# Chiral Perturbation Theory in Few-Nucleon Systems
^{†}^{†}thanks: Invited talk at the 6th Conference on the Intersections of Particle
and Nuclear Physics, Big Sky, Montana, May 1997

###### Abstract

The low-energy effective theory of nuclear physics based on chiral symmetry is reviewed. Topics discussed include the nucleon-nucleon force, few-body potentials, isospin violation, pion-deuteron scattering, proton-neutron radiative capture, pion photoproduction on the deuteron, and pion production in proton-proton collisions.

DOE/ER/41014-22-N97

Department of Physics,

University of Washington,

Seattle, WA 98195-1560

## Introduction

Although studies of its perturbative regime show that QCD is the theory of strong interactions, most of the structures of nuclear physics are apparent only at low energies where the QCD coupling constant is not small. Another expansion parameter is necessary in this energy regime, and an obvious candidate is energy itself.

The idea of a low-energy expansion is as old as nuclear physics itself. Already in the 30’s, Bethe and Peierls vkolck:bethe considered the two-nucleon system in this light. They based their approach on a previous argument about saturation due to Wigner, that the nuclear potential is of order 100 MeV and thus much larger than the deuteron binding energy of 2.2 MeV, but has a range ( 1.4 fm) much smaller than the size of the deuteron ( 4.4 fm). Bethe and Peierls reasoned that as a consequence, for distances such that , only s-waves are important and the sole effect of the potential is to provide an energy independent boundary condition at . Up to an error of , then, the system could be described by a free Schrödinger equation with the boundary condition that the logarithmic derivative of the radial wavefunction is a constant at . This constant cannot be calculated without detailed knowledge of the potential, so it was fitted to the deuteron binding energy. If that was all, nothing would have been learned. Bethe and Peierls’ point, however, was that they could then predict (with a 30% uncertainty) other processes —such as , , , and — at energies comparable to the deuteron binding energy.

This approach can be rephrased in an effective field theory (EFT) language. At typical momenta much smaller than the pion mass , the relevant degree of freedom is the nucleon, the important symmetries are parity, time-reversal and Galilean symmetry, and the appropriate expansion parameter is . (Electromagnetic processes can also be considered by adding the photon, gauge invariance, and to this list.) The most general Lagrangian involving nucleons only consists of an infinite number of terms, which are quadratic, quartic, …, in the nucleon fields with increasing number of derivatives. By dimensional reasons, derivatives come associated with inverse factors of a mass scale or greater. Nucleons are non-relativistic and the corresponding field theory has nucleon number conservation. The -matrix for the two-nucleon system is simply a sum of bubble graphs, whose vertices are the four-nucleon contact terms that appear in the Lagrangian. Formally, this is equivalent to solving a Schrödinger equation with a low-energy, effective potential consisting, schematically, of a sum , where the ’s are the coefficients of the contact terms, expected to scale as . The net effect is thus to replace the “true”, possibly complicated potential of range by a multipole expansion with moments . Life is somewhat more complicated, however, because this field theory, like any other, has to be renormalized. The bubbles are actually ultraviolet divergent, requiring regularization and absorption of the regulator dependence in renormalized parameters. It is not difficult to show vkolck:vkolck1 that the effect of renormalization is to turn the effective potential into a generalized pseudo-potential, or equivalently, turn the problem into a free one with boundary conditions at the origin which are analytic in the energy. The first, energy-independent term, parametrized by , is just the one considered by Bethe and Peierls. Much effort has been spent during the last year in trying to understand issues related to regularization and fine-tuning of this “pionless” theory vkolck:kaplan .

To the extent that there are no assumptions about the detailed dynamics of the “underlying” theory, the effective theory cannot be wrong and it is useful as long as . One may ask how strong this restriction is, however, when we consider physics of more than two nucleons. Clearly, we can always apply the pionless theory to sufficiently low-energy scattering situations, where we can control the momenta of the initial and final nucleons. But one noble goal of nuclear physics is to understand nuclei themselves. Let me use as a measure of a typical momentum of a nucleon of mass in a nucleus with nucleons and binding energy . (Other quantities such as charge radii give similar estimates.) is then about 0.3 for H, 0.5 for H, 0.8 for He, …, and 1.2 for symmetric nuclear matter in equilibrium. The same argument that justified the use of a pionless theory for the deuteron now suggests that understanding the binding of typical nuclei (He and heavier) requires explicit inclusion of pions, but not of heavier mesons such as the rho.

Now, we are in luck because QCD does explain the special role of the pion, and in the process, provides a rationale to treat pion effects systematically. This procedure goes by the name of Chiral Perturbation Theory (PT); it will be described briefly in the next section, and exemplified even more briefly in the simplest case of at most one nucleon in the following section. I then come to the main portion of this review, where we tackle nuclear forces and external probes of light nuclei.

## Effective Chiral Lagrangian

Why is the pion special? It is one of the nicest features of QCD that it provides a scenario where the lightness of the pion results from its (pseudo-) Goldstone boson nature. Here for simplicity I will limit myself to the case of two quark flavors.

If the quark masses were zero (“chiral limit”), the QCD Lagrangian would be invariant under transformations of the group of independent rotations of the quarks’ left- and right-handed components. When acting on quark bilinears, this chiral symmetry is equivalent to . A quick look at the hadronic spectrum convinces us that the chiral limit can only be phenomenologically relevant if there is spontaneous breakdown of chiral symmetry down to its diagonal subgroup, the ( for bilinears) of isospin. Although the mechanism of spontaneous breaking is not sufficiently understood at the present for a detailed derivation, we know that the effective QCD potential as a function of four quark bilinears has to have roughly a mexican hat shape, i.e. symmetry with minima in a “chiral circle” away from the origin.

Goldstone’s theorem assures us there is in the spectrum, as a consequence, a (pseudo-)scalar boson, which corresponds to excitations in the coset space . We call the radius of this “circle” , which is a function of that ends up being the pion decay constant MeV. At sufficiently low energies it is convenient to assign a field to the pion in the effective Lagrangian. An infinitesimal chiral transformation is of the form . symmetry of the dynamics implies that the Lagrangian will have a piece that is a function of only through derivatives of on the circle, which are non-linear. The Lagrangian, in principle completely determined by , has an infinite number of terms with arbitrarily high pion self-interactions, but without a pion mass term.

We know, however, that not all quark masses are zero. Quark masses generate two terms in the QCD Lagrangian. One term, with , is the fourth component of an vector and therefore breaks explicitly down to of isospin. It causes a tilt of the effective potential in the direction determined by the small parameter . The effective Lagrangian will acquire then a piece that breaks explicitly in the same way as . This piece is another infinite set of terms that do depend on in an isospin invariant way, but are all proportional to powers of . In particular, . The other quark mass term, with , is the third component of another vector and further breaks down to . Likewise, the effective Lagrangian will inherit yet another infinite set of terms, this time that break isospin as and are, in principle, of order relative to the isospin conserving chiral breaking effects. Why isospin breaking is in fact much smaller in most observables will be explained later.

QCD therefore has all the ingredients to provide a rationale not only for the special role of the pion, but also for a systematic treatment of its effects: pion interactions are weak at low energies due to (approximate) chiral symmetry.

We can now formulate an EFT for momenta along the same lines of the pionless case. The extra degrees of freedom —besides non-relativistic nucleons and photons— are obviously pions and also non-relativistic delta isobars, since the delta-nucleon mass difference is of the order of the momenta we want to consider. The new and very important symmetry is approximate . The expansion parameter is expected to be —besides . The most general Lagrangian with these ingredients has schematically the form

(1) | |||||

Here the ’s are parameters assumed to be natural, i.e. of —or with a positive integer, in the case of isospin breaking operators originating in the quark mass difference. stands for a nucleon or delta isobar, and for a covariant derivative. The interactions are naturally grouped in sets of common index but arbitrary values of . For non-electromagnetic interactions, we find that only because of chiral symmetry.

Consider an arbitrary irreducible contribution to a process involving nucleons and an arbitrary number of pions and photons, all with momenta of order . It can be represented by a Feynman diagram with continuous nucleon lines, loops, separately connected pieces, and vertices from , whose connected pieces cannot be all split by cutting only nucleon lines. ( for ; for . The reason to consider irreducible diagrams and will be discussed in the section on few-nucleon systems.) It is easy to show vkolck:weinberg1 that this contribution is typically of , where

(2) |

Since is bounded from below (0) and from above (), the chiral symmetry constraint implies that for strong interactions. Leading contributions come from tree diagrams built out of and coincide with current algebra. Perturbation theory in can be carried out by considering contributions from ever increasing .

Note that this approach is:

(i) systematic. It is a perturbation in the number of loops, derivatives/fermion fields, and many-nucleon effects.

(ii) consistent with QCD. The only (very important) QCD inputs are confinement (color singlet fields), symmetries (chiral, …), and naturalness. QCD can be represented by a point in the space of renormalized parameters (at some renormalization scale) of the effective theory. An explicit solution of QCD (such as from simulations on a lattice) would provide knowledge of the exact position of this point, and then the effective theory would be completely predictive. Until such a solution is found —or as a test of QCD after it is found— we can recourse to fitting low-energy experiments in order to determine the region in parameter space allowed on phenomenological grounds. Even in this case the theory is predictive, because to any given order the space of parameters is finite-dimensional. After a finite number of experimental results are used, an infinite number of others can be predicted up to an accuracy depending on the order of the expansion. In practice, because the number of parameters grows rapidly with the order, model-dependent estimates of parameters based on specific dynamic ideas —such as saturation by tree-level resonance exchange— are sometimes used.

(iii) a theory. It is applicable in principle to all low-energy phenomena. I will in the next section mention some of the highlights of this program for processes with at most one nucleon, and then most of the rest of the paper is devoted to nuclear physics per se.

## Meson and One-nucleon Sectors

In the case of strong mesonic processes, with increasing in steps of two. Since Weinberg discovered this systematic generalization of current algebra vkolck:weinberg2 and Gasser and Leutwyler implemented it vkolck:gasser1 , many processes (including weak interactions) involving pions (and kaons) have been examined, typically to . The most thoroughly studied process has been scattering, where a calculation was carried out vkolck:bijnens ; fits to other data plus resonance saturation show convergence and good agreement with phase shifts through energies of more than 100 MeV above threshold. Many good reviews exist on the mesonic sector; see for example Ref. vkolck:meson .

For processes where one nucleon is present, , where (with ) increases in steps of one. (Convergence can be expected to be slower compared to purely mesonic interactions.) Thanks to this power counting, a low-energy nucleon can in a very definite sense be pictured as a static, point-like object (up to corrections in powers of ), surrounded by: i) an inner cloud which is dense but of short range , so that we can expand in its relative size ; ii) an outer cloud of long range but sparse, so that we can expand in its relative strength .

The first attempt at including the nucleon in PT vkolck:gasser2 had limited success because it did not fully explore the non-relativistic nature of the nucleon and did not consider the delta isobar explicitly. This was remedied in the work of Jenkins and Manohar vkolck:jenkins , while a more complete analysis of the role of the delta in one-nucleon process has been carried out recently vkolck:hemmert . However, most calculations have been limited to the threshold region where the delta contribution is relatively unimportant and a “deltaless” theory useful; see for example Ref. vkolck:nucleon for an extensive review.

Here, for illustration, I mention explicitly the case of pion photoproduction at threshold which has received a lot of attention lately. At threshold the amplitude for a photon of polarization incident on a nucleon of spin is . This process has been studied up to in the deltaless theory in Ref. vkolck:bernard1 (where references to the experimental papers can be found; see also Ref. vkolck:korkmaz ). Results for from a fit constrained by resonance saturation are presented in Table 1.

() | Experiment | ||||
---|---|---|---|---|---|

\tableline | 34.0 | 26.4 | 28.9 | 28.2 | 27.9 0.5. |

28.8 0.7. | |||||

34.0 | 31.5 | 32.9 | 32.7 | 31.4 1.3. | |

32.2 1.2. | |||||

0 | 3.58 | 0.96 | 1.16 | 1.31 0.08. | |

1.32 0.08. | |||||

0 | 0 | 3.7 | 2.13 | ? |

We can observe that the big size of the charged pion channels result from a non-vanishing contribution (the Kroll-Ruderman term). Convergence and agreement with experimental values are pretty good. For the neutral pion channels convergence is less apparent, but the absolute values much smaller. The result for is in relatively good agreement with the recent results from Mainz and Saskatoon. To the same order there is a prediction for the reaction, which would be important to test. If isospin symmetry breaking is neglected, there are only three independent amplitudes; if we use the three measured amplitudes, their uncertainties limit extraction of to the range . Newer, more accurate data for the charged channels were presented at this conference by Korkmaz vkolck:korkmaz . Isospin breaking entered the calculation of Table 1, albeit in an incomplete form. A direct measurement of could not only check the consistency of the calculation, but also provide a new isospin violating observable. This however requires a deuteron target, and a description of in the same framework. So, even from the point of view of nucleon properties we are led naturally into the study of light nuclei.

## Nuclear Physics

A non-trivial new element enters the theory when we consider systems of more than one nucleon vkolck:weinberg1 . Because nucleons are heavy, contributions from intermediate states that differ from the initial state only in the energy of nucleons are enhanced by infrared quasi-divergences. These are linked to the existence of small energy denominators of , which generate contributions larger than what would be expected from Eq. (2). The latter is still correct for the class of sub-diagrams —called irreducible— that do not contain intermediate states with small energy denominators. For an -nucleon system these are -nucleon irreducible diagrams, the sum of which we call the potential . When we consider external probes all with , the sum of irreducible diagrams forms the kernel to which all external particles are attached. A generic diagram contributing to a full amplitude will consist of irreducible diagrams sewed together by states of small energy denominators. These irreducible diagrams might have more than one connected piece, hence the introduction of in Eq. (2). The infrared enhancement requires that we sum diagrams to all orders in the amplitude, creating the possibility of the existence of shallow bound states (nuclei). For an -nucleon system, this is equivalent to solving the Schrödinger equation with the potential . The amplitude for a process with external probes is then where () is the wavefunction of the initial (final) nuclear state calculated with the potential .

Because our expansion is still valid for the potential and the kernel, the picture of a nucleon as a mostly static object surrounded by an inner and an outer cloud leads to remarkable nuclear physics properties that we are used to, but would remain otherwise not understood from the viewpoint of QCD.

### Nuclear Forces

If we put a few non-relativistic nucleons together, each nucleon will not be able to distinguish details of the others’ inner clouds. The region of the potential associated with distances of can be expanded in delta-functions and their derivatives as Bethe and Peierls did. The outer cloud of range yields non-analytic contributions to the potential, but being sparse, it mostly produces the exchange of one pion, with progressively smaller two-, three-, …- pion exchange contributions.

For the two-nucleon system, , with as in the one-nucleon case. A calculation of all the contributions up to was carried out in Ref.vkolck:ordonez . In leading order, , the potential is simply static one-pion exchange and momentum-independent contact terms vkolck:weinberg1 . corrections vanish due to parity and time-reversal invariance. corrections include several two-pion exchange diagrams (including virtual delta isobar contributions), recoil in one-pion exchange, and several contact terms that are quadratic in momenta. At a few more two-pion exchange diagrams have to be considered. As in the pionless case, regularization and renormalization are necessary. It is not straightforward to implement dimensional regularization in this non-perturbative context, so we used an overall gaussian cut-off, and performed calculations with the cut-off parameter taking values 500, 780 and 1000 MeV. Cut-off independence means that for each cut-off value a set of (bare) parameters can be found that fits low-energy data. A sample of the results for the lower, more important partial waves is presented in Fig. 1 and for the deuteron quantities in Table 2. (See vkolck:ordonez for more details and reference to experiments and phase shift analyses.)

Deuteron | (MeV) | |||
---|---|---|---|---|

quantities | 500 | 780 | 1000 | Experiment |

\tableline (MeV) | 2.15 | 2.24 | 2.18 | 2.224579(9) |

() | 0.863 | 0.863 | 0.866 | 0.857406(1) |

(fm) | 0.246 | 0.249 | 0.237 | 0.2859(3) |

0.0229 | 0.0244 | 0.0230 | 0.0271(4) | |

(%) | 2.98 | 2.86 | 2.40 |

The fair agreement of this first calculation and data up to laboratory energies of 100 MeV or so suggests that this may become an alternative to other, more model-dependent approaches to the two-nucleon problem. Further examination of regularization effects, fine-tuning in the channel, and different aspects of two-pion exchange in this context can be found in Refs. vkolck:kaplan ; vkolck:2pion .

Perhaps more impressive is that we can get some insight into other aspects of nuclear forces. Let us look for the new forces that appear in systems with more than two nucleons. The dominant potential, at , is the two-nucleon potential of lowest order that appeared in the two-nucleon case. We can easily verify that a three-body potential will arise at , a four-body potential at , and so on. It is (approximate) chiral symmetry therefore that implies that -nucleon forces are expected to obey a hierarchy of the type , with denoting the contribution per -plet. If we estimate MeV, we can guess MeV, MeV, and so on. This is in accord with detailed few-nucleon calculations using more phenomenological potentials. The explicit three-body potential at (from the delta isobar) and was derived in Ref.vkolck:vkolck2 .

We can also look at isospin-dependent effects. Within the context of PT it can be shown vkolck:vkolck3 that isospin is an accidental symmetry, in the sense that it does not appear in the low-energy EFT in lowest order, and therefore is typically not an effect, but . In the case of nuclear forces, we find moreover a hierarchy among different types of components. It is standard to call class I the strongest forces that are isospin symmetric, class II weaker forces that are isospin violating but charge symmetric, class III even weaker forces that are charge symmetry breaking but symmetric under permutation of particles in isospin space, and class IV the weakest, remaining forces. In the chiral expansion, one indeed finds vkolck:vkolck3 that higher class forces appear at higher orders: , where denotes the contribution of the leading class potential. This qualitatively explains, for example, the observed isospin structure of the two-nucleon Coulomb-corrected scattering lengths, . Precise calculations of simultaneous electromagnetic and strong isospin violation in the nuclear potential have also been carried out vkolck:vkolck4 .

Despite these successful fits and insights, the main advantage of the method of EFT lies in its concomitant application to many other processes, which might yield more predictive statements. I now discuss some of these.

### Nuclear Probes

Before plunging into hard results, let me point out another generic result of the chiral expansion. As a result of the factor in Eq. (2), we see immediately —in an effect similar to few-nucleon forces— that external low-energy probes (’s, ’s) will tend to interact predominantly with a single nucleon, simultaneous interactions with more than one nucleon being suppressed by powers of . Again, this is a well-known result that arises naturally here.

This is of course what allows extraction, to a certain accuracy, of one-nucleon parameters from nuclear experiments. More interesting from the nuclear dynamics perspective are, however, those processes where the leading single-nucleon contribution vanishes by a particular choice of experimental conditions, for example the threshold region. In this case the two-nucleon contributions, especially in the relatively large deuteron, can become important.

#### at threshold.

This is perhaps the most direct way to check the consistency of PT in few-nucleon systems and in pion-nucleon scattering. Here the lowest-order, contributions to the kernel vanish because the pion is in an s-wave and the target is isoscalar. The term comes from the (small) isoscalar pion-nucleon seagull, related in lowest-order to the pion-nucleon isoscalar amplitude . contributions come from corrections to pion-nucleon scattering and two-nucleon diagrams, which involve besides also the much larger isovector amplitude . Weinbergvkolck:weinberg3 has estimated these various contributions to the pion-deuteron scattering length, finding agreement with previous, more phenomenological calculations, which have been used to extract .

#### at threshold.

This offers a chance of a precise postdiction. Here it is the transverse nature of the real outgoing photon that is responsible for the vanishing of the lowest-order, contribution to the kernel. The single-nucleon magnetic contributions come at (tree level), (one loop), etc. The first two-nucleon term is an one-pion exchange at long discovered to give a smaller but non-negligible contribution. There has been a longstanding discrepancy of a few percent between these contributions and experiment. At there are further one-pion exchange, two-pion exchange, and short-range terms. Park, Min and Rho vkolck:park1 calculated the two-pion exchange diagrams in the deltaless theory and used resonance saturation to estimate the other terms. With wavefunctions from the Argonne V18 potential and a cut-off MeV, they found the excellent agreement with experiment shown in Table 3. The total cross-section changes by .3 % if the cut-off is decreased to 500 MeV. (See vkolck:park1 where reference to experiment can be found.)

\tableline305.6 | 321.7 | 336.0 | 3340.5 |
---|

#### at threshold.

As emphasized earlier, this reaction offers the possibility to test a prediction arising from a combination of two-nucleon contributions and the neutral pion single-neutron amplitude. Here, it is the neutrality of the outgoing s-wave pion that ensures that the leading terms vanish. The single-scattering contribution is given by the same mechanisms described earlier, with due account of p-waves and Fermi motion inside the deuteron. The first two-nucleon term enters at , a correction appears at , and so on. At threshold the amplitude for a photon of polarization incident on the deuteron of spin is . Results for up to have been obtained vkolck:beane and are shown in Table 4. They correspond to the Argonne V18 potential and a cut-off MeV. Other realistic potentials and cut-offs from 650 to 1500 MeV give the same result within 5%, while a model-dependent estimate vkolck:wilhelm of some terms suggests a 10% or larger error from the neglected higher orders in the kernel itself. The single-scattering amplitude depends on in such a way that in units of . Thus, some sensitivity to survives the large two-nucleon contribution at .

\tableline0.36 | 1.90 | 0.25 | 1.79 |
---|

Some old Saclay data gave on its latest reanalysis, but a new precise measurement is called for. A test of the above prediction will come from new Saskatoon data, currently under analysis vkolck:korkmaz . An electroproduction experiment will also be carried out in Mainz vkolck:bernstein .

#### close to threshold.

This reaction has attracted a lot of interest because of the failure of standard phenomenological mechanisms in reproducing the small cross-section near threshold. It involves larger momenta of , so the the relevant small parameter here is the not so small . It is therefore not a good testing ground for the above ideas. But is still , so at least in some formal sense we can perform a low-energy expansion. In Ref.vkolck:cohen the chiral expansion was adapted to this reaction and the first few contributions estimated. Again, the lowest order terms all vanish. The formally leading non-vanishing terms —an impulse term and a similar diagram from the delta isobar— are anomalously small and partly cancel. The bulk of the cross-section must then arise from contributions that are relatively unimportant in other processes. One is isoscalar pion rescattering for which two sets of PT parameters were used: “ste” from a sub-threshold expansion of the amplitude and “cl” from an one-loop analysis of threshold parameters. Others are two-pion exchange and short-range terms, which were modeled by heavier-meson exchange: pair diagrams with and exchange, and a coupling, among other, smaller terms. Two potentials —Argonne V18 and Reid93— were used. Results are shown in Fig. 2 together with IUCF and Uppsala data. Other PT studies of this reaction can be found in Ref. vkolck:pppppizero , while Ref.vkolck:park2 presents a related analysis of the axial-vector current.

The situation here is clearly unsatisfactory, and presents therefore a unique window to the nuclear dynamics. Work is in progress, for example, on a similar analysis for the other, not so suppressed channels vkolck:carocha .

## Conclusions

Mesonic PT is now a mature subject, where the validity of the approach from both phenomenological and internal consistency standpoints has been demonstrated. PT has also passed several tests in systems with one nucleon, even though some issues —such as delta isobar effects— remain to be fully investigated.

In nuclear physics, only the very initial steps of a systematic chiral expansion have been attempted so far, despite the amount of information available. I have tried to argue that the first results are very auspicious. The chiral expansion has the basic ingredients of nuclear forces, as evidenced by a quantitative fit to two-nucleon data and by the qualitative insights into the size of few-body and isospin-violating forces. It provides a consistent framework for scattering on the nucleon and on light nuclei, which in turn offers a handle on nucleon parameters (as for pion-deuteron scattering and pion photoproduction), successful quantitative postdictions (such as in radiative neutron-proton capture), and quantitative predictions (such as in pion photoproduction). And best of all, it has open problems such as pion production in the reaction. There is still a lot to be done: consistent potential/kernel calculations, the above processes away from threshold, many other processes, extension to and nuclear matter, to mention just a few topics. Perhaps PT will then fulfill the role of a long-lacking theory for nuclear physics based on QCD.

#### Acknowledgements.

I am grateful to my collaborators for helping making this research program possible. This manuscript benefitted from communications with E. Korkmaz and T.-S. Park, and criticism by P. Bedaque. This research was supported by the DOE grant DE-FG03-97ER41014.