Quantum Chromodynamics (QCD) in the \( u/d \) sector has approximate chiral symmetry but this symmetry is broken in two ways:
$$
\begin{equation*}
SU(2)_L \times SU(2)_R \approx SU(2)_V \times SU(2)_A \rightarrow SU(2)_V,
\end{equation*}
$$
that is, in the QCD ground state axial symmetry is broken while isospin symmetry is intact.
We obtain 3 Goldstone bosons: the pion family!
The chiral effective Lagrangian is given by an infinite series of terms with increasing number of derivatives and/or nucleon fields, with the dependence of each term on the pion field prescribed by the rules of broken chiral symmetry. Applying this Lagrangian to \( NN \) scattering generates an unlimited number of Feynman diagrams, which may suggest again an untractable problem. However, Weinberg showed that a systematic expansion of the nuclear amplitude exists in terms of \( (Q/\Lambda_\chi)^\nu \), where \( Q \) denotes a momentum or pion mass, \( \Lambda_\chi \approx 1 \) GeV is the chiral symmetry breaking scale, and \( \nu \geq 0 \). For a given order \( \nu \), the number of contributing terms is finite and calculable; these terms are uniquely defined and the prediction at each order is model-independent. By going to higher orders, the amplitude can be calculated to any desired accuracy.
The scheme just outlined has become known as chiral perturbation theory. Therefore, we want to describe the low-energy scenario of QCD by an Effective Field Theory (EFT). The steps to take:
The starting point for the derivation of the \( NN \) interaction is an effective chiral Lagrangian
$$
{\cal L}={\cal L}_{\pi N}+{\cal L}_{\pi\pi}+{\cal L}_{NN},
$$
which is given by
a series of terms of increasing chiral dimension,
$$
{\cal L}_{\pi N} = {\cal L}_{\pi N}^{(1)}+
{\cal L}_{\pi N}^{(2)}+{\cal L}_{\pi N}^{(3)} + \ldots ,
$$
$$
{\cal L}_{\pi\pi}={\cal L}_{\pi\pi}^{(2)}+ \ldots ,
$$
$$
{\cal L}_{NN} ={\cal L}_{NN}^{(0)}+{\cal L}_{NN}^{(2)}+{\cal L}_{NN}^{(4)} + \ldots,
$$
where the superscript refers to the number of derivatives or pion mass insertions
(chiral dimension). Good review: Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006).
Common to apply the heavy baryon (HB) formulation of chiral perturbation theory in which the relativistic Lagrangian is subjected to an expansion in terms of powers of \( 1/M_N \) (kind of a nonrelativistic expansion), the lowest order of which is
$$
\begin{align*}
\widehat{\cal L}^{(1)}_{\pi N} & = \bar{N} \left(i {D}_0 - \frac{g_A}{2}\boldsymbol{\sigma} \cdot \boldsymbol{u}\right) N \\
&
\approx \bar{N} \left[ i \partial_0 - \frac{1}{4f_\pi^2} \boldsymbol{\tau} \cdot ( \boldsymbol{\pi}\times\partial_0 \boldsymbol{\pi})- \frac{g_A}{2f_\pi} \boldsymbol{\tau} \cdot (\boldsymbol{\sigma} \cdot \boldsymbol{\nabla} )\boldsymbol{\pi} \right] N + \ldots
\end{align*}
$$
For the parameters that occur in the leading order Lagrangian,
we apply \( M_N=938.919 \) MeV, \( m_\pi = 138.04 \) MeV,
\( f_\pi = 92.4 \) MeV, and
\( g_A = g_{\pi NN} \; f_\pi/M_N = 1.29 \),
which is equivalent to
\( g_{\pi NN}^2/4\pi = 13.67 \).
The chiral NN force has the general form
$$
\begin{equation*}
V_{\rm 2N} = V_{\pi} + V_{\rm cont},
\end{equation*}
$$
where \( V_{\rm cont} \) denotes the short-range terms represented by \( NN \) contact interactions
and \( V_{\pi} \) corresponds to the long-range part associated with the pion-exchange contributions
Both \( V_{\pi} \) and \( V_{\rm cont} \) are determined within the low-momentum expansion.
Notice that the nucleon kinetic energy contributes to \( \mathcal{L}^{(2)} \). The above terms determine the nuclear potential up to N2LO (with the exception of the NN contact terms at NLO) in the limit of exact isospin symmetry.
Consider now pion-exchange contributions to the potential
$$
\begin{equation*}
V_{\pi} = V_{1\pi} + V_{2\pi} + V_{3\pi} + \ldots \,,
\end{equation*}
$$
where one-, two- and three-pion exchange (3PE) contributions
\( V_{1\pi} \), \( V_{2\pi} \) and \( V_{3\pi} \) can be written in the low-momentum expansion as
$$
\begin{align*}
\tag{1}
V_{1\pi} &= V_{1\pi}^{(0)} + V_{1\pi}^{(2)} + V_{1\pi}^{(3)} + V_{1\pi}^{(4)} +\ldots \\
V_{2\pi} &= V_{2\pi}^{(2)} + V_{2\pi}^{(3)} + V_{2\pi}^{(4)} + \ldots \\
V_{3\pi} &= V_{3\pi}^{(4)} + \ldots \,.
\end{align*}
$$
Here, the superscripts denote the corresponding chiral order and the ellipses refer to
\( (Q/\Lambda)^5 \)- and higher order terms.
Contributions due to the exchange of four- and more pions are further suppressed:
\( n \)-pion exchange diagrams start to contribute at the order \( (Q/\Lambda)^{2n-2} \).
Notice further that in addition to isopin-invariant contributions there are
isospin–breaking corrections.
The static 1PE potential at N3LO has the form
$$
\begin{equation*}
V_{1\pi}^{(0)} + V_{1\pi}^{(2)} + V_{1\pi}^{(3)} + V_{1\pi}^{(4)}
= -\biggl(\frac{g_A}{2F_\pi}\biggr)^2 \, ( 1 + \delta )^2 \,
\tau_1 \cdot \tau_2 \,
\frac{\vec{\sigma}_1 \cdot\vec{q}\,\vec{\sigma}_2\cdot\vec{q}}
{\vec q \, ^2 + M_\pi^2}\,.
\end{equation*}
$$
The 2PE contributions are convenient to express as \( V_{2 \pi} \) in the form:
$$
\begin{align*}
V_{2 \pi} &= V_C + \tau_1 \cdot \tau_2 \, W_C + \left[
V_S + \tau_1 \cdot \tau_2 \, W_S \right] \, \vec \sigma_1 \cdot \vec \sigma_2
+ \left[ V_T + \tau_1 \cdot \tau_2 \, W_T \right]
\, \vec \sigma_1 \cdot \vec q \, \vec \sigma_2 \cdot \vec q \\
&+ \left[
V_{LS} + \tau_1 \cdot \tau_2 \, W_{LS} \right] \, i ( \vec \sigma_1 + \vec \sigma_2 )
\cdot ( \vec q \times \vec k )
\\ &+ \left[
V_{\sigma L} + \tau_1 \cdot \tau_2 \, W_{\sigma L} \right] \, \vec \sigma_1
\cdot (\vec q \times \vec k ) \vec \sigma_2 \cdot (\vec q \times \vec k ) \,,
\nonumber
\end{align*}
$$
where the superscripts \( C \), \( S \), \( T \), \( LS \) and \( \sigma L \) of the scalar functions
\( V_C \), \( \ldots \), \( W_{\sigma L} \) refer to central, spin-spin, tensor, spin-orbit and
quadratic spin-orbit components, respectively.
The first non-vanishing 3NF contribution appears at order \( \nu = 3 \), i.e. at N2LO. The contribution from graph (a)
$$
\begin{equation*}
\tag{2}
V^{\rm (3)}_{\rm 2 \pi}=\sum_{i \not= j \not= k} \frac{1}{2}\left(
\frac{g_A}{2 F_\pi} \right)^2 \frac{( \vec \sigma_i \cdot \vec q_{i}
)
(\vec \sigma_j \cdot \vec q_j )}{(\vec q_i\, ^2 + M_\pi^2 ) ( \vec
q_j\, ^2 + M_\pi^2)} F^{\alpha \beta}_{ijk} \tau_i^\alpha
\tau_j^\beta \,,
\end{equation*}
$$
where \( \vec q_i \equiv \vec p_i \, ' - \vec p_i \); \( \vec p_i \)
(\( \vec p_i \, ' \)) is the initial (final) momentum of the nucleon \( i \) and
$$
\begin{equation*}
F^{\alpha \beta}_{ijk} = \delta^{\alpha \beta} \left[ - \frac{4 c_1
M_\pi^2}{F_\pi^2} + \frac{2 c_3}{F_\pi^2}
\vec q_i \cdot \vec q_j \right] + \sum_{\gamma} \frac{c_4}{F_\pi^2} \epsilon^{\alpha
\beta \gamma} \tau_k^\gamma
\vec \sigma_k \cdot [ \vec q_i \times \vec q_j ]\,.
\end{equation*}
$$
The contributions from the remaining graphs (b) and (c) take the form
$$
\begin{align*}
\tag{3}
V^{\rm (3)}_{1 \pi, \; \rm cont} = - \sum_{i \not= j \not= k} \frac{g_A}{8
F_\pi^2} \, D \, \frac{\vec \sigma_j \cdot \vec q_j }{\vec q_j^2+ M_\pi^2}\left( \tau_i \cdot \tau_j \right) (\vec \sigma_i \cdot \vec q_j ),
\quad \quad \quad
V^{\rm (3)}_{\rm cont} = \frac{1}{2} \sum_{j \not= k} E \, ( \tau_j \cdot \tau_k ),
\end{align*}
$$
where \( D \) and \( E \) are the corresponding low-energy constants from the Lagrangian of order \( \nu=1 \).
| Chiral order | 2N force | 3N force | 4N force |
|---|---|---|---|
| \( \nu = 0 \) | \( V_{1 \pi}+V_{\rm cont} \) | \( - \) | \( - \) |
| \( \nu = 1 \) | \( - \) | \( - \) | \( - \) |
| \( \nu = 2 \) | \( V_{1 \pi} + V_{2 \pi} + V_{\rm cont} \) | \( - \) | \( - \) |
| \( \nu = 3 \) | \( V_{1 \pi} + V_{2 \pi} \) | \( V_{2 \pi} + V_{1 \pi, \; \rm cont} + V_{\rm cont} \) | \( - \) |
| \( \nu = 4 \) | \( V_{1 \pi} + V_{2 \pi} + V_{3 \pi} + V_{\rm cont} \) | Established | work in progress |