# Two-pion exchange as a leading-order contribution in chiral effective field
theory

Chinmay Mishra  Department of Physics and Astronomy, University of Tennessee,
Knoxville, Tennessee 37996, USA  A. Ekström  Department of Physics, Chalmers
University of Technology, SE-412 96 Göteborg, Sweden  G. Hagen  Physics
Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Department of Physics and Astronomy, University of Tennessee, Knoxville,
Tennessee 37996, USA  T. Papenbrock  Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA  Physics Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  L. Platter
Department of Physics and Astronomy, University of Tennessee, Knoxville,
Tennessee 37996, USA  Physics Division, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831, USA  Institut für Kernphysik, Technische Universität
Darmstadt, 64289 Darmstadt, Germany

###### Abstract

Pion exchange is the central ingredient to nucleon-nucleon interactions used
in nuclear structure calculations, and one pion exchange (OPE) enters at
leading order in chiral effective field theory. In the
${}^{2S+1}L_{J}={{}^{1}S_{0}}$ partial wave, however, OPE and a contact term
needed for proper renormalization fail to produce the qualitative, and
quantitative, features of the scattering phase shifts. Cutoff variation also
revealed a surprisingly low breakdown momentum $\Lambda_{\text{b}}\approx 330$
MeV in this partial wave. Here we show that potentials consisting of OPE, two
pion exchange (TPE), and a single contact address these problems and yield
accurate and renormalization group (RG) invariant phase shifts in the
${}^{1}S_{0}$ partial wave. We demonstrate that a leading-order potential with
TPE can be systematically improved by adding a contact quadratic in momenta.
For momentum cutoffs $\Lambda\lesssim 500$ MeV, the removal of relevant
physics from TPE loops needs to be compensated by additional contacts to keep
RG invariance. Inclusion of the $\Delta$ isobar degree of freedom in the
potential does not change the strong contributions of TPE.

##  I Introduction

Ever since its introduction by Yukawa [1], boson exchange has been central to
the theory of nuclear interactions. In quantum chromodynamics, chiral symmetry
is spontaneously and explicitly broken, and the pion emerges as the
corresponding pseudo Nambu-Goldstone boson. Pion exchange, together with
contact interactions that account for unknown short-range physics, thus are
the ingredients in a chiral effective field theory (EFT) description of the
nucleon-nucleon interaction [2, 3, 4, 5, 6, 7, 8, 9].

In chiral EFT, and within the commonly employed Weinberg power counting, the
leading-order contributions to the nucleon-nucleon interaction consist of OPE
and one contact each in the ${}^{1}S_{0}$ and ${}^{3}S_{1}$ partial waves. At
next-to-leading order (NLO) in Weinberg counting the leading TPE contributions
as well as contacts quadratic in momenta enter.

Statistical analyses of higher-order chiral EFT predictions for nucleon-
nucleon scattering data infer a breakdown momentum $\Lambda_{\text{b}}\approx
600-700$ MeV, and that higher chiral orders yield systematical improvements in
powers of $Q/{\rm min}(\Lambda,\Lambda_{\rm b})$ beyond the leading-order
results [10, 11]. Here $Q$ is the low-momentum scale of interest, e.g., the
external momentum in nucleon-nucleon scattering, and $\Lambda$ is the cutoff
employed in the regularization of the theory. While this looks encouraging,
there are well-known challenges [12], and we mention two of them.

First, recent results [13, 14, 15] show that leading-order chiral EFT
potentials predict light-mass nuclei that are unstable with respect to breakup
into $\alpha$ particles and lighter-mass clusters, raising questions about
what should be expected from a nuclear EFT at leading order. Of course, the
lack of any spin-orbit contributions at leading order in the Weinberg power
counting would also presumably make well-known nuclear shell-structure a
subleading effect.

Figure 1: (Color online) Nucleon-nucleon phase shifts in the ${}^{1}S_{0}$
partial wave versus the laboratory scattering energy $E$ for various chiral
interactions. The momentum cutoff is set to $4$ GeV and in all cases a single
contact $V_{\rm ct}^{(0)}$ is adjusted to match the phase shift to that of the
reference potential (high-precision Idaho-N3LO) by Entem and Machleidt [16]
(solid black line) at $E_{0}=15$ MeV. The dotted blue line shows phase shift
obtained for OPE, the dashed orange line for OPE plus the leading TPE, the
dash-dot-dotted green line for OPE plus subleading TPE, and the dash-dotted
red line for the interaction $V_{\pi,\pi\pi}$ consisting of OPE and leading
plus subleading TPE (see text for details).

Second, the leading-order description of nucleon-nucleon phase shifts in the
${}^{1}S_{0}$ partial wave are problematic in the Weinberg power counting [17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 14], see Ref. [32] for
a recent review. The combination of OPE and a single contact fails
qualitatively to capture the pronounced peak at $60-70$ degrees in the phase
shift at about 8 MeV of laboratory scattering energy, see blue dotted line in
Fig. 1. For all results in this figure, the ${}^{1}S_{0}$ contact is adjusted
to the reference phase shifts of the potential [16] (shown as a solid black
line) at a laboratory scattering energy $E=15$ MeV. This energy matches the
relevant scale of pion physics as $m_{\pi}^{2}/m_{N}\approx 20$ MeV using the
pion and nucleon mass $m_{\pi}$ and $m_{N}$, respectively. The OPE phase
shifts are much too attractive beyond the matching point and clearly fail to
capture the characteristics of the reference phase shifts. Also, the slope of
the OPE phase shift has the wrong sign.

Another problem concerns the breakdown momentum $\Lambda_{\text{b}}$. The
analysis of the ${}^{1}S_{0}$ phase shifts by Lepage [33] showed that a
potential consisting of OPE plus leading and subleading contacts leads to
$\Lambda_{\text{b}}\approx 330$ MeV in the ${}^{1}S_{0}$ partial wave. Later
analyses exploring perturbative inclusion of subleading TPE contributions [22,
34] confirmed this finding and estimated the breakdown momentum to be even
lower, i.e., $\Lambda_{\rm b}\approx 200$ MeV. This questions whether leading-
order chiral EFT based on OPE physics is consistent with the general
assumption that the breakdown momentum is somewhere between $\sim 500$ and
$\sim 1000$ MeV.

Several researchers addressed the shortcoming of too attractive ${}^{1}S_{0}$
phase shifts by adding effective-range corrections [35, 14], energy-dependent
potentials generated by di-baryon fields [29], or separable potentials [31] to
the OPE as leading-order contributions. These approaches improve the phase
shifts at a cost of introducing additional parameters. The promotion of TPE to
leading order in chiral EFT was proposed already a decade ago by Birse [25],
however, a rigorous analysis has (to the best of our knowledge) never been
carried out.

In this paper we show that chiral physics in the form of TPE, presumably a
higher-order correction in the Weinberg power counting, remedies the
shortcomings of highly attractive ${}^{1}S_{0}$ phase shifts and without
increasing the number of parameters. As Fig. 1 shows, the inclusion of TPE at
leading order significantly improves the accuracy of the ${}^{1}S_{0}$ phase
shifts. We will also show that the estimated breakdown momentum in this
approach is consistent with expectations from chiral EFT.

Phenomenology connects TPE with the strong mid-range attraction of the
nucleon-nucleon force [36, 37] which is also attributed to the $f_{0}(500)$
resonance [38] or the sigma meson whose width and mass was determined model-
independently in Ref. [39]. The latter is a central ingredient of relativistic
mean-field theories [40, 41, 42, 43, 44] and of alternative proposals to
chiral EFT which include the effects of the $f_{0}(500)$ resonance at leading
order via the sigma meson [45] or the dilaton [46, 47], i.e. the Nambu-
Goldstone boson of a broken and hidden scale symmetry.

In the Weinberg power counting TPE enters at NLO, but its strongest
contributions involving the pion-nucleon coupling constants $c_{i}$, with
$i=1,3,4$, enter at next-to-next-to-leading order (NNLO). It is known that
these subleading TPE contributions are crucial for quantitatively reproducing
the ${}^{1}S_{0}$ phase shifts, see, e.g., Refs. [21, 48]. We note, however,
that the role of TPE is somewhat obscured in chiral potentials. First, an
additional contact potential quadratic in momentum also enters in the
${}^{1}S_{0}$ partial wave at NLO in the Weinberg power counting. Second,
several popular potentials [16, 49, 50, 51, 52] employ relatively low momentum
cutoffs and thereby truncate some parts of the TPE strength.

This manuscript is ordered as follows: In Sec. II we give the explicit form of
the OPE and TPE potentials, discuss their relative strengths, and show results
for phase shifts in the ${}^{1}S_{0}$ partial wave obtained using the proposed
leading-order potential where we have promoted TPE. In Sec. III we show that a
subleading contact that is quadratic in momenta systematically improves upon
these results. In Sec. IV we show that the inclusion of $\Delta$ isobar
degrees of freedom does not alter our conclusions about promoting TPE to
leading order. We end with a summary in Sec. V.

##  II Chiral OPE and TPE potentials

Canonical chiral EFT descriptions of the nuclear interaction employ a power
counting for the pion-nucleon potential as done in chiral perturbation theory.
The OPE enters at leading order, and subleading contributions are presumably
suppressed by powers of $g_{A}m_{\pi}/(4\pi f_{\pi})\ll 1$, where
$g_{A}\approx 1.28$ is the axial-vector constant, $m_{\pi}\approx 140$ MeV is
the pion mass, and $f_{\pi}\approx 92$ MeV is the pion-decay constant. The OPE
potential is given by

$V_{\rm OPE}(\mathbf{q})=-{g_{A}^{2}\over
4f_{\pi}^{2}}\boldsymbol{\tau}_{1}\cdot\boldsymbol{\tau}_{2}\frac{(\boldsymbol{\sigma}_{1}\cdot\mathbf{q})(\boldsymbol{\sigma}_{2}\cdot\mathbf{q})}{m_{\pi}^{2}+q^{2}}\
.$ (1)

Here, $\mathbf{q}=\mathbf{p}^{\prime}-\mathbf{p}$ is the momentum transfer.
The Pauli matrices of nucleon $j$ in spin and isospin space are denoted as
$\mathbf{\sigma}_{j}$ and $\mathbf{\tau}_{j}$, respectively. The TPE
potentials considered in this work can be written as

$\displaystyle V_{\rm TPE}(\mathbf{q})$ $\displaystyle=$ $\displaystyle
V_{C}(q)+\boldsymbol{\tau}_{1}\cdot\boldsymbol{\tau}_{2}W_{C}(q)$ (2)
$\displaystyle+$
$\displaystyle\left[V_{T}(q)+\boldsymbol{\tau}_{1}\cdot\boldsymbol{\tau}_{2}W_{T}(q)\right](\boldsymbol{\sigma}_{1}\cdot\mathbf{q})(\boldsymbol{\sigma}_{2}\cdot\mathbf{q})$
$\displaystyle+$
$\displaystyle\left[V_{S}(q)+\boldsymbol{\tau}_{1}\cdot\boldsymbol{\tau}_{2}W_{S}(q)\right]\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}\
.$

The leading TPE potential, i.e., the contributions that enter at NLO in the
Weinberg power counting is given by [6, 7, 48]

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

while the subleading contributions that enter at NNLO are [6, 7, 48]

$\displaystyle V_{C}$ $\displaystyle=-{3g_{A}^{2}\over 16\pi
f_{\pi}^{4}}\left[2m_{\pi}^{2}(2c_{1}-c_{3})-c_{3}q^{2}\right]\tilde{w}^{2}A(q)\
,$ (4) $\displaystyle W_{T}$ $\displaystyle=-{W_{S}\over
q^{2}}=-{g_{A}^{2}\over 32\pi f_{\pi}^{4}}c_{4}w^{2}A(q)\ .$

Here, we used the short-hands

$\displaystyle w$ $\displaystyle\equiv\sqrt{4m_{\pi}^{2}+q^{2}}\ ,$ (5)
$\displaystyle L(q)$ $\displaystyle\equiv{w\over q}\log{w+q\over 2m_{\pi}}\ ,$
(6) $\displaystyle\tilde{w}$ $\displaystyle\equiv\sqrt{2m_{\pi}^{2}+q^{2}}\ ,$
(7) $\displaystyle A(q)$ $\displaystyle\equiv{1\over 2q}{\rm arctan}{q\over
2m_{\pi}}\ ,$ (8)

with $q\equiv|\mathbf{q}|$, and the pion-nucleon constants $c_{i}$ are of the
order of $m_{N}^{-1}$ with $m_{N}$ denoting the nucleon mass $m_{N}$. In what
follows we use $c_{1}=-0.74$ GeV-1, $c_{3}=-3.61$ GeV-1, and $c_{4}=2.44$
GeV-1 from Ref. [10], obtained from Roy-Steiner relations [53, 54]. We note
here that we have neglected any relativistic corrections (proportional to
$m_{N}^{-1}$) to the TPE at NNLO, and we have omitted any polynomial
contributions.

We evaluated the potentials in the spin-singlet / isospin-triplet partial wave
and show their magnitudes in Fig. 2 comparing the OPE, leading TPE, and
subleading TPE contributions. As expected, OPE is a dominant contribution,
while the subleading TPE potential of Eq. (4) cannot be neglected around
momentum transfers of about 1 fm-1. Clearly, at momentum transfers of the
order of the pion mass ($m_{\pi}\approx 0.7$ fm-1) or the Fermi momentum in
nuclear matter at saturation ($k_{F}\approx 1.35$ fm-1), the placement of this
TPE potential at NNLO does not reflect its actual strength.

Figure 2: (Color online) Pion-exchange contributions to the nucleon-nucleon
potentials in the spin-singlet / isospin-triplet partial wave at various
orders in Weinberg power counting as a function of momentum transfer. The OPE
potential (black solid line) is leading order in the Weinberg power counting,
the leading TPE (dashed blue line) is NLO, and the subleading TPE (dash-dotted
red line) enters at NNLO.

Let us study the impact of TPE added to OPE in the ${}^{1}S_{0}$ phase shifts.
We define a nucleon-nucleon potential in this partial wave that is of the form

$V(\mathbf{p}^{\prime},\mathbf{p})\equiv V_{\pi,\pi\pi}+V_{\rm{ct}}\ .$ (9)

Here, $V_{\pi,\pi\pi}$ consists of OPE and the leading and subleading TPE,
while $V_{\rm ct}$ denotes the contact potential to be specified. At leading
order and NLO, the contact potentials are given by

$V_{\rm ct}^{(0)}(\mathbf{p^{\prime}},\mathbf{p})=\tilde{C}\ ,$ (10)

and

$V_{\rm ct}^{(2)}(\mathbf{p^{\prime}},\mathbf{p})=C(p^{\prime 2}+p^{2})\ ,$
(11)

respectively. The potentials are regularized via

$\displaystyle V\to f({p^{\prime}}^{2}/\Lambda^{2})Vf(p^{2}/\Lambda^{2})$ (12)

using $p\equiv|\mathbf{p}|$, $p^{\prime}\equiv|\mathbf{p}^{\prime}|$,

$f(x)=e^{-x^{n}}\ ,$ (13)

and $n=3$. For TPE loop diagrams we use a spectral function regulator with a
cutoff of 700 MeV as introduced in [55, 56].

At leading order we adjust the low-energy constant (LEC) $\tilde{C}$ such that
the phase shift at the laboratory energy $E_{0}=15$ MeV reproduces the value
from the high-precision Idaho-N3LO potential [16], which we take as a
reference throughout this work. The precision of this reference is sufficient
for our purposes, as the phase shifts of this potential are virtually
indistinguishable from a recent partial-wave analysis of nucleon-nucleon
scattering data [57].

The ${}^{1}S_{0}$ phase shifts for our leading-order potential are shown as
the red dash-dotted line in Fig. 1. For comparison, we also show the results
from potentials with other combinations of pion exchanges; such as the sum of
OPE and leading TPE (orange dashed line) and OPE plus subleading TPE (green
dash-dot-dotted line). The chiral potential $V_{\pi,\pi\pi}$ stands out
through its accuracy for scattering energies below and above the energy
$E_{0}$ used for renormalizing the contact LEC. This suggests that the
combination $V_{\pi,\pi\pi}$ of OPE and TPE should be taken as the leading-
order contribution from chiral physics in the ${}^{1}S_{0}$ partial wave. This
is the main result of this paper. It is consistent with the anticipation
obtained from Fig. 2. Although the combination of OPE and leading TPE (shown
as the orange dashed line) brings the phase shift closer to reference it fails
to reproduce the characteristic decrease of the phase shifts with increasing
energy, i.e., a lack of increasing repulsion with $E$, and the amplitude zero
is nowhere near $E\approx 250$ MeV. In contrast, using $V_{\pi,\pi\pi}$ leads
to a leading-order nucleon-nucleon potential that captures the main
qualitative and quantitative features of the ${}^{1}S_{0}$ phase shifts.

We note that the phase shifts presented in Fig. 1 include chiral physics plus
a single LEC that accounts for unknown short-range physics. Figure 3
demonstrates that the potential $V_{\pi,\pi\pi}+V_{\rm ct}^{(0)}$ yields RG
invariant ${}^{1}S_{0}$ phase shifts as the cutoff $\Lambda$ is increased. For
a comparison, the reference phase shifts are shown as black stars.

Figure 3: (Color online) ${}^{1}S_{0}$ phase shifts as a function of the
momentum regulator cutoff $\Lambda$ at laboratory energies of $E=4$ (dashed),
$15$ (solid), $75$ (dash-dotted), and $225$ MeV (dotted) for the potential
$V_{\pi,\pi\pi}+V_{\rm ct}^{(0)}$. The black stars show the reference phase
shifts at these energies.

Figure 3 also shows that the phase shifts are strongly cutoff dependent for
$\Lambda\lesssim 750$ MeV. This can be understood as follows: In coordinate
space, the OPE potential exhibits a $1/r^{3}$ behavior at short distances,
where $r$ denotes the two-nucleon relative distance. The TPE potential
exhibits a significantly more singular $1/r^{6}$ short-distance behavior with
a typical momentum scale $k_{\pi\pi}\sim 115$ MeV set by the chiral couplings
employed in the TPE potential [34]. The cutoff needs therefore to be
significantly larger than $k_{\pi\pi}$ to reach convergence. Alternatively,
the relevant scale of the TPE interaction can also naively be estimated by

$\Lambda_{\text{TPE}}\equiv\sqrt{2m_{\pi}m_{N}}\approx 510~{}\mbox{MeV}\ .$
(14)

Cutoffs lower than $\Lambda_{\rm TPE}$ therefore remove physics from the TPE.
This effect is highlighted in Fig. 4. As before, we adjusted the leading-order
contact $V_{\rm ct}^{(0)}$ to reproduce the reference phase shift at
$E_{0}=15$ MeV. We clearly see the effects of removing TPE physics for cutoffs
$\Lambda\lesssim 500$ MeV. Indeed, comparison with Fig. 1 shows that the phase
shifts at a cutoff of $\Lambda=475$ MeV are close to those of OPE plus a
contact. As we will see below, adding the subleading contact $V_{\rm
ct}^{(2)}$ will restore RG invariance in this case.

Figure 4: (Color online) Phase shifts as a function of laboratory energy of
the interaction $V_{\pi,\pi\pi}+V_{\rm ct}^{(0)}$ at low momentum cutoffs as
indicated and compared to the reference (solid black line). The phase shifts
become increasing repulsive as the cutoff is increased. For $E>15$ MeV they
cross over and become more repulsive than the reference for cutoffs in the
range $500\lesssim\Lambda\lesssim 520$ MeV.

Figure 4 also allows us to estimate the breakdown momentum in this partial
wave as the momentum regulator cutoff value for which the phase shift
predictions are closest to the reference [33]. Following this strategy we
infer $\Lambda_{\text{b}}\approx 500-520$ MeV. This is significantly larger
than what was found when only OPE physics is included at leading order [33,
22, 34]. It is also in line with expectations from neglecting the physics of
more massive exchange mesons.

##  III Systematic improvements

We propose to employ higher-order contacts to systematically improve upon the
results of our leading-order potential $V_{\pi,\pi\pi}+V_{\rm ct}^{(0)}$.
Thus, the contact (11) enters as the subleading correction [33]. This
introduces a new LEC, $C$, and we adjust $\tilde{C}$ and $C$ such that the
reference phase shift and its slope is reproduced at the laboratory energy
$E_{0}=15$ MeV. In our numerical work, we used the secant slope between
$E_{0}$ and a second point just below this energy rather than the exact
tangent slope. We work at a cutoff of $\Lambda=800$ MeV.

The question then arises whether one should treat the subleading correction
perturbatively, or not. In the non-perturbative approach, the full potential
is iterated to solve the Lippmann Schwinger equation, while the perturbative
approach is linear in the subleading correction. We followed both approaches
and found similar results for the phase shifts.

Let us discuss the non-perturbative approach. A simultaneous fit of the two
LECs $(\tilde{C},C)$ to the phase shift and its slope at $E_{0}$ is somewhat
challenging. Instead, we first calibrated $\tilde{C}$ and subsequently
determined $C$ such that the reference phase shift is reproduced at $E_{0}$.
Repeating this procedure for various values of $\tilde{C}$ yields a one-
parameter curve $C(\tilde{C})$. Interestingly, we found that this is a
quadratic function to very high accuracy. We then moved along this parabola
and determined the point where the slope of the phase shift agrees with the
reference as well. The resulting phase shifts are shown as a dashed blue line
in Fig. 5 and compared to our leading-order results (red dash-dotted line).
The improvement is clearly visible.

Figure 5: (Color online) Systematic improvements of the ${}^{1}S_{0}$ phase
shifts by adding the subleading contact $V_{\rm ct}^{(2)}$ to the potential
$V_{\pi,\pi\pi}+V_{\rm ct}^{(0)}$. The cutoff is $800$ MeV. Results for the
potentials $V_{\pi,\pi\pi}+V_{\rm ct}^{(0)}$ and $V_{\pi,\pi\pi}+V_{\rm
ct}^{(0)}+V_{\rm ct}^{(2)}$ are shown as a dash-dotted red line and as blue
lines, respectively. The dashed and the dotted blue line correspond to the
non-perturbative and the perturbative inclusion of the subleading contact
$V_{\rm ct}^{(2)}$, respectively, and the solid black line shows the
reference.

Figure 6 shows the absolute differences between our phase shifts and the
reference on a log-log plot. The systematic power-law improvement from the
quadratic contact is evident. We also note that $C\Lambda_{\rm
b}^{2}/\tilde{C}\approx-4$, and this is consistent with expectations from
naive dimensional analysis where this dimensionless number should be ${\cal
O}(1)$ in size.

Figure 6: (Color online) Log-log plot of absolute differences to the reference
phase shifts versus the energy difference to the matching point. The cutoff is
$800$ MeV. Results for the potentials $V_{\pi,\pi\pi}+V_{\rm ct}^{(0)}$ and
$V_{\pi,\pi\pi}+V_{\rm ct}^{(0)}+V_{\rm ct}^{(2)}$ are shown as dash-dotted
red and as blue lines, respectively. The dashed and the dotted blue line
correspond to the non-perturbative and the perturbative inclusion of the
subleading contact $V_{\rm ct}^{(2)}$, respectively.

In a second approach, we treated the contact (11) in perturbation theory. Our
leading-order theory is adjusted to the reference phase shift at $E_{0}$ and
the corresponding LECs are $(\tilde{C},C)=(\tilde{C}_{0},0)$. In a
perturbative approach the LECs become $(\tilde{C}_{0}+\delta\tilde{C},\delta
C)$ in presence of the potential $V_{\rm ct}^{(2)}$. We expand the phase shift
as

$\displaystyle\delta(E)$
$\displaystyle\approx\delta(E){\rvert}_{0}+\frac{\partial\delta(E)}{\partial\tilde{C}}{\biggr{\rvert}}_{0}\delta\tilde{C}+\frac{\partial\delta(E)}{\partial
C}{\biggr{\rvert}}_{0}\delta C$ (15)

Here, the subscript $0$ implies that all functions are evaluated at
$(\tilde{C}_{0},0)$. We determine the derivatives numerically. Keeping the
phase shift at $E_{0}$ unchanged implies

$\displaystyle\frac{\partial\delta(E_{0})}{\partial\tilde{C}}{\biggr{\rvert}}_{0}\delta\tilde{C}+\frac{\partial\delta(E_{0})}{\partial
C}{\biggr{\rvert}}_{0}\delta C=0\ ,$ (16)

and, thus, a linear relation between $\delta\tilde{C}$ and $\delta C$. A one-
parameter search along this line yields the optimal point that also reproduces
the slope of the reference phase shifts at $E_{0}$. We then compute the
resulting phase shifts using Eq. (15). The corresponding results are shown as
blue dotted lines in Figs. 5 and 6. We see that the perturbative and non-
perturbative solutions phase shifts are close to each other, and that both
yield a power-law improvement of our leading-order results. We also note that
$\Lambda_{\rm b}^{2}\delta C/(\tilde{C}_{0}+\delta C)\approx-0.03$, and this
is smaller in magnitude than expected from naive dimensional analysis.

We also revisited the low cutoffs at and below the scale
$\Lambda_{\text{TPE}}$ in Eq. (14) and employed the subleading contact $V_{\rm
ct}^{(2)}$ in Eq. (11) non-perturbatively by matching the phase shift and its
slope to the reference at a laboratory energy of 15 MeV. The results are shown
in Fig. 7 and comparison with Fig. 4 shows that RG invariance is restored also
at low momentum cutoffs.

Figure 7: (Color online) The ${}^{1}S_{0}$ phase shifts as a function of the
laboratory energy $E$ with the interaction $V_{\pi,\pi\pi}+V_{\rm
ct}^{(0)}+V_{\rm ct}^{(2)}$ at low cutoffs as indicated and compared to the
reference (solid black line). The additional contact $V_{\rm ct}^{(2)}$
restores the RG invariance that was lacking in Fig. 4.

##  IV The role of $\Delta$ isobar degrees of freedom

The strong contributions of the subleading TPE in chiral EFT, i.e., the
relatively large values of the pion-nucleon LECs $c_{i}$, are usually
attributed to “resonance saturation.” Consequently, these couplings become
more natural in size when the $\Delta$ isobar degrees of freedom are included.
Notably, such chiral EFTs have TPE terms that involve a $\Delta$ excitation
already at NLO in the Weinberg power counting. The corresponding expressions
for the potentials (2) were derived by Kaiser  _et al._ [58] and we use those
published by Krebs  _et al._ [59] in their Eqs. (2.5) to (2.8). The chiral
potential we employ thus consists of the OPE [given in Eq. (1)], leading TPE
[given in Eq. (3)], the contact (10), and the leading $\Delta$ contributions
to TPE. In our numerical implementation this potential is regularized with
$n=4$ in the regulator (13) and spectral-function regularization was used with
a cutoff of $\tilde{\Lambda}=$700 MeV.

We repeated the calculations presented above and found very similar results
regarding the quality of the phase shifts in the ${}^{1}S_{0}$ partial wave, a
breakdown momentum $\Lambda_{\text{b}}\approx 500$ MeV, and a systematic
power-law improvement when the contact (11) quadratic in momenta is included.
An example is shown in Fig. 8, to be compared with Fig. 5.

Figure 8: (Color online) The ${}^{1}S_{0}$ phase shifts as a function of the
laboratory energy $E$ for the $V_{\pi,\pi\pi}^{\Delta}$ interaction, i.e., our
leading-order potential with $\Delta$ isobars included in the leading TPE
potential. The dash-dotted red line uses a single contact $V_{\rm ct}^{(0)}$
to reproduce the value of the reference phase shift at $E_{0}=15$ MeV. The
dashed blue line shows the phase shift when leading and subleading contacts
$V_{\rm ct}^{(0)}+V_{\rm ct}^{(2)}$ are fit to reproduce the value and the
slope of the reference phase shift at $E_{0}$. The cutoff is $800$ MeV, and
the reference phase shifts are shown as a solid black line. The results are
close to those shown in Fig. 5.

##  V Summary and discussion

We analyzed chiral EFT in the ${}^{1}S_{0}$ partial wave and propose to
promote leading and subleading TPE to accompany OPE along with a single
contact $V_{\rm ct}^{(0)}$ at leading order. Naturally, power counting in EFT
applies to observables or amplitudes. Nevertheless, the promotion of TPE to
leading order is inspired by the unexpectedly large matrix elements of the TPE
potential that formally enter at NNLO in the Weinberg power counting. We find
that our leading-order interaction cures two problems with the standard
approach. First, the phase shifts in the ${}^{1}S_{0}$ partial wave are
accurately reproduced and RG invariant for scattering energies of the order
$m_{\pi}^{2}/m_{N}$ and for large cutoffs. Second, the estimated breakdown
momentum $\Lambda_{\text{b}}\approx 500$ MeV in ${}^{1}S_{0}$ is consistent
with assumptions from chiral EFT. We also showed that adding a contact
quadratic in momenta as a subleading correction leads to a systematic power-
law improvement of the phase shifts. We find that including the leading
$\Delta$ contributions to TPE as a new leading-order contribution achieves the
same. We expect that additional and smaller pion contributions and higher-
order contacts to yield additional and systematic power-law improvements. We
also pointed out that chiral EFTs with cutoffs below about $500$ MeV are
really “two-pion-less” EFTs as they cut off parts of the TPE; in such cases
higher-order contacts are needed for maintaining proper RG invariance.

###### Acknowledgements.

We thank Daniel Phillips for insightful and useful discussions. We also thank
the participants of the INT program “Nuclear Forces for Precision Nuclear
Physics (21-1b)” for many useful exchanges and discussions. This material is
based upon work supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics under award numbers DE-FG02-96ER40963 and DE-
SC0018223 (NUCLEI SciDAC-4 collaboration), and contract No. DE-AC05-00OR22725
with UT-Battelle, LLC (Oak Ridge National Laboratory), the Swedish Research
Council grant number 2020-005127, the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement number 758027), by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Projektnummer 279384907 – CRC 1245. and by the National
Science Foundation under Grant Nos. PHY-1555030 and PHY-2111426.

## References

  * Yukawa [1935] H. Yukawa, On the interaction of elementary particles. i, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series 17, 48 (1935).
  * Weinberg [1990] S. Weinberg, Nuclear forces from chiral lagrangians, Phys. Lett. B 251, 288 (1990).
  * Weinberg [1991] S. Weinberg, Effective chiral lagrangians for nucleon-pion interactions and nuclear forces, Nuclear Physics B 363, 3 (1991).
  * Ordóñez and van Kolck [1992] C. Ordóñez and U. van Kolck, Chiral lagrangians and nuclear forces, Phys. Lett. B 291, 459 (1992).
  * van Kolck [1994] U. van Kolck, Few-nucleon forces from chiral Lagrangians, Phys. Rev. C 49, 2932 (1994).
  * Kaiser  _et al._ [1997] N. Kaiser, R. Brockmann, and W. Weise, Peripheral nucleon-nucleon phase shifts and chiral symmetry, Nucl. Phys. A 625, 758 (1997).
  * Epelbaum  _et al._ [2000] E. Epelbaum, W. Glöckle, and U.-G. Meißner, Nuclear forces from chiral lagrangians using the method of unitary transformation ii: The two-nucleon system, Nucl. Phys. A 671, 295 (2000).
  * Epelbaum  _et al._ [2009] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Modern theory of nuclear forces, Rev. Mod. Phys. 81, 1773 (2009).
  * Hammer  _et al._ [2020] H. W. Hammer, S. König, and U. van Kolck, Nuclear effective field theory: status and perspectives, Rev. Mod. Phys. 92, 025004 (2020), arXiv:1906.12122 [nucl-th] .
  * Reinert  _et al._ [2018] P. Reinert, H. Krebs, and E. Epelbaum, Semilocal momentum-space regularized chiral two-nucleon potentials up to fifth order, The European Physical Journal A 54, 86 (2018).
  * Wesolowski  _et al._ [2019] S. Wesolowski, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, Exploring bayesian parameter estimation for chiral effective field theory using nucleon–nucleon phase shifts, Journal of Physics G: Nuclear and Particle Physics 46, 045102 (2019).
  * Furnstahl  _et al._ [2021] R. J. Furnstahl, H. W. Hammer, and A. Schwenk, Nuclear Structure at the Crossroads, Few-Body Systems 62, 72 (2021).
  * Carlsson  _et al._ [2016] B. D. Carlsson, A. Ekström, C. Forssén, D. F. Strömberg, G. R. Jansen, O. Lilja, M. Lindby, B. A. Mattsson, and K. A. Wendt, Uncertainty analysis and order-by-order optimization of chiral nuclear interactions, Phys. Rev. X 6, 011019 (2016).
  * Yang  _et al._ [2021] C.-J. Yang, A. Ekström, C. Forssén, and G. Hagen, Power counting in chiral effective field theory and nuclear binding, Phys. Rev. C 103, 054304 (2021).
  * Maris  _et al._ [2021] P. Maris, E. Epelbaum, R. J. Furnstahl, J. Golak, K. Hebeler, T. Hüther, H. Kamada, H. Krebs, U.-G. Meißner, J. A. Melendez, A. Nogga, P. Reinert, R. Roth, R. Skibiński, V. Soloviov, K. Topolnicki, J. P. Vary, Y. Volkotrub, H. Witała, and T. Wolfgruber (LENPIC Collaboration), Light nuclei with semilocal momentum-space regularized chiral interactions up to third order, Phys. Rev. C 103, 054001 (2021).
  * Entem and Machleidt [2003] D. R. Entem and R. Machleidt, Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory, Phys. Rev. C 68, 041001 (2003).
  * Kaplan  _et al._ [1998] D. B. Kaplan, M. J. Savage, and M. B. Wise, A new expansion for nucleon-nucleon interactions, Physics Letters B 424, 390 (1998).
  * Frederico  _et al._ [1999] T. Frederico, V. S. Timoteo, and L. Tomio, Renormalization of the one pion exchange interaction, Nucl. Phys. A 653, 209 (1999), arXiv:nucl-th/9902052 .
  * Nogga  _et al._ [2005] A. Nogga, R. G. E. Timmermans, and U. v. Kolck, Renormalization of one-pion exchange and power counting, Phys. Rev. C 72, 054006 (2005).
  * Valderrama and Arriola [2005] M. P. Valderrama and E. R. Arriola, Renormalization of the deuteron with one pion exchange, Phys. Rev. C 72, 054002 (2005).
  * Pavón Valderrama and Arriola [2006] M. Pavón Valderrama and E. R. Arriola, Renormalization of the $\mathit{NN}$ interaction with a chiral two-pion-exchange potential: Central phases and the deuteron, Phys. Rev. C 74, 054001 (2006).
  * Birse [2006] M. C. Birse, Power counting with one-pion exchange, Phys. Rev. C 74, 014003 (2006).
  * Shukla  _et al._ [2008] D. Shukla, D. R. Phillips, and E. Mortenson, Chiral potentials, perturbation theory and the ${}^{1}s_{0}$ channel of NN scattering, Journal of Physics G: Nuclear and Particle Physics 35, 115009 (2008).
  * Yang  _et al._ [2008] C.-J. Yang, C. Elster, and D. R. Phillips, Subtractive renormalization of the $\mathit{NN}$ scattering amplitude at leading order in chiral effective theory, Phys. Rev. C 77, 014002 (2008).
  * Birse [2010] M. C. Birse, Deconstructing 1S0 nucleon-nucleon scattering, European Physical Journal A 46, 231 (2010).
  * Long and Yang [2012a] B. Long and C.-J. Yang, Short-range nuclear forces in singlet channels, Phys. Rev. C 86, 024001 (2012a).
  * Long and Yang [2012b] B. Long and C.-J. Yang, Renormalizing chiral nuclear forces: Triplet channels, Phys. Rev. C 85, 034002 (2012b).
  * Valderrama and Phillips [2015] M. P. Valderrama and D. R. Phillips, Power counting of contact-range currents in effective field theory, Phys. Rev. Lett. 114, 082502 (2015).
  * Sánchez Sánchez  _et al._ [2018] M. Sánchez Sánchez, C.-J. Yang, B. Long, and U. van Kolck, Two-nucleon ${}^{1}s_{0}$ amplitude zero in chiral effective field theory, Phys. Rev. C 97, 024001 (2018).
  * Odell  _et al._ [2019] D. Odell, A. Deltuva, J. Bonilla, and L. Platter, Renormalization of a finite-range inverse-cube potential, Phys. Rev. C 100, 054001 (2019).
  * Sánchez Sánchez  _et al._ [2020] M. Sánchez Sánchez, N. A. Smirnova, A. M. Shirokov, P. Maris, and J. P. Vary, Improved description of light nuclei through chiral effective field theory at leading order, Phys. Rev. C 102, 024324 (2020).
  * van Kolck [2020] U. van Kolck, The problem of renormalization of chiral nuclear forces, Frontiers in Physics 8, 79 (2020).
  * Lepage [1997] G. P. Lepage, How to renormalize the Schrodinger equation, in  _8th Jorge Andre Swieca Summer School on Nuclear Physics_ (1997) arXiv:nucl-th/9706029 .
  * Pavón Valderrama [2011] M. Pavón Valderrama, Perturbative renormalizability of chiral two-pion exchange in nucleon-nucleon scattering, Phys. Rev. C 83, 024003 (2011).
  * Epelbaum  _et al._ [2015] E. Epelbaum, A. M. Gasparyan, J. Gegelia, and H. Krebs, 1S0 nucleon-nucleon scattering in the modified Weinberg approach, Eur. Phys. J. A 51, 71 (2015).
  * Wiringa  _et al._ [1995] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Accurate nucleon-nucleon potential with charge-independence breaking, Phys. Rev. C 51, 38 (1995).
  * Machleidt [2001] R. Machleidt, High-precision, charge-dependent Bonn nucleon-nucleon potential, Phys. Rev. C 63, 024001 (2001).
  * Peláez [2016] J. R. Peláez, From controversy to precision on the sigma meson: A review on the status of the non-ordinary f0(500) resonance, Physics Reports 658, 1 (2016).
  * Caprini  _et al._ [2006] I. Caprini, G. Colangelo, and H. Leutwyler, Mass and width of the lowest resonance in QCD, Phys. Rev. Lett. 96, 132001 (2006), arXiv:hep-ph/0512364 .
  * Walecka [1986] J. D. Walecka, The relativistic nuclear many-body problem, in  _New Vistas in Nuclear Dynamics_, NATO ASI Series (Series B: Physics), Vol. 139, edited by P. J. Brussaard and J. H. Koch (Springer, Boston, MA, 1986) pp. 229–271.
  * Reinhard [1989] P.-G. Reinhard, The relativistic mean-field description of nuclei and nuclear dynamics, Reports on Progress in Physics 52, 439 (1989).
  * Gambhir  _et al._ [1990] Y. K. Gambhir, P. Ring, and A. Thimet, Relativistic mean field theory for finite nuclei, Annals of Physics 198, 132 (1990).
  * Serot and Walecka [1992] B. D. Serot and J. D. Walecka, Relativistic nuclear many-body theory, in  _Recent Progress in Many-Body Theories_, edited by T. L. Ainsworth, C. E. Campbell, B. E. Clements, and E. Krotscheck (Springer, Boston, MA, 1992) pp. 49–92.
  * Sugahara and Toki [1994] Y. Sugahara and H. Toki, Relativistic mean-field theory for unstable nuclei with non-linear $\sigma$ and $\omega$ terms, Nuclear Physics A 579, 557 (1994).
  * Donoghue [2006] J. F. Donoghue, Sigma exchange in the nuclear force and effective field theory, Physics Letters B 643, 165 (2006).
  * Crewther and Tunstall [2015] R. J. Crewther and L. C. Tunstall, $\mathrm{\Delta}I=1/2$ rule for kaon decays derived from QCD infrared fixed point, Phys. Rev. D 91, 034016 (2015).
  * Ma and Rho [2020] Y.-L. Ma and M. Rho, Towards the hadron–quark continuity via a topology change in compact stars, Progress in Particle and Nuclear Physics 113, 103791 (2020).
  * Machleidt and Entem [2011] R. Machleidt and D. Entem, Chiral effective field theory and nuclear forces, Physics Reports 503, 1 (2011).
  * Ekström  _et al._ [2013] A. Ekström, G. Baardsen, C. Forssén, G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, W. Nazarewicz, T. Papenbrock, J. Sarich, and S. M. Wild, Optimized chiral nucleon-nucleon interaction at next-to-next-to-leading order, Phys. Rev. Lett. 110, 192502 (2013).
  * Ekström  _et al._ [2015] A. Ekström, G. R. Jansen, K. A. Wendt, G. Hagen, T. Papenbrock, B. D. Carlsson, C. Forssén, M. Hjorth-Jensen, P. Navrátil, and W. Nazarewicz, Accurate nuclear radii and binding energies from a chiral interaction, Phys. Rev. C 91, 051301 (2015).
  * Entem  _et al._ [2015] D. R. Entem, N. Kaiser, R. Machleidt, and Y. Nosyk, Peripheral nucleon-nucleon scattering at fifth order of chiral perturbation theory, Phys. Rev. C 91, 014002 (2015).
  * Jiang  _et al._ [2020] W. G. Jiang, A. Ekström, C. Forssén, G. Hagen, G. R. Jansen, and T. Papenbrock, Accurate bulk properties of nuclei from $a=2$ to $\infty$ from potentials with $\mathrm{\Delta}$ isobars, Phys. Rev. C 102, 054301 (2020).
  * Hoferichter  _et al._ [2015] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, Matching pion-nucleon roy-steiner equations to chiral perturbation theory, Phys. Rev. Lett. 115, 192301 (2015).
  * Hoferichter  _et al._ [2016] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, Roy-steiner-equation analysis of pion-nucleon scattering, Phys. Rept. 625, 1 (2016).
  * Epelbaum  _et al._ [2004a] E. Epelbaum, W. Gloeckle, and U.-G. Meissner, Improving the convergence of the chiral expansion for nuclear forces. 1. Peripheral phases, Eur. Phys. J. A 19, 125 (2004a), arXiv:nucl-th/0304037 .
  * Epelbaum  _et al._ [2004b] E. Epelbaum, W. Glöckle, and U.-G. Meißner, Improving the convergence of the chiral expansion for nuclear forces - ii: Low phases and the deuteron, Eur. Phys. J. A 19, 401 (2004b).
  * Pérez  _et al._ [2013] R. N. Pérez, J. E. Amaro, and E. R. Arriola, Coarse-grained potential analysis of neutron-proton and proton-proton scattering below the pion production threshold, Phys. Rev. C 88, 064002 (2013).
  * Kaiser  _et al._ [1998] N. Kaiser, S. Gerstendörfer, and W. Weise, Peripheral nn-scattering: role of delta-excitation, correlated two-pion and vector meson exchange, Nucl. Phys. A 637, 395 (1998).
  * Krebs  _et al._ [2007] H. Krebs, E. Epelbaum, and U. G. Meißner, Nuclear forces with $\mathrm{\Delta}$ excitations up to next-to-next-to-leading order, part i: Peripheral nucleon-nucleon waves, Eur. Phys. J. A 32, 127 (2007).