Nuclear Forces
Ruprecht Machleidt (2014), Scholarpedia, 9(1):30710. | doi:10.4249/scholarpedia.30710 | revision #137821 [link to/cite this article] |
Nuclear forces (also known as nuclear interactions or strong forces) are the forces that act between two or more nucleons. They bind protons and neutrons (“nucleons”) into atomic nuclei. The nuclear force is about 10 millions times stronger than the chemical binding that holds atoms together in molecules. This is the reason why nuclear reactors produce about a million times more energy per kilogram fuel as compared to chemical fuel like oil or coal. However, the range of the nuclear force is short, only a few femtometer (1 fm $ = 10^{-15}$ m), beyond which it decreases rapidly. That is why, in spite of its enormous strength, we do not feel anything of this force on the atomic scale or in everyday life. The development of a proper theory of nuclear forces has occupied the minds of some of the brightest physicists for seven decades and has been one of the main topics of physics research in the 20th century. The original idea was that the force is caused by the exchange of particles lighter than nucleons known as mesons, and this idea gave rise to the birth of a new sub-field of modern physics, namely, (elementary) particle physics. The modern perception of the nuclear force is that it is a residual interaction (similar to the van der Waals force between neutral atoms) of the even stronger force between quarks, which is mediated by the exchange of gluons and holds the quarks together inside a nucleon.
Historical perspective
After the discovery of the neutron by Chadwick in 1932 (Chadwick, 1932), it was clear that the atomic nucleus is made up from protons and neutrons. In such a system, electromagnetic forces cannot be the reason why the constituents of the nucleus are sticking together. Indeed, the repulsive electrical Coulomb force between the protons should blow the nucleus apart. Therefore, the concept of a new strong nuclear force was introduced. In 1935, the first theory for this new force was developed by the Japanese physicist Hideki Yukawa (Yukawa,1935), who suggested that the nucleons would exchange particles between each other and this mechanism would create the force. Yukawa constructed his theory in analogy to the theory of the electromagnetic interaction where the exchange of a (massless) photon is the cause of the force. However, in the case of the nuclear force, Yukawa assumed that the “force-makers” (which were eventually called “mesons”) carry a mass of a fraction of the nucleon mass. A massive particle limits the effect of the force to a finite range, since the uncertainty principle allows massive particles to travel only a finite distance. The meson predicted by Yukawa was finally found in 1947 in cosmic ray and in 1948 in the laboratory and called the pion. Yukawa was awarded the Nobel Prize in 1949. In the 1950’s and 60’s more mesons were found in accelerator experiments and the meson theory of nuclear forces was extended to include many mesons. These models became known as one-boson-exchange models, which is a reference to the fact that the different mesons are exchanged singly in this model. The one-boson-exchange model is very successful in explaining essentially all properties of the nucleon-nucleon interaction at low energies. Also, in the 1970’s and 80’s, meson models were developed that went beyond the simple single-particle exchange mechanism. These models included, in particular, the explicit exchange of two pions with all its complications. Well-known models of the latter kind are the Paris (Lacombe et al.,1980) and the Bonn potential (Machleidt et al., 1987).
Since these meson models were quantitatively very successful, it appeared that they were the solution of the nuclear force problem. However, with the discovery (in the 1970’s) that the fundamental theory of strong interactions is quantum chromodynamics (QCD) and not meson theory, all “meson theories” had to be viewed as models, and the attempts to derive a proper theory of the nuclear force had to start all over again.
The problem with a derivation of nuclear forces from QCD is two-fold. First, each nucleon consists of three quarks such that the system of two nucleons is already a six-body problem. Second, the force between quarks, which is created by the exchange of gluons, has the feature of being very strong at the low energy scale that is characteristic of nuclear physics. This extraordinary strength makes it difficult to find “converging” mathematical solutions. Therefore, during the first round of new attempts, the models proposed were just "QCD-inspired". The positive aspect of these models is that they try to explain nucleon structure (which consists of three quarks) and nucleon-nucleon interactions (six-quark systems) on an equal footing. Some of the gross features of the two-nucleon force, like the “hard core” (see below) are explained successfully in such models. However, from a critical point of view, it must be noted that these quark-based approaches are yet another set of models and not a theory. Alternatively, one may try to solve the six-quark problem with brute computing power, by putting the six-quark system on a four dimensional lattice of discrete points which represents the three dimensions of space and one dimension of time. This method has become known as lattice QCD and is making progress. However, such calculations are computationally very expensive and cannot be used as a standard nuclear physics tool.
Around 1990, a major breakthrough occurred when the nobel laureate Steven Weinberg applied the concept of an effective field theory (EFT) to low-energy QCD (Weinberg, 1979, 1991). He simply wrote down the most general theory that is consistent with all the properties of low-energy QCD, since that would make this theory identical to low-energy QCD. A particularly important property is the so-called chiral symmetry, which is “spontaneously” broken. Massless particles observe chiral symmetry, which means that their spin and momentum are either parallel (“right-handed”) or anti-parallel (“left-handed”) and remain so forever. Since the quarks, which nucleons are made of (“up” and “down” quarks), are almost massless, approximate chiral symmetry is a given. Naively, this symmetry should have the consequence that one finds in nature mesons of the same mass, but with positive and negative parity. However, this is not the case and such failure is termed a “spontaneous” breaking of the symmetry. According to a theorem first proven by Jeffrey Goldstone, the spontaneous breaking of a symmetry creates a particle, which in this case is the pion. Thus, the pion becomes the main player in the creation of the nuclear force. The interaction of pions with nucleons is weak as compared to the interaction of gluons with quarks. Therefore, pion-nucleon processes can be calculated without problem. Moreover, this effective field theory can be expanded in terms of powers of momentum over "scale", where scale denotes the “chiral symmetry breaking scale” which is about 1 GeV (= $10^9$ eV). This scheme is also known as chiral perturbation theory (ChPT) and allows to calculate the various terms that make up the nuclear force systematically power by power, or order by order. Another advantage of the chiral EFT approach is its ability to generate not only the force between two nucleons, but also many-nucleon forces, on the same footing. In modern theoretical nuclear physics, the chiral EFT approach is becoming increasingly popular and is applied with great success.
Properties of the nuclear force
Some properties of nuclear interactions can be deduced from the properties of nuclei. Nuclei exhibit a phenomenon known as saturation: the volume of nuclei increases proportionally to the number of nucleons. This property suggests that the nuclear (central) force is of short range (a few fm) and strongly attractive at that range, which explains nuclear binding. But the nuclear force has also a very complex spin-dependence. Evidence of this property first came from the observation that the deuteron (the proton-neutron bound state, the smallest atomic nucleus) deviates slightly from a spherical shape: it has a non-vanishing quadrupole moment. This suggests a force that depends on the orientation of the spins of the nucleons with regard to the vector joining the two nucleons (a tensor force). In heavier nuclei, a shell structure has been observed which, according to a proposal by M. G. Mayer and J. H. D. Jensen, can be explained by a strong force between the spin of the nucleon and its orbital motion (the spin-orbit force). More clear-cut evidence for the spin-dependence is extracted from scattering experiments where one nucleon is scattered off another nucleon, with distinct spin orientations. In such experiments, the existence of the nuclear spin-orbit and tensor forces has clearly been established. Scattering experiments at higher energies (more than 200 MeV) provide evidence that the nucleon-nucleon interaction turns repulsive at short inter-nucleon distances (smaller than 0.5 fm, the hard core).
Besides the force between two nucleons (2NF), there are also three-nucleon forces (3NFs), four-nucleon forces (4NFs), and so on. However, the 2NF is much stronger than the 3NF, which in turn is much stronger than the 4NF, and so on. In exact calculations of the properties of light nuclei based upon the “elementary” nuclear forces, it has been shown that 3NFs are important. Their contribution is small, but crucial. The need for 4NFs for explaining nuclear properties has not (yet) been established.
Nuclear forces are approximately charge-independent meaning that the force between two protons, two neutrons, and a proton and a neutron are nearly the same (in the same quantum mechanical state) when electromagnetic forces are ignored.
Phenomenological approaches
Traditionally, research on the nuclear force has proceeded along two lines: phenomenology and theory. We start with phenomenology. At very low energy, nucleon-nucleon scattering can be described by the so-called effective range expansion which, in its simplest version, has only two parameters (Bethe, 1949), see Brown and Jackson (1976) for a text-book presentation. In modern times, this expansion has been recovered from an EFT which does not include the pion ('pionless EFT'), see, e.g., Bedaque and van Kolck (2002) for an introduction. At intermediate energies, the spin-dependence of the nuclear force becomes important. To produce a general expression for the $NN$ potential (that includes spin and momentum dependences), Okuba and Marshak (1958) imposed the following symmetries,
- Translational invariance
- Galilean invariance
- Rotational invariance
- Space reflection invariance
- Time reversal invariance
- Invariance under the interchange of particle 1 and 2
- Isospin symmetry
- Hermiticity
and obtained: \begin{equation} \begin{array}{llll} V & = & \:\, V_C \:\, + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_C & \boldsymbol{central} \\ %\nonumber \\ &+& \left[ \, V_S \:\, + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_S \,\:\, \right] \, \vec\sigma_1 \cdot \vec \sigma_2 & \boldsymbol{spin-spin} \\ %\nonumber \\ &+& \left[ \, V_{LS} + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_{LS} \right] \, \vec L \cdot \vec S & \boldsymbol{spin-orbit} \\ %\nonumber \\ &+& \left[ \, V_T \:\:\, + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_T \,\:\: \right] \, S_{12}(\hat r) & \boldsymbol{tensor} \\ %\nonumber \\ &+& \left[ \, V_{\sigma L} + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_{\sigma L} \, \right] \, Q_{12} & \boldsymbol{\sigma-L} \\ %\nonumber \\ &+& \left[ \, V_{\sigma p} + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_{\sigma p} \, \right] \, \vec \sigma_1 \cdot \vec p \,\,\: \vec \sigma_2 \cdot \vec p \;\;\;\;\; & \boldsymbol{\sigma-p} \end{array} \tag{1} \end{equation} with \begin{eqnarray} S_{12}(\hat r) & \equiv & 3 \vec \sigma_1 \cdot \hat r \,\,\: \vec \sigma_2 \cdot \hat r - \vec \sigma_1 \cdot \vec \sigma_2 \,, \tag{2} \\ Q_{12} & \equiv & \frac12 \left[ \vec \sigma_1 \cdot \vec L \,\,\: \vec \sigma_2 \cdot \vec L + \vec \sigma_2 \cdot \vec L \,\,\: \vec \sigma_1 \cdot \vec L \right] \,, \tag{3} \end{eqnarray} and \begin{equation} \begin{array}{llll} \vec r &\equiv& \vec r_1 - \vec r_2 & \mbox{relative coordinate},\\ \hat r &\equiv& \vec r/r & \mbox{unit vector for relative coordinate},\\ \vec p &\equiv& \frac12 (\vec p_1 - \vec p_2) & \mbox{relative momentum},\\ \vec L &\equiv& \vec L_1+\vec L_2 = \vec r \times \vec p = -i \vec r \times \vec \nabla & \mbox{total orbital angular momentum in position space},\\ \vec S &\equiv& \frac12 (\vec\sigma_1+\vec\sigma_2) & \mbox{total spin}, \end{array} \tag{4} \end{equation} where $\vec r_{1,2}$, $\vec p_{1,2}$, $\vec L_{1,2}$, $\vec \sigma_{1,2}$, and $\boldsymbol{\tau}_{1,2}$ denote position, momentum, angular momentum, spin, and isospin, respectively, of nucleon 1 and 2. The $V_i$ and $W_i$, with $i=C,S,LS,T,\sigma L, \sigma p$, are functions of $r^2, p^2$, and $L^2$ only, i.e. \begin{eqnarray} V_i & = & V_i (r^2,p^2,L^2) \,, \\ W_i & = & W_i (r^2,p^2,L^2) \,. \end{eqnarray} Charge-independence or isospin invariance requires that the potential is a scalar in the isospin space of the two nucleons. The only such scalars are 1 and $\boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2$, which explains the isospin structures in Eq. (1).
If energy is conserved in the scattering process ("on shell"), then there are only five independent terms and the $\sigma p$ term can be expressed as a combination of the other five terms. Note, however, that when a potential is applied in a scattering equation (Schroedinger or Lippmann-Schwinger equation) the potential goes off shell.
Potentials which are based upon the operator structure Eq. (1) with functions $V_i$ and $W_i$ chosen such as to fit the $NN$ data or phase shifts are called phenomenological potentials. To keep things simple, most phenomenological potentials do not include all six terms. A minimal set for a realistic potential is the central, spin-spin, spin-orbit and tensor term.
Historically noteworthy examples for phenomenological $NN$ potentials are:
- Gammel and Thaler (1957), first semi-quantitative $NN$ potential, hard-core.
- Hamada and Johnston (1962), first quantitative $NN$ potential, hard core.
- Reid (1968), first quantitative soft core potential, very popular in the 1970's.
- Argonne V14 potential (Wiringa et al., 1984), based upon a set of 14 operators.
- Argonne V18 potential (Wiringa et al., 1995), based upon a set of 18 operators, charge-dependent, high-precision (low $\chi^2$ for fit of $NN$ data).
The meson theory of nuclear forces
Yukawa's idea of 1935
In 1935, Yukawa (1935) introduced the concept of massive particle exchange to explain the nuclear force. He constructed an analogy to classical electrodynamics. In electrodynamics, the Coulomb potential \begin{equation} \phi(r)=\frac{q}{4\pi} \, \frac{1}{r} \end{equation} is the solution of Poisson's equation \begin{equation} \nabla^2\phi(\vec r) = -{q} \, \delta^{(3)}({\vec r}) \,. \end{equation} When adding a mass term to this equation (and flipping the sign on the r.h.s. to adjust for scalar coupling), \begin{equation} (\nabla^2 -m^2) \varphi(\vec r) = g \, \delta^{(3)}({\vec r}) \,, \end{equation} the solution becomes \begin{equation} \varphi(r)=-\frac{g}{4\pi} \, \frac{e^{-m r}}{r} \,, \end{equation} which is the scalar field generated by one nucleon. A second nucleon, with also coupling $g$, at a distance $r$ from the first one will be exposed to the interaction energy \begin{equation} V(r)=-\frac{g^2}{4\pi} \, \frac{e^{-m r}}{r} \,, \end{equation} which is known as the Yukawa potential. The exponential in this expression, that is due to the mass $m$ of the meson, restricts the potential to a finite range, which is the essential point. For $m \rightarrow 0$ we are back to the form of the Coulomb potential.
The one-boson-exchange model
Yukawa's original derivation was done for scalar bosons. When finally a real meson was discovered in 1947/48, it turned out to be pseudo-scalar with mass around 138 MeV and was dubbed the $\pi$-meson or pion. Consequently, in the 1950's, the attempts to derive the nuclear force focused on theories that included only pions. These 'pion theories' had many problems and little success--for reasons we understand today: pion dynamics is constrained by chiral symmetry, a concept that was unknown in the 1950's. In the early 1960's, heavier (non-strange) mesons were found in experiment, notably the vector (spin-1) mesons $\rho(770)$ and $\omega(782)$. Because of the problems with the pion theories, theoreticians were now happy to extend meson theory by including more and different species of mesons. This led to the one-boson-exchange (OBE) models, which were started in the 1960's and turned out to be very successful for the two-nucleon interaction.
Let's first address the question of which mesons to consider. When deriving the nuclear force, one has generally more confidence in the predictions for the longer ranged parts. Since the range, $R_\alpha$, of the force created by a meson is inversely proportional to the meson mass, $m_\alpha$, i.e., \begin{equation} R_\alpha \sim \frac{1}{m_\alpha} \,, \tag{5} \end{equation} one starts with the lightest mesons and moves up to mesons with masses in the order of the nucleon mass. This includes essentially six mesons (PDG, 2012), namely, $\pi(138)$, $\eta(548)$, $\sigma(500)$, $\rho(770)$, $\omega(782)$, $a_0(980)$, where the numbers in parentheses are the masses in MeV. As it turns out, $\eta$ and $a_0$ are not very important (Machleidt, 1989) and, so, we will focus here on
- the pseudo-scalar isovector pion $(0^-,1)$,
- the scalar isoscalar sigma $(0^+,0)$,
- the vector isoscalar omega $(1^-,0)$,
- the vector isovector rho $(1^-,1)$,
where the parenthetical information, $(J^P,I)$, summarizes spin $J$, parity $P$, and isospin $I$ for each particle.
Yukawa's original considerations used classical field theory. A more proper derivation should be based upon quantum field theory, as we will use here. In the one-boson-exchange (OBE) model, the mesons are exchanged singly as shown in the Feynman diagram of Figure 4. The contributions to the $NN$ potential from the various mesons are derived in Figure 5 to Figure 8. In these derivations, we always start from an appropriate interaction Lagrangian for meson-nucleon coupling, which is designed with guidance from symmetry principles (the Lagrangian must be a Lorentz scalar). Concerning the Lagrangians for the vector mesons $\omega$ and $\rho$ (Figure 7 and Figure 8), we note that they may have both a vector and a tensor coupling (with coupling constants $g_v$ and $f_v$, respectively): \begin{equation} {\cal L}_{vNN}=-g_{v}\bar{\psi}\gamma^{\mu}\psi\varphi^{(v)}_{\mu} -\frac{f_{v}}{4M} \bar{\psi}\sigma^{\mu\nu}\psi(\partial_{\mu}\varphi_{\nu}^{(v)} -\partial_{\nu}\varphi_{\mu}^{(v)}) \,. \tag{6} \end{equation} These two couplings are similar to the interaction of a photon with a nucleon. The first is analogous to the coupling of the Dirac current to the electromagnetic vector potential, while the second one corresponds to the Pauli coupling of the anomalous magnetic moment. The analogy is not accidental; the vector-meson dominance model (VDM) for the electromagnetic form factor of the nucleon explains the close relationship. In the VDM one assumes that the photon couples to the nucleon through a vector boson, which explains the extended structure of the nucleon electromagnetic form factor. In the strict interpretation of this model, the $\rho$ coupling constants ratio, $f_\rho/g_\rho$, should be 3.7, and from dispersion analysis one obtains even $f_\rho/g_\rho \approx 6$. In any case, the tensor coupling of the $\rho$ is much larger than its vector coupling, which is why we omitted the vector coupling in Figure 8. For the $\omega$ meson, it is the other way around: the vector dominance model suggests a $\omega$ coupling constants ratio $f_\omega/g_\omega = -0.12$. Since this is close to zero, the $\omega$ is given no tensor coupling in most meson models (cf. Figure 7).
The full propagator for vector bosons is \begin{equation} P_v = i \frac{-g_{\mu\nu}+q_{\mu} q_{\nu}/m_{v}^{2}}{q^{2}-m_{v}^{2}} \,, \tag{7} \end{equation} where in Figure 7 and Figure 8 we dropped the $q_{\mu} q_{\nu}/m_{v}^{2}$-term. Due to nuclear current conservation, this term vanishes on-shell, but not off-shell. The off-shell effect of this term was examined by Holinde and Machleidt (1975) and was found to be unimportant.
It is customary to multiply the vertices with cutoffs, which suppress high-momentum components to ensure the convergence of the scattering equation. A simple form for these cutoffs is \begin{equation} \left( \frac{\Lambda_\alpha^2 - m_\alpha^2}{\Lambda_\alpha^2+{\vec q}^2} \right)^{n_\alpha} \,, \end{equation} where the cutoff mass $\Lambda_\alpha$ is typically chosen in the range 1.3 - 2.0 GeV. The multiplication by these form factors is not explicitly shown in our derivations. The calculation is performed in momentum space and in the center-of-mass (CMS) system of the two interacting nucleons, where $p_1=(E, \vec p)$ and $p_2=(E,-\vec p)$ in the initial states; and $p_1'=(E', {\vec p}~')$ and $p_2'=(E',-{\vec p}~')$ in the final states. Moreover, \begin{equation} \begin{array}{llll} \vec q &\equiv& {\vec p}~' - \vec p & \mbox{ is the momentum transfer},\\ \vec k &\equiv& \frac12 ({\vec p}~' + \vec p) & \mbox{ the average momentum},\\ \vec L & = & -i(\vec q \times \vec k) = -i({\vec p}~' \times \vec p) & \mbox{the total orbital angular momentum in momentum space}, \\ S_{12}(\hat q) & \equiv & 3 \vec \sigma_1 \cdot \hat q \,\,\: \vec \sigma_2 \cdot \hat q - \vec \sigma_1 \cdot \vec \sigma_2 & \mbox{the tensor operator in momentum space}. \end{array} \tag{8} \end{equation}
The characteristic properties of the contributions from the various mesons derived in Figure 5 to Figure 8 are summarized in Table 1.
The OBE $NN$ potential (OBEP) is defined as the sum over the contributions from the four mesons discussed and, typically, a few more, e.g., \begin{equation} V_{\rm OBE} = \sum_{\alpha=\pi,\sigma,\omega,\rho,\eta,a_0} V_\alpha \,. \tag{9} \end{equation}
Meson | Central | Spin-Spin | Tensor | Spin-Orbit |
---|---|---|---|---|
$\boldsymbol{\pi(138)}$ | --- | weak, long-ranged | strong, long-ranged | --- |
$\boldsymbol{\sigma(500)}$ | strong, attractive, intermediate-ranged | --- | --- | moderate, intermediate-ranged |
$\boldsymbol{\omega(782)}$ | strong, repulsive, short-ranged | --- | --- | strong, short-ranged, coherent with $\sigma$ |
$\boldsymbol{\rho(770)}$ | --- | weak, short-ranged, coherent with $\pi$ | moderate, short-ranged, opposite to $\pi$ | --- |
OBEP in position space
The momentum-space potentials derived in the previous section can be Fourier transformed, \begin{equation} V_\alpha ({\vec r})=\frac{1}{(2\pi)^{3}}\int d^{3}q \; e^{i{\vec q \cdot \vec r}} \; V_\alpha ({\vec q}) \,, \end{equation} to obtain the following equivalent position space potentials: \begin{eqnarray} V_{\pi}({\vec r}) & = & \frac{1}{3}\frac{f_{\pi NN}^{2}}{4\pi} m_{\pi} \left\{ \left[Y(m_{\pi}r)-\frac{4\pi}{m_{\pi}^{3}}\delta^{(3)}({\vec r}) \right] {\vec \sigma}_{1} \cdot {\vec \sigma}_{2} + \left(1+\frac{3}{m_{\pi}r}+\frac{3}{(m_{\pi}r)^{2}}\right)Y(m_{\pi}r) S_{12}(\hat r) \right\} \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \,, \\ \\ \\ \\ \\ V_{\sigma}({\vec r}) & = & -\frac{g^{2}_{\sigma}}{4\pi} m_{\sigma} \left\{ \left[ 1-\frac{1}{4} \left( \frac{m_{\sigma}}{M} \right)^{2} \right] Y(m_{\sigma}r) +\frac{1}{4M^{2}} \left[ {\vec \nabla}^{2}Y(m_{\sigma}r)+Y(m_{\sigma}r){\vec \nabla}^{2} \right] \right. \nonumber \\ & & \left. +\frac{1}{2} \left( \frac{m_{\sigma}}{M} \right)^{2} \left( \frac{1}{m_{\sigma}r}+\frac{1}{(m_{\sigma}r)^{2}} \right) Y(m_{\sigma}r) { \vec L \cdot \vec S} \right\} \,, \\ \\ \\ \\ \\ V_{\omega}({\vec r}) & = & \frac{g^{2}_{\omega}}{4\pi} m_{v} \left[ Y(m_{\omega}r) -\frac{3}{2} \left(\frac{m_{\omega}}{M} \right)^{2} \left(\frac{1}{m_{\omega}r}+\frac{1}{(m_{\omega}r)^{2}}\right)Y(m_{\omega}r) {\vec L \cdot \vec S} \right] \,, \\ \\ \\ \\ \\ V_{\rho}({\vec r}) & = & \frac{1}{12} \frac{f_{\rho}^{2}}{4\pi} \left( \frac{m_{\rho}}{M} \right)^{2} m_{\rho} \left\{ 2 \left[Y(m_{\rho}r)-\frac{4\pi}{m_{\rho}^{3}}\delta^{(3)}({\vec r}) \right] \vec \sigma_{1} \cdot \vec \sigma_{2} - \left(1+\frac{3}{m_{\rho}r}+\frac{3}{(m_{\rho}r)^{2}} \right)Y(m_{\rho}r) S_{12}(\hat r) \right\} \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \,, \end{eqnarray} with the "Yukawa function" \begin{eqnarray} Y(x) & = & e^{-x}/x\\ \end{eqnarray} and $S_{12}(\hat r)$ given in Eq. (2).
We can now better understand the success of the OBE model by summarizing its properties (cf. Figure 9):
- The pseudo-scalar pion with a mass of about 138 MeV is the lightest meson and provides the longest-ranged part of the $NN$ potential and the essential part of the tensor force.
- The vector meson $\boldsymbol{\rho}$ (770 MeV) cuts down the tensor force created by the pion at short distances to arrive at a tensor force of realistic strength.
- The scalar $\boldsymbol{\sigma}$ boson of about 500 MeV provides the intermediate-range attraction necessary for nuclear binding.
- The vector meson $\boldsymbol{\omega}$ (782 MeV) produces a strong repulsive central force of short range (repulsive core) and the essential part of the nuclear spin-orbit force.
This takes care of all the empirical properties of the nuclear force discussed in an earlier section and, therefore, allows for a quantitative description of the $NN$ system.
Relativistic OBEPs
The first OBEPs ever developed were derived along the lines we followed above: one starts from the Feynman amplitude of an OBE and then performs a $Q/M$ expansion up to terms of order $(Q/M)^2$. The motivation for this procedure was twofold: First, one wanted to see in a simple way what force components (e.g., central, spin-spin, tensor, spin-orbit, ...) were created by the exchange of different mesons. Second, early researchers preferred a local potential in position space, i.e., an analytic expression for the potential that is a function of the relative distance between the two nucleons, $\vec r$. For this it is necessary that the Fourier transform of the momentum space expressions can be performed analytically.
However, there is no need to perform calculations in position space. Equally well and often in a more elegant way, calculations of $NN$ scattering and nuclear bound states can be carried out in momentum space. Locality is not an issue and presents no advantages in momentum space. The original relativistic Feyman amplitudes of OBE are functions of $p'$ and $p$ and the relativistic OBEP is defined as \begin{equation} V_{\rm OBE}^{\rm rel}(p',p) = i \sum_{\alpha} F_\alpha(p',p) \end{equation} with $F_\alpha(p',p)$ as in Figure 4, evaluated in full beauty and without approximations. Two-nucleon scattering is described covariantly by the Bethe-Salpeter equation (Salpeter and Bethe, 1951), for which $V_{\rm OBE}^{\rm rel}$ is input. Since the four-dimensional Bethe-Salpeter (BS) equation is difficult to solve (Fleischer and Tjon, 1975), relativistic three-dimensional (3D) reductions of the BS equation are frequently used, which are more amenable to numerical solution. It is common to the derivation of all relativistic three-dimensional equations that the time component of the relative momentum is fixed in some covariant way, so that it no longer appears as a separate variable. Thus, \begin{equation} V_{\rm OBE}^{\rm rel}(p',p) \longmapsto V_{\rm OBE}^{\rm rel3D}(\vec p',\vec p) \,. \end{equation} Relativistic 3D OBEPs have been developed by Franz Gross (1969), (Gross and Stadler, 2008); the Bonn group (Erkelenz, 1974), (Holinde and Machleidt, 1975), (Machleidt, 2001); and others.
Beyond the OBE approximation
The OBE model is a great simplification of the complicated scenario of a meson theory for the $NN$ interaction. Therefore, in spite of the quantitative success of the OBEPs, we should be concerned about the approximations involved in the model. Major concerns include:
- The scalar isoscalar $\sigma$ 'meson' of about 500 MeV.
- The neglect of all non-iterative diagrams.
- The role of meson-nucleon resonances.
Two pions, when 'in the air', can interact strongly. When in a relative $P$-wave $(L=1)$, they form a proper resonance, the $\rho$ meson. They can also interact in a relative $S$-wave $(L=0)$, which gives rise to the $\sigma$ boson. Whether the $\sigma$ is a proper resonance is controversial, even though the Particle Data Group lists an $f_0(500)$ or $\sigma(500)$ meson, but with a width 400-700 MeV. What's for sure is that two pions have correlations, and if one doesn't believe in the $\sigma$ as a two pion resonance, then one has to take these correlations into account. There are essentially two approaches to take care of the two-pion exchange contribution to the $NN$ interaction (which generates the intermediate range attraction): dispersion theory and field theory.
The dispersion-theoretic picture is described schematically in Figure 10. In this approach one assumes that the total diagram (a) can be analysed in terms of two 'halves' (b). The hatched ovals stand for all possible processes which two pions and a nucleon can undergo. This is made more explicit in (d) and (e). The hatched boxes represent all possible baryon intermediate states including the nucleon. (Note that there are also crossed exchanges which are not shown.) The shaded circle stands for $\pi \pi$ scattering. Quantitatively, these processes are taken into account by using empirical information from $\pi N$ and $\pi \pi$ scattering (e. g. phase shifts) which represents the input for such a calculation. Dispersion relations then provide an on-shell $NN$ amplitude, which --- with some kind of plausible prescription --- is represented as a potential. The Paris potential (Lacombe et al., 1980) is constructed along these lines complemented by one-pion-exchange and $\omega$-exchange. For further details, we refer the interested reader to a pedagogical article written by Vinh Mau (1979).
A field-theoretic model for the $2\pi$-exchange contribution is shown in Figure 11. The model includes contributions from isobars as well as from $\pi \pi$ correlations. This can be understood in analogy to the dispersion relations picture. In general, only the lowest-lying $\pi N$ resonance, the so-called $\Delta$ isobar (spin 3/2, isospin 3/2, mass 1232 MeV), is taken into account. The contributions from other resonances have proven to be small for the low-energy $NN$ processes under consideration. A field-theoretic model treats the $\Delta$ isobar as an elementary (Rarita-Schwinger) particle. The six upper diagrams of Figure 11 represent uncorrelated $2\pi$ exchange. The crossed (non-iterative) two-particle exchanges (second diagram in each row) are important. They guarantee the proper (very weak) isospin dependence due to characteristic cancelations in the isospin dependent parts of box and crossed box diagrams. Furthermore, their contribution is about as large as the one from the corresponding box diagrams (iterative diagrams); therefore, they are not negligible. In addition to the processes discussed, also correlated $2\pi$ exchange has to be included (lower two rows of Figure 11). Quantitatively, these contributions are about as large as those from the uncorrelated processes.
Besides the contributions from two pions, there are also contributions from the combination of other mesons. The combination of $\pi$ and $\rho$ is particularly significant, Figure 12. This contribution is repulsive and important to suppress the 2$\pi$ exchange contribution at high momenta (or small distances), which is too strong by itself.
The most developed meson-theoretic model for the $NN$ interaction is the Bonn Full Model (Machleidt et al., 1987), which includes all the diagrams displayed in Figure 11 and Figure 12 plus single $\pi$ and $\omega$ exchange. Besides this, the Bonn Full Model also contains non-iterative graphs of $\pi$ and $\sigma$ and $\pi$ and $\omega$ exchanges. However, the combined contribution of the latter two groups of diagrams is small and not significant.
Having highly sophisticated models at hand, like the Paris and the Bonn potentials, allows to check the approximations made in the simple OBE model. As it turns out, the highly complicated 2$\pi$ exchange contributions to the $NN$ interaction tamed by the $\pi\rho$ diagrams can well be simulated by the single scalar isoscalar boson, the $\sigma$, with a mass around 550 MeV. Retroactively, this fact provides justification for the simple OBE model.
Noteworthy examples
Noteworthy examples for meson-theoretic $NN$ potentials are:
- Bryan and Scott (1964, 1967, 1969), position-space OBEPs, among the first.
- Early Bonn potentials (Erkelenz, 1974), (Holinde and Machleidt, 1975), relativistic momentum-space OBEPs.
- Early Nijmegen potential (Nagels et al., 1978), position-space OBEP.
- Paris potential (Lacombe et al., 1980), position-space potential, $2\pi$ from dispersion theory plus $\pi$ and $\omega$.
- Bonn Full Model, (Machleidt et al, 1987), most comprehensive meson-theoretic model for the $NN$ interaction ever developed.
- High-precision Nijmegen potential (Stoks et al., 1994), charge-dependent position-space OBEP, $\chi^2/{\rm datum} \approx 1$ for fit of $NN$ data.
- High-precision Bonn potential (CD-Bonn) (Machleidt, 2001), charge-dependent momentum-space OBEP, $\chi^2/{\rm datum} \approx 1$ for fit of $NN$ data.
- Gross and Stadler (2008), relativistic OBEP using the Gross equation, accurate fit of $np$ data.
For a pedagogical review of the meson-theory of nuclear forces, see (Machleidt, 1989).
This could have been the happy end of the theory of nuclear forces. However, with the rise of QCD to the ranks of the authoritative theory of strong interactions, meson theory is demoted to the lower level of a model (even though a beautiful one), and we have to start all over again---in the next section.
QCD and the nuclear force
Quantum chromodynamics (QCD) is the theory of strong interactions. It deals with quarks, gluons and their interactions and is part of the Standard Model of Particle Physics. QCD is a non-Abelian gauge field theory with color $SU(3)$ the underlying gauge group. The non-Abelian nature of the theory has dramatic consequences. While the interaction between colored objects is weak at short distances or high momentum transfer (asymptotic freedom), it is strong at long distances ($\geq 1$fm) or low energies, leading to the confinement of quarks into colorless objects, the hadrons. Consequently, QCD allows for a perturbative analysis at large energies, whereas it is highly non-perturbative in the low-energy regime. Nuclear physics resides at low energies and the force between nucleons is a residual color interaction similar to the van der Waals force between neutral molecules. Therefore, in terms of quarks and gluons, the nuclear force is a very complicated problem that, nevertheless, can be attacked with brute computing power on a discretized, Euclidean space-time lattice (known as lattice QCD). In a recent study (Beane et al., 2006), the neutron-proton scattering lengths in the singlet and triplet $S$-waves have been determined in fully dynamical lattice QCD, with a smallest pion mass of 354 MeV. This result is then extrapolated to the physical pion mass with the help of chiral perturbation theory. The pion mass of 354 MeV is still too large to allow for reliable extrapolations, but the feasibility has been demonstrated and more progress can be expected for the near future. In a lattice calculation of a very different kind, the nucleon-nucleon ($NN$) potential was studied (Hatsuda, 2012). The central part of the potential shows a repulsive core plus attraction of intermediate range. This is a very promising result, but it must be noted that also in this investigation still rather large pion masses are being used. In any case, advanced lattice QCD calculations are under way and continuously improved. However, since these calculations are very time-consuming and expensive, they can only be used to check a few representative key-issues. For everyday nuclear structure physics, a more efficient approach is needed.
The chiral effective field theory approach to nuclear forces
The efficient approach is an effective field theory (EFT). For the development of an EFT, it is crucial to identify a separation of scales. In the hadron spectrum, a large gap between the masses of the pions and the masses of the vector mesons, like $\rho(770)$ and $\omega(782)$, can clearly be identified. Thus, it is natural to assume that the pion mass sets the "soft scale", $Q \sim m_\pi$, and the rho mass the "hard scale", $\Lambda_\chi \sim m_\rho$, also known as the chiral-symmetry breaking scale. This is suggestive of considering an expansion in terms of the soft scale over the hard scale, $Q/\Lambda_\chi$. Concerning the relevant degrees of freedom, we noticed already that, for the ground state and the low-energy excitation spectrum of an atomic nucleus as well as for conventional nuclear reactions, quarks and gluons are ineffective degrees of freedom, while nucleons and pions are the appropriate ones (this may include also low-lying nucleon resonances, s. below). To make sure that this EFT is not just another phenomenology, it must have a firm link with QCD. The link is established by having the EFT observe all relevant symmetries of the underlying theory. This requirement is based upon a "folk theorem" by Weinberg (Weinberg, 1979):
If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition, and the assumed symmetry principles.
In summary, the EFT program consists of the following steps:
- Identify the soft and hard scales, and the degrees of freedom appropriate for (low-energy) nuclear physics.
- Identify the relevant symmetries of low-energy QCD and investigate if and how they are broken.
- Construct the most general Lagrangian consistent with those symmetries and symmetry breakings.
- Design an organizational scheme that can distinguish between more and less important contributions: a low-momentum expansion.
- Guided by the expansion, calculate Feynman diagrams for the problem under consideration to the desired accuracy.
Important papers on the subject are (Weinberg, 1991), (Ordonez et al., 1994), (Kaiser et al., 1997), (Epelbaum et al., 2009). A review with a very pedagogical introduction can be found in (Machleidt and Entem, 2011).
Note that we are presenting here the chiral EFT which scales with the $\rho$-mass and includes pion degrees of freedom. However, the $\rho$-mass is not the only possible (hard) scale. Depending on what energies we are interested in, other scales may be more appropriate. For example, if we wish to focus on a nuclear scenario at very low energy (< $m_\pi$), then it is suggestive to choose the pion-mass, $m_\pi$, as scale. Such EFT can, of course, not have pion degrees of freedom anymore and is, therefore, known as pionless EFT. It consists of contact terms, only. Thus, this pionless EFT may appear forbiddingly simplistic at first glance, but, as it turns out, it has some surprisingly intriguing features, particularly, with regard to renormalization and power counting (s. below). Since it is beyond the scope of this article to discuss the pionless EFT in detail, we like to refer the interested reader to the excellent review articles by Bedaque and van Kolck (2002) and van Kolck (2007).
In the following subsections, we will elaborate on the above-listed steps, one by one. Since we discussed the first step already, we will address now step two.
Symmetries of low-energy QCD and their breakings
The QCD Lagrangian reads ${\cal L}_{\rm QCD} = \bar{q} (i \gamma^\mu {\cal D}_\mu - {\cal M})q - \frac14 {\cal G}_{\mu\nu,a} {\cal G}^{\mu\nu}_{a} \tag{10}$ with the gauge-covariant derivative \begin{equation} {\cal D}_\mu = \partial_\mu -ig\frac{\lambda_a}{2} {\cal A}_{\mu,a} \tag{11} \end{equation} and the gluon field strength tensor (note that for $SU(N)$ group indices, we use Latin letters, $\ldots,a,b,c,\ldots,i,j,k,\dots$, and, in general, do not distinguish between subscripts and superscripts) \begin{equation} {\cal G}_{\mu\nu,a} = \partial_\mu {\cal A}_{\nu,a} -\partial_\nu {\cal A}_{\mu,a} + g f_{abc} {\cal A}_{\mu,b} {\cal A}_{\nu,c} \,. \tag{12} \end{equation} In the above, $q$ denotes the quark fields and ${\cal M}$ the quark mass matrix. Further, $g$ is the strong coupling constant and ${\cal A}_{\mu,a}$ are the gluon fields. The $\lambda_a$ are the Gell-Mann matrices and the $f_{abc}$ the structure constants of the $SU(3)_{\rm color}$ Lie algebra $(a,b,c=1,\dots ,8)$; summation over repeated indices is always implied. The gluon-gluon term in the last equation arises from the non-Abelian nature of the gauge theory and is the reason for the peculiar features of the color force.
The masses of the up $(u)$, down $(d)$, and strange (s) quarks are (PDG, 2012): \begin{eqnarray} m_u &=& 2.3\pm 0.7 \mbox{ MeV} , \tag{13} \\ m_d &=& 4.8\pm 0.7 \mbox{ MeV} , \tag{14} \\ m_s &=& 95\pm 5 \mbox{ MeV} . \tag{15} \end{eqnarray} These masses are small as compared to a typical hadronic scale, i.e., a scale of low-mass hadrons which are not Goldstone bosons, e.g., $m_\rho=0.78 \mbox{ GeV} \approx 1 \mbox{ GeV}$.
It is therefore of interest to discuss the QCD Lagrangian in the limit of vanishing quark masses: \begin{equation} {\cal L}_{\rm QCD}^0 = \bar{q} i \gamma^\mu {\cal D}_\mu q - \frac14 {\cal G}_{\mu\nu,a} {\cal G}^{\mu\nu}_{a} \,. \end{equation} Defining right- and left-handed quark fields, \begin{equation} q_R=P_Rq \,, \;\;\; q_L=P_Lq \,, \end{equation} with \begin{equation} P_R=\frac12(1+\gamma_5) \,, \;\;\; P_L=\frac12(1-\gamma_5) \,, \end{equation} we can rewrite the Lagrangian as follows: \begin{equation} {\cal L}_{\rm QCD}^0 = \bar{q}_R i \gamma^\mu {\cal D}_\mu q_R +\bar{q}_L i \gamma^\mu {\cal D}_\mu q_L - \frac14 {\cal G}_{\mu\nu,a} {\cal G}^{\mu\nu}_{a} \, . \end{equation} This equation reveals that the right- and left-handed components of massless quarks do not mix in the QCD Lagrangian. For the two-flavor case, this is $SU(2)_R\times SU(2)_L$ symmetry, also known as chiral symmetry. However, this symmetry is broken in two ways: explicitly and spontaneously.
Explicit symmetry breaking
The mass term $- \bar{q}{\cal M}q$ in the QCD Lagrangian Eq. (10) breaks chiral symmetry explicitly. To better see this, let's rewrite ${\cal M}$ for the two-flavor case, \begin{eqnarray} {\cal M} & = & \left( \begin{array}{cc} m_u & 0 \\ 0 & m_d \end{array} \right) \nonumber \\ & = & \frac12 (m_u+m_d) \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) + \frac12 (m_u-m_d) \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \nonumber \\ & = & \frac12 (m_u+m_d) \; I + \frac12 (m_u-m_d) \; \tau_3 \,. \tag{16} \end{eqnarray} The first term in the last equation in invariant under $SU(2)_V$ (isospin symmetry) and the second term vanishes for $m_u=m_d$. Thus, isospin is an exact symmetry if $m_u=m_d$. However, both terms in Eq. (16) break chiral symmetry. Since the up and down quark masses [Eqs. (13) and (14)] are small as compared to the typical hadronic mass scale of $\sim 1$ GeV, the explicit chiral symmetry breaking due to non-vanishing quark masses is very small.
Spontaneous symmetry breaking
A (continuous) symmetry is said to be spontaneously broken if a symmetry of the Lagrangian is not realized in the ground state of the system. There is evidence that the (approximate) chiral symmetry of the QCD Lagrangian is spontaneously broken---for dynamical reasons of nonperturbative origin which are not fully understood at this time. The most plausible evidence comes from the hadron spectrum.
From chiral symmetry, one naively expects the existence of degenerate hadron multiplets of opposite parity, i.e., for any hadron of positive parity one would expect a degenerate hadron state of negative parity and vice versa. However, these 'parity doublets' are not observed in nature. For example, take the $\rho$-meson which is a vector meson of negative parity ($J^P=1^-$) and mass 776 MeV. There does exist a $1^+$ meson, the $a_1$, but it has a mass of 1230 MeV and, therefore, cannot be perceived as degenerate with the $\rho$. On the other hand, the $\rho$ meson comes in three charge states (equivalent to three isospin states), the $\rho^\pm$ and the $\rho^0$, with masses that differ by at most a few MeV. Thus, in the hadron spectrum, $SU(2)_V$ (isospin) symmetry is well observed, while axial symmetry is broken: $SU(2)_R\times SU(2)_L$ is broken down to $SU(2)_V$.
A spontaneously broken global symmetry implies the existence of (massless) Goldstone bosons. The Goldstone bosons are identified with the isospin triplet of the (pseudoscalar) pions, which explains why pions are so light. The pion masses are not exactly zero because the up and down quark masses are not exactly zero either (explicit symmetry breaking). Thus, pions are a truly remarkable species: they reflect spontaneous as well as explicit symmetry breaking. Goldstone bosons interact weakly at low energy. They are degenerate with the vacuum and, therefore, interactions between them must vanish at zero momentum and in the chiral limit ($m_\pi \rightarrow 0$).
Chiral effective Lagrangians
The next step in the EFT program is to build the most general Lagrangian consistent with the (broken) symmetries discussed above. An elegant formalism for the construction of such Lagrangians was developed by Callan et al. (1969), who worked out the group-theoretical foundations of non-linear realizations of chiral symmetry. It is characteristic for these non-linear realizations that, whenever functions of the Goldstone bosons appear in the Langrangian, they are always accompanied with at least one space-time derivative. The Lagrangians given below are built upon the Callan et al. (1969) formalism.
As discussed, the relevant degrees of freedom are pions (Goldstone bosons) and nucleons. Since the interactions of Goldstone bosons must vanish at zero momentum transfer and in the chiral limit ($m_\pi \rightarrow 0$), the low-energy expansion of the Lagrangian is arranged in powers of derivatives and pion masses. The hard scale is the chiral-symmetry breaking scale, $\Lambda_\chi \approx 1$ GeV. Thus, the expansion is in terms of powers of $Q/\Lambda_\chi$ where $Q$ is a (small) momentum or pion mass. This is the essence of chiral perturbation theory (ChPT).
The effective Lagrangian can formally be written as, \begin{equation} {\cal L_{\rm eff}} = {\cal L}_{\pi\pi} + {\cal L}_{\pi N} + {\cal L}_{NN} + \, \ldots \,, \end{equation} where ${\cal L}_{\pi\pi}$ deals with the dynamics among pions, ${\cal L}_{\pi N}$ describes the interaction between pions and a nucleon, and ${\cal L}_{NN}$ contains two-nucleon contact interactions which consist of four nucleon-fields (four nucleon legs) and no meson fields. The ellipsis stands for terms that involve two nucleons plus pions and three or more nucleons with or without pions, relevant for nuclear many-body forces (cf. last two terms of Eq. (20), below). The individual Lagrangians are organized as follows: \begin{equation} {\cal L}_{\pi\pi} = {\cal L}_{\pi\pi}^{(2)} + {\cal L}_{\pi\pi}^{(4)} + \ldots \,, \end{equation} \begin{equation} {\cal L}_{\pi N} = {\cal L}_{\pi N}^{(1)} + {\cal L}_{\pi N}^{(2)} + {\cal L}_{\pi N}^{(3)} + \ldots , \end{equation} and \begin{equation} \tag{17} {\cal L}_{NN} = {\cal L}^{(0)}_{NN} + {\cal L}^{(2)}_{NN} + {\cal L}^{(4)}_{NN} + \ldots \,, \end{equation} where the superscript refers to the number of derivatives or pion mass insertions (chiral dimension) and the ellipsis stands for terms of higher dimensions.
Above, we have organized the Lagrangians by the number of derivatives or pion-mass insertions. This is the standard way, appropriate particularly for considerations of $\pi$-$\pi$ and $\pi$-$N$ scattering. As it turns out, for interactions among nucleons, it is sometimes more useful to consider the so-called index of the interaction, \begin{equation} \Delta \equiv d + \frac{n}{2} - 2 \, , \tag{18} \end{equation} where $d$ is the number of derivatives or pion-mass insertions and $n$ the number of nucleon field operators (nucleon legs). We will now write down the Lagrangian in terms of increasing values of the parameter $\Delta$ and we will do so using the so-called heavy-baryon (HB) formalism which we indicate by a "hat".
The lowest-index Lagrangian reads, \begin{eqnarray} \widehat{\cal L}^{\Delta=0} &=& \frac{1}{2} \partial_\mu \boldsymbol{\pi} \cdot \partial^\mu \boldsymbol{\pi} - \frac{1}{2} m_\pi^2 \boldsymbol{\pi}^2 \nonumber \\ && + \frac{1-4\alpha}{2f_\pi^2} (\boldsymbol{\pi} \cdot \partial_\mu \boldsymbol{\pi}) (\boldsymbol{\pi} \cdot \partial^\mu \boldsymbol{\pi}) - \frac{\alpha}{f_\pi^2} \boldsymbol{\pi}^2 \partial_\mu \boldsymbol{\pi} \cdot \partial^\mu \boldsymbol{\pi} +\; \frac{8\alpha-1}{8f_\pi^2} m_\pi^2 \boldsymbol{\pi}^4 \nonumber \\ && + \bar{N} \left[ i \partial_0 - \frac{g_A}{2f_\pi} \; \boldsymbol{\tau} \cdot ( \vec \sigma \cdot \vec \nabla ) \boldsymbol{\pi} - \frac{1}{4f_\pi^2} \; \boldsymbol{\tau} \cdot ( \boldsymbol{\pi} \times \partial_0 \boldsymbol{\pi}) \right] N \nonumber \\ && + \bar{N} \left\{ \frac{g_A(4\alpha-1)}{4f_\pi^3} \; (\boldsymbol{\tau} \cdot \boldsymbol{\pi}) \left[ \boldsymbol{\pi} \cdot ( \vec \sigma \cdot \vec \nabla ) \boldsymbol{\pi} \right] + \frac{g_A\alpha}{2f_\pi^3} \; \boldsymbol{\pi}^2 \left[ \boldsymbol{\tau} \cdot ( \vec \sigma \cdot \vec \nabla ) \boldsymbol{\pi} \right] \right\} N \nonumber \\ && -\frac{1}{2} C_S \bar{N} N \bar{N} N -\frac{1}{2} C_T (\bar{N} \vec \sigma N) \cdot (\bar{N} \vec \sigma N) \; + \; \ldots \,, \tag{19} \end{eqnarray}
and higher-index Lagrangians are, \begin{eqnarray} \widehat{\cal L}^{\Delta=1} &=& \bar{N} \left\{ \frac{{\vec \nabla}^2}{2M_N} -\frac{ig_A}{4M_Nf_\pi} \boldsymbol{\tau} \cdot \left[\vec \sigma \cdot \left( \stackrel{\leftarrow}{\nabla} \partial_0 \boldsymbol{\pi} - \partial_0 \boldsymbol{\pi} \stackrel{\rightarrow}{\nabla} \right) \right] \right. \nonumber \\ && \left. - \frac{i}{8M_N f_\pi^2} \boldsymbol{\tau} \cdot \left[ \stackrel{\leftarrow}{\nabla} \cdot ( \boldsymbol{\pi} \times \vec\nabla \boldsymbol{\pi} ) - ( \boldsymbol{\pi} \times \vec\nabla \boldsymbol{\pi} ) \cdot \stackrel{\rightarrow}{\nabla} \right] \right\} N \nonumber \\ && + \bar{N} \left[ 4c_1m_\pi^2 -\frac{2 c_1}{f_\pi^2} \, m_\pi^2\, \boldsymbol{\pi}^2 \, + \, \left( c_2 - \frac{g_A^2}{8M_N}\right) \frac{1}{f_\pi^2} (\partial_0 \boldsymbol{\pi} \cdot \partial_0 \boldsymbol{\pi}) \right. \nonumber \\ && \left. + \, \frac{c_3}{f_\pi^2}\, (\partial_\mu \boldsymbol{\pi} \cdot \partial^\mu \boldsymbol{\pi}) - \, \left( c_4 + \frac{1}{4M_N} \right) \frac{1}{2f_\pi^2} \epsilon^{ijk} \epsilon^{abc} \sigma^i \tau^a (\partial^j \pi^b) (\partial^k \pi^c) \right] N \nonumber \\ && - \frac{D}{4f_\pi} (\bar{N}N) \bar{N} \left[ \boldsymbol{\tau} \cdot ( \vec \sigma \cdot \vec \nabla ) \boldsymbol{\pi} \right] N -\frac12 E (\bar{N}N) (\bar{N} \boldsymbol{\tau} N) \cdot (\bar{N} \boldsymbol{\tau} N) \; + \; \ldots \,, \tag{20} \\ \widehat{\cal L}^{\Delta=2} &=& \; {\cal L}^{(4)}_{\pi\pi} \; + \; \widehat{\cal L}^{(3)}_{\pi N} \; + \; \widehat{\cal L}^{(2)}_{NN} \; + \; \ldots \,, \tag{21} \\ \widehat{\cal L}^{\Delta=4} &=& \; \widehat{\cal L}^{(4)}_{NN} \; + \; \ldots \,, \tag{22} \end{eqnarray} where the ellipses represent terms that are irrelevant for the derivation of nuclear forces up to fourth order. The Lagrangian $\widehat{\cal L}^{(3)}_{\pi N}$ can be found in (Fettes et al., 2000) and the $NN$ contact Lagrangians are given below. The pion fields are denoted by $\boldsymbol{\pi}$ and the heavy baryon nucleon field by $N$ ($\bar{N}=N^\dagger$). Furthermore, $g_A$, $f_\pi$, $m_\pi$, and $M_N$ are the axial-vector coupling constant, pion decay constant, pion mass, and nucleon mass, respectively. Numerical values for these quantities will be given later. The $c_i$ are low-energy constants (LECs) from the dimension two $\pi N$ Lagrangian and $\alpha$ is a parameter that appears in the expansion of the pion fields, see (Machleidt and Entem, 2011) for more details. Results are independent of $\alpha$. The $\pi NN$ coupling constant, $f_{\pi NN}$, used in the derivation of the one-pion-exchange potential in Figure 5, is related to the above quantities by \begin{equation} \frac{f_{\pi NN}}{m_\pi} = \frac{g_A}{2 f_\pi} \,, \tag{23} \end{equation} cf. Eq. (33) below.
The lowest order (or leading order) $NN$ Lagrangian has no derivatives and reads \begin{equation} \widehat{\cal L}^{(0)}_{NN} = -\frac{1}{2} C_S \bar{N} N \bar{N} N -\frac{1}{2} C_T (\bar{N} \vec \sigma N) \cdot (\bar{N} \vec \sigma N) \, , \tag{24} \end{equation} where $C_S$ and $C_T$ are unknown constants which are determined by a fit to the $NN$ data.
The second order $NN$ Lagrangian can be stated as follows, \begin{eqnarray} \widehat{\cal L}^{(2)}_{NN} &=& -C'_1 \left[(\bar{N} \vec \nabla N)^2+ (\overline{\vec \nabla N} N)^2 \right] -C'_2 (\bar{N} \vec \nabla N)\cdot (\overline{\vec \nabla N} N) %\nonumber \\ && -C'_3 \bar{N} N \left[\bar N \vec \nabla^2 N+\overline{\vec \nabla^2 N} N \right] \nonumber \\ && -i C'_4 \left[ \bar N \vec \nabla N \cdot (\overline{\vec \nabla N} \times \vec \sigma N) + \overline{(\vec \nabla N)} N \cdot (\bar N \vec \sigma \times \vec \nabla N) \right] \nonumber \\ && -i C'_5 \bar N N(\overline{\vec \nabla N} \cdot \vec \sigma \times \vec \nabla N) -i C'_6 (\bar N \vec \sigma N)\cdot (\overline{\vec \nabla N} \times \vec \nabla N) \nonumber \\ && -\left(C'_7 \delta_{ik} \delta_{jl}+C'_8 \delta_{il} \delta_{kj} +C'_9 \delta_{ij} \delta_{kl}\right) %\nonumber \\ && \times \left[\bar N \sigma_k \partial_i N \bar N \sigma_l \partial_j N + \overline{\partial_i N} \sigma_k N \overline{\partial_j N} \sigma_l N \right] \nonumber \\ && -\left(C'_{10} \delta_{ik} \delta_{jl}+C'_{11} \delta_{il} \delta_{kj}+C'_{12} \delta_{ij} \delta_{kl}\right) \bar N \sigma_k \partial_i N \overline{\partial_j N} \sigma_l N \nonumber \\ && -\left(\frac{1}{2} C'_{13} (\delta_{ik} \delta_{jl}+ \delta_{il} \delta_{kj}) %\nonumber \\ && +C'_{14} \delta_{ij} \delta_{kl} \right) \left[\overline{\partial_i N} \sigma_k \partial_j N + \overline{\partial_j N} \sigma_k \partial_i N\right] \bar N \sigma_l N \, . \tag{25} \end{eqnarray} Similar to $C_S$ and $C_T$ of Eq. (24), the $C'_i$ of Eq. (25) are unknown constants which are fixed by a fit to the $NN$ data. Obviously, these contact Lagrangians blow up quite a bit with increasing order, which is why we do not give $\widehat{\cal L}^{(4)}_{NN}$ explicitly here. The $NN$ contact potentials that emerge from these Lagrangians are given below.
Power counting
Effective Langrangians have infinitely many terms, and an unlimited number of Feynman graphs can be calculated from them. Therefore, we need a scheme that makes the theory manageable and calculable. This scheme which tells us how to distinguish between large (important) and small (unimportant) contributions is chiral perturbation theory (ChPT).
In ChPT, graphs are analyzed in terms of powers of small external momenta over the large scale: $(Q/\Lambda_\chi)^\nu$, where $Q$ is generic for a momentum (nucleon three-momentum or pion four-momentum) or a pion mass and $\Lambda_\chi \sim 1$ GeV is the chiral symmetry breaking scale (hadronic scale, hard scale). Determining the power $\nu$ has become known as power counting.
The nuclear potential is assembled from irreducible graphs. By definition, an irreducible graph is a diagram that cannot be separated into two by cutting only nucleon lines. Following the Feynman rules of covariant perturbation theory, a nucleon propagator is $Q^{-1}$, a pion propagator $Q^{-2}$, each derivative in any interaction is $Q$, and each four-momentum integration $Q^4$. This is also known as naive dimensional analysis. Applying then some topological identities, one obtains for the power of an irreducible diagram involving $A$ nucleons \begin{equation} \nu = -2 +2A - 2C + 2L + \sum_i \Delta_i \, , \tag{26} \end{equation} with \begin{equation} \Delta_i \equiv d_i + \frac{n_i}{2} - 2 \, , \tag{27} \end{equation} where $C$ denotes the number of separately connected pieces and $L$ the number of loops in the diagram; $d_i$ is the number of derivatives or pion-mass insertions and $n_i$ the number of nucleon fields (nucleon legs) involved in vertex $i$; the sum runs over all vertices $i$ contained in the diagram under consideration. Note that $\Delta_i \geq 0$ for all interactions allowed by chiral symmetry. Purely pionic interactions have at least two derivatives ($d_i\geq 2, n_i=0$); interactions of pions with a nucleon have at least one derivative ($d_i\geq 1, n_i=2$); and nucleon-nucleon contact terms ($n_i=4$) have $d_i\geq0$. This demonstrates how chiral symmetry guarantees a low-energy expansion.
The power formula Eq. (26) allows to predict the leading orders of connected multi-nucleon forces. Consider a $m$-nucleon irreducibly connected diagram ($m$-nucleon force) in an $A$-nucleon system ($m\leq A$). The number of separately connected pieces is $C=A-m+1$. Inserting this into Eq. (26) together with $L=0$ and $\sum_i \Delta_i=0$ yields $\nu=2m-4$. Thus, two-nucleon forces ($m=2$) start at $\nu=0$, three-nucleon forces ($m=3$) at $\nu=2$ (but they happen to cancel at that order), and four-nucleon forces at $\nu=4$ (they don't cancel). More about this in the next subsection.
For later purposes, we note that for an irreducible $NN$ diagram ($A=2$, $C=1$), the power formula collapses to the very simple expression \begin{equation} \nu = 2L + \sum_i \Delta_i \,. \tag{28} \end{equation}
In summary, the chief point of the ChPT expansion is that, at a given order $\nu$, there exists only a finite number of graphs. This is what makes the theory calculable. The expression $(Q/\Lambda_\chi)^{\nu+1}$ provides a rough estimate of the relative size of the contributions left out and, thus, of the accuracy at order $\nu$. In this sense, the theory can be calculated to any desired accuracy and has predictive power.
The hierarchy of nuclear forces: Overview
Chiral perturbation theory and power counting imply that nuclear forces emerge as a hierarchy controlled by the power $\nu$, Figure 13.
In lowest order, better known as leading order (LO, $\boldsymbol{\nu = 0}$), the $NN$ potential is made up by two momentum-independent contact terms ($\sim Q^0$), represented by the four-nucleon-leg graph with a small dot shown in the first row of Figure 13. Furthermore, there is the static one-pion exchange (1PE), second diagram in the first row of the figure. This is, of course, a rather rough approximation to the two-nucleon force (2NF), but accounts already for some important features. The 1PE provides the tensor force, necessary to describe the deuteron, and it explains $NN$ scattering in peripheral partial waves of very high orbital angular momentum. At this order, the two contacts, which contribute only in $S$-waves, provide the short- and intermediate-range interaction, which is somewhat crude.
In the next order, $\nu=1$, all contributions vanish due to parity and time-reversal invariance.
Therefore, the next-to-leading order (NLO) is $\boldsymbol{\nu=2}$. Two-pion exchange (2PE) occurs for the first time ("leading 2PE") and, thus, the creation of a more sophisticated description of the intermediate-range interaction is starting here (Ordonez et al., 1996). Since the loop involved in each 2PE-diagram implies already $\nu=2$ [cf. Eq. (28)], the vertices must have $\Delta_i = 0$. Therefore, at this order, only the lowest order $\pi NN$ and $\pi \pi NN$ vertices are allowed which is why the leading 2PE is rather weak. Furthermore, there are seven new contact terms of ${\cal O}(Q^2)$, shown in the figure by the four-nucleon-leg graph with a solid square, which contribute in $S$ and $P$ waves. The operator structure of these contacts include a spin-orbit term besides central, spin-spin, and tensor terms. Thus, essentially all spin-isospin structures necessary to describe the two-nucleon force phenomenologically have been generated. The main deficiency at this stage of development is an insufficient intermediate-range attraction.
This problem is finally fixed at order three ($\boldsymbol{\nu=3}$), next-to-next-to-leading order (NNLO). The 2PE involves now the two-derivative $\pi\pi NN$ seagull vertices (proportional to the $c_i$ LECs) denoted by a large solid dot in Figure 13. These vertices represent correlated 2PE as well as intermediate $\Delta(1232)$-isobar contributions. It is well-known from the meson phenomenology of nuclear forces that these two contributions are crucial for a realistic and quantitative 2PE model. Consequently, the 2PE now assumes a realistic size and describes the intermediate-range attraction of the nuclear force about right (Ordonez et al., 1996). Moreover, first relativistic corrections come into play at this order. There are no new contacts, because contacts appear only at even orders.
The reason why we talk of a hierarchy of nuclear forces is that two- and many-nucleon forces are created on an equal footing and emerge in increasing number as we go to higher and higher orders. At NNLO, the first set of nonvanishing three-nucleon forces (3NFs) occur, cf. column '3N Force' of Figure 13 (van Kolck, 1994). In fact, at the previous order, NLO, irreducible 3N graphs appear already, however, it can be shown that these diagrams all cancel. Since nonvanishing 3NF contributions happen first at order $(Q/\Lambda_\chi)^3$, they are very weak as compared to the 2NF which starts at $(Q/\Lambda_\chi)^0$.
More 2PE is produced at $\boldsymbol{\nu =4}$, next-to-next-to-next-to-leading order (N$^3$LO), of which we show only a few symbolic diagrams in Figure 13. Two-loop 2PE graphs show up for the first time (Kaiser, 2001) and so does three-pion exchange (3PE) which necessarily involves two loops (Kaiser, 2000). The 3PE is negligible at this order. Most importantly, 15 new contact terms $\sim Q^4$ arise and are represented by the four-nucleon-leg graph with a solid diamond. They include a quadratic spin-orbit term and contribute up to $D$-waves.
Mainly due to the larger number of contact terms, a quantitative description of the two-nucleon interaction up to about 300 MeV lab. energy is possible, at N$^3$LO (Entem and Machleidt, 2003), (Epelbaum et al., 2005), (Machleidt and Entem, 2011). This is demonstrated in Figure 14, where we show the order by order improvement of the $NN$ phase shift predictions from LO to N$^3$LO. Table 2 quantifies this order by order improvement by providing the $\chi^2$/datum for the fit of $NN$ data from NLO to N$^3$LO.
$T_{\rm lab}$ bin (MeV) | number of $np$ data | N$^3$LO | NNLO | NLO |
---|---|---|---|---|
0--100 | 1058 | 1.05 | 1.7 | 4.5 |
100--190 | 501 | 1.08 | 22 | 100 |
190--290 | 843 | 1.15 | 47 | 180 |
0--290 | 2402 | 1.10 | 20 | 86 |
Moreover, there are more 3NF contributions at N$^3$LO, and four-nucleon forces (4NFs) start at this order. Since the leading 4NFs come into existence one order higher than the leading 3NFs, 4NFs are weaker than 3NFs. Thus, ChPT provides a straightforward explanation for the empirically known fact that 2NF $\gg$ 3NF $\gg$ 4NF $\ldots$.
Comparison with conventional meson theory
We have now two approaches at hand that can both describe the $NN$ interaction quantitatively, which is a non-trivial result. It is then natural to ask, in which way the two approaches differ. There is a clear and revealing answer.
In chiral EFT, the nuclear potential is expanded in terms of increasing powers of small momenta, $(Q/\Lambda_\chi)^\nu$ (Figure 13). In meson theory, the expansion is in terms of Yukawas, $Y(m_\alpha r)$, of increasing masses $m_\alpha$, corresponding to decreasing ranges $1/m_\alpha$ (cf. Figure 9).
Since both approaches describe the same complicated object quantitatively, they should be equivalent to a large extent. This is demonstrated in Figure 15. First, there is a 1PE in both cases, which is trivial. The 2PE may look diagrammatically quite different, but the figure shows the correspondence between the contributions. The main difference is that, in chiral EFT, the 2PE is build up order by order, while in conventional meson theory it comes as one set. Finally, the short-range contributions appear graphically very different with heavy boson exchange (like $\omega$-exchange) in meson theory and contacts in chiral EFT. However, since $Q \ll m_\omega \approx \Lambda_\chi$, the propagator of a heavy-meson can be expanded into a power series generating contacts of increasing order, as demonstrated in Figure 16.
Although the two approaches can be regarded as equivalent, there are arguments why chiral EFT may be perceived as superior. Chiral EFT
- is more closely connected to QCD via chiral symmetry;
- comes with an organizational scheme (power counting) that allows to estimate the accuracy of the predictions (at a given order);
- generates two- and many-body forces on an equal footing.
Two-nucleon forces: Doing the math
Here we will fill in the mathematical details we left out when presenting the overview over the chiral hierarchy. Up to N$^3$LO, the various irreducible diagrams, which were shown in Figure 13 and define the chiral $NN$ potential order by order, are given by: \begin{eqnarray} V_{\rm LO} & = & V_{\rm ct}^{(0)} + V_{1\pi}^{(0)} \tag{29} \\ V_{\rm NLO} & = & V_{\rm LO} + V_{\rm ct}^{(2)} + V_{1\pi}^{(2)} + V_{2\pi}^{(2)} \tag{30} \\ V_{\rm NNLO} & = & V_{\rm NLO} + V_{1\pi}^{(3)} + V_{2\pi}^{(3)} \tag{31} \\ V_{{\rm N}^3{\rm LO}} & = & V_{\rm NNLO} + V_{\rm ct}^{(4)} + V_{1\pi}^{(4)} + V_{2\pi}^{(4)} + V_{3\pi}^{(4)} \tag{32} \end{eqnarray} where the superscript denotes the order $\nu$ of the low-momentum expansion. Contact potentials carry the subscript "ct" and pion-exchange potentials can be identified by an obvious subscript.
The charge-independent 1PE potential reads \begin{equation} V_{1\pi} ({\vec p}~', \vec p) = - \frac{g_A^2}{4f_\pi^2} \: \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \: \frac{ \vec \sigma_1 \cdot \vec q \,\, \vec \sigma_2 \cdot \vec q} {q^2 + m_\pi^2} \,, \tag{33} \end{equation} Numerical vales for the constants are $f_\pi=92.4$ MeV, $g_A=1.29$; and $m_\pi=138$ MeV for the average pion mass (cf. Figure 5 and Eq. (23)). Since higher order corrections contribute only to mass and coupling constant renormalizations and since, on shell, there are no relativistic corrections, the on-shell 1PE has the same form as in Eq. (33) at all orders.
The two zero-order contact terms at LO are \begin{equation} V_{\rm ct}^{(0)}({\vec p}~',\vec{p}) = C_S + C_T \, \vec{\sigma}_1 \cdot \vec{\sigma}_2 \, . \tag{34} \end{equation}
To state the mathematical expressions for 2PE contributions, we use the following scheme: \begin{eqnarray} V_{2\pi}^{(\nu)}({\vec p}~', \vec p) & = & \:\, V_C^{(\nu)} \:\, + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_C^{(\nu)} \nonumber \\ &+& \left[ \, V_S^{(\nu)} \:\, + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_S^{(\nu)} \,\:\, \right] \, \vec\sigma_1 \cdot \vec \sigma_2 \nonumber \\ &+& \left[ \, V_{LS}^{(\nu)} + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_{LS}^{(\nu)} \right] \, \left(-i \vec S \cdot (\vec q \times \vec k) \,\right) \nonumber \\ &+& \left[ \, V_T^{(\nu)} \:\, + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_T^{(\nu)} \,\:\, \right] \, \vec \sigma_1 \cdot \vec q \,\, \vec \sigma_2 \cdot \vec q \nonumber \\ &+& \left[ \, V_{\sigma L}^{(\nu)} + \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 \, W_{\sigma L}^{(\nu)} \, \right] \, \vec\sigma_1\cdot(\vec q\times \vec k\,) \,\, \vec \sigma_2 \cdot(\vec q\times \vec k\,) \, , \tag{35} \end{eqnarray} where ${\vec p}~'$ and $\vec p$ denote the final and initial nucleon momenta in the CMS, respectively; moreover, \begin{equation} \begin{array}{llll} \vec q &\equiv& {\vec p}~' - \vec p & \mbox{ is the momentum transfer},\\ \vec k &\equiv& \frac12 ({\vec p}~' + \vec p) & \mbox{ the average momentum},\\ \vec S &\equiv& \frac12 (\vec\sigma_1+\vec\sigma_2) & \mbox{ the total spin}, \end{array} \tag{36} \end{equation} and $\vec \sigma_{1,2}$ and $\boldsymbol{\tau}_{1,2}$ are the spin and isospin operators, respectively, of nucleon 1 and 2. For on-energy-shell scattering, $V_i$ and $W_i$ $(i=C,S,LS,T,\sigma L)$ can be expressed as functions of $q$ and $k$, only (with $q\equiv |\vec q|$ and $k\equiv |\vec k|$).
Using the above scheme, the contribution from the five NLO 2PE diagrams can be stated in an amazingly compact form, namely, \begin{eqnarray} W_C^{(2)} &=&-{L(q)\over384\pi^2 f_\pi^4} \left[4m_\pi^2(5g_A^4-4g_A^2-1) +q^2(23g_A^4 -10g_A^2-1) + {48g_A^4 m_\pi^4 \over w^2} \right], \tag{37} \\ V_T^{(2)} &=& -{1\over q^2} V_{S}^{(2)} \; = \; -{3g_A^4 L(q)\over 64\pi^2 f_\pi^4} \,, \tag{38} \end{eqnarray} where \begin{equation} L(q) \equiv {w\over q} \ln {w+q \over 2m_\pi} \end{equation} and \begin{equation} w \equiv \sqrt{4m_\pi^2+q^2} \,. \end{equation}
Note that all 2$\pi$ loops involve a four-dimensional integral, which is divergent. Thus regularization is required. All 2$\pi$ contributions given in this subsection are obtained by applying dimensional regularization (DR). For a pedagogical introduction into DR, see Appendix A.2 of Scherrer (2003).
The seven NLO contact terms are: \begin{eqnarray} V_{\rm ct}^{(2)}({\vec p}~',\vec{p}) &=& C_1 \, q^2 + C_2 \, k^2 \nonumber \\ &+& \left( C_3 \, q^2 + C_4 \, k^2 \right) \vec{\sigma}_1 \cdot \vec{\sigma}_2 \nonumber \\ &+& C_5 \left( -i \vec{S} \cdot (\vec{q} \times \vec{k}) \right) \nonumber \\ &+& C_6 \, ( \vec{\sigma}_1 \cdot \vec{q} )\,( \vec{\sigma}_2 \cdot \vec{q} ) \nonumber \\ &+& C_7 \, ( \vec{\sigma}_1 \cdot \vec{k} )\,( \vec{\sigma}_2 \cdot \vec{k} ) \,. \tag{39} \end{eqnarray} The coefficients $C_i$ used here in the contact potential are, of course, related to the coefficients $C_i'$ that occur in the Lagrangian $\widehat{\cal L}^{(2)}_{NN}$, Eq. (25), but, the exact relationship is unimportant.
The NNLO 2PE is represented by the following expressions (Kaiser et al., 1997): \begin{eqnarray} V_C^{(3)} &=&{3g_A^2 \over 16\pi f_\pi^4} \left\{ {g_A^2 m_\pi^5 \over 16M_N w^2} -\left[2m_\pi^2( 2c_1-c_3)-q^2 \left(c_3 +{3g_A^2\over16M_N}\right) \right] \widetilde{w}^2 A(q) \right\} \,, \tag{40} \\ W_C^{(3)} &=& {g_A^2\over128\pi M_N f_\pi^4} \left\{ 3g_A^2 m_\pi^5 w^{-2} - \left[ 4m_\pi^2 +2q^2-g_A^2(4m_\pi^2+3q^2) \right] \widetilde{w}^2 A(q) \right\} \,,\\ V_T^{(3)} &=& -{1 \over q^2} V_{S}^{(3)} \; = \; {9g_A^4 \widetilde{w}^2 A(q) \over 512\pi M_N f_\pi^4} \,, \\ W_T^{(3)} &=&-{1\over q^2}W_{S}^{(3)} =-{g_A^2 A(q) \over 32\pi f_\pi^4} \left[ \left( c_4 +{1\over 4M_N} \right) w^2 -{g_A^2 \over 8M_N} (10m_\pi^2+3q^2) \right] \,, \tag{41} \\ V_{LS}^{(3)} &=& {3g_A^4 \widetilde{w}^2 A(q) \over 32\pi M_N f_\pi^4} \,,\\ W_{LS}^{(3)} &=& {g_A^2(1-g_A^2)\over 32\pi M_N f_\pi^4} w^2 A(q) \,, \tag{42} \end{eqnarray} with \begin{equation} A(q) \equiv {1\over 2q}\arctan{q \over 2m_\pi} \end{equation} and \begin{equation} \widetilde{w} \equiv \sqrt{2m_\pi^2+q^2} \,. \end{equation} This contribution to the 2PE is the crucial one, because it provides an intermediate-range attraction of proper strength. The iso-scalar central potential, $V_C^{(3)}$, is strong and attractive due to the LEC $c_3$, which is negative and of large magnitude. Via resonance saturation, $c_3$ is associated with $\pi$-$\pi$ correlations ('$\sigma$ meson') and virtual $\Delta$-isobar excitations, which create the most crucial contributions to 2PE in the frame work of conventional meson theory (Machleidt et al., 1987). First relativistic $1/M_N$ corrections come also into play at this order; they are included in the above potential expressions.
The contacts at N$^3$LO are: \begin{eqnarray} V_{\rm ct}^{(4)}(\vec{p'},\vec{p}) &=& D_1 \, q^4 + D_2 \, k^4 + D_3 \, q^2 k^2 + D_4 \, (\vec{q} \times \vec{k})^2 \nonumber \\ &+& \left( D_5 \, q^4 + D_6 \, k^4 + D_7 \, q^2 k^2 + D_8 \, (\vec{q} \times \vec{k})^2 \right) \vec{\sigma}_1 \cdot \vec{\sigma}_2 \nonumber \\ &+& \left( D_9 \, q^2 + D_{10} \, k^2 \right) \left( -i \vec{S} \cdot (\vec{q} \times \vec{k}) \right) \nonumber \\ &+& \left( D_{11} \, q^2 + D_{12} \, k^2 \right) ( \vec{\sigma}_1 \cdot \vec{q} )\,( \vec{\sigma}_2 \cdot \vec{q}) \nonumber \\ &+& \left( D_{13} \, q^2 + D_{14} \, k^2 \right) ( \vec{\sigma}_1 \cdot \vec{k} )\,( \vec{\sigma}_2 \cdot \vec{k}) \nonumber \\ &+& D_{15} \left( \vec{\sigma}_1 \cdot (\vec{q} \times \vec{k}) \, \, \vec{\sigma}_2 \cdot (\vec{q} \times \vec{k}) \right) . \tag{43} \end{eqnarray}
The 2PE potential at N$^3$LO, $V_{2\pi}^{(4)}$, is very involved, which is why will not give its expressions here. It can be found in (Machleidt and Entem, 2011). The N$^3$LO 3PE contributions, $V_{3\pi}^{(4)}$, are negligible.
The two-nucleon system is characterized by large scattering lengths and a shallow bound states (the deuteron), which cannot be calculated by perturbation theory. Therefore, the $NN$ potential must be inserted into a Schroedinger or Lippmann-Schwinger (LS) equation to obtain the $NN$ amplitude, \begin{equation} \widehat{T}({\vec p}~',{\vec p})= \widehat{V}({\vec p}~',{\vec p})+ \int d^3p''\: \widehat{V}({\vec p}~',{\vec p}~'')\: \frac{M_N} {{ p}^{2}-{p''}^{2}+i\epsilon}\: \widehat{T}({\vec p}~'',{\vec p}) \, , \tag{44} \end{equation} where the definitions \begin{equation} \widehat{V}({\vec p}~',{\vec p}) \equiv \frac{1}{(2\pi)^3} \sqrt{\frac{M_N}{E_{p'}}}\: {V}({\vec p}~',{\vec p})\: \sqrt{\frac{M_N}{E_{p}}} \tag{45} \end{equation} and \begin{equation} \widehat{T}({\vec p}~',{\vec p}) \equiv \frac{1}{(2\pi)^3} \sqrt{\frac{M_N}{E_{p'}}}\: {T}({\vec p}~',{\vec p})\: \sqrt{\frac{M_N}{E_{p}}} \,, \tag{46} \end{equation} are used, with $E_{p}\equiv \sqrt{M_N^2 + {p}^2}$. Iteration of $\widehat V$ in the LS equation, Eq. (44), requires cutting $\widehat V$ off for high momenta to avoid infinities. This is consistent with the fact that ChPT is a low-momentum expansion which is valid only for momenta $Q < \Lambda_\chi \approx 1$ GeV. Therefore, the potential $\widehat V$ is multiplied with a regulator function $f(p',p)$, \begin{equation} {\widehat V}(\vec{ p}~',{\vec p}) \longmapsto {\widehat V}(\vec{ p}~',{\vec p}) \, f(p',p) \end{equation} with, for example, \begin{equation} f(p',p) = \exp[-(p'/\Lambda)^{2n}-(p/\Lambda)^{2n}] \,. \tag{47} \end{equation} Typical choices for the cutoff parameter $\Lambda$ that appears in the regulator are $\Lambda \approx 0.5 \mbox{ GeV} < \Lambda_\chi \approx 1$ GeV.
It is pretty obvious that results for the $\widehat T$-matrix may then depend sensitively on the regulator and its cutoff parameter, which is undesirable. The removal of such regulator dependence is known as renormalization. Note that renormalizability is crucial for the validity of an EFT. The quantitative chiral $NN$ potentials currently in use (Entem and Machleidt, 2003) apply two regularization schemes: In the derivation of the potential, dimensional regularization is used (where the cutoff is taken to infinity), while in the LS equation, Eq. (44), the regulator Eq. (47) is applied with a finite cutoff. This scheme has produced useful $NN$ potentials, but the way regularization and renormalization is handled is controversial. In spite of almost two decades of research by a large variety of theoretical physicists, there is still no consensus in the community on how to conduct the renormalization of chiral EFT based nuclear forces in a satisfactory way. In this context, the pionless EFT has turned out to be enlightening, since it allows for a more transparent renormalization procedure (because it consists of contacts only and does not include pion loops).
A related unresolved issue is the proper counting of the powers of the low-energy expansion. Notice that the power given in Eq. (26) is the one of the perturbatively calculated potential. However, the quantity of physical relevance is the $\widehat T$-matrix, which is obtained by iterating the potential in the LS equation, Eq. (44), which may change the power. Also here the pionless theory has been helpful since, due to its simplicity, it allows for analytic solutions of the LS equation revealing the power explicitly. The rather involved issues of renormalization and (modified) power counting are beyond the scope of this introductory article. The interested reader is referred to the reviews by Bedaque and van Kolck (2002) and Machleidt and Entem (2011) for a more detailed discussion and a comprehensive list of references.
Three-nucleon forces
In microscopic calculations of nuclear structure and reactions, the 2NF makes, of course, the largest contribution. However, from ab-initio studies it is well-known that certain few-nucleon reactions and nuclear structure issues require 3NFs for their precise microscopic explanation. In short, we need 3NFs. As noted before, an important advantage of the EFT approach to nuclear forces is that it creates two- and many-nucleon forces on an equal footing (cf. Figure 13).
For a 3NF, we have $A=3$ and $C=1$ and, thus, Eq. (26) implies \begin{equation} \nu = 2 + 2L + \sum_i \Delta_i \,. \tag{48} \end{equation} We will use this equation to analyze 3NF contributions order by order.
The lowest possible power is obviously $\nu=2$ (NLO), which is obtained for no loops ($L=0$) and only leading vertices ($\sum_i \Delta_i = 0$). As it turns out, this contribution vanishes.
The first non-vanishing 3NF appears at NNLO $\boldsymbol{(\nu=3)}$. The power $\nu=3$ is obtained when there are no loops ($L=0$) and $\sum_i \Delta_i = 1$, i.e., $\Delta_i=1$ for one vertex while $\Delta_i=0$ for all other vertices. There are three topologies which fulfill this condition, known as the 2PE, 1PE, and contact graphs, Figure 17 (van Kolck, 1994), (Epelbaum et al., 2002).
The 2PE 3N-potential is derived to be \begin{equation} V^{\rm 3NF}_{\rm 2PE} = \left( \frac{g_A}{2f_\pi} \right)^2 \frac12 \sum_{i \neq j \neq k} \frac{ ( \vec \sigma_i \cdot \vec q_i ) ( \vec \sigma_j \cdot \vec q_j ) }{ ( q^2_i + m^2_\pi ) ( q^2_j + m^2_\pi ) } \; F^{ab}_{ijk} \; \tau^a_i \tau^b_j \tag{49} \end{equation} with $\vec q_i \equiv \vec{p_i}' - \vec p_i$, where $\vec p_i$ and $\vec{p_i}'$ are the initial and final momenta of nucleon $i$, respectively, and \begin{equation} F^{ab}_{ijk} = \delta^{ab} \left[ - \frac{4c_1 m^2_\pi}{f^2_\pi} + \frac{2c_3}{f^2_\pi} \; \vec q_i \cdot \vec q_j \right] + \frac{c_4}{f^2_\pi} \sum_{c} \epsilon^{abc} \; \tau^c_k \; \vec \sigma_k \cdot [ \vec q_i \times \vec q_j] \; . \tag{50} \end{equation} There are great similarities between this force and derivations of 2PE 3NFs from conventional meson theory (Fujita and Miyazawa, 1957; Coon et al., 1979).
The other two 3NF contributions are easily derived by taking the last two terms of the $\Delta=1$ Langrangian, Eq. (20), into account. The 1PE contribution is \begin{equation} V^{\rm 3NF}_{\rm 1PE} = -D \; \frac{g_A}{8f^2_\pi} \sum_{i \neq j \neq k} \frac{\vec \sigma_j \cdot \vec q_j}{ q^2_j + m^2_\pi } ( \boldsymbol{\tau}_i \cdot \boldsymbol{\tau}_j ) ( \vec \sigma_i \cdot \vec q_j ) \tag{51} \end{equation} and the 3N contact potential reads \begin{equation} V^{\rm 3NF}_{\rm ct} = E \; \frac12 \sum_{j \neq k} \boldsymbol{\tau}_j \cdot \boldsymbol{\tau}_k \; . \tag{52} \end{equation} These 3NF terms involve the two new parameters $D$ and $E$, which do not appear in the 2N problem. There are many ways to pin these two parameters down. Using the triton binding energy and the $nd$ doublet scattering length $^2a_{nd}$ is one possibility. One may also choose the binding energies of $^3$H and $^4$He or an optimal over-all fit of the properties of light nuclei. Once $D$ and $E$ are fixed, the results for other 3N, 4N, etc. observables are predictions.
The 3NF at NNLO has been applied in calculations of few-nucleon reactions, structure of light- and medium-mass nuclei, and nuclear and neutron matter with a good deal of success. Yet, the problem with the underprediction of the analyzing power of nucleon-deuteron and $p$-$^3$He scattering, which has become known as the '$A_y$ puzzle', is not resolved by this 3NF. Furthermore, the spectra of light nuclei leave room for improvement. Therefore, 3NFs of higher orders are needed for at least two reasons: to hopefully resolve outstanding problems in microscopic structure and reactions and for consistency with the 2NF (recall that a precise 2NF is of order N$^3$LO).
The next order is N$^3$LO $\boldsymbol{(\nu=4)}$, where we are faced with a very large number of loop diagrams (Figure 18). For those loops, $L$ is one and, therefore, all $\Delta_i$ have to be zero to ensure $\nu=4$. Thus, these one-loop 3NF diagrams can include only leading order vertices, the parameters of which are fixed from $\pi N$ and $NN$ analysis. There are five loop topologies. In Figure 18 we show one sample diagram for each topology. Note, however, that each topology consists of many diagrams such that the total number of diagrams is between 50 and 100, depending on how the diagrams are represented (Bernard et al., 2011). Preliminary applications of the 3N potentials derived from these diagrams indicate that the N$^3$LO 3NF is fairly weak and does not solve the $A_y$ puzzle (Witala et al., 2013).
Since we are dealing with a perturbation theory, it is natural to turn to the next order when looking for further improvements. Thus, we proceed to order N$^4$LO $\boldsymbol{(\nu=5)}$. The loop contributions that occur at this order (Figure 19) are obtained by replacing in the N$^3$LO loops one vertex by a $\Delta_i=1$ vertex (with LEC $c_i$)}, which is why these loops may be more sizable than the N$^3$LO loops. Again, there are five loop topologies, each of which consists of many diagrams. In addition, we have three 'tree' topologies (Figure 20) which include a new set of 3N contact interactions [graph (c)]. Contact terms are typically simple (as compared to loop diagrams) and their coefficients are unconstrained (except for naturalness). The N$^4$LO 3NF terms include all possible spin-isospin-momentum structures that a 3NF can have. Thus, there is hope that the 3NF at N$^4$LO may provide the missing pieces in the 3NF puzzle. However, a problem is how to deal with the explosion of 3NF contributions that emerge at N$^3$LO and N$^4$LO.
Four-nucleon forces
For connected ($C=1$) $A=4$ diagrams, Eq. (26) yields \begin{equation} \nu = 4 + 2L + \sum_i \Delta_i \,. \tag{53} \end{equation}
Therefore, a connected 4NF appears for the first time at $\nu = 4$ (N$^3$LO), with no loops and only leading vertices, Figure 21 (Epelbaum, 2007). This 4NF includes no new parameters and does not vanish. Some graphs in Figure 21 appear to be reducible (iterative). Note, however, that these are Feynman diagrams, which are best analyzed in terms of time-ordered perturbation theory. The various time-orderings include also some irreducible topologies (which are, by definition, 4NFs). Or, in other words, the Feynman diagram minus the reducible part of it yields the (irreducible) contribution to the 4NF.
Assuming a good rate of convergence, a contribution of order $(Q/\Lambda_\chi)^4$ is expected to be rather small. Thus, ChPT predicts 4NF to be essentially insignificant, consistent with experience. Still, nothing is fully proven in physics unless we have performed explicit calculations. The leading 4NF has been applied in a calculation of the $^4$He binding energy, where it was found to contribute about 0.1 MeV. This is small as compared to the full $^4$He binding energy of 28.3 MeV.
Introducing $\Delta$-isobar degrees of freedom
The lowest excited state of the nucleon is the $\Delta(1232)$ resonance or isobar (a $\pi$-$N$ $P$-wave resonance with both spin and isospin 3/2) with an excitation energy of $\Delta M=M_\Delta - M_N = 293$ MeV. Because of its strong coupling to the $\pi$-$N$ system and low excitation energy, it is an important ingredient for models of pion-nucleon scattering in the $\Delta$-region and pion production from the two-nucleon system at intermediate energies, where the particle production proceeds prevailingly through the formation of $\Delta$ isobars. At low energies, the more sophisticated conventional models for the 2$\pi$-exchange contribution to the $NN$ interaction include the virtual excitation of $\Delta$'s, which in these models accounts for about 50% of the intermediate-range attraction of the nuclear force---as demonstrated by the Bonn potential (Machleidt et al., 1987).
Because of its relatively small excitation energy, it is not clear from the outset if, in an EFT, the $\Delta$ should be taken into account explicitly or integrated out as a heavy degree of freedom. If it is included, then $\Delta M \sim m_\pi$ is considered as another small expansion parameter, besides the pion mass and small external momenta. This scheme has become known as the small scale expansion (SSE). Note, however, that this extension is of phenomenological character, since $\Delta M$ does not vanish in the chiral limit.
In the chiral EFT discussed so far (also known as the $\Delta$-less theory), the effects due to $\Delta$ isobars are taken into account implicitly. Note that the dimension-two LECs, $c_i$, have unnaturally large values. The reason for this is that the $\Delta$-isobar (and some meson resonances) contribute considerably to the $c_i$---a mechanism that has become known as resonance saturation. Therefore, the explicit inclusion of the $\Delta$ [the so-called $\Delta$-full theory (Ordonez et al., 1996), (Kaiser et al., 1998)] takes strength out of these LECs and moves this strength to a lower order. As a consequence, the convergence of the expansion improves, which is another motivation for introducing explicit $\Delta$-degrees of freedom. In the $\Delta$-less theory, the subleading 2PE and 3PE contributions to the 2NF are larger than the leading ones. The promotion of large contributions by one order in the $\Delta$-full theory fixes this problem (Ordonez et al., 1996).
In analogy to the $NN\pi$ coupling, which is proportional to $g_A/2f_\pi$ and includes one derivative, the $N\Delta\pi$ coupling is proportional to $h_A/2f_\pi$ and includes one derivative. The LECs of the $\pi N$ Lagrangian are usually extracted in the analysis of $\pi$-$N$ scattering data and clearly come out differently in the $\Delta$-full theory as compared to the $\Delta$-less one. While in the $\Delta$-less theory the magnitude of the LECs $c_3$ and $c_4$ is about 3-5 GeV$^{-1}$, they turn out to be around 1 GeV$^{-1}$ in the $\Delta$-full theory.
In the 2NF, the virtual excitation of $\Delta$-isobars requires at least one loop and, thus, the contribution occurs first at $\nu=2$ (NLO), see Figure 22. The consistency between the $\Delta$-full and $\Delta$-less theories has been verified by showing that the contributions due to intermediate $\Delta$-excitations, expanded in powers of $1/\Delta M$, can be absorbed into a redefinition of the LECs of the $\Delta$-less theory. The corresponding shift of the LECs $c_3,c_4$ is given by \begin{equation} c_3=-2c_4=-\frac{h_A^2}{9\Delta M} \,. \tag{54} \end{equation} Using $h_A=3g_A/\sqrt{2}$ (large $N_c$ value), almost all of $c_3$ and an appreciable part of $c_4$ is explained by the $\Delta$ resonance.
Several studies have confirmed that a large amount of the intermediate-range attraction of the 2NF is shifted from NNLO to NLO with the explicit introduction of the $\Delta$-isobar. However, it is also found that the NNLO 2PE potential of the $\Delta$-less theory provides a very good approximation to the NNLO potential in the $\Delta$-full theory.
The $\Delta$ isobar also changes the 3NF scenario, see Figure 23. The leading 2PE 3NF is promoted to NLO. In the $\Delta$-full theory, this term has the same mathematical form as the corresponding term in the $\Delta$-less theory, Eqs. (49) and (50), provided one chooses $c_1=0$ and $c_3$, $c_4$ according to Eq.~(54). Note that the other two NLO 3NF terms involving $\Delta$'s vanish as a consequence of the antisymmetrisation of the 3N states. The $\Delta$ contributions to the 3NF at NNLO vanish at this order, because the subleading $N\Delta \pi$ vertex contains a time-derivative, which demotes the contributions by one order. However, substantial 3NF contributions are expected at N$^3$LO from one-loop diagrams with one, two, or three intermediate $\Delta$-excitations, which correspond to diagrams of order N$^4$LO, N$^5$LO, and N$^6$LO, respectively, in the $\Delta$-less theory.
To summarize, the inclusion of explicit $\Delta$ degrees of freedom does certainly improve the convergence of the chiral expansion by shifting sizable contributions from NNLO to NLO. On the other hand, at NNLO the results for the $\Delta$-full and $\Delta$-less theory are essentially the same. Note that the $\Delta$-full theory consists of the diagrams involving $\Delta$'s plus all diagrams of the $\Delta$-less theory. Thus, the $\Delta$-full theory is much more involved. Moreover, in the $\Delta$-full theory, $1/M_N$ 2NF corrections appear at NNLO (not shown in Figure 22), which were found to be uncomfortably large (Kaiser et al., 1998). Thus, it appears that up to NNLO, the $\Delta$-less theory is more manageable.
The situation could, however, change at N$^3$LO where potentially large contributions enter the picture. It may be more efficient to calculate these terms in the $\Delta$-full theory, because in the $\Delta$-less theory they are spread out over N$^3$LO, N$^4$LO and, in part, N$^5$LO. These higher order contributions are a crucial test for the convergence of the chiral expansion of nuclear forces and represent a challenging topic for future research.
Baryon-baryon interactions
All baryons interact strongly with each other. Therefore, besides interactions between nucleons, which was the topic of this article, there are many more strong baryon-baryon interactions. Traditionally, one focus has been the forces between nucleons and hyperons (strange baryons) and hyperons and hyperons. Furthermore, the interaction between a baryon and an anti-baryon has drawn considerable interest, for which the nucleon-antinucleon interaction is the most studied example. As in the case of the nucleon-nucleon interaction, the approaches that have been tried to explain the baryon-(anti)baryon interactions include: phenomenology, meson theory, quark models, lattice QCD, and effective field theory. It is not the purpose of this article to discus those other baryon-baryon interactions, but it is worthwhile pointing out that due to the quark sub-structure of hadrons, all baryon-baryon interactions are related. Thus, the nucleon-nucleon interaction discussed in this review is not an isolated object and should be viewed in the context of all baryonic interactions. A description of all baryon-baryon interactions that is consistent with all relevant underlying symmetries is a challenging subject of contemporary research. For more information on this topic, we like to refer the interested reader to the literature. Hyperon-nucleon interactions are reviewed by Rijken et al. (2013) in the meson picture and by Haidenbauer et al. (2013) in chiral EFT. For nucleon-antinucleon, see Klempt et al. (2005), and Kang et al. (2013) for a chiral EFT approach.
Conclusions
The nuclear force has been one of the most difficult problems of modern physics. Characteristic for any fundamental science problem is that it has an intrinsic as well as an extrinsic value.
The intrinsic value of the nuclear force problem is reflected by the fact that this problem has been a fundamental challenge for eight decades and has engaged a large number of physicists creating a great diversity of ideas. Thus, understanding the nature of the nuclear force is a value by itself.
The extrinsic value derives from the fact that quantitative potentials for the nuclear two- and many-body forces are the necessary input for ab initio calculations of the properties of atomic nuclei and their reactions. Nuclear forces are crucial for the understanding of any nuclear system from a microscopic standpoint.
It has been the purpose of this review article to provide detailed information on both aspects of the nuclear force problem.
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Further reading
- Henley E M and Garcia A (2007). Subatomic Physics, 3rd Edition (World Scientific, Singapore). doi:10.1142/6263.
- Ericson T and Weise W (1988). Pions in nuclei (Oxford University Press, Oxford).
External links
http://machleidt.weebly.com/lectures-on-nuclear-forces.html
See also
Bethe-Salpeter equation, Chiral perturbation theory, Lattice quantum field theory, Nucleon-nucleon scattering, Microscopic nuclear structure, Quantum chromodynamics