SNUTP 92-97

May 25, 2023
Chiral Dynamics and Heavy-Fermion Formalism in Nuclei: I. Exchange Axial Currents
Tae-Sun Park

Department of Physics and Center for Theoretical Physics

Seoul National University

Seoul 151-742, Korea and Dong-Pil Min***Permanent address: Department of Physics, Seoul National University, Seoul 151-742, Korea. and Mannque Rho

Service de Physique Théorique, CEA Saclay

91191 Gif-sur-Yvette, France
ABSTRACT

Chiral perturbation theory in heavy-fermion formalism is developed for meson-exchange currents in nuclei and applied to nuclear axial-charge transitions. Calculation is performed to the next-to-leading order in chiral expansion which involves graphs up to one loop. The result turns out to be very simple. The previously conjectured notion of “chiral filter mechanism” in the time component of the nuclear axial current and the space component of the nuclear electromagnetic current is verified to that order. As a consequence, the phenomenologically observed soft-pion dominance in the nuclear process is given a simple interpretation in terms of chiral symmetry in nuclei. In this paper we focus on the axial current, relegating the electromagnetic current which can be treated in a similar way to a separate paper. We discuss the implication of our result on the enhanced axial-charge transitions observed in heavy nuclei and clarify the relationship between the phenomenological meson-exchange description and the chiral Lagrangian description.

## 1 Introduction

By now there exist a large number of unambiguous experimental evidences  for meson-exchange currents in nuclear responses to electroweak probes. We also have available a rather satisfactory and successful theory to describe the large bulk of experimental observations . While inherently phenomenological in character, the approaches taken so far to describe meson-exchange currents have been commensurate with the ingredients that account for our progressive understanding of nuclear forces and to the extent that nucleon-nucleon interactions are now fairly accurately understood, one can have a great deal of confidence in the theoretical tool with which the effect of exchange currents is calculated. There remains however the fundamental question as to how our phenomenological understanding of nuclear forces and associated meson currents can be linked to the fundamental theory of strong interactions, QCD.

In this paper we make a first step towards answering this question by applying chiral perturbation theory (ChPT) to nuclear electroweak processes. To start with, we assume that at low energies dominated by infrared properties of QCD, the most important aspect of QCD is the spontaneously broken chiral symmetry and hence that in nuclear dynamics, it is chiral symmetry that plays a predominant role . The important role of chiral symmetry in nuclear physics was recognized early on by Chemtob and Rho for exchange currents  but this issue was recently given a stronger impetus and a more modern meaning by Weinberg in connection with nuclear forces  and by Rho  in connection with what is known “nuclear chiral filter phenomenon” (for definition, both intuitive and more rigorous, see later).

The key question addressed here is this: To what extent can nuclear processes be described by QCD or equivalently at low energies by chiral perturbation theory? Weinberg approaches this issue by studying nuclear many-body forces. We propose here to do the same by looking at nuclear response functions responding to slowly varying electroweak fields . We suggest that chiral perturbation theory can be made – under certain conditions specified below – considerably more powerful and predictive for response functions than for nuclear forces. In calculating nuclear forces to loop orders in chiral perturbation theory, one encounters a plethora of counter terms to renormalize the theory, most of which are not accessible by experiments ; furthermore there are contact four-fermion interactions in the Lagrangian – most of which are again unknown parameters – that have to be carefully examined and treated. In principle, this may be feasible, perhaps with the help of lattice QCD calculations but in practice it may not be possible to make clear and useful predictions because of many uncontrolled parameters. While the tree order chiral theory justifies à posteriori the current nuclear physics practice of using two-body static forces , it appears that chiral symmetry will be unable to make any truly significant statement on the structure of nuclear forces for sometime to come. A major new development will be required before one can make a prediction that goes beyond the accuracy of the phenomenological approach which has been strengthened by the wealth of experimental data. On the contrary, as we will show in this paper, when the formalism is applied to nuclear response functions, in particular, to exchange currents, it can make a highly nontrivial and potent prediction. This is because nuclear short-range correlations generated by nuclear interactions at short distance – which while poorly understood of their mechanism, are nonetheless operative in nuclear medium – screen all the contact interactions, both intrinsic and induced and consequently all of the four-fermion (and higher) counter term contributions, effectively “filtering” off the ill-understood short-range operators: Given phenomenological information on nuclear wave functions at short distance, the short-range suppression helps in simplifying nuclear response functions. In addition, in certain kinematic conditions, higher order chiral corrections are found to be naturally suppressed. The suppression of the many-fermion counter term contributions at the one-loop order that we are studying is, as will be stated more precisely later, a consequence of the fact that such terms occurring at high orders in the chiral expansion reflect the degrees of freedom that enter directly neither in nuclear forces nor in nuclear currents at the chiral order considered. The combination of these two phenomena lead to the “chiral filtering” proposed previously . In this paper, we will establish this chiral filtering to one-loop order in chiral perturbation theory. This will provide, in our opinion, the very first compelling explanation of the pion-exchange dominance observed in axial charge transitions (considered in this paper) as well as in radiative np capture or in threshold electrodisintegration of the deuteron.

As stressed by Weinberg, chiral perturbation theory is useful in nuclear physics only for “irreducible” diagrams that are by choice free of infrared divergences. This means that both in nuclear forces and in exchange currents, reducible graphs are to be taken care of by a Schrödinger equation or its relativistic generalization with the irreducible graphs entering as potentials. This also implies that in calculating exchange currents in ChPT, we are to use the wave functions so generated to calculate matrix elements to obtain physical amplitudes. This is of course the standard practice in the theory of meson-exchange currents but it is also in this sense that ChPT is predictive in nuclei. Clearly this precludes what one might call “fully consistent chiral perturbation theory” where nuclear forces, nuclear currents and wave functions are all calculated to the same order of chiral perturbation expansion. Such a calculation even if feasible is likely to make no sense. A little thought would persuade the reader that it is a futile exercise.

We will here focus on the irreducible diagrams contributing to exchange currents. We will calculate next-to-leading order terms in the chiral counting involving one-loop graphs. In doing this, we will employ the recently developed heavy fermion formalism (HFF) . The standard ChPT  arranges terms in power of and/or of where is four-derivative acting on the Goldstone boson (viz, pion) field, the pion mass ( MeV) and GeV, the chiral expansion scale. It has been established that this expansion works well at low energies for such processes as scattering. However the situation is different when baryons are involved. The dynamically generated masses of the baryons are of and hence when the baryon field is acted upon by time-derivative, it gives an term. Therefore a straightforward derivative expansion fails. (Incidentally this is also the reason why a chiral Lagrangian describing pion interactions well with low-order derivative terms does not necessarily describe well skyrmion properties.) The HFF circumvents this difficulty in rearranging the derivative expansion. Indeed the principal virtue of the HFF is that it provides a consistent chiral expansion in where is four-derivative on pion field or pion mass or space derivative on baryon field; it avoids time derivative on baryon field which is of order which is not small. The standard ChPT involving baryons  can in principle be arranged to give a similar expansion. However it requires a laborious reshuffling of terms avoided in the HFF. The distinct advantage of the HFF is that the multitude of diagrams that appear in such calculations as ours in the standard ChPT involving baryons get reduced to a handful of manageable terms, thus alleviating markedly the labor involved. We will see that there is an enormous simplification in the number of terms and in their expressions. The potential disadvantage might be that the HFF is not fully justified for the mass corresponding to that of the nucleon and hence higher order “” corrections may have to be systematically included. We will examine the class of approximations we make in the calculation by looking at the next order terms. It turns out that the leading “” correction is absent in our calculation. We shall discuss this matter in the concluding section.

While the procedure is practically the same, the resulting expression for electromagnetic (EM) current is somewhat more involved. We will therefore not treat it here although we shall give a general treatment of the theory applicable to both axial and EM currents. The detailed analysis on the EM currents, together with an application to threshold np capture, will be reported in a separate paper . Both currents are intricately connected even at low energy through current algebras and we will need some vertices involving the EM current.

It is perhaps obvious but we should stress that for both vector and axial-vector currents, relevant symmetries (i.e, conserved vector current and partially conserved axial-vector current) are preserved to the chiral order considered since both nuclear forces and currents are treated on the same footing with the same effective Lagrangian. More on this point later.

This paper is organized as follows. In Section 2, we state our basic assumption in applying ChPT to nuclear dynamics. In Section 3, we describe the effective chiral Lagrangian with which we develop heavy-fermion formalism including “” corrections. We also define the relevant kinematics we will consider. The chiral counting rules are given in Section 4. In Section 5, the renormalization of n-point vertices that enter in the calculation is detailed. For the sake of making this paper as self-contained as possible and to define notations, we also list the renormalized quantities for the pion and the nucleon following from the Lagrangian. Readers familiar with renormalization of heavy-fermion chiral Lagrangian could proceed directly to subsection 5.4. Two-body exchange currents are calculated in Section 6. Both momentum-space and coordinate-space formulas are given. Numerical analyses are described in Section 7. In Section 8, we explain why there are no other graphs that can contribute to the same order and point out in what circumstances they can show up in physical observables. Concluding remarks including those on the observed enhanced axial-charge transitions in heavy nuclei are made in Section 9. The Appendices A-I list all the formulas needed in the calculation.

This paper is written in as a self-contained way as possible so as to be readable by those who are not familiar with the recent development in the field. Some of the material are quite standard and readily available in the literature. Most of them however serve as a check of our calculation.

## 2 Strategies in Nuclear Physics

We wish to calculate operators effective in nuclei for transitions induced by the vector and axial vector currents of electroweak interactions, denoted respectively by and associated with the electroweak fields and . In principle there will be -body currents for in -body systems. Here we will focus only on one- and two-body currents, ignoring those with . The reasons for so doing are given in the literature [2, 4] but we will later show that -body currents for are suppressed to the order considered for long wavelength probes.

The diagrams we wish to calculate are generically given by Fig. 1. They correspond to the standard definition of single-particle and two-particle exchange currents entering in the description of nuclear response functions to the external electroweak fields. These have been calculated before in terms of phenomenological Lagrangians. Here we wish to do so using chiral perturbation theory (ChPT), starting with a chiral Lagrangian that is supposed to model QCD at low energies. Following the chiral counting rule we will derive later, we will restrict our consideration to one-loop order, which corresponds to going to the next order in the chiral counting to the leading soft-pion limit. Although one-loop calculations have been done before for nucleon properties  and for infinite nuclear matter , they have up to date not been performed in finite nuclear processes. We believe this work constitutes the first attempt to implement consistently chiral symmetry in nuclear processes.

In dealing with divergences encountered in calculating loop graphs, in particular the loops involving two-pion exchange, we will need a certain prescription for handling operators that are short-ranged in coordinate space. This prescription does not follow from chiral symmetry alone and will have to be justified on a more general ground. Specifically, we argue that consistency with ChPT demands that zero-range interactions be “killed” by nuclear short-range correlations:

• Firstly, the zero-range operators that come from finite counter terms appearing in four-fermion interactions figure neither directly nor indirectly – but importantly – in the successful phenomenological nucleon-nucleon potentials and hence must represent the degrees of freedom unimportant for the length scale involved. In fact, one can show (see Appendix I) that the counter terms we need to introduce (denoted later) cannot arise, unlike in the better understood - scattering , from an approximation of taking an infinite mass limit of the strong interaction resonances such as the vector mesons , etc. which have a scale comparable to the chiral scale and play an important role in boson-exchange potential models.

• Secondly, ChPT by its intrinsic limitation cannot possibly provide a nuclear force that can account for the interactions shorter-ranged than two pion or one vector-meson range at most. Thus the truly short-range interaction known to be present in the nucleon-nucleon interaction must involve elements that are not calculable by means of finite-order chiral expansion even if such an expansion existed. Thus it would be inconsistent to put a part of such interactions into the currents in the context of ChPT without a similar account in the nuclear force. It is known that even to one-loop order, the number of counter terms is so large in the calculation of nuclear forces that it is highly unlikely that one can make a meaningful prediction based strictly on low-order chiral perturbation expansion . As suggested in Ref., one should implement ChPT calculations with phenomenological informations whenever available.

• Applied to the “irreducible diagrams” that enter in the definition of exchange currents, ChPT screens out the short-range part of the interaction which originates from dynamics of possibly non-chiral origin. When the matrix elements of the operators arising from the irreducible graphs are calculated with wave functions suitably computed in the presence of two-nucleon potentials, the short-range correlation built into nuclear wavefunctions must therefore suppress strongly interactions that occur at an internuclear distance fm, automatically “killing” the function interactions associated with finite counter terms. This fact will be kept in mind when we derive two-body operators in coordinate space.

We will present, at several places in the paper, arguments to justify the above procedure which purports to establish that the only unknown parameters in the theory must be (a) negligible in magnitude and (b) further suppressed by nuclear correlations when embedded in nuclear medium.

There is nothing very much new in our calculation of the one-body operators except for its consistency with chiral invariance. As for the two-body exchange currents, our results are new. There are two graphs to consider: One-pion exchange (Fig. 2a) and two-pion exchange (Fig. 2b). Both involve one-loop order graphs. Note that we are to calculate only “irreducible graphs.”

## 3 Effective Chiral Lagrangian

We begin with the effective chiral Lagrangian that consists of pions and nucleons involving lowest derivative terms relegating the role of other degrees of freedom such as vector mesons and nucleon resonances to a later publication#1#1#1While the vector mesons and the nucleon resonances (in particular, the ) play an important role in nuclear phenomenology – and they can be easily implemented in ChPT at least in low orders, they are unimportant for the process we discuss in this paper. It is not difficult to see which processes require such degrees of freedom but we will not pursue this matter, for a treatment of such processes goes beyond the framework of ChPT.,

 L0 = ¯¯¯¯¯N[iγμ(∂μ+Γμ)−m+igAγμγ5Δμ]N−12Ca(¯¯¯¯¯NΓaN)2 (1) +F24Tr(∇μΣ†∇μΣ)+12M2F2Tr(Σ)+⋯+L\tiny CT,

where MeV is the nucleon mass, is the axial coupling constant and MeV is the pion decay constant. The ellipsis stands for higher derivative and/or symmetry-breaking terms which will given later as needed. We have written the Lagrangian with the renormalized parameters , , and with suitable counter terms to be specified later. Under the chiral SU(2)SU(2) transformation#2#2#2We are using a slightly unconventional notation of Ref.  which we will follow in this paper. This facilitates checking our results on single-nucleon properties against those derived in  using standard (relativistic) chiral perturbation expansion. The more familiar transformation of the chiral field used in the literature is gotten by replacing by . We are also working with the exponentiated (Sugawara) form of chiral Lagrangian instead of Weinberg’s  used previously. They are of course equivalent. For the rest we will follow the Bjorken-Drell metric and convention., the chiral field transforms as ( SU(2)) and the covariant derivative of the chiral field transforms as does,

 ∇μΣ = ∂μΣ−i(Vμ+Aμ)Σ+iΣ(Vμ−Aμ) (2) → g\tiny R∇μΣg†\tiny L

where the external gauge fields and transform locally

 Vμ+Aμ→V′μ+A′μ=g\tiny R(Vμ+Aμ)g†\tiny R−i∂μg\tiny R⋅g†% \tiny R, Vμ−Aμ→V′μ−A′μ=g\tiny L(Vμ−Aμ)g†\tiny L−i∂μg\tiny L⋅g†% \tiny L.

In our work, only the electroweak () external fields will be considered. The Lagrangian of course has global invariance in the absence of the pion mass term. Non-linear realization of chiral symmetry is expressed in terms of and defined with

 ξ→g\tiny RξU†=Uξg†\tiny L.

Now nucleon field transforms as , and covariant derivatives of nucleon field and chiral field transform as nucleon field does, and where#3#3#3 We have defined two covariant derivatives involving chiral fields, and . We can express one in terms of the other, but it is convenient as done frequently in the literature to use for the meson sector and for the meson-nucleon sector.

 DμN = (∂μ+Γμ)N, Γμ = 12[ξ†,∂μξ]−i2ξ†(Vμ+Aμ)ξ−i2ξ(Vμ−Aμ)ξ†, Δμ = 12ξ†(∇μΣ)ξ†=12{ξ†,∂μξ}−i2ξ†(Vμ+Aμ)ξ+i2ξ(Vμ−Aμ)ξ†. (3)

The can be expressed as a complicated local function of , and . The explicit form of is not needed for our discussion.

Note that we have included the four-fermion non-derivative contact term studied recently by Weinberg. We will ignore possible four-fermion contact terms involving derivatives (except for counter terms encountered later) and quark mass terms since they are not relevant to the chiral order (in the sense defined precisely later) that we are working with. The explicit chiral symmetry breaking is included minimally in the form of the pion mass term. Higher order symmetry breaking terms do not play a role in our calculation.

### 3.1 Heavy-fermion formalism

For completeness – and to define our notations, we sketch here the basic element of the heavy-fermion formalism (HFF) applied to nuclear systems as developed by Jenkins and Manohar wherein the nucleon is treated as a heavy fermion. As stressed in Introduction, the relativistic formulation of ChPT works well when only mesons are involved but it does not work when baryons are involved since while space derivatives on baryons fields can be arranged to appear on the same footing as four-derivatives on pion fields, the time derivative on baryon fields picks up a term of order of the chiral symmetry breaking scale and hence cannot be used in the chiral counting. This problem is avoided in the HFF. To set up the HFF, the fermion momentum is written as

 pμ=mvμ+kμ (4)

where is the velocity with , and is the small residual momentum. (In the practical calculation that follows, we will choose the heavy-fermion rest frame .) We define heavy fermion field for a given four-velocity , by#4#4#4Another familiar field redefinition is , with . This definition gives exactly the same physics to the lowest order in expansion.

 Bv(x)=eimv⋅xN(x). (5)

The field is divided into two parts which are eigenstates of ,

 Bv=B(+)v+B(−)v≡1+⧸v2Bv+1−⧸v2Bv≡P+Bv+P−Bv. (6)

As defined, can be identified as positive (negative) energy solution. As will be justified in the following subsection, the negative energy solution is suppressed for large baryon mass and its contribution can subsequently be incorporated as higher-order corrections in the inverse mass expansion. Thus to the leading order, the fermion loops can be ignored. With the neglect of the negative energy solutions, we have a useful relation for gamma matrices sandwiched between spinors which holds for any ,

 ¯¯¯¯BvΓBv=¯¯¯¯Bv⧸vΓBv=¯¯¯¯BvΓ⧸vBv=¯¯¯¯Bv12{Γ,⧸v}Bv. (7)

It follows from this identity that

 ¯¯¯¯Bvγ5Bv=0,  ¯¯¯¯BvγμBv=vμ¯¯¯¯BvBv. (8)

Let us define spin operators by

 ¯¯¯¯Bvγ5γμBv=−2¯¯¯¯BvSμvBv (9)

or explicitly

 Sμv=14γ5[⧸v,γμ]. (10)

The spin operators have the following identities,

 {Sμv,Sνv} = 12(vμvν−gμν), (11) [Sμv,Sνv] = iϵμναβvαSβ   % with   ϵ0123=1. (12)

From the anti-commutation rule, we have

 Sv⋅Sv = 14(1−d)=−3−2ϵ4≃−34, (13) SαvSμvSvα = 14(d−3)Sμv≃14Sμv, (14) (q⋅Sv)2 = 14[(q⋅v)2−q2] (15)

where is the dimension of the space-time, and we have defined . Between spinors, we have the approximate relations

 ¯¯¯¯BvSμvBv ≃ (12→σ⋅→v,12→σ), (16) ¯¯¯¯Bv[S0v,→Sv]Bv ≃ −i2→v×→σ (17)

with the usual Pauli spin matrices. We see that and are suppressed by a factor of where is the characteristic small momentum scale for processes with small three-velocity. Since

 [Sμv,Sνv] = i4(σμν+⧸vσμν⧸v) = i2σμν+(vμSνv−vνSμv)γ5

where , we also have

 ¯¯¯¯Bv(i2σμν)Bv = ¯¯¯¯Bv[Sμv,Sνv]Bv (18) ¯¯¯¯Bv(σμνγ5)Bv = 2i¯¯¯¯Bv(vμSνv−vνSμ)Bv. (19)

We are now in position to write down the chiral Lagrangian (1) in HFF. The nucleon part of the Lagrangian becomes

 ¯¯¯¯¯N(i⧸∂−m)N=¯¯¯¯Bviv⋅∂Bv (20)

and the corresponding nucleon propagator is#5#5#5 Although we do not actually encounter it in our calculation, it might be worthwhile to point out one technical subtlety. The infinitesimal is inserted to define the singularity structure of the propagator. When we encounter a dimensional loop integral we first perform the Wick-rotation to put it in the Euclidean metric. In doing this, we assume that the first and third quadrants (in the plane of real vs. imaginary ) contain no poles. If we can take the flow direction of the loop-momentum to be the direction of the fermion momentum, there is no problem. However for some graphs it is impossible to do this. For instance consider a two-nucleon box diagram. In this case, we have one fermion line in which the loop-momentum flow direction is opposite to that of the fermion arrow. In this case, the fermion propagator is of the form

 iS(mv+k)=iv⋅k+i0+. (21)

Our chiral Lagrangian (1) expressed in terms of the heavy-fermion field to leading (i.e, zeroth) order in takes the form

 L0 = ¯¯¯¯Bv[ivμ(∂μ+Γμ)+2igASμvΔμ]Bv−12Ca(¯¯¯¯BvΓaBv)2 (22) +F24Tr(∇μΣ†∇μΣ)+12M2F2Tr(Σ).

In practical calculations, the chiral field or is expanded in power of the pion field. The explicit form resulting from such expansion as well as the vector and axial-vector currents calculated via Noether’s theorem are given in Appendix A.

### 3.2 1m corrections

As mentioned, the HFF is based on simultaneous expansion in the chiral parameter and in “”. We have so far considered leading-order terms in , namely, . We now discuss corrections following closely the discussion of Grinstein . We choose to do this in a perhaps more general way than needed for our purpose. Consider the following Lagrangian

 L = ¯¯¯¯¯N[i⧸D−m+γμγ5Aμ]N−12Ca(¯¯¯¯¯NΓaN)2 = ¯¯¯¯B[i⧸D−m(1−⧸v)+γμγ5Aμ]B−12Ca(¯¯¯¯BΓaB)2

where we have included Weinberg’s four-fermion contact term with an arbitrary hermitian operator which we assume to contain no derivatives. We have also introduced an arbitrary “axial” field which we take to be hermitian and free of gamma matrices. (Here and in what follows, we shall omit the subscript in , and .) The equation of motion satisfied by is

 [G−m(1−⧸v)]B=0 (23)

with

 G≡g−CaΓa(¯¯¯¯BΓaB),   g≡i⧸D+γμγ5Aμ. (24)

Multiplying (23) on the left by , we obtain

 P−GB−2mB(−)=0

 B(−)=12mP−GB=12mP−GB(+)+O(1m2). (25)

Now multiplying to (23), we get

 P+GB=0

which gives

 P+[G+12mGP−G]B(+)=O(1m2). (26)

Given this, it is now a simple matter to write down the Lagrangian that gives rise to the equation of motion to the desired order #6#6#6We remind the reader that one should not insert the solution of into the original Lagrangian, since in Lagrangian approach, what is important is the form, not the value. However in Hamiltonian approach, the insertion of the solution for into the original Hamiltonian is allowed.. The result correct to is

 L = ¯¯¯¯B[g+12mgP−g]B (27) −12Ca{¯¯¯¯B[Γa+12m(ΓaP−g+gP−Γa)]B}2 +12mCaCb(¯¯¯¯BΓaB)(¯¯¯¯BΓaP−ΓbB)(¯¯¯¯BΓbB)

with defined in (24). To put this into a more standard form, we use the identities

 P+gP+ = P+12{⧸v,g}P+, P+gP−g′P+ = P+{P−,g}g′P+=P+g{P−,g′}P+, P+gP−gP+ = P+[12{⧸v,g2}−14{g,g{⧸v,g}}]P+. (28)

One can show from these identities that

 (¯¯¯¯B(+)ΓaB(+))⊗¯¯¯¯B(+)ΓaP−=0 (29)

for any . This allows us to simplify the Lagrangian further to the form

 L = ¯¯¯¯B(iv⋅D+2S⋅A)B−12Ca(¯¯¯¯BΓaB)2+12m¯¯¯¯B(−D2+(v⋅D)2 (30) +[Sμ,Sν][Dμ,Dν]−(v⋅A)2−2i{v⋅A,S⋅D})B,

for general allowed by symmetries. In our case, , so our term Lagrangian is

 δL = 12m¯¯¯¯B(−D2+(v⋅D)2+[Sμ,Sν][Dμ,Dν]+g2A(v⋅Δ)2+2gA{v⋅Δ,S⋅D})B (31) + O(1m2).

While Eq.(31) is the first “” correction, it is not the entire story to the order considered. One can see that it is also the next order in the chiral counting in derivatives and it is expected in any case, independently of the inverse baryon mass corrections. Thus in a practical sense, the coefficients that appear in each term are parameters rather than fixed by chiral symmetry in HFF. We also note that in the derivation given above, neither corrections to the quartic fermion term nor sixth-order fermion terms are generated by the expansion. This of course does not mean that such terms cannot contribute on a general ground. We have however checked that no such terms arise from exchanges of single heavy mesons that are formally “integrated out” which means that our calculations are not affected by such terms, at least to the order we are concerned with.

## 4 Counting Rules

In this section, we rederive and generalize somewhat Weinberg’s counting rule using HFF. Although we do not consider explicitly the vector-meson degrees of freedom, we include them here in addition to pions and nucleons. Much of what we obtain turn out to be valid in the presence of vector mesons. Now in dealing with them, their masses – which are comparable to the chiral scale – will be regarded as heavy compared to the momentum probed – say, scale of external three momenta or .

In establishing the counting rule, we make the following key assumptions: Every intermediate meson (whether heavy or light) carries a four-momentum of order of . In addition we assume that for any loop, the effective cut-off in the loop integration is of order of . We will be more precise as to what this means physically when we discuss specific processes, for this clarifies the limitation of the chiral expansion scheme.

An arbitrary Feynman graph can be characterized by the number of external – both incoming and outgoing – nucleon (vector-meson) lines, the number of loops, the number of internal nucleon (pion, vector-meson) lines. Each vertex can in turn be characterized by the number of derivatives and/or of factors and the number of nucleon (vector-meson) lines attached to the vertex. Now for a nucleon intermediate state of momentum where , we acquire a factor since

 SF(mv+k)=1v⋅k=O(Q−1). (32)

An internal pion line contributes a factor since

 Δ(q2;m2π)=1q2−m2π=O(Q−2)

while a vector-meson intermediate state contributes as one can see from its propagator

 ΔF(q2;m2V)=1q2−m2V≃1−m2V=O(Q0) (33)

where represents a generic mass of vector mesons. Finally a loop contributes a factor because its effective cut-off is assumed to be of order of . We thus arrive at the counting rule that an arbitrary graph is characterized by the factor with

 ν=−IN−2Iπ+4L+∑idi (34)

where the sum over runs over all vertices of the graph. Using the identities, , and , we can rewrite the counting rule

 ν=2−EN+2EH2+2L+∑iνi,     νi≡di+ni+2hi2−2. (35)

We recover the counting rule derived by Weinberg  if we set .

The situation is different depending upon whether or not there is external gauge field (i.e., electroweak field) present in the process. In its absence (as in nuclear forces), is non-negative

 di+ni+2hi2−2≥0. (36)

This is guaranteed by chiral symmetry . This means that the leading order effect comes from graphs with vertices satisfying

 di+ni+2hi2−2=0. (37)

Examples of vertices of this kind are: with , , , with , etc where denotes vector-meson fields.

In scattering or in nuclear forces, and , and so we have . The leading order contribution corresponds to , coming from three classes of diagrams; one-pion-exchange, one-vector-meson-exchange and four-fermion contact graphs. In scattering, and , we have and the leading order comes from nucleon Born graphs, seagull graphs and one-vector-meson-exchange graphs.#7#7#7We note here that scalar glueball fields play only a minor role in scattering because the vertex () acquires an additional power.

In the presence of external fields, the condition becomes 

 (di+ni+2hi2−2)≥−1. (38)

The difference from the previous case comes from the fact that a derivative is replaced by a gauge field. The equality holds only when or . We will later show that this is related to the “chiral filter” phenomenon. The condition (38) plays an important role in determining exchange currents. Apart from the usual nucleon Born terms which are in the class of ‘‘reducible” graphs and hence do not enter into our consideration, we have two graphs that contribute in the leading order to the exchange current: the ‘‘seagull” graphs and ‘‘pion-pole” graphs #8#8#8These are standard jargons in the literature. See [2, 4]., both of which involve a vertex with . On the other hand, a vector-meson-exchange graph involves a vertex. This is because at the vertex. Therefore vector-exchange graphs are suppressed by power of . This counting rule is the basis for establishing the chiral filtering even when vector mesons are present (see Appendix I). Thus the results we obtain without explicit vector mesons are valid more generally.

## 5 Renormalization in Heavy-Fermion Formalism

In this section, we discuss renormalization in heavy fermion formalism. Most of the renormalized quantities that we will write down here have been obtained by others in standard ChPT . We rederive them for completeness and as a check of our renormalization procedure.

For reasons stated above, fermion loops are suppressed in HFF. Our basic premise is that antiparticle solutions should be irrelevant to physical processes in large-mass and low-energy situations. Their effects can however be systematically taken into account in “” expansion.

We shall denote “bare” quantities by and the corresponding “renormalized (to their physical values)” quantities by , respectively, for nucleon mass, pion mass, pion decay constant ( MeV) and axial coupling constant ().

### 5.1 Dimensional regularization

We adopt the dimensional regularization scheme to handle ultraviolet singularities in our loop calculations. It has the advantage of avoiding power divergences like where is the cut-off mass. In dimensions, all the infinities are absorbed in . When heavy fields are involved, somewhat different parametrization schemes and integral formulas are needed. The relevant ones for our calculation are

 1AnB = 2nΓ(n+1)Γ(n)∫∞0dλλn−1(2λA+B)n+1, (39) 1AnBC = 2nΓ(n+2)Γ(n)∫10dz∫∞0dλλn−1[2λA+zB+(1−z)C]n+2 (40)

and

 Γ(n)∫∞0dλλk(λ2+□2)−n=12□1+k−2nΓ(n−k+12)Γ(k+12). (41)

This integral is singular when

 n−k+12=0,−1,−2,⋯

so must be kept finite until the integration in is performed. Some relevant integral identities needed in this paper are given in Appendix C.

As is customary in the dimensional regularization, we introduce an arbitrary mass scale . After renormalization, the results should, of course, be independent of the scale . Here some comments are in order regarding the one-loop renormalization scheme. First, all the divergences of our theory can be classified in two classes by their degrees of divergence: quadratic and logarithmic divergences. The quadratic divergences are removed by counter terms that are of exactly the same form as the lowest chiral-order Lagrangian, as given by (22). Thus these quadratic singularities can be absorbed into the renormalization process of the basic quantities, namely, . To take care of the logarithmic singularities, we include counter terms that are higher chiral order by than . As it is an arduous task  to write down all possible counter terms, we shall write down only the counter terms needed for the calculation. All the quadratic divergences can be written in terms of and all the logarithmic divergences in terms of defined by

 Δ(M2) = μ4−di∫ddl(2π)d1l2−M2=−M216π2Γ(−1+ϵ)(M24πμ2)−ϵ (42) = M216π2(1ϵ+1+Γ′(1)−lnM24πμ2)+O(ϵ), η = 116π2Γ(ϵ)(M24πμ2)−ϵ=116π2(1ϵ+Γ′(1)−lnM24πμ2)+O(ϵ) (43)

where and Note that both and are scale-dependent and singular. But the coefficients of the counter terms (written down below) are also scale-dependent and singular. To remove the scale dependence and singularity, one must adjust the coefficients of the counter terms. This procedure however is not unique since finite parts of the coefficients of the counter terms are totally arbitrary. There are many ways to eliminate this non-uniqueness leading to a variety of “subtraction schemes.” In this paper, we use the scheme whereby renormalization is made at the on-shell point for the nucleon and at zero four-momentum for the pion and the current. Thus the quantities , and the coefficients of the counter terms are defined at zero momentum of the axial current and the pion.

To make our discussion on renormalization streamlined, we list below all the counter terms needed in our work (for the meson sector, see ) to which we will refer back as required;

 L\tiny CT = ¯¯¯¯B[−δm+(ZN−1)iv⋅D+ic1F2(v⋅D)3]B (44) + igAc2F2¯¯¯¯BSμvνvα(ΔμDνDα−←DνΔμDα+←Dν←DαΔμ)B + ic3F2¯¯¯¯Bvα[Dβ,[Dα,Dβ]]B + ic4F2¯¯¯¯B[Sα,Sβ]{v⋅D,[Dα,Dβ]}B + ic5F2¯¯¯¯B[v⋅Δ,[v⋅D,v⋅Δ]]B + ic6F2¯¯¯¯B[S⋅Δ,[v⋅D,S⋅Δ]]B + {−gA8F4d(1)4¯¯¯¯BτaDμB⋅¯¯¯¯B[v⋅Δ,τa]SμB −gA4F4d(2)4(¯¯¯¯B[Sμ,Sν]DμB⋅¯¯¯¯Bv⋅ΔSνB+¯¯¯¯B[Sμ,Sν]v⋅ΔDμB⋅¯BSνB) + h.c.}

with

 ZN = 1+3(d−1)g2A4Δ(M2)F2, δm = 3g2AM332πF2, c1 = 32g2Aη+cR1, c2 = d−33g2Aη+cR2, c3 = −16[1+(d+1)g2A]η+cR3, c4 = −2g2Aη+cR4, c5 = η+cR5, c6 = −(d−3)g4Aη+cR6, d(1)4 = κ(1)4+[(d−1)g2A−2]η, d(2)4 = κ(2)4−8g2Aη (45)

where and are finite renormalized constants that we will refer to as “finite counter terms.” Since these finite counter terms and the finite parts of loop contributions are scale-independent, our final results also are scale-independent and regular. The chiral counting is immediate from the counter-term Lagrangian. We should mention for later purpose that the two-derivative four-fermion counter terms proportional to cannot be gotten from single low-mass resonance exchanges and hence do not figure in long-range as well as intermediate-range NN potentials.

We should note here that although the above counter-term Lagrangian contains the isospin matrix , chiral invariance of the Lagrangian is preserved. This can be verified by noting that

 (R−1τaR)ij(R−1τaR)kl=(τa)ij(τa)kl (46)

where and is the SU(2) matrix for chiral transformation, .#9#9#9Actually this equation can be simply understood if one uses an representation. In this case, is an orthogonal real matrix and , . Also note that is hermitian while and are antihermitian and where is the antihermitian operator defined by .

### 5.2 Pion properties to one loop

Since due to pair suppression fermion loops can be ignored, renormalization in the pion properties is rather simple. The wavefunction renormalization , renormalized mass and pion decay constant (here as well as in what follows renormalized at with ) are given by #10#10#10 We have not put in the counter terms that appear at in the pure meson sector, so the terms of Gasser and Leutwyler  are missing from our expression. Since the meson sector proper does not concern us here, we will leave out such terms from now on.

 Zπ = 1−23Δ(M2)F2, (47) M2 = \tiny oM2[1−Δ(M2)2F2], (48) F = \tiny oF[1+Δ(M2)F2] (49)

with defined in Eq.(42).
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