11institutetext: Grupo de Física Nuclear e IUFFyM, Universidad de Salamanca,
E-37008 Salamanca, Spain 22institutetext: Departamento de Física, Universidad
de Murcia, E-30071 Murcia, Spain

# Non-perturbative methods for $NN$ singular interactions

D.R. Entem 11<EMAIL_ADDRESS>J.A. Oller 22<EMAIL_ADDRESS>
###### Abstract

Chiral Effective Field Theory ($\chi$EFT) has been extensively used to study
the $NN$ interaction during the last three decades. In Effective Field
Theories (EFTs) the renormalization is performed order by order including the
necessary counter terms. Due to the strong character of the $NN$ interaction a
non-perturbative resummation is needed. In this work we will review some of
the methods proposed to completely remove cutoff dependencies. The methods
covered are renormalization with boundary conditions, renormalization with one
counter term in momentum space (or equivalently substractive renormalization)
and the exact $N/D$ method. The equivalence between the methods up to one
renormalization condition will be checked showing results in the $NN$ system.
The exact $N/D$ method allows to go beyond the others, and using a toy model
it is shown how it can renormalize singular repulsive interactions.

##  1 Introduction

Chiral Effective Field Theory ($\chi$EFT) has already a long history applied
to the $NN$ system. It was first proposed by Weinberg Weinberg (1990, 1991) in
the 1990’s motivated by the success of Chiral Perturbation Theory ($\chi$PT)
in the pion-pion Gasser and Leutwyler (1983a, b, 1985) and pion-nucleon Gasser
_et al._ (1988) sectors, constraining the interactions to those consistent
with Chiral symmetry. Weinberg realized that they have substantial differences
with the $NN$ sector since in the last one the interaction is stronger and
reducible diagrams have a nucleon mass enhancement which breaks the Chiral
expansion. For this reason he proposed to use the rules of $\chi$PT to build
the potential instead of the scattering amplitude, and perform a non-
perturbative resummation of reducible diagrams using a Schrödinger or
Lippmann-Schwinger equation. This is a similar situation, although for
different reasons, to the study of an electron-proton system. Here the
electromagnetic interaction is perturbative, but in order to study the
Hydrogen atom, a non-perturbative resummation of reducible diagrams with the
lowest-order one photon exchange interaction is needed. In the $NN$ system
also bound states appear, as the deuteron or heavier nuclei, and this non-
perturbative resummation is unavoidable, at least in some channels.

Of course, the two systems are essentially different. While the
electromagnetic scattering is perturbative and renormalizable, the strong $NN$
interaction is non-perturbative and non-renormalizable. Here we will refer as
a renormalizable theory to a theory that has a finite number of divergent
amplitudes that can be renormalized by a finite number of renormalization
conditions, which is the usual concept of renormalizability for a fundamental
theory. Perturbation theory and standard renormalization techniques have shown
an unprecedented accuracy (1990) (editor). Due to the success of QED in the
1950’s there was a big effort to build the strong interaction in terms of
nucleons and pions. Unfortunately this effort was unsuccessful and it started
a long history for the study of $NN$ interactions Machleidt and Slaus (2001);
Machleidt (2017).

As mentioned before, $\chi$PT has been shown to be very successful
constraining the couplings between pions, or nucleons and pions, to those
consistent with Chiral symmetry. For example, the pseudo-scalar $\pi N$
coupling is not allowed by Chiral symmetry and the pseudo-vector coupling
arises naturally. However this brings us to the problem in the 1950’s, the
pseudo-vector coupling is non-renormalizable. The way out of this dilemma is
to consider the theory of nucleons and pions, not as a fundamental theory, but
as an Effective Field Theory (EFT) valid only in the low energy regime.

In the EFT framework one requires renormalizability in a different way. One in
principle has an infinite number of divergent amplitudes, but it also has an
infinite number of terms that can absorb the divergences. Weinberg’s
assumption Weinberg (1979) is that if one includes all possible terms allowed
by symmetry requirements, then the theory can be renormalized and produces the
most general description of the system for the energy regime where is valid.

In practice one has to deal with a finite number of amplitudes and a finite
number of terms, and the theory is organized by a power counting rule, which
defines the diagrams and the terms to be included at a certain order in the
expansion. The first power counting rule for the $NN$ system, introduced by
Weinberg Weinberg (1990), is based on naive dimensional analysis (NDA), which
in the case of a perturbative calculation, fulfills this requirement. In non-
perturbative calculations the situation concerning the power counting to be
assigned to $NN$ contact terms is not clear and there has been a lot of
discussion about different power counting rules Kaplan  _et al._ (1998a);
Birse (2006); Pavón Valderrama (2011a, b); Long and Yang (2011, 2012a, 2012b);
Epelbaum and Meißner (2013); Epelbaum and Gegelia (2009); Machleidt and Entem
(2010); Marji  _et al._ (2013); Epelbaum  _et al._ (2020). However, although
NDA could not be the most efficient and it is not clear if it can be
inconsistent in non-perturbative calculations Kaplan  _et al._ (1996); Nogga
_et al._ (2005), nowadays is probably the most popular. This manuscript is not
intended to discuss the different power counting rules and NDA will be used in
the evaluation of the finite-range part of the $NN$ potential.

$\chi$EFT at lowest order has contributions from two contact interactions and
the one-pion-exchange interaction. These interactions are singular so that
they can not be included in an Schrödinger or Lippmann-Schwinger equation
without regularization since they don’t vanish for infinite momentum. This is
a common situation in nuclear physics and the way to proceed usually is to
include a regulator function that kills the interaction at short range or high
momenta. The regulator function depends on a cut-off scale parameter $\Lambda$
which in many applications in nuclear physics is on the GeV scale. When
singular interactions are present a strong dependence on the cut-off is
unavoidable if no further constrains are used. In the EFT program one uses
$\Lambda$ dependent contact interactions fixing some low energy observable to
achieve regulator independence on the observables. At higher orders
irreducible loop diagrams appear and the contact terms are also used here to
absorb the infinities generated by such diagrams. NDA allows to do it.

There is a big discussion about how this program should be implemented. There
are mainly two different points of view. On one hand one takes $\Lambda$
bigger than the low-energy scales of the EFT ($\sim m_{\pi}$ in $\chi$EFT) an
smaller than the high energy scale of the EFT ($\sim 4\pi f_{\pi}\sim 1$ GeV
in $\chi$EFT). Then one varies the parameter $\Lambda$ in this window, always
fixing the same low energy constraints, and finds an approximate cut-off
independence in a rather small interval of possible values of the cut-off.
This approach has been shown to be phenomenologically very fruitful and the
high-precision $NN$ potentials in $\chi$EFT are based on this point of view.

On the other hand, one can follow the standard renormalization procedure and
take $\Lambda$ to infinity, so removing completely the cut-off dependence.
This is the way QED is renormalized and has been shown to be very successful
in this renormalizable theory. However in $\chi$EFT there is a serious
problem, since the non-perturbative resummation of reducible diagrams can not
be performed analytically, and one has to use numerical methods to sum them.

Which is the correct approach is something under debate and there have been
arguments in favor and against both approaches. We don’t want to enter on this
discussion here and refer the reader to some recent review van Kolck (2020).
Nonetheless, the scattering amplitudes are constrained by unitarity and
analyticity which allows us to introduce a new method Entem and Oller (2017);
Oller and Entem (2019). The former relies on a derivation of the exact
discontinuity along the left-hand cut of a two-body scattering amplitude
together with the implementation of unitarity and analyticity by employing the
$N/D$ method Chew and Mandelstam (1960).

In this work we only want to give an overview of some of the approaches that
have been developed to do the non-perturbative resummation with singular
interactions using the second approach. Unfortunately, these rely on numerical
solutions of scattering problem and there is no proof of the exact convergence
of the result in the limit $\Lambda\to\infty$. However, as we will see, quite
different approaches agree for high values of the $\Lambda$ parameter, which
gives confidence about the convergence of the result for the low energy
regime. Also a toy model with an exact solution will be employ to check the
agreement of renormalization of singular interactions with the underlying
theory.

The paper is organized as follows. In Section 2 we write down the $\chi$EFT
potentials up to NNLO to be used in all calculations. In Section 3 we
introduce the method of renormalization with boundary conditions performed in
coordinate space. The problem of renormalization of singular repulsive
interactions will be presented using a toy model. In Section 4 we turn into
momentum space and show how one can renormalize with one counter term and the
equivalence with the previous approach and introduce substractive
renormalization. In Section 5 we present the more recent approach based on the
$N/D$ method, but considering the non-perturbative resummation also along the
left-hand cut. We will show the equivalence with previous approaches and show
how it can go beyond. We will use a toy model to check the agreement of
renormalization of singular interactions with the underlying theory. We will
end in Section 6 with a summary and conclusions.

##  2 $\chi$EFT potentials up to NNLO

$\chi$EFT potentials were derived long time ago. The first contributions were
given by Weinberg Weinberg (1990, 1991) and very soon after the first nuclear
potentials were obtained by Ordoñez and van Kolck Ordóñez and van Kolck
(1992); Ordóñez  _et al._ (1994, 1996) in coordinate space up to NNLO and
regularized by a cut-off function. After that, momentum space potentials using
dimensional regularization for loop diagrams were developed Kaiser  _et al._
(1997); Epelbaoum  _et al._ (1998); Epelbaum  _et al._ (2000) also up to NNLO.
However it was not until 2003 that $\chi$EFT reached high precision when the
first chiral potential at N3LO was developed by the Idaho group Entem and
Machleidt (2003); Machleidt and Entem (2011) that was able to describe the
$NN$ scattering data with a $\chi^{2}/{\rm d.o.f}$ of the order of one,
similar to what the high-precision potentials of the 90’s had achieved Stoks
_et al._ (1994); Wiringa  _et al._ (1995); Machleidt  _et al._ (1996);
Machleidt (2001). Very soon after the Bochum group developed an N3LO potential
Epelbaum  _et al._ (2005) and during the last years calculations have gone up
to N4LO Entem  _et al._ (2015) and potentials at this order have been
developed Entem  _et al._ (2017); Reinert  _et al._ (2018).

Since different potentials used sometimes slightly different power countings,
we give here the potentials up to NNLO that are going to be used in all the
present calculations. Here we only give finite range interactions, zero range
interactions given by the contact terms, will be included by renormalization
conditions.

We write the potential in momentum space in the usual form

$\displaystyle V({\vec{p}}^{\prime},\vec{p})$ $\displaystyle=$
$\displaystyle\>\,V_{C}\>\,+\vec{\tau}_{1}\cdot\vec{\tau}_{2}\,W_{C}$ (1)
$\displaystyle+$
$\displaystyle\left[\,V_{S}\>\,+\vec{\tau}_{1}\cdot\vec{\tau}_{2}\,W_{S}\,\>\,\right]\,\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}$
$\displaystyle+$
$\displaystyle\left[\,V_{LS}+\vec{\tau}_{1}\cdot\vec{\tau}_{2}\,W_{LS}\right]\,\left(-i\vec{S}\cdot(\vec{q}\times\vec{k})\,\right)$
$\displaystyle+$
$\displaystyle\left[\,V_{T}\>\,+\vec{\tau}_{1}\cdot\vec{\tau}_{2}\,W_{T}\,\>\,\right]\,\vec{\sigma}_{1}\cdot\vec{q}\,\,\vec{\sigma}_{2}\cdot\vec{q}$
$\displaystyle+$ $\displaystyle\left[\,V_{\sigma
L}+\vec{\tau}_{1}\cdot\vec{\tau}_{2}\,W_{\sigma
L}\,\right]\,\vec{\sigma}_{1}\cdot(\vec{q}\times\vec{k}\,)\,\,\vec{\sigma}_{2}\cdot(\vec{q}\times\vec{k}\,)\,,$

At LO there is only the one-pion exchange interaction given by

$\displaystyle W_{T}^{\rm
LO}=-\frac{g_{A}^{2}}{4f_{\pi}^{2}}\frac{1}{q^{2}+m_{\pi}^{2}}\,,$ (2)

where $\vec{q}$ is the momentum transfer.

At NLO one-loop contributions with lowest-order $\pi N$ vertexes appear. The
contribution is given by

$\displaystyle W_{C}^{\rm NLO}$ $\displaystyle=$ $\displaystyle-{L(q)\over
384\pi^{2}f_{\pi}^{4}}\left[4m_{\pi}^{2}(5g_{A}^{4}-4g_{A}^{2}-1)+q^{2}(23g_{A}^{4}-10g_{A}^{2}-1)+{48g_{A}^{4}m_{\pi}^{4}\over
w^{2}}\right]\,,$ (3) $\displaystyle V_{T}^{\rm NLO}$ $\displaystyle=$
$\displaystyle-{1\over q^{2}}V_{S}^{\rm NLO}\;=\;-{3g_{A}^{4}\over
64\pi^{2}f_{\pi}^{4}}L(q)\,,$ (4)

with

$\displaystyle w$ $\displaystyle=$ $\displaystyle\sqrt{4m_{\pi}^{2}+q^{2}}\,,$
(5) $\displaystyle L(q)$ $\displaystyle=$ $\displaystyle{w\over
q}\ln\bigg{(}\frac{w+q}{2m_{\pi}}\bigg{)}\,,$ (6)

At NNLO one-loop diagrams with one vertex of the NLO $\pi N$ Lagrangian
appears and give

$\displaystyle V_{C}^{\rm NNLO}$ $\displaystyle=$
$\displaystyle{3g_{A}^{2}\over 16\pi
f_{\pi}^{4}}\bigg{[}\frac{g_{A}^{2}m_{\pi}^{5}}{16m_{N}w^{2}}-\bigg{\\{}2m_{\pi}^{2}(2c_{1}-c_{3})-q^{2}\left(c_{3}+\frac{3g_{A}^{2}}{16m_{N}}\right)\bigg{\\}}\tilde{w}^{2}A(q)\bigg{]}\,,$
(7) $\displaystyle W_{C}^{\rm NNLO}$ $\displaystyle=$
$\displaystyle{g_{A}^{2}\over 128\pi
m_{N}f_{\pi}^{4}}\bigg{[}\frac{3g_{A}^{2}m_{\pi}^{5}}{w^{2}}-\bigg{\\{}4m_{\pi}^{2}+2q^{2}-g_{A}^{2}(4m_{\pi}^{2}+3q^{2})\bigg{\\}}\tilde{w}^{2}\bigg{\\}}A(q)\bigg{]}\,,$
(8) $\displaystyle V_{T}^{\rm NNLO}$ $\displaystyle=$ $\displaystyle-{1\over
q^{2}}V_{S}^{\rm NNLO}={9g_{A}^{4}\over 512\pi
m_{N}f_{\pi}^{4}}\tilde{w}^{2}A(q)\,,$ (9) $\displaystyle W_{T}^{\rm NNLO}$
$\displaystyle=$ $\displaystyle-{1\over q^{2}}W_{S}^{\rm
NNLO}=-{g_{A}^{2}\over 32\pi
f_{\pi}^{4}}\bigg{[}\bigg{(}c_{4}+\frac{1}{4m_{N}}\bigg{)}w^{2}-\frac{g_{A}^{2}}{8m_{N}}(10m_{\pi}^{2}+3q^{2})\bigg{]}A(q)\,,$
(10) $\displaystyle V_{LS}^{\rm NNLO}$ $\displaystyle=$
$\displaystyle{3g_{A}^{4}\over 32\pi m_{N}f_{\pi}^{4}}\tilde{w}^{2}A(q)\,,$
(11) $\displaystyle W_{LS}^{\rm NNLO}$ $\displaystyle=$
$\displaystyle{g_{A}^{2}(1-g_{A}^{2})\over 32\pi
m_{N}f_{\pi}^{4}}w^{2}A(q)\,,$ (12)

with

$\displaystyle\tilde{w}$ $\displaystyle=$
$\displaystyle\sqrt{2m_{\pi}^{2}+q^{2}}\,,$ (13) $\displaystyle A(q)$
$\displaystyle=$ $\displaystyle{1\over
2q}\arctan\left(\frac{q}{2m_{\pi}}\right)\,.$ (14)

##  3 Renormalization with boundary conditions

In this section we will overview the so called renormalization with boundary
conditions. This was already used in molecular physics where singular
interactions are usually used. The method applied to the $NN$ system was
introduce in Refs. Nogga  _et al._ (2005); Pavón Valderrama and Arriola
(2006a, b). In Ref. Pavón Valderrama and Arriola (2006a) the uncoupled case is
worked out, while the coupled case is covered in Ref. Pavón Valderrama and
Arriola (2006b). Here we only reproduce a few details about the method and
some results for comparison with other approaches. We refer the interested
reader to these publications, where many details and results for all partial
waves can be found.

We will consider for simplicity just the uncoupled case. It starts with the
Schrödinger equation in some partial wave which is written as

$\displaystyle\bigg{[}-\frac{1}{2m}\frac{d^{2}}{dr^{2}}+\frac{l(l+1)}{2m}\frac{1}{r^{2}}+V(r)\bigg{]}u_{El}(r)=Eu_{El}(r)$
(15)

When the potential is regular or diverges slower than $1/r^{2}$ at $r\to 0$,
near the origin we have the two well known independent solutions $u_{l}(r)\sim
r^{l+1}$ and $u_{l}(r)\sim r^{-l}$. The first solution is regular at the
origin while the second gives a non-normalizable wave function and so only the
first one is considered to give physical states. This is the standard way to
solve the Schrödinger equation.

However when we have a potential that fulfills

$\displaystyle\lim_{r\to 0}|r^{2}V(r)|=\infty$ (16)

the dominant term at the origin is not the centrifugal term any more and the
approximate solutions at the origin are not the previous ones.

Now we have two possibilities. The first one is

$\displaystyle\lim_{r\to 0}r^{2}V(r)=+\infty$ (17)

so we have short range repulsion. In this case the equation can still be
solved. There are two solutions, one convergent to zero and one divergent.
Again, the divergent is left out, and the solution is well defined.

However in the case

$\displaystyle\lim_{r\to 0}r^{2}V(r)=-\infty$ (18)

we have two oscillatory solutions, with oscillations with lower and lower wave
lengths when one approaches the origin. Now the limit of the wave function is
not well defined at the origin and the Schrödinger equation can not be solved
in the standard way.

The idea of the method is to fix this limit at some energy (usually zero
energy) by fixing some low energy observable (usually scattering length,
scattering volume, $\ldots$) and then get the solution at a different
(low)energy using the orthogonality between the wave functions with different
energies. This is fulfilled with the condition

$\displaystyle\frac{u^{\prime}_{E_{1}l}(r_{c})}{u_{E_{1}l}(r_{c})}=\frac{u^{\prime}_{E_{2}l}(r_{c})}{u_{E_{2}l}(r_{c})}$
(19)

at some short-range point $r_{c}$. Notice that since the Schrödinger equation
is a linear second order differential equation the logarithmic derivative at
$r_{c}$ defines uniquely the solution up to the normalization constant.

If we consider the equation at zero energy111We consider here the case
$\lim_{r\to\infty}rV(r)=0$. the asymptotic solution for $l=0$ is

$\displaystyle\lim_{r\to\infty}u_{00}(r)\to N\bigg{(}1-\frac{r}{a}\bigg{)}$
(20)

where $N$ is a normalization constant and $a$ is the scattering length. The
method integrates-in the equation at zero energy with the boundary conditions
Eq. (20) and using the relation Eq. (19) integrates-out the equation to obtain
the wave function at a non-zero energy. From the asymptotic behavior at
$r\to\infty$ the phase-shift can be evaluated.

We first consider the singlet ${}^{1}S_{0}$ partial wave where the LO
interaction without contact terms is regular. The potential is given by

$\displaystyle V^{\rm LO}_{1S_{0}}(r)$ $\displaystyle=$
$\displaystyle-\frac{g_{A}^{2}m_{\pi}^{3}}{16\pi f_{\pi}^{2}}\frac{e^{-x}}{x}$
(21)

with $x\equiv m_{\pi}r$.

For regular interactions one of course recover the original solution if one
fix the scattering length to the solution of the regular interaction. This is
shown in Fig. 1 where we show results for the potential Eq. (21) using the
parameters of Table 1. The purple dots show the solution with standard regular
boundary conditions and the purple line the result fixing the scattering
length to the solution of the potential with a value $a=-0.91299$ fm.

Table 1: Parameters used in the calculations. $\hbar c$ | 197.326963 | MeV$\cdot$fm
---|---|---
$g_{A}$ | 1.26 |
$m_{\pi}$ | 138.039 | MeV
$f_{\pi}$ | 92.4 | MeV
$m_{N}$ | 938.919 | MeV
$c_{1}$ | -0.74 | GeV-1
$c_{3}$ | -3.61 | GeV-1
$c_{4}$ | 2.44 | GeV-1

However the method can generate different solutions that will differ by a zero
range interaction in the potential as we will see later. We can fix the
scattering length to the Granada value $a=-23.735$ fm and we get the gold
line. We include for comparison the result in pionless EFT fixing the same
value of the scattering length with a blue line and the Granada $np$ phase-
shift analysis Pérez  _et al._ (2013) by the solid dots with error bars.

Figure 1: Phase shift in the ${}^{1}S_{0}$ $NN$ partial wave. Black dots with
error bands correspond to the results from the Granada $np$ phase-shift
analysis Pérez  _et al._ (2013). The purple dots shows the result for OPE with
regular boundary conditions, and the purple line fitting to $a=-0.91299$ fm.
The blue line corresponds to the effective range expansion at lowest order
using the scattering length from the Granada analysis $a=-23.735$ fm. The
gold, green and red lines give the result for renormalization with boundary
condition of OPE, NLO and NNLO $\chi$EFT potentials, respectively.

However if we consider the potentials at NLO or NNLO the interactions are
singular and attractive so one low energy parameter has to be fixed. The
$r$-space representation of the potentials of Sec. 2 was also given in Ref.
Kaiser  _et al._ (1998). Since the potentials are divergent in the momentum
transfer, the Fourier transform has to be done using a spectral representation
with the necessary subtractions to remove the divergency that only affect
zero-range interactions.

We give here the potentials at NLO and NNLO in the ${}^{1}S_{0}$ partial wave
for completeness

$\displaystyle V^{\rm NLO}_{{}^{1}S_{0}}(r)$ $\displaystyle=$
$\displaystyle-\frac{m_{\pi}^{5}}{128\pi^{3}f_{\pi}^{4}}\frac{1}{x^{4}}\bigg{[}(g_{A}^{4}(59+4x^{2})-10g_{A}^{2}-1)xK_{0}(2x)$
(22)
$\displaystyle+(g_{A}^{4}(59+36x^{2})-2g_{A}^{2}(5+x^{2})-1)K_{1}(2x)\bigg{]}$
$\displaystyle V^{\rm NNLO}_{{}^{1}S_{0}}(r)$ $\displaystyle=$
$\displaystyle\frac{m_{\pi}^{6}}{1024\pi^{2}m_{N}f_{\pi}^{4}}\frac{e^{-2x}}{x^{6}}\bigg{[}g_{A}^{4}(360+720x+644x^{2}+328x^{3}+96x^{4}+9x^{5})$
$\displaystyle+8g_{A}^{2}(24c_{1}m_{N}x^{2}(x+1)^{2}-(12+24x+20x^{2}+8x^{3}+x^{4})$
$\displaystyle-8c_{4}m_{N}(3+6x+5x^{2}+2x^{3})+12c_{3}m_{N}(6+12x+10x^{2}+4x^{3}+x^{4}))\bigg{]}$

At the origin these potential go as

$\displaystyle V^{\rm NLO}_{{}^{1}S_{0}}(r)$ $\displaystyle\to$
$\displaystyle-\frac{1}{256\pi^{3}f_{\pi}^{4}}\frac{59g_{A}^{4}-10g_{A}^{2}-1}{r^{5}}=-\frac{R_{\rm
NLO}^{4}}{r^{5}}$ (24) $\displaystyle V^{\rm NNLO}_{{}^{1}S_{0}}(r)$
$\displaystyle\to$
$\displaystyle\frac{3}{128\pi^{2}m_{N}f_{\pi}^{4}}\frac{15g_{A}^{4}+4g_{A}^{2}(6c_{3}m_{N}-2c_{4}m_{N}-1)}{r^{6}}=-\frac{R_{\rm
NNLO}^{5}}{r^{6}}$ (25)

with $R_{\rm NLO}=0.767$ fm and $R_{\rm NNLO}=1.056$ fm.

Using these potentials we obtain in Fig. 1 the green line for the NLO case and
the red line for the NNLO case. The convergence in $r_{c}\to 0$ will be shown
in the next section comparing with substractive renormalization. As mentioned
before a careful study of the $NN$ system in the framework of $\chi$EFT up to
NNLO was performed in Refs. Pavón Valderrama and Arriola (2006a, b).

We end this section considering the toy model inspired in the one proposed in
Ref. Epelbaum  _et al._ (2018). There a singular repulsive interactions at
long range is regulated by short range interactions which showed a terrible
convergence due to the repulsive character of the interaction. Here we want to
show how a repulsive singular interaction can be also renormalized. For that
we consider a long range regular attractive interaction and a two-pion
inspired singular repulsion given by the potentials 1 and 2 in Eq. (26). The
short range interaction 3 and 4 are added to end with a regular interaction at
short range so it can be solved and we will consider as the underlying theory.
The expressions are

$\displaystyle V(r)$ $\displaystyle=$ $\displaystyle
V_{1}(r)+V_{2}(r)+V_{3}(r)+V_{4}(r)$ (26) $\displaystyle V_{1}(r)$
$\displaystyle=$ $\displaystyle-\alpha\frac{e^{-m_{\pi}r}}{r}$ (27)
$\displaystyle V_{2}(r)$ $\displaystyle=$
$\displaystyle\alpha_{1}\frac{e^{-2m_{\pi}r}}{r^{3}}$ (28) $\displaystyle
V_{3}(r)$ $\displaystyle=$
$\displaystyle-\alpha_{1}(m_{2}-2m_{\pi})\frac{e^{-m_{1}r}}{r^{2}}$ (29)
$\displaystyle V_{4}(r)$ $\displaystyle=$
$\displaystyle-\alpha_{1}\frac{e^{-m_{2}r}}{r^{3}}$ (30)

Once we add $V_{2}(r)$ the potential is singular repulsive if $V_{4}(r)$ is
not included, however at short distances the full potential tends to

$\displaystyle V(r)$ $\displaystyle\to$
$\displaystyle\frac{1}{2}\frac{-2\alpha+\alpha_{1}(2m_{\pi}-m_{2})(2m_{\pi}-2m_{1}+m_{2})}{r}$
(31)

We use the parameters of Table 2.

Table 2: Parameters used in the toy model when applied in the repulsive singular case. $m_{\pi}$ | 138.5 | MeV
---|---|---
$m_{1}$ | 1000 | MeV
$m_{1}$ | 1200 | MeV
$m_{N}$ | 938.919 | MeV
$\alpha$ | 0.1 |
$\alpha_{1}$ | 5.0 | GeV-2

Figure 2: Phase shift for the toy model Eq. (26) in $S$ wave. The purple line
shows the regular solution for $V_{1}(r)$ and the gold line fixing the
scattering length parameter to the solution of the full potential. The green
and red lines show the results summing up to $V_{2}(r)$ and $V_{3}(r)$,
respectively, while the black line shows the result of the full theory.

Although there is no power counting here, in the following we will refer as
order $n$ to the result considering the sum of the potentials up to $n$
($\sum_{i=1}^{n}V_{i}(r)$).

As order 1 is regular it has a regular solution showed by the purple line in
Fig. 2. The scattering length is $a=-1.50659$ fm, far from the solution of the
full theory $a=-0.615182$ fm. Being regular we can renormalize it with
boundary conditions and we get the gold line which shows the correct low
energy behavior, to compare with the black line which shows the full theory.
However orders 2 and 3 are singular repulsive and can not be renormalized in
$r$ space. The results are given by the green line for order 2 and the red
line for order 3. Clearly the description of the system is much worse than the
renormalization of the order 1 potential. However we will come back to this
issue on Section 5.

Here only scattering states have been considered, however the method can also
be used to study bound states. An example is the application of the method to
study quarkonium models done in Refs.Segovia  _et al._ (2012a, b).

##  4 Renormalization with one counter term

In this section we are going to see how one can performed equivalent
calculations in momentum space to those in the previous section. The starting
point is the Lippmann-Schwinger equation in momentum space given by

$\displaystyle T(p^{\prime},p;k)$ $\displaystyle=$ $\displaystyle
V(p^{\prime},p)+\frac{m_{N}}{2\pi^{2}}\int_{0}^{\infty}d\tilde{p}\,V(p^{\prime},\tilde{p})\frac{\tilde{p}^{2}}{k^{2}-\tilde{p}^{2}+i\epsilon}T(\tilde{p},p;k)$
(32)

where $V(p^{\prime},p)$ represents the potential projected in one uncoupled
partial wave.

The solution of this equation is obtain numerically, only special cases can be
performed analytically. In particular pionless $\chi$EFT is one of such cases.
A general procedure to solve these equation for contact interactions is given
in Refs. Oller (2020); Phillips  _et al._ (1998). If one considers the lowest
order in $S$ waves the potential is just a constant $C$ and the equation is

$\displaystyle T(p^{\prime},p;k)$ $\displaystyle=$ $\displaystyle
C+\frac{m_{N}}{2\pi^{2}}\int_{0}^{\infty}d\tilde{p}\,C\frac{\tilde{p}^{2}}{k^{2}-\tilde{p}^{2}+i\epsilon}T(\tilde{p},p;k)$
(33)

To solve the equation one has first to regularize the integral introducing a
cutoff scale $\Lambda$, for example introducing a sharp cutoff in the
integration momentum

$\displaystyle T_{\Lambda}(p^{\prime},p;k)$ $\displaystyle=$ $\displaystyle
C(\Lambda)+\frac{m_{N}}{2\pi^{2}}\int_{0}^{\Lambda}d\tilde{p}\,C(\Lambda)\frac{\tilde{p}^{2}}{k^{2}-\tilde{p}^{2}+i\epsilon}T_{\Lambda}(\tilde{p},p;k)$
(34)

where the solution of the original equation is obtained in the limit
$\Lambda\to\infty$. The solution is just

$\displaystyle T_{\Lambda}(p^{\prime},p;k)$ $\displaystyle=$
$\displaystyle\frac{C(\Lambda)}{1-C(\Lambda)I(k,\Lambda)}$ (35)

with

$\displaystyle I(k,\Lambda)$ $\displaystyle=$
$\displaystyle\frac{m_{N}}{2\pi^{2}}\int_{0}^{\Lambda}dq\frac{q^{2}}{k^{2}-q^{2}+i\epsilon}=\frac{m_{N}}{2\pi^{2}}(-\Lambda+\frac{k}{2}\log\frac{\Lambda+k}{\Lambda-k})-i\frac{m_{N}}{4\pi}k$
(36)

Since the integral is linearly divergent, the loop function Eq. (36) diverges
for $\Lambda\to\infty$ and we don’t have a meaningful result for a fix value
of $C$ in that limit. This is the same problem as in usual perturbative
renormalization. The way to solve it is to fix some quantity to its physical
value and remove the infinities with the constants of the Lagrangian. For this
reason we already include a $\Lambda$ dependence on the coupling $C$ which is
given by the zero order Lagrangian. Now we fix the $T$ matrix at some scale
$\mu$

$\displaystyle T(\mu)$ $\displaystyle=$ $\displaystyle
T_{\Lambda}(p^{\prime},p;\mu)=\frac{C(\Lambda,\mu)}{1-C(\Lambda,\mu)I(\mu,\Lambda)}$
(37)

which implies

$\displaystyle C^{-1}(\Lambda,\mu)=T^{-1}(\mu)+I(\mu,\Lambda)$ (38)

The solution is now

$\displaystyle T_{\Lambda}(p^{\prime},p;k)$ $\displaystyle=$
$\displaystyle\frac{1}{T^{-1}(\mu)+I(\mu,\Lambda)-I(k,\Lambda)}$ (39)

with

$\displaystyle
I(\mu,\Lambda)-I(k,\Lambda)=\frac{m_{N}}{2\pi^{2}}(\frac{\mu}{2}\log\frac{\Lambda+\mu}{\Lambda-\mu}-\frac{k}{2}\log\frac{\Lambda+k}{\Lambda-k})-i\frac{m_{N}}{4\pi}(\mu-k)$
(40)

and the result is finite in the limit $\Lambda\to\infty$

$\displaystyle\lim_{\Lambda\to\infty}I(\mu,\Lambda)-I(k,\Lambda)$
$\displaystyle=$ $\displaystyle-i\frac{m_{N}}{4\pi}(\mu-k)$ (41)
$\displaystyle\lim_{\Lambda\to\infty}T_{\Lambda}(p^{\prime},p;k)$
$\displaystyle=$
$\displaystyle\frac{1}{T^{-1}(\mu)-i\frac{m_{N}}{4\pi}(\mu-k)}=\frac{T(\mu)}{1+m_{N}T(\mu)(ik-i\mu)/4\pi}$
(42)

In the case $\mu=0$ one recovers the result given by Weinberg Weinberg (1991)
and with an imaginary scale the result of Ref. Kaplan  _et al._ (1998b) in
dimensional regularization.

This is the procedure of renormalization with a counterterm, one regularizes
the Lippmann-Schwinger Equation, then fixes the value of the contact term to
some low energy observable, and finally takes the limit $\Lambda\to\infty$ to
obtain regularization independent results. The problem is that once pions are
included the process can not be performed analytically and only numerical
solutions are possible. Also, one can not use the Fredholm theorem to prove
that the solution exists, since the kernel is not square integrable with
singular interactions in that limit. However, there is also no proof that the
limit does not exist.

The idea is to consider a potential

$\displaystyle V(p^{\prime},p;\Lambda)$ $\displaystyle=$ $\displaystyle V_{\rm
ct}(p^{\prime},p;\Lambda)+V_{\pi}(p^{\prime},p;\Lambda)$ (43)

where $V_{\rm ct}(p^{\prime},p;\Lambda)$ are contact terms222These are zero-
range interactions in the $\Lambda\to\infty$ limit and
$V_{\pi}(p^{\prime},p;\Lambda)$ finite range interactions in $\chi$EFT. To be
more specific at lowest order in the ${}^{1}S_{0}$ partial wave333Notice that
we have included contact contributions from OPE in the contact interaction.

$\displaystyle V_{\rm ct}(p^{\prime},p;\Lambda)$ $\displaystyle=$
$\displaystyle C_{{}^{1}S_{0}}(\Lambda)f(p^{\prime},p;\Lambda)$ (44)
$\displaystyle V_{\pi}(p^{\prime},p;\Lambda)$ $\displaystyle=$
$\displaystyle-\frac{g_{A}^{2}}{4f_{\pi}^{2}}\frac{m_{\pi}^{2}}{2p^{\prime}p}Q_{0}(z)f(p^{\prime},p;\Lambda)$
(45) $\displaystyle Q_{0}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}\frac{z+1}{z-1}\bigg{)}$ (46) $\displaystyle
z$ $\displaystyle=$ $\displaystyle\frac{p^{\prime
2}+p^{2}+m_{\pi}^{2}}{2p^{\prime}p}$ (47)

being $f(p^{\prime},p;\Lambda)$ a regulator function.

This program was performed for the ${}^{1}S_{0}$ $NN$ partial wave in the
context of $\chi$EFT up to N3LO in Ref. Entem  _et al._ (2008) and for higher
partial waves in Ref. Zeoli  _et al._ (2013).

A very interesting way to perform this program was proposed in Frederico  _et
al._ (1999). For a review and the application of multiple subtractions see
Batista  _et al._ (2017) and references there in. Again here we don’t give all
the details of the method and only reproduce those necessary to implement
renormalization with one counter term to compare with other methods. The
derivations in similar notation as ours can be found in Yang  _et al._ (2008)
with slightly different conventions. We are going to do it also in the simple
case of the ${}^{1}S_{0}$ partial wave.

Consider a potential of the type Eq. (43) with one contact term

$\displaystyle V(p^{\prime},p;\Lambda)$ $\displaystyle=$
$\displaystyle(C+V_{\pi}(p^{\prime},p))\theta(\Lambda-p^{\prime})\theta(\Lambda-p)$
(48)

with $\theta(x)$ the step Heaviside function. Then the Lippmann-Schwinger
equation is

$\displaystyle T_{\Lambda}(p^{\prime},p;k)$ $\displaystyle=$ $\displaystyle
V_{\pi}(p^{\prime},p)+C+\frac{m_{N}}{2\pi^{2}}\int_{0}^{\Lambda}d\tilde{p}\,\tilde{p}^{2}\left(\frac{V_{\pi}(p^{\prime},\tilde{p})+C}{k^{2}-\tilde{p}^{2}+i\epsilon}\right)T_{\Lambda}(\tilde{p},p;k)$
(49)

The method fix $T_{\Lambda}(0,0;0)=T(0,0;0)=\frac{a}{4\pi m_{N}}$ and makes
the subtraction $T_{\Lambda}(p^{\prime},0;0)-T_{\Lambda}(0,0;0)$ to remove $C$
from the equation finding

$\displaystyle T_{\Lambda}(p^{\prime},0;0)$ $\displaystyle=$ $\displaystyle
V_{\pi}(p^{\prime},0)+\frac{a}{4\pi
m_{N}}+\frac{m_{N}}{2\pi^{2}}\int_{0}^{\Lambda}d\tilde{p}\,\tilde{p}^{2}\left(\frac{V_{\pi}(p^{\prime},\tilde{p})-V_{\pi}(0,\tilde{p})}{-\tilde{p}^{2}}\right)T_{\Lambda}(\tilde{p},0;0)\quad\quad$
(50)

This equation is solved finding $T_{\Lambda}(p^{\prime},0;0)$. Using
$T_{\Lambda}(0,p;0)=T_{\Lambda}(p,0;0)$ an equation for the fully off-shell
$T$ matrix at zero energy is found

$\displaystyle T_{\Lambda}(p^{\prime},p;0)-T_{\Lambda}(0,p;0)$
$\displaystyle=$ $\displaystyle V_{\pi}(p^{\prime},p)-V_{\pi}(0,p)$ (51)
$\displaystyle+\frac{m_{N}}{2\pi^{2}}\int_{0}^{\Lambda}d\tilde{p}\,\tilde{p}^{2}\left(\frac{V_{\pi}(p^{\prime},\tilde{p})-V_{\pi}(0,\tilde{p})}{-\tilde{p}^{2}}\right)T_{\Lambda}(\tilde{p},p;0)$

once the $T_{\Lambda}(p^{\prime},p;0)$ is found, the $T$ matrix at non-zero
energy is obtained solving

$\displaystyle T_{\Lambda}(p^{\prime},p;k)$ $\displaystyle=$ $\displaystyle
T_{\Lambda}(p^{\prime},p;0)+\frac{m_{N}}{2\pi^{2}}\int_{0}^{\Lambda}d\tilde{p}\,T_{\Lambda}(p^{\prime},p;0)\frac{k^{2}}{k^{2}-\tilde{p}^{2}+i\epsilon}T_{\Lambda}(\tilde{p},p;k)$
(52)

which is similar to a Lippmann-Schwinger equation being
$T_{\Lambda}(p^{\prime},p;0)$ the potential but with a different propagator.

Figure 3: ${}^{1}S_{0}$ $NN$ phase-shifts for the NLO case (left) and NNLO
(right) as explained in the text. The blue line shows the result of
renormalization with boundary conditions as a function of the short-range
boundary condition point $r_{c}$. The gold line shows the results for
renormalization with one counter term and a gaussian regulator function Eq.
(53) as a function of the scale $hc/\Lambda$. The red line shows the result of
substractive renormalization as explained in the text.

In Fig. 3 we show results for the NLO and NNLO cases considered previously
compared with the result of renormalization with boundary conditions. We
represent the ${}^{1}S_{0}$ phase-shift for onshell momentum $k=400$ MeV as a
function of the boundary condition point $r_{c}$ and the scale
$\frac{hc}{\Lambda}$ for the momentum space calculation. The blue line is the
result of the calculation in coordinate space, the gold line is the result of
renormalization with one counter term using

$\displaystyle f(p^{\prime},p;\Lambda)=e^{-\frac{p^{\prime
2}+p^{2}}{\Lambda^{2}}}$ (53)

and the red line the result of substractive renormalization. All of them agree
in the limit $r_{c}\to 0$ or $\Lambda\to\infty$. In Fig, 4 we compare the
results in coordinate space (dots) with substractive renormalization (solid
lines) for LO (gold), NLO (green) and NNLO (red).

Figure 4: Same color code as in Fig. 1 (gold, green and red for LO, NLO and
NNLO respectively) showing with dots the result of renormalization with
boundary conditions and solid line the result with substractive
renormalization.

In Ref. Entem  _et al._ (2008) the inclusion of a second contact term to fix
the effective range was investigated and no way was found to fix it with high
cutoff values, which is in agreement with the findings in pionless EFT.

##  5 Non-perturbative calculations with the $N/D$ method

In this section we introduce a recent method developed by the authors Oller
and Entem (2019) to calculate the non-perturbative resummation of reducible
diagrams using the $N/D$ method.

The $N/D$ method uses the analytical properties of the onshell $T$-matrix.
There is always the so called right-hand-cut (RHC) or physical cut due to
intermediate $NN$ states, which starts at the $NN$ threshold or zero onshell
relative $NN$ momentum. However, for interactions in $\chi$EFT there is also a
left-hand-cut (LHC) which is generated by pion exchanges due to the poles of
pion propagators.

The $N/D$ method writes the onshell $T$-matrix in partial waves in the form

$\displaystyle T(A)$ $\displaystyle=$ $\displaystyle-\frac{N(A)}{D(A)}$ (54)

where $A\equiv k^{2}$ is the square of the onshell momentum and $N(A)$ has
only LHC and $D(A)$ RHC. We have introduced a minus sign since we use a
different sign convention for $T(A)$ as it is usual in $N/D$ method
calculations, so $N(A)$ and $D(A)$ are the same as the usual case.

Unitarity on the RHC implies the condition ($A>0$)

$\displaystyle{\rm Im}D(A)$ $\displaystyle=$ $\displaystyle-\rho(A)N(A)$ (55)

with $\rho(A)=\frac{m_{N}\sqrt{A}}{4\pi}$ the phase-space factor.

The input of the method is the LHC discontinuity of the $T$-matrix that we
denote by $2i\Delta T(A)=2i{\rm Im}T(A)$. This implies the relation on the LHC
($A<0$)

$\displaystyle{\rm Im}N(A)$ $\displaystyle=$ $\displaystyle-D(A)\Delta T(A)$
(56)

The general expressions of the $N/D$ method with $2n$ subtractions were given
in Ref. Guo  _et al._ (2014). For regular interactions no subtractions are
needed and we can solve

$\displaystyle D(A)$ $\displaystyle=$ $\displaystyle
1-\frac{A}{\pi}\int_{0}^{\infty}d\omega_{R}\frac{\rho(\omega_{R})N(\omega_{R})}{(\omega_{R}-A)\omega_{R}}$
(57) $\displaystyle N(A)$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\int_{-\infty}^{L}d\omega_{L}\frac{D(\omega_{L})\Delta(\omega_{L})}{(\omega_{L}-A)}$
(58)

Notice that really we made a subtraction in $D(A)$ since $D(A)$ or $N(A)$ has
to be fixed at some point.

Now we can make subtractions to fix low energy parameters of the effective
range expansion

$\displaystyle
k\cot\delta=-\frac{1}{a}+\frac{1}{2}rk^{2}+\sum_{i=2}v_{i}k^{2i}$ (59)

We obtain Entem and Oller (2017) with one subtraction

$\displaystyle D(A)$ $\displaystyle=$ $\displaystyle
1+i{a}\sqrt{A}-i\frac{m_{N}}{4\pi^{2}}\int_{-\infty}^{L}d\omega_{L}\frac{D(\omega_{L})\Delta(\omega_{L})}{\omega_{L}}\frac{A}{\sqrt{A}+\sqrt{\omega_{L}}}$
(60) $\displaystyle N(A)$ $\displaystyle=$
$\displaystyle-\frac{4\pi{a}}{m_{N}}-\frac{A}{\pi}\int_{-\infty}^{L}d\omega_{L}\frac{D(\omega_{L})\Delta(\omega_{L})}{(\omega_{L}-A)\omega_{L}}$
(61)

However we can also obtain equations to fix the effective range $r$, the next
effective range parameter $v_{2}$, etc. As in Ref. Entem and Oller (2017) we
denote by $N/D$01444In general $N/D_{mn}$ refers to the $N/D$ dispersion
relations with $m(n)$ subtractions in $N(D)$. the regular case, $N/D$11 one
subtraction to fix $a$, $N/D$12 two subtractions and $N/D$22 three
subtractions. Here we will only make up to three subtractions.

In order to solve the $N/D$ method an integration in an infinite interval for
$\omega_{L}$ is needed. One can use a cut-off there but we use a mapping of
gaussian points in the interval $(-\infty,L)$ and we check convergence with
the number of gaussian points, so no cutoff scale is introduced.

In order to solve the $N/D$ method we need to know $\Delta(A)$. To illustrate
how the LHC arises let’s consider the onshell OPE potential in the
${}^{1}S_{0}$ partial wave, which is given by

$\displaystyle
V_{\pi}^{{}^{1}S_{0}}(A)=-\frac{g_{A}^{2}}{4f_{\pi}^{2}}\frac{m_{\pi}^{2}}{4A}\log\bigg{(}\frac{m_{\pi}^{2}+4A}{m_{\pi}^{2}}\bigg{)}$
(62)

where again we have not included the contact contribution. When we do the
analytical continuation to the complex $A$ plane we have the cut due to the
$\log$ function. So for

$\displaystyle m_{\pi}^{2}+4A<0\Rightarrow A<-\frac{m_{\pi}^{2}}{4}\equiv L$
(63)

the $\log$ cut makes a discontinuity in the imaginary part. The imaginary part
on the cut (as the limit from the upper half plane) is given by

$\displaystyle\Delta V_{\pi}^{{}^{1}S_{0}}(A)=-\frac{\pi
g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}A}\theta(L-A)$ (64)

Notice that here we use a different prescription (with a minus sign
difference) than in Ref. Entem and Oller (2017); Oller and Entem (2019) to use
the same prescription for the potential as in previous sections. Also notice
that any polynomial in external momenta generated by contact (counter) terms
does not generate any LHC.

In Ref. Oller and Entem (2019) the integral equation to obtain $\Delta T(A)$
in the LHC for the non-perturbative resummation of reducible diagrams was
derived. To see how the method works let’s consider the onshell once-iterated
OPE in the ${}^{1}S_{0}$ partial wave given by

$\displaystyle V^{{}^{1}S_{0}}_{{\rm it}\pi}(k,k)$ $\displaystyle\equiv$
$\displaystyle\frac{m_{N}}{2\pi^{2}}\int_{0}^{\infty}dp\,V^{{}^{1}S_{0}}_{\pi}(k,p)\frac{p^{2}}{k^{2}-p^{2}+i\epsilon^{\prime}}V^{{}^{1}S_{0}}_{\pi}(p,k)$
(65) $\displaystyle=$
$\displaystyle-\frac{m_{N}}{4\pi^{2}}\int_{-\infty}^{\infty}dp\,V^{{}^{1}S_{0}}_{\pi}(k,p)\frac{p^{2}}{p^{2}-A-i\epsilon^{\prime}}V^{{}^{1}S_{0}}_{\pi}(p,k)$

where we have used $V^{{}^{1}S_{0}}_{\pi}(k,-p)=V^{{}^{1}S_{0}}_{\pi}(k,p)$
and $k^{2}=A$. Notice that for $A<0$ there is no pole for the intermediate
$NN$ propagator and we can omit the $-i\epsilon^{\prime}$ prescription.

To do the analytical continuation we write the potential as

$\displaystyle V^{{}^{1}S_{0}}_{\pi}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle-\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\frac{1}{p^{\prime}p}\bigg{\\{}\log\left[(p^{\prime}+p)^{2}+m_{\pi}^{2}\right]-\log\left[(p^{\prime}-p)^{2}+m_{\pi}^{2}\right]\bigg{\\}}$
(66)

Now we consider the onshell momentum imaginary with an small positive real
part $k=i\bar{k}+\epsilon$, so we approach the LHC from the upper half plane
$A=-\bar{k}^{2}+i\epsilon$.

Naively one would say that the iterated OPE has no imaginary part since

$\displaystyle V^{{}^{1}S_{0}}_{\pi}(i\bar{k},p)$ $\displaystyle=$
$\displaystyle-\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\frac{1}{i\bar{k}p}\bigg{\\{}\log\left[(p+i\bar{k})^{2}+m_{\pi}^{2}\right]-\log\left[(p-i\bar{k})^{2}+m_{\pi}^{2}\right]\bigg{\\}}$
(67)

is real and of course the intermediate $NN$ propagator is also real, so the
integral is real. However this is not the analytical continuation of
$V^{{}^{1}S_{0}}_{{\rm it}\pi}(k,k)$ since the integration crosses the cuts of
$V^{{}^{1}S_{0}}_{\pi}(i\bar{k},p)$ for $\bar{k}>m_{\pi}$. In order to do the
analytical continuation we have to use contour deformations to avoid the cut
of the integrand.

The branch points of $V^{{}^{1}S_{0}}_{\pi}(k,p)$ are given by $p=-k\pm
im_{\pi}$ and $p=k\pm im_{\pi}$ and the cuts are represented in Fig. 5 for the
case of real $k$ (left) and $k=i\bar{k}+\epsilon$ (right). There we see that
for real $k$ we don’t cross any cuts but for imaginary $k$ we cross them if
$\bar{k}>m_{\pi}$. So we have to deform the contour to avoid these cuts as
shown in the Figure.

Figure 5:  Cuts of $V^{{}^{1}S_{0}}_{\pi}(k,p)$ in the $p$ complex plane for
$k$ real (left) and $k=i\bar{k}+\epsilon$ (right).

We first perform the integral in the contour with positive real part. There we
can use $p=\epsilon-\delta+i\nu$ with $\nu\in(0,\bar{k}-m_{\pi})$ when we go
up and $p=\epsilon+\delta+i\nu$ with $\nu\in(\bar{k}-m_{\pi},0)$ when we go
down, always with positive $\delta<\epsilon$. The $\log$ functions are then,
in the limit $\delta\to 0^{+}$,

$\displaystyle\lim_{\delta\to
0^{+}}\log\big{[}(i\bar{k}+\epsilon+p)^{2}+m_{\pi}^{2}\big{]}$
$\displaystyle=$
$\displaystyle\log\big{[}m_{\pi}^{2}-(\bar{k}+\nu)^{2}+i\epsilon\big{]}$ (68)
$\displaystyle\to$ $\displaystyle\ln\rho+i\pi$ $\displaystyle\lim_{\delta\to
0^{+}}\log\big{[}(i\bar{k}+\epsilon-p)^{2}+m_{\pi}^{2}\big{]}$
$\displaystyle=$ $\displaystyle\lim_{\delta\to
0^{+}}\log\big{[}m_{\pi}^{2}-(\bar{k}-\nu)^{2}\pm i\delta\big{]}$ (69)
$\displaystyle=$
$\displaystyle\ln\big{[}(\bar{k}-\nu)^{2}-m_{\pi}^{2}\big{]}\pm
i\pi=\ln\rho^{\prime}\pm i\pi$

where the $\pm$ sign comes from the path going up $+$ and going down $-$ and
we define $\rho=(\bar{k}+\nu)^{2}-m_{\pi}^{2}$ and
$\rho^{\prime}=(\bar{k}-\nu)^{2}-m_{\pi}^{2}$. Notice that
$\bar{k}-\nu>m_{\pi}>0$.

The integrals on the contour are

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{m_{N}}{4\pi^{2}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\int_{0}^{\bar{k}-m_{\pi}}id\nu\,\frac{\nu^{2}}{(\bar{k}+\nu)(\bar{k}-\nu)}\frac{(\ln\rho-\ln\rho^{\prime})^{2}}{\nu^{2}\bar{k}^{2}}$
(70) $\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{m_{N}}{4\pi^{2}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\int^{0}_{\bar{k}-m_{\pi}}id\nu\,\frac{\nu^{2}}{(\bar{k}+\nu)(\bar{k}-\nu)}\frac{(\ln\rho-\ln\rho^{\prime}+i2\pi)^{2}}{\nu^{2}\bar{k}^{2}}$
(71) $\displaystyle I_{1}+I_{2}$ $\displaystyle=$
$\displaystyle\frac{m_{N}}{\pi}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\int_{0}^{\bar{k}-m_{\pi}}d\nu\,\frac{1}{(\bar{k}+\nu)(\bar{k}-\nu)}\frac{(\ln\rho-\ln\rho^{\prime})+i\pi}{\bar{k}^{2}}$
(72)

Doing the same for the other contour we obtain

$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle-\frac{m_{N}}{4\pi^{2}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\int_{0}^{\bar{k}-m_{\pi}}id\nu\,\frac{\nu^{2}}{(\bar{k}+\nu)(\bar{k}-\nu)}\frac{(\ln\rho-\ln\rho^{\prime}+i2\pi)^{2}}{\nu^{2}\bar{k}^{2}}$
(73) $\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle-\frac{m_{N}}{4\pi^{2}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\int^{0}_{\bar{k}-m_{\pi}}id\nu\,\frac{\nu^{2}}{(\bar{k}+\nu)(\bar{k}-\nu)}\frac{(\ln\rho-\ln\rho^{\prime})^{2}}{\nu^{2}\bar{k}^{2}}$
(74) $\displaystyle I_{3}+I_{4}$ $\displaystyle=$
$\displaystyle\frac{m_{N}}{\pi}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\int_{0}^{\bar{k}-m_{\pi}}d\nu\,\frac{1}{(\bar{k}+\nu)(\bar{k}-\nu)}\frac{(\ln\rho-\ln\rho^{\prime})+i\pi}{\bar{k}^{2}}$
(75)

The imaginary part of the iterated OPE is then 555Again notice that the
integral on the real $p$ axis is real.

$\displaystyle{\rm Im}\big{[}V^{{}^{1}S_{0}}_{{\rm it}\pi}(A)\big{]}$
$\displaystyle=$ $\displaystyle{\rm Im}\big{[}I_{1}+I_{2}+I_{3}+I_{4}\big{]}$
(76) $\displaystyle=$
$\displaystyle\frac{2m_{N}}{\bar{k}^{2}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\int_{0}^{\bar{k}-m_{\pi}}d\nu\,\frac{1}{(\bar{k}+\nu)(\bar{k}-\nu)}$
$\displaystyle=$
$\displaystyle\frac{2m_{N}}{\bar{k}^{2}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\frac{1}{2\bar{k}}\log\bigg{(}\frac{2\bar{k}}{m_{\pi}}-1\bigg{)}$
$\displaystyle=$
$\displaystyle-\frac{m_{N}}{A\sqrt{-A}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\log\bigg{(}\frac{2\sqrt{-A}}{m_{\pi}}-1\bigg{)}\theta(4L-A)$

where at the end we include the Heaviside step function to make explicit the
condition $\bar{k}>m_{\pi}$.

The integral equation to obtained $\Delta T(A)$ was deduced in Oller and Entem
(2019) using similar techniques and the properties of the half-offshell
$T$-matrix. We write here Eq. (5.47) for the uncoupled case666We use a
different notation than in Oller and Entem (2019), so is more in tune with
previous sections

$\displaystyle\Delta\hat{T}(\nu,\bar{k})$ $\displaystyle=$
$\displaystyle\Delta\hat{V}(\nu,\bar{k})-\theta(\bar{k}-m_{\pi})\theta(\bar{k}-2m_{\pi}-\nu)\frac{m_{N}}{2\pi^{2}}$
(77)
$\displaystyle\int_{\nu+m_{\pi}}^{\bar{k}-m_{\pi}}d\nu_{1}\,\frac{\nu_{1}^{2}}{\bar{k}^{2}-\nu_{1}^{2}}S(\nu_{1})\Delta\hat{V}(\nu,\nu_{1})\Delta\hat{T}(\nu_{1},\bar{k})$

with

$\displaystyle S(\nu_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{(\nu_{1}+i\epsilon)^{2l+2}}+\frac{1}{(\nu_{1}-i\epsilon)^{2l+2}}$
(78) $\displaystyle 2\Delta\hat{V}(\nu_{1},\nu_{2})$ $\displaystyle=$
$\displaystyle\nu_{1}^{l+1}\nu_{2}^{l+1}\lim_{\epsilon\to 0}\lim_{\delta\to
0}\big{[}{\rm Im}V(i\nu_{1}+\epsilon-\delta,i\nu_{2}+\epsilon)$ (79)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-{\rm
Im}V(i\nu_{1}+\epsilon+\delta,i\nu_{2}+\epsilon)\big{]}$ $\displaystyle
2\Delta\hat{T}(\nu,\bar{k})$ $\displaystyle=$
$\displaystyle\nu^{l+1}\bar{k}^{l+1}\lim_{\epsilon\to 0}\lim_{\delta\to
0}\big{[}{\rm Im}T(i\nu+\epsilon-\delta,i\bar{k}+\epsilon)$ (80)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-{\rm
Im}T(i\nu+\epsilon+\delta,i\bar{k}+\epsilon)\big{]}$

where the onshell $T$-matrix is given by

$\displaystyle\Delta T(A)$ $\displaystyle=$
$\displaystyle-\frac{\Delta\hat{T}(-\bar{k},\bar{k})}{\bar{k}^{2l+2}}$ (81)

Notice that the integral equation Eq. (77) is always finite and no cutoff
scale is introduced.

If we want to get the once-iterated OPE we only need to calculate the integral
term in Eq. (77) changing $\Delta\hat{T}(\nu_{1},\bar{k})$ by
$\Delta\hat{V}(\nu_{1},\bar{k})$. This is done in appendix A. Also from Eq.
(77) one can demonstrate the Eqs. (8-9) from Entem and Oller (2017) since it
is easy to show that Eq. (9) is just Eq. (77) for the case of OPE in the
${}^{1}S_{0}$ partial wave.

The method can be used for any interaction that has a spectral decomposition
since the structure of the LHC is the same. If we consider the spectral
decomposition

$\displaystyle V(q)$ $\displaystyle=$
$\displaystyle\int_{\mu_{0}}^{\infty}d\mu\,\frac{2\mu\eta(\mu^{2})}{q^{2}+\mu^{2}}$
(82)

for the $S$ waves we have

$\displaystyle\Delta\hat{V}(\nu_{1},\nu_{2})$ $\displaystyle=$
$\displaystyle-{\rm
sign}(\nu_{1}-\nu_{2})\theta(|\nu_{1}-\nu_{2}|-\mu_{0})\frac{\pi}{2}\int_{\mu_{0}}^{|\nu_{1}-\nu_{2}|}d\mu\,\mu\eta(\mu^{2})$
(83)

For the ${}^{1}S_{0}$ partial wave we get for the NLO contribution

$\displaystyle\Delta\hat{V}_{\rm NLO}^{{}^{1}S_{0}}(\nu_{1},\nu_{2})$
$\displaystyle=$ $\displaystyle-\theta(|\nu_{1}-\nu_{2}|-2m_{\pi}){\rm
sign}(\nu_{1}-\nu_{2})\frac{1}{1536\pi
f_{\pi}^{4}}\frac{\nu_{12}}{\hat{\omega}(\nu_{12})}\bigg{\\{}$ (84)
$\displaystyle(10+52g_{A}^{2}-158g_{A}^{4})m_{\pi}^{4}+\frac{1}{2}(197g_{A}^{4}-46g_{A}^{2}-7)m_{\pi}^{2}\nu_{12}^{2}$
$\displaystyle-\frac{1}{4}(59g_{A}^{4}-10g_{A}^{2}-1)\nu_{12}^{4}+6(5g_{A}^{4}+2g_{A}^{2}+1)m_{\pi}^{4}\hat{L}(\nu_{12})\bigg{\\}}$

with $\nu_{12}=|\nu_{1}-\nu_{2}|$ and

$\displaystyle\hat{\omega}(\nu_{12})$ $\displaystyle\equiv$
$\displaystyle\sqrt{\nu_{12}^{2}-4m_{\pi}^{2}}$ (85)
$\displaystyle\hat{L}(\nu_{12})$ $\displaystyle\equiv$
$\displaystyle\frac{\hat{\omega}(\nu_{12})}{\nu_{12}}\log\bigg{(}\frac{\nu_{12}+\hat{\omega}(\nu_{12})}{2m_{\pi}}\bigg{)}$
(86)

And for the NNLO contribution

$\displaystyle\Delta\hat{V}^{{}^{1}S_{0}}_{\rm NNLO}(\nu_{1},\nu_{2})$
$\displaystyle=$ $\displaystyle-\theta(|\nu_{1}-\nu_{2}|-2m_{\pi}){\rm
sign}(\nu_{1}-\nu_{2})\frac{g_{A}^{2}}{3840f_{\pi}^{4}}\bigg{\\{}$ (87)
$\displaystyle\bigg{(}480c_{1}-336c_{3}-128c_{4}-\frac{4}{M_{N}}-\frac{25g_{A}^{2}}{4M_{N}}\bigg{)}m_{\pi}^{5}$
$\displaystyle+\bigg{(}-720c_{1}+360c_{3}-\frac{30}{M_{N}}+\frac{30g_{A}^{2}}{M_{N}}\bigg{)}m_{\pi}^{4}\nu_{12}$
$\displaystyle+\bigg{(}120c_{1}-120c_{3}+40c_{4}+\frac{20}{M_{N}}-\frac{95g_{A}^{2}}{2M_{N}}\bigg{)}m_{\pi}^{2}\nu_{12}^{3}$
$\displaystyle+\bigg{(}18c_{3}-6c_{4}-\frac{3}{M_{N}}+\frac{45g_{A}^{2}}{4M_{N}}\bigg{)}\nu_{12}^{5}\bigg{\\}}$

For the ${}^{1}S_{0}$ $NN$ partial wave the potentials in $\chi$EFT are
singularly attractive. They can be renormalized with one subtraction, however
as shown in Ref. Entem and Oller (2017) additional solutions with more
subtractions can be found. Here we consider the toy model proposed first in
Eq. (26) but in a singular attractive case by using $\alpha_{1}=-3$ GeV-2. Now
the situation is the opposite to the previous case, order 1 is regular but
order 2 is singular attractive. In Fig. 6 we show the result for the order 2
case. Again the full result is given by black dots for the $r$ space
calculation and a black solid line for the $N/D$ method. Similarly the magenta
line and dots refer to the results of the regular potential $V_{1}(r)$ in $r$
space and with the $N/D$ method, in this order. The gold line and dots give
the result of the $N/D$ method and boundary conditions, respectively fixing
$a$ and considering the order 2 case. $N/D$ allows to make more subtractions
and we perform a calculation with two and three subtractions. The case $N/D$12
does not show converge and we don’t include it in the Figure. The red line
give the result with three subtractions $N/D$22, fixed in terms of $a$, $r$
and $v_{2}$, which agrees very well with the full theory. The low-energy
parameters in the effective range expansion are taken from the result of the
full theory.

Figure 6: Phase shifts for the toy model as in Eq. (26) but in the singular
attractive case using $\alpha_{1}=-3$ GeV-2. The black (purple) line and dots
show the result of the full (regular $V_{1}(r)$) potential for the $N/D$ and
$r$ space calculations, respectively. The gold line and dots are the same for
the case of considering the singular attractive interaction at order 2 with
one renormalization condition. The red line gives the result for the $N/D$22
calculation with three renormalization conditions. The low energy constants
are those of the full theory.

Figure 7: Phase shifts for the toy model as in Eq. (26). The dots shows the
result in coordinate space, the full lines are the result of $N/D$ method, and
the dashed lines the effective range expansion up to $k^{2n}$. The dashed
gold, green, red and black are the results for of the effective range
expansion for $n=0,1,2,6$, respectively. For the coordinate space we have in
purple and black the regular solution for $V_{1}(r)$ and $V(r)$, respectively,
and in gold fixing $a$ and using only $V_{1}(r)$. For the $N/D$ method we have
in a black line the solution of the full theory, and in gold, green and red
the solution using $V_{1}(r)$ with one, two and three subtractions
respectively.

One of the main problems of the previous methods is that they can not
renormalize the singular repulsive case. Only the regular solution is
possible, as shown in Fig. 2 for our toy model, and the results including the
singular repulsive interactions does not even have the correct low-energy
behavior. However the exact $N/D$ method allows the renormalization of such
cases as we discuss next.

Let’s start showing results for the toy model proposed in Eq. (26). In Figure
7 we show the results for the order 1 case. Again the black line is the full
result. The dots correspond to the results in $r$ space, while the lines shows
the results for the regular case, one, two and three subtractions in purple,
gold, green and red lines, respectively. Since order 1 is regular any number
of subtractions can be made. We also include for comparison with dashed lines
the result of the effective range expansion with only $a$, adding $r$, adding
$v_{2}$ and up to $v_{6}$ in gold, green, red and black lines. Notice that the
parameters of the effective range expansion are not fitted to phase shifts,
they are the exact values of the full theory given in Tab. 3. We can see that
the effective range expansion is only valid for very low energies, however the
renormalized results at order 1 give a good description at higher energies,
although for $k\sim 250$ MeV sizable discrepancies with the full theory are
seen. The reason why the effective range expansion is not good is clear, at
around $k\sim 150$ MeV, $k\cot\delta$ becomes singular and the expansion is
not valid any more as can be seen from the dashed-black line that goes up to
$k^{12}$ in the expansion. Only considering the finite range interactions, we
can go beyond this point.

Another interesting point that can be seen on Fig. 7 is that adding
subtraction allows to go to higher energies. The green line with two
subtractions start to show differences with the full theory at the scale of
the figure for $k\sim 150-200$ MeV, while the red line can go up to $k\sim
200-250$ MeV. This is the same idea that was illustrated in Fig. 5 of Entem
and Oller (2017).

However now we consider the order 2 case which adds a singular repulsive
interaction. The results are given in Fig. 8 with the same color codes as in
Fig. 7. Being a singular repulsive case, the regular solution is the one
obtained in coordinate space as can be seen comparing the dots and line in
purple. Also we can not fix $a$ as in the previous renormalization methods and
so the gold line does not appear. However with the $N/D$ method we can use
additional subtractions. The calculation shows that we don’t have convergence
fixing $a$ and $r$ and the green line is absent. The result fixing up to
$v_{2}$ converges and is shown with the red line, which shows a very good
agreement with the full theory (black line).

Figure 8: Same as in Fig. 7, but for the order 2 case. The red line is the
result of the $N/D$ method with a singular repulsive interaction
$V_{1}(r)+V_{2}(r)$ with three subtractions. The low energy constants are not
fitted to the phase shifts, they are the exact low energy constants of the
full theory given in Table 3.

In Table 3 we give the effective range expansion parameters for the different
cases. They give a feeling of the accuracy of the calculation at low energies.
Finally in Fig. 9 we show instead the phase-shift, $k\cot(\delta)$ to show the
breaking of the effective range expansion in the toy model. The black solid
line is the result of the full theory. The dashed lines shows the result of
Eq. (59) cutting the sum at an order $k^{2n}$. The gold, green and red short
dashed lines corresponds to $n=0,1,2$ respectively. Then the cases $n=3,4,5$
are shown as red dashed lines with longer dashed lengths. The last case shown
is $n=6$ in a black dashed line. The red solid line corresponds to the result
renormalizing with three-subtractions the singular repulsive interaction at
order 2.

Table 3: Effective range parameters as defined in Eq. (59) for the toy model Eq. (26). |  $a$ (fm) |  $r$ (fm) |  $v_{2}$ (fm3) |  $v_{3}$ (fm5) |  $v_{4}$ (fm7) |  $v_{5}$ (fm9) |  $v_{6}$ (fm11)
---|---|---|---|---|---|---|---
$N/D$01 |  |  |  |  |  |  |
order 1 | -1.50659 | 8.42932 | -5.62505 | 22.3554 | -113.278 | 647.596 | -3979.14
order 2 | -0.407820 | 55.2322 | 112.088 | 532.816 | 2114.65 | 10562.8 | 38255.68
$N/D$11 |  |  |  |  |  |  |
order 1 | -0.615182 | 28.6632 | 22.3775 | 79.7037 | -4.42830 | 984.991 | -4097.91
$N/D$12 |  |  |  |  |  |  |
order 1 | -0.615182 | 28.1482 | 18.9220 | 63.7131 | -52.0605 | 859.823 | -4412.90
$N/D$22 |  |  |  |  |  |  |
order 1 | -0.615182 | 28.1482 | 19.0066 | 64.7974 | -47.5274 | 871.750 | -4384.37
order 2 | -0.615182 | 28.1482 | 19.0066 | 64.7579 | -47.9885 | 870.161 | -4388.18
Full | -0.615182 | 28.1482 | 19.0066 | 64.7583 | -47.9860 | 870.170 | -4388.16

Figure 9: $k\cot(\delta)$ effective range expansion for the toy model. The
black solid line is the result of the full theory. The dashed lines shows the
result of Eq. (59) cutting the sum at an order $k^{2n}$. The gold, green and
red short dashed lines corresponds to $n=0,1,2$ respectively. Then the cases
$n=3,4,5$ are shown as red dashed lines with longer dashed lengths. The last
case shown is $n=6$ in a black dashed line. The red solid line shows the
result of $N/D_{22}$ at order 2.

Now the open question is to know a priory when a solution exists or not in
terms of the number of subtractions, which is indeed an interesting problem in
mathematical physics. It is true that here we don’t proof the convergence of
the solution, we just rely on numerical convergence, but the agreement with
the full theory is compeling.

##  6 Summary and conclusion

In this work we have made a brief review of some of the methods used to remove
the cut-off dependence when singular interactions are used. The methods are
renormalization with boundary conditions, renormalization in momentum space
with one counter term, and equivalently substractive renormalization, and the
exact $N/D$ method with subtractions. We showed that the methods are
equivalent for singular repulsive potentials, or singular attractive
potentials with a renormalization condition. However, the $N/D$ method allows
to make more subtractions or, analogously, implement more renormalization
conditions. In particular we made a toy model with the property that removing
the short range part of the potential is singular, but the full model is
regular and can be solved using standard techniques. We study the singular
attractive and repulsive cases, showing that the exact $N/D$ method with
multiple subtractions can describe the full theory in the low energy regime.
The convergence of the solutions are only checked numerically and it is still
open to strictly demonstrate that these solutions exist. However, the
agreement with the full theory is compeling, giving hope to renormalize the
$NN$ interaction in the framework of $\chi$EFT.

###### Acknowledgements.

DRE wants to thank E. Ruíz-Arriola for fruitful discussions about the
renormalization with boundary conditions and with one counter term. This work
has been funded by Ministerio de Ciencia e Innovación under Contract No.
PID2019-105439GB-C22/AEI/10.13039/501100011033, and
PID2019-106080GB-C22/AEI/10.13039/501100011033, and by EU Horizon 2020
research and innovation program, STRONG-2020 project, under grant agreement No
824093.

##  Appendix A Iterated OPE

Here we obtain the once iterated OPE from Eq. (77) as

$\displaystyle\Delta\hat{V}^{{}^{1}S_{0}}_{{\rm it}\pi}(-\bar{k},\bar{k})$
$\displaystyle=$
$\displaystyle-\theta(\bar{k}-m_{\pi})\frac{m_{N}}{2\pi^{2}}\int_{-\bar{k}+m_{\pi}}^{\bar{k}-m_{\pi}}d\nu_{1}\,\frac{\nu_{1}^{2}}{\bar{k}^{2}-\nu_{1}^{2}}S(\nu_{1})$
(88)
$\displaystyle\Delta\hat{V}^{{}^{1}S_{0}}_{\pi}(-\bar{k},\nu_{1})\Delta\hat{V}^{{}^{1}S_{0}}_{\pi}(\nu_{1},\bar{k})$

We start calculating

$\displaystyle 2\Delta\hat{V}^{{}^{1}S_{0}}_{\pi}(\nu_{1},\nu_{2})$
$\displaystyle=$ $\displaystyle\nu_{1}\nu_{2}\lim_{\epsilon\to
0}\lim_{\delta\to 0}\big{[}{\rm
Im}V^{{}^{1}S_{0}}_{\pi}(i\nu_{1}+\epsilon-\delta,i\nu_{2}+\epsilon)$ (89)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-{\rm
Im}V^{{}^{1}S_{0}}_{\pi}(i\nu_{1}+\epsilon+\delta,i\nu_{2}+\epsilon)\big{]}$

We have that

$\displaystyle
V^{{}^{1}S_{0}}_{\pi}(i\nu_{1}+\epsilon\mp\delta,i\nu_{2}+\epsilon)$
$\displaystyle=$
$\displaystyle-\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\frac{1}{(i\nu_{1}+\epsilon-\delta)(i\nu_{2}+\epsilon)}$
(90)
$\displaystyle\bigg{\\{}\log\big{[}(i\nu_{1}+\epsilon\mp\delta+i\nu_{2}+\epsilon)^{2}+m_{\pi}^{2}\big{]}$
$\displaystyle-\log\big{[}(i\nu_{1}+\epsilon\mp\delta-i\nu_{2}-\epsilon)^{2}+m_{\pi}^{2}\big{]}\bigg{\\}}$
$\displaystyle\sim$
$\displaystyle\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\frac{1}{\nu_{1}\nu_{2}}$
$\displaystyle\bigg{\\{}\log\big{[}m_{\pi}^{2}-(\nu_{1}+\nu_{2})^{2}+i{\rm
sing}(\nu_{1}+\nu_{2})\epsilon\big{]}$
$\displaystyle-\log\big{[}m_{\pi}^{2}-(\nu_{1}-\nu_{2})^{2}\mp i{\rm
sing}(\nu_{1}-\nu_{2})\delta\big{]}\bigg{\\}}$

and so

$\displaystyle\Delta\hat{V}^{{}^{1}S_{0}}_{\pi}(\nu_{1},\nu_{2})$
$\displaystyle=$ $\displaystyle{\rm
sing}(\nu_{1}-\nu_{2})\theta(|\nu_{1}-\nu_{2}|-m_{\pi})\frac{\pi
g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}$ (91)

From the limits we have

$\displaystyle\nu_{1}>-\bar{k}+m_{\pi}$ $\displaystyle\Rightarrow$
$\displaystyle\nu_{1}+\bar{k}>m_{\pi}>0$ (92)
$\displaystyle\nu_{1}<\bar{k}-m_{\pi}$ $\displaystyle\Rightarrow$
$\displaystyle\bar{k}-\nu_{1}>m_{\pi}>0$ (93)

So we have

$\displaystyle\Delta\hat{V}^{{}^{1}S_{0}}_{{\rm it}\pi}(-\bar{k},\bar{k})$
$\displaystyle=$
$\displaystyle-\theta(\bar{k}-m_{\pi})\frac{m_{N}}{\pi^{2}}\int_{-\bar{k}+m_{\pi}}^{\bar{k}-m_{\pi}}d\nu_{1}\,\frac{1}{\bar{k}^{2}-\nu_{1}^{2}}\bigg{(}\frac{\pi
g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}$ (94) $\displaystyle=$
$\displaystyle-\theta(\bar{k}-m_{\pi})\frac{m_{N}}{\bar{k}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\log\bigg{(}\frac{2\bar{k}}{m_{\pi}}-1\bigg{)}$

And finally

$\displaystyle\Delta V^{{}^{1}S_{0}}_{{\rm
it}\pi}(A)=-\frac{\Delta\hat{V}^{{}^{1}S_{0}}_{{\rm
it}\pi}(-\bar{k},\bar{k})}{\bar{k}^{2}}$ $\displaystyle=$
$\displaystyle\theta(\bar{k}-m_{\pi})\frac{m_{N}}{\bar{k}^{3}}\bigg{(}\frac{g_{A}^{2}m_{\pi}^{2}}{16f_{\pi}^{2}}\bigg{)}^{2}\log\bigg{(}\frac{2\bar{k}}{m_{\pi}}-1\bigg{)}$
(95)

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