On-Shell Approach to Pion-Nucleon Physics

James V. Steele1, Hidenaga Yamagishi 2 and Ismail Zahed1

1Department of Physics, SUNY, Stony Brook, New York 11794, USA;24 Chome 11-16-502, Shimomeguro, Meguro, Tokyo, Japan. 153

(July 23, 1997)

We discuss an on-shell approach to pion-nucleon physicsthat is consistent order by order in a 1/fπ expansion withthe chiral reduction formula, crossing, and relativistic unitar-ity. A number of constraints between the on-shell low-energyparameters are derived at tree level in the presence of thepion-nucleon sigma term, and found to be in fair agreementwith experiment. We analyze the nucleon form factors, andthe πN → πN scattering amplitude to one-loop, as well asπN → ππN to tree level. We use the latter to derive a newconstraint for the pion-nucleon sigma term at threshold. Wecompare our results to both relativistic and non-relativisticchiral perturbation theory, and discuss the convergence char-acter of the expansion in light of experiment.

I. INTRODUCTION

Pion-nucleon interactions have been extensively inves-tigated using dispersion relations and chiral symmetry.Most of these studies are built around unphysical pointssuch as the soft pion limit [1] or the chiral limit [2]. Atypical example is the pion-nucleon sigma term — thefraction of the nucleon mass due to the explicit breakingof chiral SU(2) × SU(2). The scattering amplitude isanalytically continued to the unphysical Cheng-Dashenpoint [3], and chiral perturbation theory (ChPT) is ap-plied [4].

An important exception to the above is Weinberg’s on-shell formula for pion-nucleon scattering [5], which alsoyields the Tomozawa-Weinberg relations for the S-wavescattering lengths [6]. Recently, we have been able to ex-tend this result to processes involving an arbitrary num-ber of on-shell pions and nucleons [7]. A number of iden-tities using the chiral reduction formula were derived —one of which was applied to ππ scattering and shown tobe in good agreement with the data well beyond thresh-old [8].

This paper applies the results of the chiral reductionformula to the nucleon sector, allowing for an on-shelldetermination of the pion-nucleon sigma term and πNscattering. We start by introducing a model in sectionII that is uniquely specified by the form of the symme-try breaking in QCD to tree level. This model can beused to ensure Lorentz invariance, causality, and pos-itivity while at the same time enforce the constraintsbrought about by the chiral reduction formula. Thestrategy involved in this calculation compared to those

of ChPT is presented in section III. In section IV, wederive an axial Ward identity and discuss the deviationfrom the Goldberger-Treiman relation. In section V, werecall Weinberg’s relation for πN scattering and use themeasured S-wave scattering lengths to predict the pion-nucleon sigma term, the pion-nucleon coupling, and theinduced pseudoscalar coupling to tree level. In sectionVI, we discuss the one-loop on-shell corrections to thevector, axial, and scalar form factors, and critically ex-amine the character of the convergence. We then evaluatethe one-loop corrections to πN scattering in section VII.Finally, we look at πN → ππN to tree level and find anadditional way to determine the pion-nucleon sigma termin section VIII. Our conclusions are summarized in sec-tion IX. Details about the Feynman rules and the loopexpansion are found in the appendices.

II. MODEL

There are two kinds of chiral models possible for theπN system. The first is a Skyrme-type model, where thenucleon is a chiral soliton [9]. Since solitons often ac-company spontaneous symmetry breaking, this is a nat-ural approach. Also, if vector mesons (particularly theomega) are included in chiral Lagrangians, avoiding soli-ton solutions is more difficult than having them.

However, there are two difficulties in this model. One isthat the semiclassical expansion does not commute withthe chiral limit. As a result, the S-wave πN scatteringlengths are not compatible with the Tomozawa-Weinbergrelation to leading order [10]. Similarly, the nucleon axialcharge gA is small to leading order (about half of exper-iment), and yields a different sign for gA − 1 [10] fromthat obtained with the Adler-Weisberger sum rule. Thismeans that a quantitative comparison with experimentis usually difficult, unless a calculational scheme beyondthe semiclassical expansion is developed.

The other difficulty is more fundamental. In QCD, nu-cleon operators qqq and meson operators qq exist, whichare mutually local. This has not been shown in Skyrme-type models [11].

We will therefore adopt the other type of model, wherepions and nucleons are taken to be independent. For theSU(2) × SU(2) symmetric part of πN interactions, wetake the standard non-linear sigma model as the effectiveLagrangian gauged with vector and axial-vector externalsources.

1

L1 =f2π

4Tr[(iDµU + {aµ, U})

((iDµU)† +

{aµ, U †

})]+Ψ (i/∂ + /v + /aγ5) Ψ−m0

(ΨRUΨL + ΨLU

†ΨR

)+

1

2(gA − 1)ΨR

(i /DU + {/a, U}

)U †ΨR

−1

2(gA − 1)ΨLU

†(i /DU + {/a, U}

)ΨL (1)

where U is a chiral field, Ψ = (ΨR,ΨL) is the nucleonfield, /∂ = γµ∂µ, vµ = vaµτ

a/2, and DµU = ∂µU−i[vµ, U ].In the low-energy limit, matrix elements calculated from(1) are essentially unique, given that the isospin of thenucleon is 1

2 [12]. Higher derivative (1,1) terms at treelevel lead to pathologies such as acausality or lack of pos-itivity, so they will not be considered.

Ignoring isospin breaking and strong CP violation, theterm which explicitly breaks chiral symmetry must be ascalar-isoscalar. The simplest non-trivial representationof SU(2) × SU(2) which contains such a term is (2, 2).This is the same representation as the quark mass termmqq in QCD which generates both the pion mass and thesigma term. Therefore we take1

L2 =1

4f2πm

2π Tr

(U + U †

)−m2π

ΛΨΨ

− cm2π

4ΛTr(U + U †

)ΨRUΨL + h.c. (2)

with c and Λ arbitrary constants. A bilinear form inΨ in (2) has been retained, again in analogy with mqq.We assume that Λ is non-vanishing as mπ → 0, so that(2) vanishes in the chiral limit. Scalar and pseudoscalarexternal fields can be added to eq. (2) by taking

m2πTrU → Tr

[(m2

π + s− iτapa)U]

m2πΨΨ→ Ψ(m2

π + s− iτapaγ5)Ψ

and similarly for TrU †. The nucleon mass is definedas mN ≡ m0 + σπN with the pion-nucleon sigma termσπN = (1 + c)m2

π/Λ as read from the Lagrangian to treelevel.

Noether’s theorem implies that the symmetry breakingterm must be non-derivative, otherwise the vector andaxial vector currents will not transform as (3, 1) + (1, 3).The only other term allowable in the (2, 2) representationthen is ΨRU

2ΨL + h.c. which is a linear combinationof the terms already included in eq. (2). Therefore ourstarting point for the loop-expansion L1+2 is essentiallyunique.

The currents used in this paper can be written downby functional differentiation of the action I ≡

∫d4xL1+2

with respect to the external sources. In particular, thepion field is just

1An earlier version of this work [13] used the specific casec = 0.

πa(x) =1

fπ

δI

δpa(x)

= −ifπ

4Tr(τa(U − U †)

)+

1

fπΛΨiγ5τ

aΨ

+ ic

4fπΛTr(τa(U − U †)

)ΨRUΨL + h.c. (3)

which reduces to the free incoming pion field πin(x) asx0 → −∞. This choice is just the gauge covariant versionof the PCAC pion field, also defined in terms of the axialcurrent Aa

µ ≡ δI/δaaµ as ∂µAa

µ = fπm2ππ

a.The one-pion reduced axial current can be defined as

the part of the full axial current that contains no πin part.The most convenient definition is

jaAµ = Aaµ + fπ∂µπ

a (4)

= gAΨγµγ5τa

2Ψ +

1

Λ∂µ(Ψiγ5 τ

aΨ)

−c

fπΛ∂µ(πaΨΨ

)+O(π3).

with the expansion to leading order in the PCAC pionfield given in the last two lines. We note that the PCACpion field is uniquely defined off-shell within the prescrip-tions of [7], and so is jAµ. The vector Va

µ and scalar σcurrents are similarly defined through

Vaµ(x) = i

f2π

8Tr([τa, U †]∂µU

)+ h.c.+ Ψγµ

τa

2Ψ

−1

4(gA − 1)ΨLγµU

†[τa, U ]ΨL

+1

4(gA − 1)ΨRγµ[τa, U ]U †ΨR

σ(x) =1

m2πfπL2 .

The Feynman rules for L1+2 that are used throughoutthis paper are in Appendix A.

III. STRATEGY

We adopt an on-shell loop expansion in 1/fπ whichcan be thought of as a semi-classical expansion with√h ∼ 1/fπ. It includes pions and nucleon loops beyond

tree-level, and is consistent order by order with the iden-tities following from the chiral reduction formula. Thisexpansion applies equally well to the non-linear sigma-model and QCD as thoroughly discussed in [7]. We re-call that in both cases, the physical pion decay constantfπ shows up through the asymptotic condition of PCACon the axial-vector current. Therefore, it is a good ex-pansion parameter when the master formula approach isapplied to these two cases.

All scattering amplitudes will be reduced by the iden-tities derived in [7], and then expanded to one-loop using

2

the Feynman diagrams from (1-2). This way, reparam-eterization invariance (in the sense of Nishijima-Gursey[14]) and vector as well as axial-vector current identitiesare guaranteed to one-loop. If we were to just use (1-2)without the chiral reduction formula, then ππ scatteringsubdiagrams in, for example πN → πN or πN → ππNappear to break reparameterization invariance. In [7] wehave checked that conventional ChPT fulfills the perti-nent identities following from the chiral reduction for-mula in the mesonic sector. We are not aware of suchchecks in the nucleon sector2. This work and others tofollow will provide for these checks in our approach.

In our approach broken chiral symmetry and relativis-tic unitarity will be addressed for each process individ-ually directly on-shell. This procedure is conceptuallyclear, since on-shell renormalization implies that quanti-ties mN , gA, Λ, σπN , fπ, and mπ are fixed once and forall at tree level, thereby including all powers of the quarkmasses and QCD scale. (In constrast to ChPT where thechiral logarithms are assessed in these quantities.)

The ultraviolet finite and non-diagrammatic formula-tion extensively discussed in [7] will be presented else-where [15]. To make our exposition in line with currentexpositions using ChPT, we will use diagrams. A BPHZ(momentum) subtraction scheme will be used through-out. This is to enforce the number of subtraction con-stants commensurate with the number of divergences.Dimensional regularization is not appropriate, since weneed to evaluate nucleon loops in the axial form factors.These are in general quadratically divergent, requiringtwo subtraction constants as opposed to one by dimen-sional regularization.

Our strategy is essentially the same as for ordinaryrenormalizable theories. No constants other than thoserequired by the divergences will be considered. Thismakes our approach minimal in comparison to ChPTwhere all possible constants required by symmetry andpower counting are used. This is appealing in that lessconstants need to be fixed. Although ChPT is more gen-eral (and generalized ChPT [16] even more so), the excessof constants there require additional assumptions such asresonance saturation [17] to fix them. In any case, whichis the better approach will be dictated by comparisonwith experiment.

Below, we will show that to one-loop our results forthe form factors reduce to those obtained by Gasser,Sainio and Svarc [4] (GSS) in the context of relativis-tic chiral perturbation theory when the nucleon is takenoff mass shell (Λ → ∞). On mass shell, however, anumber of relations are already observed at tree level

2In [4] it was shown that Weinberg’s relation for the par-ticular reaction πN → πN holds to leading order in ChPT,thereby confirming the reparameterization invariance of theirresults to the order quoted.

in reasonable agreement with experiment, emphasizingthe importance of (broken) chiral symmetry. What isundoubtedly important in our approach is that the pion-nucleon sigma term is included at tree level along withthe pion mass term. Since both terms originate fromthe same quark mass term in QCD, they naturally gotogether. This term — along with the on-shell renormal-ization scheme — allow the pions and nucleons to stay onmass shell to all orders. A nonzero nucleon scalar formfactor and Goldberger-Treiman discrepancy at tree levelare also consequences of this.

Finally, as is well known, the loop expansion in thepionic sector is equivalent to a momentum expansion [4].This is no longer true in the pion-nucleon sector. Toovercome this, heavy baryon chiral perturbation the-ory (HBChPT) was proposed [18], where an expan-sion in 1/mN is made3 [19,20]. However, the relativis-tic one-loop calculations contain terms which behave aslnmπ/mN and are not able to be expanded in this way.Also in this limit relativistic unitarity is lost, makingcomparison with experiments difficult. Lacking a sat-isfactory theoretical resolution of these issues, we willmaintain a relativistic approach throughout. Compari-son with HBChPT will be made directly by expandingthe on-shell results.

At this point we note that the unitarity bounds aremore stringent than simple power counting based on amomentum expansion. For instance in ππ scattering theunitarity bound is saturated for k2 ≤ 5.2m2

π whereas inπN scattering the bound is k2 ≤ 3.8m2

π, indicating inboth cases that the expansion parameter should in factbe closer to k2/4πf2

π instead of k2/(4π)2f2π . Throughout,

we will work in the kinematical regime where tree contri-butions are greater than one-loop, but within the unitarybounds. All the loop corrections discussed in this workare on the order of 10− 30% of the tree level result withthe exception of the 1/Λ terms which are small enoughto be sensitive to the input parameters. These terms willbe assessed in as many ways as possible.

IV. AXIAL WARD IDENTITY

The matrix element of the axial-vector current betweennucleon states of momentum pi and implicit spin depen-dence si can be decomposed as

〈N(p2)|jaAµ(0)|N(p1)〉 =

= u(p2)(γµγ5G1(t) + (p2 − p1)µγ5G2(t)

) τa2u(p1) (5)

with t = (p1−p2)2 and G1 and G2 are free of pion poles.From (3-4), we also have ∂µjAµ = fπ(2+m2

π)π. Hence,

3Since the nucleon is off mass shell in these approaches, theexpansion is more in terms of a ‘bare’ nucleon mass.

3

〈N(p2)|πa(0)|N(p1)〉 =

=1

fπ

1

m2π − t

(2mN G1(t) + tG2(t)

)u(p2)iγ5

τa

2u(p1) (6)

By definition, eq. (6) is also equal to

gπNN(t)1

m2π − t

u(p2)iγ5τau(p1)

and leads to the following Ward identity

fπgπNN(t) = mNG1(t) +t

2G2(t) (7)

where gπNN = gπNN(m2π) is the pion-nucleon coupling

constant. Extrapolating from t = m2π to t = 0 gives the

standard Goldberger-Treiman relation gAmN ∼ fπgπNN ,with G1(0) ≡ gA.

Relation (7) is exact. The Goldberger-Treiman dis-crepancy is just given by

fπ(gπNN (m2

π)− gπNN(0))≡ −∆πN

We stress again that gπNN(t) is physically accessible atboth t = m2

π and t = 0, making the above discrepancymeasurable. Substituting (4) at tree level into (5) givesG1(t) = gA and G2(t) = −2/Λ. Therefore using (7) wefind ∆πN = m2

π/Λ to tree level. We choose to renormal-ize Λ on-shell such that this is true to all orders in theloop expansion. So

fπgπNN = gAmN −m2π

Λ≡ G (8)

and m2π/Λ is exactly the Goldberger-Treiman discrep-

ancy.

V. WEINBERG’S RELATION

One way to determine G and therefore attain a valuefor gπNN is to use pion-nucleon scattering at thresh-old. The scattering amplitude iT fulfills a basic Wardidentity established by Weinberg [5] and reproduced bythe master formula approach [7]. Taking (k1, a) as theincoming pion, and (k2, b) as the outgoing pion, withp1 + k1 = p2 + k2, the formula is

iT = iTV + iTS + iTAA

iTV = −1

f2π

kµ1 εbac〈N(p2)|Vc

µ(0)|N(p1)〉conn.

iTS = −i

fπm2πδab〈N(p2)|σ(0)|N(p1)〉conn.

iTAA = −1

f2π

kµ1 kν2

∫d4x e−ik1·x

×〈N(p2)|T ∗jaAµ(x)jbAν(0)|N(p1)〉conn. .

The isospin structure is decomposed as T ba = δabT + +iεbacτcT − to give

T + = T +S + T +

AA T − = T −V + T −AA.

The amplitudes T ± can be calculated to tree level usingthe Feynman rules in Appendix A. At threshold they are

T + =σπN

f2π

−∆

2πN

f2πmN

−G2

f2π

m2π/mN

4m2N −m

2π

(9)

T − =mπ

2f2π

(1− g2

A

)+G2

f2π

2mπ

4m2N −m

2π

. (10)

Here the pion-nucleon sigma term σπN and theGoldberger-Treiman discrepancy ∆πN appear. These ex-pressions reduce to the Tomozawa-Weinberg formulas forΛ→∞, showing the corrections are small.

Experimentally, the threshold amplitudes are ex-pressed in terms of the S-wave scattering lengths T ± =4π(1 + mπ/mN )a±. The Karlsruhe-Helsinki phase shiftanalysis gives (a−, a+) = (9.2±0.2,−0.8±0.4)×10−2/mπ

[21]. The same group now has new data from PSI[22] which reduces many of the inconsistencies for lowpion energies and finds a positive value a+ = 0 —4 × 10−3/mπ. Furthermore, pionic atoms give a+ =2 ± 1 × 10−3/mπ [23]. Therefore we take the weightedmean a+ = 1.5(9)×10−3/mπ. The accuracy of a− is lessof an issue, so we will take the value given above. Thesegive ∆πN = −54±10 MeV and σπN = 14±1 MeV. Usingeq. (8), this value of G gives gπNN = 13.45± 0.15, veryclose to the experimental value 13.4 taken in the Parisand Bonn potentials [24]. Our on-shell tree level cal-culation favors a positive value for a+ and smaller thannormal value for σπN . This is opposite to what [25] finds,meriting a one-loop evaluation as carried out in sectionVII.

We can also determine the value for the induced pseu-doscalar coupling constant which has been experimen-tally measured by two groups

gp ≡ mµG2(−0.88m2µ) =

{8.2± 2.4 ref. [26]

8.7± 1.9 ref. [27]

from muon capture in hydrogen. Using eq. (4-6) and thedefinition of G2 from the full-axial vector current

〈N(p2)|Aaµ(0)|N(p1)〉 =

= u(p2) (γµγ5G1(t) + (p2 − p1)µγ5G2(t))τa

2u(p1)

gives a relation between G2 and G2

G2(t) =1

m2π − t

(2mNG1(t) +m2

πG2(t)). (11)

Using eq. (8) we find to tree level

gp =2mµG

m2π + 0.88m2

µ

' 9.0

which is at most 10% higher than the experimental value.

4

VI. FORM FACTORS

We now calculate the vector form factor in order togain insight into the loop corrections. It can be decom-posed as

〈N(p2)|Vaµ(0)|N(p1)〉 =

= u(p2)

(γµF1(t) +

i

2mNσµν(p2 − p1)νF2(t)

)τa

2u(p1)

The tree level result just gives the charge F1(t) = 1.Including the one-loop form of the vector current, wefind

F1(t) = 1 +2(g2

A − 1)

f2π

(tcV2 + Jππ22 (t)

)+G2

f2π

[2ΓπN3 (t)− JNN (t) + 8ΓNπ3 (t)−m2

πΓπN (t)

+ 8m2N

(ΓπN4 (t) + 4ΓNπ4 (t)− 4ΓNπ1 (t) + ΓNπ(t)

)]

F2(t) = −8m2

NG2

f2π

[ΓπN4 (t) + ΓNπ(t) + 4ΓNπ4 (t)− 4ΓNπ1 (t)

]with the Γ’s representing loop integrals defined in Ap-pendix A and G = gAmN − m2

π/Λ as in the previoussection. An overlined function denotes a subtraction att = 0. Other than the subtraction constants, this re-duces to the ChPT result of GSS for Λ→∞. This is tobe expected from the form of the Lagrangian used andis a good check on the calculation. Notice that the coef-ficients of the terms group into factors of G/fπ ≡ gπNNto the order we are calculating by eq. (8).

Another check is that the charge F1(0) = 1 is not renor-malized by loop corrections from the strong force. Thiscan be shown to be true in dimensional regularization [4].However, for the rest of this paper we will instead adoptthe BPHZ renormalization scheme. This just amounts tosubtracting the Taylor series of divergent loop integralsup to the degree of divergence and replacing the subtrac-tion by arbitrary constants. This has two advantages: 1)we do not need to construct the most general Lagrangianto obtain the constants, 2) we obtain only the minimalamount of constants consistent with the symmetries ofthe theory. In other words, if a diagram is not divergent,we do not do any subtractions on it.

The only constant to one-loop, cV2 , can be fixed by thevector charge radius of the nucleon 〈r2〉V1 = 6F ′1(0) =0.578 fm2 [28] giving 2(g2

A − 1)cV2 = 7.30 × 10−3. Thisaccounts for about one third of the radius with the restbeing given by the loop integrals. This also agrees withChPT.

The first difference with ChPT is that there is no sub-traction constant for the Pauli form factor F2. In fact,the experimental value for F2(0) is just the difference in

the anomalous magnetic moments κp − κn ≡ κv = 3.71;and a numerical calculation shows this is indeed valid toabout 3% as first shown by GSS. The most general La-grangian, however, allows an extra subtraction constantfor F2 which is not needed [4].

We can also check the consistency of our results withthose of heavy baryon chiral perturbation theory by tak-ing Λ → ∞ and expanding in mπ/mN and t/m2

N . Theconsistency of the finite parts of the vector form factorwith the relativistic theory has been shown by [19]. How-ever, the opening of the two pion threshold at t = 4m2

π

which is also seen in the data for the vector form factor[28] is lost in HBChPT. In addition, extra divergences inthe 1/mN expansion can develop. If these divergencescannot be absorbed into the already existing divergencestructure, the procedure of taking the limit of a large nu-cleon mass at the level of the Lagrangian could be differ-ent than taking this limit after calculating all amplitudeswith a finite nucleon mass.

One such case does occur in the slope of the Pauliform factor F2(t). The full one-loop calculation can beworked out from the explicit form of the loop integrals inAppendix A to give (µ = mπ/mN )

F ′2(0) =G2

3(4πfπ)2

∫ 1

0

dyy3[y2 + 4(1− y)2

][y2 + µ2(1− y)]

2

=G2

(4πfπ)2m2N

(π

3µ+ 2 lnµ2 +

29

6

)+O(µ).

which is consistent with GSS in the Λ → ∞ limit. Thelnµ2 cannot be expanded in 1/mN . In principle, thisshould be taken into account in HBChPT by an addi-tional subtraction constant.

Taking gA = 1.265 and ∆πN = −54 MeV as in the lastsection, the magnetic radius 〈r2〉V2 = 6F ′2(0)/κv is 0.21fm2 to one-loop. The terms to O(1) give 0.31 fm2 andthe 1/µ term alone gives the HBChPT result of 0.51 fm2.The empirical value is 0.77 fm2 [28]. Ironically, the firstterm in the µ expansion gives the result closest to exper-iment. However, this term can only be singled out in anon-relativistic expansion which then would require theadditional subtraction constant to absorb the logarithmicsingularities mentioned above.

There are no corrections of order m2N/(4πfπ)2 to F1(0)

since it is protected by a non-renormalization condition.In addition, F2(0) had no contribution from tree level andthe one-loop value was shown to be close to the experi-mental value. Therefore, further verdict on the conver-gence of this expansion requires a two-loop calculation.

Extending the results of the previous sections requiresthe axial-vector form factors to one-loop.

G1(t) = gA −gAG

2

f2π

[2ΓπN3 (t)− JNN (t)−m2

πΓπN (t)]

G2(t) = −2

Λ+

2gAmNG2

m2πf

2π

[2ΓπN3 (m2

π)− JNN(m2π)

5

−m2πΓπN (m2

π)

]−G2

f2π

[4gAmNΓπN6 (t)

+2

Λ

(JNN (t) +m2

πΓπN (t))− 8I(t)

]with

I(t) =G

m2π − t

JNN (t) +1

ΛJNN (t)

and an underlined function denotes a subtraction att = m2

π. The function I(t) is from the nucleon loop.It is doubly subtracted in order to satisfy the consis-tency relation 〈0|jAµ|π〉 = 0 stating that the one-pionreduced axial current truly does not contain asymptoticpion fields. Both G1(t) and G2(t) have subtraction con-stants which are fixed by the on-shell renormalizationprescription for gA and Λ as discussed at the end of sec-tion IV. Therefore the one-loop corrections to G1(t) areof order t/(4πfπ)2. In addition to this correction, G2 hasan additional constant correction of order Λ/mN . Wehave fixed Λ ∼ 3mπ in the last section. Therefore thisis about a 50% correction to the tree level result. Thecorresponding correction to gπNN(t) and G2(t) is on theorder of a few percent.

The induced pseudoscalar coupling constant is now

gp =2mµ

m2π + 0.88m2

µ

[G+

G2

f2π

(F(−0.88m2

µ)−F(m2π))]

with

F(t) = G[JNN (t) +m2

πΓπN (t)]

+ 4m2πI(t)

− 2gAmN

[ΓπN3 (t) +m2

πΓπN6 (t)]

and G = gAmN − m2π/Λ as before. Taking mµ = 106

MeV, this result gives gp ' 8.85 to one-loop. This isabout a 1% correction to the tree result. A plot of G2(t)for spacelike t is shown in fig. 1. The dotted line is thepion pole prediction 2mNG1(t)/(m2

π − t) with G1(t) =gA[1− t/M2

A]−2 and MA = (0.96± 0.03) GeV [29]. Thesolid line is from including the one-loop form for G2(t)for ∆πN = −54 MeV and the dashed-dotted line is theone-loop form for all of G2(t). The data [30] is not preciseenough to distinguish between the results.

It should also be noted that taking G2(t) = 0 in (11)along with the linear approximation for G1(t) = gA(1 +r2At/6) reproduces the Adler-Dothan-Wolfenstein result

[31]

gp =2mµgπNNfπ

m2π + 0.88m2

µ

−1

3gAmNmµr

2A

Taking the proper empirical dipole form for G1 gives lessthan a 1% correction, and including the G2 contributionto one-loop gives about a 5% correction, comparable tothe r2

A term above. Again more precise data is neededbefore any statements can be made.

0.00 0.05 0.10 0.15 0.20−t (GeV

2)

0.01

0.10

G2(

t) (

MeV

−1 )

FIG. 1. The pseudoscalar form factor for spacelike t. Thedotted line is the pion pole prediction and the solid line in-cludes the one-loop form of G2. The dashed-dotted line isfrom using the one-loop form for both G1 and G2.

Finally, the scalar form factor cannot be directly mea-sured, but is important in that its value at the Cheng-Dashen point may be tied to the pion-nucleon scatteringdata by dispersion analysis [28]. It is defined as

〈N(p2)|σ(0)|N(p1)〉 = FS(t)u(p2)u(p1)

FS(t) = −1 + c

fπΛ−

3

2f3π

(gAG+

m2π

Λ

)Jππ(t)

−3G2

f3π

[mN

(ΓNπ(t)− 2ΓNπ1 (t)

)+

1 + c

Λ

(JNN (t) +m2

πΓπN (t))]

(12)

The fact that we have kept σπN to tree level in the La-grangian shows up here as the leading piece in FS(t).Since we renormalize this quantity on-shell, no subtrac-tion constants appear here. Other than this fact, weagree with GSS for Λ → ∞. Note that, unlike in thevector form factors, the coefficients of the terms do notgroup exclusively into factors of G/fπ showing differentfactors from just the naive gπNN .

Defining σ(t) = −m2πfπFS(t) as used by other authors

with σ(0) = σπN , we can use eq. (12) to give a predic-tion for the scalar form factor at the Cheng-Dashen pointt = 2m2

π [3]. The value of the sigma term obtained fromelastic πN scattering (at t = 2m2

π) as compared to thevalue from the baryon mass spectrum (at t = 0) is about20 MeV larger [32]. A numerical evaluation shows forσπN = 20 MeV that the difference σ(2m2

π) − σ(0) = 5.3MeV and is not large enough to account for this discrep-ancy. This observation is similar to GSS.

6

VII. ONE-LOOP πN SCATTERING

In order to calculate πN scattering to one-loop, weonly need the two-axial-vector correlator to one-loopsince the vector and scalar form factors were evaluatedin the previous section. Defining

T ± = A± +1

2(/k1 + /k2)B±,

and using the Mandelstam variables s = (p1 + k1)2 andt = (k1 − k2)2, the tree and one-loop result for the formfactors can be written as

A+S =

σ(t)

f2π

A−V = −s− u

8f2πmN

F2(t)

B−V =1

2f2π

[F1(t) + F2(t)] .

The rest of the tree result comes from the Born terms of〈N |T ∗jAjA|N〉 .

A+AA,tree =

gAG

f2π

A−AA,tree = 0

B+AA,tree = −

G2

f2π

(1

s−m2N

−1

u−m2N

)B−AA,tree = −

g2A

2f2π

−G2

f2π

(1

s−m2N

+1

u−m2N

)with G = 2G−gAmN . Its one-loop contribution is quotedin Appendix B.

Analyzing the divergence structure of the one-loop am-plitude shows that it contains six independent subtrac-tion constants for the total crossing symmetric amplitude

f4πA

+div. = mN t c1 +m3

N c2

f4πA−div. = mN (s− u) c3 f4

πB+div. = (s− u) c4

f4πB−div. = t c5 +m2

N c6.

These six constants are in one-to-one correspondencewith the six renormalized constants of GSS: cr1−4, br2,and br5. The five additional finite constants in [4] haveno counterpart here. This is a direct consequence of ourminimality assumption: only taking into account the di-vergent constants.

The constants may be fixed at subthreshold s = u,t = 0 by fixing the following constants defined in [28,33]

a+00 = (−1.28± 0.24)/mπ b+00 = (−3.54± 0.06)/m3

π

a−00 = (−8.83± 0.10)/m2π b−00 = (10.36± 0.10)/m2

π

a+01 = (1.14± 0.02)/m3

π b−01 = (0.24± 0.01)/m4π.

Since we will be only looking at reactions in the forwarddirection (t = 0) below, the coefficients of t, a+

01 andb−01, will not be discussed further. Using the conven-tional model for the inclusion of the ∆(1232) [28] (wetake g2

∆/4π = 17.7 GeV−2 as in [34] and Z = 12 as in

the original Rarita-Schwinger paper for the non-pole ∆terms [35]), the contribution of the ∆ included in theexperimental subthreshold values is

a+∆00 = −1.11/mπ b+∆

00 = −4.86/m3π

a−∆00 = −11.34/m2

π b−∆00 = 11.62/m2

π.

One can see that these contributions are large and needto be taken into account for a proper fit. We have analyt-ically checked that for any value of Z and indeed even forthe case where the non-pole terms are neglected the finalresult for the scattering lengths presented below is iden-tical. The only difference is that part of the strength ofthe ∆ contribution is shifted from subthreshold to thresh-old; the overall difference between the subthreshold andthreshold remaining the same. Our choice Z = 1/2 ismerely for convenience since for this value the ∆ contri-bution vanishes at threshold.

Moving to threshold, a prediction on the scatteringlengths can be made to one-loop. Taking gπNN =G/fπ = 13.3, a reasonable result is found for σπN = 53MeV

a+1 loop ' 2.1× 10−3/mπ a−1 loop ' 12× 10−2/mπ.

The value for a+ is close to the weighted average dis-cussed in section V whereas a− is somewhat large. Theresults are very sensitive to the ∆ contribution. TakingσπN = 45 MeV gives a+ = −8.7 × 10−3/mπ withoutchanging a−. Taking σπN = 0 and G = gAmN give(a+, a−) = (−4.8, 10)× 10−2/mπ similar to [36]. A morerecent calculation [25] finds a+ = −10× 10−3/mπ. Thisshows that the contribution of the 1/Λ terms to one-loopreally makes a large difference.

Both scattering lengths come from large cancellationsbetween the constants that were fixed at subthresholdand the loop contribution. This cancellation is neededdue to the close proximity of the tree result to exper-iment. The large contribution from the ∆ clouds thepredictability, but our one-loop analysis seems to favora value of σπN close to the commonly accepted value of45 ± 8 MeV [32] and a small but positive a+ scatteringlength. Both of these results are in contrast to [25] andrely on the non-zero value of σπN and the Goldberger-Treiman discrepancy at tree level. The value for a− isabout 20% off from the experimental extrapolation fromthe Karlsruhe-Helsinki data.

The ability to fix gA and gπNN independently fromexperiment allows for a satisfactory starting point to thesensitive prediction of the πN scattering lengths and canalso lead to an estimation of σπN . The above analysis,however, showed an extreme sensitivity of the thresholdresults in πN scattering and call for further study in thefuture.

7

VIII. πN → ππN AND THE PION-NUCLEONSIGMA TERM

As a final estimate on the value of σπN , we turn to theprocess πa(k1)N(p1) → πb(k2)πc(k3)N(p2). The scat-tering amplitude iT3π fulfills an identity which can bederived by chiral reduction from the master formula ap-proach [7] similar to what was done for πN scattering.Defining the Mandelstam variables for a three body pro-cess [37]

s = (p1 + k1)2 s1 = (p2 + k3)2 s2 = (k2 + k3)2

t1 = (p1 − p2)2 t2 = (k1 − k2)2,

the identity is

iT3π = {iTπ + iTA + iTSA + iTV A}+ 2 perms + iTAAA

iTπ =i

f2π

(t2 −m2π)δab〈N(p2)|πc(0)|N(p1)〉

iTA =1

2f3π

(k2 − k1)µδab〈N(p2)|jcAµ(0)|N(p1)〉

iTSA = −im2

π

f2π

kµ3 δab

∫d4xe−i(k1−k2)·x

×〈N(p2)|T ∗σ(x)jcAµ(0)|N(p1)〉

iTV A =1

f3π

kµ1 kν3 εabe

∫d4x e−i(k1−k2)·x

×〈N(p2)|T ∗Veµ(x)jcAν(0)|N(p1)〉

iTAAA = −1

f3π

kµ1 kν2k

λ3

∫d4x1d

4x2 e−ik1·x1+ik2·x2

×〈N(p2)|T ∗jaAµ(x1)jbAν(x2)jcAλ(0)|N(p1)〉

with “perms” meaning a permutation of (k1, a; −k2, b;−k3, c). The structure of the chiral reduction formulaimmediately shows that the amplitude depends on σπN(through the appearance of the scalar current σ in theTSA term) allowing for an alternative way to fix its value.

At threshold, the amplitude can be decomposed as (see[38])

iT cba =~σ · ~k1

2mN

[D1(τbδac + τcδab) +D2τ

aδbc].

The tree contribution to the threshold amplitudes are

f3πD1 =

3

2

(mN +mπ

mN + 2mπ

)[mN + 2mπ

2mN +mπG+ 2∆πN

]−

1

4(2mN +mπ)

[gA +

3∆πN

mN + 2mπ

]+ σπN

[gA +

mπG

(mN +mπ)(2mN +mπ)

]+

1

4(2mN + 3mπ)

[gA

4−

2mNG

(2mN +mπ)(mN +mπ)

−3∆πN

mN + 2mπ

]

+g2A

2(G− 2∆πN )−

g3Amπ

2

−G

(mN +mπ)(2mN +mπ)

(g2Am

2π

2− gAmπG

+2(mN + 2mπ)G2

2mN +mπ

)

f3πD2 = −

3

2

[mN + 2mπ

2mN +mπG+ 2∆πN

]+mπ

2

[gA +

3∆πN

mN + 2mπ

]+ σπN

[gA −

G

2

4mN + 5mπ

2m2N + 2mNmπ −m2

π

]−

1

2(2mN + 3mπ)

[gA

4−

2mNG

(2mN +mπ)(mN +mπ)

−3∆πN

mN + 2mπ

]

+g2A

2(G− 2∆πN ) + g3

A(mN +mπ)

−gAGG

2

4mN + 5mπ

2m2N + 2mNmπ −m2

π

−2G

(2mN +mπ)(mN +mπ)

(g2A

4(6m2

N + 9mNmπ +m2π)

−G2

2

4mN + 5mπ

2mN +mπ

).

The brackets section off, in order, the contribution ofthe first four terms in the chiral reduction formula. Thecontribution from TAAA is given by the remaining terms.This calculation is in agreement with [39] if we take Λ→∞. Note the explicit dependence on σπN in the thresholdamplitudes. Taking ∆πN = −54 MeV for the propervalue of gπNN , σπN = 53 MeV as in the previous section,and defining Di = Di/2mN , we find

D1 = 2.78 fm3 D2 = −6.59 fm3

whereas experimentally [40]

Dexp1 = 1.82± 0.09 fm3 Dexp

2 = −7.30± 0.24 fm3.

Although both Di’s depend on σπN , only D1 is sensitiveto it, decreasing to D1 = 2.5 fm3 for σπN = 14 MeV.Therefore experiment seems to favor a smaller σπN .

The large 30% discrepancy in D1 reflects on the diffi-culty in extracting the threshold parameters. A differentfit in the literature [41] gives (D1, D2) = (2.26,−9.05)fm3. Furthermore, large corrections occur in ChPT from

8

higher order terms bringing the convergence of this pa-rameter into question. The one-loop corrections to theDi’s will be presented elsewhere.

IX. CONCLUSIONS

We have introduced a minimal model for πN dynamicsthat embodies uniquely at tree level the main features ofbroken chiral symmetry with on-shell pions and nucleonsto all orders. Using this on-shell expansion and a BPHZsubtraction scheme, we have shown how the chiral re-duction formula be enforced with a minimal number ofparameters. All of our results are consistent with data.

With this simple model we have analyzed the axialWard identity derived in section IV as well as a Wardidentity for πN scattering originally derived by Weinbergand a new chiral reduction formula for πN → ππN . Wehave presented a one-loop calculation of the nucleon formfactors and πN scattering and shown their equivalence toChPT in the Λ→∞ limit.

For finite Λ, this model has the additional feature ofallowing room for gA, mN , gπNN , and σπN to be fixedto their phenomenological values — all at tree level in a1/fπ loop expansion. In particular, an estimate can bemade on the value of σπN from various processes. Termsproportional to σπN are certainly important in the scalarform factor FS(t), the πN scattering length a+, and thethreshold parameter D1 from the πN → ππN process.

The only hindrance in nailing down a more stringentprediction on σπN comes from determining the contri-bution of nucleonic resonances such as the ∆(1232) atthe point where the divergent constants are fixed. Anideal situation would be to find an amplitude with thedivergences constrained by current conservation and yetstill dependent on σπN for an unambiguous determina-tion of the pion-nucleon sigma term. Photo-productionγN → πN may be such a case.

On-shell renormalization along with the approach ofusing a minimal amount of parameters dictated purely bythe divergences increases the predictability of the modeldue to fewer constraints needed to fix the constants. Inparticular, there are no constants in F2(t) or FS(t) asopposed to one each in ChPT and there are six constantsin πN scattering as opposed to eleven in ChPT.

The analysis of πN scattering shows that all six ofthe subtraction constants in the amplitude can be fixedat subthreshold. Including the ∆(1232) contribution, thescattering lengths can be predicted with reasonable accu-racy. The value of σπN as constrained from πN scatteringgoes from being on the lower side of the canonically ac-cepted value of 45±8 MeV [32] at tree level to within thepredicted accuracy at one-loop, showing an improvementupon adding loop corrections as expected.

We have kept a relativistic formulation in order tomaintain relativistic unitarity. Indeed, many HBChPTcalculations, although formulated with a heavy baryon

Lagrangian, tend to start with the relativistic Feynmanrules and only after evaluation of the amplitude take thenon-relativistic limit. This is not only more natural butkeeps from missing terms as could happen from the non-relativistic formulation.

The convergence of a relativistic calculation cru-cially depends on the appearance of the constant termsm2N/(4πfπ)2. Although such terms do appear, in all the

cases considered here they are always accompanied by adivergent subtraction constant. A mere redefinition ofthis arbitrary constant removes such terms from the ex-pansion and rectifies the convergence. Whether this isa general feature of the loop expansion employed heremerits further investigation.

ACKNOWLEDGMENTS

This work was supported in part by the US DOE grantDE-FG02-88ER40388.

9

APPENDIX A

The Feynman rules from L1+2 needed in this paper areones with external current lines and internal (loop) pionlines. We take the transformation Ψi → ξiψi (i = R,L)

with U = ξRξ†L and choose ξR = ξ†L. The rules for the ψ

nucleon fields are

�

= �1 + c

f��

Va�

= ��a

2

P "

jaA�

=

�gA � +

2

�P�

� 5

�a

2

� b

= �i

f 2��

5�b

Va� b

= �gA

2f��abc� c � 5

jaA� b-P

=i

2f�

�i�

abc� c +2(1 + c)

�P��

ab

!

b c

�

= �1

f��bc

k1 k2

b c

Va�

= i�abc(k1 + k2)�

k; a

=1

f�

gA/k +

2m2�

�

! 5

�a

2

k1; a k2; b

=i

2f 2�

/k2 � /k1

2i�abc� c +

2(1 + c)m2�

��ab!

c d

jaA� jbA�

= �ig����a(c�d)b � 2�ab�cd

�

All loop processes in this paper can be expressed interms of the following general Feynman parameter inte-grals. Here dk is shorthand for d4k/(2π)4 properly reg-ularized and [p]a = (k + p)2 −m2

a. For two propagators,the integrals are

−i

∫dk

(1; kµ; kµkν)

[−p]a[0]b≡ (J ; pµJ1; pµpνJ21 + gµνJ22)

with Ji = Jabi (p2),

(J ; J1; J21

)= −

∫ 1

0

dx

(4π)2lnhJ(p2)

hJ(p20)

(1; x; x2)

J22 = −

∫ 1

0

dx

2(4π)2

[hJ(p2) ln

hJ(p2)

hJ(p20)

+ x(1− x)(p2 − p20)

]

hJ (s) = m2ax +m2

b(1− x) − s x(1− x),

and p20 is the subtraction point. The number of bars

above a function denote how many terms of its Taylorseries are subtracted at the chosen point. Note thatJaa = 2Jaa1 .

For three propagators, since there are only two par-ticles which play a role in the loops (nucleon and pion),two of the propagators will certainly have the same mass.

−i

∫dk

(1; kµ; kµkν ; kµkνkρ)

[0]a[−p]b[−q]b≡

(Γ; QµΓ1 + PµΓ2;

gµνΓ3 +QµQνΓ4 +Q(µPν)Γ5 + PµPνΓ6

)

with (Q,P ) = (p + q, p − q), Γi = Γab(p2, q2, P 2). Thefinite integrals are

Γi = −1

(4π)2

∫ 1

0

dx dy yαi

hΓ

with

α = 1 α1 =y

2α2 =

y

2(2x− 1)

α4 =y2

4α5 =

y2

4(2x− 1) α6 =

y2

4(2x− 1)2

and

hΓ = m2a(1− y) +m2

by − p2xy(1− y)

− q2(1− x)y(1− y)− P 2x(1− x)y2.

The divergent function can be subtracted to give

Γ3 = −1

2(4π)2

∫ 1

0

dx dy y lnhΓ

hΓ,0

with Γi(m2N ,m

2N , P

2) = Γi(P2) and Γi(s,m

2N , P

2) =Γi(s, P

2) shorthand notation used in this paper.Finally we need the following functions for four prop-

agators

−i

∫dk

(1; kµ; kµkν)

[0]π[−p]N [−q]N [−r]N≡ (G; pµG1 + qµG2 + rµG3)

with Gi = GπNi (p, q, r). All four integrals are finite

Gi =1

(4π)2

∫ 1

0

dx dy dz y2zβi

h2G

with

β = 1 β1 = xyz β2 = (1− x)yz β3 = (1− z)y

For this paper only the form GπNi (p1 + q1, p1, p2) withp2

1 = p22 = m2

N , p1 + q1 = p2 + q2 and q21 = q2

2 = m2π was

used, for which GπN1 = GπN2 and

hG = m2π(1− y) +m2

Ny2 − (s−m2

N )y(1− y)(1− z)

−m2πy

2z(1− z)− tx(1− x)(yz)2

with s = (p1 + q1)2 and t = (p1 − p2)2.

10

APPENDIX B

The one-loop form for the two axial-vector correlator 〈N |T ∗jAjA|N〉 is quoted below. This is needed for πNscattering. Using the notation ∆πN = gAmN −G we find:

a � � � b c d� e f

FIG. 2. One-loop diagrams for πN scattering. The graphs with a star also have a mirror image diagram which must betaken into account and all graphs except for those in the last line require the addition of a crossed diagram.

Writing A± = A±(s, t, u)±A±(u, t, s) and B± = B±(s, t, u)∓B±(u, t, s) to take into account the crossed diagrams,we only quote the contribution from the direct diagrams shown in fig. 2. The self energy and form factor contributionsof fig. 2a can be written succinctly with the use of the axial-vector nucleon form factors with one off-shell nucleon leg[15]. With the understanding of only taking this to order 1/f4

π, it can be written as

A±2a(s, t, u) =1

4f2π

√s−mN√

sfs.e.(

√s)[g(√s,mN ,m

2π)]2

+ (√s→ −

√s)

B±2a(s, t, u) = −1

4f2π

1√sfs.e.(

√s)[g(√s,mN ,m

2π)]2

+ (√s→ −

√s)

with g = (√s+mN )g1 +m2

πg2 − (s−m2N)g3,

fs.e.(√s) =

1√s−mN

+3

4f2π

(gA√s+ G)2

(√s−mN )2

[mNJπN (s)−

√s JπN1 (s)

]−

3m2N

2f2π

(gA√s+ G)2

√s−mN

[JπN ′(m2

N )− JπN1′(m2

N )]

g1(√s,mN , t) = gA +

g2A − 4

4f2π

(gA√s+ G

)[mNJπN (s)−

√s JπN1 (s)

]+gAG

2f2π

(gA√s+ G

)[− 2ΓπN3 (s, t) + JNN(t) +m2

πΓπN (s, t)− (s−m2N )ΓπN1 (s, t)− (

√s−mN )2ΓπN2 (s, t)

]

g2(√s,mN , t) = −

2

Λ+gA − 2(1 + c)

2f2πΛ

(gA√s+ G

) [mNJπN (s)−

√sJπN1 (s)

]−

G

f2πΛ

(gA√s+ G

)[JNN (t) +m2

πΓπN (s, t)− (√s−mN )2ΓπN1 (s, t)− (s−m2

N )ΓπN2 (s, t)]

−gAG

2f2π

(gA√s+ G

) [(√s−mN )

(2ΓπN5 (s, t)− ΓπN1 (s, t)− ΓπN2 (s, t)

)+ 2(√s+mN )ΓπN6 (s, t)

]+

4G

f2π

(gA√s+ G

)[ G

m2π − t

JNN (t) +1

ΛJNN (t)

]

g3(√s,mN , t) =

gAG

2f2π

(gA√s+ G

)[2(√s−mN )ΓπN4 (s, t) + 2(

√s+mN )ΓπN5 (s, t)

+ (√s−mN )ΓπN1 (s, t) + (

√s−mN)ΓπN2 (s, t)

],

and G = 2G− gAmN . It can be checked by the Ward identity in eq. (7) that g(mN ,mN ,m2π) ≡ 2fπgπNN exactly to

one loop therefore showing a simple way in which the on-shell values are maintained. The other graphs in fig. 2 give

11

f4πA

+2b(s, t, u) =

3

16g2AmN

[g2A(s−m2

N ) + 4G2]JπN (s)

+3

16g2A

[2gAs(G− 2∆πN) +mN

(g2As+ (G− 2∆πN )2

)]JπN1 (s)

+ 3gAGG2[−JNN(m2

π)−m2πΓπN (s) + (s+ 3m2

N)ΓπN1 (s) + (s−m2N )ΓπN2 (s)

]+ 3g2

AmNG2(s−m2

N )ΓπN1 (s) +3

2g2AG

2mN (s− u)ΓπN4 (t)− 6mNG4ΓπN1 (t)

+3

2gAGG

2[JNN (t) +m2

πΓπN (t)]

+ 3mNG4(s−m2

N )[2GπN1 + GπN3

]f4πB

+2b(s, t, u) =

3

8g3AmN (G− 2∆πN )JπN (s) +

3

16g2A

(g2As+ (G− 2∆πN )2

)JπN1 (s)

+3

2g2AG

2[−JNN(m2

π)−m2πΓπN (s) + (s−m2

N )(ΓπN1 (s) + ΓπN2 (s)

)]+

3

4g2AG

2[2ΓπN3 (t)− JNN (t)−m2

πΓπN (t)]

+ 6gAmN GG2ΓπN1 (s)

+ 3G4[(s−m2

N )GπN3 −m2πG

πN − ΓNN(m2π ,m

2π, t)

]A−2b(s, t, u) = −

1

3A+

2b(s, t, u) B−2b(s, t, u) = −1

3B+

2b(s, t, u)

f4πA

+2c(s, t, u) =

1

2mN

[(s−m2

N + 2σ2πN )JπN (s)− (s−m2

N − 2σ2πN )JπN1 (s)

]f4πA−2c(s, t, u) =

1

4(s−m2

N )[mNJπN(s)− (mN − 4σπN )JπN1 (s)

]f4πB

+2c(s, t, u) = −

1

2

[2m2

NJπN (s)− (s+m2

N + 2σ2πN)JπN1 (s)

]f4πB−2c(s, t, u) = −

1

4

[2mN (mN − 2σπN )JπN (s)− (s+m2

N − 4mNσπN )JπN1 (s)]

f4πA

+2d(s, t, u) = −

1

2gAmN

(gA(s−m2

N )− 2σπN G)JπN (s)

−1

2gA((gAmN − 4∆πN )(s−m2

N )− gAσπN (s+m2N) + 4mN∆πNσπN

)JπN1 (s)

+ 2σπNG2(s−m2

N )ΓπN2 (s)− 2σπNG2(JNN (m2

π) +m2πΓπN (s)

)f4πB

+2d(s, t, u) =

1

2gAmN

(gAσπN + 4∆πN

)JπN (s)

−1

2gA(gA(s−m2

N −mNσπN ) + 2∆πN (2mN + σπN ))JπN1 (s)

+ 2G2(JNN (m2

π) +m2πΓπN(s)

)− 2G2(s−m2

N )(ΓπN1 (s) + ΓπN2 (s)

)+ 4mNσπNG

2ΓπN1 (s)

A−2d(s, t, u) = 0 B−2d(s, t, u) = 0

f4πA

+2e(s, t, u) = (t− 2m2

π)[gAGJππ(t) + 2mNG

2(ΓπN (t)− 2ΓπN1 (t)

)]f4πA

+2f (s, t, u) = σπN (t− 2m2

π)Jππ(t)

In the Λ → ∞ limit we reproduce all the finite parts of GSS. This shows that calculation of the πN scatteringamplitude using the Ward identity with external fields is equivalent to a calculation with the pion vertices from theLagrangian without mention of external fields or the Ward identity. Whether this holds for πN → ππN will bediscussed elsewhere.

The relation between the two calculations can be made more transparent by use of diagrams. The vector andscalar form factor contribution to the πN Ward identity, along with the contact interactions from 〈N |T ∗jAjA|N〉,

12

are equivalent to the the graphs from the pion calculation which contain two external pions meeting at one point.Diagramatically this is just:

�

+

V

+

jA jA

+

jA jA

= + + +

with the left-hand side containing the proper coefficient given by the Ward identity. The other possible graphs fromthe pion calculation are in one-to-one correspondence with the graphs of the same topology from the two axial-vectorcorrelator.

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