Lambda_{R}^{2}$ is small, we still think such type of solutions are
acceptable for the evaluation of physics at lower energies.

As a comparison we plot the “spectral function”, i.e.,
${\mathrm{Im}_{L}f(s)}/{s}$, obtained here together with those discussed in
section 2.2, in Fig. 3.

Figure 3: left: $N/D$ fit to the $S_{11}$ phase shift with
$\mathcal{O}(p^{2})$ $\chi$PT input; right: a comparison of different
${\mathrm{Im}f(s)}/{s}$ in different situations.

Comparing Eq. (39) with that in table 1, and different $l.h.c$ contributions
drawn in Fig. 3, we observe that the $\mathcal{O}(p^{2})$ calculation
overestimates the $l.h.c$ contribution comparing with that of $Case$ II. As a
result, the pole contribution to the phase shift has to be increased and hence
the pole location has to move towards to the right direction in $s$ plane
closer to the $\pi N$ threshold. But of course, such discussions are only
meaningful under the condition that the effects of spurious branch cut and the
spurious pole around $s=\Lambda^{2}_{R}$ cancel each other.

If all the spurious contributions are ignored,555Because of two excuses:
firstly they cancel each other; secondly, they are from distant places as they
are associated with the cutoff at $s=\Lambda^{2}_{R}$ anyway. It is desirable
to know the possible origin of spurious effects since it leaves the hope to
isolate and to remove them. On the contrary, it is difficult to cure the
similar problem in Padé approximation[27].  then we may ask a question. The
calculation here and that in Ref. [3] use the same ‘data’ sample and $l.h.c.$
while in Ref. [3] the pole locates at $\sqrt{s}=0.86\pm 0.05-i(0.13\pm 0.08)$
GeV. Comparing with Eq.(39) it is found that there exists a rather large
systematic error in determining the pole location. One possible reason to
explain this may come from the fact that in Ref. [3] a truncation of $l.h.c.$
is performed while in here there is no truncation on the left, see Eq. (7). In
fact in the calculation made in table 4 of Ref. [3], it is found that when
sending the cutoff $s_{c}$ to $\infty$ the pole location moves upwards to
$\sqrt{s}=0.91-i0.21$ GeV, i.e., getting closer to Eq. (39).

Finally, we have also tested the $\mathcal{O}(p^{3})$ inputs (at tree level
only) and found that the outputs are similar to the situation found in the
$\mathcal{O}(p^{2})$ case, so we no longer discuss the results here anymore.

###  4.2 $N/D$ calculation using phenomenological models

In the above section we have made discussions on the problem encountered when
using $\chi$PT results to estimate the $l.h.c$. The higher order terms
appeared in the chiral lagrangian describing $\pi$N interactions are obtained
by integrating out heavy degrees of freedom like the $\rho$ meson in
$t$-channel and $N^{*}$ resonances in $s$ and $u$-channels, etc.. 666In the
meson–meson scattering lagrangian, the LECs at $\mathcal{O}(p^{4})$ level are
known to be saturated by heavy degrees of freedom [28]. In meson–baryon
system, systematic studies on this point are not known to the authors. The ill
singularities at $s=0$ in partial wave chiral amplitudes come at least partly
from integrating out heavy degrees of freedom. To see this more clearly, let
us write down an effective interaction lagrangian responsible for $t$-channel
$\rho$ meson exchange,

$\displaystyle\mathcal{L}_{t}=g_{\rho}\vec{\rho}^{\mu}\cdot(\partial_{\mu}\vec{\pi}\times\vec{\pi})+g_{\rho}\bar{N}\dfrac{1}{2}\vec{\tau}\cdot(\gamma^{\mu}\vec{\rho}_{\mu}+\dfrac{\kappa}{2m_{N}}\sigma^{\mu\nu}\partial_{\mu}\vec{\rho}_{\nu})N\
,$ (40)

where $g_{\rho}$ and $\kappa$ are resonance coupling constants, $\vec{\tau}$
are Pauli matrices, $\vec{\rho}_{\mu}$, $\vec{\pi}$ and $N$ refer to $\rho$
resonance, pion and nucleon, respectively. For $S_{11}$ channel, the $\rho$
exchange contribution to the invariant amplitude, Eq. (19), can be obtained:

$A=\frac{g_{\rho}^{2}\kappa(u-s)}{2m_{N}(t-m_{\rho}^{2})}\
,\,\,\,B=\frac{2g_{\rho}^{2}(\kappa+1)}{t-m_{\rho}^{2}}\ .$ (41)

Now, if a $1/m_{\rho}$ expansion is made, at leading order we have

$A=\frac{g_{\rho}^{2}\kappa(s-u)}{2m_{N}m_{\rho}^{2}}\
,\,\,\,B=-\frac{2g_{\rho}^{2}(\kappa+1)}{m_{\rho}^{2}}\ .$ (42)

Comparing with Eq. (22), we find that the $\rho$ meson exchange only
contributes to $c_{4}$ term [29]. As we already know from the discussions made
in section 3.1, the $c_{4}$ term will cause an $s^{-3/2}$ singularity after
partial wave projection. This is avoidable, if we do not make a $1/m_{\rho}$
expansion in the beginning. It can be seen that all these resonance exchange
amplitudes contain singularity of $s^{-1/2}$ type at most when $s=0$.
Therefore partial wave projections and chiral expansions do not commute, which
can be checked directly by evaluating $t$-channel $\rho$ exchange
contributions by making partial wave projections of Eqs. (25) and (28).

We further make an asymptotic expansion of the $\rho$ exchange contributions
to $T(S_{11})$ in the vicinity of $s=0$ and find that the first two most
singular terms are of type

$\displaystyle\frac{a+bs}{\sqrt{s}}\ ,$ (43)

which are not of type $\mathcal{O}(s^{-5/2})$ obtained if the $\rho$
propagator were expanded beforehand. Similar things happen if we introduce,
for example, a $u$-channel as well as $s$-channel $S_{11}$ resonance exchange.
In this situation one can prove that expanding the $N^{*}$ propagator in the
full amplitudes leads contributions to $c_{3}$ and $c_{4}$ terms in Eq. (22).
Not making a chiral expansion beforehand the resonance exchange contribution
to the partial wave amplitude can be expanded at $s=0$ and similar results as
Eq. (43) are again obtained, so do the $P_{11}$ resonance exchange
contributions.

Explicit expressions for resonance contributions to parameters, e.g., $a$ and
$b$ defined in Eq. (43) are obtainable. However, another obscure problem
occurs here. These coefficients depend on the mass parameters of the exchanged
resonances and do not vanish as the resonance mass gets large, which seems to
contradict the general expectation from the decoupling theorem [30][31].
777The reason behind this phenomenon is that as $s\to 0$ the interval of
partial wave integration diverges, make any “heavy” mass scale not heavy
anymore and cannot be naively integrated out. Without a deeper understanding
on this problem, we point out that the sign of contributions from different
sources can be different. For example, the $t$-channel $\rho$ exchange
contribution to parameter $a$ is
$a(\rho)=-\frac{g_{\rho}^{2}\kappa\left(m_{N}^{2}-m_{\pi}^{2}\right)}{64\pi
m_{N}}$; the $\frac{1}{2}^{+}$ baryon exchange contribution is
$a(N^{*+})=\frac{\left(g_{N^{*}}\right)^{2}\left(-m_{N}^{2}+m_{\pi}^{2}\right)\left(3m_{N}^{2}+\left(m_{N^{*}}\right)^{2}\right)}{128\pi
F^{2}m_{N^{*}}}$; whereas the $\frac{1}{2}^{-}$ resonance exchange
contribution is
$a(N^{*-})=\frac{\left(g_{N^{*}}\right)^{2}\left(m_{N}^{2}-m_{\pi}^{2}\right)\left(3m_{N}^{2}+\left(m_{N^{*}}\right)^{2}\right)}{128\pi
F^{2}m_{N^{*}}}$, which is different in sign as comparing with that of the
first two contributions. Hence, a conspiracy theory of cancellation is assumed
to overcome the problem of too large resonance contributions to parameter $a$,
or more accurately, $T(s)$ near $s=0$. In practice, we therefore use the
$\mathcal{O}(p^{1})$ $\chi$PT result plus a polynomial background as the input
$\mathrm{Im}_{L}T$, i.e.:

$\mathrm{Im}_{L}T(s)=\mathrm{Im}_{L}T^{(1)}(s)+\mathrm{Im}_{L}[\dfrac{a+bs}{\sqrt{s}}]\
,$ (44)

where $a$ and $b$ are simply two free parameters without relating to resonance
parameters anymore. The fit gives $N(s_{0})=0.57$, $a=-2.39$ GeV and $b=-6.27$
GeV-1, and one second sheet pole is found with $\sqrt{s}=0.95-0.25i$ GeV
without sizable spurious branch cut contributions.

Since the mass of the $\rho$ meson is fixed we also tried the case of
$\mathcal{O}(p^{1})$ $\chi$PT results plus the $\rho$ meson exchange term and
a polynomial. That is,

$\mathrm{disc}\,T(s)=\mathrm{disc}\,T^{(1)}(s)+\mathrm{disc}\,T^{\rho}(s)+\mathrm{d\mathrm{\mathrm{}}isc}\,[\dfrac{a+bs}{\sqrt{s}}]\
.$ (45)

In this case the $\rho$ meson exchange produces an extra arc in $s$ plane [9],
see Fig. 4.

Figure 4:  The $l.h.c.$ caused by $t$-channel $\rho$ meson exchange (circular
arc [9]); $u$-channel exchange (line segment from $c_{L}$ to $c_{R}$ ). The
branch point $d$ satisfies $|d|=m_{N}^{2}-m_{\pi}^{2}$ [9]

We get ${N(s_{0})=0.61,}$ $a=-7.88$ GeV, $b=-8.00$ GeV-1, and one second sheet
pole is found located at

$\displaystyle\sqrt{s}=0.90-0.20i\mathrm{GeV}\ .$ (46)

These solutions are not stable – there exist other solutions but with similar
behaviours. In all cases the contributions from the spurious branch cuts are
negligible as comparing with the results from the ${\cal O}(p^{2})$ case, and
the $N^{*}(890)$ pole location remains stable. It is noticed that the
contribution from the arc cut generated by $t$-channel $\rho$ exchange is very
small, e.g., it only contributes 1.5∘ at $\sqrt{s}=1.16$ GeV.

The “spectral” function in this case is plotted in Fig. 5. Different
contributions to the phase shift according to the PKU decomposition are
plotted in Fig. 6.

Figure 5: Comparison among different “spectral” functions. The singular
behaviors of $T(s)$ at $s=0$ are $\mathcal{O}(s^{-5/2})$,
$\mathcal{O}(s^{-1/2})$ and $\mathcal{O}(s^{0})$ for $\mathcal{O}(p^{2})$
$\chi$PT , model Eq. (45) and $Case$ II, respectively. Figure 6: Fit results
using Eq. (45). Phase shift decomposition: only contributions from physical
ingredients are plotted including their summation ‘Total’. It clearly
demonstrates that spurious contributions cancel each other, otherwise curve
‘Total’ cannot get close to the data.

Before closing the discussions on numerical calculations we would like to
stress that major physical outputs rely very little on the choice of the
cutoff parameter $\Lambda_{R}^{2}=1.48$GeV2. For example, setting
$\Lambda_{R}^{2}=2.0,2.5$GeV2 in model Eq. (45), lead to almost the same pole
location at $\sqrt{s}=0.897-i0.193$GeV.

In above, we have made a rather long and exhaustive analyses which has to be
stopped somewhere with some regrets. One is that all the calculations made in
this paper are performed at tree level only. At loop level, there are of
course dynamical cut contributions, like the circular cut. The latter is
estimated in Ref. [1] using the complete $\mathcal{O}(p^{3})$ $\chi$PT input
and it is found that the sign of the circular cut contribution may vary
depending on the choice of cutoff parameter, but always remains small in
magnitude: e.g., at $\sqrt{s}=1.16$ GeV its contribution to the phase shift is
0.2∘ when $s_{c}=0.32$ GeV2, and -1.7∘ when $s_{c}=-0.08$ GeV2. Since when
evaluating the circle there is no problem like what happens at $s=0$, we think
this estimation on the smallness of the circular cut contribution is at least
qualitatively reasonable. See Fig. 7, the cutoff parameter $s_{c}=0.32$GeV2
corresponds to evaluating the $l.h.c.$ region covered by the green dashed
circle, which can be estimated by chiral perturbation theory; whereas
$s_{c}=-0.08$GeV2 corresponds to that covered by the red dotted circle, which
is required by best fit. The estimations made in Refs. [1][3] pointed out that
the region where $\chi$PT calculation can be safely used is not enough to
generate the $N^{*}(890)$,888More precisely, the pole position is not stable.
For example when taking $s_{c}=0.32$ GeV2 the $N^{*}(890)$ degrades into two
deep virtual poles. i.e., certain help from the contribution in the region
$s\in(-\infty,0.32)$ is needed.999Taking $s_{c}=0.32$ GeV2 will cause
disasters in other channels as well [1]. The singularity in the “spectral”
function at $s=0$ seems to be helpful. It is realised that the rescue task is
easily fulfilled by looking at Fig. 5. The fit Case II only contains a weak
singularity at $s=0$, i.e., $T(0)\propto\mathrm{const}$ while its contribution
in the segment $(0.32,s_{L}]$ is much weaker than the $\mathcal{O}(p^{2})$
ones, but it still affords a pole. The real situation should be much more
optimistic.

Figure 7: Region of $l.h.c.$ being used in Ref. [1]. The cutoff parameter
$s_{c}=0.32$GeV2 corresponds to evaluating the $l.h.c.$ region covered by the
green dashed circle, which can be estimated by chiral perturbation theory;
whereas $s_{c}=-0.08$GeV2 corresponds to that covered by red dotted circle,
which is required by best fit.

It is noticed that the model Eq. (44) behaves quite like $\mathcal{O}(p^{2})$
$\chi$PT results in the region $(0.32,s_{L})$ and it is expected continuously
to work in the region $(-\Lambda^{2}_{L},-\epsilon)\cup(+\epsilon,0.32)$,
where $\Lambda^{2}_{L}$ is estimated to be around $R=m_{N}^{2}-m_{\pi}^{2}$
for example. We make a test by setting $\epsilon\simeq 0.05$ GeV2 and cut off
the peak around $s=0$ in the ‘spectral’ function when
$s\in(-\epsilon,+\epsilon)$. The $N^{*}(890)$ emerges stubbornly with a
location $\sqrt{s}=0.89-0.24i$ GeV.

Hence, we conclude that the $l.h.c.$ contributions in total to the phase shift
is sizable, based on which the $N^{*}(890)$ survives with a rather stable pole
location. Considering the level of accuracy of our calculations, we do not try
to give statistical error bars here in this paper.

##  5 Discussions and conclusions

It may be somewhat amazing to claim something new in a field under extensive
studies for more than half a century. However, according to our studies, a
subthreshold broad resonance has well chance of being existed if the $s$-wave
phase shift steadily rise above the threshold as a convex curve. The
discussion made in the end of the last section suggests that the kinematical
singularity structure at $s=0$ plays a rather important role. This is not
surprising. One even finds examples in extreme cases that a pole can be
generated totally for kinematic reasons. For example, in $J=0,I=2$ channel of
$\pi\pi$ scatterings, there exists a virtual pole which can be understood from
pure kinematical reasons and it brings important contributions to the phase
shift [6], and can be proved to exist rigorously [32]. Another example is the
companionate virtual state of the nucleon, which can also be explained for
pure kinematical reasons [3].

It is also interesting to notice that the $N^{*}(890)$ state may be related to
the lowest lying $1/2^{-}$ baryon states suggested by Azimov dated back in
1970 [33] named as $N^{\prime}$ there, having been searched for desperately
since then [34]. Contrary to the original proposal that the lowest lying
nucleon counterpart lie above the $\pi N$ threshold, or at least lie above the
nucleon mass (on the 1st sheet), the pole position named $N^{*}(890)$
determined in Refs. [1][2][3] as well as in this paper, easily escapes of all
bounds and limits set up previously [34]. The mass (width) difference may be
explained by a familiar mechanism that when a strong coupling gradually be
turned on, the pole will move from the real axis above the threshold to the
left of $s$ plane off the real axis. Nevertheless, it may also be possible
that the $N^{*}(890)$ ‘resonance’ be a virtual pole on $k$ plane, as already
suggested in Ref. [3]. In the latter situation, the ‘width’ of $N^{*}(890)$
does not need to have any relations with particle decays. There still remains
a lot of work to be done to identify $N^{\prime}$ and $N^{*}(890)$. We plan to
investigate this question in future.

The $N/D$ calculations discussed in this paper are of rather complicated
dynamical ones, however the production representation has been shown to be
useful in providing us a simple and pictorial way of understanding the essence
of the $N/D$ calculations: in the language of this work, the evidence on the
existence of $N^{*}(890)$ seems to be partly from the peculiar singularity
structure of the background integral defined in Eqs. (10). It reads that if
$T(s=0)$ does not vanish (or does not vanish fast enough), then an $s$-wave
subthreshold resonance exist, in the most attractive channels.101010Similar
observations are obtained, from a different point of view. [35]. This may even
be a rather universal phenomenon, if the background contributions are
universally negative, as suggested by quantum scattering theory [36] and
repeatedly verified by calculations in quantum field theories. [7] [4][6]

It is apparent that the existence of a light $1/2^{-}$ nucleon state is
crucial for the completion and establishment of the lowest lying $1/2^{-}$
octet baryons as suggested in Ref. [33], if ever it exists111111For more
information on the status of the octet baryons, see the talk of Igor
Strakovsky given at $EHS-2019$, York, UK, December 2019.. It will definitely
improve our understanding of strong interaction physics as well. For example,
the $N^{*}(890)$ state, if exists, will definitely force us to rethink the
possible physics behind $f_{0}(500)$ and $K(700)$. Another question it raises
is how to interpret spontaneous chiral symmetry breaking more properly. The
textbook explanation on this point is that the axial charge $Q_{A}$ commutes
with the strong interaction hamiltonian, hence if chiral symmetry were not
broken parity doublets would appear in nature. But what was observed as the
lowest lying $1/2^{-}$ nucleon is $N^{*}(1535)$, the non-degeneracy of its
mass comparing with the nucleon mass therefore indicates that chiral symmetry
is spontaneously broken. The emergence of $N^{*}(890)$ may also bring new
thinking on the related physics.

Finally, it is also pointed out in this paper, that there exist two virtual
poles located on the real axis, outside but very close to the $u$ channel cut
$[c_{L},c_{R}]$. Their existence is proven relying on the validity of chiral
expansions up to all orders.

The authors would like to thank Zhi-hui Guo and De-Liang Yao for helpful
discussions at various stage of this work, and Ulf-G. Meißner for a careful
reading of the manuscript and useful suggestions. Especially we thank Igor
Strakovsky at George Washington University, for very interesting information
on the $1/2^{-}$ octet baryons. This work is support in part by National
Nature Science Foundations of China under contract number 11975028 and
10925522, and by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - Project-ID 196253076 -TRR 110.

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##  Appendix A Phenomena of the cut $[c_{L},c_{R}]$ induced by $u$ channel
nucleon exchanges

The $u$ channel nucleon pole exchange diagram will contribute to the partial
wave amplitude a cut $\in[c_{L},c_{R}]$, with
$c_{L}=\frac{(m_{N}^{2}-m_{\pi}^{2})^{2}}{m_{N}^{2}}$ and
$c_{R}=m_{N}^{2}+2m_{\pi}^{2}$. The point $s=c_{L}$ is reached when
$z_{s}=-1$, $u=m_{N}^{2}$ and $s=c_{R}$ is reached when $z_{s}=+1$,
$u=m_{N}^{2}$ (see Eq. (27)). More precisely, at branch points $c_{L}$ and
$c_{R}$ one gets the leading term of the partial wave amplitude:

$\displaystyle{s\to{c_{L}}:}$
$\displaystyle\,\,\,T(s)\to-\frac{g^{2}m_{N}^{4}}{16\pi
F^{2}\left(4m_{N}^{2}-m_{\pi}^{2}\right)}\ln\frac{s-c_{L}}{c_{L}-c_{R}}\ ,$
$\displaystyle{s\to{c_{R}}:}$
$\displaystyle\,\,\,T(s)\to\frac{g^{2}m_{N}^{2}(m_{N}^{2}+2m_{\pi}^{2})}{\pi
F^{2}(4m_{N}^{2}-m_{\pi}^{2})}\ln\frac{c_{R}-c_{L}}{s-c_{R}}\ ,$ (47)

which are solely from the $u$ channel nucleon pole term, and are hence exact,
i.e., receiving no chiral corrections. Based on Eq. (A) one further finds

$\begin{split}&s\to c_{L},\quad S\simeq
A_{c_{L}}+B_{c_{L}}\ln\frac{s-c_{L}}{c_{L}-c_{R}}\ ,\\\ &s\to c_{R},\quad
S\simeq A_{c_{R}}+B_{c_{R}}\ln\frac{s-c_{R}}{c_{R}-c_{L}}\ ,\end{split}$ (48)

in which the coefficients read

$\begin{split}&A_{c_{L}}=A_{c_{R}}=1+\frac{g^{2}m_{N}m_{\pi}}{8\pi
F^{2}}+O\left(m_{\pi}^{3}\right)\ ,\\\
&B_{c_{L}}=B_{c_{R}}=\frac{g^{2}m_{N}m_{\pi}}{16\pi
F^{2}}+O\left(m_{\pi}^{3}\right)\ .\end{split}$ (49)

These equations are obtained from Born term calculations, $A_{c_{L}}$
($B_{c_{L}}$) and $A_{c_{R}}$ ($B_{c_{R}}$) differ at $O(m_{\pi}^{3})$ level.
It is worth emphasizing that $A_{c_{L}}$ and $A_{c_{R}}$ may receive chiral
corrections, but $B_{c_{L}}$ and $B_{c_{R}}$ do not, since the latter is
related to the residue of the $u$ channel nucleon pole.

One important conclusion one can draw is that $S(c_{L}),S(c_{R})\to-\infty$
which are exact (correct at least to any order of chiral expansions) and are
immune of any loop corrections. Remember that $S=+1$ at $s_{L}$ and $s_{R}$ by
definition, and $S(s)$ is real when $s\in(s_{L},c_{L})\cup(c_{R},s_{R})$, one
finds that there have to be two $S$ matrix zeros: one below $c_{L}$ and
another above $c_{R}$, on the real axis.121212In a $K$-matrix unitarization,
the $S$ matrix no longer diverges at $s=c_{L},c_{R}$, the two virtual poles
however still exist, and no major conclusions change. Their locations (denoted
by $v_{L}$ and $v_{R}$) are:

$\begin{split}&v_{L}=c_{L}-(c_{R}-c_{L})e^{-A_{c_{L}}/B_{c_{L}}}\ ,\\\
&v_{R}=c_{R}+(c_{R}-c_{L})e^{-A_{c_{R}}/B_{c_{R}}}\ .\end{split}$ (50)

Even more surprisingly, these two virtual poles in total give a large
contribution to the phase shift. E.g., they give roughly $50^{\circ}$ at
$\sqrt{s}=1.16$GeV, which seems to completely destroy the picture presented in
Figs. 3 and 6. The solution to this apparent paradox is rather tricky, which
we discuss in the following.

In the derivation of the production representation, Eqs. (10) and (12), it is
generally assumed that the branch cut singularity structure of $S^{{cut}}$
($=\exp\\{2i\rho(s)f(s)\\}$) and $f(s)$ (or $\ln{S^{cut}}$) being the same.
This is not always being true: when $S^{cut}$ is real and negative at certain
point, $\ln S^{cut}$ will have to be discontinuous when the sign of the
imaginary part of $S^{cut}$ changes. This situation indeed happens in the
present situation. Recall that at $O(p^{1})$ level

$\begin{split}T^{(1)}=&-\frac{g^{2}\left(-2m_{N}^{2}m_{\pi}^{2}-m_{N}^{2}s-m_{\pi}^{2}s+s^{2}\right)}{32\pi
F^{2}\left(s-m_{N}^{2}\right)}+\frac{-m_{N}^{2}-m_{\pi}^{2}+s}{32\pi F^{2}}\\\
&+\frac{m_{N}\left(-m_{N}^{2}+m_{\pi}^{2}+s\right)}{32\pi
F^{2}\sqrt{s}}-\frac{g^{2}m_{N}\left(-m_{N}^{4}+m_{N}^{2}m_{\pi}^{2}+m_{N}^{2}s+2m_{\pi}^{2}s\right)}{32\pi
F^{2}\left(s-m_{N}^{2}\right)\sqrt{s}}\\\
&+\frac{g^{2}m_{N}^{2}s^{2}\left(-m_{N}^{2}-m_{\pi}^{2}+s\right)}{16\pi
F^{2}\left(s-c_{L}\right)^{2}\left(s-c_{R}\right)^{2}}\left(\frac{m_{N}^{2}}{s}\left(s-c_{L}\right)\left(\log\frac{s-c_{L}}{s-c_{R}}+\log\frac{m_{N}^{2}}{s}\right)-s\rho(s)^{2}\right)\\\
&-\frac{g^{2}m_{N}^{3}s^{2}\left(-m_{N}^{2}+m_{\pi}^{2}+s\right)}{16\pi
F^{2}\left(s-c_{L}\right)^{2}\left(s-c_{R}\right)^{2}\sqrt{s}}\left(\left(s-c_{R}\right)\left(\log\frac{s-c_{L}}{s-c_{R}}+\log\frac{m_{N}^{2}}{s}\right)-s\rho(s)^{2}\right)\
.\end{split}$ (51)

It is seen from the above expressions that the cut
$L\in(-\infty,0)\cup(c_{L},c_{R})$. There are two sources contributing to the
imaginary part of $T(s)$ when $s\in(-\infty,0)$: one comes from the
kinematical $\sqrt{s}$ while another comes from the logarithmic function; when
$s\in(c_{L},c_{R})$ $\mathrm{Im}T$ solely comes from the $u$ channel nucleon
pole exchange. On $(c_{L},c_{R})$ the imaginary part of $T^{(1)}$ reads:

$\displaystyle\mathrm{Im}T^{(1)}(s)=\frac{g^{2}m_{N}^{2}s^{2}}{16F^{2}\left(s-c_{L}\right)^{2}\left(s-c_{R}\right)^{2}}$
$\displaystyle\times\left[\dfrac{m_{N}^{2}}{s}(s-c_{L})\left(-m_{N}^{2}-m_{\pi}^{2}+s\right)-\dfrac{m_{N}}{\sqrt{s}}(s-c_{R})\left(-m_{N}^{2}+m_{\pi}^{2}+s\right)\right]\
,$ (52)

from which it is seen that the $\mathrm{Im}T^{(1)}$ develops a zero at
$s=s_{c}\simeq m_{N}^{2}-\frac{m_{\pi}^{4}}{2m_{N}^{2}}$ and changes sign when
$s$ crosses $s_{c}$. It is important to realize that Eq. (A) is immune of
chiral perturbation corrections.

Figure 8: Real (left) and imaginary (right) part of $S^{cut}(s)$ when $s$ lies
in $(c_{L},c_{R})$. The dot-dashed line comes from N/D solution of Eq. (44),
the yellow solid line is obtained from $O(p^{1})$ $\chi$PT results.

From Fig. 8, it is found that the imaginary part of $S^{cut}$ vanishes at one
point $s_{c}$ which is close to $m_{N}^{2}$ meanwhile the real part of
$S^{cut}$ is negative. In Fig. 8 $S^{cut}$ is calculated by

$S^{cut}={S^{phys}\over\prod_{p}S^{p}\times S^{v_{L}}\times S^{v_{R}}}$ (53)

where the newly found virtual poles are included, and $S^{phys}$ are
approximated by both the unitary amplitude obtained by fitting Eq. (44) and
the pure $O(p^{1})$ perturbation amplitude. 131313The ‘$O(p^{1})$ calculation’
in Fig. 8 is done via Eq. (53) in the following way: $S^{phys}$ is calculated
in $O(p^{1})$, ‘poles’ are picked up by searching for zeros of $S^{phys}$,
though they are actually not second sheet poles for lacking of unitarity. From
Fig. 8 it is seen that at $s=s_{c}$, $\ln S^{cut}$ has to develop a
discontinuity and hence a branch cut emerges crossing $s_{c}$. It is
numerically checked that the cut is an arc on the complex $s$-plane in $N/D$
solutions. It should be emphasized here that this unexpected additional cut
does not pollute $S^{cut}$, since across the cut the discontinuity of $\ln
S^{cut}$ is $2i\pi$ and has no influence to the value of an exponential.
Though not producing any trouble in the analyticity structure of $S^{cut}$,
the additional cut of $\ln S^{cut}$, or the dispersive representation of
function $f$ defined in Eq. (10) has to be changed since the integration
contour has to be modified.

The problem found above is rather severe since the distorted contour may
depend on numerics and hence being impossible to control. However, it can be
overcome by the following consideration. Define

$\bar{f}(s)=\dfrac{\ln-S^{cut}}{2i\rho(s)}-\dfrac{\pi}{2\bar{\rho}(s)}$ (54)

where the function $\bar{\rho}(s)$ is the ‘deformed’ $\rho(s)$ with its cut
$\in[s_{L},s_{R}]$, while the cut of the latter is defined on
$(-\infty,s_{L}]\cup[s_{R},+\infty)$. Notice that $\bar{\rho}(s)$ and
$\rho(s)$ are identical in the physical region. Function $\bar{f}(s)$ is
identical to $f(s)$ when $s$ in the physical region as well but differs in cut
alignment. Particularly $\bar{f}$ no longer contains the arc cut of $f$, since
$s_{c}$ is not a branch point of $\bar{f}$ anymore. However, $\bar{f}$
contains an additional cut induced by $\bar{\rho}$ which is absent in $f$.
Both the two terms on the $r.h.s.$ of Eq. (54) contain cuts on
$[s_{L},s_{R}]$, but the two cuts cancel each other when $s\in[c_{R},s_{R}]$.
Hence the left cut of $\bar{f}$ on the real axis is actually
$[-\infty,c_{R}]$, comparing with the left cut of $f$ on the real axis:
$[-\infty,s_{L}]\cup[c_{L},c_{R}]$. The dispersive integral representation of
$\bar{f}$ can be written as:

$\begin{split}\bar{f}(s)=&\dfrac{s}{\pi}\int_{L}\dfrac{\mathrm{Im}[\ln{-S^{cut}(s^{\prime})}/(2i\rho(s^{\prime}))]}{s^{\prime}(s^{\prime}-s)}ds^{\prime}\\\
&{+\dfrac{s}{\pi}\int_{s_{L}}^{c_{L}}\dfrac{\mathrm{Im}[\ln{-S^{cut}(s^{\prime})}/(2i\rho(s^{\prime}))]}{s^{\prime}(s^{\prime}-s)}ds^{\prime}}\\\
&{-s\int_{s_{L}}^{c_{R}}\dfrac{\mathrm{Im}[1/(2\bar{\rho}(s^{\prime}))]}{s^{\prime}(s^{\prime}-s)}ds^{\prime}}\
.\end{split}$ (55)

In the first term on the $r.h.s.$ of the above equation, the integration
domain $L=(-\infty,s_{L}]\cup[c_{L},c_{R}]$, i.e., the same as that in Eq.
(10). Actually the first term is identical to $f$ defined in Eq. (10) as in
the integrand $S^{cut}$ can actually be replaced by $S^{phys}$. To prove this
it is realized that, firstly, on $(-\infty,s_{L}]$,
$\mathrm{Im}[\ln{-S^{cut}}/(2i\rho)]=-\ln{|S^{cut}|}/(2\rho)$, and
$|S^{cut}|=|S^{phys}|$. Secondly, the integral on $[c_{L},c_{R}]$ can be
recast as

$\begin{split}&\dfrac{s}{\pi}\int_{c_{L}}^{c_{R}}\dfrac{\mathrm{Im}[\ln{-S^{cut}(s^{\prime})}/(2i\rho(s^{\prime}))]}{s^{\prime}(s^{\prime}-s)}ds^{\prime}=\dfrac{s}{2i\pi}\int_{c_{L}}^{c_{R}}\dfrac{\ln[S^{cut}_{+}/S^{cut}_{-}]}{2i\rho(s^{\prime})s^{\prime}(s^{\prime}-s)}ds^{\prime}\\\
&=\dfrac{s}{2i\pi}\int_{c_{L}}^{c_{R}}\dfrac{\ln[S^{phys}_{+}/S^{phys}_{-}]}{2i\rho(s^{\prime})s^{\prime}(s^{\prime}-s)}ds^{\prime}=\dfrac{s}{\pi}\int_{c_{L}}^{c_{R}}\dfrac{\mathrm{Im}[\ln{S^{phys}(s^{\prime})}/(2i\rho(s^{\prime}))]}{s^{\prime}(s^{\prime}-s)}ds^{\prime}\
,\end{split}$ (56)

where $S_{\pm}\equiv S(s\pm i\epsilon)$. Hence we complete the proof.

The second integral in Eq. (55) is analytically integrable once it is realized
that $S^{cut}$ is positive definite along $[s_{L},c_{L}]$, and the third
integral is also integrable. The sum of the two integrals gives a contribution
to the phase shift exactly canceled by the contribution from the two virtual
poles, when their positions are taken at $c_{L}$ and $c_{R}$. If the two
virtual pole locations are fixed by Eq. (50), the net effects of the sum of
virtual poles and the additional cut are vanishingly small (e.g., of order of
${10^{-3}}$ degrees at $\sqrt{s}=1.16$GeV). Therefore, The calculation using
Eq. (10) is still valid with high accuracy, when one ignores the existence of
the two virtual poles. Further, it is worth emphasizing that after such a
surgery, there no longer exists the unwanted cut in $\ln S^{cut}$ crossing
$s_{c}$, as checked by numerical analyses. Moreover, it may be worth pointing
out that in numerical analyses there may appear additional cuts in $\ln
S^{cut}$ as well on the $s$ plane in distance, which is caused by the peculiar
analyticity property of the logarithmic function. But it is easy to prove that
it is not hazardous and can be simply ignored.

The discussions made in this appendix can be extended to higher partial waves
as well and some very interesting results appear, which will be presented
elsewhere.