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Chiral Dynamics and Heavy-Fermion Formalism

arXiv:hep-ph/9301295v1 1 Feb 1993SNUTP 92-97July 10, 2018Chiral Dynamics and Heavy-Fermion Formalismin Nuclei: I. Exchange Axial CurrentsTae-Sun ParkDepartment of Physics and Center for Theoretical PhysicsSeoul National UniversitySeoul 151-742, KoreaandDong-Pil Min∗and Mannque RhoService de Physique Th´eorique, CEA Saclay91191 Gif-sur-Yvette, FranceABSTRACTChiral perturbation theory in heavy-fermion formalism is developed for meson-exchange currents in nuclei and applied to nuclear axial-charge transitions.

Cal-culation is performed to the next-to-leading order in chiral expansion whichinvolves graphs up to one loop. The result turns out to be very simple.

Thepreviously conjectured notion of “chiral filter mechanism” in the time componentof the nuclear axial current and the space component of the nuclear electromag-netic current is verified to that order. As a consequence, the phenomenologicallyobserved soft-pion dominance in the nuclear process is given a simple interpre-tation in terms of chiral symmetry in nuclei.In this paper we focus on theaxial current, relegating the electromagnetic current which can be treated in asimilar way to a separate paper.

We discuss the implication of our result onthe enhanced axial-charge transitions observed in heavy nuclei and clarify therelationship between the phenomenological meson-exchange description and thechiral Lagrangian description.∗Permanent address: Department of Physics, Seoul National University, Seoul 151-742, Korea.

1IntroductionBy now there exist a large number of unambiguous experimental evidences [1] formeson-exchange currents in nuclear responses to electroweak probes. We also have avail-able a rather satisfactory and successful theory to describe the large bulk of experimentalobservations [2].

While inherently phenomenological in character, the approaches taken sofar to describe meson-exchange currents have been commensurate with the ingredients thataccount 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 ofconfidence in the theoretical tool with which the effect of exchange currents is calculated.There remains however the fundamental question as to how our phenomenological under-standing of nuclear forces and associated meson currents can be linked to the fundamentaltheory of strong interactions, QCD.In this paper we make a first step towards answering this question by applying chiralperturbation theory (ChPT) to nuclear electroweak processes. To start with, we assumethat at low energies dominated by infrared properties of QCD, the most important aspect ofQCD is the spontaneously broken chiral symmetry and hence that in nuclear dynamics, it ischiral symmetry that plays a predominant role [3].

The important role of chiral symmetryin nuclear physics was recognized early on by Chemtob and Rho for exchange currents[4] but this issue was recently given a stronger impetus and a more modern meaning byWeinberg in connection with nuclear forces [5] and by Rho [6] in connection with what isknown “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 bedescribed by QCD or equivalently at low energies by chiral perturbation theory? Weinbergapproaches this issue by studying nuclear many-body forces.

We propose here to do thesame by looking at nuclear response functions responding to slowly varying electroweakfields [7]. We suggest that chiral perturbation theory can be made – under certain conditionsspecified below – considerably more powerful and predictive for response functions than fornuclear 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 arenot accessible by experiments [8]; furthermore there are contact four-fermion interactions inthe Lagrangian – most of which are again unknown parameters – that have to be carefullyexamined and treated. In principle, this may be feasible, perhaps with the help of latticeQCD calculations but in practice it may not be possible to make clear and useful predictionsbecause of many uncontrolled parameters.

While the tree order chiral theory justifies `aposteriori the current nuclear physics practice of using two-body static forces [5], it appearsthat chiral symmetry will be unable to make any truly significant statement on the structureof nuclear forces for sometime to come. A major new development will be required beforeone can make a prediction that goes beyond the accuracy of the phenomenological approach1

which has been strengthened by the wealth of experimental data. On the contrary, as wewill show in this paper, when the formalism is applied to nuclear response functions, inparticular, to exchange currents, it can make a highly nontrivial and potent prediction.This is because nuclear short-range correlations generated by nuclear interactions at shortdistance – which while poorly understood of their mechanism, are nonetheless operativein nuclear medium – screen all the contact interactions, both intrinsic and induced andconsequently all of the four-fermion (and higher) counter term contributions, effectively“filtering” offthe ill-understood short-range operators: Given phenomenological informationon nuclear wave functions at short distance, the short-range suppression helps in simplifyingnuclear response functions.

In addition, in certain kinematic conditions, higher order chiralcorrections are found to be naturally suppressed. The suppression of the many-fermioncounter term contributions at the one-loop order that we are studying is, as will be statedmore precisely later, a consequence of the fact that such terms occurring at high ordersin the chiral expansion reflect the degrees of freedom that enter directly neither in nuclearforces nor in nuclear currents at the chiral order considered.The combination of thesetwo phenomena lead to the “chiral filtering” proposed previously [10].

In this paper, wewill establish this chiral filtering to one-loop order in chiral perturbation theory.Thiswill provide, in our opinion, the very first compelling explanation of the pion-exchangedominance observed in axial charge transitions (considered in this paper) as well as inradiative np capture or in threshold electrodisintegration of the deuteron.As stressed by Weinberg, chiral perturbation theory is useful in nuclear physics onlyfor “irreducible” diagrams that are by choice free of infrared divergences. This means thatboth in nuclear forces and in exchange currents, reducible graphs are to be taken care of bya Schr¨odinger equation or its relativistic generalization with the irreducible graphs enteringas potentials.

This also implies that in calculating exchange currents in ChPT, we are to usethe 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 isalso in this sense that ChPT is predictive in nuclei. Clearly this precludes what one mightcall “fully consistent chiral perturbation theory” where nuclear forces, nuclear currents andwave functions are all calculated to the same order of chiral perturbation expansion.

Sucha calculation even if feasible is likely to make no sense. A little thought would persuade thereader that it is a futile exercise.We will here focus on the irreducible diagrams contributing to exchange currents.

Wewill 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) [11].The standard ChPT [12] arranges terms in power of (∂/Λχ) and/or of (mπ/Λχ) where ∂isfour-derivative acting on the Goldstone boson (viz, pion) field, mπ the pion mass (≈140MeV) and Λχ ≈1 GeV, the chiral expansion scale.It has been established that thisexpansion works well at low energies for such processes as ππ scattering.However thesituation is different when baryons are involved. The dynamically generated masses of the2

baryons are of O(Λχ) and hence when the baryon field is acted upon by time-derivative, itgives an O(1) term. Therefore a straightforward derivative expansion fails.

(Incidentallythis 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 HFFcircumvents this difficulty in rearranging the derivative expansion.

Indeed the principalvirtue of the HFF is that it provides a consistent chiral expansion in Q/Λχ where Q isfour-derivative on pion field or pion mass or space derivative on baryon field; it avoids timederivative on baryon field which is of order Λχ which is not small. The standard ChPTinvolving baryons [13] can in principle be arranged to give a similar expansion.

Howeverit requires a laborious reshuffling of terms avoided in the HFF. The distinct advantageof the HFF is that the multitude of diagrams that appear in such calculations as ours inthe standard ChPT involving baryons get reduced to a handful of manageable terms, thusalleviating markedly the labor involved.

We will see that there is an enormous simplificationin the number of terms and in their expressions. The potential disadvantage might be thatthe HFF is not fully justified for the mass corresponding to that of the nucleon and hencehigher order “1/m” corrections may have to be systematically included.

We will examinethe class of approximations we make in the calculation by looking at the next order terms.It turns out that the leading “1/m” correction is absent in our calculation. We shall discussthis matter in the concluding section.While the procedure is practically the same, the resulting expression for electromag-netic (EM) current is somewhat more involved.

We will therefore not treat it here althoughwe 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 npcapture, will be reported in a separate paper [14]. Both currents are intricately connectedeven at low energy through current algebras and we will need some vertices involving theEM current.It is perhaps obvious but we should stress that for both vector and axial-vectorcurrents, relevant symmetries (i.e, conserved vector current and partially conserved axial-vector current) are preserved to the chiral order considered since both nuclear forces andcurrents are treated on the same footing with the same effective Lagrangian.

More on thispoint later.This paper is organized as follows. In Section 2, we state our basic assumption inapplying ChPT to nuclear dynamics.In Section 3, we describe the effective chiral La-grangian with which we develop heavy-fermion formalism including “1/m” corrections.

Wealso define the relevant kinematics we will consider. The chiral counting rules are given inSection 4.

In Section 5, the renormalization of n-point vertices that enter in the calculationis detailed. For the sake of making this paper as self-contained as possible and to define no-tations, we also list the renormalized quantities for the pion and the nucleon following fromthe Lagrangian.

Readers familiar with renormalization of heavy-fermion chiral Lagrangiancould proceed directly to subsection 5.4. Two-body exchange currents are calculated in3

Section 6.Both momentum-space and coordinate-space formulas are given.Numericalanalyses are described in Section 7. In Section 8, we explain why there are no other graphsthat can contribute to the same order and point out in what circumstances they can showup in physical observables.

Concluding remarks including those on the observed enhancedaxial-charge transitions in heavy nuclei are made in Section 9. The Appendices A-I list allthe formulas needed in the calculation.This paper is written in as a self-contained way as possible so as to be readable bythose who are not familiar with the recent development in the field.

Some of the materialare quite standard and readily available in the literature. Most of them however serve as acheck of our calculation.2Strategies in Nuclear PhysicsWe wish to calculate operators effective in nuclei for transitions induced by the vectorand axial vector currents of electroweak interactions, denoted respectively by Vµ and Aµassociated with the electroweak fields Vµ and Aµ.

In principle there will be n-body currentsfor N ≥n > 1 in N-body systems. Here we will focus only on one- and two-body currents,ignoring those with n > 2.

The reasons for so doing are given in the literature [2, 4] but wewill later show that n-body currents for n > 2 are suppressed to the order considered forlong wavelength probes.The diagrams we wish to calculate are generically given by Fig. 1.

They correspondto the standard definition of single-particle and two-particle exchange currents entering inthe description of nuclear response functions to the external electroweak fields. These havebeen calculated before in terms of phenomenological Lagrangians.

Here we wish to do sousing chiral perturbation theory (ChPT), starting with a chiral Lagrangian that is supposedto model QCD at low energies. Following the chiral counting rule we will derive later, we willrestrict our consideration to one-loop order, which corresponds to going to the next orderin the chiral counting to the leading soft-pion limit.

Although one-loop calculations havebeen done before for nucleon properties [15] and for infinite nuclear matter [16], they haveup to date not been performed in finite nuclear processes. We believe this work constitutesthe first attempt to implement consistently chiral symmetry in nuclear processes.In dealing with divergences encountered in calculating loop graphs, in particularthe loops involving two-pion exchange, we will need a certain prescription for handlingoperators that are short-ranged in coordinate space.

This prescription does not follow fromchiral 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” bynuclear short-range correlations:• Firstly, the zero-range operators that come from finite counter terms appearing infour-fermion interactions figure neither directly nor indirectly – but importantly – in4

the successful phenomenological nucleon-nucleon potentials and hence must representthe degrees of freedom unimportant for the length scale involved. In fact, one canshow (see Appendix I) that the counter terms we need to introduce (denoted κ(1,2)4later) cannot arise, unlike in the better understood π-π scattering [17], from an ap-proximation of taking an infinite mass limit of the strong interaction resonances suchas the vector mesons ρ, ω etc.

which have a scale comparable to the chiral scale andplay an important role in boson-exchange potential models.• Secondly, ChPT by its intrinsic limitation cannot possibly provide a nuclear forcethat 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 presentin the nucleon-nucleon interaction must involve elements that are not calculable bymeans of finite-order chiral expansion even if such an expansion existed.

Thus it wouldbe inconsistent to put a part of such interactions into the currents in the context ofChPT without a similar account in the nuclear force. It is known that even to one-looporder, the number of counter terms is so large in the calculation of nuclear forces thatit is highly unlikely that one can make a meaningful prediction based strictly on low-order chiral perturbation expansion [8].

As suggested in Ref. [9], one should implementChPT calculations with phenomenological informations whenever available.• Applied to the “irreducible diagrams” that enter in the definition of exchange cur-rents, ChPT screens out the short-range part of the interaction which originates fromdynamics of possibly non-chiral origin.

When the matrix elements of the operatorsarising from the irreducible graphs are calculated with wave functions suitably com-puted in the presence of two-nucleon potentials, the short-range correlation built intonuclear wavefunctions must therefore suppress strongly interactions that occur at aninternuclear distance ≤0.6 fm, automatically “killing” the δ function interactionsassociated with finite counter terms. This fact will be kept in mind when we derivetwo-body operators in coordinate space.We will present, at several places in the paper, arguments to justify the above procedurewhich 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 embeddedin nuclear medium.There is nothing very much new in our calculation of the one-body operators exceptfor its consistency with chiral invariance.As for the two-body exchange currents, ourresults 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 calculateonly “irreducible graphs.”5

3Effective Chiral LagrangianWe begin with the effective chiral Lagrangian that consists of pions and nucleonsinvolving lowest derivative terms[13] relegating the role of other degrees of freedom such asvector mesons and nucleon resonances ∆to a later publication#1,L0=N [iγµ(∂µ + Γµ) −m + igAγµγ5∆µ] N −12CaNΓaN2+ F 24 Tr∇µΣ†∇µΣ+ 12M2F 2Tr(Σ) + · · · + LCT,(1)where m ≃939MeV is the nucleon mass, gA ≃1.25 is the axial coupling constant and F ≃93MeV 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 therenormalized parameters m, gA, F and M with suitable counter terms LCT to be specifiedlater.Under the chiral SU(2)×SU(2) transformation#2, the chiral field Σ = exp(i⃗τ·⃗πF )transforms as Σ →gRΣg†L (gR, gL ∈SU(2)) and the covariant derivative of the chiral fieldtransforms as Σ does,∇µΣ=∂µΣ −i(Vµ + Aµ)Σ + iΣ(Vµ −Aµ)→gR ∇µΣ g†L(2)where the external gauge fields Vµ = ⃗Vµ · ⃗τ2 and Aµ = ⃗Aµ · ⃗τ2 transform locallyVµ + Aµ →V′µ + A′µ = gR(Vµ + Aµ)g†R −i∂µgR · g†R,Vµ −Aµ →V′µ −A′µ = gL(Vµ −Aµ)g†L −i∂µgL · g†L.In our work, only the electroweak (SU(2) × U(1)) external fields will be considered. TheLagrangian of course has global SU(2) × SU(2) invariance in the absence of the pion massterm.

Non-linear realization of chiral symmetry is expressed in terms of ξ =√Σ = exp(i⃗τ·⃗π2F )and U = U(ξ, gL, gR) defined with ξξ →gRξU † = Uξg†L.#1While the vector mesons and the nucleon resonances (in particular, the ∆) play an important role innuclear phenomenology – and they can be easily implemented in ChPT at least in low orders, they areunimportant for the process we discuss in this paper. It is not difficult to see which processes require suchdegrees of freedom but we will not pursue this matter, for a treatment of such processes goes beyond theframework of ChPT.#2We are using a slightly unconventional notation of Ref.

[13] which we will follow in this paper. Thisfacilitates checking our results on single-nucleon properties against those derived in [13] using standard(relativistic) chiral perturbation expansion.

The more familiar transformation of the chiral field used in theliterature is gotten by replacing Σ by Σ†. We are also working with the exponentiated (Sugawara) form ofchiral Lagrangian instead of Weinberg’s [5] used previously.

They are of course equivalent. For the rest wewill follow the Bjorken-Drell metric and convention.6

Now nucleon field N transforms as N →UN, and covariant derivatives of nucleon field andchiral field transform as nucleon field does, DµN →UDµN and ∆µ →U∆µU † where#3DµN=(∂µ + Γµ)N,Γµ=12hξ†, ∂µξi−i2ξ†(Vµ + Aµ)ξ −i2ξ(Vµ −Aµ)ξ†,∆µ=12ξ† (∇µΣ) ξ† = 12nξ†, ∂µξo−i2ξ†(Vµ + Aµ)ξ + i2ξ(Vµ −Aµ)ξ†. (3)The U can be expressed as a complicated local function of ξ, gL and gR.

The explicit formof U is not needed for our discussion.Note that we have included the four-fermion non-derivative contact term studied re-cently by Weinberg[5]. We will ignore possible four-fermion contact terms involving deriva-tives (except for counter terms encountered later) and quark mass terms since they are notrelevant 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 massterm.

Higher order symmetry breaking terms do not play a role in our calculation.3.1Heavy-fermion formalismFor completeness – and to define our notations, we sketch here the basic element of theheavy-fermion formalism (HFF)[18] applied to nuclear systems as developed by Jenkins andManohar[11] 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 doesnot work when baryons are involved since while space derivatives on baryons fields can bearranged to appear on the same footing as four-derivatives on pion fields, the time derivativeon baryon fields picks up a term of order of the chiral symmetry breaking scale and hencecannot be used in the chiral counting.

This problem is avoided in the HFF. To set up theHFF, the fermion momentum is written aspµ = mvµ + kµ(4)where vµ is the 4−velocity with v2 = 1, and kµ is the small residual momentum.

(In thepractical calculation that follows, we will choose the heavy-fermion rest frame vµ = (1,⃗0). )We define heavy fermion field Bv(x) for a given four-velocity vµ, by#4Bv(x) = eim v·xN(x).

(5)The field Bv is divided into two parts which are eigenstates of ̸v,Bv = B(+)v+ B(−)v≡1 + ̸v2Bv + 1 −̸v2Bv ≡P+ Bv + P−Bv. (6)#3 We have defined two covariant derivatives involving chiral fields, ∇µΣ and ∆µ.

We can express one interms of the other, but it is convenient as done frequently in the literature to use ∇µΣ for the meson sectorand ∆µ for the meson-nucleon sector.#4Another familiar field redefinition is Bv(x) = eimγ·v v·xN(x), with v2 = 1. This definition gives exactlythe same physics to the lowest order in1m expansion.7

As defined, B(+)v(B(−)v) can be identified as positive (negative) energy solution. As will bejustified in the following subsection, the negative energy solution is suppressed for largebaryon mass and its contribution can subsequently be incorporated as higher-order correc-tions in the inverse mass expansion.

Thus to the leading order, the fermion loops can beignored. With the neglect of the negative energy solutions, we have a useful relation forgamma matrices sandwiched between spinors which holds for any Γ,BvΓBv = Bv̸vΓBv = BvΓ̸vBv = Bv12{Γ, ̸v}Bv .

(7)It follows from this identity thatBvγ5Bv = 0,BvγµBv = vµBvBv. (8)Let us define spin operators Sµv byBvγ5γµBv = −2BvSµv Bv(9)or explicitlySµv = 14γ5 [̸v, γµ] .

(10)The spin operators have the following identities,{Sµv , Sνv }=12(vµvν −gµν) ,(11)[Sµv , Sνv ]=iǫµναβvαSβwithǫ0123 = 1 . (12)From the anti-commutation rule, we haveSv · Sv=14(1 −d) = −3 −2ǫ4≃−34 ,(13)Sαv Sµv Svα=14(d −3)Sµv ≃14Sµv ,(14)(q · Sv)2=14h(q · v)2 −q2i(15)where d is the dimension of the space-time, d = gµµ and we have defined ǫ = (4 −d)/2.Between spinors, we have the approximate relationsBv Sµv Bv≃12⃗σ · ⃗v, 12⃗σ,(16)BvhS0v, ⃗SviBv≃−i2⃗v × ⃗σ(17)with ⃗σ the usual Pauli spin matrices.

We see that S0v andhS0v, ⃗Sviare suppressed by afactor of ⃗v = OQmwhere Q is the characteristic small momentum scale for processes withsmall three-velocity. Since[Sµv , Sνv ]=i4 (σµν + ̸vσµν̸v)=i2σµν + (vµSνv −vνSµv ) γ58

where σµν = i2 [γµ, γν], we also haveBv i2σµνBv=Bv [Sµv , Sνv ] Bv(18)Bv (σµνγ5) Bv=2iBv (vµSνv −vνSµ) Bv. (19)We are now in position to write down the chiral Lagrangian (1) in HFF.

The nucleonpart of the Lagrangian becomesN(i̸∂−m)N = Bviv · ∂Bv(20)and the corresponding nucleon propagator S(mv + k) is#5iS(mv + k) =iv · k + i0+ . (21)Our chiral Lagrangian (1) expressed in terms of the heavy-fermion field to leading (i.e,zeroth) order in1m takes the formL0=Bv [ivµ(∂µ + Γµ) + 2igASµv ∆µ] Bv −12CaBvΓaBv2+ F 24 Tr∇µΣ†∇µΣ+ 12M2F 2Tr(Σ).

(22)In practical calculations, the chiral field Σ or ξ is expanded in power of the pion field. Theexplicit form resulting from such expansion as well as the vector and axial-vector currentscalculated via Noether’s theorem are given in Appendix A.3.21m correctionsAs mentioned, the HFF is based on simultaneous expansion in the chiral parameterand in “1/m”.

We have so far considered leading-order terms in 1/m, namely, O((1/m)0).We now discuss 1m corrections following closely the discussion of Grinstein [19]. We choose todo this in a perhaps more general way than needed for our purpose.

Consider the followingLagrangianL=N [i̸D −m + γµγ5Aµ] N −12Ca(NΓaN)2=B [i̸D −m(1 −̸v) + γµγ5Aµ] B −12Ca(BΓaB)2#5 Although we do not actually encounter it in our calculation, it might be worthwhile to point out onetechnical subtlety. The infinitesimal i0+ is inserted to define the singularity structure of the propagator.When we encounter a d−dimensional loop integral we first perform the Wick-rotation to put it in theEuclidean metric.

In doing this, we assume that the first and third quadrants (in the plane of real l0 vs.imaginary l0) contain no poles. If we can take the flow direction of the loop-momentum to be the directionof the fermion momentum, there is no problem.

However for some graphs it is impossible to do this. Forinstance 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 isof the form1v · k −i0+ =1v · k + i0+ + 2iπδ(v · k).9

where we have included Weinberg’s four-fermion contact term with Γa an arbitrary her-mitian operator which we assume to contain no derivatives. We have also introduced anarbitrary “axial” field Aµ which we take to be hermitian and free of gamma matrices.

(Hereand in what follows, we shall omit the subscript v in Bv, Bv and Sµv .) The equation ofmotion satisfied by B is[G −m(1 −̸v)] B = 0(23)withG ≡g −CaΓa(BΓaB),g ≡i̸D + γµγ5Aµ.

(24)Multiplying (23) on the left by P−, we obtainP−G B −2mB(−) = 0which leads toB(−) =12m P−G B =12m P−G B(+) + O 1m2. (25)Now multiplying P+ to (23), we getP+ G B = 0which givesP+G + 12mG P−GB(+) = O 1m2.

(26)Given this, it is now a simple matter to write down the Lagrangian that gives rise to theequation of motion to the desired order #6. The result correct to O(1/m) isL=Bg + 12mg P−gB−12CaBΓa + 12m (Γa P−g + g P−Γa)B2+ 12mCaCbBΓaB BΓa P−ΓbB BΓbB(27)with g defined in (24).

To put this into a more standard form, we use the identitiesP+ g P+=P+12 {̸v, g} P+ ,P+ g P−g′ P+=P+ { P−, g} g′ P+ = P+ g P−, g′ P+ ,P+ g P−g P+=P+12n̸v, g2o−14 {g, g {̸v, g}}P+ . (28)One can show from these identities thatB(+) Γa B(+)⊗B(+) Γa P−= 0(29)#6We remind the reader that one should not insert the solution of B(−) into the original Lagrangian, sincein Lagrangian approach, what is important is the form, not the value.

However in Hamiltonian approach,the insertion of the solution for B(−) into the original Hamiltonian is allowed.10

for any Γa = {1, γ5, γµ, γµγ5, σµν}. This allows us to simplify the Lagrangian further tothe formL=B (iv · D + 2S · A) B −12CaBΓaB2 + 12mB−D2 + (v · D)2+ [Sµ, Sν] [Dµ, Dν] −(v · A)2 −2i {v · A, S · D}B,(30)for general Γa allowed by symmetries.

In our case, Aµ = igA∆µ = igA 12ξ†(∇µΣ)ξ†, so our1m term Lagrangian isδL=12mB−D2 + (v · D)2 + [Sµ, Sν] [Dµ, Dν] + g2A(v · ∆)2 + 2gA {v · ∆, S · D}B+ O 1m2. (31)While Eq.

(31) is the first “1/m” correction, it is not the entire story to the orderconsidered. One can see that it is also the next order in the chiral counting in derivativesand it is expected in any case, independently of the inverse baryon mass corrections.

Thusin a practical sense, the coefficients that appear in each term are parameters rather thanfixed by chiral symmetry in HFF. We also note that in the derivation given above, neitherO(1/m) corrections to the quartic fermion term nor sixth-order fermion terms are generatedby the (1/m) expansion.

This of course does not mean that such terms cannot contributeon a general ground. We have however checked that no such terms arise from exchanges ofsingle heavy mesons that are formally “integrated out” which means that our calculationsare not affected by such terms, at least to the order we are concerned with.4Counting RulesIn this section, we rederive and generalize somewhat Weinberg’s counting rule[5]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 outto be valid in the presence of vector mesons.

Now in dealing with them, their masses –which are comparable to the chiral scale Λχ – will be regarded as heavy compared to themomentum probed Q – say, scale of external three momenta or mπ.In establishing the counting rule, we make the following key assumptions: Everyintermediate meson (whether heavy or light) carries a four-momentum of order of Q. Inaddition we assume that for any loop, the effective cut-offin the loop integration is of orderof Q. We will be more precise as to what this means physically when we discuss specificprocesses, for this clarifies the limitation of the chiral expansion scheme.An arbitrary Feynman graph can be characterized by the number EN(EH) of external– both incoming and outgoing – nucleon (vector-meson) lines, the number L of loops, thenumber IN(Iπ, IH) of internal nucleon (pion, vector-meson) lines.

Each vertex can in turnbe characterized by the number di of derivatives and/or of mπ factors and the number ni11

(hi) of nucleon (vector-meson) lines attached to the vertex. Now for a nucleon intermediatestate of momentum pµ = mvµ + kµ where kµ = O(Q), we acquire a factor Q−1 sinceSF(mv + k) =1v · k = O(Q−1).

(32)An internal pion line contributes a factor Q−2 since∆(q2; m2π) =1q2 −m2π= O(Q−2)(33)while a vector-meson intermediate state contributes Q0 (∼O(1)) as one can see from itspropagator∆F(q2; m2V ) =1q2 −m2V≃1−m2V= O(Q0)(34)where mV represents a generic mass of vector mesons. Finally a loop contributes a factor Q4because its effective cut-offis assumed to be of order of Q.

We thus arrive at the countingrule that an arbitrary graph is characterized by the factor Qν withν = −IN −2Iπ + 4L +Xidi(35)where the sum over i runs over all vertices of the graph. Using the identities, Iπ +IH +IN =L + V −1, IH = 12Pi hi −EH2and IN = 12Pi ni −EN2 , we can rewrite the counting ruleν = 2 −EN + 2EH2+ 2L +Xiνi,νi ≡di + ni + 2hi2−2.

(36)We recover the counting rule derived by Weinberg [5] if we set EH = hi = 0.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), νi isnon-negativedi + ni + 2hi2−2 ≥0.

(37)This is guaranteed by chiral symmetry [5]. This means that the leading order effect comesfrom graphs with vertices satisfyingdi + ni + 2hi2−2 = 0 .

(38)Examples of vertices of this kind are: πkNN with k ≥1 (di = 1, ni = 2, hi = 0),hNN (di = 0, ni = 2, hi = 1), (NΓN)2 (di = 0, ni = 4, hi = 0), hπk with k ≥1 (di =1, ni = 0, hi = 1), etc where h denotes vector-meson fields.In NN scattering or in nuclear forces, EN = 4 and EH = 0, and so we have ν ≥0. Theleading order contribution corresponds to ν = 0, coming from three classes of diagrams;one-pion-exchange, one-vector-meson-exchange and four-fermion contact graphs.In πN12

scattering, EN = 2 and EH = 0, we have ν ≥1 and the leading order comes from nucleonBorn graphs, seagull graphs and one-vector-meson-exchange graphs.#7In the presence of external fields, the condition becomes [6]di + ni + 2hi2−2≥−1 . (39)The difference from the previous case comes from the fact that a derivative is replacedby a gauge field.

The equality holds only when hi = 0, ni = 2 or hi = 0, ni = 0. Wewill later show that this is related to the “chiral filter” phenomenon.

The condition (39)plays an important role in determining exchange currents. Apart from the usual nucleonBorn terms which are in the class of “reducible” graphs and hence do not enter into ourconsideration, we have two graphs that contribute in the leading order to the exchangecurrent: the “seagull” graphs and “pion-pole” graphs #8, both of which involve a vertexwith νi = −1.On the other hand, a vector-meson-exchange graph involves a νi = +1vertex.

This is because di = 1, hi = 2 at the Jµhh vertex. Therefore vector-exchangegraphs are suppressed by power of Q2.

This counting rule is the basis for establishing thechiral filtering even when vector mesons are present (see Appendix I). Thus the results weobtain without explicit vector mesons are valid more generally.5Renormalization in Heavy-Fermion FormalismIn this section, we discuss renormalization in heavy fermion formalism.

Most of therenormalized quantities that we will write down here have been obtained by others in stan-dard ChPT [13]. We rederive them for completeness and as a check of our renormalizationprocedure.For reasons stated above, fermion loops are suppressed in HFF.

Our basic premiseis that antiparticle solutions should be irrelevant to physical processes in large-mass andlow-energy situations. Their effects can however be systematically taken into account in“1/m” expansion.We shall denote “bare” quantities byom,oM,oF,ogA and the corresponding “renor-malized (to their physical values)” quantities by m =om + δm, M, F, gA, respectively,for nucleon mass, pion mass, pion decay constant (≃93 MeV) and axial coupling constant(≃1.25).5.1Dimensional regularizationWe adopt the dimensional regularization scheme to handle ultraviolet singularities inour loop calculations.

It has the advantage of avoiding power divergences like δ(0) ∼Λ4cut#7We note here that scalar glueball fields χ play only a minor role in πN scattering because the χππ vertex(di = 2, ni = 0, hi = 1) acquires an additional Q power.#8These are standard jargons in the literature. See [2, 4].13

where Λcut is the cut-offmass. In d = 4−2ǫ dimensions, all the infinities are absorbed in 1ǫ .When heavy fields are involved, somewhat different parametrization schemes and integralformulas are needed.

The relevant ones for our calculation are1AnB=2n Γ(n + 1)Γ(n)Z ∞0dλλn−1(2λA + B)n+1 ,(40)1AnBC=2n Γ(n + 2)Γ(n)Z 10dzZ ∞0dλλn−1[2λA + zB + (1 −z)C]n+2(41)andΓ(n)Z ∞0dλλk(λ2 + ✷2)−n = 12✷1+k−2nΓn −k + 12Γk + 12. (42)This integral is singular whenn −k + 12= 0, −1, −2, · · ·(43)so ǫ must be kept finite until the integration in λ is performed.

Some relevant integralidentities needed in this paper are given in Appendix C.As is customary in the dimensional regularization, we introduce an arbitrary massscale µ. After renormalization, the results should, of course, be independent of the scale µ.Here some comments are in order regarding the one-loop renormalization scheme.

First, allthe 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 counterterms that are of exactly the same form as the lowest chiral-order Lagrangian, L0 as given by(22).

Thus these quadratic singularities can be absorbed into the renormalization processof the basic quantities, namely, gA, F, M, m. To take care of the logarithmic singularities,we include counter terms that are higher chiral order by Q2 than L0. As it is an arduoustask [13] to write down all possible counter terms, we shall write down only the counterterms needed for the calculation.

All the quadratic divergences can be written in terms of∆(M2) and all the logarithmic divergences in terms of η defined by∆(M2)=µ4−diZddl(2π)d1l2 −M2 = −M216π2 Γ(−1 + ǫ) M24πµ2!−ǫ=M216π2 1ǫ + 1 + Γ′(1) −ln M24πµ2!+ O(ǫ),(44)η=116π2 Γ(ǫ) M24πµ2!−ǫ=116π2 1ǫ + Γ′(1) −ln M24πµ2!+ O(ǫ)(45)where ǫ = (4 −d)/2 and Γ′(1) ≃−0.577215. Note that both ∆and η are scale-dependentand 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 adjustthe coefficients of the counter terms.

This procedure however is not unique since finiteparts of the coefficients of the counter terms are totally arbitrary. There are many ways to14

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 nucleonand at zero four-momentum for the pion and the current. Thus the quantities gA, F andthe coefficients of the counter terms are defined at zero momentum of the axial current andthe pion.To make our discussion on renormalization streamlined, we list below all the counterterms needed in our work (for the meson sector, see [12]) to which we will refer back asrequired;LCT=B−δm + (ZN −1)iv · D + i c1F 2 (v · D)3B+igAc2F 2 B Sµvνvα∆µDνDα−←Dν ∆µDα+←Dν←Dα ∆µB+i c3F 2 Bvα hDβ, [Dα, Dβ]iB+i c4F 2 BhSα, Sβi{v · D, [Dα, Dβ]} B+i c5F 2 B [v · ∆, [v · D, v · ∆]] B+i c6F 2 B [S · ∆, [v · D, S · ∆]] B+−gA8F 4 d(1)4BτaDµB · B [v · ∆, τa] SµB−gA4F 4 d(2)4B [Sµ, Sν] DµB · Bv · ∆SνB + B [Sµ, Sν] v · ∆DµB · BSνB+ h.c.}(46)withZN=1 + 3(d −1)g2A4∆(M2)F 2,δm=3g2AM332πF 2 ,c1=32g2Aη + cR1 ,c2=d −33g2Aη + cR2 ,c3=−16h1 + (d + 1)g2Aiη + cR3 ,c4=−2g2Aη + cR4 ,c5=η + cR5 ,c6=−(d −3)g4Aη + cR6 ,d(1)4=κ(1)4+h(d −1)g2A −2iη,d(2)4=κ(2)4−8g2Aη(47)where cRi(i = 1, 2, · · · , 6) and κ(1,2)4are finite renormalized constants that we will referto as “finite counter terms.” Since these finite counter terms and the finite parts of loop15

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 mentionfor later purpose that the two-derivative four-fermion counter terms proportional to d(1,2)4cannot be gotten from single low-mass resonance exchanges and hence do not figure inlong-range as well as intermediate-range NN potentials.We should note here that although the above counter-term Lagrangian contains theisospin matrix τ, chiral invariance of the Lagrangian is preserved.

This can be verified bynoting thatR−1τaRijR−1τaRkl = (τa)ij (τa)kl(48)where (i, j, k, l = 1, 2) and R is the SU(2) matrix for chiral transformation, B →RB.#9 Alsonote that Sµ is hermitian while ∆µ and [Sµ, Sν] are antihermitian and γ0D†µγ0 =←Dµ≡←∂µ−Γµ where Γµ is the antihermitian operator defined by Dµ = ∂µ + Γµ.5.2Pion properties to one loopSince due to pair suppression fermion loops can be ignored, renormalization in thepion properties is rather simple. The wavefunction renormalization Zπ, renormalized massM and pion decay constant F (here as well as in what follows renormalized at q2 = 0 withM ̸= 0) are given by [12]#10Zπ=1 −23∆(M2)F 2,(49)M2=oM2"1 −∆(M2)2F 2#,(50)F=oF"1 + ∆(M2)F 2#(51)with ∆(M2) defined in Eq.

(44).5.3Nucleon properties to one loopOne-loop graphs for nucleon propagator are given in Fig. 3.

Fig. 3b vanishes due toisospin symmetry, so only Fig.

3a survives to contribute to the nucleon self-energy Σ,Σ(v · k) = −3 g2AF 2 SαSβZllαlβv · (l + k) (l2 −M2)(52)where (and in what follows)Zl≡µ4−diZddl(2π)d . (53)#9Actually this equation can be simply understood if one uses an O(3) representation.

In this case, R isan orthogonal real matrix and (τa)ij = −iǫaij, a, i, j = 1, 2, 3.#10 We have not put in the counter terms that appear at O(Q4) in the pure meson sector, so the Lri terms ofGasser and Leutwyler [12] are missing from our expression. Since the meson sector proper does not concernus here, we will leave out such terms from now on.16

From this, we getZN = 1 + Σ′(0) = 1 + 3(d −1)g2A4∆(M2)F 2(54)where the prime on Σ stands for derivative with respect to v · k. In our case confined toirreducible graphs, there is no nucleon pole, so we can set v · k = 0 in the denominator ofthe nucleon propagator. For an off-shell nucleon, we haveΣ(v · k) = (d −1) 3g2A4F 2 h(v · k)(55)with the function h(v · k) defined byZllαlβv · (l + k) (l2 −M2) = gαβh(v · k) + vαvβ(· · ·)(56)where (· · ·) stands for a function that does not concern us.

The evaluation of h(v · k) isdescribed in Appendix D. For (v · k)2 ≤M2, we haveΣ(y) = 3(d −1)g2A4F 2∆(M2) y −3g2A2F 2 η y3 + 3g2A4F 2 (M2 −y2) h0(y),(57)where y = v · k and η is a singular quantity given by (45) and h0 is a finite function, theexplicit form of which is given in Appendix D. We note that the above equation containsthe 1ǫ divergence in the coefficient of (v · k)3 as well as in the coefficient of (v · k) (i.e, in∆(M2)). This additional singularity arises also in the conventional method.

See the paperby Gasser et. al.[13].

The counter term needed to remove this divergence, as given in (46),isΣCT(y) = δm −(ZN −1) y + c1F 2 y3 . (58)In order to regularize the propagator subject to the condition Σ(0) = Σ′(0) = 0, we chooseZN=1 + 3(d −1)g2A4∆(M2)F 2,δm=3g2AM332πF 2 ,c1=32g2Aη + cR1 .The result isΣ(y) = δm + cR1F 2 y3 + 3g2A4F 2M2 −y2h0(y).

(59)Here the finite constant cR1 is in principle to be determined from experiments. To see itsphysical meaning, we should look at a process involving an off-shell nucleon.

For instance,when v · k = ±M, we haveΣ(±M) = ± cR1F 2 M3 + δm. (60)Finally, the1m correction is readily seen to beδΣ(k) = −12mhk2 −(v · k)2i.

(61)17

5.4Renormalization of 3-Point Vertex FunctionsIn this subsection, we shall calculate three-point vertex functions to one-loop order,in particular, JµNN and πNN given in Fig. 4, where Jµ = Aµ(Vµ) denotes the axial vector(vector) current.

We treat the vector current simultaneously since some vertices involvingit figure in our calculation. Each vertex function is a sum of contributions from tree graphs,one-loop graphs, wavefunction renormalization, higher-order counter term insertion andO1mcorrections, if needed.

The tree-graph contribution to JµNN isΓµ,aANN(tree)=gA τa Sµ,Γµ,aV NN(tree)=τa2 vµ. (62)Wavefunction renormalization produces multiplicative coefficients Z, i.e., ZN for Γµ,aANN andZ12π ZN for ΓaπNN etc.Unless noted otherwise we will always set the momentum flow of all pions and currentsto be outgoing.

The current-off-shell-nucleon couplings that we will consider are of the typeN(mv + k)→N(mv + k −q) + Jµa (q),N(mv + k)→N(mv + k −q) + πa(q)with the relevant momenta indicated in the parentheses.For completeness, we list in Appendix F all the contributions to the three-pointfunctions of each Feynman graph.5.4.1Axial vertex function Γµ,aANNFor off-shell nucleons, we findΓµ,aANN = gA τa Sµ"1 + ∆(M2)F 2+ 3(d −1)g2A4F 2∆(M2) + d −34g2AF 2 h3(v · k, v · q)#(63)where h3(v · k, v · q) ≡1v·q [h(v · k −v · q) −h(v · k)] is evaluated in Appendix D. The singu-larity in the above equation is removed by the counter term contribution[Γµ,aANN]CT = gA τa Sµ−c22F 2 h3(v · k)2 −3v · k v · q + (v · q)2i(64)withc2 = d −33g2Aη + cR2(65)where cR2 is a finite renormalized coupling constant. Adding the counter term contributionto the loop contribution, we obtain a renormalized axial coupling constant by gA = gA(k =0, q = 0) where gA(k, q) is defined byΓµ,aANNR(k, q) ≡gA(k, q) τa Sµ.

(66)18

Physically gA(k, q) is just the axial charge form factor for the incoming nucleon of momen-tum mvµ + kµ and the axial current carrying the momentum qµ.Explicitly it is givenbygA(k, q)gA= 1 + g2A4F 2 h3(v · k, v · q) −cR22F 2h3(v · k)2 −3v · k v · q + (v · q)2i(67)where h3(v · k, v · q) is a finite function defined byh3(v · k, v · q) ≡h3(v · k, v · q) + ∆(M2) −23ηh3(v · k)2 −3v · k v · q + (v · q)2i(68)andgA =ogA"1 + ∆(M2)F 21 + d2g2A#≃1.25(69)where we have equated the renormalized gA to the experimental value.Finally, the1mcorrection isδΓµ,aANN = −12mgA τa vµ (2k −q) · S.(70)For on-shell nucleons, v · k = v · q = 0 or in Breit frame, v · k = 12v · q, so we findgA(k, q) = gA.(71)Note that we have neither momentum dependence nor 1m corrections for an on-shell nucleonor in Breit frame. This means that there will be no one-loop correction to the πNN vertexin the exchange currents calculated below.Given an experimental axial charge form factor of the off-shell nucleon, one can fixthe constant cR2 .

The vertex ΓaπNN can be obtained by a direct calculation or by means ofLehmann-Symanzik-Zimmermann (LSZ) formulation: Both give the same resultΓaπNNR(k, q) = −igA(k, q)Fτa q · S.(72)5.4.2Vector vertex function Γµ,aV NNThe full form of Γµ,aV NN is rather involved,Γµ,aV NN = τa2vµ F V1 + 1m [q · S, Sµ] F V2 −qµF V3(73)whereF V1=1 −1F 2hf1(q2) −∆(M2)i+ g2AF 2 ⊗(d −1)34∆(M2) −14h3(v · k, v · q) −B2(k, q)+2h(v · q)2 −q2i ∂∂q2 B2(k, q) + 4v · qB1(k, q),F V2=4m g2AF 2 B0(k, q),F V3=v · qF 2 f3(q2) + g2AF 2(d + 1) −2(v · q)2 −q2 ∂∂q2B1(k, q)(74)19

with the functions fi, hi and Bi given explicitly in Appendix B, D and E, respectively.Although the above equations appear to have quadratic divergences, they actually haveonly logarithmic divergences as can be seen below :F V1=1 −94(2v · k −v · q) g2AF 2M6π +η6F 2 q2 + d + 16g2AF 2 ηq2+d −12η g2AF 2h3(v · k)2 −3v · k v · q + (v · q)2i+ · · · ,F V2=−4m M16πg2AF 2 + 2m g2AF 2 (2v · k −v · q)η + · · · ,F V3=h1 + (d + 1)g2Ai v · q6F 2 η + · · ·(75)where the ellipsis denotes finite and O(Qn)|n>2 terms. Quadratic divergences disappearbecause of EM gauge invariance.

We see that Γµ,aV NN(k = q = 0) = 12τa vµ. The remaininglogarithmic divergences are removed by the counter term in Eq.

(46) of the form[Γµ,aV NN]CT=−c12F 2 τavµ h3(v · k)2 −3v · q v · k + (v · q)2i+c32F 2 τaq2 vµ −v · q qµ+c42F 2 τa v · (2k −q) [q · S, Sµ](76)withc3=−16h1 + (d + 1)g2Aiη + cR3 ,c4=−2g2Aη + cR4 . (77)When v · k = v · q = 0, we have F V3 = 0 andF V1 (q2)=1 + cR3F 2 q2 −q216π2F 2"1 + 3g2A2K0(q2) −2(1 + 2g2A)K2(q2)#,(78)F V2 (q2)=−g2Am4πF 2Z 10dzqM2 −z(1 −z)q2 + 1(79)where we have added the1m correction appearing in the second term of F V2#11.

It is easyto see in (78) that the counter term constant cR3 can be related directly to the isovectorcharge radius of the nucleon. We will give the precise relation later.

Ki(q2) are the finitepieces of the functions fi(q2) defined in Appendix B,K0(q2)=Z 10dz ln"1 −z(1 −z) q2M2#,K2(q2)=Z 10dz z(1 −z) ln"1 −z(1 −z) q2M2#. (80)#11Although the loop contribution to the Pauli form factor F V2is finite and hence requires no infinitecounter term, there is a finite counter term contributing to it which we did not – but should – include in ourformula.

As pointed out by Bernard et al. [20], the finite counter term can be considered as coming fromthe ρ exchange as in vector dominance picture.

This point will be addressed more precisely in [14].20

5.5Renormalization of 4-Point Vertex FunctionIn this section, we study the 4-point vertex functions denoted Γµ,abπA and Γµ,abπVto oneloop as given in Fig. 5, corresponding to the processN(mv + k) →N(mv + k −qa −qb) + Jµa (qa) + πb(qb)where the isospin components of the current and the pion are denoted by the subscripts aand b, respectively.

Here v · k represents how much off-shell the incoming nucleon is andv · (k −qa −qb) the same for the outgoing nucleon. For tree graphs, we haveΓµ,abπA (tree)=−12F ǫabcτc vµ.

(81)The full formulas for non-vanishing graphs for off-shell nucleons are given in AppendixG. Here we limit ourselves only to the on-shell nucleon case.

For axial-charge transitions,only the six graphs Fig.5 (a) −(f) survive. Figures 5 (g), (h) are proportional to Sµ, sosuppressed for the time component and Figures 5 (i) −(n) are proportional to v · S = 0.Figures 5 (o) −(r) do not contribute to the axial current.

(We have included them for laterpurpose, see [14]. )Adding the loop contributions and tree graphs with their wavefunction renormaliza-tion constant, we haveΓµ,abπA=ǫabcτc Γµ,−πA + iδab Γµ,+πA(82)withΓµ,−πA=−vµ2F1 −1F 2h1 + (d −1)g2Af1(q2) −8g2A(q · S)2f2(q2) −v · qa hA0 (v · qa)i−2g2AF 3 [q · S, Sµ] B0(q2)+ g4AF 31 −d4{qb · S, Sµ} hA4 (v · qa) + d −34[qb · S, Sµ] hS4 (v · qa),(83)Γµ,+πA=−(qa + 3qb)µ 4g2A3F 31 −d4+ 2(q · S)2 ∂∂q2B0(q2) + vµF 3 v · qa hS0 (v · qa)−3 g4AF 31 −d4{qb · S, Sµ} hS4 (v · qa) + d −34[qb · S, Sµ] hA4 (v · qa)(84)where q = qa + qb, f1(q2) = f1(q2) −f1(0) and hS,Ai(y) = 12 [hi(y) ± hi(−y)].

The integralsdefining the functions fi(q2) for i = 0, 1, 2, 3 are listed and evaluated in Appendix B. (hi’sare defined in Appendix D and B0 in Appendix E.) The log divergences contained in thesevertices are removed by the counter term contribution,hΓµ,abπAiCT=−ǫabc1F [Γµ,cV NN]CT21

+ ǫabcτcc5F 2 vµ(v · qa)2+c62F 2v · qaF(iδab [Sµ, qb · S] −ǫabcτc {Sµ, qb · S})(85)withc5=η + cR5 ,c6=−(d −3)g4Aη + cR6(86)where cRi are renormalized finite constants listed in Eq. (46) and [Γµ,cV NN]CT is given by (76).With soft momentum, we have v · qa = 0 for which we obtain a surprisingly simpleexpression, viz,Γµ=0,abπA=−ǫabcτc12F F V1 (t)(87)where t ≡q2 = (qa + qb)2 and F V1 is the isovector Dirac form factor Eq.(78).

The one-looprenormalization of the πANN vertex corresponds to the isovector charge radius obtainablefrom the form factor F V1 for which the finite counter term cR3 plays a key role. We see thatΓµ,abπA and Γµ,aANN are related, respectively, to Γµ,aV NN and Γµ,abπV calculated in Appendix G. Thatthe AµπNN vertex for a soft pion is simply given by F V1has of course been understoodsince a long time via current algebra and also in terms of the ρ-meson exchange.Finally for the1m correction, one can readily obtain the corrections to the verticesδΓµ,abπA=−ǫabcτc14mF (2kµ −qµ −vµ(2v · k −v · q) + 2 [q · S, Sµ])−iδab vµ v · qag2A4mF ,δΓµ,abπV=−iδab v · qagA2mF Sµ(88)where k is the residual momentum of the incoming nucleon and k −q = k −qa −qb is that ofthe outgoing nucleon.

An important point to note from Eq. (88) is that for the case v·qa = 0and on-shell nucleons, we have no contribution from1m corrections to the time componentof the axial current and the space component of the vector current.

This is the basis for thepair suppression we will exploit in the application to axial-charge transitions in nuclei.The complete list of the four-point functions involving the vector and axial-vectorcurrents needed here and in [14] is given in Appendix G.6Two-Body Exchange CurrentsSo far we have computed one-loop corrections to the graphs involving one nucleon.They are extractable from experimental data on nucleon properties.In this section weincorporate the above corrections into – and derive – two-body exchange currents in heavy-fermion chiral perturbation theory. As shown previously [7], the time component of the22

axial-vector current (and also the space component of the vector current [14]) in the long-wavelength limit is best amenable to chiral perturbation loop calculations.#12 We will workout the computation to one loop order corresponding to the next-to-leading order in thechiral counting rules as derived in Section 4. The process of interest isN(p1) + N(p2) →N(p′1) + N(p′2) + Jµa (k),where we have indicated the relevant kinematics with q2 = p′2 −p2, q1 = p′1 −p1 and theenergy-momentum conservation q1 + q2 + k = 0.

The process is soft in the sense thatv · k ≃v · qi = v · (p′i −pi) ≃O Q2m!≃0where m is the nucleon mass or chiral scale. This kinematics markedly simplifies the calcu-lation.

Clearly this kinematics does not hold, say, for energetic real photons.It is convenient to classify graphs by the current vertex involved. The graphs thatcontain JµπNN play a dominant role since JµπNN ∼vµ for the axial current (andJµπNN ∼Sµ for the vector current).

The graphs which contain JµNN (and JµππNNfor the vector current) can be ignored to the relevant order because the role of the vectorand axial currents is interchanged. In what follows, we discuss the axial current only.

Theargument for the vector current goes almost in parallel and will be detailed in [14].What we are particularly interested in is the time component of the axial current,with axial-charge transitions in nuclei in mind.This is where the “pion dominance” isparticularly cleanly exhibited. The space component is also interesting both theoreticallyand phenomenologically.

Theoretically Gamow-Teller transitions – observed to be quenched– represent the other side of the coin relating to the chiral filter phenomenon discussed above,namely that chiral symmetry alone or more precisely soft mechanisms associated with itcannot make statements on this quantity [21]. Empirically the quenching phenomenon isclosely associated with the missing strength of giant Gamow-Teller resonances in nuclei.Since the treatment of the space part of the axial current requires going beyond chiralperturbation theory, we will not pursue this issue any further in this paper.For convenience, we define an “axial-charge” operator M by#13⃗Aµ=0 ≡gA4F 2 ⃗M.

(89)#12This is the crucial point in using ChPT in nuclei that quantifies the general discussion given in Section2. Since this point is often misunderstood by nuclear physics community, we would like to stress it oncemore although to some it may sound obvious and repetitive.

The chiral counting on which our analysis isbased is meaningful only if Q2m2 << 1 where Q is the characteristic momentum or energy scale involved in theprocess. Therefore ChPT cannot describe processes that involve energy or momentum scale exceeding thatcriterion.

This means that processes probing short-distance interactions are not accessible by finite-orderChPT. In particular two-body currents describing an internuclear distance r12 ≤0.6 fm cannot be probedby the expansion we are using.

We argued before – and will make use of the fact – that there is a naturalcut-offprovided by short-range nuclear interactions that go beyond the strategy of ChPT which allows ameaningful use of the small Q expansion.#13 The operator M is an isovector but in what follows we will not explicitly carry the isospin index.23

We decompose M into M = Mtree + Mloop where Mtree denotes the axial charge operatorcoming from the one-pion-exchange tree graph (i.e, the soft-pion term) and Mloop is whatcomes from loop corrections. Further we decompose Mloop intoMloop = M1π + M2π(90)where M1π denotes the loop correction to the one-pion-exchange axial charge operator(also referred in the literature to as “seagull graph”) and M2π the contribution from two-pion-exchange graphs and tree graphs involving four-fermion contact terms with counter-term insertions.

We will later argue that the latter does not contribute. One-loop graphsinvolving four-fermion contact interactions, while allowed in the relevant chiral order, donot however contribute either.6.1Results in momentum spaceAs stated above, the non-zero contributions to the time component of the axial cur-rent come only from the graphs that contain a JµπNN vertex.The tree seagull graphsupplemented with a vertex form factor – and properly renormalized – leads to#14⃗Aµtree + ⃗Aµ1π = i⃗τ1 × ⃗τ2gA2F 2 vµ q2 · S21M2 −q22F V1 (q21) + (1 ↔2)(91)where F V1 is the Dirac isovector form factor of the nucleon.

To the order considered, there areno further corrections. The present formalism allows us to calculate within the scheme theform factor F V1 .

The tree-graph (or soft-pion) contribution corresponds to taking F V1 = 1.The difference (F V1 −1) is given by Figs. 5(a) −(f).

Taking kµ →0, we encounter twospin-isospin operators,T (1)≡−2i⃗τ1 × ⃗τ2 q · S1 + (1 ↔2) ≃i⃗τ1 × ⃗τ2 ⃗q · (⃗σ1 + ⃗σ2),(92)T (2)≡2(⃗τ1 + ⃗τ2) [q · S2, S1 · S2] + (1 ↔2) ≃i(⃗τ1 + ⃗τ2) ⃗q · ⃗σ1 × ⃗σ2(93)with qµ ≡qµ2 ≃−qµ1 . With the help of these operators, we can rewrite (91) asM1π = −T (1)1M2 −q2hF V1 (q2) −1i(94)with Mtree = −T (1)1M2−q2.As for two-pion-exchange and four-fermion contact interaction contributions, the rel-evant diagrams are those given in Fig.6(a) −(k) and their symmetrized ones.

Before going#14We denote particle indices by i = 1, 2 without expliciting heavy fermion fields. For instance, S1 should beunderstood as the spin operator sandwiched between Bv and Bv of particle 1 with velocity v. In this section,q2 is the four-momentum squared of the pion but we are concerned with the situation whereq0|q| << 1, sothe static approximation q2 ≈−|q|2 will be made in practical calculations and also in Fourier-transformingto coordinate space later.

In fact, the static approximation is not only natural for the chiral counting butalso essential for suppression of n-body forces and currents for n > 2. More on this later.24

into any details, one can readily see that each graph in Fig. 6 contributes a term of at leastO(Q).

This can be shown both in the conventional method and in HFF by observing thattheir contributions vanish if we set M = qµi = kµ = 0. This assures us that our countingrule is indeed correct.

Therefore we can neglect all the graphs proportional to Sµ since theaxial-charge operator involves S0 ∼O(Q/mN) as stated before. Figures (f), (g), (h) and(j) belong to this class.

Now Fig. (e) is identically zero because of the isospin symmetryand Fig.

(i) is proportional to v · S = 0.The graph (k), involving time-ordered pionpropagators, are the so-called “recoil graphs” [4] which as we shall argue in Section 8 willbe cancelled by similar recoil terms in reducible graphs. So we are left with only the fourgraphs (a), (b), (c) and (d) to calculate.

Without any further approximation than usingHFF, the full expression of the four graphs comes out to be⃗Aµ(a)=−(2⃗τ2 −i⃗τ1 × ⃗τ2) gA8F 4 (vµ q2 · S)1 f0(q22),⃗Aµ(b)=(2⃗τ2 + i⃗τ1 × ⃗τ2) gA8F 4 (q2 · S vµ)1 f0(q22),⃗Aµ(c)=(−2⃗τ1 −2⃗τ2 + i⃗τ1 × ⃗τ2) g3A2F 4 (vµ Sα)1SβSν2 Iν,αβ(q2),⃗Aµ(d)=(2⃗τ1 + 2⃗τ2 + i⃗τ1 × ⃗τ2) g3A2F 4 (Sα vµ)1SνSβ2 Iν,αβ(q2),(95)with f0(q2) given in detail in Appendix B and Iµ,αβ(q) defined byIµ,αβ(q) =Zl(l + q)µlαlβv · l v · (l + q) (l2 −M2) [(l + q)2 −M2]. (96)This integral is evaluated in Appendix E. Using the conditions v · qi = 0 and kµ ≃0, wecan rewrite them in a symmetrized form⃗Aµ(a + b)=vµ16π2gA8F 4hK0(q2) −16π2ηiT (1),(97)⃗Aµ(c + d)= −vµ16π2g3A16F 4nh−(d −1)16π2η + 3K0(q2) + 2K1(q2)iT (1)−8hK0(q2) −16π2ηiT (2)o,h ⃗AµiCT=−vµgA16F 4d(1)4 T (1) + d(2)4 T (2).

(98)where qµ ≡12(q2 −q1)µ andh ⃗AµiCT is the contribution from the counter-term Lagrangian(46). The Ki(q2) are finite pieces of the functions fi(q2) defined in Appendix B, i.e.,K0(q2) =Z 10dz ln"1 −z(1 −z) q2M2#,K1(q2) =Z 10dz−z(1 −z)q2M2 −z(1 −z)q2 .The expressions (97) and (98) contain singularities in η, which are removed by the counterterm contribution withd(1)4=κ(1)4+h(d −1)g2A −2iη,d(2)4=κ(2)4−8g2Aη(99)25

where the renormalized constants κ4’s are finite and scale-independent.The resulting two-body axial-charge operator including finite counter-term contribu-tions isM2π=116π2F 2("−3g2A −24K0(q2) −12g2AK1(q2)#T (1) + 2g2AK0(q2)T (2))−14F 2κ(1)4 T (1) + κ(2)4 T (2). (100)The two-body axial-charge operator due to loop correction is then the sum of (94) and (100)Mloop = M1π + M2π.

(101)As it stands, the constants κ4’s are the only unknowns in the theory as they cannot bedetermined from nucleon-nucleon interactions as mentioned before. They could in principlebe extracted from two-nucleon processes like N +N →N +N +π but they appear as higher-order corrections and it is inconceivable to obtain an information on these presumably smallconstants from such processes.

However as argued above, we expect the constants κ(i)4tobe numerically small and what is more significant, when we go to coordinate space as weshall do below to apply the operator to finite nuclei, they become δ functions and will becompletely suppressed as we discussed in Section 2. In momentum space, such constantterms have also to be removed as done for the celebrated Lorentz-Lorenz effect (or moregenerally for the Landau-Migdal g′0) in pion-nuclear scattering [22].

It should be stressedthat once the constant counter terms are removed, no unknown parameters enter at next tothe leading order in the chiral expansion in nuclei. It is also noteworthy that to the orderconsidered, the loop contributions are renormalization-scale independent.6.2Going to coordinate spaceApplications in nuclear transitions are made more readily in configuration space.Furthermore considerations based on ranges of nucleon-nucleon interactions which seemnecessary for rendering chiral symmetry meaningful in nuclei are more transparent in thisspace.Therefore we wish to Fourier-transform the operators (97) and (98) into a formsuitable for calculations with realistic nuclear wave functions.

In doing this, we will treatthe pion propagator in static approximation, namely, q2 ≈−|q|2. In Appendix B, we showhow the highly oscillating integrals involved in the calculation can be converted into inte-grals of smooth functions by performing the Fourier transform before doing the parametricintegration.

Since the spin-isospin operators T (1) and T (2) contain ⃗q – which is a derivativeoperator in configuration space, it is convenient to define˜T (1)=⃗τ1 × ⃗τ2 ˆr · (→σ 1 +→σ 2),˜T (2)=(⃗τ1 + ⃗τ2) ˆr · (→σ 1 ×→σ 2). (102)26

Writing Eqs. (94) and (100) in coordinate space which we will denote by˜M to distinguishfrom the momentum-space expression, we obtain – modulo δ function terms mentionedabove – the principal result of this paper:˜Mtree(r)=˜T (1) ddr−14πre−Mr,(103)˜M1π(r)=cR3M2F 2˜Mtree+˜T (1)16π2F 2ddr(−1 + 3g2A2hK0(r) −˜K0(r)i+ (2 + 4g2A)hK2(r) −˜K2(r)i),(104)˜M2π(r)=116π2F 2ddr(−"3g2A −24K0(r) + 12g2AK1(r)#˜T (1) + 2g2AK0(r) ˜T (2)), (105)˜Mloop(r)=˜M1π(r) + ˜M2π(r).

(106)As defined, Mnπ are nπ exchange corrections to the soft-pion (tree) term. Mloop is thereforethe total loop correction we wish to calculate.

The explicit forms of the functions Ki(r)and ˜Ki(r) are given in Appendix B.As noted above, the constant cR3 can be extracted from the isovector Dirac form factorof the nucleon, i.e.,cR3M2F 2 = M26 ⟨r2⟩V1 ≃0.04784. (107)It is interesting to separate what we might call “long wavelength contribution” from˜M1π,˜M1π(r) = δsoft ˜Mtree(r) + (short range part)whereδsoft = cR3M2F 2 +M216π2F 2"1 + 3g2A22 −π√3−(1 + 2g2A)179 −π√3#≃0.051(108)and compare this one-loop prediction for δsoft #15 to what one would expect from thephenomenological dipole form factorF V1 (q2) = Λ2Λ2 −q2!2(109)with Λ = 840 MeV.

This form factor leads to the following one-pion exchange contributionto Mloop, corresponding to (104):˜Mdipole1π=˜T (1)M24πxπY1(xπ) Λ2Λ2 −M2!2−1#15It is worth noting that this contribution is generic in the sense that it is more or less model-independent:It is of the same form and magnitude whether it is given by chiral one-loop graphs or by the vector dominance(see Appendix I) or by the phenomenological dipole form factor. This may have to do with the fact thatit is controlled entirely by chiral symmetry.

It is curious though that this longest wavelength effect is anenhancement rather than quenching usually associated with form factors.27

−14π12 Λ2Λ2 −M2!Λ2 + Λ2Λ2 −M2!2 1r2 + Λre−Λr. (110)Identifying the first term of (110) with the first term of (104), we see that δsoft correspondsto Λ2Λ2 −M2!2−1 ≃0.0571.It is remarkable – and pleasing – that the one-loop calculation of δsoft is so close to theempirical value.

Furthermore the remaining term in (104) involving the functions K′i cor-responds – and when applied to the process of interest, is numerically close – to the second(short-ranged) term in (110).7Numerical ResultsIn order to get a qualitative idea of the size involved, let us first look at the magnitudeof the relevant terms given in momentum space. For this purpose we set q2 ≈−|q|2 ∼−Q2,where Q is taken to be a characteristic small momentum scale probed in the process which wetake to be of order of mπ at most.

For convenience we shall factor out the tree contributionfrom the expression (101) and write it asM = Mtree(1 + δM + O(Q3))(111)where δM is the chiral correction of O(Q2) that we have computed (relative to the treecontribution). We obtainδM = δ1π + δ2πwhere, setting ⟨T (1)⟩= ⟨T (2)⟩in nuclear medium #16 and dropping the κ4’s #17δ1π≈Q24F 2"−23F 2⟨r2⟩V1 + 1 + 3g2A8π2K0(Q2) −1 + 2g2A2π2K2(Q2)#,δ2π≈−Q2 + M24F 2"5g2A + 216π2K0(Q2) −g2A8π2 K1(Q2)#.

(112)For Q ∼mπ ≈140 MeV, gA = 1.25 and F ≈93 MeV, we get|δ1π| ∼0.045,|δ2π| ∼7.5 × 10−3. (113)This is consistent with the notion that at the relevant scale Q, the chiral correction remainssmall.#16 One can show in fermi-gas model, Wigner’s SU(4) supermultiplet model or even jj-coupling shell modelof nucleus with one particle outside of closed core, ⟨T (1)⟩= ⟨T (2)⟩.

This relation will be assumed in allnumerical calculations that follow. We would like to thank Kuniharu Kubodera for his help on this relation.#17Dropping the constant terms in momentum space is not fully justified unless all other terms of the samenature are removed as well.

This problem is avoided in coordinate space. We give only the absolute valuesfor the δnπ for the same reason.

See below for more on this matter.28

We now turn to a more realistic estimate of the chiral correction appropriate to theactual situation in finite nuclei. Calculating nuclear transition matrix elements in momen-tum space is cumbersome and delicate.

There are several reasons for this. The most seriousproblem is the implementation of the short-range correlation.

In the well-studied case asin the π-nuclear scattering, we know how to proceed, obtaining the celebrated Lorentz-Lorenz effect. Roughly the argument goes as follows [22].

Consider a term of the form⃗q2/(⃗q2 + M2) that figures in the p-wave pion-nuclear scattering amplitude, or more specif-ically in the interaction between the particle-hole states excited by the pion. Rewrite thisas 1 −M2/(⃗q2 + M2).

Removing the constant 1 corresponds to suppressing a δ functionin coordinate space and leads to the Lorentz-Lorenz factor. Note that this procedure ofaccounting for short-range correlations can even change the sign.

Unfortunately our casedoes not lend itself to a simple treatment of this kind because of the nonanalytic termscoming from the loop contributions: there is no economical way of “removing δ functions”from them. This task is much simpler and more straightforward in coordinate space.Let us therefore turn to the coordinate space operators (103) and (106).

In Fig. 7, weplot ˜Mtree (103) and ˜Mloop (106) as function of the internuclear distance r = |⃗r1−⃗r2| setting˜T (1) = ˜T (2) = 1.

Some of the important features discussed in the preceding sections canbe seen in this plot. While negligible at large distance, say, r > 1 fm, the loop correctionsget progressively significant at shorter distances and at r ∼0.4 fm, they are comparable tothe soft-pion result.

There is nothing surprising or disturbing about this feature at shortdistances.At shorter distances which are probed by the momentum scale approachingthe chiral scale, there is no reason to ignore the degrees of freedom integrated out fromthe theory. Low-order calculations with higher chiral-order degrees of freedom eliminatedcannot possibly describe the short-distance physics properly.

This may be construed as asign that ChPT is not predictive in nuclei. We claim that this is not so.

The point is thatas long as the scale Q probed by experiments is much less than the chiral scale, truncatinghigher chiral-order and shorter wavelength degrees of freedom as done in ChPT can bemeaningful provided short-range nuclear correlations are implemented in the way discussedabove.Calculations of the nuclear matrix elements with sophisticated wave functions infinite (light and heavy) nuclei – and comparison with experimental data – will be madeand reported in a separate paper.Here for our purpose of getting a semi-quantitativeidea, the fermi-gas model as used by Delorme [23] will suffice.One could incorporateaccurate correlation functions – and this will be done for specific transitions in finite nuclei.Here we will not do so. We shall instead take the simplest correlation function, namelyˆg(r, d) = θ(r −d) with the cut-offdistance d ≈0.7 fm as used by Towner [24].

Since this isa rather crude approach, we will consider the range of d values between 0.5 and 0.7 fm.Specifically we are interested in the ratio of the matrix elements ⟨Mloop⟩/⟨Mtree⟩29

which in fermi-gas model takes the form (see Appendix H)R(d, ρ) ≡⟨Mloop⟩⟨Mtree⟩=R ∞d dr r [j1(pFr)]2 ˜Mloop(r)R ∞d dr r [j1(pFr)]2 ˜Mtree(r)(114)where pF and ρ =23π2 p3F are, respectively, the fermi-momentum and density of the sys-tem, j1(x) =sin xx2−cos xxand˜Mloop(r) ≡˜M1π(r) +˜M2π(r).Note that w(pF, r) ≡4πr [j1(pFr)]2 /p2F can be viewed as a weighting function. Since this calculation is straightfor-ward, we shall not go into details here.

For completeness, however, we sketch the calculationin Appendix H.In Fig. 8 are plotted the functions w(pF, r) ˜Mtree(r) and w(pF, r) ˜Mloop(r) with ˜T (1) =˜T (2) = 1 for pF ≃1.36 fm−1 corresponding to nuclear matter density.

The ratio R(d, ρ) isplotted in Figure 9 for d = 0.5, 0.7 fm. For d = 0.7 fm which was used by Towner[24],the loop correction is at most of the order of 10% of the soft-pion term at nuclear matterdensity.

There are two important points to note in the result. The first is that separatelythe loop corrections to the one-pion term (i.e, M1π) and the two-pion term (i.e, M2π) canbe substantial but the sum is small.

The second point is that the resulting loop contributionhas a remarkably weak density dependence. The first is a consequence of chiral symmetryreminiscent of the tree-order cancellation in linear σ model of the nucleon pair term andthe σ-exchange term in the S-wave πN scattering amplitude.

The second observation has asignificant ramification on the mass dependence of axial-charge transitions in heavy nucleito which we will return shortly.8Other ContributionsHere we briefly discuss what other graphs could potentially contribute and the reasonwhy they are suppressed in our calculation. Consider the two-body graphs given by Figure6(k) where the pion propagators are time-ordered.

They belong to what one calls “recoilgraphs” in the literature [4]. To O(Q2) relative to the soft-pion term, these graphs – andmore generally all recoil graphs including one-pion exchange – do not contribute.

The reasonis identical to the suppression of three-body forces as discussed by Weinberg[9]: the graphsin Fig. 10 are exactly cancelled by the recoil corrections to the iterated one-pion exchangegraphs that are included in the class of reducible graphs.

Thus to the extent that the staticapproximation is used in defining the one-pion exchange potential, these graphs should notbe included as corrections. Incidentally this justifies the standard practice of ignoring recoilgraphs in calculating exchange contributions in both weak and electromagnetic processesin nuclei.We have ignored in our calculation three-body and higher-body contributions suchas Figure 11.

The reason for ignoring these graphs is identical to that used for proving thesuppression of three-body and other multi-body forces [9]. As in nuclear forces, they cancontribute at O(Q3) relative to the soft-pion term[8].30

An interesting question to ask is in what situations the approximations that justifydropping the graphs considered here break down in nuclei. It is clear that the static ap-proximation – one of the essential ingredients of the heavy-fermion formalism – must breakdown when the energy transfer involved is large.

Imagine that one is exciting a ∆resonancein nuclei by electroweak field. The energy transfer is of the order of 300 MeV, so the staticapproximation for the pion propagator involving a π∆N vertex cannot be valid.

In suchcases, one would expect that multi-body forces and currents suppressed in this work couldbecome important. This suggests specifically that in electron scattering from nuclei withsufficiently large energy transfer, n-body currents (for n > 2) will become progressivelymore important in heavier nuclei.

Combined with the dropping mass effect (i.e, “Brown-Rho” scaling mentioned below), one expects a large deviation from the standard mean-fielddescription used currently.9Conclusions and DiscussionsWe have used heavy-baryon chiral perturbation formalism to calculate the leadingcorrections to the soft-pion axial-charge operator in nuclei. Exploiting short-range suppres-sion of the counter terms and other short-range components of the two-body operator, wehave shown that the chiral filter mechanism holds in nuclear matter with a possible uncer-tainty of no more than 10%, thus confirming the dominance of the soft-pion exchange.

Ina separate paper, we will show that the same holds in electromagnetic responses in nuclei.Since the currents (both vector and axial-vector) are calculated consistently with the symme-tries involved, they are fully consistent with nuclear forces that are calculated to the samechiral order: Ward-Takahashi identities will be formally satisfied although in practice ap-proximations made for calculations may disturb them. The final consistency will of coursehave to be checked `a posteriori case-by-case.Taking this result as a statement of chiral symmetry of QCD in nuclei, what can onelearn from this concerning the phenomenological models popular in nuclear physics whereone uses exchanges of all the low-lying bosons in fitting nucleon-nucleon scattering (suchas the Bonn potential) as well as calculating the exchange currents?

Suppose we denotethe axial-charge two-body operator from one-pion exchange with form factors by A1π, one-heavy-meson exchange with form factors by AH, the axial current form factor by AF F , allcalculated within a phenomenological model, then our result implies that for the model tobe consistent with chiral symmetry, then the total must sum toAtotal = A1π + AH + AF F + · · · ≈Asoft(1 + δ),|δ| << 1(115)where Asoft is the soft-pion term as defined in this paper and δ is the next-to-leading termof O(Q2). Our calculation illustrates how individually significant terms conspire to give asmall O(Q2) correction which is insensitive to nuclear density.31

One other outcome of our result is that while a subset of graphs can have a substantialdensity dependence, the small net chiral correction from the totality of the graphs doesnot have an appreciable nuclear density dependence, at least in fermi-gas model. We seeno reason why this weak density dependence should not persist in more realistic nuclearmodels.

Thus assuming that n-th order chiral corrections for n ≥3 (relative to the leadingsoft-pion term) are not anomalously large, we come to the conclusion that meson-exchangeaxial-charge contributions to nuclear matrix elements cannot be substantially enhanced inheavy nuclei over that in light nuclei. The question arises then as to what could be theexplanation for Warburton’s recent observation that while the mesonic effect is about 50%in light nuclei, it is required to be 100% in heavy nuclei such as in lead region [25].

Onesuggestion [26] was that the parameters of the basic chiral Lagrangian have to be modifiedin the presence of nuclear matter consistent with trace anomaly of QCD [27]. It predictedthat hadron masses and pion decay constants that appear in the single-particle and one-pion exchange two-body operators are scaled by a universal factor Φ that depends on matterdensity.

Another suggestion [24, 28] was that exchanges of heavy mesons σ, ρ, ω, a1 etc couldbecome important in heavy nuclei while relatively unimportant in light systems. The lattermechanism relied on nucleon-antinucleon pair terms in phenomenological Lagrangians.

Bothmechanisms seemed to qualitatively account for the enhancement.We wish to understand the possible link, if any, between the chiral Lagrangian ap-proach and the phenomenological approach that includes pair terms involving heavy mesons.Since within the chiral approach developed in this paper the pair is naturally suppressedas required by chiral symmetry and multi-body currents are also suppressed as discussedabove, the scaling mass effect of [26, 27] is the only plausible mechanism left within low-order chiral expansion for the medium enhancement noted by Warburton. Needless to say,we cannot rule out – though we deem highly unlikely – the possibility that higher order chi-ral terms supply the needed density effect.

Incorporating the possible 10% loop correctioncalculated above in the two-body operator and the scaling factor Φ = m∗N/mN ≈0.8, onegets in the scheme of Ref. [26] the enhancement in heavy nuclei (at nuclear matter density)ǫMEC ≈2.1 which is reasonable in the lead region compared with the experimental value2.01 ± 0.05 [25].

Within the scheme, this is the entire story and a surprisingly simple one.Of course more detailed finite nuclei calculations will be needed to make a truly meaningfultest of the theory.In the phenomenological approach studied by the authors in [24] and [28], there is nofundamental reason to suppress N ¯N pairs, so that heavy mesons could contribute throughthe pair term. However the exchange of heavy mesons, particularly that of vector mesons, issuppressed by short-range correlations in nuclear wave functions.

Furthermore in the modelof Towner[24], a large cancellation takes place in the sum such that the relation (115) seemsto hold well[29]: Towner finds δ < 10% over a wide range of nuclei. It is naturally temptingto suggest that Towner’s model gives a result close to ours because it is consistent with chiralsymmetry, at least to the same order of chiral expansion as ours, with higher-order terms32

implicit in Towner’s model which need not be consistent with ChPT somehow cancellingout#18. There is however one aspect that needs to be clarified: in the models of [24] and [28],there is a pair term associated with a scalar meson (σ) exchange.

In the chiral Lagrangianused in this paper, there is no equivalent scalar field. We have however the scalar field χassociated with the trace anomaly of QCD which plays a role in the Brown-Rho scaling[27].We believe that these two effects are roughly related in the sense discussed in Ref.

[30].In this sense we would say that the pair term involving the σ meson is simulating thedensity-dependence of the nucleon mass in the one-body axial-charge operator. There is nomechanism in Ref.

[24, 28], however, for the density dependence of Ref. [26] in the two-bodyoperator.

We suggest that this can be generated by taking three-body terms with an N ¯Npair coupled to a σ-exchange.An obvious omission in our treatment of the axial current is the space component ofthe current governing Gamow-Teller transitions in nuclei (and the time part of the electro-magnetic current in [14]). The reason for this was already stated at several points in thepaper: this part of the current is not dominated by a soft-pion exchange and indeed as notedmany years ago [21] it is rather the very short-ranged part of nuclear interactions (roughlyequivalent to the removal of the δ functions associated with the counter terms in the spin-isospin channel) that plays an important role, e.g.

in quenching the axial-vector couplingconstant from the free-space value gA = 1.25 to g∗A ≈1 in nuclear matter. (For a simi-lar situation with the isoscalar axial-charge transition mediated by a neutral weak currentwhere soft-pion exchange is forbidden, see Ref.[31].) Furthermore, three-body operators forthe space component of the axial current may not be negligible.

For instance, as one cansee from Appendix A, the three-body Gamow-Teller operator involving one nucleon with anAµππNN vertex with the pions absorbed by two other nucleons is not trivially suppressedas it is for the axial-charge operator. This suggests that low-order chiral perturbation the-ory may have little to say about this aspect of nuclear interactions.

It is intriguing that innuclei, both chiral and non-chiral aspects of QCD seem to coexist in the same low-energydomain. This makes QCD in nuclei quite different from and considerably more intricatethan QCD in elementary particles studied by particle physicists.Finally we mention a few additional issues we have not treated in this paper but weconsider to be important topics for future studies.• It would be interesting to see what two-loop (and hopefully higher-loop) and corre-sponding chiral corrections do to the chiral filter phenomenon.

Two-loop calculationsare in general a horrendous task but the situation in nuclear axial-charge transitions#18 The following observation may be relevant to our argument that the counter terms κ(i)4must be ignored.Suppose one constructs a purely phenomenological theory based on meson-exchange picture by fitting ex-periments but conform to the symmetries of the strong interactions. Towner’s model is one such example.One can convince oneself that in such a model, it is not possible to generate counter terms of the κ4 type ininfinite mass limit.

Therefore if such terms existed, then they must be due to degrees of freedom that arenot relevant at the accuracy required.33

might be considerably simpler than in other processes.• It would be important to see whether ChPT is predictive for processes involving largermomentum transfers as well as large energy transfer. From our experience with theelectrodisintegration of the deuteron at large momentum transfers where the naivesoft-pion approximation seems still to work fairly well, we conjecture that the chiralfilter mechanism holds still in some channels even in processes probed at shorterdistances or at larger momentum transfers.

But as mentioned above, large energytransfer electron scattering might require multi-body currents in heavy nuclei.• In this paper, we worked with an effective Lagrangian in which all other degrees offreedom than pions and nucleons have been integrated out. It would be important toreformulate ChPT using a Lagrangian that contains vector mesons incorporated `a lahidden gauge symmetry (HGS) [32]#19 and also nucleon resonances (such as ∆).

Asmentioned before, we believe that the chiral filter argument presented in this paperis not modified in the presence of these resonances in the Lagrangian. In AppendixI, we show that the presence of vector mesons does not modify our prediction on theaxial-charge operator.

Furthermore it is not difficult to see that the baryon resonances– in particular the ∆resonances – do not contribute to the axial-charge transitionsto the order considered. However as is known for Gamow-Teller transitions in nuclei[21], certain processes in nuclei might require, even at zero momentum transfer, anexplicit role of some of these heavier particles.

As recently shown by Harada andYamawaki [34], vector mesons introduced via HGS can easily be quantized, so theirimplementation in ChPT would pose no great difficulty.• A systematic higher-loop chiral perturbation approach using the same heavy-fermionformalism to kaon-nuclear interactions and kaon condensation has not yet been workedout. This is an important issue for hypernuclear physics, relativistic heavy-ion physicsand stellar collapse [35].• Finally if the parameters of effective chiral Lagrangians scale as a function of nuclearmatter density as suggested by Brown and Rho[27], then one expects that as matterdensity increases, many-body currents will become increasingly important even atsmall energy transfer.

This was already noticed in [26] where the soft-pion exchangecharge operator became stronger in heavier nuclei. We already noted that this effectwill show up more prominently in nuclear electromagnetic responses with large energytransfer.

Future accurate experiments in electron scattering offnuclei will test thisprediction. Of course this issue has to be treated together with many-body forces thatenter into such processes.#19As stressed by Georgi, the HGS is an approach most suited to a systematic chiral counting when vectormesons are explicitly present.

See [33].34

AcknowledgmentsWe are grateful for useful discussions with G.E. Brown, K. Kubodera, U.-G. Meissner,D.O.

Riska, I.S. Towner and E.K.

Warburton. One of us (DPM) wishes to acknowledge thehospitality of Service de Physique Th´eorique of CEA Saclay where part of his work was done.The work of TSP and DPM is supported in part by the Korean Science and EngineeringFoundation through the Center for Theoretical Physics, Seoul National University.35

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Appendix A: Chiral Lagrangian Eq. (22) ExpandedFor completeness we expand L0 in powers of the pion field and in external fields.

We willgroup L0 by the number of external gauge field, L0 = L00 + L10 + L20,L00=12(∂µ⃗π)2 −12M2⃗π2 −16F 2h⃗π2∂µ⃗π · ∂µ⃗π −(⃗π · ∂µ⃗π)2i+ M24! F 2⃗π22+ Biv · ∂B+ B−vµ4F 2⃗τ · ⃗π × ∂µ⃗π −gAF Sµ⃗τ ·∂µ⃗π +16F 2⃗π ⃗π · ∂µ⃗π −∂µ⃗π ⃗π2B−12CaBΓaB2 + · · · ,(A.1)L10=⃗Vµ ·⃗π × ∂µ⃗π −13F 2⃗π × ∂µ⃗π ⃗π2−F ⃗Aµ ·∂µ⃗π −23F 2∂µ⃗π ⃗π2 −⃗π ⃗π · ∂µ⃗π+ 12Bvµ⃗Vµ + 2gASµ ⃗Aµ·⃗τ +12F 2⃗π ⃗τ · ⃗π −⃗τ ⃗π2B+ 12Bvµ ⃗Aµ + 2gASµ⃗Vµ· 1F ⃗τ × ⃗π −16F 3⃗τ × ⃗π ⃗π2B + · · · ,(A.2)L20=12F 2 ⃗A2µ + F⃗π · ⃗Vµ × ⃗Aµ + 12h⃗π2 ⃗V2µ −⃗A2µ−(⃗π · ⃗Vµ)2 + (⃗π · ⃗Aµ)2i+ · · · .

(A.3)From the (A.2), we extract Noether currents,⃗Aµ=−F∂µ⃗π −23F 2∂µ⃗π ⃗π2 −⃗π ⃗π · ∂µ⃗π+ 12B2gASµ⃗τ +12F 2⃗π ⃗τ · ⃗π −⃗τ ⃗π2+ vµ 1F ⃗τ × ⃗π −16F 3⃗τ × ⃗π ⃗π2B + · · · ,(A.4)⃗V µ=⃗π × ∂µ⃗π −13F 2⃗π × ∂µ⃗π ⃗π2+ 12Bvµ⃗τ +12F 2⃗π ⃗τ · ⃗π −⃗τ ⃗π2+ 2gASµ 1F ⃗τ × ⃗π −16F 3⃗τ × ⃗π ⃗π2B + · · · . (A.5)Appendix B: Functions fi(q2)The functions fi(q2) (i = 0, 1, 2, 3) figuring in subsections (5.4.2) and (5.5) are definedbyf0(q2)=Zl1(l2 −M2) [(l + q)2 −M2],12gαβf1(q2) + qαqβf2(q2)=Zl(l + q)αlβ(l2 −M2) [(l + q)2 −M2]f3(q2)=2f2(q2) + 12f0(q2)(B.1)38

where we have defined f3(q2) throughZllα (2l + q)β(l2 −M2) [(l + q)2 −M2] ≡gαβf1(q2) + qαqβf3(q2). (B.2)Here and in what follows, the mass M could be thought of as the pion mass M. One canverify that∂f1(q2)∂q2= f2(q2).After performing the parametric integration, we havef0(q2)=η −116π2 K0(q2),f1(q2)=∆(M2) −η6q2 −q216π22K2(q2) −12K0(q2),f2(q2)=−η6 +116π2 K2(q2),f3(q2)=η6 +116π22K2(q2) −12K0(q2)(B.3)whereη =116π2 Γ(ǫ) M24πµ2!−ǫand K(q2)’s are finite functions given explicitly byK0(q2)=Z 10dz ln"1 −z(1 −z) q2M2#= −2 + σ lnσ + 1σ −1,(B.4)K1(q2)=Z 10dz−z(1 −z)q2M2 −z(1 −z)q2 = 1 −σ2 −12σlnσ + 1σ −1,(B.5)K2(q2)=Z 10dz z(1 −z) ln"1 −z(1 −z) q2M2#= −49 + σ26 + σ(3 −σ2)12lnσ + 1σ −1,(B.6)withσ ≡ 4M2 −q2−q2!

12. (B.7)We should note that all the functions given above are positive definite for negative q2 andvanish when q2 = 0.

For −q2 ≪M2, we haveK0(q2)=K1(q2) = 16−q2M2 + O q4M4!,K2(q2)=130−q2M2 + O q4M4!.39

In the chiral limit (−q2 ≫M2), they simplify toK0(q2)=ln−q2M2 −2,K1(q2)=1,K2(q2)=16ln−q2M2 −518.In order to go to the r−space, we must Fourier-transform Ki(q2) and M2/(M2 −q2) Ki(q2). This involves an integration#20 of highly oscillating functions.

Instead of in-troducing a regulating function which kills contributions from large ⃗q2 and performing atricky numerical calculation, we transform the problem into an integral of a smooth functionwith the use of residue calculation. The point is that we do the parametric integration (ofvariable z) at the last step.

To see how this work, let us rewrite Ki(q2),K0(−Q2)=Z 10dx ln 1 + Q2E2!,K1(−Q2)=Z 10dxQ2Q2 + E2 ,K2(−Q2)=Z 10dx 1 −x24ln 1 + Q2E2!. (B.8)where Q = |q| =p−q2 and E = E(x) = 2M/√1 −x2.#21First we Fourier-transformalgebraically the integrands of the above integrals and then do the parametric integralnumerically.

The Fourier transform of K1 becomes an elementary residue calculation witha pole at Q = iM,K1(r) = δ(r) −14πrZ 10dx E2 e−Er(B.9)while K0 and K2 are somewhat involved due to the logarithmic function. We rewrite thelogarithmic functions (of K0 and K2) into a simple pole form by integration by partZ 10dx"1, 1 −x24#ln 1 + Q2E2!=Z 10dx2x21 −x2"1, 14 −x212#Q2Q2 + E2 .

(B.10)With the above equation andZd3q(2π)3 ei q·rQ2Q2 + E2Λ2Q2 + Λ2 =14πrΛ2Λ2 −E2Λ2e−Λr −E2e−Er,(B.11)we obtain the expressions for K(r, Λ) defined byKi(r, Λ) ≡Zd3q(2π)3 ei q·r Ki(−Q2)Λ2Q2 + Λ2 ,(B.12)#20Recall that Ki(q2) does not go to zero when ⃗q goes to infinity.#21 We have made the change of variable, x = 2z −1, to render the expressions more symmetric.40

K0(r, Λ)=14πrZ 10dx2x21 −x2Λ2Λ2 −E2Λ2e−Λr −E2e−Er,K2(r, Λ)=14πrZ 10dx2x21 −x2 14 −x212!Λ2Λ2 −E2Λ2e−Λr −E2e−Er. (B.13)Here the parameter Λ is introduced to regularize the integrals near the origin of the config-uration space.

When Λ = ∞, we get the expressions for the Ki(r) needed in the paper,Ki(r) ≡limΛ→∞Ki(r, Λ). (B.14)The above integrals are non-singular even near x = 1, since E increases so as to make theintegrals regular.

However the expressions contain highly singular terms near the origin ofconfiguration space when Λ goes to infinity. To see this, noteK0(r) = −14πrZ 10dx2x21 −x2 E2 e−Er + limΛ→∞14πr Λ2K0(q2 = Λ2) e−Λr,andlimΛ→∞14πr Λ2e−Λr = δ(r).Now K0(Λ2) goes to infinity logarithmically when Λ2 goes to infinity.So, roughly, thesecond term of (22) behaves in the limit of infinite Λ aslimΛ2→∞ln Λ2M2 δ(r).The mathematical reason for this behavior is not hard to see : For large Q2, both K0 andK2 increase logarithmically with a gentle slope.

Thus they can be viewed as a constant atlarge Q2 with their value increasing to infinity. The constant behavior leads to the δ(r)function.

This term singular at the origin is of course “killed” by the short-range correlationpresent in nuclear wave functions.In terms of the function so defined, we can immediately obtain ˜Ki(r) defined by˜Ki(r) ≡Zd3q(2π)3 ei q·r Ki(−Q2)M2Q2 + M2 = Ki(r, M). (B.15)In summary we have the expressions for Ki(r) (valid for r > 0)K0(r)=−14πrZ 10dx2x21 −x2 E2e−Er,K2(r)=−14πrZ 10dx2x21 −x2 14 −x212!E2e−Er(B.16)and the expressions for ˜Ki(r) (valid in the whole region)˜K0(r)=14πrZ 10dx2x21 −x2M2M2 −E2M2e−Mr −E2e−Er,˜K2(r)=14πrZ 10dx2x21 −x2 14 −x212!M2M2 −E2M2e−Mr −E2e−Er.

(B.17)41

Appendix C: Integral IdentitiesIn this Appendix, we list some useful identities for the integrals we need to evaluate.Consider the following integral without imposing the condition v2 = 1,Iα(v, M2) ≡Zllαv · l (l2 −M2),then Lorentz covariance implies that its most general form isIα(v, M2) = vαI0(v2, M2) .Now multiplying vα to both sides, we get∆(M2) = v2I0(v2, M2)orZllαv · l (l2 −M2) = vαv2 ∆(M2) . (C.1)Using this expression, we obtain the following identities by successive differentiation withrespect to v,Zllαlβ(v · l)2 (l2 −M2)=−gαβv2 −2 vαvβ(v2)2∆(M2)(C.2)Zllαlβlµ(v · l)3 (l2 −M2)=4vαvβvµ(v2)3−vµgαβ + vαgβµ + vβgαµ(v2)2∆(M2)(C.3)and so on.Appendix D: Functions h(v · k)’sIn this Appendix, we define – and give explicit forms of – the function h(v · k) thatfigures in the self-energy for the off-shell nucleon discussed in section (4.3).

The basic oneis h0(v · k),h0(v · k)=Zl1v · (l + k) (l2 −M2)=2yη +18π2πM∗+ 2y −2M∗sin−1 yM,for|y| ≤M,(D.1)=2yη +18π22y −2 ˜M sinh−1 yM−θ(y −M)2iπ ˜M, for |y| ≥M(D.2)where y ≡v · k, M∗≡pM2 −y2, ˜M ≡py2 −M2 and −π2 ≤sin−1(x) ≤π2 . We observethat the imaginary part appears only when v · k ≥M.

The even part of h0 has a verysimple formhS0 (y) ≡h0(y) + h0(−y)2= −18πqM2 −y2. (D.3)42

For special values of y, we have h0(0) = −M8π, h′0(0) = 2η, h0(M) = −h0(−M) =2M1−2ǫη.The finite function h0(y) is defined byh0(y) = h0(y) −2yη. (D.4)We now examine the function h(v · k) defined byZllα lβv · (l + k) (l2 −M2) ≡gαβh(v · k) + vαvβ(· · ·).

(D.5)If we multiply the above equation by gαβ and vβ, we obtain the following identityh(y) =1d −1hy ∆(M2) + (M2 −y2)h0(y)i. (D.6)Note that the even part has a very simple form,hS(y) ≡h(y) + h(−y)2= −124πM2 −y2 32 .

(D.7)Let us define h3(v · k, v · q)Zllαlβv · (l + k −q) v · (l + k) (l2 −M2) = gαβh3(v · k, v · q) + vαvβ(· · ·)orh3(v · k, v · q) =1v · q [h(v · k −v · q) −h(v · k)] . (D.8)When the nucleon is on-shell, that is, v ·k = v ·q = 0, then h3 becomes h3(0, 0) = −∆(M2).More generally, for small momentum, we haveh3(v·k, v·q) = −∆(M2)−M16π(2v·k−v·q)+ 23ηh3(v · k)2 −3v · k v · q + (v · q)2i+· · · (D.9)where the ellipsis denotes finite and higher momentum terms.

Finally consider h4(v · q)defined byZllα lβ(v · l)2 v · (l −q) (l2 −M2) ≡gαβh4(v · q) + vαvβ(· · ·). (D.10)The h4 is a somewhat complicated function,h4(y)=h(−y) −h(0)y2+ ∆(M2)y=M16π + 23v · q η + · · ·(D.11)with h(y) given by (D.6) and the ellipsis again denotes finite and higher momentum terms.43

Appendix E: Integrals for Two-Pion Exchange CurrentsConsider the integrals of the formZll’sv · (l + k) (l2 −M2) [(l + q)2 −M2]which figure in two-pion exchange currents. For most of the cases, we do not need the termsproportional to vµ as they appear multiplied by the spin operator Sµ and vanish.

To utilizethis, we assume that the spin operator is multiplied to the numerator. Now we haveZl(l + q)α lβv · (l + k) (l2 −M2) [(l + q)2 −M2]=gαβ + 2qαqβ ∂∂q2B0(k, q),(E.1)Zl(l + q)α lβ (2l + q)µv · (l + k) (l2 −M2) [(l + q)2 −M2]=gαβ + 2qαqβ ∂∂q2[qµB1(k, q) + vµB2(k, q)]+qα gβµ + qβgαµB1(k, q) +qαgβµ −qβgαµB0(k, q)(E.2)where we have neglected terms proportional to vα or vβ.

After some algebra, we can getthe following relations,q2B1(k, q) + v · qB2(k, q) = h(v · k) −h(v · k −v · q),B2(k, q) = f1(q2) + v · q [B0(k, q) −B1(k, q)] −2v · kB0(k, q).When v · k = v · q = 0, they become elementary functions,B0(q2)=−116πZ 10dzqM2 −z(1 −z)q2,B1(q2)=0,B2(q2)=f1(q2)(E.3)For small but nonzero momentum, they becomeB0(k, q)=−M16π +v · k −12v · qη + · · · ,B1(k, q)=v · q6 η + · · · ,B2(k, q)=∆(M2) + (2v · k −v · q) M16π −"2(v · k)2 −2v · k v · q + 23(v · q)2 + q26#η+ · · · . (E.4)For two pion exchange graphs, we need to evaluateIν,αβ ≡Zl(l + q)ν lα lβv · l v · (l + q) (l2 −M2) [(l + q)2 −M2].

(E.5)In evaluating this function, we neglect terms proportional to vν, vα or vβ because theyvanish when multiplied by SνSαSβ. With the parametrization explained in the text, wehave, in the limit of v · q = 0,Iν,αβ(q) = −132π2 (−qνgαβ + qαgνβ + qβgνα)hK0(q2) −16π2ηi−116π2qνqαqβq2K1(q2), (E.6)44

where K1(q2) is defined at (B.5),K1(q2) =Z 10dz−z(1 −z)q2M2 −z(1 −z)q2 . (E.7)For completeness, we also list results for vector currents for which we need the fol-lowing integralsZl(l + q)α lβ (2l + q)µv · l v · (l + q) (l2 −M2) [(l + q)2 −M2].

(E.8)We first look at its low momentum behavior,−vµ M8πgαβ −qα gβµ −qβ gαµη + Oq2. (E.9)In the limit of v · q = 0, we haveZl(l + q)α lβ (2l + q)µ(v · l)2 (l2 −M2) [(l + q)2 −M2]=−vµ8πgαβ + 2qαqβ ∂∂q2 Z 10dzqM2 −z(1 −z)q2 −qα gβµ −qβ gαµf0(q2).

(E.10)Again we dropped terms proportional to vα or vβ.Appendix F: Three-point Vertices (Figure 4)In this section as well as in the next, we classify graphs into Class A, Class V andClass AV. The graphs in Class V appear only with the vector current while the graphsin Class AV appear both for the vector and axial-vector currents.

The graphs in ClassA involving the axial-vector current do not figure in three-point vertices. We define twooperators for the graphs in Class AV: T µ1 = vµ and T2 = 2gASµ.

We write the expressionsonly for the axial-vector current for graphs in Class AV. The expressions for the vectorcurrent is obtained by interchanging T µ1 and T µ2 .

Fig.4c and 4d vanish because they areproportional to v · S.Class AV#22Γµ,aANN(a)=τa2T µ2F 2 ∆(M2),(F.1)Γµ,aANN(b)=τa2g2AF 2 SαT µ2 Sα h3(v · k, v · q),(F.2)iΓaπNN(a)=gA3F 3 τa q · S ∆(M2),(F.3)iΓaπNN(b)=d −34τa q · S g3AF 3 h3(v · k, v · q),(F.4)#22The figure label a, b, c... is given in the parenthesis.45

Class VΓµ,aV NN(e)=−τa2F 2hvµf1(q2) + v · q qµf3(q2)i,(F.5)Γµ,aV NN(f)=2τag2AF 2S · S + 2(q · S)2 ∂∂q2[qµB1(k, q) + vµB2(k, q)]+ {q · S, Sµ} B1(k, q) + [q · S, Sµ] B0(k, q)}(F.6)Γµ,abcV ππ (g)=iǫabc(qc −qb)µ 53∆(M2)F 2,(F.7)Γµ,abcV ππ (h)=iǫabc1F 2h−(qc −qb)µf1(q2a) + qµa (q2c −q2b)f3(q2a)i. (F.8)Here the index a labels the isospin of the photon with four-momentum qa, the indices b andc the isospin of the pions with their momenta qb and qc and the momentum conservation isqa + qb + qc = 0.Appendix G: Four-point Vertices (Figure 5)Here we define qµ = qµa + qµb .

For other notations, see Appendix F. Figures 5(i) −(n)vanish because they are proportional to v · S. Here we restrict ourselves to the case ofon-shell nucleons, v · k = v · (qa + qb) = 0.Class AΓµ,abπA (a)=12F 3 ǫabcτc vµf1(q2),(G.1)Γµ,abπA (b)=−iδab(2qa + 6qb)µ 2g2A3F 3S · S + 2(q · S)2 ∂∂q2B0(q2)−ǫabcτc2g2AF 3vµS · S + 2(q · S)2 ∂∂q2f1(q2) + [q · S, Sµ] B0(q2),(G.2)Class AVΓµ,abπA (c)=−512ǫabcτcT µ1F 3 ∆(M2),(G.3)Γµ,abπA (d)=g2A2F 3 ǫabcτc SαT µ1 Sα ∆(M2),(G.4)Γµ,abπA (e + f)=T µ14F 3n4iδab v · qa hS0 (v · qa) + ǫabcτch∆(M2) −2v · qahA0 (v · qa)io,(G.5)Γµ,abπA (g + h)=(−3iδab + ǫabcτc) g3A2F 3 Sα qb · S T µ2 Sα h4(v · qa)+(−3iδab −ǫabcτc) g3A2F 3 Sα T µ2 qb · S Sα h4(−v · qa),(G.6)46

Class VΓµ,abπV (o + p + q)=gA2F 3 v · qa(−2iδab −ǫabcτc)Zll · S (2l + qa)µv · l (l2 −M2) [(l + qa)2 −M2]−(2iδab −ǫabcτc)Zll · S (2l −qa)µv · l (l2 −M2) [(l −qa)2 −M2],(G.7)Γµ,abπV (r)=−2iδabg3AF 3Zl(l + qa) · S qb · S l · S(2l + qa)µv · (l + qa) v · l (l2 −M2) [(l + qa)2 −M2](G.8)whereB0(q2)=−116πZ 10dzqM2 −z(1 −z)q2,(G.9)hS0 (v · q)=12 [h0(v · q) + h0(−v · q)] = −18πqM2 −(v · q)2,(G.10)hA0 (v · q)=12 [h0(v · q) −h0(−v · q)] . (G.11)Now we study low-momentum expansion for on-shell nucleons with v · k = v · (qa + qb) = 0.To second order in external momentum;Class AΓµ,abπA (a)=12F 3 ǫabcτc vµ∆(M2) −η6q2 + · · ·,Γµ,abπA (b)=−iδab(qa + 3qb)µ g2AF 3 M′−ǫabcτc2g2AF 3vµ S · S∆(M2) −vµ η6hS · S q2 + 2(q · S)2i−[q · S, Sµ] M′+ · · · .Class AVΓµ,abπA (c)=−512ǫabcτcT µ1F 3 ∆(M2),Γµ,abπA (d)=g2A2F 3 ǫabcτc SαT µ1 Sα ∆(M2),Γµ,abπA (e + f)=T µ14F 3n−8iδab v · qa M′ + ǫabcτch∆(M2) −4(v · qa)2 ηio· · · ,Γµ,abπA (g + h)=(−3iδab + ǫabcτc) g3A2F 3 Sα qb · S T µ2 SαM′ + 23 v · qa η+(−3iδab −ǫabcτc) g3A2F 2 Sα T µ2 qb · S SαM′ −23 v · qa η+ · · · .Class VΓµ,abπV (o + p + q)=gAF 3 v · qa4iδabM′Sµ + ǫabcτc(Sµv · qa + vµqa · S)η + · · · ,Γµ,abπV (r)=−2iδabg3AF 3−2vµM′Sαqb · S Sα −(qa · S qb · S Sµ −Sµ qb · S qa · S)η+ · · · .We have used the notations M′ =M16π and η =116π2M24πµ2−ǫ Γ(ǫ).47

Appendix H: Fermi-Gas Model for Two-Body Axial-ChargeOperatorLet |F⟩be the ground state of Fermi-gas model whose fermi-momentum is pF and|ph⟩= b†pbh|F⟩be the one-particle (labeled by p) one-hole (labeled by h) excited state,where bα(b†α) is the annihilation(creation) operator of a fermion state characterized by α.Consider the matrix element ⟨ph|M|F⟩(or its effective one body operator Meff),⟨ph|M|F⟩= ⟨p|Meff|h⟩=Xβ∈F1gβ⟨p, β|M|h, β⟩(H.1)where gβ is defined bynbα, b†βo= gβδα,β. In computing this, it is convenient to define theantisymmetrized wave function |α, β⟩in terms of the simple two-particle state |α, β)|α, β⟩= |α, β) −|β, α)√2,(H.2)so the matrix element ⟨p, β|M|h, β⟩is of the form⟨p, β|M|h, β⟩= (p, β|M|h, β) −(p, β|M|β, h).

(H.3)The first term is the Hartree term and the second the Fock term. Rewriting |α⟩as |pαmαtα⟩where pα is the momentum of the state labeled by α and mα (tα) the third component ofthe spin (isospin) of the state α, we may write the axial charge operator as(1′, 2′|M|1, 2) = (t′1m′1, t′2m′2|hT (1)φ1(q) + T (2)φ2(q)i|t1m1, t2m2)(H.4)where T (1) = i⃗τ1 ×⃗τ2(→σ 1 +→σ 2) · q, T (2) = i (⃗τ1 +⃗τ2)→σ 1 ×→σ 2 · q and q = p′2 −p2 = p1 −p′1,q ≡|q|.It is trivial to see that the Hartree term must vanish.

Thus we are to calculate theFock term. First we shall show that the matrix element of T (1) is equal to that of T (2)when summed over spin and isospin of occupied states.

Doing the sum, we getXmβXtβ(mptp, mβtβ|T (1)|mβtβ, mhth) = −4 ⟨mptp|⃗τ ⊗→σ|mhth⟩· q. (H.5)For T (2), we simply interchange the spin and isospin operators and get the same result.

Itthen follows that the effective one-body operator of the axial-charge operator becomes⟨p|Meff|h⟩= 4⃗τph→σ phZ|p|

(H.7)48

In order to give a meaning to this expression, we have to account for short-range correlations.Otherwise we can get erroneous results. For instance, a constant in momentum space (say,φ(q) = constant) gives a contribution whereas it should be suppressed in reality.

One wayto assure a correct behavior at short-distance is to subtract the constant as one does forthe Lorentz-Lorenz effect in π-nuclear scattering.However this procedure is not alwayspracticable if one is dealing with non-polynomial terms. It is therefore preferable to go tocoordinate space by Fourier-transform.

For this, define f(r) byf(r) =Zdq(2π)3 eiq·r φ(q) . (H.8)UsingZ|p|

(H.9)Introducing a correlation function ˆg(r, d) where d is a parameter of ˆg, we get the finalexpression⟨ph|M|F⟩= ⃗τph→σ ph · ˆph8p2FπZ ∞ddr r j1(pFr)j1(phr) ddrf(r). (H.10)In the numerical results discussed in the next, we have used the simplest correlation function,ˆg(r, d) = θ(r −d).Appendix I: The Role of Vector MesonsIn this Appendix, we describe briefly the role that vector mesons play in the axial-charge transitions in chiral perturbation theory.

We will in particular establish that vectormesons can contribute only at O( 1m2 ) and hence their contributions are suppressed to thechiral order we are concerned with. For simplicity, we shall consider the vector field Vµ only.The axial vector field a1µ could also be included but we shall leave it out since it plays evenless significant role in our case.

LetVµ = taV aµ ,Tr(tatb) = 12δab(I.1)denote the spin-1 field. The index a and b are (1, 2, 3) for SU(2) with ⃗t = ⃗τ2, and (0, 1, 2, 3)for U(2) with t0 = 12.

The a = (1, 2, 3) components correspond to the ρ mesons and a = 0to the ω meson. We write the relevant part of the Lagrangian as#23L=¯N [γµ (i∂µ + gVµ + gAγ5 i∆µ) −m] N + F 22 ⟨i∆µ i∆µ⟩+ 14M2F 2⟨Σ⟩#23We have not included the axial-vector field a1, although it is not difficult to do so.

For the axial-chargeprocess we are considering the a1 field does not play an important role. For the Gamow-Teller operator,however, the axial field may not be ignorable.49

+12M2V ⟨Vµ −ig Γµ2⟩−14⟨VµνV µν⟩+ Lan(I.2)whereVµν = ∂µVν −∂νVµ −ig [Vµ, Vν] ,(I.3)and Γµ (∆µ) were given in Section 3 and explicitly take the formiΓµ=⃗t ·⃗Vµ + 1F ⃗π × ⃗Aµ −12F 2⃗π × ∂µ⃗π + · · ·,i∆µ=⃗t ·⃗Aµ + 1F ⃗π × ⃗Vµ −1F ∂µ⃗π + · · ·(I.4)where Vµ (Aµ) is the external vector (axial-vector) field and the ellipsis denotes termsinvolving more than three fields. Here Lan is an “anomalous parity” piece involving thetotally antisymmetric ǫ tensor which we do not explicit here as it does not contribute.

Alsofour-fermion interaction terms do not figure in the discussion. In (I.2), the constants gand MV can be identified as the V NN coupling constant and the mass of the V mesonrespectively.

We are using the short-hand notation⟨X⟩≡2 Tr(X)(I.5)for any X. This convention is convenient due to the normalization of ta, ⟨XY ⟩= XaYa forany X = taXa and Y = taYa.Before proceeding, let us note a few characteristics of this Lagrangian:• It is vector gauge-invariant (or hidden gauge invariant) provided that gVµ transformsas iΓµ does, Vµ →UVµU † −ig∂µU · U †.It is also invariant under (global) chiraltransformation apart from the pion mass term.• It has vector-meson dominance.• When MV goes to infinity, we recover our previous chiral Lagrangian involving onlyπ’s and nucleons N.• There is a V γ mixing but the mixing is trivial in the sense that the photon fieldappears only as an external (non-propagating) field.For the reasons spelled out in the main text, we wish to transform the Lagrangian toa form appropriate for heavy fermion formalism.

Including the “1/m” terms, we haveL=¯B (iv · D + 2gA S · i∆) B + F 22 ⟨i∆µ i∆µ⟩+ 14M2F 2⟨Σ⟩+12m¯B−D2 + (v · D)2 + [Sµ, Sν] [Dµ, Dν] −g2A(v · i∆)2 −2igA {v · i∆, S · D}B+12M2V ⟨Vµ −ig Γµ2⟩−14⟨VµνV µν⟩+ Lan(I.6)50

where Dµ = ∂µ −igVµ.Now let us calculate the tree-order contribution of the vectormesons to the two-body axial charge operator.Three types of graphs contribute.Therelevant graphs are given in Figure 11.First we find that the graph (c) does not contribute. To see this, note that G-paritydoes not allow the couplings ρρAµ and ωωAµ.The coupling for Aµρω is of the formǫµναβωναρβ coming from the anomalous-parity term Lan of (I.6).

In the figure (c), eachvector meson brings in vµ as one can see in (I.6), so that we have ǫµναβvαvβ = 0.Working out the graphs (a) and (b) #24 we get⃗Aµ(1)=i⃗τ1 × ⃗τ2gA2F 2M2VM2V −q211M2 −q22q2 · S2 −v · q2mNS2 · P2⊗ vµ1 +1mN[Sµ1 , q1 · S1] −v1 · q1M2Vqµ1!+ (1 ↔2)(I.8)⃗Aµ(2)=(⃗τ1 + ⃗τ2) gA2F 2M2VM2V −q22vµ4mN v2 · S2 +1mN[S1 · S2, q2 · S2] −v2 · q2M2Vq2 · S2!+ (1 ↔2)(I.9)where we have used the KSRF relation M2V = 2g2F 2 and definedP µi=12(pi + p′i)µ,vµi=vµ +1mN(P µi −vµ v · Pi)(I.10)and qi = p′i −pi, i = 1, 2. Now noting that v · q ≃vi · qj = OQ2mNand S0i = OQmN, wehave (setting q2 = −q1 ≡q)⃗A0(1)=i⃗τ1 × ⃗τ2gA2F 2M2VM2V −M2 1M2 −q2 −1M2V −q2!q · S2"1 + O Q2m2N!#+ (1 ↔2)(I.11)⃗A0(2)=(⃗τ1 + ⃗τ2) gA2F 2"O Q2m3N!+ O Q3m2NM2V!#.

(I.12)The leading part of Equation (I.11) is nothing but the one-pion exchange current with one-loop radiative corrections (91) expressed now in terms of a vector-dominated Dirac formfactor F V1 . (In fact, with the Lagrangian (I.6), the soft-pion contribution corresponds to(I.11) in the limit MV →∞.) There is no further correction to what has already beenobtained with our Lagrangian given in its full glory in Appendix A.

This corroborates our#24In the spin-1 propagatorDµνab (q) =δabq2 −M 2V−gµν + qµqνM 2V. (I.7)the term proportional to1M2V qµqν is a correction term of OQ2M2Vrelative to the leading term (∝gµν).

Itis further suppressed in the vector-meson exchange between two nucleons since the V NN vertex function isproportional to vµ and v · q = OQ2m. So effectively this term is of orderQ3mM2V and hence can be dropped.51

argument that the counter terms κ(1,2)4cannot come from one vector-meson exchange in thelimit mV →∞. This also establishes our assertion that vector mesons do not modify ourresult on the “chiral filter mechanism.”For completeness, we give the corresponding axial-charge operator in coordinate space˜MVMDtree + ˜MVMD1π= ˜T (1) (1 + δsoft)14πrM + 1re−Mr −MV + 1re−MV r(I.13)where δsoft =M2M2V −M2 and we have dropped the terms of order m−2N .

This is the vector-dominated form of M1π, in place of (104): The second term of vector-meson range in (I.13)is the counterpart to the shorter-ranged loop correction in (104). In fermi-gas model (I.13)predicts roughly the same quenching as the loop calculation (104).52

FIGURE CAPTIONSFigure 1Generic nuclear electroweak currents up to two body. The solid line representsthe nucleon, the blob with a cross the coupling of electroweak fields and theshaded blob without cross stands for the strong interactions.Figure 2Two-body exchange currents: (a) One-pion exchange; (b) two-pion exchange.The solid blob represents a strong-interaction vertex and the shaded blob witha cross the vertex involving an external field and strong interactions.

The solidline represents the nucleon and broken line the pion.Figure 3One-loop graphs contributing to the nucleon self-energy Σ. As in Figure 2, thesolid line represents the nucleon, the broken line the pion.Figure 4One-loop graphs contributing to the three-point GµNN vertex where Gµ = Aµ(Vµ) is the external axial-vector (vector) field, the encircled cross representingthe field coupling.

Here and in Fig. 5, vector-field couplings are also drawn forcomparison and for later use in [14].Figure 5One-loop graphs contributing to the four-point GµπNN vertex.

For axial-chargetransitions, only the graphs (a)-(f) contribute.Figure 6One-loop graphs contributing to two-body two-pion exchange currents ((a) −(h)), four-fermion-field contact interaction currents ((i) −(j)) and “recoil” cur-rent (k). The pion propagator appearing in (a) −(j) is the Feynman one whilethat in (k) is a time-ordered one.

Only the graphs (a), (b), (c) and (d) survivefor the axial-charge operator.53

Figure 74πr2 ˜Mtree (solid line) and 4πr2 ˜Mloop (broken line) defined in Eqs. (103) and(106) vs. r in fm.

Here and in Fig. 8, we have set ˜T (1) = ˜T (2) = 1.Figure 8r[j1(pF r)]2 ˜Mtree (solid line) and r[j1(pF r)]2 ˜Mloop (broken line) vs. r with pF ≈1.36 fm−1 (corresponding to nuclear matter density).

See the caption for Fig.7.Figure 9The ratios of the matrix elements⟨MX⟩⟨Mtree⟩in fermi-gas model vs. ρ/ρ0 for d =0.5, 0.7 fm for X = 1π, 2π, 1π + 2π corresponding to one-loop correction to theone-pion exchange graph, one-loop two-pion exchange graph and the sum of thetwo, respectively.Figure 10Three-body currents: a) Genuine three-body current with Feynman pion propa-gators; b) “recoil” three-body currents with time-ordered pion propagators; theellipsis stands for other time-orderings and permutations. Both (a) and (b) areof order O(Q3) relative to the leading soft-pion term.Figure 11Vector-meson contribution with the Lagrangian (I.6) to the two-body axialcharge operator.V and V ′ stand for vector mesons of mass MV .For theaxial current, V = ρ and V ′ = ω.54

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