Modern Theory of Nuclear Forces

Modern Theory of Nuclear Forces

Abstract

Effective field theory allows for a systematic and model-independent derivation of the forces between nucleons in harmony with the symmetries of Quantum Chromodynamics. We review the foundations of this approach and discuss its application for light nuclei at various resolution scales. The extension of this approach to many-body systems is briefly sketched.

Commissioned article for Reviews of Modern Physics

1

I QCD and Nuclear Forces

Within the Standard Model of particle physics, the strong interactions are described by Quantum Chromodynmics (QCD). QCD is a fascinating theory with many intriguing manifestations. Its structure and interactions are governed by a local non-abelian gauge symmetry, namely SU(3). Its fundamental degrees of freedom, the quarks (the matter fields) and gluons (the force carriers), have never been observed in isolation (confinement). The strong coupling constant exhibits a very pronounced running and is of order one in the typical energy scales of nuclear physics. The bound states made from the basic constituents are the hadrons, the strongly interacting particles. The particle spectrum shows certain regularities that can be traced back to the flavor symmetries related to the fermions building up these states. More precisely, there are six quark flavors. These can be grouped into two very different sectors. While the light quarks () are almost massless and thus have to be treated relativistically, bound states made from heavy quarks allow for a precise non-relativistic treatment. In what follows, we will only consider the light quarks at low energies, where perturbation theory in is inapplicable (this regime is frequently called “strong QCD”). A further manifestation of strong QCD is the appearance of nuclei, shallow bound states composed of protons, neutrons, pions or strange particles like hyperons. The resulting nuclear forces that are responsible for the nuclear binding are residual color forces, much like the van der Waals forces between neutral molecules. It is the aim of this article to provide the link between QCD and its symmetries, in particular the spontaneously and explicitely broken chiral symmetry, and the nuclear forces which will allow to put nuclear physics on firm theoretical grounds and also gives rise to a very accurate calculational scheme for nuclear forces and the properties of nuclei.

This review is organized as follows: In this section, we briefly discuss some of the concepts underlying the chiral effective field theory of the nuclear forces and make contact to ab initio lattice simulations of two-baryon systems as well as to more phenomenological approaches. Sec. II deals with the foundations and applications of nuclear EFT and should be considered the central piece of this review. In particular, tests of these forces in few-nucleon systems are discussed. Attempts to tackle nuclear matter and finite nuclei are considered in sec. III. We end with a short summary and outlook.

i.1 Chiral symmetry

First, we must discuss chiral symmetry in the context of QCD. Chromodynamics is a non-abelian gauge theory with flavors of quarks, three of them being light () and the other three heavy (). Here, light and heavy refers to a typical hadronic scale of about 1 GeV. In what follows, we consider light quarks only (the heavy quarks are to be considered as decoupled). The QCD Lagrangian reads

(1.1)

where we have absorbed the gauge coupling in the definition of the gluon field and color indices are suppressed. The three-component vector collects the quark fields, . As far as the strong interactions are concerned, the different quarks have identical properties, except for their masses. The quark masses are free parameters in QCD - the theory can be formulated for any value of the quark masses. In fact, light quark QCD can be well approximated by a fictitious world of massless quarks, denoted in Eq. (1.1). Remarkably, this theory contains no adjustable parameter - the gauge coupling merely sets the scale for the renormalization group invariant scale . Furthermore, in the massless world left- and right-handed quarks are completely decoupled. The Lagrangian of massless QCD is invariant under separate unitary global transformations of the left- and right-hand quark fields, the so-called chiral rotations, , leading to conserved left- and conserved right-handed currents by virtue of Noether’s theorem. These can be expressed in terms of vector () and axial-vector () currents

(1.2)

Here, , and the are Gell-Mann’s flavor matrices. The singlet axial current is anomalous, and thus not conserved. The actual symmetry group of massless QCD is generated by the charges of the conserved currents, it is . The subgroup of generates conserved baryon number since the isosinglet vector current counts the number of quarks minus antiquarks in a hadron. The remaining group is often referred to as chiral . Note that one also considers the light and quarks only (with the strange quark mass fixed at its physical value), in that case, one speaks of chiral and must replace the generators in Eq. (I.1) by the Pauli-matrices. Let us mention that QCD is also invariant under the discrete symmetries of parity (), charge conjugation () and time reversal (). Although interesting in itself, we do not consider strong violation and the related -term in what follows, see e.g. Peccei:2006as ().

The chiral symmetry is a symmetry of the Lagrangian of QCD but not of the ground state or the particle spectrum – to describe the strong interactions in nature, it is crucial that chiral symmetry is spontaneously broken. This can be most easily seen from the fact that hadrons do not appear in parity doublets. If chiral symmetry were exact, from any hadron one could generate by virtue of an axial transformation another state of exactly the same quantum numbers except of opposite parity. The spontaneous symmetry breaking leads to the formation of a quark condensate in the vacuum , thus connecting the left- with the right-handed quarks. In the absence of quark masses this expectation value is flavor-independent: . More precisely, the vacuum is only invariant under the subgroup of vector rotations times the baryon number current, . This is the generally accepted picture that is supported by general arguments Vafa:1983tf () as well as lattice simulations of QCD (for a recent study, see Giusti:2007cn ()). In fact, the vacuum expectation value of the quark condensate is only one of the many possible order parameters characterizing the spontaneous symmetry violation - all operators that share the invariance properties of the vacuum qualify as order parameters. The quark condensate nevertheless enjoys a special role, it can be shown to be related to the density of small eigenvalues of the QCD Dirac operator (see Banks:1979yr () and more recent discussions in Leutwyler:1992yt (); Stern:1998dy ()), . For free fields, near . Only if the eigenvalues accumulate near zero, one obtains a non-vanishing condensate. This scenario is indeed supported by lattice simulations and many model studies involving topological objects like instantons or monopoles.

Before discussing the implications of spontaneous symmetry breaking for QCD, we briefly remind the reader of Goldstone’s theorem Goldstone:1961eq (); Goldstone:1962es (): to every generator of a spontaneously broken symmetry corresponds a massless excitation of the vacuum. These states are the Goldstone bosons, collectively denoted as pions in what follows. Through the corresponding symmetry current the Goldstone bosons couple directly to the vacuum,

(1.3)

In fact, the non-vanishing of this matrix element is a necessary and sufficient condition for spontaneous symmetry breaking. In QCD, we have eight (three) Goldstone bosons for () with spin zero and negative parity – the latter property is a consequence that these Goldstone bosons are generated by applying the axial charges on the vacuum. The dimensionful scale associated with the matrix element Eq. (1.3) is the pion decay constant (in the chiral limit)

(1.4)

which is a fundamental mass scale of low-energy QCD. In the world of massless quarks, the value of differs from the physical value by terms proportional to the quark masses, to be introduced later, . The physical value of is MeV, determined from pion decay, .

Of course, in QCD the quark masses are not exactly zero. The quark mass term leads to the so-called explicit chiral symmetry breaking. Consequently, the vector and axial-vector currents are no longer conserved (with the exception of the baryon number current)

(1.5)

However, the consequences of the spontaneous symmetry violation can still be analyzed systematically because the quark masses are small. QCD possesses what is called an approximate chiral symmetry. In that case, the mass spectrum of the unperturbed Hamiltonian and the one including the quark masses can not be significantly different. Stated differently, the effects of the explicit symmetry breaking can be analyzed in perturbation theory. As a consequence, QCD has a remarkable mass gap - the pions (and, to a lesser extent, the kaons and the eta) are much lighter than all other hadrons. To be more specific, consider chiral . The second formula of Eq. (1.5) is nothing but a Ward-identity (WI) that relates the axial current with the pseudoscalar density ,

(1.6)

Taking on-shell pion matrix elements of this WI, one arrives at

(1.7)

where the coupling is given by . This equation leads to some intriguing consequences: In the chiral limit, the pion mass is exactly zero - in accordance with Goldstone’s theorem. More precisely, the ratio is a constant in the chiral limit and the pion mass grows as if the quark masses are turned on.

There is even further symmetry related to the quark mass term. It is observed that hadrons appear in isospin multiplets, characterized by very tiny splittings of the order of a few MeV. These are generated by the small quark mass difference and also by electromagnetic effects of the same size (with the notable exception of the charged to neutral pion mass difference that is almost entirely of electromagnetic origin). This can be made more precise: For , QCD is invariant under isospin transformations: , with a unitary matrix. In this limit, up and down quarks can not be disentangled as far as the strong interactions are concerned. Rewriting of the QCD quark mass term allows to make the strong isospin violation explicit:

where the first (second) term is an isoscalar (isovector). Extending these considerations to , one arrives at the eighfold way of Gell-Mann and Ne’eman that played a decisive role in our understanding of the quark structure of the hadrons. The flavor symmetry is also an approximate one, but the breaking is much stronger than it is the case for isospin. From this, one can directly infer that the quark mass difference must be much bigger than .

The consequences of these broken symmetries can be analyzed systematically in a suitably tailored effective field theory (EFT), as discussed in more detail below. At this point, it is important to stress that the chiral symmetry of QCD plays a crucial role in determining the longest ranged parts of the nuclear force, which, as we will show, is given by Goldstone boson exchange between two and more nucleons. This was already stressed long ago, see e.g Brown:1970th () (and references therein) but only with the powerful machinery of chiral effective field theory this connection could be worked out model-independently, as we will show in what follows.

i.2 Scales in nuclear physics

To appreciate the complexity related to a theoretical description of the nuclear forces, it is most instructive to briefly discuss the pertinent scales arising in this problem. This can most easily be visualized by looking at the phenomenological central potential between two nucleons, as it appears e.g. in meson-theoretical approaches to the nuclear force, see Fig. 1. The longest range part of the interaction is the one-pion exchange (OPE) that is firmly rooted in QCD’s chiral symmetry. Thus, the corresponding natural scale

Figure 1: Schematic plot of the central nucleon-nucleon potential. The longest range contribution is the one-pion-exchange, the intermediate range attraction is described by two-pion exchanges and other shorter ranged contributions. At even shorter distances, the NN interaction is strongly repulsive.

of the nuclear force problem is the Compton wavelength of the pion

(1.9)

where MeV is the charged pion mass. The central intermediate range attraction is given by exchange (and shorter ranged physics). Finally, the wavefunctions of two nucleons do not like to overlap, which is reflected in a short-range repulsion that can e.g. be modelled by vector meson exchange. From such considerations, one would naively expect to be able to describe nuclear binding in terms of energy scales of the order of the pion mass. However, the true binding energies of the nuclei are given by much smaller energy scales, between 1 to 9 MeV per nucleon. Another measure for the shallow nuclear binding is the so called binding-momentum . In the deuteron, , with MeV the nucleon mass and MeV the deuteron binding energy. The small value of signals the appearance of energy/momentum scales much below the pion mass. The most dramatic reflection of the complexity of the nuclear force problem are the values of the S-wave neutron-proton scattering lengths,

(1.10)

Thus, to properly set up an effective field theory for the forces between two (or more) nucleons, it is mandatory to deal with these very different energy scales. If one were to treat the large S-wave scattering lengths perturbatively, the range of the corresponding EFT would be restricted to momenta below MeV. To overcome this barrier, one must generate the small binding energy scales by a non-perturbative resummation. This can e.g. be done in a theory without explicit pion degrees of freedom, the so-called pion-less EFT. In such an approach, the limiting hard scale is the pion mass. To go further, one must include the pions explicitely, as it is done in the pion-full or chiral nuclear EFT. The relation between these different approaches is schematically displayed in Fig. 2.

Figure 2: Scales in the two-nucleon problem and the range of validity of the corresponding EFTs as explained in the text. Here is the hard scale related to spontaneous chiral symmetry breaking, with , with MeV the mass of the rho meson.

A different and more formal argument that shows the breakdown of a perturbative treatment of the EFT with two or more nucleons is related to the pinch singularities in the two-pion exchange diagram in the static limit as will be discussed later in the context of the explicit construction of the chiral nuclear EFT.

In addition, if one extends the considerations to heavier nuclei or even nuclear matter, the many-body system exhibits yet another scale, the Fermi momentum , with at nuclear matter saturation density. This new scale must be included in a properly modified EFT for the nuclear many-body problem which is not a straightforward exercise as we will show below. It is therefore not astonishing that the theory for heavier nuclei is still in a much less developed stage that the one for the few-nucleon problem. These issues will be taken up in Sec. III.

For more extended discussions of scales in the nuclear force problem and in nuclei, we refer to Friar:1995dt (); Friar:1996zw (); Kaiser:2006tu (); Delfino:2007zu ().

i.3 Conventional approaches to the nuclear force problem

Before discussing the application of the effective field theory approach to the nuclear force problem, let us make a few comments on the highly successful conventional approaches. First, we consider the two-nucleon case. Historically, meson field theory and dispersion relations have laid the foundations for the construction of a two-nucleon potential. All these approaches incorporate the long-range one-pion exchange as proposed by Yukawa in 1935 Yukawa:1935xg () which nowadays is firmly rooted in QCD. Dispersion relations can be used to construct the two-pion exchange contribution to the nuclear force as pioneered at Paris Cottingham:1973wt () and Stony Brook Jackson:1975be (). For a review, see e.g. Machleidt:2001rw (). In the 1990ties, the so-called high-precision potentials have been developed that fit the large basis of and elastic scattering data with a . One of these is the so-called CD-Bonn potential Machleidt:2000ge () (which was developed at Moscow, Idaho). Besides one-pion, and vector-meson exchanges, it contains two scalar–isoscalar mesons in each partial wave up to angular momentum with the mass and coupling constant of the second fine-tuned in any partial wave. The hadronic vertices are regulated with form factors with cut-offs ranging from 1.3 to 1.7 GeV. Similarly, in the Nijmegen I,II potentials one–pion exchange is supplemented by heavy boson exchanges with adjustable parameters which are fitted for all (low) partial waves separately Stoks:1994wp (). The Argonne V18 (AV18) potential starts from a very general operator structure in coordinate space and has fit functions for all these various operators Wiringa:1994wb (). While these various potentials give an accurate representation of the nucleon-nucleon phase shifts and of most deuteron properties, the situation becomes much less satisfactory when it comes to the much smaller but necessary three-nucleon forces. Such three-body forces are needed to describe the nuclear binding energies and levels, as most systematically shown by the Urbana-Argonne group Pieper:2001mp (). Systematic studies of the dynamics and reactions of systems with three or four-nucleons further sharpen the case for the necessity of including three-nucleon forces (3NFs), see e.g. Gloeckle:1995jg (). The archetype of a 3NF is due to Fujita and Miyazawa (FM) Fujita:1957zz (), who extended Yukawa’s meson exchange idea by sandwiching the pion-nucleon scattering amplitude between nucleon lines, thus generating the 3NF of longest range. In fact, the work of Fujita and Miyazawa has been the seed for many meson-theoretical approaches to the three-nucleon force like the families of Tucson-Melbourne TM1 (); Coon:1974vc (), Brazilian Coelho:1984hk () or Urbana-Illinois Pudliner:1997ck (); Pieper:2001ap () 3NFs.

While the conventional approach as briefly outlined here as enjoyed many successes and is frequently used in e.g. nuclear structure and reaction calculations, it remains incomplete as there are certain deficiencies that can only be overcome based on EFT approaches. These are: (i) it is very difficult - if not impossible - to assign a trustworthy theoretical error, (ii) gauge and chiral symmetries are difficult to implement, (iii) none of the three-nucleon forces is consistent with the underlying nucleon-nucleon interaction models/approaches and (iv) the connection to QCD is not at all obvious. Still, as we will show later, there is a very natural connection between these models and the forces derived from EFT by mapping the complicated physics of the short-distance part of any interaction at length scales to the tower of multi-fermion contact interactions that naturally arise in the EFT description (see Sec. II.2).

i.4 Brief introduction to effective field theory

Effective field theory (EFT) is a general approach to calculate the low-energy behavior of physical systems by exploiting a separation of scales in the system (for reviews see e.g. Georgi:1994qn (); Manohar:1996cq (); Burgess:2007pt ()). Its roots can be traced to the renormalization group Wilson-83 () and the intuitive understanding of ultraviolet divergences in quantum field theory Lepage-89 (). A succinct formulation of the underlying principle was given by Weinberg Weinberg:1978kz (): If one starts from the most general Lagrangian consistent with all symmetries of the underlying interaction, one will get the most general S-matrix consistent with these symmetries. Together with a power counting scheme that specifies which terms are required at a desired accuracy leads to a predictive paradigm for a low-energy theory. The expansion is typically in powers of a low-momentum scale which can be the typical external momentum over a high-momentum scale . However, what physical scales and are identified with depends on the considered system. In its most simple setting, consider a theory that is made of two particle species, the light and the heavy ones with . Consider now soft processes in which the energies and momenta are of the order of the light particle mass (the so-called soft scale). Under such conditions, the short-distance physics related to the heavy particles can never be resolved. However, it can be represented by light-particle contact interactions with increasing dimension (number of derivatives). Consider e.g. heavy particle exchange between light ones in the limit that while keeping the ratio fixed, with the light-heavy coupling constant. As depicted in Fig. 3, one can represent such exchange diagrams by a sum of local operators of the light fields with increasing number of derivatives. In a highly symbolic notation

(1.11)

with the squared invariant momentum transfer.

Figure 3: Expansion of a heavy-particle exchange diagram in terms of local light-particle operators. The solid and dashed lines denote light and heavy particles, respectively. The filled circle and square denote insertions with zero and two derivatives, in order. The ellipses stands for operators with more derivatives.

In many cases, the corresponding high-energy theory is not known. Still, the framework of EFT offers a predictive and systematic framework for performing calculations in the light particle sector. Denote by a typical energy or momentum of the order of and by the hard scale where the EFT will break down. In many cases, this scale is set by the masses of the heavy particles not considered explicitely. In such a setting, any matrix element or Greens function admits an expansion in the small parameter Weinberg:1978kz ()

(1.12)

where is a function of order one (naturalness), a regularization scale (related to the UV divergences appearing in the loop graphs) and the denotes a collection of coupling constants, often called low-energy constants (LECs). These parameterize (encode) the unknown high-energy (short-distance) physics and must be determined by a fit to data (or can be directly calculated if the corresponding high-energy theory is known/can be solved). The counting index in general depends on the fields in the effective theory, the number of derivatives and the number of loops. This defines the so-called power counting which allows to categorize all contributions to any matrix element at a given order. It is important to stress that must be bounded from below to define a sensible EFT. In QCD e.g. this is a consequence of the spontaneous breaking of its chiral symmetry. The contributions with the lowest possible value of define the so-called leading order (LO) contribution, the first corrections with the second smallest allowed value of the next-to-leading order (NLO) terms and so on. In contrast to more conventional perturbation theory, the small parameter is not a coupling constant (like, e.g., in Quantum Electrodynamcis) but rather one expands in small energies or momentum, where small refers to the hard scale . The archetype of such a perturbative EFT is chiral perturbation theory that exploits the strictures of the spontaneous and explicit chiral symmetry breaking in QCD Gasser:1983yg (); Gasser:1984gg (). Here, the light degrees of freedom are the pions, that are generated through the symmetry violation. Heavier particles like e.g. vector mesons only appear indirectly as they generate local four-pion interactions with four, six, derivatives. For a recent review, see Ref. Bernard:2006gx (). Of course, the pions also couple to heavy matter fields like e.g. nucleons, that can also be included in CHPT, as reviewed by Bernard Bernard:2007zu ().

So far, we have made the implicit assumption of naturalness, which implies e.g. that the scattering length is of natural size as e.g. in CHPT, where the scale is set by fm. This also implies that there are no bound states close to the scattering thresholds. In many physical systems and of particular interest here, especially in the two-nucleon system, this is not the case, but one rather has to deal with unnaturally large scattering lengths (and also shallow bound states). To be specific, let us consider nucleon-nucleon scattering at very low energies in the channel, cf. Eq. (1.10). For such low energies, even the pions can be considered heavy and are thus integrated out. To construct an EFT that is applicable for momenta , one must retain all terms in the scattering matrix. This requires a non-perturbative resummation and is most elegantly done in the power divergence scheme of Kaplan, Savage and Wise Kaplan:1998tg (); Kaplan:1998we (). This amounts to summing the leading four-nucleon contact term to all orders in and matching the scale-dependent LEC to the scattering length. This leads to the T-matrix

(1.13)

where the expansion around the large scattering length is made explicit. All other effects, like e.g. effective range corrections, are treated perturbatively. This compact and elegant scheme is, however, not sufficient for discussing nuclear processes with momenta . We will come back to this topic when we give the explicit construction of the chiral nuclear EFT in sec. II.2. It is important to stress that such EFTs with unnaturally large scattering length can exhibit universal phenomena that can be observed in physical systems which differ in their typical energy scale by many orders of magnitude, for a review see Braaten and Hammer Braaten:2004rn (). We also remark that there are many subtleties in constructing a proper EFT, but space forbids to discuss these here. Whenever appropriate and/or neccessary, we will mention these in the following sections and provide explicit references.

i.5 First results from lattice QCD

Lattice QCD (LQCD) is a promising tool to calculate hadron properties ab initio from the QCD Lagrangian on a discretized Euclidean space-time. This requires state-of-the-art high performance computers and refined algorithms to analyse the QCD partition function by Monte Carlo methods. Only recently soft- and hardware developments have become available that allow for full QCD simulations at small enough quark masses (corresponding to pion masses below 300 MeV), large enough volumes (corresponding to spatial dimensions larger than 2.5 fm) and sufficiently fine lattice spacing (fm) so that the results are not heavily polluted by computational artefacts and can really be connected to the physical quark masses by sensible chiral extrapolations.

For the nuclear force problem, there are two main developments in LQCD to be reviewed here. These concern the extraction of hadron-hadron scattering lengths from unquenched simulations and the first attempts to construct a nuclear potential. These are groundbreaking studies, but clearly at present one has not yet achieved an accuracy to obtain high-precision predictions for nuclear properties. We look very much forward to the development of these approaches in the years to come.

The first exploratory study of the nucleon-nucleon scattering lengths goes back to Fukugita et al. Fukugita:1994na (); Fukugita:1994ve () in the quenched approximation. They make use of an elegant formuala, frequently called the “Lüscher formula”, that relates the S-wave scattering length between two hadrons and to the energy shift of the two-hadron state at zero relative momentum confined in spatial box of size . It is given by Hamber:1983vu (); Luscher:1986pf (); Luscher:1990ux ()

(1.14)

with the reduced mass and and . A generalization of this formalism was given by Beane et al.Beane:2003da () utilizing methods developed for the so-called pionless nuclear EFT (EFT with contact interactions, for a review see e.g. Bedaque:2002mn ()). It reads

(1.15)

which gives the location of all energy eigenstates in the box. Here, is the S-wave phase shift. The sum over all three-vectors of integers is such that and the limit is implicit. In the limit , Eq. (I.5) reduces to the Lüscher formula, Eq. (1.14). On the other hand, for large scattering length, , the energy of the lowest state is given by

(1.16)

with , and is evaluated at the energy . Within this framework, in Ref. Beane:2006mx () the first fully dynamical simulation of the neutron-proton scattering lengths was performed, with a lowest pion mass of 354 MeV. This mass is still too high to perform a precise chiral extrapolation to the physical pion mass, but this calculation clearly demonstrates the feasilbilty of this approach (see also sec. II.8). This scheme can also be extended to hyperon-nucleon interactions, see Beane:2003yx (). A first signal for and scattering was reported in Ref. Beane:2006gf (). For a recent review on these activities of the NPLQCD collaboration, we refer the reader to Beane:2008dv ().

Another interesting development was initiated and carried out by Aoki, Hatsuda and Ishii Ishii:2006ec (). They have generalized the two-center Bethe-Salpeter wavefunction approach of the CP-PACS collaboration Aoki:2005uf (), which offers an alternative to Lüscher’s formula, to the two-nucleon (NN) system. Given an interpolating field for the neutron and for the proton, the NN potential can be defined from the properly reduced 6-quark Bethe-Salpeter amplitude . The resulting Lippmann-Schwinger equation defines a non-local potential for a given, fixed separation . Performing a derivative expansion, the central potential at a given energy is extracted from

(1.17)

Monte Carlo Simulations are then performed to generate the 6-quark Bethe-Salpeter amplitude in a given spin and isospin state of the two-nucleon system on a large lattice, in the quenched approximation for pion masses between 380 and 730 MeV. Despite these approximations, the resulting effective potential extracted using Eq. (1.17) shares the features of the phenomenological NN potentials - a hard core (repulsion) at small separation surrounded by an attractive well at intermediate and larger distances, see Fig. 4.

Figure 4: Effective potential in the channel for three different quark masses in quenched LQCD after Ref. Ishii:2007xz (). The dashed line is the asymptotic OPEP for MeV, GeV and . We are grateful to Dr. N. Ishii for providing us with the data.

Furthermore, the asymptotic form of this potential has exactly the form of the OPE, provided one rescales the formula

(1.18)

with the pion and nucleon masses used in the simulations but keeping the pion-nucleon coupling at its physical value, . These interesting results have led to some enthusiastic appraisal, see e.g. Wilczek:2007 (). However, it is important to stress that the so-calculated potential is not unique, especially its properties at short distances, since it depends on the definition of the interpolating nucleon fields. Furthermore, the quenched approximation is known to have uncontrolled systematic uncertainties as it does not even define a quantum field theory. In this context, the authors of Ref. Aoki:2005uf () report on the numerical absence of the large-distance-dominating -exchange from the flavor-singlet hair-pin diagram. Still, one would like to see this promising calculation repeated with dynamical quarks of sufficiently small masses. In Ref. Nemura:2008sp () this framework was used to study the interaction. Interestingly, the central potential of the interaction looks very similar to the central potential. It would be interesting to extend these calculations to other hyperon-nucleon channels and also study the effects of SU(3) symmetry breaking. We will come back to these issues in the context of an three-flavor chiral EFT in sec. II.6. For recent developments in this scheme concerning the calculation of the tensor force, the energy dependence of the NN potential and preliminary results for full QCD (2+1 flavors), see the talks by Aoki, Ishii, and Nemura at the Lattice 2008 conference Latt08conf ().

i.6 Observables and not-so observable quantities

There is an extensive literature, primarily from the sixties and seventies, on the role of off-shell physics in nuclear phenomena (see, e.g., Ref. SRIVASTAVA75 () and references therein). This includes not only few-body systems (e.g., the triton) and nuclear matter, but interactions of two-body systems with external probes, such as nucleon-nucleon bremstrahlung and the electromagnetic form factors of the deuteron. The implicit premise was that there is a true underlying potential governing the nucleon-nucleon force, so that its off-shell properties can be determined. Indeed, the nuclear many-body problem has traditionally been posed as finding approximate solutions to the many-particle Schrödinger equation, given a fundamental two-body interaction that reproduces two-nucleon observables.

In contrast, effective field theories are determined completely by on-energy-shell information, up to a well-defined truncation error. In writing down the most general Lagrangian consistent with the symmetries of the underlying theory, many-body forces arise naturally. Even though they are usually suppressed at low energies, they enter at some order in the EFT expansion. These many-body forces have to be determined from many-body data. The key point is, however, that no off-energy-shell information is needed or experimentally accessible. A fundamental theorem of quantum field theory states that physical observables (or more precisely, S-matrix elements) are independent of the choice of interpolating fields that appear in a Lagrangian Haag58 (); COLEMAN69 (). Equivalently, observables are invariant under a change of field variables in a field theory Lagrangian (or Hamiltonian):

(1.19)

where is a local polynomial of the field and its derivatives and is an arbitray counting parameter. Newly generated contributions to observables have to cancel separately at each order in . This “equivalence theorem” holds for renormalized field theories. In an EFT, one exploits the invariance under field redefinitions to eliminate redundant terms in the effective Lagrangian and to choose the most convenient or efficient form for practical calculations POLITZER80 (); GEORGI91 (); KILIAN94 (); SCHERER95a (); ARZT95 (). Since off-shell Green’s functions and the corresponding off-shell amplitudes do change under field redefinitions, one must conclude that off-shell properties are unobservable.

Several recent works have emphasized from a field theory point of view the impossibility of observing off-shell effects. In Refs. FEARING98 (); FEARING99 (), model calculations were used to illustrate how apparent determinations of the two-nucleon off-shell T-matrix in nucleon-nucleon bremstrahlung are illusory, since field redefinitions shift contributions between off-shell contributions and contact interactions. Similarly, it was shown in Ref. SCHERER95b () that Compton scattering on a pion cannot be used to extract information on the off-shell behavior of the pion form factor. The authors of Refs. FHK99 (); CFM96 () emphasized the nonuniqueness of chiral Lagrangians for three-nucleon forces and pion production. Field redefinitions lead to different off-shell forms that yield the same observables within a consistent power counting. In Ref. KSW99 (), an interaction proportional to the equation of motion is shown to have no observable consequence for the deuteron electromagnetic form factor, even though it contributes to the off-shell T-matrix.

In systems with more than two nucleons, one can trade off-shell, two-body interactions for many-body forces. This explains how two-body interactions related by unitary transformations can predict different binding energies for the triton ARNAN73 () if many-body forces are not consistently included. These issues were discussed from the viewpoint of unitary transformations in Refs. POLYZOU90 () and AMGHAR95 (). The extension to many-fermion systems in the thermodynamic limit was considered in Furnstahl:2000we (). The effects of field redefinitions were illustrated using the EFT for the dilute Fermi gas Hammer:2000xg (). If many-body interactions generated by the field redefinitions are neglected, a Coester line similar to the one observed for nuclear matter COESTER70 () is generated. Moreover, the connection to more traditional treatments using unitary transformations was elucidated. The question of whether occupation numbers and momentum distributions of nucleons in nuclei are observables was investigated in Ref. Furnstahl:2001xq (). Field redefinitions lead to variations in the occupation numbers and momentum distributions that imply the answer is negative. The natural size of the inherent ambiguity (or scheme dependence) in these quantities is determined by the applicability of the impulse approximation. Only if the impulse approximation is well justified, the ambiguity is small and these quantities are approximately scheme independent. This has important implications for the interpretation of (,) experiments with nuclei. Whether the stark difference in occupation numbers between nonrelativistic and relativistic Brueckner calculations can be explained by this ambiguity is another interesting question Jaminon:1990zz ().

Ii EFT for Few-Nucleon Systems: Foundations and Applications

ii.1 EFT with contact interactions and universal aspects

In nuclear physics, there are a number of EFTs which are all useful for a certain range of systems (cf. Fig. 2). The simplest theories include only short range interactions and even integrate out the pions. At extremely low energies, is given by the NN scattering lengths and one can formulate a perturbative EFT in powers of of the typical momentum divided by . Since the NN scattering lengths are large this theory has a very limited range of applicability. It is therefore useful to construct another EFT with short-range interactions that resums the interactions generating the large scattering length. This so-called pionless EFT can be understood as an expansion around the limit of infinite scattering length or equivalently around threshold bound states. Its breakdown scale is set by one-pion exchange, , while . For momenta of the order of the pion mass , pion exchange becomes a long-range interaction and has to be treated explicitly. This leads to the chiral EFT whose breakdown scale is set by the chiral symmetry breaking scale and will be discussed in detail below.

The pionless theory relies only on the large scattering length and is independent of the mechanism responsible for it. It is very general and can be applied in systems ranging from ultracold atoms to nuclear and particle physics. It is therefore ideally suited to unravel universal phenomena driven by the large scattering length such as limit cycle physics Mohr:2005pv (); Braaten:2003eu () and the Efimov effect Efimov-70 (). For recent reviews of applications to the physics of ultracold atoms, see Refs. Braaten:2004rn (); Braaten:2006vd (). Here we consider applications of this theory in nuclear physics.

The pionless EFT is designed to reproduce the well known effective range expansion. The leading order Lagrangian can be written as:

(2.1)

where the dots represent higher-order terms suppressed by derivatives and more nucleon fields. The Pauli matrices operate in spin (isospin) space, respectively. The contact terms proportional to () correspond to two-nucleon interactions in the () NN channels. Their renormalized values are related to the corresponding large scattering lengths and in the spin-triplet and spin-singlet channels, respectively. The exact relation, of course, depends on the renormalization scheme. Various schemes can be used, such as a momentum cutoff or dimensional regularization. Convenient schemes that have a manifest power counting at the level of individual diagrams are dimensional regularization with PDS subtraction, where poles in 2 and 3 spatial dimensions are subtracted Kaplan:1998we (), or momentum subtractions schemes as in Gegelia:1998xr (). However, a simple momentum cutoff can be used as well.

Since the scattering lengths are set by the low-momentum scale , the leading contact interactions have to be resummed to all orders Kaplan:1998we (); vanKolck:1998bw (). The nucleon-nucleon scattering amplitude in the () channels is obtained by summing the so-called bubble diagrams with the () interactions shown in Fig. 5.


Figure 5: The bubble diagrams with the contact interaction or contributing to the two-nucleon scattering amplitude.

This summation gives the exact solution of the Lippmann-Schwinger equation for the or interactions. Higher order derivative terms which are not shown explicitly in Eq. (2.1) reproduce higher order terms in the effective range expansion. Since these terms are natural and their size is set by , their contribution at low energies is suppressed by powers of and can be treated in perturbation theory. The subleading correction is given by the effective range and the corresponding diagrams are illustrated in Fig. 6.


Figure 6: Diagrams for the inclusion of higher order contact interactions.

The renormalized S-wave scattering amplitude to next-to-leading order in a given channel then takes the form

(2.2)

where is the relative momentum of the nucleons and the dots indicate corrections of order for typical momenta . The pionless EFT becomes very useful in the two-nucleon sector when external currents are considered and has been applied to a variety of electroweak processes. These calculations are reviewed in detail in Refs. Beane:2000fx (); Bedaque:2002mn (). More recently Christlmeier and Grießhammer have calculated low-energy deuteron electrodisintegration in the framework of the pionless EFT Christlmeier:2008ye (). For the double differential cross sections of the reaction at excellent agreement was found with a recent experiment at S-DALINAC Ryezayeva:2008zz ().2 The double-differential cross section for an incident electron energy MeV and is shown in Fig. 7.

Figure 7: Double-differential cross sections of the H reaction with errors (hatched bands) extracted from the experiment. The gray bands and dashed lines are calculations in pionless EFT and a potential model. Figure courtesy of H.W. Grießhammer.

The data were used to precisely map the response which governs the reaction relevant to big-bang nucleosythesis. Finally, the reaction near threshold was studied by Ando Ando:2007in ().

We now proceed to the three-nucleon system. Here it is convenient (but not mandatory) to rewrite the theory using so-called “dimeron” auxilliary fields Kaplan:1996nv (). We need two dimeron fields, one for each S-wave channel: (i) a field with spin (isospin) 1 (0) representing two nucleons interacting in the channel (the deuteron) and (ii) a field with spin (isospin) 0 (1) representing two nucleons interacting in the channel Bedaque:1999ve ():

(2.3)

where are spin and are isospin indices while , , , and are the bare coupling constants. This Lagrangian goes beyond leading order and already includes the effective range terms. The coupling constants , , , are matched to the scattering lengths and effective ranges in the two channels (). Alternatively, one can match to the position of the bound state/virtual state pole in the -matrix instead of the scattering length which often improves convergence Phillips:1999hh (). The two quantities are related through:

(2.4)

where . The term proportional to constitutes a Wigner- symmetric three-body interaction. It only contributes in the spin-doublet S-wave channel. When the auxilliary dimeron fields and are integrated out, an equivalent form containing only nucleon fields is obtained. At leading order when the effective range corrections are neglected, the spatial and time derivatives acting on the dimeron fields are omitted and the field is static. The coupling constants and , are then not independent and only the combination enters in observables. This combination can then be matched to the scattering length or pole position.

The simplest three-body process to consider is neutron-deuteron scattering below the breakup threshold. In order to focus on the main aspects of renormalization, we suppress all spin-isospin indices and complications from coupled channels in the three-nucleon problem. This corresponds to a system of three spinless bosons with large scattering length. If the scattering length is positive, the bosons form a two-body bound state analog to the deuteron which we call dimeron. The leading order integral equation for boson-dimeron scattering is shown schematically in Fig. 8.


Figure 8: The integral equation for the boson-dimeron scattering amplitude. The single (double) line indicates the boson (dimeron) propagator.

For total orbital angular momentum , it takes the following form:

(2.5)

where the inhomogeneous term reads

(2.6)

Here, determines the strength of the three-body force which enters already at leading order and is a UV cutoff introduced to regularize the integral equation. The magnitude of the incoming (outgoing) relative momenta is () and . The on-shell point corresponds to and the phase shift can be obtained via . For and , Eq. (2.5) reduces to the STM equation first derived by Skorniakov and Ter-Martirosian Skorniakov:1957aa (). It is well known that the STM equation has no unique solution Danilov:1961aa (). The regularized equation has a unique solution for any given (finite) value of the ultraviolet cutoff but the amplitude in the absence of the three-body force shows an oscillatory behavior on . Cutoff independence of the amplitude is restored by an appropriate “running” of which turns out to be a limit cycle Bedaque:1998kg (); Bedaque:1998km ():

(2.7)

where is a dimensionful three-body parameter generated by dimensional transmutation. Adjusting to a single three-body observable allows to determine all other low-energy properties of the three-body system. Note that the choice of the three-body parameter is not unique and there are other definitions more directly related to experiment Braaten:2004rn (). Because in Eq. (2.7) vanishes for certain values of the cutoff it is possible to eliminate the explicit three-body force from the equations by working with a fixed cutoff that encodes the dependence on . This justifies tuning the cutoff in the STM equation to reproduce a three-body datum and using the same cutoff to calculate other observables as suggested by Kharchenko Kharchenko:1973aa (). Equivalently, a subtraction can be performed in the integral equation Hammer:2000nf (); Afnan:2003bs (). In any case, one three-body input parameter is needed for the calculation of observables. A comprehensive study of the range corrections to the three-boson spectrum was carried out in Ref. Platter:2008cx (). The authors showed that all range corrections vanish in the unitary limit due to the discrete scale invariance. While the corrections proportional to vanish trivially, this includes also the corrections proportional to where is the binding momentum of the Efimov state fixed by the chosen renormalization condition. Moreover, they have calculated the corrections to the Efimov spectrum for finite scattering length. The range corrections are negligible for the shallow states but become important for the deeper bound states.

The integral equations for the three-nucleon problem derived from the Lagrangian (2.3) are a generalization of Eq. (2.5). (For their explicit form and derivation, see e.g. Ref. Bedaque:2002yg ().) For S-wave nucleon-deuteron scattering in the spin-quartet channel only the spin-1 dimeron field contributes. This integral equation has a unique solution for and there is no three-body force in the first few orders. The spin-quartet scattering phases can therefore be predicted to high precision from two-body data Bedaque:1997qi (); Bedaque:1998mb (). In the spin-doublet channel both dimeron fields as well as the the three-body force in the Lagrangian (2.3) contribute Bedaque:1999ve (). This leads to a pair of coupled integral equations for the T-matrix. Thus, one needs a new parameter which is not determined in the 2N system in order to fix the (leading) low-energy behavior of the 3N system in this channel. The three-body parameter gives a natural explanation of universal correlations between different three-body observables such as the Phillips line, a correlation between the triton binding energy and the spin-doublet neutron-deuteron scattering length Phillips68 (). These correlations are purely driven by the large scattering length independent of the mechanism responsible for it. As a consequence, they occur in atomic systems such as He atoms as well Braaten:2004rn ().

Higher-order corrections to the amplitude including the ones due to 2N effective range terms can be included perturbatively. This was first done at NLO for the scattering length and triton binding energy in Efimov:1991aa () and for the energy dependence of the phase shifts in Hammer:2000nf (). In Refs. Bedaque:2002yg (); Griesshammer:2004pe (), it was demonstrated that it is convenient to iterate certain higher order range terms in order to extend the calculation to NLO. Here, also a subleading three-body force was included as required by dimensional analysis. More recently, Platter and Phillips showed using the subtractive renormalization that the leading three-body force is sufficient to achieve cutoff independence up to NLO in the expansion in Platter:2006ev (). The results for the spin-doublet neutron-deuteron scattering phase shift at LO Bedaque:1999ve (), NLO Hammer:2000nf (), and NLO Platter:2006ad () are shown in Fig. 9.

Figure 9: Phase shifts for neutron-deuteron scattering below the deuteron breakup at LO (dash-dotted line), NLO (dashed line), and NLO (solid line). The filled squares and circles give the results of a phase shift analysis and a calculation using AV18 and the Urbana IX three-body force, respectively. Figure courtesy of L. Platter.

There is excellent agreement with the available phase shift analysis and a calculation using a phenomenological NN interaction. Whether there is a suppression of the subleading three-body force or simply a correlation between the leading and subleading contributions is not fully understood. The extension to 3N channels with higher orbital angular momentum is straightforward Gabbiani:1999yv () and three-body forces do not appear until very high orders. A general counting scheme for three-body forces based on the asymptotic behavior of the solutions of the leading order STM equation was proposed in Griesshammer:2005ga (). A complementary approach to the few-nucleon problem is given by the renormalization group where the power counting is determined from the scaling of operators under the renormalization group transformation Wilson-83 (). This method leads to consistent results for the power counting Barford:2004fz (); Birse:2008wt (); Ando:2008jb (). Universal low-energy properties of few-body systems with short-range interactions and large two-body scattering length were reviewed in Braaten:2004rn (). (See also Efimov:1981aa () for an early work on this subject.) Three-body calculations with external currents are still in their infancy. However, a few exploratory calculations have been carried out. Universal properties of the triton charge form factor were investigated in Ref. Platter:2005sj () and neutron-deuteron radiative capture was calculated in Refs. Sadeghi:2005aa (); Sadeghi:2006aa (). This opens the possibility to carry out accurate calculations of electroweak reactions at very low energies for astrophysical processes.

The pionless approach has also been extended to the four-body sector Platter:2004qn (); Platter:2004zs (). In order to be able to apply the Yakubovsky equations, an equivalent effective quantum mechanics formulation was used. The study of the cutoff dependence of the four-body binding energies revealed that no four-body parameter is required for renormalization at leading order. As a consequence, there are universal correlations in the four-body sector which are also driven by the large scattering length. The best known example is the Tjon line: a correlation between the triton and alpha particle binding energies, and , respectively. Of course, higher order corrections break the exact correlation and generate a band.

Figure 10: The Tjon line correlation as predicted by the pionless theory. The grey circles and triangles show various calculations using phenomenological potentials Nogga:2000uu (). The squares show the results of chiral EFT at NLO for different cutoffs while the diamond gives the NLO result Epelbaum:2000mx (); Epelbaum:2002vt (). The cross shows the experimental point.

In Fig. 10, we show this band together with some some calculations using phenomenological potentials Nogga:2000uu () and a chiral EFT potential with explicit pions Epelbaum:2000mx (); Epelbaum:2002vt (). All calculations with interactions that give a large scattering length must lie within the band. Different short-distance physics and/or cutoff dependence should only move the results along the band. This can for example be observed in the NLO results with the chiral potential indicated by the squares in Fig. 10 or in the few-body calculations with the low-momentum NN potential carried out in Ref. Nogga:2004ab (). The potential is obtained from phenomenological NN interactions by intergrating out high-momentum modes above a cutoff but leaving two-body observables (such as the large scattering lengths) unchanged. The results of few-body calculations with are not independent of but lie all close to the Tjon line (cf. Fig. 2 in Ref. Nogga:2004ab ()). The studies of the four-body system in the pionless theory were extendend further in Ref. Hammer:2006ct (). Here the dependence of the four-body bound state spectrum on the two-body scattering length was investigated in detail and summarized in a generalized Efimov plot for the four-body spectrum.

The question of whether a four-body parameter has to enter at leading order was reanalyzed by Yamashita et al. Yama06 (). Within the renormalized zero-range model, they found a strong sensitivity of the deepest four-body energy to a four-body subtraction constant in their equations. They motivated this observation from a general model-space reduction of a realistic two-body interaction close to a Feshbach resonance. The results of Ref. Platter:2004qn () for the He tetramer that include a four-body parameter were also reproduced. Yamashita et al. concluded that a four-body parameter should generally enter at leading order. They argued that four-body systems of He atoms and nucleons (where this sensitivity is absent Platter:2004qn (); Platter:2004zs (); Nogga:2004ab ()) are special because repulsive interactions strongly reduce the probability to have four particles close together. However, the renormalization of the four-body problem was not explicitly verified in their calculation. Another drawback of their analysis is the focus on the deepest four-body state only. Therefore, their findings could be an artefact of their particular regularization scheme. Another recent study by von Stecher and collaborators vStech08 () confirmed the absence of a four-body parameter for shallow states while some sensitivity was found for the deepest four-body state.

The pionless theory has also been extended to more than four particles using it within the no-core shell model approach. Here the expansion in a truncated harmonic oscillator basis is used as the ultraviolet regulator of the EFT. The effective interaction is determined directly in the model space, where an exact diagonalization in a complete many-body basis is performed. In Ref. Stetcu:2006ey (), the excited state of He and the Li ground state were calculated using the deuteron, triton, alpha particle ground states as input. The first excited state in He is calculated within 10% of the experimental value, while the Li ground state comes out at about 70% of the experimental value in agreement with the 30 % error expected for the leading order approximation. These results are promising and should be improved if range corrections are included. Finally, the spectrum of trapped three- and four-fermion systems was calculated using the same method Stetcu:2007ms (). In this case the harmonic potential is physical and not simply used as an ultraviolet regulator.

ii.2 Chiral EFT for few nucleons: foundations

The extension of the previously discussed EFT with contact interactions to higher energies requires the inclusion of pions as explicit degrees of freedom. The interaction between pions and nucleons can be described in a systematic way using chiral perturbation theory. In contrast, the interaction between the nucleons is strong and leads to nonperturbative phenomena at low energy such as e.g. shallow-lying bound states. This breakdown of perturbation theory can be linked to the fact that the interaction between the nucleons is not suppressed in the chiral limit contrary to the pion and pion-nucleon interactions. Moreover, an additional enhancement occurs for Feynman diagrams involving two and more nucleons due to the appearance of the so-called pinch singularities in the limit of the infinite nucleon mass. Although such infrared singularities disappear if one keeps the nucleon mass at its physical value, they do generate large enhancement factors which destroy the chiral power counting. This can be more easily understood utilizing the language of time-ordered perturbation theory. Consider, for example, the two-pion exchange box diagram shown in Fig. 11.

Figure 11: Representation of the two-pion exchange Feynman diagram in terms of time-ordered graphs. Solid and dashed lines represent nucleons and pions, respectively.

While all intermediate states in the first two time-ordered graphs, often referred to as irreducible, involve at least one virtual pion and thus lead to energy denimonators of the expected size, , the remaining reducible diagrams involve an intermediate state with nucleons only which produces unnaturally small energy denominators of the order . Clearly, the enhanced reducible time-ordered diagrams are nothing but the iterations of the Lippmann-Schwinger equation with the kernel which contains all possible irreducible diagrams and defines the nuclear Hamiltonian. It is free from infrared enhancement factors and can be worked out systematically using the machinery of chiral perturbation theory as suggested in Weinberg’s seminal work Weinberg:1990rz (); Weinberg:1991um ().3 This natural reduction to the quantum mechanical -body problem is a welcome feature for practical calculations as it allows to apply various existing few-body techniques such as e.g. the Faddeev-Yakubovsky scheme, the no-core shell model, Green’s function Monte Carlo and hyperspherical harmonics methods. On the other hand, the framework offers a systematic and perturbative scheme to derive nuclear forces and current operators in harmony with the chiral symmetry of QCD. The expansion parameter is given by the ratio where is the soft scale associated with the pion mass and/or external nucleon momenta and is the pertinent hard scale. For a given connected irreducible diagram with nucleons, pion loops and vertices of type , the power of the soft scale which determines its importance can be obtained based on naive dimensional analysis (i.e. assuming classical scaling dimensions for various operators in the effective Lagrangian):

(2.8)

Here, is the number of nucleon field operators and the number of derivatives and/or insertions of . The spontaneously broken chiral symmetry of QCD guarantees . As a consequence, the chiral dimension is bounded from below, and only a finite number of diagrams contribute at a given order. In addition, Eq. (2.8) provides a natural explanation to the dominance of the two-nucleon interactions and the hierarchy of nuclear forces observed in nuclear physics. In particular, it implies that two-, three-, and four-nucleon forces start to contribute at orders , and , respectively. Notice that as argued in Ref. Weinberg:1991um (), the nucleon mass should be counted as (which implies that ) in order to maintain consistency with the appearance of shallow-lying bound states.4 Notice further that according to this counting rule, the momentum scale associated with the real pion production is treated as the hard scale, , and needs not be explicitly kept track of, see also Mondejar:2006yu () and section II.5 for a related discussion. Clearly, such a framework is only applicable at energies well below the pion production threshold. We also emphasize that the validity of the naive dimensional scaling rules for few-nucleon contact operators has been questioned in Refs. Nogga:2005hy (); Birse:2005um (). We will come back to this issue in section II.3.

Before discussing the chiral expansion of the nuclear forces it is important to clarify the relation between the underlying chiral Lagrangian for pions and nucleons and the nuclear Hamiltonian we are finally interested in. The derivation of the nuclear potentials from field theory is an old and extensively studied problem in nuclear physics. Different approaches have been developed in the fifties of the last century in the context of the so-called meson theory of nuclear forces, see e.g. the review article Phillips:1959aa (). In the modern framework of chiral EFT, the most frequently used methods besides the already mentioned time-ordered perturbation theory are the ones based on S-matrix and the unitary transformation. In the former scheme, the nuclear potential is defined through matching the amplitude to the iterated Lippmann-Schwinger equation Kaiser:1997mw (). In the second approach, the potential is obtained by applying an appropriately choosen unitary transformation to the underlying pion-nucleon Hamiltonian which eliminates the coupling between the purely nucleonic Fock space states and the ones which contain pions, see Epelbaum:1998ka () for more details. We stress that both methods lead to energy-independent interactions as opposed by the ones obtained in time-ordered perturbation theory. The energy independence of the potential is a welcome feature which enables applications to three- and more-nucleon systems.

We are now in the position to discuss the structure of the nuclear force at lowest orders of the chiral expansion. The leading-order (LO) contribution results, according to Eq. (2.8), from two-nucleon tree diagrams constructed from the Lagrangian of lowest dimension , , which has the following form in the heavy-baryon formulation Jenkins:1990jv (); Bernard:1992qa ():

(2.9)

where , and denote the large component of the nucleon field, the nucleon four-velocity and the covariant spin vector, respectively. The brackets denote traces in the flavor space while and refer to the chiral-limit values of the pion decay and the nucleon axial vector coupling constants. The low-energy constants (LECs) and determine the strength of the leading NN short-range interaction. Further, the unitary matrix in the flavor space collects the pion fields,

(2.10)

where denotes the isospin Pauli matrix. The covariant derivatives of the nucleon and pion fields are defined via and . The quantity with involves the explicit chiral symmetry breaking due to the finite light quark masses, . The constant is related to the value of the scalar quark condensate in the chiral limit, , and relates the pion mass to the quark mass via . For more details on the notation and the complete expressions for the pion-nucleon Lagrangian including up to four derivatives/-insertions the reader is referred to Fettes:2000gb (). Expanding the effective Lagrangian in Eqs. (2.9) in powers of the pion fields one can easily verify that the only possible connected two-nucleon tree diagrams are the one-pion exchange and the contact one, see the first line in Fig. 12,

Figure 12: Chiral expansion of the two-nucleon force up to NLO. Solid dots, filled circles, squares and diamonds denote vertices with , , and , respectively. Only irreducible contributions of the diagrams are taken in to account as explained in the text.

yielding the following potential in the two-nucleon center-of-mass system (CMS):

(2.11)

where the superscript of denotes the chiral order , are the Pauli spin matrices, is the nucleon momentum transfer and () refers to initial (final) nucleon momenta in the CMS. Further, MeV and denote the pion decay and the nucleon axial coupling constants, respectively.

The first corrections to the LO result are suppressed by two powers of the low-momentum scale. The absence of the contributions at order can be traced back to parity conservation which forbids vertices with one spatial derivative and vertices with two derivatives (i. e. ). The next-to-leading-order (NLO) contributions to the 2NF therefore result from tree diagrams with one insertion of the -interaction and one-loop diagrams constructed from the lowest-order vertices, see Fig. 12. The relevant terms in the effective Lagrangian read Gasser:1987rb ()

(2.12)

where , and denote further LECs and is the nucleon mass in the chiral limit. The ellipses in the pion and pion-nucleon Lagrangians refer to terms which do not contribute to the nuclear force at NLO. In the case of the nucleon-nucleon Lagrangian only a few terms are given explicitly. The complete reparametrization-invariant set of terms can be found in Epelbaum:2000kv (). The NLO contributions to the two-nucleon potential have been first considered in Ordonez:1993tn (); Ordonez:1995rz () utilizing the framework of time-ordered perturbation theory. The corresponding energy-independent expressions have been worked out in Friar:1994zz () using the method described in Friar:1977xh () and then re-derived in Kaiser:1997mw () using an S-matrix-based approach and, independently, in Epelbaum:1998ka (); Epelbaum:1999dj () based on the method of unitary transformation. The one-pion () exchange diagrams at NLO do not produce any new momentum dependence. Apart from renormalization of various LECs in Eq. (2.11), one obtains the leading contribution to the Goldberger-Treiman discrepancy Epelbaum:2002gb ()

(2.13)

where the ellipses refer to higher-order terms. Similarly, loop diagrams involving NN short-range interactions only lead to (-dependent) shifts in the LO contact terms. The remaining contributions to the 2NF due to higher-order contact interactions and two-pion exchange have the form:

(2.14)

where and the LECs can be written as linear combinations of in Eq. (2.12). The loop function is defined in the spectral function regularization (SFR) Epelbaum:2003gr (); Epelbaum:2003xx () as

(2.15)

where we have introduced the following abbreviations: and . Here, denotes the ultraviolet cutoff in the mass spectrum of the two-pion-exchange potential. If dimensional regularization (DR) is employed, the expression for the loop function simplifies to

(2.16)

In addition to the two-nucleon contributions, at NLO one also needs to consider three-nucleon diagrams shown in the first line of Fig. 13.

Figure 13: Chiral expansion of the three-nucleon force up to NLO. Diagrams in the first line (NLO) yield vanishing contributions to the 3NF if one uses energy-independent formulations as explained in the text. The five topologies at NLO involve the two-pion exchange, one-pion-two-pion-exchange, ring, contact-one-pion exchange and contact-two-pion-exchange diagrams in order. Shaded blobs represent the corresponding amplitudes. For remaining notation see Fig. 12.

The first diagram does not involve reducible topologies and, therefore, can be dealt with using the Feynman graph technique. It is then easy to verify that its contribution is shifted to higher orders due to the additional suppression by the factor of caused by the appearance of time derivative at the leading-order vertex, the so-called Weinberg-Tomozawa vertex. The two remeining diagrams have been considered by Weinberg Weinberg:1990rz (); Weinberg:1991um () and later by Ordonez and van Kolck Ordonez:1992xp () using the energy-dependent formulation based on time-ordered perturbation theory. In this approach, it was shown that the resulting 3NF cancels exactly (at the order one is working) against the recoil correction to the 2NF when the latter is iterated in the dynamical equation. In energy-independent approaches such as e.g. the method of unitary transformation which are employed in most of the existing few-nucleon calculations one observes that the irreducible contributions from the last two diagrams in the first line of Fig. 13 are suppressed by the factor and thus occur at higher orders Epelbaum:2000kv (), see also Coon:1986kq (); Eden:1996ey (). Consequently, there is no 3NF at NLO in the chiral expansion.

The contributions at next-to-next-to-leading order (NLO) involve one-loop diagrams with one insertion of the subleading vertices of dimension , see Fig. 12. The corresponding Lagrangians read: