# The Fundamental Nature of Low-Energy Pion-Nucleon Interactions

Gerald A. Miller  Department of Physics, University of Washington, Seattle, WA
98195-1560, USA

###### Abstract

A historical review of pion-nucleon interactions at low energy is presented,
with aims toward an introductory, pedagogic approach focusing on issues
germane to current research. These topics include the use of chiral effective
field theory, the sigma term, qualitative understanding of low-energy phase
shifts, the physics of the $\Delta$ resonance, and isospin violation. A theme
is that low-energy pion-nucleon scattering is directly related to QCD via
effective field theory. Another theme is that modern needs regarding the pion-
nucleon sigma term and the study of isospin violation requires a better low-
energy data set.

##  I Introduction

The pion is the lightest of the strongly interacting particles, with a mass
only about 1/7 of that of the nucleon. The large Compton wavelength of
$1/m_{\pi}\approx 1.4$ fm provides the longest-ranged contribution to the
nucleon-nucleon interaction and is therefore an an element of all models, from
the ancient to the newest, of the nucleon- nucleon interaction. The one pion
exchange potential is crucially responsible for the binding of nuclei
Kaiser:1997mw ; Kaiser:2001jx . Moreover, the presence of the pion as a
significant component of the nuclear wave function is reinforced by the
dominance of the pion in meson exchange corrections to a variety of nuclear
properties. This was discussed long ago Riska:1970jxh ; Riska:1972zz ;
Brown:1973qgs  and recently King:2020wmp . Moreover, the pion cloud of the
nucleon presents an important component of the nucleon sea by causing the
asymmetry of the nucleon sea, $\overline{d}>\overline{u}$, Alberg:2017ijg ;
Alberg:2021nmu  as verified by the SeaQuest experiment SeaQuest:2021zxb .

The smallness of the pion mass is a central feature of theories that describe
low-energy hadron physics. Knowledge of the mass hints to an approximate
symmetry of nature, chiral symmetry. A chiral transformation, in its simplest
form, changes a fermion field-operator $\psi$ to $\gamma_{5}\psi$. The
Lagrangians of QED and QCD are invariant under this transformation except for
terms involving mass, $m\overline{\psi}\psi$, and the small quark masses of
QCD are related to the small value of the pion mass GellMann:1968rz .

The remainder of this review is mainly concerned with describing the
consequences of the small pion mass for pion nucleon scattering. Sect. II is
con- cerned with the fundamental Lagrangian as expressed in both quark and
hadronic degrees of freedom. Sect. III describes chiral symmetry breaking and
the sigma term. The current importance of the sigma term and means to extract
it are also described. Sec. IV de- scribes the low-energy phase shifts. A
pedagogic de- scription of the $\Delta$ resonance is presented in Sect. V.
Isospin violation is described in Sect. VI. A brief summary appears in Sect.
VII.

##  II The Chiral Lagrangian and its Relation to QCD

The well-known QCD Lagrangian for the light up and down quark flavors is given
by

$\displaystyle{\cal
L}(x)=\overline{q}(x)\gamma^{\mu}\left(i\partial_{\mu}-g{\lambda^{a}\over
2}G^{a}_{\mu}\right)q(x)$ $\displaystyle-\overline{q}(x)m_{q}q(x)-{1\over
4}G^{a\mu\nu}G^{a}_{\mu\nu}\>,$ (1)

where $G^{a}_{\mu}$ are the gluon-field operators, $a$ represents the color,
$q$ are the quark-field operators (a column vector), $\lambda^{a}$ are the
eight matrices generating $SU(3)_{\rm color}$ and

$m_{q}=\begin{bmatrix}m_{u}&0\\\ 0&m_{d}\end{bmatrix}$ (2)

is the quark-mass matrix. The masses in the Lagrangian are known as current
quark masses. If $m_{d}=m_{u}$ the theory is invariant under isospin
rotations:

$q(x)\rightarrow e^{i{\bf a}\cdot\mbox{\boldmath$\tau$}/2}q(x)\>,$ (3)

where ${\bf a}$ is a constant vector. The relevant transformation that
interchanges the $u$ and $d$ quarks is obtained with ${\bf
a}=\pi{\bf\hat{y}}$, a rotation of $\pi$ about the $y-$ axis in isospin space,
with the $z-$ axis associated with charge. This transformation is known as
charge symmetry, a specific rotation in isospin space Miller:1990iz ;
Miller:2006tv .

In the further limit that the current quark masses vanish $m_{u}=m_{d}=0$ the
right- and left handed $(1\pm\gamma_{5})q$ decouple and the theory is
symmetric under two copies of isospin symmetry:

$q(x)\rightarrow e^{i{\bf
a}\cdot\mbox{\boldmath$\tau$}(1+\gamma_{5})/2}q(x),\quad q(x)\rightarrow
e^{i{\bf a}\cdot\mbox{\boldmath$\tau$}(1-\gamma_{5})/2}q(x)\>.$ (4)

These chiral transformations represent the global symmetry group
SU(2)${}_{R}\times$SU(2)L. The current quark masses are small enough so that
QCD respects this symmetry to a good approximation.

The conserved Noether currents related to the transformations of Eq. (4) are
the isospin currents:

$J_{\mu}^{a}(x)=\overline{q}(x)\gamma_{\mu}{\tau^{a}\over 2}q(x)$ (5)

and the axial currents

$A^{a}_{\mu}(x)=\overline{q}(x)\gamma_{\mu}\gamma_{5}{\tau^{a}\over 2}q(x)\>.$
(6)

The volume integral of $A_{0}^{a}$ is known as the axial charge $Q_{A}^{a}$.

The elegant QCD Lagrangian is not immediately applicable to understanding low-
energy pion-nucleon scattering because of the non-perturbative difficulties
associated with strong interactions at low energies. However there is another
long-standing technique: that of using effective Lagrangians that embody the
physics of QCD. The parameters of the effective Lagrangian can sometimes be
computed using lattice QCD techniques, and this aspect is discussed below.

Weinberg Weinberg:1978kz  wrote that quantum field theory has no content
beyond analyticity, unitarity, cluster decomposition and symmetry. In
particular, “if one writes the most general possible Lagrangian, including all
terms consistent with assumed symmetry principles, and then computes matrix
elements with this Lagrangian to any given order of perturbation theory, the
result will be the most general $S-$matrix consistent with analyticity,
perturbative unitarity, cluster decomposition and the assumed symmetry
principles.”.

This means that a properly constructed theory of pion-nucleon interactions
contains, in principle, the same information content as QCD. Here the
symmetries are isospin and chiral. Although approximate, using these should
give a good guide to the correct results. Weinberg Weinberg:1994tu
illustrates the suggested procedure by considering a theory of massless pions,
governed by a chiral SU(2)${}_{R}\times$SU(2)L symmetry. The Lagrangian will
be SU(2)${}_{R}\times$SU(2)L invariant provided it conserves isospin and is
constructed only from a chiral-covariant derivative of the pion field, which
may be taken to be

${D}_{\mu}\mbox{\boldmath$\pi$}={\partial_{\mu}\mbox{\boldmath$\pi$}\over
1+\mbox{\boldmath$\pi$}^{2}/4f_{\pi}^{2}}.$ (7)

Then the most general such Lagrangian consists of an infinite series of
operators:

${\cal
L}=\sum_{n=1}g_{n}({D}_{\mu}\mbox{\boldmath$\pi$}\cdot{D}_{\mu}\mbox{\boldmath$\pi$})^{n}.$
(8)

When this Lagrangian is used to compute pionic $S$-matrix elements it yields
the most general matrix elements consistent with the stated general
principles, provided that all terms of all orders in the coupling constants
$g_{n}$ are included. Weinberg explains that using the effective Lagrangian
allows a systematic way to calculate corrections to results of leading order.
One may derive an expansion in powers of (small values) of momentum divided by
a large mass scale.

The Lagrangian for pion-baryon interactions is more complicated. Displaying
only the interaction terms to the first-and second- order in powers of the
pion field, we use

$\displaystyle\mathcal{L}_{\rm int}=-\;{g_{A}\over
2f_{\pi}}\overline{\psi}\gamma_{\mu}\gamma_{5}\tau^{a}\psi\,\partial_{\mu}\pi^{a}-{1\over
4f_{\pi}^{2}}\overline{\psi}\gamma_{\mu}\tau^{a}\psi\;\epsilon^{abc}\pi^{b}\partial_{\mu}\pi^{c}$
$\displaystyle-{g_{\pi N\Delta}\over
2M}(\overline{\Delta}^{i}_{\mu}g^{\mu\nu}\psi\partial_{\nu}\pi^{i}+{\rm
H.C.})\>,$ (9)

where $\psi$ is the nucleon Dirac field of the nucleon, $\pi^{a}(a=1,2,3)$ the
chiral pion field and $M$ is the nucleon mass. In Eq. (9) $g_{A}$ denotes the
nucleon axial vector coupling and $f_{\pi}$ the pion decay constant. The
second term is the Weinberg-Tomozawa term which describes low-energy
$\pi-$nucleon scattering. In the third term $g_{\pi N\Delta}$ is the $\pi
N\Delta$ coupling constant, and the $\Delta^{i}_{\mu}$ field is a vector in
both spin and isospin space. Notice that only derivatives of the pion field
enter.

The meson factories LAMPF, SIN (now PSI) and TRIUMF began to operate in the
1907’s with a focus on pion-nucleon and pion-nucleus interactions with an
earlier motivation involving the pion as a fundamental carrier of the strong
interaction. The very close connection between the hadronic Lagrangians in use
at the time and QCD was mainly not understood during the peak of the
activities of the meson factories ericson1991 . Thus opportunities for
improved studies of the fundamental interaction involving better low-energy
pion beams were not realized or perhaps not even dreamed about. We shall see
that determination of the pion-nucleon sigma term would be greatly improved by
better quality data.

The Lagrangian of Eq. (9) enjoys remarkable success when applied to compute
pion-nucleon scattering amplitudes at threshold. Then terms involving spatial
momenta drop out. A simple discussion is provided in the book by Ericson &
Weise Ericson:1988gk . The only surviving term, expressed as a Hamiltonian
density ${\cal H}$ is given by

${\cal H}={1\over
4f_{\pi}^{2}}\mbox{\boldmath$\tau$}\cdot(\mbox{\boldmath$\pi$}\times{\partial\mbox{\boldmath$\pi$}\over\partial
t})\>.$ (10)

This interaction may be used to compute the isovector $s$-wave pion-nucleon
interaction. First take the matrix elements between in- and outgoing pion
states to find

$\langle\pi_{b}|\mbox{\boldmath$\tau$}\cdot(\mbox{\boldmath$\pi$}\times{\partial\mbox{\boldmath$\pi$}\over\partial
t})|\pi_{b}\rangle=2im_{\pi}\epsilon_{abc}\tau_{c}\>.$ (11)

This Hamiltonian density corresponds to a zero range (s-wave) pion-nucleon
potential, $V_{s}({\bf r})$ given by

$V_{s}({\bf r})={1\over 4f_{\pi}^{2}}\mbox{\boldmath$\tau$}\cdot{\bf
t}\delta({\bf r})\>,$ (12)

where $\bf t$ is the pion isospin operator. Using the Born approximation the
scattering length, $a$ is given by

$a=-{\tilde{m}_{\pi}}\int d^{3}rV_{s}({\bf r})\>,$ (13)

where $\tilde{m}_{\pi}=m_{\pi}/(1+m_{\pi}/M)$ is the pion-nucleon reduced
mass. The isospin operator operator $\mbox{\boldmath$\tau$}\cdot{\bf t}$
evaluates to 1 for an $\pi$N isospin state $3/2$ and $-2$ for an isospin state
of 1/2. Thus the two scattering lengths $a_{1,3}$ are given by

$a_{1}={\tilde{m}_{\pi}\over 4\pi f_{\pi}^{2}},\quad
a_{3}={-\tilde{m}_{\pi}\over 8\pi f_{\pi}^{2}}\>.$ (14)

The scattering length can be expressed as

$a=(a_{1}+2a_{3})/3+(a_{3}-a_{1})/3\,\mbox{\boldmath$\tau$}\cdot{\bf t}$ (15)

so the isovector combination is

$a_{1}-a_{3}=3{{\tilde{m}_{\pi}}\over 8\pi f_{\pi}^{2}}=0.234m_{\pi}^{-1}$
(16)

and the isoscalar term is given by

$a_{1}+2a_{3}=0\>.$ (17)

A more complete analysis is necessary because the second Born approximation
for the potential of Eq. (12) is divergent. This has been provided via the
detailed chiral perturbation theory analyses Fettes:1998ud ; Fettes:1998wf ;
Fettes:2000gb ; Fettes:2000bb ; Fettes:2000xg ; Fettes:2000vm ; Fettes:2001cr
that Fettes:1998ud  finds

$a_{1}-a_{3}=0.264(15)m_{\pi}^{-1},$ (18)

while Ericson:1988gk  reports

$a_{1}+2a_{3}=-0.029(9)m_{\pi}^{-1}.$ (19)

The large magnitude of the isospin-dependent term and the small magnitude of
the isospin-independent term of the scattering length, $a$, are a striking
success of the simple Lagrangian. It has been argued Weinberg ; Long:2009wq ,
in the spirit of an expansion in the pion mass, that it is the pion mass,
instead of the reduced pion mass, that should appear in the above equations.
In that case the agreement between the simple estimate Eq. (16) and empirical
extraction Eq. (18) would be perfect.

##  III Chiral Symmetry Breaking and the Sigma Term

The quark mass term of Eq. (1) is not invariant under the transformations of
Eq. (4). It may be put in the form Weinberg:1994tu :

${\cal L}_{\rm mass}=-(m_{u}+m_{d})V_{4}-(m_{u}-m_{d})A_{3}\>,$ (20)

where

$V_{4}={1\over 2}(\overline{u}u+\overline{d}d),\quad A_{3}={1\over
2}(\overline{u}u-\overline{d}d\>).$ (21)

The operators $V_{4}$ and $A_{3}$ are spatial scalars and components of
independent chiral four vectors $V_{\alpha}$ and $A_{\alpha}$. Following
Weinberg, terms must be added to the effective chiral Lagrangian with the same
transformation properties. The aim is to construct chiral symmetry breaking
terms without derivatives of the pion field. Using only the pion field, there
is no Lagrangian term in $A_{3}$ and only one term in $V_{4}$. This is the
pion mass term

$V_{4}(m_{\pi})=-{m_{\pi}^{2}\pi^{2}\over 2(1+\pi^{2}/(4f_{\pi}^{2}))}\>.$
(22)

Using pion fields and a nucleon bi-linear (with no derivatives) and assuming
the isospin limit of $m_{u}=m_{d}$ one finds a term

$\delta{\cal L}_{\rm int}=-{A\over 2}\left({1-\pi^{2}/4f_{\pi}^{2}\over
1+\pi^{2}/4f_{\pi}^{2}}\right)\overline{\psi}\,\psi,$ (23)

where $A$ is a constant proportional to $m_{u}+m_{d}$ and $\psi$ is the
nucleon doublet. This term gives a contribution to the isoscalar scattering
length (the first term of Eq. (15)):

${a_{1}+2a_{3}\over 3}={A\over 2f_{\pi}^{2}}\>.$ (24)

The term $A$ is the sigma term, $\sigma_{\pi N}$, which has recently emerged
as a quantity related to many aspects of physics, such as the origin of the
mass of ordinary matter, studies of dark matter, nuclear binding and
nucleosynthesis. The plan here is to first consider the relation of
$\sigma_{\pi N}$ to pion-nucleon scattering and then discuss its significance.
My discussion parallels that of the recent review by Alarcón Alarcon:2021dlz .

Consider the pion-nucleon scattering process:

$\displaystyle\pi^{a}(q)N(p)\to\pi^{b}(q^{\prime})N(p^{\prime}).$ (25)

where $a\ (b)$ are the isospin indices of the initial (final) pion of momentum
$q\ (q^{\prime})$, and $p\ (p^{\prime})$ the momentum of the initial (final)
nucleon. The off-shell scattering amplitude for pions of squared four-momentum
$q^{2}$ and $q^{\prime 2}$, $T^{ba}$, is given by Reya:1974gk

$\displaystyle{T^{ba}(\nu,t,q^{2},q^{\prime 2})\over K(q,q^{\prime})}=\int
d^{4}x\langle
N(p^{\prime})|q^{\prime}_{\mu}q_{\gamma}T(A^{b\mu}(x)\partial_{\gamma}A^{a\gamma}(0))+iq^{\prime}_{\mu}\delta(x_{0})[A^{b\mu}(x),\partial_{\gamma}A^{a\gamma}(0)]|N(p)\rangle
e^{iq^{\prime}\cdot x}$ $\displaystyle-\int d^{4}x\langle
N(p^{\prime})|\delta(x_{0})[A^{b0}(x),A^{a\gamma}(0)]|N(p)\rangle
e^{iq^{\prime}\cdot x}.$ (26)

where $K(q,q^{\prime})\equiv i\frac{(q^{2}-m_{\pi}^{2})(q^{\prime
2}-m_{\pi}^{2})}{m_{\pi}^{4}f_{\pi}^{2}},\,\nu=\frac{s-u}{4M}$, with $M$ the
nucleon mass, $A^{a\mu}$ the axial current, $f_{\pi}$ the weak decay constant
of the pion and $m_{\pi}$ the pion mass. The last term is not determined by
the commutation relations provided by current algebra, but instead is isolated
by taking the so-called “soft-meson limit” $q^{\prime}_{\mu},q_{\nu}\to 0$.
Then, using Eq. (III) and $\partial_{\nu}A^{a\nu}=-i[Q_{5}^{a},\mathcal{H}]$
yields

$\displaystyle T^{ba}(0,0,0,0)$ $\displaystyle=-\frac{1}{f_{\pi}^{2}}\langle
N(p^{\prime})|[Q_{A}^{b}(0),[Q_{A}^{a}(0),\mathcal{H}(0)]]|N(p)\rangle$

where $Q_{A}^{i}$ are the axial charges and $\mathcal{H}$ is the Hamiltonian
density. Only terms that break chiral symmetry enter in its commutator with
the axial charges. The double commutator of the last matrix element of Eq.
(III) is the $\pi N$ sigma commutator, $\sigma_{\pi N}$. The Jacobi identity
tells us that the sigma commutator is symmetric in the isospin indices and,
therefore, contributes only to the isoscalar part of the scattering amplitude.
The mass matrix $m_{q}$ of Eq. (2) provides the part of $\mathcal{H}$ that
does not commute with the axial charge and yields

$\displaystyle\sigma_{\pi N}=\frac{\overline{m}}{2M}\langle
N(p)|\overline{u}u+\overline{d}d|N(p)\rangle,$ (28)

where $\overline{m}\equiv\frac{m_{u}+m_{d}}{2}$, usually called the pion-
nucleon sigma term. The factor $2M$ in the denominator arises from the
covariant normalization $\langle
N(p^{\prime})|N(p)\rangle=(2\pi)^{3}2E\,\delta({\vec{p}\,^{\prime}-\vec{p}})$,
used here. Note also that $\overline{m}$ arises because here we use
$m_{d}=m_{u}$.

The sigma term appears in many areas of modern physics as explained below.
First it is necessary to understand its rather indirect relation to
scattering. The amplitude of Eq.(III) is not evaluated at measurable
kinematics, so one needs to find a way to connect the experimental data to
$T^{ba}(0,0,0,0)$. Ref. Cheng:1970mx  showed that one can expand the amplitude
in $q^{2}$ and $q^{\prime 2}$ to relate the physical amplitude to its soft
limit. One invokes the Adler consistency conditions Adler  and evaluates the
amplitude at $(\nu=0,t=2m_{\pi}^{2})$ to cancel the leading correction in
$q^{2}$ and $q^{\prime 2}$, and relate the physical amplitude to
$T^{ba}(0,0,0,0)$ plus a remainder. This kinematical point is called the
Cheng-Dashen point (with $\nu=0,\,s=m_{N}^{2},t=2m_{\pi}^{2}$) and lies in the
unphysical (subthreshold) region of $\pi N$ scattering. The ability to extract
the amplitudes depends on having very high quality scattering data at low
energies and a very precise method of handling the effects of the Coulomb
interaction. The date begins for pion kinetic energies of about only 30 MeV.
Moreover, there is no unique way to split the interaction into strong and
electromagnetic parts Gasser:2003hk , so in general models must be used.

Cheng and Dashen, assumed that the remainder was of order $m_{\pi}^{4}$, but
Brown, Pardee and Peccei showed that there were non-analytic contributions
that made this relation valid only up to order $m_{\pi}^{2}$ Brown:1971pn .
They presented a widely-used modified form of the Cheng-Dashen theorem valid
up to $\mathcal{O}(m_{\pi}^{4})$. To be more specific, one writes the $\pi N$
scattering amplitude in terms of the four Lorentz invariant amplitudes
$A^{\pm}$, $B^{\pm}$

$\displaystyle
T^{ba}=\delta^{ba}T^{+}+\frac{1}{2}\left[\tau^{b},\tau^{a}\right]T^{-}$ (29)
$\displaystyle
T^{\pm}=\overline{u}^{\prime}\left[A^{\pm}+\frac{1}{2}(\not{q}^{\prime}+\not{q})B^{\pm}\right]u$
(30)

where $T^{\pm}$ are the isoscalar and isovector parts of the amplitude. The
combination $D^{\pm}\equiv A^{\pm}+\nu B^{\pm}$ is amenable to a detailed
analysis. The original result by Cheng and Dashen states that $\Sigma_{\pi
N}\equiv f_{\pi}^{2}\overline{D}^{+}(\nu=0,t=2m_{\pi}^{2})$ and $\sigma_{\pi
N}$ are equal up to order $m_{\pi}^{2}$, with $\overline{D}^{+}$ the Born-
subtracted part of the amplitude. In Ref. Brown:1971pn  this theorem was
recast as follows

$\displaystyle\Sigma_{\pi
N}=f_{\pi}^{2}\overline{D}^{+}(\nu=0,t=2m_{\pi}^{2})$
$\displaystyle=\sigma(t=2m_{\pi}^{2})+\Delta_{R},$ (31)

where

$\displaystyle\sigma(t)=\frac{1}{2M}\langle
N(p^{\prime})|\hat{m}(\overline{u}u+\overline{d}d)|N(p)\rangle.$ (32)

This expression is useful because $\Delta_{R}$ is of order $m_{\pi}^{4}$ and
small ($\Delta_{R}\simeq 2$ MeV Bernard:1996nu ), and the scalar form factor
of the nucleon at $t=2m_{\pi}^{2}$, $\sigma(t=2m_{\pi}^{2})$, can be related
to the sigma term though dispersion relations or other methods, because
$\sigma(t=0)=\sigma_{\pi N}$. The latter was first calculated in Ref.
Pagels:1972kh .

###  III.1 Why is the sigma term important?

This sub-section presents a brief discussion explaining the current importance
of the sigma term.

####  III.1.1 Origin of the mass of ordinary matter

Recently there has been an increasing effort to understand the the proton (or
nucleon) mass in terms of QCD, see e.g. Ji:1994av ; He:1994gz ; Metz:2020vxd ;
Lorce:2021xku ; Ji:2021qgo . The usual strategy to analyze its decomposition
is to consider that the energy-momentum tensor of QCD can be related to the
mass of a hadron (in this case the nucleon) in the following way:

$\displaystyle M=\frac{1}{2M}\langle N(p)|T^{\mu}_{\ \mu}|N(p)\rangle$ (33)

where $T^{\mu}_{\ \mu}$ is the trace of the energy momentum tensor. This
trace, in its renormalized form, is given by Crewther:1972kn ;
Chanowitz:1972vd ; Collins:1976yq ; Shifman:1978zn :

$\displaystyle M=\frac{1}{2M}\langle
N(p)|\frac{\beta(\alpha_{s})}{2\alpha_{s}}G^{a\mu\eta}G^{a}_{\ \mu\eta}$
$\displaystyle+\sum_{q=u,d,s}\gamma_{m_{q}}m_{q}\overline{q}q|N(p)\rangle+\frac{1}{2M}\langle
N(p)|\sum_{q}m_{q}\overline{q}q|N(p)\rangle,$

where $\alpha_{s}$ is the strong coupling constant, $\beta(\alpha_{s})$ its
beta function, $m_{q}$ is the mass of the quark $q$ and $\gamma_{m_{q}}$ the
anomalous dimension of the mass operator. In the limit in which strange quarks
are neglected, the last term in (III.1.1) is $\sigma_{\pi N}$. This term is
renormalization scheme and scale independent, and can be interpreted as the
light quark mass contribution to the nucleon mass. This term arises from the
Higgs boson couplings to quarks, so that its accurate determination would tell
us the amount of ordinary matter mass that is generated though this mechanism
and how much is dynamically generated. Other mass decompositions include
Ji:1994av ; He:1994gz ; Metz:2020vxd ; Lorce:2021xku ; Ji:2021qgo , but all
contain the quark-mass term.

####  III.1.2 Dark matter detection

The main recent interest in the precise value of $\sigma_{\pi N}$ arises from
its role in for detecting scalar dark matter. If one assumes that dark matter
is a scalar, an effective Lagrangian that couples dark matter ($\chi$) to
quarks ($q$) is given by:

$\displaystyle\mathcal{L}_{\chi
q}=C\frac{m_{q}}{\Lambda^{3}}\overline{\chi}\chi\overline{q}q$ (35)

where $C$ is a Wilson coefficient containing information regarding energy
scales higher than the cutoff $\Lambda$.

This effective Lagrangian leads to a $\chi-$ nucleon and consequently a
$\chi-$nuclear cross section Hill:2011be  that is proportional to the square
of $\sigma_{\pi N}$ (if $u$ and $d$ quarks are sufficient) so that
uncertainties in $\sigma_{\pi N}$ are magnified in dark matter searches
Bottino ; Ellis:2008hf .These authors stressed the vital importance of an
experimental campaign to determine a better value of $\sigma_{\pi N}$.

It is noteworthy that Ref. Crivellin:2013ipa  showed how to avoid unnecessary
and uncontrolled assumptions usually made in the literature about soft SU(3)
flavor symmetry breaking in determining the two-flavor nucleon matrix elements
relevant for direct detection of weakly interacting massive particles (WIMPs).
This work was the first to allowfor an accurate assessment of hadronic
uncertainties in spin-independent WIMP-nucleon scattering and for a reliable
calculation of isospin-violating effects.

####  III.1.3 Nuclear binding and Nucleosynthesis

The $\sigma$-model Lagrangian Gell-Mann:1960mvl ; Brown:1971pn ; Noble:1979iv
contains a term that violates chiral symmetry of the form
$(m^{2}_{\sigma}-m_{\pi}^{2})/(2M)g\sigma\mbox{\boldmath$\pi$}\cdot\mbox{\boldmath$\pi$}$,
where $\sigma$ represents the field of a scalar-isoscalar interaction and $g$
is the pion-nucleon coupling constant. This terms yields a contribution to
$T^{+}$ of Eq. (29) and therefore to $\sigma_{\pi N}$. The $\sigma$-model when
enhanced by the inclusion of vector mesons incorporates the dynamics necessary
to bind nuclei: long- and medium-ranged attractive forces and short-ranged
repulsive forces.

More generally, the pion-nucleon sigma term is related to the strength of the
$\pi N$ interaction with scalar-isoscalar quantum numbers. This part of the
interaction plays a prominent role in alpha-like particle clustering
Elhatisari:2016owd . Understanding this process becomes essential in order to
explain the observed ${}^{12}\text{C}$ abundance, which is possible only
though to the formation of a $0^{+}$ resonance near the ${}^{4}\text{He
}\text{+ }{}^{8}\text{Be}$ threshold (the Hoyle state). This state is
important for nucleosynthesis.

####  III.1.4 Nuclear thermodynamics

The matrix element of QCD related to the sigma term also provides valuable
information about the properties of nuclear matter. When one studies the
restoration of chiral symmetry in nuclear matter, one of the quantities used
as order parameter in the phase transition is the chiral condensate in the
nuclear medium, $\langle\Psi|\overline{q}q|\Psi\rangle$, where $|\Psi\rangle$
is the nuclear ground state Weise:2012yv . Applying the Hellmann-Feynman
theorem,for nuclear matter of Fermi momentum $k_{F}$, to the Hamiltonian
density of QCD one finds Kaiser:2007nv

$\displaystyle\frac{\langle\Psi|\overline{q}q|\Psi\rangle}{\langle
0|\overline{q}q|0\rangle}(\rho)=1-\frac{\rho}{f_{\pi}^{2}}\left\\{\frac{\sigma_{\pi
N}}{m_{\pi}^{2}}\left(1-\frac{3k_{f}^{2}}{10m_{N}^{2}}+\frac{9k_{f}^{4}}{56m_{N}^{4}}\right)+{\partial
E(\rho)/A\over\partial m_{\pi}^{2}}\right\\}$ (36)

where $\langle 0|\overline{q}q|0\rangle$ is the quark condensate in the
vacuum, and $E$ is the energy of nuclear matter. This means that the sigma
term is important for determining the density for the transition to a chiral
symmetric regime in which $\langle\Psi|\overline{q}q|\Psi\rangle=0.$
Increasing the value of $\sigma_{\pi N}$ reduces the value of the transition
density. However, the interaction terms ${\partial E(\rho)/A\over\partial
m_{\pi}^{2}}$ are very important Kaiser:2007nv .

####  III.1.5 Strangeness content of the nucleon

The quantity $\langle N|m_{s}\overline{s}s|N\rangle\equiv 2M\sigma_{s}$ is
related to the contribution of the strange quark to the nucleon mass. This
matrix element has been related to $\sigma_{\pi N}$ though group-theoretical
arguments as follows. Consider the extended mass term of the QCD Hamiltonian:

$\displaystyle\mathcal{H}_{m}=\frac{1}{3}(m_{s}+2\overline{m})(\overline{u}u+\overline{d}d+\overline{s}s)$
$\displaystyle+\frac{1}{3}(\overline{m}-m_{s})(\overline{u}u+\overline{d}d-2\overline{s}s).$
(37)

The second term can be related to the mass splitting of the strange and non-
strange baryons at leading order in the $SU(3)$-breaking parameter
$m_{s}-\overline{m}$, using octet mass-splitting Alarcon:2012nr ;
Fernando:2018jrz :

$\displaystyle\sigma_{0}\equiv\frac{\overline{m}}{2M}\langle
N|(\overline{u}u+\overline{d}d-2\overline{s}s)|N\rangle$
$\displaystyle=\frac{\overline{m}}{m_{s}-\overline{m}}(m_{\Xi}+m_{\Sigma}-2M).$
(38)

The relation of $\sigma_{0}$ with the octet baryon mass splitting provides an
estimate of $\sigma_{s}$, assuming a known value of $\sigma_{\pi N}$:

$\displaystyle\sigma_{s}=\frac{m_{s}}{2\hat{m}}(\sigma_{\pi N}-\sigma_{0})$
(39)

Assuming $\langle N|\overline{s}s|N\rangle=0$ (Zweig rule), the value of
$\sigma_{\pi n}$ should be well approximated by $\sigma_{0}$ Cheng:1975wm .
Then using the present day-values of neutral baryon masses and the current
quark ratio Zyla:2020zbs  : $m_{s}/2\overline{m}\approx 27$. then $\sigma_{\pi
n}$ should be around $24$ MeV. Estimates of the SU(3)-breaking effects
increase this value to about 35 MeV Gasser:1980sb . If one allows for a non-
zero value of $\sigma_{s}$, then one finds $\sigma_{\pi N}\approx
0.073\,\sigma_{s}+35\,\rm MeV.$ Then using a typical value of $\sigma_{\pi
N}\approx 60$ MeV one obtains $\sigma_{s}\approx 350$ MeV, a huge value. In
contrast, lattice QCD calculations give results for $\sigma_{s}$ between 30
and 50 MeV Durr:2011mp ; Durr:2015dna ; Yang:2015uis ; Freeman:2012ry ;
Junnarkar:2013ac . Recent developments, discussed in the next section show
that $\sigma_{\pi N}$ can be about 60 MeV without a need for a huge value of
$\sigma_{s}$. Therefore the above analysis is seems dated.

The strangeness content of the nucleon, related to $\sigma_{s}$ has also been
postulated to reduce the proton spin below the value obtained in simple quark
models, and therefore play a role in the proton spin crisis; see the reviews
Hughes:1999wr ; Beck:2001yx .

###  III.2 Methods to determine the sigma term

In this section I briefly discuss some of the different approaches and
strategies that have been used to determine the sigma term. The problem in
using experimental measurements is in relating $T(0,0,,0,0)$ to measurable
kinematics in which $q^{2}={q^{\prime}}^{2}=m_{\pi}^{2}$.

The earliest attempts used the simple assumption, explained above that using
Eq. III.1.5 with $\sigma_{s}=0$ gives $\sigma_{\pi N}\approx 35$ MeV. The
second is the use of dispersion relations that allow a continuation from $\pi
N$ scattering data to the Cheng-Dashen point See the review Alarcon:2021dlz
and the original works Gasser:1988jt ; Gasser:1990ap ; Gasser:1990ce ;
Bernard:1996nu ; Hoferichter:2015dsa . Sum rules can also be used to express
$\Sigma$ in terms of pion nucleon threshold parameters Altarelli:1971kh ;
Olsson:1979ee ; Olsson:1999jt

Another method is the use of chiral perturbation theory. As a systematic way
to determine QCD Green’s functions Gasser&Leutwyler , chiral EFT is widely
used to study strong interactions at low energies. Chiral EFT is constructed
to systematically provide corrections to the low-energy theorems of QCD, so it
is a natural tool to calculate the pion-nucleon sigma term. One writes a
Lagrangian that complies with chiral symmetry, with unknown short-distance,
high-momentum transfer effects modeled by the use of low energy constants that
are determined by data. Then explicit chiral symmetry breaking is provided by
including a term of the form

$\displaystyle\mathcal{L}_{\pi N}^{(2)}\supset
c_{1}\langle\chi_{+}\rangle\overline{\psi}\psi$ (40)

where $\psi$ refers to the nucleon isodoublet, and $\langle\chi_{+}\rangle$ is
a trace in the isospin space of a matrix involving even powers of the pion
field and quark masses (see Ref. Alarcon:2012kn  for more details). The value
of $c_{1}$ is determined by using the Lagrangian to compute $\pi N$ phase
shifts fits to data, and accurate results for $c_{1}$ are obtained only if all
of the low energy constants can be determined.

Lattice QCD is well-known to allows direct, non-perturbative numerical
calculations using the QCD Lagrangiani. a numerical method to compute QCD
Green functions. Two different strategies have been used. The first one is to
calculate the sigma term through the Hellmann-Feynman theorem. For that, one
needs the quark mass dependence of the nucleon mass. Another approach is to
calculate the sigma term as the scalar form factor of the nucleon at $q^{2}=0$
Durr:2015dna ; Yang:2015uis ; Abdel-Rehim:2016won ; Bali:2016lvx ;
Alexandrou:2017qyt ; Borsanyi:2020bpd .

Ref. Friedman:2019zhc  used pionic $\pi H$ atom data to estimate the sigma
term. The strategy of this approach is to relate the observed modification of
the isovector scattering length $a_{0+}^{-}\equiv(a_{1}-a_{3})/3$ in pionic
atoms to $\sigma_{\pi N}$. They construct the relation from the leading order
chiral result for $a_{0+}^{-}$ in vacuum, given by the Weinberg-Tomozawa term

$\displaystyle
a_{0+}^{-}=-\frac{M_{\pi}m_{N}}{8\pi(m_{N}+M_{\pi})f_{\pi}^{2}}.$ (41)

One obtains the in-medium modification of $a_{0+}^{-}$ by noting that the
ratio of condensates appearing in Eq. (36) is equal to (following
GellMann:1968rz ) the ratio $\frac{f_{\pi}^{2}(\rho)}{f_{\pi}^{2}}$ then

$\displaystyle\frac{f_{\pi}^{2}(\rho)}{f_{\pi}^{2}}=\frac{\langle\Psi|\overline{q}q|\Psi\rangle}{\langle
0|\overline{q}q|0\rangle}(\rho)\approx 1-\rho\frac{\sigma_{\pi
N}}{f_{\pi}^{2}M_{\pi}^{2}}$ (42)

which modifies the free scattering length to

$\displaystyle\hat{a}_{0+}^{-}=a_{0+}^{-}\left(1-\rho\frac{\sigma_{\pi
N}}{f_{\pi}^{2}M_{\pi}^{2}}\right)^{-1}$ (43)

where $\hat{a}_{0+}^{-}$ refers to the in-medium value of $a_{0+}^{-}$. For
details see Friedman:2019zhc . This method is based on the medium modificaton
of the dominant isovector s-wave term. But the medium modification also
modifies the nuclear isoscalar s-wave term that is much larger than predicted
by the impulse approximation Miller:1980zza ; Miller:1980zzb ; Olivier:1984wm
; Chakravarti:1993cy  of the pion nucleus optical potential related to the
isoscalar s-wave term.

###  III.3 Determining the sigma term

Alarcon Alarcon:2021dlz  presented a detailed discussion of the lengthy
history of determinations and the great variation of the results. The latest
results are tabulated in the 2021 FLAG review Aoki:2021kgd . We present their
results in Figure 1.

Figure 1:  Lattice results and FLAG averages for the nucleon sigma
term,$\sigma_{\pi N}$ for the $N_{f}=2,2+1$, and 2 + 1 + 1 flavor
calculations. Determinations via the direct approach are indicated by squares
and the Feynman-Hellmann method by triangles. Results from calculations which
analyze more than one lattice data set within the Feynman-Hellmann approach
Alexandrou:2017xwd ; Procura:2006bj ; Young:2009zb ; Ren:2012aj ; Alvarez-
Ruso:2013fza ; Lutz:2014oxa ; Ren:2014vea ; Ren:2016aeo ; Ren:2017fbv ;
Lutz:2018cqo , are shown for comparison (pentagons) along with those from
recent analyses of $\pi N$ scattering Alarcon:2011zs ; Chen:2012nx ;
Hoferichter:2015dsa ; RuizdeElvira:2017stg  (circles). The results that pass
the FLAG criteria are shown in green, and are reasonably consistent.

FLAG presents its results as $N_{f}=2+1+1:\sigma_{\pi N}=64.9(1.5)(13.2)$ MeV,
a result that is larger than for the case of $N_{f}=2+1$ of $37(8)(6)$ MeV.
The meaning of 2 + 1+ 1 is that $m_{u}=m_{d}<m_{s}<m_{c}$.The latter value is
believed to be more reliable because of the smaller error bar. A weighted
average of the two results would greatly favor the more precise latter value.
The FLAG review cite a tension with the result of Hoferichter:2015dsa  of
$\sigma_{\pi N}=(59.1\pm 3.5)$ MeV of about three or four standard deviations.
A significant source of uncertainty in lattice calculations is contamination
due to excited states. A recent, calculation Gupta:2021ahb  forces the first
excited state to be that of a nucleon-pion $p$-wave energy. This procedure
results in a sigma term of $\sigma_{\pi N}=(59.6\pm 7.4)$ MeV consistent with
phenomenology. Ref. Gupta:2021ahb  also states that further calculations are
needed to confirm their result and finally resolve the tension between lattice
QCD and phenomenology. I also note that pionic atom data are used in
Hoferichter:2015dsa  and there is possibly an additional uncertainty due to
medium effects that are not included in the analyses.

The strangeness sigma term Aoki:2021kgd  is presented in Fig. 2. The result
for $N_{f}=2+1+1$ is $\sigma_{s}=41.0(8.8)\rm MeV$, and for $N_{f}=2+1$,
$\sigma_{s}=52.9(7.0)$ MeV. There is no need for a very large value of
$\sigma_{s}$.

Figure 2:  Lattice results and FLAG averages for $\sigma_{s}$ for the
$N_{f}=2,2+1$, and 2 + 1 + 1 flavor calculations. Determinations via the
direct approach are indicated by squares and the Feynman-Hellmann method by
triangles. Results from calculations which analyze more than one lattice data
set within the Feynman-Hellmann approach Young:2009zb ; Ren:2012aj ;
Lutz:2014oxa ; Ren:2014vea ; Lutz:2018cqo , are shown for comparison
(pentagons) (circles). he results that pass the FLAG criteria are shown in
green, and are reasonably consistent.

##  IV Momentum dependence of the low-energy $\pi-N$ phase shifts

A nice discussion of these phase shifts is presented in the book
Ericson:1988gk . This is updated here based on more recent data from the SAID
data base using the WI08 fit Workman:2012hx  which is an update of that of
Arndt:2006bf .

For $s$-wave scattering the isospin $1/2$ phase shifts are designated as
$S_{11}$, while those of isospin $3/2$ are termed $S_{33}$. These are shown in
Fig. 3. The range of pion lab momenta is from 0 to 350 MeV/c which corresponds
to a range in lab kinetic energy from 0 to 240 MeV, and cm momenta, c from 0
to 260 MeV/c.

These phase shifts of small magnitude are approximately linear in the cm
momentum $q=M/\sqrt{s}q_{L}$ for this values of $q$ up to about $q_{\rm cm}$
of about 100 MeV/c. There $\delta(S_{11}\approx{4.9\times
10^{-2}}\,q/\rm(MeV/c)deg$ and $\delta(S_{31})\approx{-2.9\times
10^{-2}}q/\rm(MeV/c)deg$. The combination $\delta(S_{11})+2\delta(S_{31})$
nearly vanishes, consistent with early chiral perturbation theory. Effective
range corrections must be included to describe the phase shifts for momenta
greater than about 90 MeV/c. Detailed chiral perturbation theory calculations
are presented in Refs. Fettes:1998ud ; Fettes:1998wf ; Fettes:2000gb ;
Fettes:2000xg ; Fettes:2000vm ; Fettes:2001cr . These have been updated in
Refs. Ditsche:2012fv ; Hoferichter:2015hva ; Hoferichter:2012wf ; Baru:2010x;
Baru:2011bw .

Figure 3: $S_{11}$ and $S_{33}$ phase shifts as a function of the cm momentum
of the pion beam, $q$. The points are from the single-energy fits and the
curves are energy dependent fits. The data are from the SAID data base, SAID
and are from the WI08 fit Workman:2012hx , an update of that of Arndt:2006bf .

The $p-$wave phase shifts are designated $P_{2I,2J}$. The $P_{33}$ and
$P_{11}$ phase shifts are shown in Fig. 4. They are proportional to $q^{3}$
near threshold. The $P_{33}$ channel is clearly dominant, with the phase shift
rising rapidly to $90^{\circ}$ at $q\approx 292\,\rm MeV/c$, the position of
the $\Delta$ resonance. The $P_{11}$ phase shift is small at the low energies
of relevance here. At low energies it is negative and crosses 0 at about
$q=195$ MeV/c and rises at higher energies, reflecting the broad $N^{*}$(1440)
resonance.

The $P_{33},\,P_{13}$ and $P_{31}$ phase shifts are shown in Fig.5.. The other
phase shifts remain small in the low-energy range of interest.

The most detailed treatment of low-energy phase shifts is to be found in Refs.
Ditsche:2012fv ; Hoferichter:2015hva ; Hoferichter:2012wf ; Baru:2010xn ;
Baru:2011bw . These authors found that much higher precision (and accuracy)
can be achieved by combining dispersion relations with the experimental input
that is available (either at threshold from scattering lengths or at higher
energies from phase-shift analyses). The low-energy data set for $\pi N$
scattering is scarce. So Ref. Hoferichter:2015hva  summarizes that one can do
much better by combining theory input in the form of Roy-Steiner equations,
experiment for the scattering lengths from pionic atoms, and the phase-shift
analyses only for the high-energy input.

Figure 4: $P_{33}$ and $P_{11}$ phase shifts as a function of the cm momentum
of the pion, $q$. The points are from the single-energy fits and the curves
are energy dependent fits. The data are from the SAID data base, SAID and are
from the WI08 fit Workman:2012hx , an update of that of Arndt:2006bf .  Figure
5:  $P_{33}$ and $P_{11}$ phase shifts as a function of the cm momentum of the
pion, $q$. The points are from the single-energy fits and the curves are
energy dependent fits. The data are from the SAID data base, SAID and are from
the WI08 fit Workman:2012hx , an update of that of Arndt:2006bf .

The figures in this section seem to indicate that the data set is in good
shape. However, as pointed out in Sect. III, improved knowledge of
$\sigma_{\pi N}$ requires better data. Section. VI below also points out that
understanding of isospin violation requires better low-energy data. The Muon
Scattering Experiment (MUSE) MUSE:2017dod ; Cline:2021vlw ; Cline:2021ehf
intends to obtain a better determination of the proton radius, and more
generally test lepton-universality by comparing $\mu^{\pm}p$ with $e^{\pm}p$
scattering. Pion-nucleon scattering is an important background so the
collaboration plans to measure $\pi^{\pm}$-nucleon cross sections at beam
momenta of about 115, 160, and 210 MeV/c. The angular range is 20 - 100
degrees in the lab. These measurements would greatly improve the data base.

##  V analysis of the $\Delta_{33}$ resonance

The $\Delta_{33}$ resonance (or $\Delta$) is the simplest of all resonances.
Nevertheless it has some interesting complications caused by the need to
include the crossed nucleon Born graph and the effects of renormalizing the
$\Delta$ mass caused by the need to maintain unitarity. The aim here is to
present a simple discussion to illustrate the features that are perhaps the
minimal ones needed to understand more complicated resonances. These basic
features were first included in the Cloudy Bag Model calculations
Miller:1979kg ; Theberge:1980ye  and are absent in the calculations of
Oset:1981ih ; Cheung:1983rz . See Ref. Long:2009wq  for a detailed chiral
perturbation theory analysis.

Figure 6: Irreducible diagrams in the 33 channel. (a) Crossed nucleon term (b)
Excitation of bare $\Delta$. (c) Iteration of (b) via an integral equation

The starting point is the Born graphs (irreducible diagrams) of Fig. 6 a and
b. The iteration of $\Delta$ excitation term, Fig. 6 c, via an integral
equation causes the inclusion of terms (isolated in the box) that correspond
to the dressing of the $\Delta$ and a shift in the resonance peak away from
the energy of the bare $\Delta$, $M_{\Delta}^{0}$. The iteration of the
crossed nucleon term is not shown in Fig. 6 but must be included. The crossed
graph with an intermediate $\Delta$ is not included for simplicity and because
it is small.

Detailed discussion of the methods to project amplitudes onto the 33 channel
are shown in the book Ericson:1988gk . Here I use projected amplitudes in
which the (cm) Born terms take the form

$\displaystyle\hat{V}(s)=\gamma(s)|\Delta\rangle\langle\Delta|$ (44)

with $s$ the square of the $\pi N$ cm energy,

$\displaystyle\gamma(s)={1\over 3}{M\over\sqrt{s}}{q^{2}}\left[{f_{\pi
N}^{2}\over 4\pi}{2M\over 2E(s)\omega(s)}+{f_{\pi N\Delta}^{2}\over
4\pi}\frac{2M_{\Delta}^{0}}{s-{M_{\Delta}^{0}}^{2}}\right]$

and $|\Delta\rangle$ representing the form factor arising from the finite
extent of the pion and baryon. The first term in the bracket is inversely
proportional to $M^{2}-\overline{u}$, in which $\overline{u}$ accurately
approximates the Mandelstam variable $u$ Ericson:1988gk . I use $\langle
q|\Delta\rangle=1/(1+q^{2}/M_{A}^{2})^{2}\equiv v(q)$ with $M_{A}$ the axial
mass of 1030 MeV. I use $f_{\pi N}^{2}/(4\pi)=0.08$ and $f_{\pi
N\Delta}=1.5f_{\pi N}$, obtained from the large $N_{c}$ limit. See Refs.
Siemens:2016jwj ; PhysRevD.87.054032 ; NavarroPerez:2014ovp ; Epelbaum:2008ga
.

The $K$\- matrix then takes the form

$\displaystyle\hat{K}(s)=D(s)|\Delta\rangle\langle\Delta|.$ (46)

The on-shell $K-$matrix element in a given channel is $\tan\delta(q)/q$, with
$q$ the cm momentum. Solving the integral equation:

$\displaystyle\hat{K}(s)=\hat{V}(s)+\hat{V}(s)PG(s)\hat{K}(s),$ (47)

with $P$ representing a principle-value integral, yields the result

$\displaystyle D(s)={\gamma(s)\over
1-\gamma(s)v^{2}(q)\langle\Delta|G(s)|\Delta\rangle}$ (48)

with Green’s function matrix element given by:

$\displaystyle\langle\Delta|G(s)|\Delta\rangle=P\int
dp\,{p^{4}v^{2}(p)}\,{1\over s-(E(p)+\omega(p))^{2}}.$ (49)

Defining the smooth function
$\hat{\gamma}(s)\equiv(s-{M_{\Delta}^{0}}^{2})\gamma(s)$ allows $D(s)$ to be
re-written as

$\displaystyle D(s)={\hat{\gamma}(s)\over
s-{M_{\Delta}^{0}}^{2}-\hat{\gamma}(s)v^{2}(q)\langle\Delta|G(s)|\Delta\rangle}$
(50)

This shows that the resonance position is shifted away from
$\sqrt{s}=M_{\Delta}^{0}$ by the iteration of both the nucleon crossed Born
term and the $\Delta$ excitation. The calculations of Oset:1981ih ;
Cheung:1983rz  use $\hat{K}(s)\to\gamma(s)|\Delta\rangle\langle\Delta|$, a
procedure that amounts to unitarizing the Born approximation..

Results of a numerical calculation are shown in Fig. 7. The solid (blue) curve
is the full calculation that approximately reproduces the data. The dashed
(red) curve shows result of using the Born approximation procedure of
Oset:1981ih ; Cheung:1983rz  with the present parameters. The dotted (black)
curve shows the results obtained by omitting the crossed nucleon Born term.
The figure shows that the cross Born term must be kept and that the Born
approximation to the $K$-matrix is a poor approximation.

Figure 7: $P_{33}$ phase shifts computed with the model explained in the text.
Solid (blue) curve: full calculation. Dashed(red): Unitarized Born
approximation of Oset:1981ih ; Cheung:1983rz  . Dotted: full calculation
without the crossed nucleon Born term of fig (a).

##  VI Isospin violation in pion-nucleon interactions

The basis for isospin violation in the strong interaction is the mass
difference between the up and down quarks, as written the $A_{3}$ term of Eq.
(20). All isospin violations within the standard model arise from this term
and electro-weak interactions, as stressed in Ref. Miller:1990iz .

Weinberg Weinberg:1994tu  wrote the effective interaction arising from this
term as a charge symmetry breaking (csb) contribution to the effective
interaction

$\displaystyle\delta{\cal L}_{\rm
csb}=-B\left[\overline{\psi}t_{3}\psi-{1\over 2f_{\pi}^{2}}\left({\pi_{3}\over
1+{\mbox{\boldmath$\pi$}^{2}\over 4f_{\pi}^{2}}}\right)\overline{\psi}{\bf
t}\cdot\mbox{\boldmath$\pi$}\psi\right],$ (51)

with $B\propto m_{u}-m_{d}$. This term leads to a term in the scattering
length for the process of Eq. (25):

$\displaystyle\delta a_{ba}({\rm csb})={1\over 4\pi(1+m_{\pi}/m_{N})}{B\over
2f_{\pi}^{2}}(t_{a}\delta_{3b}+t_{b}\delta_{3a}).$ (52)

Weinberg comments that the immediate consequence of Eq. (51) is that “isospin
violation never appears in a process that does not involve at least one
neutral pion.”. I believe that Weinberg meant to say that “isospin violation
from the strong interaction never appears in a process that does not involve
at least one neutral pion.” For example, the Coulomb interaction causes
isospin violating differences between $\pi^{+}d$ and $\pi^{-}d$ scattering.

Weinberg Weinberg:1977hb  used Eq. (52) estimated that the
$\pi^{0}n\to\pi^{0}n$ scattering length was 30% larger than the one for
$\pi^{0}p\to\pi^{0}p$, although both are of the order of 0.01 fm. Of course,
the related scattering experiments are not possible with current techniques.

I turn to the more general case of $\pi N$ scattering. If isospin invariance
is respected one can write the $\pi-N$ scattering amplitude as an operator in
a schematic form

$\displaystyle f=f_{0}+f_{1}{\bf t}\cdot{\bf t}_{\pi},$ (53)

where $f_{0,1}$ are operators that depend on relative momentum and nucleon
spin. There are eight reactions of the form
$\pi^{a}(q)N(p)\to\pi^{b}(q^{\prime})N(p^{\prime})$, but only three are
available to experiments Gibbs:1995dm : $\pi^{\pm}p$ elastic scattering with
amplitudes $f_{\pm}$ and $\pi^{-}p\to\pi^{0}n$ charge exchange (CEX)
scattering with amplitude $f_{rmCEX}$. The use of Eq. (53) yields the so-
called triangle identity:

$\displaystyle f_{\rm CEX}={1\over\sqrt{2}}(f_{+}-f_{-}).$ (54)

The failure of this equality is a failure of isospin invariance, but not of
charge symmetry. The $B$ term of Eq. (51) causes no difference between the
left and right-hand sides of the above equation. Thus a powerful conclusion
results: the only Standard Model effects that break the triangle identity come
from electroweak and hadronic mass-difference effects. The neutron-proton mass
difference does arises from light quark mass differences and from the
electroweak interaction, but is an indirect effect through its kinematic
influence.

If isospin symmetry is broken the term $f_{1}{\bf t}\cdot{\bf t}_{\pi}$ is
replaced by the isospin-breaking (ib) term $f_{1}({\rm ib})$ with

$\displaystyle f_{1}({\rm ib})=\alpha t_{z}t_{\pi z}+\beta(t_{x}t_{\pi
x}+t_{y}t_{\pi y}),$ (55)

with $\alpha\neq\beta$ and $\alpha-\beta\propto m_{u}-m_{d}$ if
electromagnetic effects are neglected. The $\alpha$ term causes differences
between $\pi^{+}p$ and $\pi^{-}p$ scattering, the terms proportional to
$\beta$ are responsible for charge exchange. One can get a quick glimpse of
the size the effects by examining the kinematics of the $\pi^{-}p\to\pi^{0}n$
reaction. The $\pi^{-}$ experiences a Coulomb attraction of about
$\alpha/r_{p}\approx 1.7$ MeV and the proton is lighter than the neutron by
about 1.3 MeV the sum of these two effects of 3 MeV is offset by the
$\pi^{-}-\pi^{0}$ mass difference of 4.6 MeV.

Early analyses Gibbs:1995dm ; Matsinos:1997pb  of the triangle identity in
$\pi N$ low-energy scattering data found indications for significant 6-7 % csb
effects in the strong-interaction sector. Fettes and Meissner Fettes:2000vm
have found a smaller violation using a third-order chiral perturbation theory
calculation without electromagnetic effects, and even smaller violations once
the latter were included Fettes:2001cr . They found -0.7% violation in the
$s$-wave. An interesting feature of this work is the inclusion of the effects
of the $\gamma n$ channel in the final state. A detailed comprehensive
partial-wave analysis of $\pi^{\pm}p$ elastic scattering and charge- exchange
data, covering the region from threshold to 2.1 GeV in the lab pion kinetic
energy found no compelling evidence for sizable isospin breaking effects
beyond those of the standard model.

Isospin breaking through quark mass differences and virtual photons in
scattering lengths was analyzed using covariant baryon chiral perturbation
theory Hoferichter:2009ez ; Hoferichter:2009gn . This work found that the so-
called triangle relation that vanishes in the isospin limit is violated by
about 1.5% consistent with earlier findings and inconsistent with the much
larger deviations extracted from the data at lowest pion momenta in
Gibbs:1995dm ; Matsinos:1997pb . Isospin violation in the triangle identity
was studied later in RuizdeElvira:2017stg  using cross-section data. The
conclusion was that these data are simply not precise enough to test the
triangle relation at the percent level.

An experimental study of isospin symmetry in the $\pi^{-}p\to\pi^{0}n$
reaction at momenta between 104 and 143 MeV/c was made at TRIUMF Jia:2008rt .
This kinematic region exhibits a vanishing zero-degree cross section from
destructive interference between $s$ and $p$ waves, thus yielding special
sensitivity to pion-nucleon dynamics. Their data and previous data
Fitzgerald:1986fg  disagree. The key feature is that the predictions of Gibbs
et al. Gibbs:1995dm ; Gibbs:2004jf  and Matsinos et al. Matsinos:1997pb ;
Matsinos:2006sw  based on fitting $\pi^{\pm}p$ data that constrain the term
$\alpha$ of Eq. (55) agree with the angular distributions obtained in
Jia:2008rt , but not with those obtained in Fitzgerald:1986fg . This means
that no significant isospin breaking effect is observed. The difference in
data sets was attributed to normalization uncertainties and background
subtraction.

The net result regarding tests of the triangle identity is that there isn’t
any evidence for these huge isospin-breaking effects.

One also studies charge symmetry breaking and isospin violation involving
light nuclei $\rm d,^{3}H,^{3}He,^{4}He$ as detailed in the review
Miller:1990iz . Here I focus on later experiments involving $\pi^{\pm}$
scattering on the deuteronBaru:1999ns , 3He and 3H Kudryavtsev:2001fe .

Charge symmetry breaking effects in experimental measurements have been
expressed in terms of $A_{\pi}$ for the deuteron:

$\displaystyle
A_{\pi}=\frac{d\sigma/d\Omega(\pi^{-}d)-d\sigma/d\Omega(\pi^{+}d)}{d\sigma/d\Omega(\pi^{-}d)+d\sigma/d\Omega(\pi^{+}d)},$
(56)

and for the nuclei with $A=3$:

$\displaystyle r_{1}=\frac{d\sigma/d\Omega(\pi^{+}{{}^{3}}\rm
H)}{d\sigma/d\Omega(\pi^{-}{{}^{3}}\rm He)}$ (57) $\displaystyle
r_{2}=\frac{d\sigma/d\Omega(\pi^{-}{{}^{3}}\rm
H)}{d\sigma/d\Omega(\pi^{+}{{}^{3}}\rm He)}$ (58) $\displaystyle R=r_{1}r_{2}$
(59)

Charge symmetry predicts that $A_{\pi}=1,r_{1}=1,r_{2}=1$ and $R=1$. Of course
it is necessary to remove the effects of the Coulomb interaction. These
experiments were performed at energies near and on that of the $\Delta$
resonance and one must account for the mass and width differences amongst the
four charge states of the $\Delta$ as well as the neutron-proton mass
difference and the 3He-3H mass difference.

The early history suggested evidence for a small effect, $A_{\pi}\approxeq
2\%$ at 143 MeV Masterson:1982ki . A sizable effect was reported in the 3He3H
case. For example, $r_{2}=0.7\pm 0.1$ for $T_{\pi}=256$ MeV at
$\theta=82^{\circ}$ Dhuga:1996cg .

Ref. Baru:1999ns  described the charge symmetry breaking effect on the
deuteron in terms of single and double scattering amplitudes, taking the
$\Delta$ mass splitting into account. The results of that study indicate that
the $\Delta$ mass and width differences, can along with the Coulomb
interaction are of the same order as the deviations between $A_{\pi}$ and
unity.

Ref. Kudryavtsev:2001fe  described pion scattering from the three-nucleon
system in terms of single- and double-scattering amplitudes. Coulomb
interactions and $\Delta$-mass splitting are taken into account as sources of
charge symmetry breaking. Reasonable agreement between theoretical
calculations and the experimental data was obtained for $T_{\pi}=180,220$, and
256 MeV. For these energies, it is found that the $\Delta$-mass splitting and
the differences in wave functions of the $A=3$ nuclei are the most important
contributions for charge symmetry breaking. Data for $r_{2}$ and R at
$T_{\pi}=$ 142 MeV did not agree with the predictions of the model, which may
indicate that there are additional mechanisms for charge symmetry breaking
that are important only at lower energies.

The summary of the activity for elastic pion scattering from nuclei with $A=2$
or 3 is that the $\Delta$ mass differences and Coulomb effects account for
most of the charge symmetry breaking effects. Other mechanisms involving non-
resonant charge symmetry breaking in $s-$wave $\pi^{0}$-nucleon scattering
that occurs in the intermediate state, as indicated in Eq. (55) and Eq. (52)
were not included in the calculations.

Weinberg’s statements Weinberg:1994tu  about charge symmetry breaking
involving the $\pi^{0}$ were borne out in experiments on the $np\to\pi^{0}d$
reaction Opper:2003sb  in which charge symmetry breaking predicts a symmetry
in the angular distribution about a production angle of 90∘ and in the $\rm
dd\to^{4}He\,\pi^{0}$ reaction Stephenson:2003dv ; Bacher:2019tpm  that is
forbidden by charge symmetry. Ref. Opper:2003sb  observed a forward-backward
asymmetry $[17.2\pm 8.0\rm(stat)\pm 5.5(syst)]\times 10^{-4}$ at an incident
neutron energy of 279.5 MeV. This result could be accounted for by the effects
of the mass difference between up and down quarks. Refs. Stephenson:2003dv ;
Bacher:2019tpm  measured total cross sections for neutral pion production of
$12.7\pm 2.2$ pb at 228.5 MeV and $15.1\pm 3.1$ pb at 231.8 MeV. The
uncertainty is dominated by statistical errors. These cross sections arise
fundamentally from the down-up quark mass difference and quark electromagnetic
effects that contribute in part through meson mixing (e.g.,$\pi^{0}-\eta$)
mechanisms.

The results of these experiments and the related theory are discussed in the
review Miller:2006tv . The simplest summary is that the effects of the mass
difference between up and down quarks of Eq. (20) cause the charge symmetry
breaking . Initial calculations of the manifestation of the mass difference in
terms of the necessary hadronic degrees of freedom accounted for the findings
of experiments with reasonable accuracy. The effects of the light-quark mass
difference has made its presence felt in nuclear physics.

##  VII Summary

Chiral effective field theory is a greatly useful tool to understand low-
energy pion-nucleon interactions and their connection with the fundamental
theory of QCD. The breaking of chiral symmetry via quark mass differences is
manifest in specific features of Lagrangians using hadronic interactions. The
progress in understanding the sigma term and its widespread importance is
reviewed. The extraction of the sigma term would benefit from improving the
low-energy data set. The features of the low-energy $\pi$N phase shifts are
reviewed. The complicated physics of the simplest baryon resonant state–the
$\Delta$ is explained. Recent progress in understanding isospin violation in
theory and fact is reviewed. The experimental observations of isospin
violation seem to be accounted for by the effects of light-quark mass
differences and electromagnetic effects. However, higher energy phase shifts
provide the dominant experimental information. This is a second reason for why
it would be very useful to have better low-energy pion-nucleon scattering
data.

The study of pion-nucleon scattering is equivalent to the study of the
fundamental strong interaction of QCD. Its importance remains vital to this
day.

## Acknowledgements

This work was supported by the US Department of Energy Office of Science,
Office of Basic Energy Sciences program under Grant No. DE-FG02-97ER- 41014. I
thank Avraham Gal, Ronald Gilman, Andre Walker-Loud and Martin Hoferichter for
useful discussions and Igor Strakovsky for providing Figs. 3-5.

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