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11institutetext: H. Alanazi, R. Machleidt 22institutetext: Department of
Physics, University of Idaho, Moscow, ID 83844, USA
22email<EMAIL_ADDRESS>
# The relevance of pion-exchange contributions versus contact terms in the
chiral effective field theory description of nucleon-nucleon scattering

H. Alanazi  R. Machleidt

(Received: date / Accepted: date)

###### Abstract

The standard way to demonstrate the relevance of chiral symmetry for the $NN$
interaction is to consider higher partial waves of $NN$ scattering which are
controlled entirely by chiral pion-exchanges (since contacts vanish). However,
in applications of $NN$-potentials to nuclear structure and reactions, the
lower partial waves are the important ones, making the largest contributions.
Lower partial waves are sensitive to the short-range potential, and so, when
the short-range contacts were to dominate over the chiral pion-contributions
in lower partial waves, then the predictions from “chiral potentials” would
have little to do with chiral symmetry. To address this issue, we investigate
systematically the role of the (chiral) one- and two-pion exchanges, on the
one hand, and the effect of the contacts, on the other hand, in the lower
partial waves of $NN$ scattering. We are able to clearly identify the
signature of chiral symmetry in lower partial waves. Our study has also a
pedagogical spin-off as it demonstrates in detail how the reproduction of the
lower partial-wave phase shifts comes about from the various ingredients of
the theory.

##  1 Introduction

During the past three decades, it has been demonstrated that chiral effective
field theory (chiral EFT) represents a powerful tool to deal with hadronic
interactions at low energy (see Refs. ME11 ; EHM09 ; HKK20  for recent
reviews). By construction, chiral EFT is a model-independent theory with a
firm connection to QCD via the (broken) chiral symmetry of low-energy QCD.
Moreover, the method is systematic in the sense that the various contributions
to a particular dynamical process can be arranged as an expansion in terms of
powers of the soft scale, $Q\sim m_{\pi}$, over the hard scale,
$\Lambda_{\chi}\sim m_{\rho}$: $Q/\Lambda_{\chi}$ (where $m_{\pi}$ denotes the
pion mass and $m_{\rho}$ the mass of the $\rho$-meson). Within the chiral EFT
approach, nucleon-nucleon ($NN$) interactions have been constructed up to
fifth order of the chiral expansion (see Refs. Pia15 ; Pia16 ; Car16 ; EMN17 ;
RKE18 ; Eks18  for some recent examples).

These chiral $NN$ potentials complemented by chiral three-nucleon forces have
been applied in calculations of few-nucleon reactions NRQ10 ; Viv13 ; Gol14 ;
Gir19 , the structure of light- and medium-mass nuclei Nav07a ; Rot11 ; Rot12
; Hag12a ; Hag12b ; BNV13 ; Her13 ; Hag14a ; Bin14 ; HJP16 ; Sim16 ; Sim17 ;
Mor18 ; Som19 ; Hop19 ; Rot19 , and infinite matter HS10 ; Heb11 ; Cor13 ;
Hag14b ; Cor14 ; Sam15 ; MS16 ; Dri16 ; Heb15 ; DHS19 ; SM20 ; Jia20 —with, by
and large, satisfactory results. These successes have been attributed to the
chiral symmetry foundation of the potentials MS20 .

In the chiral EFT procedure to nuclear forces, a clear dictinction is made
between the long- and short-range parts of the $NN$ potential. While the long-
range part is given by one- and multi-pion exchanges, the short-range
description is very different. Since the short-range nucleon structure cannot
be resolved at the low energy scale characteristic for traditional nuclear
physics, the short-range description consists simply of polynomials of
increasing degree, also known as contact terms, which contribute exclusively
in the lower partial waves of the $NN$ interaction.

Only the pion-contributions are ruled by chiral symmetry, while the contacts
are based on just the usual non-relativistic invariances and have nothing to
do with chiral symmetry. Therefore, the standard way to demonstrate the
relevance of chiral symmetry for the $NN$ interaction is to consider only
higher partial waves to which contacts do not make contributions and that are,
thus, controled entirely by chiral symmetry KBW97 ; Ent15a ; Ent15b .

But, how about the lower partial waves? Here, the situation is different,
because both contacts and pion-exchanges contribute. Thus, to obtain a clear
picture of the role of chiral symmetry in the lower partial waves, the chiral
pion-exchanges need to be disentagled from the contact contributions, which
may not be easy. Nevertheless, this issue is of interest for the following
reasons.

Lower partial waves are more sensitive to the short-range potential.
Therefore, one may suspect that the contact contributions are dominant and
simply override the pion-exchange contributions in lower partial waves.

In applications of $NN$-potentials to nuclear structure and reactions, the
lower partial waves make the large contributions. Thus, if chiral symmetry
would rule only the upper partial waves while the lower partial wave were
essentially governed by the contacts, then the predictions from these “chiral
potentials” for nuclear structure and reactions would have little to do with
chiral symmetry. Chiral potentials would not be much different from
phenomenological potentials.

Motivated by the above concerns, the purpose of this paper is to
systematically investigate the role of the (chiral) one- and two-pion exchange
contributions, on the one hand, and the effect of the contacts, on the other
hand, in the lower partial waves of chiral $NN$ potentials; to determine if
chiral symmetry plays an essential role in those lower waves.

Besides this main physical motivation for this study, we like to point out
that there is also a pedagogical spin-off to this work. We will demonstrate in
detail how the reproduction of the lower partial-wave phase shifts comes about
from the various ingredients of the theory.

In Sect. 2, will present the formalism for the chiral $NN$ potential, order by
order, as it matters to the investigation of the issue, which is conducted in
Sect. 3. Sect. 4 concludes the paper.

##  2 The chiral $NN$ potential

###  2.1 Power counting

In nuclear EFT, contributions are analyzed in terms of powers of small
external momenta over the large scale: $(Q/\Lambda_{b})^{\nu}$, where $Q$ is
generic for an external momentum (nucleon three-momentum or pion four-
momentum) or a pion mass and $\Lambda_{b}\sim 0.6$ GeV is the breakdown scale
appropriate specifically for the $NN$ problem EKM15 . Determining the power
$\nu$ has become known as power counting. Applying (naive) dimensional
analysis or Weinberg counting Wei90 , one obtains: for the power of a
connected irreducible diagram involving $A$ nucleons ME11 ,

$\nu=-2+2A-2C+2L+\sum_{i}{\Delta_{i}},$ (1)

with

$\Delta_{i}\equiv d_{i}+\frac{n_{i}}{2}-2,$ (2)

where $C$ denotes the number of separately connected pieces and $L$ the number
of loops in the diagram; $d_{i}$ is the number of derivatives or pion-mass
insertions and $n_{i}$ the number of nucleon fields (nucleon legs) involved in
vertex $i$; the sum runs over all vertexes $i$ contained in the diagram under
consideration. Note that $\Delta_{i}\geq 0$ for all interactions allowed by
chiral symmetry.

Since we use the heavy-baryon formalism, we encounter terms which include
factors of $Q/M_{N}$, where $M_{N}$ denotes the nucleon mass. We count the
order of such terms by the rule $Q/M_{N}\sim(Q/\Lambda_{\chi})^{2}$, for
reasons explained in Ref. Wei90 .

In this paper, we are mainly concerned with the $NN$ system $(A=2,C=~{}1)$, in
which case the power formula collapses to the very simple expression

$\nu=2L+\sum_{i}\Delta_{i}.$ (3)

###  2.2 The long-range $NN$ potential

The long-range part of the $NN$ potential is built up from pion exchanges,
which are ruled by chiral symmetry. The various pion-exchange contributions
may be analyzed according to the number of pions being exchanged between the
two nucleons:

$V_{\pi}=V_{1\pi}+V_{2\pi}+V_{3\pi}+\cdots,$ (4)

where the meaning of the subscripts is obvious and the ellipsis represents
$4\pi$ and higher pion exchanges. For each of the above terms, we have a low-
momentum expansion:

$\displaystyle V_{1\pi}$
$\displaystyle=V_{1\pi}^{(0)}+V_{1\pi}^{(2)}+V_{1\pi}^{(3)}+V_{1\pi}^{(4)}+V_{1\pi}^{(5)}+\cdots,$
(5) $\displaystyle V_{2\pi}$
$\displaystyle=V_{2\pi}^{(2)}+V_{2\pi}^{(3)}+V_{2\pi}^{(4)}+V_{2\pi}^{(5)}+\cdots,$
(6) $\displaystyle V_{3\pi}$
$\displaystyle=V_{3\pi}^{(4)}+V_{3\pi}^{(5)}+\cdots,$ (7)

where the superscript denotes the order $\nu$ of the expansion.

Order by order, the long-range $NN$ potential builds up as follows:

$\displaystyle V_{L\Omega}$ $\displaystyle\equiv V^{(0)}=V_{1\pi}^{(0)},$ (8)
$\displaystyle V_{NL\Omega}$ $\displaystyle\equiv
V^{(2)}=V_{L\Omega}+V_{1\pi}^{(2)}+V_{2\pi}^{(2)},$ (9) $\displaystyle
V_{NNL\Omega}$ $\displaystyle\equiv
V^{(3)}=V_{NL\Omega}+V_{1\pi}^{(3)}+V_{2\pi}^{(3)},$ (10) $\displaystyle
V_{N^{3}L\Omega}$ $\displaystyle\equiv
V^{(4)}=V_{NNL\Omega}+V_{1\pi}^{(4)}+V_{2\pi}^{(4)}+V_{3\pi}^{(4)},$ (11)
$\displaystyle V_{N^{4}L\Omega}$ $\displaystyle\equiv
V^{(5)}=V_{N^{3}L\Omega}+V_{1\pi}^{(5)}+V_{2\pi}^{(5)}+V_{3\pi}^{(5)},$ (12)

where L$\Omega$ stands for leading order, NL$\Omega$ for next-to-leading
order, etc..

####  2.2.1 Leading order

Table 1: Basic constants used throughout this work PDG . Quantity |  | Value
---|---|---
Axial-vector coupling constant $g_{A}$ |  | 1.29
Pion-decay constant $f_{\pi}$ |  | 92.4 MeV
Charged-pion mass $m_{\pi^{\pm}}$ |  | 139.5702 MeV
Neutral-pion mass $m_{\pi^{0}}$ |  | 134.9766 MeV
Average pion-mass $\bar{m}_{\pi}$ |  | 138.0390 MeV
Proton mass $M_{p}$ |  | 938.2720 MeV
Neutron mass $M_{n}$ |  | 939.5654 MeV
Average nucleon-mass $\bar{M}_{N}$ |  | 938.9183 MeV

At leading order (L$\Omega$, $\nu=0$), only one-pion exchange (1PE)
contributes to the long range. The charge-independent 1PE is given by

$V_{1\pi}^{(CI)}(\vec{p^{\prime}},\vec{p})=-\>\frac{g_{A}^{2}}{4f_{\pi}^{2}}\>\bm{\tau}_{1}\cdot\bm{\tau}_{2}\>\frac{{\vec{\sigma}}_{1}\cdot\vec{q}\>\>\vec{\sigma}_{2}\cdot\vec{q}}{q^{2}+m_{\pi}^{2}}\>,$
(13)

where $\vec{p^{\prime}}$ and $\vec{p}$ denote the final and initial nucleon
momenta in the center-of-mass system (CMS), respectively. Moreover,
$\vec{q}={\vec{p}}\,^{\prime}-\vec{p}$ is the momentum transfer and
$\vec{\sigma}_{1,2}$ and $\bm{\tau}_{1,2}$ are the spin and isospin operators
of nucleons 1 and 2. Parameters $g_{A},f_{\pi}$ and $m_{\pi}$ denote the
axial-vector coupling constant, pion-decay constant, and the pion mass,
respectively. See Table 1 for their values. Higher order corrections to the
1PE are taken care of by mass and coupling constant renormalizations. Note
also that, on shell, there are no relativistic corrections. Thus, we apply 1PE
in the form Eq. (13) through all orders.

The 1PE potential, Eq. (13), can be re-written as follows:

$V_{1\pi}^{(CI)}(\vec{p^{\prime}},\vec{p})=-\>\frac{g_{A}^{2}}{12f_{\pi}^{2}}\>\bm{\tau}_{1}\cdot\bm{\tau}_{2}\left(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\,-\,\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\,\frac{m_{\pi}^{2}}{q^{2}+m_{\pi}^{2}}+\,\frac{S_{12}(\vec{q})}{q^{2}+m_{\pi}^{2}}\>\right),$
(14)

with tensor operator

$S_{12}(\vec{q})=3\>\vec{\sigma}_{1}\cdot\vec{q}\>\>\vec{\sigma}_{2}\cdot\vec{q}\,-\,\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\,q^{2}\,,$
(15)

where the 1PE has been broken up into a zeroth order spin-spin contact term
(“$\delta$-function term”), a spin-spin Yukawa central force, and a tensor
piece. The 1PE tensor force is known to be strong, while the spin-spin central
force is weak.

If one takes the charge-dependence of the 1PE into account, then, in proton-
proton ($pp$) and neutron-neutron ($nn$) scattering one has

$V_{1\pi}^{(pp)}(\vec{p^{\prime}},\vec{p})=V_{1\pi}^{(nn)}(\vec{p^{\prime}},\vec{p})=V_{1\pi}(m_{\pi^{0}})$
(16)

and in $np$ scattering

$V_{1\pi}^{(np)}(\vec{p^{\prime}},\vec{p})=-V_{1\pi}(m_{\pi^{0}})+(-1)^{I+1}\>2\>V_{1\pi}(m_{\pi^{\pm}})\>,$
(17)

where $I=0,1$ denotes the total isospin of the two-nucleon system and

$V_{1\pi}(m_{\pi})=-\>\frac{g_{A}^{2}}{4f_{\pi}^{2}}\>\frac{\vec{\sigma}_{1}\cdot\vec{q}\>\vec{\sigma}_{2}\cdot\vec{q}}{q^{2}+m_{\pi}^{2}}\>.$
(18)

The charge-dependence of the 1PE is of order NL$\Omega$ ME11 , but we include
it already at order L$\Omega$ to make the comparison with the $np$ phase-shift
analyses meaningful.

####  2.2.2 Next-to-leading order

At next-to-leading order (NL$\Omega$, $\nu=2$), two-pion exchange (2PE) starts
and continues through all higher orders. The 2PE potential expressions will be
stated in terms of contributions to the momentum-space $NN$ amplitudes in the
CMS, which takes the following general form:

$\displaystyle V({\vec{p}}~{}^{\prime},\vec{p})$ $\displaystyle=$
$\displaystyle\>\,V_{C}\>\,+\bm{\tau}_{1}\cdot\bm{\tau}_{2}\,W_{C}$ (19)
$\displaystyle+$
$\displaystyle\left[\,V_{S}\>\,+\bm{\tau}_{1}\cdot\bm{\tau}_{2}\,W_{S}\,\>\,\right]\,\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}$
$\displaystyle+$
$\displaystyle\left[\,V_{LS}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}\,W_{LS}\right]\,\left(-i\vec{S}\cdot(\vec{q}\times\vec{k})\,\right)$
$\displaystyle+$
$\displaystyle\left[\,V_{T}\>\,+\bm{\tau}_{1}\cdot\bm{\tau}_{2}\,W_{T}\,\>\,\right]\,\vec{\sigma}_{1}\cdot\vec{q}\,\,\vec{\sigma}_{2}\cdot\vec{q}$
$\displaystyle+$ $\displaystyle\left[\,V_{\sigma
L}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}\,W_{\sigma
L}\,\right]\,\vec{\sigma}_{1}\cdot(\vec{q}\times\vec{k}\,)\,\,\vec{\sigma}_{2}\cdot(\vec{q}\times\vec{k}\,)\,,$

where $\vec{k}=({\vec{p}}\,^{\prime}+\vec{p})/2$ is the average momentum and
$\vec{S}=(\vec{\sigma}_{1}+\vec{\sigma}_{2})/2$ the total spin. For on-shell
scattering, $V_{\alpha}$ and $W_{\alpha}$ ($\alpha=C,S,LS,T,\sigma L$) can be
expressed as functions of $q=|\vec{q}\,|$ and
$p=|{\vec{p}}\,^{\prime}|=|\vec{p}\,|$, only.

The NL$\Omega$ 2PE is given by KBW97 ; Ent15a :

$\displaystyle W_{C}$ $\displaystyle=$
$\displaystyle{L(\tilde{\Lambda};q)\over
384\pi^{2}f_{\pi}^{4}}\left[4m_{\pi}^{2}(1+4g_{A}^{2}-5g_{A}^{4})+q^{2}(1+10g_{A}^{2}-23g_{A}^{4})-{48g_{A}^{4}m_{\pi}^{4}\over
w^{2}}\right]$ (20) $\displaystyle+\,\mbox{\rm polynomial terms of order
two}\,,$ $\displaystyle V_{T}$ $\displaystyle=$ $\displaystyle-{1\over
q^{2}}V_{S}\;=\;-{3g_{A}^{4}\over
64\pi^{2}f_{\pi}^{4}}L(\tilde{\Lambda};q)\,+\,\mbox{\rm polynomial terms of
order zero}\,,$ (21)

with $w=\sqrt{4m_{\pi}^{2}+q^{2}}$ and where the (regularized) logarithmic
loop function is given by

$L(\tilde{\Lambda};q)={w\over
2q}\ln{\frac{\tilde{\Lambda}^{2}(2m_{\pi}^{2}+q^{2})-2m_{\pi}^{2}q^{2}+\tilde{\Lambda}\sqrt{\tilde{\Lambda}^{2}-4m_{\pi}^{2}}\,q\,w}{2m_{\pi}^{2}(\tilde{\Lambda}^{2}+q^{2})}}\,.$
(22)

Note that

$\lim_{\tilde{\Lambda}\rightarrow\infty}L(\tilde{\Lambda};q)={w\over
q}\ln{\frac{w+q}{2m_{\pi}}}$ (23)

results in the logarithmic loop function of dimensional regularization.

For the explicit expressions of the polynomial terms that contribute to Eqs.
(20) and (21), see Ref. KBW97 .

####  2.2.3 Next-to-next-to-leading order

Table 2: Values of the $\pi N$ LECs as determined in Ref. Hof15 . The $c_{i}$ and $\bar{d_{i}}$ are the LECs of the second and third order $\pi N$ Lagrangians and are in units of GeV-1 and GeV-2, respectively. The uncertainties in the last digit are given in parentheses after the value. | NNL$\Omega$ | N3L$\Omega$
---|---|---
$c_{1}$ | -0.74(2) | -1.07(2)
$c_{2}$ |  | 3.20(3)
$c_{3}$ | -3.61(5) | -5.32(5)
$c_{4}$ | 2.44(3) | 3.56(3)
$\bar{d_{1}}+\bar{d_{2}}$ |  | 1.04(6)
$\bar{d_{3}}$ |  | -0.48(2)
$\bar{d_{5}}$ |  | 0.14(5)
$\bar{d}_{14}-\bar{d}_{15}$ |  | -1.90(6)

At next-to-next-to-leading order (NNL$\Omega$, $\nu=3$), we have the following
2PE contributions KBW97 ; Ent15a :

$\displaystyle V_{C}$ $\displaystyle=$ $\displaystyle{3g_{A}^{2}\over 16\pi
f_{\pi}^{4}}\left[2m_{\pi}^{2}(c_{3}-2c_{1})+c_{3}q^{2}\right](2m_{\pi}^{2}+q^{2})A(\tilde{\Lambda};q)$
(24) $\displaystyle+\,\mbox{\rm polynomial terms of order three}\,,$
$\displaystyle W_{T}$ $\displaystyle=$ $\displaystyle-{1\over
q^{2}}W_{S}=-{g_{A}^{2}\over 32\pi f_{\pi}^{4}}c_{4}w^{2}A(\tilde{\Lambda};q)$
(25) $\displaystyle+\,\mbox{\rm polynomial terms of order one}\,.$

The loop function that appears in the above expressions, regularized by
spectral-function cut-off $\tilde{\Lambda}$, is

$A(\tilde{\Lambda};q)={1\over 2q}\arctan{q(\tilde{\Lambda}-2m_{\pi})\over
q^{2}+2\tilde{\Lambda}m_{\pi}}\,,$ (26)

with

$\lim_{\tilde{\Lambda}\rightarrow\infty}A(\tilde{\Lambda};q)={1\over
2q}\arctan{q\over 2m_{\pi}}$ (27)

yielding the corresponding loop function of dimensional regularization.

The polynomial terms that occur in Eqs. (24) and (25) are given in Ref. KBW97
.

In the above expressions, the $\pi N$ low-energy constants (LECs), $c_{i}$,
from the second order $\pi N$ Lagrangian appear for the first time. We use the
very precise values as determined in the Roy-Steiner analysis by Ref. Hof15
(Table 2).

####  2.2.4 Next-to-next-to-next-to-leading order

At next-to-next-to-next-to-leading order (N3L$\Omega$, $\nu=4$), we have many
contributions which we will subdivide into separate groups.

##### Football diagram at N3L$\Omega$.

The 2PE football diagram at N3L$\Omega$ generates Kai01 :

$\displaystyle V_{C}$ $\displaystyle=$ $\displaystyle{3\over
16\pi^{2}f_{\pi}^{4}}\left[\left({c_{2}\over
6}w^{2}+c_{3}(2m_{\pi}^{2}+q^{2})-4c_{1}m_{\pi}^{2}\right)^{2}+{c_{2}^{2}\over
45}w^{4}\right]L(\tilde{\Lambda};q)\,,$ (28) $\displaystyle W_{T}$
$\displaystyle=$ $\displaystyle-{1\over q^{2}}W_{S}={c_{4}^{2}\over
96\pi^{2}f_{\pi}^{4}}w^{2}L(\tilde{\Lambda};q)\,.$ (29)

In addition to the non-polynomial contributions shown in the above equations,
there are polynomial terms of order four in the central potential and
polynomial terms of order two in the tensor (and spin-orbit) potentials, which
we do not show explicitly. This note applies to all N3L$\Omega$ expressions.

##### Leading two-loop contributions.

We state the 2PE two-loop contributions in terms of their spectral functions
Kai01 :

$\displaystyle{\rm Im}\,V_{C}$ $\displaystyle=$
$\displaystyle{3g_{A}^{4}(2m_{\pi}^{2}-\mu^{2})\over\pi\mu(4f_{\pi})^{6}}\Bigg{[}(m_{\pi}^{2}-2\mu^{2})\left(2m_{\pi}+{2m_{\pi}^{2}-\mu^{2}\over
2\mu}\ln{\mu+2m_{\pi}\over\mu-2m_{\pi}}\right)$ (30)
$\displaystyle+\,4g_{A}^{2}m_{\pi}(2m_{\pi}^{2}-\mu^{2})\Big{]}\,,$
$\displaystyle{\rm Im}\,W_{C}$ $\displaystyle=$ $\displaystyle{2\kappa\over
3\mu(8\pi
f_{\pi}^{2})^{3}}\int_{0}^{1}dx\,\Big{[}g_{A}^{2}(\mu^{2}-2m_{\pi}^{2})+2(1-g_{A}^{2})\kappa^{2}x^{2}\Big{]}$
(31)
$\displaystyle\times\Bigg{\\{}\,96\pi^{2}f_{\pi}^{2}\left[(2m_{\pi}^{2}-\mu^{2})(\bar{d}_{1}+\bar{d}_{2})-2\kappa^{2}x^{2}\bar{d}_{3}+4m_{\pi}^{2}\bar{d}_{5}\right]$
$\displaystyle+\left[4m_{\pi}^{2}(1+2g_{A}^{2})-\mu^{2}(1+5g_{A}^{2})\right]{\kappa\over\mu}\ln{\mu+2\kappa\over
2m_{\pi}}\,$ $\displaystyle+\,{\mu^{2}\over
12}(5+13g_{A}^{2})-2m_{\pi}^{2}(1+2g_{A}^{2})$
$\displaystyle-\,3\kappa^{2}x^{2}+6\kappa
x\sqrt{m_{\pi}^{2}+\kappa^{2}x^{2}}\ln{\kappa
x+\sqrt{m_{\pi}^{2}+\kappa^{2}x^{2}}\over m_{\pi}}$
$\displaystyle+\,g_{A}^{4}\left(\mu^{2}-2\kappa^{2}x^{2}-2m_{\pi}^{2}\right)$
$\displaystyle\times\left[{5\over
6}+{m_{\pi}^{2}\over\kappa^{2}x^{2}}-\left(1+{m_{\pi}^{2}\over\kappa^{2}x^{2}}\right)^{3/2}\ln{\kappa
x+\sqrt{m_{\pi}^{2}+\kappa^{2}x^{2}}\over m_{\pi}}\right]\Bigg{\\}}\,,$
$\displaystyle{\rm Im}\,V_{S}$ $\displaystyle=$ $\displaystyle\mu^{2}\,{\rm
Im}\,V_{T}={g_{A}^{2}\mu\kappa^{3}\over 8\pi
f_{\pi}^{4}}\left(\bar{d}_{15}-\bar{d}_{14}\right)\,+\,{2g_{A}^{6}\mu\kappa^{3}\over(8\pi
f_{\pi}^{2})^{3}}\int_{0}^{1}dx(1-x^{2})$ (32)
$\displaystyle\times\left[{1\over
6}-{m_{\pi}^{2}\over\kappa^{2}x^{2}}+\left(1+{m_{\pi}^{2}\over\kappa^{2}x^{2}}\right)^{3/2}\ln{\kappa
x+\sqrt{m_{\pi}^{2}+\kappa^{2}x^{2}}\over m_{\pi}}\right]\,,$
$\displaystyle{\rm Im}\,W_{S}$ $\displaystyle=$ $\displaystyle\mu^{2}\,{\rm
Im}\,W_{T}(i\mu)={g_{A}^{4}(4m_{\pi}^{2}-\mu^{2})\over\pi(4f_{\pi})^{6}}$ (33)
$\displaystyle\times\left[\left(m_{\pi}^{2}-{\mu^{2}\over
4}\right)\ln{\mu+2m_{\pi}\over\mu-2m_{\pi}}+(1+2g_{A}^{2})\mu
m_{\pi}\right]\,,$

where $\kappa=\sqrt{\mu^{2}/4-m_{\pi}^{2}}$. The above expressions involve the
$\pi N$ LECs, $\bar{d}_{i}$, from the third order $\pi N$ Lagrangian. The
values we apply for these LECs are shown in Table 2.

The momentum space amplitudes $V_{\alpha}(q)$ and $W_{\alpha}(q)$ are obtained
from the above spectral functions via the subtracted dispersion integrals,

$\displaystyle V_{C,S}(q)$ $\displaystyle=$
$\displaystyle-{2q^{6}\over\pi}\int_{2m_{\pi}}^{\tilde{\Lambda}}d\mu\,{{\rm
Im\,}V_{C,S}(i\mu)\over\mu^{5}(\mu^{2}+q^{2})}\,,$ $\displaystyle V_{T}(q)$
$\displaystyle=$
$\displaystyle{2q^{4}\over\pi}\int_{2m_{\pi}}^{\tilde{\Lambda}}d\mu\,{{\rm
Im\,}V_{T}(i\mu)\over\mu^{3}(\mu^{2}+q^{2})}\,,$ (34)

and similarly for $W_{C,S,T}$. For $\tilde{\Lambda}\rightarrow\infty$ the
above dispersion integrals yield the results of dimensional regularization,
while for finite $\tilde{\Lambda}\geq 2m_{\pi}$ we have what has become known
as spectral-function regularization (SFR) EGM04 . The purpose of the finite
scale $\tilde{\Lambda}$ is to constrain the imaginary parts to the low-
momentum region where chiral effective field theory is applicable. Thus, a
reasonable choice for $\tilde{\Lambda}$ is to keep it below the masses of the
vector mesons $\rho(770)$ and $\omega(782)$, but above the $f_{0}(500)$ [also
know as $\sigma(500)$] PDG . This suggests that the region 600-700 MeV is
appropriate for $\tilde{\Lambda}$. Consequently, we use $\tilde{\Lambda}=650$
MeV.

The subtracted dispersion integrals generate (besides the non-polynomial
contributions) polynomial terms of order four for the central potentials and
polynomial tertms of order two for the tensor potentials.

##### Leading relativistic corrections.

The relativistic corrections to the 2PE NL$\Omega$ diagrams count as
N3L$\Omega$ ($\nu=4$) and are given by ME11 :

$\displaystyle V_{C}$ $\displaystyle=$ $\displaystyle\frac{3g_{A}^{4}}{128\pi
f_{\pi}^{4}M_{N}}\bigg{[}\frac{m_{\pi}^{5}}{2w^{2}}+(2m_{\pi}^{2}+q^{2})(q^{2}-m_{\pi}^{2})A(\tilde{\Lambda};q)\bigg{]}\,,$
(35) $\displaystyle W_{C}$ $\displaystyle=$
$\displaystyle\frac{g_{A}^{2}}{64\pi
f_{\pi}^{4}M_{N}}\Bigg{\\{}\frac{3g_{A}^{2}m_{\pi}^{5}}{2w^{2}}+\big{[}g_{A}^{2}(3m_{\pi}^{2}+2q^{2})-2m_{\pi}^{2}-q^{2}\big{]}$
(36)
$\displaystyle\times(2m_{\pi}^{2}+q^{2})A(\tilde{\Lambda};q)\Bigg{\\}}\,,$
$\displaystyle V_{T}$ $\displaystyle=$
$\displaystyle-\frac{1}{q^{2}}V_{S}=\frac{3g_{A}^{4}}{256\pi
f_{\pi}^{4}M_{N}}(5m_{\pi}^{2}+2q^{2})A(\tilde{\Lambda};q)\,,$ (37)
$\displaystyle W_{T}$ $\displaystyle=$
$\displaystyle-\frac{1}{q^{2}}W_{S}=\frac{g_{A}^{2}}{128\pi
f_{\pi}^{4}M_{N}}\big{[}g_{A}^{2}(3m_{\pi}^{2}+q^{2})-w^{2}\big{]}A(\tilde{\Lambda};q)\,,$
(38) $\displaystyle V_{LS}$ $\displaystyle=$ $\displaystyle{3g_{A}^{4}\over
32\pi f_{\pi}^{4}M_{N}}\,(2m_{\pi}^{2}+q^{2})A(\tilde{\Lambda};q)\,,$ (39)
$\displaystyle W_{LS}$ $\displaystyle=$
$\displaystyle{g_{A}^{2}(1-g_{A}^{2})\over 32\pi
f_{\pi}^{4}M_{N}}\,w^{2}A(\tilde{\Lambda};q)\,.$ (40)

##### Leading three-pion exchange

The leading $3\pi$-exchange contributions that occur at N3L$\Omega$ have been
calculated in Refs. Kai00a ; Kai00b  and are found to be negligible. We,
therefore, omit them.

####  2.2.5 Subleading relativistic corrections

We also include the $1/M_{N}$ corrections of the 2PE NNL$\Omega$ diagrams.
This contribution is repulsive and proportional to $c_{i}/M_{N}$. Since we
count $Q/M_{N}\sim(Q/\Lambda_{b})^{2}$, these relativistic corrections are
formally of order N4L$\Omega$ ($\nu=5$), but we add them to our N3L$\Omega$
potential to compensate the excessive attraction generated by the football
diagram at N3L$\Omega$ EMN17 .

The result for this group of diagrams reads Kai01 ; EMN17 :

$\displaystyle V_{C}$ $\displaystyle=$
$\displaystyle{g_{A}^{2}\,L(\tilde{\Lambda};q)\over
32\pi^{2}M_{N}f_{\pi}^{4}}\Big{[}(6c_{3}-c_{2})q^{4}+4(3c_{3}-c_{2}-6c_{1})q^{2}m_{\pi}^{2}$
(41)
$\displaystyle+\,6(2c_{3}-c_{2})m_{\pi}^{4}-24(2c_{1}+c_{3})m_{\pi}^{6}w^{-2}\Big{]}\,,$
$\displaystyle W_{C}$ $\displaystyle=$ $\displaystyle-{c_{4}\over
192\pi^{2}M_{N}f_{\pi}^{4}}\left[g_{A}^{2}(8m_{\pi}^{2}+5q^{2})+w^{2}\right]q^{2}\,L(\tilde{\Lambda};q)\,,$
(42) $\displaystyle W_{T}$ $\displaystyle=$ $\displaystyle-{1\over
q^{2}}W_{S}={c_{4}\over
192\pi^{2}M_{N}f_{\pi}^{4}}\left[w^{2}-g_{A}^{2}(16m_{\pi}^{2}+7q^{2})\right]L(\tilde{\Lambda};q)\,,$
(43) $\displaystyle V_{LS}$ $\displaystyle=$
$\displaystyle{c_{2}\,g_{A}^{2}\over
8\pi^{2}M_{N}f_{\pi}^{4}}\,w^{2}L(\tilde{\Lambda};q)\,,$ (44) $\displaystyle
W_{LS}$ $\displaystyle=$ $\displaystyle-{c_{4}\over
48\pi^{2}M_{N}f_{\pi}^{4}}\left[g_{A}^{2}(8m_{\pi}^{2}+5q^{2})+w^{2}\right]L(\tilde{\Lambda};q)\,.$
(45)

###  2.3 The short-range $NN$ potential ($NN$ contact terms)

In the EFT approach, the short range interaction is described by contributions
of the contact type, which are constrained by parity, time-reversal, and the
usual invariances, but not by chiral symmetry. Only even powers of momentum
are allowed because of parity and time-reversal. Thus, the expansion of the
contact potential is formally given by

$V_{ct}=V_{ct}^{(0)}+V_{ct}^{(2)}+V_{ct}^{(4)}+V_{ct}^{(6)}+\cdots\>,$ (46)

where the superscript denotes the power or order.

In operator form, the contact potentials are given by:
Zeroth-order (leading order, L$\Omega$),

$V_{ct}^{(0)}(\vec{p^{\prime}},\vec{p})=C_{S}+C_{T}\>\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\,.$
(47)

Second order (next-to-leading order, NL$\Omega$),

$\displaystyle V_{ct}^{(2)}(\vec{p^{\prime}},\vec{p})$
$\displaystyle=C_{1}q^{2}+C_{2}k^{2}+(C_{3}q^{2}+C_{4}k^{2})\vec{\sigma_{1}}\cdot\vec{\sigma_{2}}$
$\displaystyle+C_{5}[-i\vec{S}\cdot(\vec{q}\times\vec{k})]+C_{6}(\vec{\sigma_{1}}\cdot\vec{q})(\vec{\sigma_{2}}\cdot\vec{q})$
$\displaystyle+C_{7}(\vec{\sigma_{1}}\cdot\vec{k})(\vec{\sigma_{2}}\cdot\vec{k})\,.$
(48)

Fourth order (next-to-next-to-next-to-leading order, N3L$\Omega$):

$\displaystyle V_{ct}^{(4)}(\vec{p^{\prime}},\vec{p})$
$\displaystyle=D_{1}q^{4}+D_{2}k^{4}+D_{3}q^{2}k^{2}+D_{4}(\vec{q}\times\vec{k})^{2}$
$\displaystyle+[D_{5}q^{4}+D_{6}k^{4}+D_{7}q^{2}k^{2}+D_{8}(\vec{q}\times\vec{k})^{2}]\vec{\sigma_{1}}\cdot\vec{\sigma_{2}}$
$\displaystyle+(D_{9}q^{2}+D_{10}k^{2})[-i\vec{S}\cdot(\vec{q}\times\vec{k}]$
$\displaystyle+(D_{11}q^{2}+D_{12}k^{2})(\vec{\sigma_{1}}\cdot\vec{q})(\vec{\sigma_{2}}\cdot\vec{q})$
$\displaystyle+(D_{13}q^{2}+D_{14}k^{2})(\vec{\sigma_{1}}\cdot\vec{k})(\vec{\sigma_{2}}\cdot\vec{k})$
$\displaystyle+D_{15}[\vec{\sigma_{1}}\cdot(\vec{q}\times\vec{k})\vec{\sigma_{2}}\cdot(\vec{q}\times\vec{k})]\,.$
(49)

In terms of a partial-wave decomposition, we have up to fourth order:

$\displaystyle\langle{}^{1}S_{0},p^{\prime}|V_{ct}|{}^{1}S_{0},p\rangle$
$\displaystyle=\widetilde{C}_{{}^{1}S_{0}}+C_{{}^{1}S_{0}}(p^{2}+p^{\prime
2})+\widehat{D}_{{}^{1}S_{0}}(p^{\prime 4}+p^{4})+{D}_{{}^{1}S_{0}}p^{\prime
2}p^{2},$
$\displaystyle\langle{}^{3}S_{1},p^{\prime}|V_{ct}|{}^{3}S_{1},p\rangle$
$\displaystyle=\widetilde{C}_{{}^{3}S_{1}}+C_{{}^{3}S_{1}}(p^{2}+p^{\prime
2})+\widehat{D}_{{}^{3}S_{1}}(p^{\prime 4}+p^{4})+{D}_{{}^{3}S_{1}}p^{\prime
2}p^{2},$
$\displaystyle\langle{}^{3}S_{1},p^{\prime}|V_{ct}|{}^{3}D_{1},p\rangle$
$\displaystyle=C_{{}^{3}S_{1}-{}^{3}D_{1}}p^{2}+\widehat{D}_{{}^{3}S_{1}-{}^{3}D_{1}}p^{4}+{D}_{{}^{3}S_{1}-{}^{3}D_{1}}p^{\prime
2}p^{2},$
$\displaystyle\langle{}^{1}P_{1},p^{\prime}|V_{ct}|{}^{1}P_{1},p\rangle$
$\displaystyle=C_{{}^{1}P_{1}}\>pp^{\prime}+D_{{}^{1}P_{1}}(p^{\prime
3}p+p^{\prime}p^{3}),$
$\displaystyle\langle{}^{3}P_{0},p^{\prime}|V_{ct}|{}^{3}P_{0},p\rangle$
$\displaystyle=C_{{}^{3}P_{0}}\>pp^{\prime}+D_{{}^{3}P_{0}}(p^{\prime
3}p+p^{\prime}p^{3}),$
$\displaystyle\langle{}^{3}P_{1},p^{\prime}|V_{ct}|{}^{3}P_{1},p\rangle$
$\displaystyle=C_{{}^{3}P_{1}}\>pp^{\prime}+D_{{}^{3}P_{1}}(p^{\prime
3}p+p^{\prime}p^{3}),$
$\displaystyle\langle{}^{3}P_{2},p^{\prime}|V_{ct}|{}^{3}P_{2},p\rangle$
$\displaystyle=C_{{}^{3}P_{2}}\>pp^{\prime}+D_{{}^{3}P_{2}}(p^{\prime
3}p+p^{\prime}p^{3}),$
$\displaystyle\langle{}^{3}P_{2},p^{\prime}|V_{ct}|{}^{3}F_{2},p\rangle$
$\displaystyle={D}_{{}^{3}P_{2}-{}^{3}F_{2}}p^{\prime}p^{3},$
$\displaystyle\langle{}^{1}D_{2},p^{\prime}|V_{ct}|{}^{1}D_{2},p\rangle$
$\displaystyle={D}_{{}^{1}D_{2}}p^{\prime 2}p^{2},$
$\displaystyle\langle{}^{3}D_{1},p^{\prime}|V_{ct}|{}^{3}D_{1},p\rangle$
$\displaystyle={D}_{{}^{3}D_{1}}p^{\prime 2}p^{2},$
$\displaystyle\langle{}^{3}D_{2},p^{\prime}|V_{ct}|{}^{3}D_{2},p\rangle$
$\displaystyle={D}_{{}^{3}D_{2}}p^{\prime 2}p^{2},$
$\displaystyle\langle{}^{3}D_{3},p^{\prime}|V_{ct}|{}^{3}D_{3},p\rangle$
$\displaystyle={D}_{{}^{3}D_{3}}p^{\prime 2}p^{2}.$ (50)

Notice that, in our notation, partial-wave contact LECs

  * •

$\widetilde{C}_{\alpha}$ are of zeroth order (there are two),

  * •

${C}_{\alpha}$ are of second order (there are seven), and

  * •

$\widehat{D}_{\alpha}$ and ${D}_{\alpha}$ are of fourth order (there are 15),

where $\alpha$ stands for a partial wave or a combination thereof. There exist
linear one-to-one relations between the two $\widetilde{C}_{\alpha}$ and
$C_{S}$ and $C_{T}$ of Eq. (47), the seven $C_{\alpha}$ and the seven $C_{i}$
of Eq. (48), and the 15 $\widehat{D}_{\alpha}$ and $D_{\alpha}$ and the 15
$D_{i}$ of Eq. (49). The relations can be found in Ref. ME11 .

Note that the partial-wave decomposition of $Q^{\nu}$ (where $Q$ is either the
momentum transfer $q$ or the average momentum $k$) has an interesting
property. For even $\nu$,

$Q^{\nu}=f_{\frac{\nu}{2}}(\cos\theta)\,,$ (51)

where $f_{m}$ stands for a polynomial of degree $m$ and $\theta$ is the CMS
scattering angle. The partial-wave decomposition of $Q^{\nu}$ for a state of
orbital-angular momentum $L$ involves the integral

$I^{(\nu)}_{L}=\int_{-1}^{+1}Q^{\nu}P_{L}(\cos\theta)d\cos\theta=\int_{-1}^{+1}f_{\frac{\nu}{2}}(\cos\theta)P_{L}(\cos\theta)d\cos\theta\,,$
(52)

where $P_{L}$ is a Legendre polynomial. Due to the orthogonality of the
$P_{L}$,

$I^{(\nu)}_{L}=0\hskip 14.22636pt\mbox{for}\hskip
14.22636ptL>\frac{\nu}{2}\,.$ (53)

Consequently, contact terms of order zero contribute only in $S$-waves, while
second order terms can contribute up to $P$-waves, fourth order terms up to
$D$-waves, etc..

###  2.4 Scattering equation and regularization

The potential $V$ is, in principal, an invariant amplitude (with relativity
taken into account perturbatively) and, thus, satisfies a relativistic
scattering equation, like, e. g., the Blankenbeclar-Sugar (BbS) equation BS66
, which reads explicitly,

${T}({\vec{p}}~{}^{\prime},{\vec{p}})={V}({\vec{p}}~{}^{\prime},{\vec{p}})+\int\frac{d^{3}p^{\prime\prime}}{(2\pi)^{3}}\>{V}({\vec{p}}~{}^{\prime},{\vec{p}}~{}^{\prime\prime})\>\frac{M_{N}^{2}}{E_{p^{\prime\prime}}}\>\frac{1}{{p}^{2}-{p^{\prime\prime}}^{2}+i\epsilon}\>{T}({\vec{p}}~{}^{\prime\prime},{\vec{p}})\,,$
(54)

with $E_{p^{\prime\prime}}\equiv\sqrt{M_{N}^{2}+{p^{\prime\prime}}^{2}}$. The
advantage of using a relativistic scattering equation is that it automatically
includes relativistic kinematical corrections to all orders. Thus, in the
scattering equation, no propagator modifications are necessary when moving up
in the orders.

Defining

$\displaystyle\widehat{V}(\vec{p^{\prime}},\vec{p})\equiv\frac{1}{(2\pi)^{3}}\sqrt{\frac{M_{N}}{E_{p^{\prime}}}}\>V(\vec{p^{\prime}},\vec{p})\sqrt{\frac{M_{N}}{E_{p}}}$
(55)

and

$\displaystyle\widehat{T}(\vec{p^{\prime}},\vec{p})\equiv\frac{1}{(2\pi)^{3}}\sqrt{\frac{M_{N}}{E_{p^{\prime}}}}\>T(\vec{p^{\prime}},\vec{p})\sqrt{\frac{M_{N}}{E_{p}}}\,,$
(56)

the BbS equation collapses into the usual, nonrelativistic Lippmann-Schwinger
(LS) equation,

$\displaystyle\widehat{T}(\vec{p^{\prime}},\vec{p})$
$\displaystyle=\widehat{V}(\vec{p^{\prime}},\vec{p})+\int{d^{3}\>p^{\prime\prime}}\>\widehat{V}(\vec{p^{\prime}},\vec{p^{\prime\prime}})\>\frac{M_{N}}{p^{2}-p^{\prime\prime
2}+i\epsilon}\>\widehat{T}(\vec{p^{\prime\prime}},\vec{p})\,.$ (57)

Iteration of $\widehat{V}$ in the LS equation, Eq. (57), requires cutting
$\widehat{V}$ off for high momenta to avoid infinities. This is consistent
with the fact that ChPT is a low-momentum expansion which is valid only for
momenta $Q\leq\Lambda_{b}\approx 0.6$ GeV. Therefore, the potential
$\widehat{V}$ is multiplied with a regulator function $f(p^{\prime},p)$,

$\widehat{V}(\vec{p^{\prime}},\vec{p})\longmapsto\widehat{V}(\vec{p^{\prime}},\vec{p})\>f(p^{\prime},p)\,,$
(58)

with

$f(p^{\prime},p)=exp[-(p^{\prime}/\Lambda)^{2n}\>-(p/\Lambda)^{2n}].$ (59)

For the chiral potentials applied in this investigation, we use $\Lambda=500$
MeV EMN17 .

##  3  Relevance of contact terms versus pion exchanges in lower partial waves

[height=12cm, width=8cm]figph1a.pdf

Figure 1: Chiral expansion of $np$ scattering as represented by the phase
shifts in the $J=0,1$ states. Five orders ranging from L$\Omega$ to
N4L$\Omega$ are shown as denoted. The solid dots and open circles are results
from the Nijmegen multienergy $np$ phase shift analysis Sto93  and GWU single-
energy $np$ analysis SP07 , respectively. (From Ref. EMN17 )

-0.75cm

Figure 2: Same as Fig. 1, but for $J=2$, $\epsilon_{1}$, $\epsilon_{2}$, and
${}^{3}D_{3}$.

In Ref. EMN17 , $NN$ potentials through all orders from L$\Omega$ to
N4L$\Omega$ were constructed. There are improvements in the reproduction of
the empirical phase shifts as the orders increase and an excellent agreement
is achieved at orders N3L$\Omega$ and N4L$\Omega$, Figs. 1 and 2. Similar
results have been obtained by other researchers in the field Pia15 ; Pia16 ;
Car16 ; RKE18 ; Eks18 ; EKM15 . Note that such fits involve two contacts at
L$\Omega$ which contribute in $S$-waves, nine contacts at NL$\Omega$ and
NNL$\Omega$ which contribute up to $P$-waves, and 24 contacts at N3L$\Omega$
and N4L$\Omega$ which contribute up to $D$-waves [cf. Eq. (50)]. It is the
purpose of this study to analyze in detail the role of these contacts in the
fits of the $L\leq 2$ phase shifts up to N3L$\Omega$.

As discussed, the nuclear force consists essentially of two parts: the short
range (Sect. 2.3) and the long range (Sect. 2.2). In chiral EFT, the long-
range is represented by one- and multi-pion exchanges, and the short range is
described by contact terms. The lower partial waves are particularly sensitive
to the short range and, in fact, at N3L$\Omega$, four contact terms contribute
to each $S$-wave, two to each $P$-wave, and one to each $D$-wave, Eq. (50).
There are no contact contributions in $F$ and higher partial waves—at
N3L$\Omega$.

As explained in the Introduction, since lower partial waves are sensitive to
the short range potential, one may suspect that the contact contributions are
dominant and simply override the pion-exchange contributions in lower partial
waves which, on the other hand, are the most important ones in applications of
the potentials to nuclear structure and reactions.

To shed light on this issue, we will now systematically investigate the role
of those contacts versus pion exchange in those lower partial waves of $NN$
scattering.

In our $NN$ potential construction, the $\pi N$ LECs are not fit-parameters;
they are held fixed at their values determined in $\pi N$ scattering, Table 2.
Therefore, the LECs of the $NN$ contacts are the only fit parameters available
to optimize the reproduction of the $NN$ data (below 300 MeV laboratory
energy). In this investigation, we will use the contact LECs to fit specific
$NN$ low-energy parameters or phase shifts. We will consider various
scenarios, namely, using contacts only or using contacts together with pion
contributions of increasing chiral order. The failure to reproduce the $NN$
data by contacts only and the improvements that occur when (chiral) pion
contributions are added will reveal the relevance of chiral symmetry in those
lower partial waves. To obtain maximum insight into the role that contact
terms can play, we will not follow here the rule that contact and pion
contributions should be of the same order. In fact, we may, for example,
consider contact contributions up to fourth order alone or with just the
(lowest order) 1PE or low-order 2PE added, to demonstrate what contacts can
maximally achieve or not achieve. For contacts and pion exchanges, we consider
orders up to N3L$\Omega$ (fourth order).

To keep it simple at the start, we begin with the partial waves that have only
one contact, namely, $D$-waves and, then, proceed to the more elaborate cases,
$P$ and $S$-waves.

###  3.1 $D$-waves

To demonstrate the relevance of the pion exchange contributions (versus
contacts) in $D$-waves, we consider the following cases for which we introduce
the short notation given in parenthesis.

  * •

Contact contribution only (ct1).

  * •

L$\Omega$ pion exchange (i. e., 1PE) only and no contact term (L0).

  * •

L$\Omega$ 1PE plus contact term (L1).

  * •

NL$\Omega$ pion exchanges only, no contact term (NL0).

  * •

NL$\Omega$ pion exchanges plus contact term (NL1).

  * •

NNL$\Omega$ pion exchanges only, no contact term (NNL0).

  * •

NNL$\Omega$ pion exchanges plus contact term (NNL1).

  * •

N3L$\Omega$ pion exchanges plus contact term (NNNL1).

Our short notation (given in parenthesis) is designed such that the letters
always indicate the order of the pion exchanges included and the integer
states the number of contacts involved (from the contacts available for the
given partial wave). Note that in $D$-waves, there is only one (fourth order)
contact per partial wave available, Eq. (50). When we include this contact
term, we fit it to the empirical phase-shift at 50 MeV laboratory energy as
determined in the Nijmegen phase-shift analysis Sto93 . The values for the
contact LECs so obtained are listed in Table 3.

Note that the chiral 2PE expressions at orders NL$\Omega$, Eqs. (20) and (21),
and NNL$\Omega$, Eqs. (24) and (25), include polynomial terms up to order
$Q^{3}$ KBW97 , which do not contribute in $D$-waves [cf. Eq. (53) and text
below the equation]. Therefore, in the cases of L1, NL1, and NNL1, the $Q^{4}$
contacts are not renormalized and represent the true corrections needed on top
of the non-polynomial parts of the pion-exchanges, denoted by L0, NL0, and
NNL0, respectively (Table 3 and Fig. 3).

The situation is different at N3L$\Omega$. The subtracted dispersion
integrals, Eq. (34), generate—besides the non-polynomial parts—polynomial
terms up to fourth order. Moreover, the other N3L$\Omega$ 2PE contributions
(Sect. 2.2.4) and the contributions of Sect. 2.2.5 also include polynomial
terms of $\mathcal{O}(Q^{4})$. Thus, the fourth order contact term, we
introduce to fit the phase shift at 50 MeV, includes a compensation for the
fourth order polynomial terms generated by the 2PE contributions. Therefore,
in the case of NNNL1 of Table 3, the contact LECs are “renormalized”. It is
not just the correction needed besides the non-polynominal 2PE contribution at
N3L$\Omega$ to fit the $D$-wave phase shifts at 50 MeV, and should not be
interpreted that way. In fact, the large size of the NNNL1 contact LECs shown
in Table 3 indicate that the fourth order polynomial terms generated by
N3L$\Omega$ pion contributions can be sizable.

The phase shifts up to 300 MeV predicted for the various cases are shown in
Fig. 3. Next we will discuss those phase shifts partial wave by partial wave.

Table 3: Contact LECs used for $D$-waves [cf. Eq. (50)] in units of $10^{4}$ GeV-6. case | $D_{{}^{1}D_{2}}$ | $D_{{}^{3}D_{2}}$ | $D_{{}^{3}D_{3}}$
---|---|---|---
ct1 | -3.2575 | -5.7202 | -1.0130
L1 | -1.6165 | -0.0578 | -1.7843
NL1 | -1.2045 | -0.3464 | -2.3773
NNL1 | -0.2068 | 0.2023 | -1.3345
NNNL1 | -2.088 | -3.3804 | -1.4764

(a)  (b)

(c)  (d)

(e)  (f)

Figure 3: $D$-wave phase shifts of neutron-proton scattering for the various
cases discussed in the text. Solid dots and open circles as in Fig. 1.

####  3.1.1 The ${}^{1}D_{2}$-wave

We start with the left ${}^{1}D_{2}$ frame in Fig. 3. When only the contact
term is applied and no pion-exchanges (curve ct1) then the phase shift
increases dramatically with energy indicating that the contact contribution is
of very short range and completely inadequate to describe this $D$-wave. 1PE
is weak (curve L0). Adding the contact to 1PE brings the phase shift up, but
too much since obviously the contact is dominant. When 2PE contributions are
added (right ${}^{1}D_{2}$ frame), the description improves with increasing
order. While the NL$\Omega$ 2PE is weak and, therefore, does not lead to much
improvement (cf. NL0 and NL1), the NNL$\Omega$ 2PE is known to provide a
realistic intermediate-range attraction and together with the contact leads to
a quantitative description (curve NNL1), and so does NNNL1. The conclusion is
that the contact alone can by no means describe ${}^{1}D_{2}$. The strong
intermediate-range attraction provided by chiral 2PE at NNL$\Omega$ and
N3L$\Omega$ is crucial. As the small contact LEC in the case of NNL1 reveals
(Table 3), the contact contribution is minor, while chiral 2PE rules. This
example demonstrates that even when a contact term is involved, chirality can
still be the major factor and show a clear signature.

####  3.1.2 The ${}^{3}D_{2}$-wave

Also in the ${}^{3}D_{2}$-wave, the contact contribution alone (cf. ct1 curve
in the left ${}^{3}D_{2}$ frame in Fig. 3) leads to a dramatically wrong
description. In this particular partial wave, the 1PE (L0 curve) happens to
play a dominant role, because the matrix element of the tensor operator is 2
in this state which, in addition, is mutiplied by (-3) from the
$\bm{\tau}_{1}\cdot\bm{\tau}_{2}$ factor, resulting in an overall factor of
(-6) for the pion tensor potential. As the L0 curve reveals, this large tensor
contribution alone, essentially, explains the ${}^{3}D_{2}$-wave. 2PE
contributions play only a minor role (cf. right ${}^{3}D_{2}$ frame), because
the (mainly) central forces provided by 2PE are small as compared to the huge
tensor force contribution from 1PE in this particuler wave. This scenario
leaves little room for contact contributions. One-pion-exchange, the most
pronounced expression of chiral symmetry, rules this wave.

####  3.1.3 The ${}^{3}D_{3}$-wave

The cases ct1, L0, and L1 are inadequate similarly to what we have seen in
${}^{1}D_{2}$. The 2PE contributions at NL$\Omega$ and NNL$\Omega$ without and
with contact contribution (NL0, NNL0 and NL1, NNL1, respectively) do not lead
to much improvement. Finally, with NNNL1 a more realistic result starts to
develop. A quantitative description has to wait for N4L$\Omega$ as
demonstrated in the ${}^{3}D_{3}$ frame of Fig. 2. In any case, the contact
alone cannot describe the ${}^{3}D_{3}$ wave, since the contact contribution
is too short-ranged. Substantial intermediate-range attraction is needed,
which only chiral 2PE can provide.

####  3.1.4 The ${}^{3}D_{1}$-wave

Since the ${}^{3}D_{1}$ wave is coupled to ${}^{3}S_{1}$, we will discuss it
in conjunction with the coupled ${}^{3}S_{1}$-${}^{3}D_{1}$-$\epsilon_{1}$
system, below.

####  3.1.5 $D$-waves summary

Contacts alone can not reproduce $D$-waves (cf. all the ct1 cases in the left
column of Fig. 3), because of the short-range nature of contact contributions,
which are ill-suited for $D$-waves. The strong intermediate-range attraction
provided by chiral 2PE at NNL$\Omega$ and N3L$\Omega$ is crucial, unless the
1PE tensor force is dominant, which also is a reflection of chiral symmetry.
For exact fits, contact corrections are needed, but they are very small. Thus,
in spite of contributions from contacts, chirality makes the largest imprint
on $D$-waves.

The $D$-waves are, in fact, an interesting case. On the one hand, they are not
so peripheral that the (very long-ranged) 1PE is dominant and, on the other
hand, their orbital angular momentum is large enough to prevent them from
being too sensitive to the (short-ranged) contact potential. Thus, the
$D$-waves are a true window on the intermediate range. Consequently, they test
the reality of the (internediate-ranged) 2PE as produced by chiral symmetry.
In particular, the ${}^{1}D_{2}$-wave demonstrates that this test is passed
well.

###  3.2 $P$-waves

In $P$-waves, we have two contacts available per partial wave; one is of order
two, $C_{\alpha}$, and the other one is of order four, $D_{\alpha}$ [cf. Eq.
(50)]. We then consider the following cases with the short notation given in
parenthesis.

  * •

One contact contribution and nothing else (ct1).

  * •

Two contact contributions (ct2).

  * •

L$\Omega$ pion exchange (i. e., 1PE) only and no contact term (L0).

  * •

L$\Omega$ 1PE plus one contact term (L1).

  * •

L$\Omega$ 1PE plus two contact terms (L2).

  * •

NL$\Omega$ pion exchanges plus one contact term (NL1).

  * •

NL$\Omega$ pion exchanges plus two contact terms (NL2).

  * •

NNL$\Omega$ pion exchanges plus one contact term (NNL1).

  * •

NNL$\Omega$ pion exchanges plus two contact terms (NNL2).

  * •

N3L$\Omega$ pion exchanges plus two contact terms (NNNL2).

The values for the contact LECs used in the various cases are listed in Table
4.

As mentioned, the chiral 2PE expressions at orders NL$\Omega$ and NNL$\Omega$
include polynomial terms of order $Q^{2}$ and the 2PE expressions at order
N3L$\Omega$ include polynomial terms up to order $Q^{4}$. $\mathcal{O}(Q^{2})$
and $\mathcal{O}(Q^{4})$ polynomial terms do not vanish in $P$-waves [Eq.
(53)]. These terms are absorbed by the second and fourth order contact terms.
Therefore, the minimal number of contacts to be applied at NL$\Omega$ and
NNL$\Omega$ is one (second order) contact and two (second and fourth order)
contacts at N3L$\Omega$. Thus, the contact LECs shown in Table 4 for NL1,
NNL1, and NNNL2 are not just the corrections needed besides the genuine 2PE
contributions and their size does not reflect the size of “what is missing”.
However, in the cases NL2 and NNL2, the second contact included, $D_{\alpha}$
(fourth order contact), is not renormalized (since NL$\Omega$ and NNL$\Omega$
2PE does not generate $Q^{4}$ polynomials) and, therfore, reflects a true
fourth order correction.

The phase shifts up to 300 MeV that result from the various $P$-wave cases are
shown in Fig. 4, which we will discuss now.

Table 4: Contact LECs used in $P$-waves [cf. Eq. (50)]. Second order contacts, $C_{\alpha}$, are in units of 104 GeV-4, while fourth order contacts, $D_{\alpha}$, are in units of 104 GeV-6. | ${}^{1}P_{1}$ | ${}^{3}P_{0}$ | ${}^{3}P_{1}$ | ${}^{3}P_{2}$
---|---|---|---|---
case | $C_{{}^{1}P_{1}}$ | $D_{{}^{1}P_{1}}$ | $C_{{}^{3}P_{0}}$ | $D_{{}^{3}P_{0}}$ | $C_{{}^{3}P_{1}}$ | $D_{{}^{3}P_{1}}$ | $C_{{}^{3}P_{2}}$ | $D_{{}^{3}P_{2}}$
ct1 | 6.5533 | 0 | -0.4631 | 0 | 4.3248 | 0 | -0.3256 | 0
ct2 | 2.17 | -5.0 | -0.874 | 10.0 | 1.4127 | -5.0 | -0.4766 | 1.6
L1 | 0.1349 | 0 | 0.8463 | 0 | -0.1732 | 0 | -0.2302 | 0
L2 | 0.1613 | 0.95 | 0.8531 | -0.55 | -0.1480 | 1.58 | -0.3300 | 1.1
NL1 | 0.2295 | 0 | 1.3228 | 0 | -0.4607 | 0 | -0.2203 | 0
NL2 | 0.2664 | 1.45 | 1.3234 | -0.03 | -0.4352 | 1.2 | -0.3203 | 1.1
NNL1 | 0.1821 | 0 | 1.1415 | 0 | -0.7851 | 0 | -0.6333 | 0
NNL2 | 0.1912 | 0.3 | 1.1495 | -0.95 | -0.8133 | -0.58 | -0.6251 | -0.1
NNNL2 | 0.1933 | 9.72 | 1.1883 | 4.92 | -0.8105 | 4.74 | -0.7464 | 5.95

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 4: $P$-wave phase shifts of neutron-proton scattering for the various
cases discussed in the text. Solid dots and open circles as in Fig. 1.

When we apply only one contact, we use the order two one and fit it to the
empirical phase-shift at 50 MeV laboratory energy as determined in the
Nijmegen phase-shift analysis Sto93 . When both contacts are involved, we fit
the empirical phase-shifts at 50 MeV and 150 MeV (if possible).

Obviously, with just one contact term and no pion contributions (cases ct1 of
the left column of Fig. 4) the description is grossly wrong in all $P$-waves.
Adding the second contact does not lead to any improvement in ${}^{1}P_{1}$
and ${}^{3}P_{1}$ and, in fact, in these two cases it is not possible to fit
the phase shift at 150 MeV. The ${}^{3}P_{0}$ and ${}^{3}P_{2}$ partial waves
improve with the second contact, but are not any close to a quantitative
description. Adding 1PE (L0) together with one or two contacts (L1, L2) brings
about considerable improvement in most $P$-waves. Turning to the frames of the
right column of Fig. 4 where the 2PE exchanges of various orders are added, we
observe order by order improvement. ${}^{1}P_{1}$ is described well in the
cases of NNL1 and NNL2, while the other partial waves assume a quantitative
character only when the powerfull 2PE at N3L$\Omega$ is added (case NNNL2).

In summary, contacts alone are inadequate to describe $P$-waves. 1PE brings
improvement, but strong chiral 2PE is needed for a quantitative description of
$P$-waves. Thus, a clear signature of chiral symmetry can be identified in
$P$-waves.

A note is in place on ${}^{3}P_{2}$, since it is coupled with ${}^{3}F_{2}$
and $\epsilon_{2}$ through the contact LEC $D_{{}^{3}P_{2}-^{3}F_{2}}$, Eq.
(50). We found that the latter parameter has only a weak effect on the
${}^{3}P_{2}$ phase shift and, therefore, we decided to leave it out of our
considerations. We kept it at zero.

###  3.3 The ${}^{1}S_{0}$-wave

Table 5: Columns two to five show the contact LECs used in the ${{}^{1}S_{0}}$ wave [cf. Eq. (50)]. The zeroth order contact $\widetilde{C}_{{}^{1}S_{0}}$ is in units of $10^{4}$ GeV-2; the second order contact $C_{{}^{1}S_{0}}$ in units of $10^{4}$ GeV-4; and fourth order contacts $\widehat{D}_{{}^{1}S_{0}}$ and $D_{{}^{1}S_{0}}$ in units of $10^{4}$ GeV-6. Column six and seven display the $np$ scattering length, $a_{np}$, and effective range, $r_{np}$, in the ${{}^{1}S_{0}}$ state.  case | $\widetilde{C}_{{}^{1}S_{0}}$ | $C_{{}^{1}S_{0}}$ | $\widehat{D}_{{}^{1}S_{0}}$ | $D_{{}^{1}S_{0}}$ |  $a_{np}$ (fm) |  $r_{np}$ (fm)
---|---|---|---|---|---|---
ct1 | -0.063985 | 0 | 0 | 0 | -23.74 | 0.69
ct2 | 0.475799 | 4.0 | 0 | 0 | -23.74 | 2.37
ct3 | -0.158301 | 2.0 | -6.0 | 0 | -23.74 | 2.66
L1 | -0.109340 | 0 | 0 | 0 | -23.74 | 1.73
L2 | -0.130919 | 1.33 | 0 | 0 | -23.74 | 2.70
NL2 | -0.146214 | 1.815 | 0 | 0 | -23.74 | 2.70
NNL2 | -0.152032 | 2.36 | 0 | 0 | -23.74 | 2.70
NNNL4 | -0.139563 | 2.417 | -2.332 | -16.74 | -23.74 | 2.70

(a)  (b)

Figure 5: ${}^{1}S_{0}$ phase shifts of neutron-proton scattering for the
various cases discussed in the text. Solid dots and open circles as in Fig. 1.

In the ${}^{1}S_{0}$ wave, we have available a total of four contact terms
[cf. Eq. (50)], namely, one zeroth order contact,
$\widetilde{C}_{{}^{1}S_{0}}$, one second order contact, $C_{{}^{1}S_{0}}$,
and two fourth order contacts, $\widehat{D}_{{}^{1}S_{0}}$ and
$D_{{}^{1}S_{0}}$. When we use only one contact, we pick the zeroth order one
and fit it to the $np$ ${}^{1}S_{0}$ scattering length, $a_{np}=-23.74$ MeV.
When we apply two contacts, we fit, besides the scattering length, the
${}^{1}S_{0}$ $np$ effective range parameter, $r_{np}=2.70\pm 0.05$ MeV. With
three parameters, we also try to reproduce (if possible) the empirical phase-
shift at 50 MeV laboratory energy as determined in the Nijmegen phase-shift
analysis Sto93  and, with four parameters, the phase shift at 150 MeV is
included in the fit. We consider the following cases with the short notation
given in parenthesis.

  * •

One contact contribution, and nothing else (ct1).

  * •

Two contact contributions (ct2).

  * •

Three contact contributions (ct3).

  * •

L$\Omega$ 1PE and no contact term (L0).

  * •

L$\Omega$ 1PE plus one contact term (L1).

  * •

L$\Omega$ 1PE plus two contact terms (L2).

  * •

NL$\Omega$ pion exchanges plus two contact terms (NL2).

  * •

NNL$\Omega$ pion exchanges plus two contact terms (NNL2).

  * •

N3L$\Omega$ pion exchanges plus four contact terms (NNNL4).

The values for the contact LECs are listed in Table 5 and the phase shifts up
to 300 MeV that result from the various ${}^{1}S_{0}$ cases are shown in Fig.
5, which we will discuss now.

When only one contact term is used (fit to $a_{np}$) and no pion contributions
(case ct1), then the ${}^{1}S_{0}$ phase shifts for intermediate energies are
far above the data. Adding more contacts (cases ct2 and ct3) moves those
predictions below the data. The prediction with four contacts is essentially
the same as with three contacts and, therefore, not shown. Clearly, contacts
alone cannot describe the ${}^{1}S_{0}$ wave at intermediate energies, no
matter how many contacts one is using. 1PE alone (L0) is small and adding to
it one or two contacts (cases L1 and L2) brings about predictions that are
very similar to the coresponding cases with contacts alone (ct1 and ct2) and,
again, adding more contacts does essentially not change anything. Thus, in
${}^{1}S_{0}$, 1PE is obviously of very limited relevance, except for the
effective range parameter, $r_{np}$, which is improved by 1PE (cf. Table 5).
The strong part of 1PE is its tensor force, which does not contribute in
singlet states where only the (weak) central force has a presence. The
momentum-space 1PE includes also a constant term/contact term [see Eq. (14)],
which converts into a $\delta(\vec{r})$-function in position space. The L0
case includes the $\delta(\vec{r})$-function contribution.

We now turn to the frame on the right of Fig. 5, where the 2PE exchanges of
the various orders are added in. The NL$\Omega$ 2PE (curve NL2) does not
create any improvement over the L2 case. However 2PE at NNL$\Omega$ (curve
NNL2) leads to an excellent reproduction of the ${}^{1}S_{0}$ phase shifts up
to 300 MeV. Adding more contacts beyond two in the cases of NL$\Omega$ and
NNL$\Omega$ does not improve the description, which is why we do not show
these cases. The NNNL4 case creates further subtle refinements.

We remind the reader again of the fact that the chiral 2PE expressions at
orders NL$\Omega$ and NNL$\Omega$ include polynomial terms of order $Q^{0}$
and $Q^{2}$, and the 2PE expressions at order N3L$\Omega$ include polynomial
terms up to order $Q^{4}$, which are always compensated by contacts of the
same order. Therefore, in the case of the ${}^{1}S_{0}$ wave, the minimal
number of contacts to be applied at NL$\Omega$ and NNL$\Omega$ is two (zeroth
and second order) and four (of orders zero, two, and four) at N3L$\Omega$.
Thus, the contact LECs shown in Table 5 for NL2, NNL2, and NNNL4 are
renormalized numbers whose size does not necessarily reflect the size of what
is missing beyond the genuine pion exchange contributions.

In summary, contacts alone are inadequate to describe the ${}^{1}S_{0}$-wave
at intermediate energies. The strong chiral 2PE that starts at NNL$\Omega$ is
needed for a quantitative description of the ${}^{1}S_{0}$-wave. There is a
clear signature of chiral symmetry in ${}^{1}S_{0}$-wave.

###  3.4 The coupled ${}^{3}S_{1}$-${}^{3}D_{1}$-$\epsilon_{1}$ system

Table 6: Columns two to seven show the contact LECs used in the
${}^{3}S_{1}-^{3}D_{1}$ waves [cf. Eq. (50)]. The $\widetilde{C}_{\alpha}$ of
the zeroth order contact are given in units of $10^{4}$ GeV-2; the
$C_{\alpha}$ of second order in $10^{4}$ GeV-4; and $\widehat{D}_{\alpha}$ and
$D_{\alpha}$ of fourth order in $10^{4}$ GeV-6. Column eight and nine display
the triplet scattering length, $a_{t}$, and effective range, $r_{t}$,
respectively, in the ${{}^{3}S_{1}}$ state.

case | $\widetilde{C}_{{}^{3}S_{1}}$ | $C_{{}^{3}S_{1}}$ | $\widehat{D}_{{}^{3}S_{1}}$ | $D_{{}^{3}S_{1}}$ | ${D}_{{}^{3}D_{1}}$ | $C_{{{}^{3}S_{1}}-{{}^{3}D_{1}}}$ |  $a_{t}$ (fm) |  $r_{t}$ (fm)
---|---|---|---|---|---|---|---|---
ct1 | -0.077103 | 0 | 0 | 0 | 0 | 0 | 5.42 | 0.68
ct5 | -0.1311 | 2.0 | -0.5 | 0 | 27.0 | -1.25 | 5.42 | 1.76
L1 | -0.06366 | 0 | 0 | 0 | 0 | 0 | 5.42 | 1.59
L5 | -0.13345 | 0.4 | -0.7 | 0 | -2.0 | 0.41 | 5.42 | 1.73
NL2 | -0.136835 | -0.39 | 0 | 0 | 0 | 0 | 5.42 | 1.76
NL5 | -0.1255 | -0.5 | -2.3 | 0 | -2.3 | 0.1 | 5.42 | 1.73
NNL2 | -0.10002 | -0.335 | 0 | 0 | 0 | 0 | 5.42 | 1.75
NNL5 | -0.14875 | 0.4 | -0.1 | 0 | -1.4 | 0.4 | 5.42 | 1.74
NNNL8a | -0.159635 | 0.8233 | -4.319 | -19.17 | -5.59 | 0.503 | 5.42 | 1.75

a In the case of NNNL8, besides the six parameters given,
$\widehat{D}_{{}^{3}S_{1}-^{3}D_{1}}=1.162$ and
$D_{{}^{3}S_{1}-^{3}D_{1}}=1.759$. In all other cases,
$\widehat{D}_{{}^{3}S_{1}-^{3}D_{1}}=D_{{}^{3}S_{1}-^{3}D_{1}}=0$.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 6: ${}^{3}S_{1}$, ${}^{3}D_{1}$, and $\epsilon_{1}$ phase parameters of
neutron-proton scattering for the various cases discussed in the text. Solid
dots and open circles as in Fig. 1.

In the coupled ${}^{3}S_{1}$-${}^{3}D_{1}$-$\epsilon_{1}$ system, we have
available a total of eight contact terms [cf. Eq. (50)]; namely, four for
${}^{3}S_{1}$ ($\widetilde{C}_{{}^{3}S_{1}}$, $C_{{}^{3}S_{1}}$,
$\widehat{D}_{{}^{3}S_{1}}$, $D_{{}^{3}S_{1}}$), one for ${}^{3}D_{1}$
(${D}_{{}^{3}D_{1}}$), and three for the ${{}^{3}S_{1}}$-${{}^{3}D_{1}}$
transition potential ($C_{{{}^{3}S_{1}}-{{}^{3}D_{1}}}$,
$\widehat{D}_{{{}^{3}S_{1}}-{{}^{3}D_{1}}}$,
${D}_{{{}^{3}S_{1}}-{{}^{3}D_{1}}}$). When we use only one of the eight
contacts, we pick the zeroth order one, $\widetilde{C}_{{}^{3}S_{1}}$, and fit
it to the ${}^{3}S_{1}$ scattering length, $a_{t}=5.42$ MeV. When we apply the
two contacts $\widetilde{C}_{{}^{3}S_{1}}$ and $C_{{}^{3}S_{1}}$, we fit,
besides the scattering length, the ${}^{3}S_{1}$ effective range parameter,
$r_{t}=1.75\pm 0.02$ MeV. Using the three ${}^{3}S_{1}$ parameters
$\widetilde{C}_{{}^{3}S_{1}}$, $C_{{}^{3}S_{1}}$, and
$\widehat{D}_{{}^{3}S_{1}}$, we try to also reproduce (if possible) the
empirical ${}^{3}S_{1}$ phase-shift at 50 MeV laboratory energy as determined
in the Nijmegen phase-shift analysis Sto93 . Besides the three contact LECs
mentioned, we will, in some cases, also include $D_{{}^{3}D_{1}}$ and
$C_{{}^{3}S_{1}-^{3}D_{1}}$, which affect the ${}^{3}D_{1}$ phase shift and
the $\epsilon_{1}$ parameter, respectively. To prevent our investigation from
becoming too involved, we do not vary the LECs $D_{{}^{3}S_{1}}$,
$\widehat{D}_{{}^{3}S_{1}-^{3}D_{1}}$, and $D_{{}^{3}S_{1}-^{3}D_{1}}$ at
orders up to NNL$\Omega$ and keep them at zero. Thus, up to NNL$\Omega$, we
will be experimenting with maximally five contacts in the
${}^{3}S_{1}$-${}^{3}D_{1}$-$\epsilon_{1}$ system.

We consider the following cases with the short notation given in parenthesis.

  * •

One contact contribution, and nothing else (ct1).

  * •

Five contact contributions (ct5).

  * •

L$\Omega$ pion exchange (i. e., 1PE) plus one contact term (L1).

  * •

L$\Omega$ 1PE plus five contact terms (L5).

  * •

NL$\Omega$ pion exchanges plus two contact terms (NL2).

  * •

NL$\Omega$ pion exchanges plus five contact terms (NL5).

  * •

NNL$\Omega$ pion exchanges plus two contact terms (NNL2).

  * •

NNL$\Omega$ pion exchanges plus five contact terms (NNL5).

  * •

NNNL$\Omega$ pion exchanges plus eight contact terms (NNNL8).

The values for the contact LECs used in the various cases are listed in Table
6, and the corresponding phase shifts up to 300 MeV are shown in Fig. 6.

When only one contact term is used (fit to $a_{t}$) and no pion contributions
(case ct1), then the ${}^{3}S_{1}$ phase shifts at intermediate energies are
substantially above the data and $r_{t}$ is off by about 1 fm. Adding one more
contact (case ct2, not shown), gets $r_{t}$ correct, but moves the phase
shifts at intermediate energies far below the data, very similar to the case
ct5 that is shown in Fig. 6. In fact, adding more contacts to the coupled
system under consideration does not change the ${}^{3}S_{1}$ phase shifts up
to the maximum of five contacts. Clearly, contacts alone cannot describe the
${}^{3}S_{1}$ wave at intermediate energies, no matter how many contacts one
is using. However, adding 1PE (case L1) makes a big difference, getting the
${}^{3}S_{1}$ phase shifts almost right and finally perfect with more contacts
(L5). This is quite in contrast to ${}^{1}S_{0}$, where 1PE has little
influence and where 1PE plus contacts never lead to a reproduction of the
phase shifts. The reason for this is that, in the coupled
${}^{3}S_{1}$-${}^{3}D_{1}$ state, the 1PE tensor force contributes strongly
which is crucial for the correct description of this coupled system.

We now turn to the ${}^{3}S_{1}$ frame on the right of Fig. 6, where the 2PE
exchanges of the various orders are added, and we see that 2PE does not make
much difference.

Turning to the ${}^{3}D_{1}$ phase shifts, we see again that contacts alone
cannot get this partial wave right. The contact contribution is too short-
ranged for this partial wave as clearly seen by the very small contribution at
low energies and too strong a contribution above 150 MeV. Adding 1PE gets it
right at low energies, but requires short-ranged corrections at higher
energies. This can be done by contacts (case L5) or by 2PE contributions of
higher order together with moderate contacts (right ${}^{3}D_{1}$ frame).

Finally, we turn to the $\epsilon_{1}$ parameter, which is interesting,
because it is proportional to the ${}^{3}S_{1}$-${}^{3}D_{1}$ transition
potential created exclusively by the tensor force. 1PE generates a (too)
strong tensor force (L1) which, when damped by a short ranged contact, gets it
about right (L5). The 2PE of the various orders do also generate more or less
tensor force contributions which require short-range contact corrections to
get it right.

Thus, qualitatively, 1PE plus a short-range correction is all that is needed
for the ${}^{3}S_{1}$-${}^{3}D_{1}$ system. Interestingly, the chiral 2PE
contributions are not important in this case. The deeper reason for this is
that the iteration of the 1PE tensor force in this coupled system generates a
2PE contribution that is so strong that it makes other 2PE contributions
insignificant.

Because of the polynomial terms that accompany chiral 2PE contributions, in
the case of the coupled ${}^{3}S_{1}$-${}^{3}D_{1}$ system, the minimal number
of contacts to be applied at NL$\Omega$ and NNL$\Omega$ is three, namely,
$\widetilde{C}_{{}^{3}S_{1}}$, $C_{{}^{3}S_{1}}$, and
$C_{{}^{3}S_{1}-^{3}D_{1}}$. In the case of N3L$\Omega$ it is eight.

In summary, contacts alone are inadequate to describe the
${}^{3}S_{1}$-${}^{3}D_{1}$-$\epsilon_{1}$ system at intermediate energies.
Crucial is the 1PE which, for good reasons, is called the Leading Order of the
chiral expansion.

##  4 Summary and conclusions

The most characteristic feature in the design of chiral $NN$ potentials is
that the long-range part of the potential is described by one- and multi-pion
exchanges which are ruled by chiral symmetry. In contrast, the short-range
part consists simply of polynomial terms (“contact” terms), since the short-
range nucleon structure cannot be resolved at low energies.

In the lower partial waves of $NN$ scattering, which are the dominant ones for
predictions of observables of nuclear structure and reactions, contacts as
well as pion-exchanges contribute. But, since lower waves are more sensitive
to the short-range, the contacts may be squashing the pion-exchange
contributions, thus, diminishing the role of chiral symmetry for those
predictions.

Hence, the purpose of this study was to investigate the role of the contacts,
on the one hand, and the effect of the pion exchanges, on the other hand, in
the lower partial waves of chiral $NN$ potentials.

We have shown in detail, what contact terms alone can achieve. This is
displayed by the brown ct curves in the left frames of Figs. 3 to 6, which all
demonstrate that contacts alone are totally inadequate and do not catch
anything of the nature of the nuclear force in those partial waves. Adding
(chiral) 1PE yields semi-realistic results in some specific partial-wave
states, where the tensor force plays an outstanding role. Such cases are the
${}^{3}D_{2}$ state and the ${}^{3}S_{1}$-${}^{3}D_{1}$-$\epsilon_{1}$ system
that is coupled through the tensor force. Chiral 2PE at NL$\Omega$ is
generally weak and, therefore, does not bring about much improvement. However,
the NNL$\Omega$ 2PE is strong, creating a realistic intermediate range
attraction that cannot be simulated by contacts.

This fact is also reflected in the $\chi^{2}$ calculations for the fit of the
$NN$ data conducted in Ref. EMN17 . While the $\chi^{2}$/datum at NL$\Omega$
comes out to be 51.5, at NNL$\Omega$ it is 6.3, even though in both cases the
number of contact terms is the same. The improvement in the $\chi^{2}$ is due
to an improvement of the chiral 2PE at NNL$\Omega$. Obviously, the contacts
cannot substitute the chiral terms.

For very low energies, the so-called pionless EFT has been developed HKK20 ,
which consists only of contact terms and does not include any pion-exchange
contributions. In view of our rather poor results for most of our “contacts
only” fits, one may wonder, how well the pionless EFT is doing in describing
$NN$ scattering. It needs to be explained that the pionless EFT is meant to be
used only for momenta less than the pion mass, say, 100 MeV/c CMS momentum or
less, which is equivalent to a laboratory energy of about 20 MeV. Moreover,
the pionless theory is mostly used only for $S$-waves. Fitting the $S$-waves
at low energies is then understood as reproducing the two effective range
parameters. As shown in Table 5, cases ct2 and ct3, the ${}^{1}S_{0}$ $a_{np}$
and $r_{np}$ can be reproduced o.k. with two or three contacts. Concerning
${}^{3}S_{1}$, the ct5 case shown in Table 6 demonstrates a perfect
description of $a_{t}$ and $r_{t}$. Note that for just two contacts in
${}^{3}S_{1}$, the same result is obtained (not shown in the table). Also the
${}^{1}S_{0}$ and ${}^{3}S_{1}$ phase shifts up to 25 MeV (or even 50 MeV) are
reasonably close to the empirical ones, Figs. 5 and 6. But, what our fits also
show is that, when one moves above about 50 MeV, contacts only are inadequate
and pion-contributions are needed in a decisive way.

In this context, it should be noted that, in the pionless theory, better fits
above 50 MeV can probably be achieved if one allows, e. g., for different
parametrizations in different spin-isospin channels. In the present work, we
have not attempted this, but see, e. g., Ref. CRS99 .

Finally, we also note that fitting phase shifts is not everything. The
ultimate purpose of nuclear forces (in the context of nuclear structure) is to
bind nuclei (with the proper binding energies). Thus, to judge the failure or
sucess of different contributions to nuclear forces, it would be interesting
to study what (combination of) contributions are needed to bind nuclei
properly, where the analysis should be subdivided into the consideration of
light, intermediate, and heavy nuclei. First attempts to examine light nuclei
in terms of pionless forces have been started HKK20 , but so far there have
not been any comprehensive inquiries. This subject represents a very
attractive topic for future research.

In conclusion, despite the fact that contact and pion-exchange contributions
are entangled in the all important lower partial waves of an $NN$ potential,
we were able to disentangle them. We managed to identify and pin down many
characteristic signatures of chiral symmetry that are crucial for the
quantitaive description of the nuclear force in those low angular momentum
states. However, that does not imply that contacts are totally useless. For
the accurate fit of $NN$ quantities, like, the effective range parameters, the
phase shifts at low energies, and the deuteron binding energy, contacts are
needed. They play a subtle role and are like the “dot over the i”.

###### Acknowledgements.

This work was supported in part by the U.S. Department of Energy under Grant
No. DE-FG02-03ER41270.

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