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SMALL MOMENTUM EVOLUTION OF THE EXTENDED

arXiv:hep-ph/9212257v1 14 Dec 1992SMALL MOMENTUM EVOLUTION OF THE EXTENDEDDRELL–HEARN–GERASIMOV SUM RULE *V´eronique BernardCentre de Recherches Nucl´eaires et Universit´e Louis Pasteur de StrasbourgPhysique Th´eorique, BP 20 Cr, 67037 Strasbourg Cedex 2, FranceandNorbert KaiserPhysik Department T30Technische Universit¨at M¨unchen, James Franck StraßeD-8046 Garching, GermanyandUlf-G. Meißner†Universit¨at BernInstitut f¨ur Theoretische PhysikSidlerstr. 5, CH–3012 Bern, SwitzerlandABSTRACTWe investigate the momentum dependence of the extended Drell-Hearn-Gerasimovsum rule.

An economical formalism is developed which allows to express the extendedDHG sum rule in terms of a single virtual Compton amplitude in forward direction.Rigorous results for the small momentum evolution are derived from chiral perturbationtheory within the one-loop approximation. Furthermore, we evaluate some higher ordercontributions arising from ∆(1232) intermediate states and relativistic corrections.BUTP–92/51December 1992CRN 92–53* Work supported in part by Deutsche Forschungsgemeinschaft and by SchweizerischerNationalfonds.† Heisenberg Fellow.0

I.INTRODUCTIONMany years ago, Drell and Hearn [1] and Gerasimov [2] (DHG) suggested a sum rulefor spin-dependent Compton scattering. It expresses the squared anomalous magneticmoment of the nucleon in terms of a dispersive integral over the difference of the totalphotonucleon absorption cross sections σ1/2(ω) and σ3/2(ω) for the scattering of circularpolarized photons on polarized nucleons.

The subscripts λ = 1/2 and λ = 3/2 denotethe total γN helicity, corresponding to states with photon and nucleon spin antiparallelor parallel. Experimentally, this sum rule has never been tested directly since up to nowno measurements of the helicity cross sections have been performed.

However, modelsfor the photoabsorption cross sections [3,4,5] do indicate its approximate validity (on aqualitative level). One can now extend this sum rule to virtual photons with k2 < 0 thefour-momentum transfer of the virtual photon* since the corresponding helicity crosssections can be parametrized in terms of the spin-dependent nucleon structure functions.The recent data of the European Muon Collaboration [6] taken in the scaling region oflarge |k2| ≃10 GeV2 suggest not only that the pertinent sum rule behaves as 1/k2 forlarge |k2|, but also that the sign is opposite to the DHG sum rule for real photons (whichin standard notations is negative).

Therefore the integralI(k2) =Z ∞ωthrdωωσ1/2(ω, k2) −σ3/2(ω, k2)(1.1)with ω the virtual photon energy in the nucleon rest frame must change its sign betweenthe photon point (k2 = 0) and the EMC region, k2 ≃−10 GeV2. A recent model pre-dicts this turnover to happen at k2 ≃−0.8 GeV2 [7] and it explains this value mainly interms of the low-energy contribution of the ∆(1232) resonance to the pertinent photoab-sorption cross sections.

Notice that the model of ref. [7] as well as the phenomenologicalanalysis of ref.

[5] seem to indicate a negative slope of Ip(k2) in the vicinity of the photonpoint, k2 ≃0.Here, we wish to add some new insight into the momentum dependence of the in-tegral I(k2) in the region of small k2 where small means that√−k2 does not exceed afew pion masses. Our model–independent analysis is based on the fact that at low ener-gies, the interactions of hadrons are governed by chiral symmetry and gauge invariance(when external photons are involved).

One can systematically solve the chiral Ward-Takahashi identities of QCD via an expansion in external momenta and quark masses,which are considered small against the scale of chiral symmetry breaking, Λχ ≃1 GeV.This method is called chiral perturbation theory. It uses the framework of an effectivelagrangian of the asymptotically observed fields.

The low-energy expansion correspondsto an expansion in pion loops. In the presence of baryons, a complication arises.

Thenucleon (baryon) mass in the chiral limit is comparable to the chiral scale Λχ and thusonly baryon three-momenta can be considered small [8]. One can, however, restore theexact one-to-one correspondence between the loop and low-energy expansion using a* It is customary to set k2 = −Q2 and only use Q2.

We will not do this in the following.1

non-relativistic formulation of baryon chiral perturbation theory [9]. The nucleon isconsidered as a very heavy (static) source and in that case, all momenta involved aresmall therefore restoring the consistent power counting.

In what follows, we will usethe non-relativistic version of baryon CHPT which was systematically investigated inref. [10] as well as the relativistic formulation as spelled out in detail in ref.[8].

This willallow us to extract the leading term in the chiral expansion of I(k2) and to calculatethe derivative of I(k2) around k2 ≃0. This is the region where CHPT applies.

Fur-thermore, following the suggestion of Jenkins and Manohar [11], we will also add the∆(1232) resonance to non-relativistic baryon CHPT. The ∆(1232) is the lowest nucleonexcitation and its closeness to the nucleon mass, m∆−m ≃2.1 Mπ, might indicatesubstantial contributions from it (this is also supported by phenomenological models).In fact, using these various approximation schemes, we will get a band of values for theslope of I(k2).

Our most important result, however, is that independent of the schemewe are using, we find that I(k2) increases as |k2| increases (around k2 ≃0). This newresult should serve as a constraint for all model builders and should eventually be seenin refined phenomenological analyses or directly from the data (when they will becomeavailable).The paper is organized as follows.

In section II, we spell out an economical formal-ism to calculate I(k2) in terms of a single function which posseses a right-handed cutstarting at the single pion production threshold. This method is considerably simplerthan the one recently proposed by Meyer [12] whose formalism involves half-off-shell nu-cleon form factors.

In section III, we use CHPT to calculate I(k2) for the proton and theneutron at small k2, in the extreme non-relativistic and the fully relativistic formulation.The contribution of loops involving the ∆(1232) isobar in the non-relativistic approachis also discussed. The numerical results and conclusions are presented in section IV.II.SPIN–DEPENDENT COMPTON SCATTERING: FORMALISMIn this section, we outline the formalism necessary to describe the scattering ofpolarized (virtual) photons on polarized nucleons (protons and neutrons).

Denote byp and k the four-momenta of the nucleon and photon, respectively. It is convenient towork with the two lorentz invariants k2 and ω = p · k/m, with m the nucleon mass.The spin of the photon and nucleon can couple to the values 1/2 and 3/2 with thecorresponding photoabsorption cross sections denoted by σ1/2(ω, k2) and σ3/2(ω, k2), inorder.

* In what follows, we are interested in the extended Drell-Hearn-Gerasimov sumrule, i.e. the integralI(k2) =Z ∞ωthrdωωσ1/2(ω, k2) −σ3/2(ω, k2)(2.1)* For the definition of these cross sections see ref.

[13] (chap.2). We omit the tilde overthe symbol σ used in that book.2

with k2 ≤0 and the threshold photon energy ωthr due to single pion electroproductionis given byωthr = Mπ + M 2π −k22m(2.2)where Mπ denotes the pion mass. For real photons, the expression (2.1) becomes thecelebrated DHG sum ruleI(0) =Z ∞ωthrdωωσ1/2(ω, 0) −σ3/2(ω, 0)= −πe2κ22m2 .

(2.3)Here, κ is the anomalous magnetic moment of the proton or the neutron and we usestandard units, e2/4π = 1/137.036. The DHG sum rule is derived under the assumptionthat the spin-dependent forward Compton amplitude for real photons f2(ω2) satisfies anunsubtracted dispersion relation which guarantees that the right-hand side of eq.(2.3)converges.

In what follows, we will make use of the same assumption for virtual photons.To set the scale for I(k2), let us give the numerical values for the proton and the neutron,Ip(0) = −0.526 GeV−2 ,In(0) = −0.597 GeV−2 . (2.4)Our main concern will be the k2 evolution of the extended DHG sum rule, in particulararound the origin k2 ≃0.

The interest in that comes from the relation of the helicitycross sections to the spin-dependent nucleon structure functions G1(ω, k2) and G2(ω, k2).Following the notations of Ioffe et al. [13]*, one can show thatσ1/2(ω, k2) −σ3/2(ω, k2) =4πe22mω + k2ωmG1(ω, k2) + k2mω G2(ω, k2).

(2.5)The relation of these structure functions to the spin-dependent virtual Compton ampli-tudes in forward direction S1,2(ω, k2) is standard2π Gi(ω, k2) = Im Si(ω, k2) ,(i = 1, 2)(2.6)which follows from the optical theorem. Furthermore, crossing symmetry implies thatS1(ω, k2) and G2(ω, k2) are even functions under (ω →−ω) whereas S2(ω, k2) andG1(ω, k2) are odd.

In fact, for our purpose one does not need the information on bothamplitudes S1(ω, k2) and S2(ω, k2) but only the particular combination entering eq. (2.5).In order to isolate this relevant combination one contracts the antisymmetric (in µ ↔ν)part of the virtual Compton tensor in forward direction with polarization vectors ǫ′µ andǫν for the outgoing and incoming virtual photon, respectively.

If we choose the gauge* We use a different normalization for the nucleon spinor, ¯uu = 1 instead of ¯uu = 2m.3

conditions ǫ · p = ǫ′ · p = ǫ · k = ǫ′ · k = 0 for the polarization vectors and work in thenucleon rest-frame pµ = (m, 0, 0, 0) we obtainǫ′µ T µν(a) ǫν =i2m2 χ†⃗σ · (⃗ǫ ′ ×⃗ǫ )ωS1(ω, k2) + ω2m S2(ω, k2)−⃗σ · ⃗k ⃗k · (⃗ǫ ′ ×⃗ǫ )S2(ω, k2)mχ=iω2m2 χ†⃗σ · (⃗ǫ ′ ×⃗ǫ )χS1(ω, k2) + k2mω S2(ω, k2)(2.7)where χ is a conventional two-component (Pauli) spinor. In eq.

(2.7) we have exploitedthe fact that under the chosen gauge ⃗ǫ ′×⃗ǫ is parallel to ⃗k and ⃗k 2 = ω2−k2. Obviously, weare projecting out the particular combination of S1(ω, k2) and S2(ω, k2) whose imaginarypart enters the extended DHG sum rule I(k2).

In analogy to the real photon case wecall this combinationf2(ω2, k2) =e28πm2S1(ω, k2) + k2mω S2(ω, k2). (2.8)Here, we indicated already that f2(ω2, k2) is an even function of ω which follows fromthe (ω →−ω) crossing properties of S1,2(ω, k2) [13].

The odd amplitude ω f2(ω2, k2)can now be expressed in terms of a single function A(s, k2) as follows2π(s −m2 −k2) f2(ω2, k2) = e2A(s, k2) −A(2m2 + 2k2 −s, k2). (2.9)Here, we introduced the Mandelstam variable s = (p + k)2 which is related to ω viaω = (s−m2 −k2)/2m.

The function A(s, k2) appearing in eq. (2.9) can always be chosensuch that it has only a right-handed cut starting at the single pion production thresholds = (m+Mπ)2.

Under the assumption that f2(ω2, k2) fulfills an unsubtracted dispersionrelation (in ω) or equivalently that A(s, k2) fulfills a once-subtracted dispersion relation(in s, subtracted at an arbitrary point s0) we can make use of the previous equationsand calculate the extended DHG sum rule I(k2) asI(k2) = 8πZ ∞(m+Mπ)2 dsIm f2(ω2, k2)s −m2= 4e2Z ∞(m+Mπ)2 dsImA(s, k2)(s −m2)(s −m2 −k2)= 4πe2k2A(m2 + k2, k2) −A(m2, k2). (2.10)This equation is our basic result.

It is completely general and allows one to calculatethe extended DHG sum rule I(k2) from a single function A(s, k2) which can be easilycomputed from the virtual Compton tensor in forward direction. To repeat it, eq.

(2.10)was derived under the assumption that A(s, k2) obeys a once-subtracted dispersion4

relation. That this is not a too strong assumption e.g.

can be seen from the fact thatin the relativistic formulation of baryon CHPT to one-loop A(s, k2) indeed has thisanalytical property. However, a general proof for this is not yet available.

In this sensethe situation is analogous to f2(ω2, 0) where the validity of an unsubtracted dispersionrelation can not yet be proven in general. In the following section, we will use CHPT(in the one–loop approximation) to evaluate A(s, k2) and to calculate I(k2) for k2 inthe vicinity of zero (this is where CHPT applies).III.CHIRAL EXPANSIONAt low energies, any QCD Green function can be systematically expanded in powersof small momenta and quark (pion) masses.

This is done within the framework of aneffective chiral lagrangian of the asymptotically observed fields, here the nucleons, pionsand photons. The low-energy expansion amounts to an expansion in (pion) loops of theeffective theory.

In the presence of baryons, a complication arises due to the baryonmass which is non-vanishing in the chiral limit and therefore adds a new scale to thetheory. In that case there is in general no strict one-to-one correspondence between thelow energy and loop expansion.

Stated differently, there is no guarantee that all next-to-leading order corrections at order q3 (with q denoting a generic small momentum) aregiven completely by the one loop graphs. All calculations performed so far, however,indicate that the leading non-analytic terms (in the quark masses) which arise dueto infrared singularities in the chiral limit of vanishing pion mass are indeed produced.Furthermore one also gets in the one loop approximation an infinite tower of higher orderterms [8] which spoil the one-to-one mapping between low-energy and loop expansion.To overcome these difficulties, it was recently proposed to use a heavy fermion effectivefield theory, i.e.

considering the baryons as very heavy [9] and to expand the theory ininverse powers of the baryon mass. In that case, the n-loop contributions are suppressedby relative powers of q2n (with q a genuine small momentum) and a consistent countingscheme emerges.

Furthermore, in this framework one can easily couple in the ∆(1232)resonance since one does not encounter the usual problems with the relativistic spin-3/2 particle [11]. Nevertheless, we have to stress that the baryon mass m comparableto the chiral symmetry breaking scale Λχ is not very large.

Therefore, an expansionin powers of Mπ/m is a priori not to be expected to converge very fast. Such Mπ/msuppressed contributions are partly resummed in the relativistic approach.

Of coursethe evaluation of all Mπ/m corrections is necessary to judge the quality of the chiralexpansion. Furthermore, once the spin–3/2 decuplet is included, one has an extra non–vanishing scale in the chiral limit (the average octet–decuplet mass splitting) whichagain complicates the low energy structure.The basic πNγ lagrangian in the relativistic formulation of baryon CHPT to leadingorder (O(q)) readsL = L(1)πN + L(2)ππL(1)πN = ¯Ψ(i̸D −m + gA2 ̸uγ5)ΨL(2)ππ = F 24 Tr[∇µU∇µU † + M 2π(U + U †)](3.1)5

where U = exp[i⃗τ · ⃗π/F] embodies the Goldstone bosons, u =√U and uµ = iu†∇µUu†with ∇µ the pertinent covariant derivative. The isospinor Ψ contains the proton andneutron fields.

The superscript (i) denotes the chiral power of the corresponding terms,it counts derivatives and meson masses. The construction of this effective lagrangian isunique.

Let us point out that it contains four parameters. These are the pion decayconstant F, the axial-vector coupling gA and the nucleon mass (all in the chiral limit)and the leading term in the quark mass expansion of the pion mass, Mπ =√2 ˆmB.Here, ˆm = 12(mu + md) is the average light quark mass and B = −< 0|¯uu|0 > /F 2is the order parameter of the spontaneous chiral symmetry breaking.

Calculating treediagrams with this effective lagrangian, one reproduces the well-known current algebraresults. To restore unitarity, one has to consider pion loops in additon.

To give allcorrections at next-to-leading order in the chiral expansion one has to work out all oneloop diagrams constructed from the vertices in L and furthermore one has to add thetree graph contribution from the most general chirally symmetric counterterm lagrangianL(2)πN + L(3)πN + L(4)ππ. For the (spin-dependent) Compton tensor under consideration here,however, no such counterterm can contribute.

As stressed in ref. [10], we are dealingwith a pure loop effect (within the one-loop approximation).As already noted, in eq.

(3.1) the troublesome nucleon mass term appears. In theextreme non-relativistic limit, it can be eliminated in the following way.

Decompose thebaryon four momentum as pµ = mvµ + lµ with vµ the four-velocity (v2 = 1) and lµ asmall off-shell momentum (v·l ≪m, ) and write Ψ in terms of eigenstates of the velocityprojection operatorΨ = e−im v·x(H + h)(3.2)with ̸vH = H and ̸vh = −h. Eliminating now the ”small” component h via its equationof motion, one ends up withL(1)πN = ¯H(iv · D + gAS · u)H + O(1/m)(3.3)Here, Sµ = i2γ5σµνvν is the covariant spin operator which obeys S · v = 0.

The nucleonmass term has disappeared allowing for a consistent chiral power counting scheme. Allone loop contributions are order q3.

Furthermore, one has to expand the tree contribu-tions from the vertices of eq. (3.1) in 1/m appropriately to collect all terms up to andincluding order q3.

For a more detailed discussion of these topics, see ref.[10]. One canfurthermore add the ∆(1232), which is a spin-3/2 field, very easily in the extreme non-relativistic limit.

For details on the couplings of the ∆(1232) see the appendix. Here,we just note that the mass splitting m∆−m stays finite in the chiral limit.

Thereforeloops with intermediate ∆(1232) states will count as order q4 and higher (since thecounterterm contributions start only at order q5).Let us now turn to the calculation of I(k2) for small k2. In Fig.1.a we show thepertinent Feynman diagrams which contribute in the heavy mass limit (with interme-diate nucleons only).

We work in the Coulomb gauge ǫ′ · v = ǫ · v = 0 which is veryeconomical in the calculation of photon-nucleon processes since most diagrams (those6

with an isolated photon-nucleon vertex) are then identical to zero. The integral I(k2)takes the formI(k2) = I(0) + ˜I(k2)(3.4)with I(0) = −πe2κ2/2m2 the DHG sum rule value for real photons.

In the heavy massformulation of baryon CHPT the leading term of the chiral expansion of ˜I(k2) is givencompletely by the one loop graphs in Fig.1a. All higher order corrections to ˜I(k2) aresuppressed by further powers of the pion mass Mπ and k2.

Some (but not all) of thesecorrections will be generated from loop diagrams with ∆(1232) intermediate states orin the relativistic version of baryon CHPT. The leading term of the chiral expansion of˜I(k2) can be given in closed from˜I(k2) = e2g2A4πF 2−1 +r1 + 4ρ lnr1 + ρ4 +√ρ2= e2g2A48πF 2 ρ + O(ρ2)(3.5)with ρ = −k2/M 2π > 0.

We see that the slope of I(k2) at k2 = 0 is negative and singularin the chiral limit, i.e it diverges like 1/M 2π. This behaviour is a direct consequenceof the chiral structure of QCD which governs the low-energy strong interaction phe-nomena.

Furthermore, ˜I(k2) is equal for both proton and neutron (within the O(q3)approximation to the virtual Compton tensor). We should also add here that presentlythe usual DHG sum rule value I(0) for real photons can not be obtained through adispersive integral like eq.

(2.10) within the one-loop approximation of CHPT. In theheavy mass formulation this term arises from real 1/m2 suppressed tree graphs involv-ing the anomalous magnetic moment κ (in the chiral limit).

In the relativistic versionof baryon CHPT the anomalous magnetic moment of the nucleon is generated from oneloop diagrams and it is non-vanishing in the chiral limit. In order to obtain a termproportional to κ2 like I(0) one necessarily has to go to the level of two-loop graphs.This problem of how I(0) can be obtained from a dispersion relation for loop amplitudesdoes, however, not affect our discussion of the k2 dependence of I(k2).

Extending theeffective lagrangian to the ∆(1232) resonance as spelled out in the appendix we have tocalculate the diagrams of Fig.1b. These amount to some higher order (qn, n ≥1) correc-tions to eq.

(3.5) which we include because of the phenomenological importance of thisresonance (a complete evaluation of all O(q) corrections to I(k2) corresponding to O(q4)for the virtual Compton tensor goes beyond the scope of this paper). A straightforwardcalculation gives for the sum of nucleon and ∆(1232) one-loop diagrams˜I(k2) = e2g2A4πF 2r√r2 −1lnr +pr2 −1−Z 10dxrpr2 −1 −ρx(1 −x)lnrp1 + ρx(1 −x)+sr21 + ρx(1 −x) −1(3.6)7

with r = (m∆−m)/Mπ ≃2.1. Obviously, ˜I(0) = 0 in agreement with the celebratedlow-energy theorem of Low, Gell-Mann and Goldberger [15].

As a check one can showthat in the limit m∆−m →∞one recovers the result of eq.(3.5). Again there is nosplitting between proton and neutron sum rules, i.e.˜I(k2) = ˜Ip(k2) = ˜In(k2).

Theslope of the extended DHG sum rule at the photon point is given asI′(0) = −e2g2A48πF 2M 2πr2√r2 −1 −r ln(r +√r2 −1)(r2 −1)3/2. (3.7)In the relativistic formulation matters are different.First one has to calculatemany more Feynman diagrams.

These generate some of the Mπ/m suppressed higherorder corrections and naturally lead to a splitting between proton and neutron for themomentum dependence of the extended DHG sum rule, i.e. ˜Ip(k2) ̸= ˜In(k2).

What isconceptually most important is that in the relativistic version of baryon CHPT one canindeed show that the amplitude function A(s, k2) obeys a once–subtracted dispersionrelation. Using now the definitions of the various loop functions as given in ref.

[14]extended to k2 ≤0, the following expressions can be deduced for ˜Ip(k2) and ˜In(k2)˜Ip(k2) = e2g2Am24πF 2k2Z 10dxZ 10dy1 −3m2k2 yln M 2π(1 −y) + m2y2 + k2y(y −1)M 2π(1 −y) + m2y2−2y ln M 2π(1 −y) + m2y2 + k2xy(xy −1)M 2π(1 −y) + m2y2 + k2x(x −1)y2 +32y(1 −y)[k2 −(2m2 + k2)y]M 2π(1 −y) + m2y2−2k2xy(1 −y)2M 2π(1 −y) + m2y2 + k2x(x −1)(1 −y)2 + y2[k2(xy + x2y −12) + m2(y −1)M 2π(1 −y) + m2y2 + k2x(x −1)y2+y2[m2(1 −y) −k2x2y]M 2π(1 −y) + m2y2 + k2xy(xy −1) +2m2k2(1 −x)xy4[M 2π(1 −y) + m2y2 + k2xy(xy −1)]2+ y4[k4x2(1 −x)(y −12) + m2k2( 12 −32x + 2x2 −xy)][M 2π(1 −y) + m2y2 + k2x(x −1)y2]2−32k2m2y3(1 −y)2[M 2π(1 −y) + m2y2]2(3.8)˜In(k2) = e2g2Am24πF 2k2Z 10dxZ 10dy2(1 −y) ln M 2π(1 −y) + m2y2 + k2x(y −1)(1 −x + xy)M 2π(1 −y) + m2y2 + k2x(x −1)(1 −y)2+ 2y ln M 2π(1 −y) + m2y2 + k2xy(xy −1)M 2π(1 −y) + m2y2 + k2x(x −1)y2 +y2[−2m2 + k2(2x −1)]M 2π(1 −y) + m2y2 + k2x(x −1)y2+2m2y2M 2π(1 −y) + m2y2 + k2xy(xy −1) +k2(1 −x)y4[k2x2 + m2(1 −4x)][M 2π(1 −y) + m2y2 + k2x(x −1)y2]2+4m2k2x(1 −x)y4[M 2π(1 −y) + m2y2 + k2xy(xy −1)]2. (3.9)As an important analytical check we can again verify that ˜Ip(0) = ˜In(0) = 0 and onecan show that in the limit m →∞both ˜Ip(k2) and ˜In(k2) tend to ˜I(k2) as given in8

eq.(3.5). With this we have collected all formulae necessary to study I(k2) for both theproton and the neutron.IV.RESULTS AND DISCUSSIONFirst, we must fix parameters.

Throughout, we use F = 93 MeV, Mπ = 139.57MeV, m = 938.27 MeV and gA = 1.26. In the case of the ∆(1232) resonance, we use theSU(4) relation among coupling constants gπN∆= 3gπN/√2 with gπN = gA m/F givenby the Goldberger-Treiman relation.

The mass splitting between nucleon and ∆(1232)has a value of m∆−m = 293 MeV.Consider now the proton. We will first discuss the slope of Ip(k2) at the photonpoint, k2 = 0.

In the heavy mass limit with only intermediate nucleon states we finddIp(k2)dk2k2=0= −e2g2A48πF 2M 2π= −5.7 GeV−4(4.1)This value is decreased by 16% when the ∆(1232) resonance is included in the oneloop graphs as inspection of eq. (3.7) reveals.

Therefore the ∆(1232) does not play amajor role in determining the slope of Ip(k2) in our approach. Much more drastic isthe effect of the relativistic Mπ/m suppressed terms.

In the fully relativistic calculationwhere many (but not all) of such terms are included we find I′p(0) = −2.2 GeV−4 forthe proton and I′n(0) = −1.7 GeV−4 for the neutron.In Fig.2, we show ˜Ip(k2) for−k2 ≤0.25 GeV2. In the heavy mass limit half of the value of Ip(0) (in magnitude) isreached at k2 ≃0.06 GeV2.

The crossover where Ip(k2) goes from negative to positivevalues takes place at k2 ≃−0.15 GeV2. This is a very low value compared to previousphenomenological analysis but compared to the pion mass scale M 2π it is already quitelarge, k2 ≃= −7.7 M 2π.

Therefore one can no longer trust the one loop approximation inthat region of k2 where the sign change of Ip(k2) takes place. Including some higher orderchiral corrections through loops with ∆(1232) resonances, the momentum dependenceof ˜Ip(k2) becomes softer and the corresponding numbers decrease by roughly 30%.

Thezero of Ip(k2) is now shifted to a higher value of k2 ≃−0.23 GeV2. In the relativisticformulation of CHPT where in addition to the leading terms also many higher ordercorrections are included, ˜Ip(k2) is much smaller than in the case of infinite nucleonmass.

This phenomenon, that higher order relativistic correction are quite large wasalso observed in previous calculations of the nucleon electromagnetic polarizabilities[14]. However, since the Mπ/m corrections generated in the one–loop approximation ofrelativistic baryon CHPT are by no means complete, one can not draw any conclusionsabout the convergence of the chiral expansion at the moment.In summary, we have presented a novel formalism to calculate the momentum de-pendence of the extended DHG sum rule at finite k2 ≤0.

A single amplitude functionA(s, k2) which enters the spin-dependent virtual Compton tensor in forward directionis sufficient to evaluate I(k2), as long as A(s, k2) fulfills a once–subtracted dispersionrelation. We have used baryon chiral perturbation theory to investigate the behaviourof the extended DHG sum rule I(k2) in the vicinity of k2 = 0.

We could give a (ratherwide) range of values for the slope I′p(0). Eventually, this prediction will be tested ex-perimentally, at present we consider it as a constraint following from the chiral structureof QCD which will be useful for phenomenological analysis and model-building.9

APPENDIX: THE ∆(1232) IN THE HEAVY MASS FORMULATIONHere, we discuss briefly the description of the ∆(1232) resonance in the heavymass formulation following ref.[11]. To leading order (up to O(q)) the relevant effectivelagrangian reads (we write down only those terms which are actually needed for ourpurpose)L(1)πN∆= −i ¯T µa v · Dab T bµ + δm ¯T µaT aµ + 3gA2√2( ¯T µauaµH + ¯HuaµT µa) .

(A.1)The Rarita-Schwinger spinor T aµ with a an isospin index and µ a Lorentz index incor-porates the four charge states of the ∆(1232) as followsT 1µ =1√2∆++ −∆0/√3∆+/√3 −∆−µ,T 2µ =i√2∆++ + ∆0/√3∆+/√3 + ∆−µ,T 3µ = −r23∆+∆0µ. (A.2)Furthermore in the heavy mass limit this field is subject to the constraint vµT µa = 0.In (A.1) δm = m∆−m stands for the mass splitting of nucleon and ∆(1232) anduaµ = i2Tr(τ au†∇µUu†) = −∂µπa/F −eǫa3bAµ πb/F +.

. .

gives rise to the chiral couplingsof pions and photons to the N∆system.We already exploited the SU(4) relationgπN∆= 3gπN/√2 with gπN = gAm/F between the πN∆and πNN coupling constant.The empirical information on the ∆→πN decay width confirms that this relation holdsvery well within a few percent. In the heavy mass limit the propagator of the ∆(1232)readsP µν =iv · l −δmvµvν −gµν −43SµSν(A.3)where Sµ is the covariant spin operator of heavy mass approach satisfying v ·S = 0.

Letus finally remark that this formulation of ∆(1232) couplings is completely equivalentto the usual isobar model as discussed in ref. [16] for the special choice vµ = (1, 0, 0, 0).This corresponds to the standard non-relativistic description.ACKNOWLEDGEMENTSThe work of (UGM) was supported through funds provided by a Heisenberg fellow-ship.10

REFERENCES1. S.D.

Drell and A.C. Hearn, Phys. Rev.

Lett. 16 (1966) 908.2.

S.B. Gerasimov, Sov.

J. Nucl. Phys.

2 (1966) 430.3. I. Karliner, Phys.

Rev. D7 (1973) 2717.4.

R.L. Workman and R.A. Arndt, Phys.

Rev. D45 (1992) 1789.5.

V. Burkert and Z. Li, CEBAF preprint, CEBAF–PR–92–0176. J. Ashman et al., Phys.

Lett. B206 (1988) 364.7.

V. Burkert and B.L. Ioffe, CEBAF preprint, CEBAF–PR–92–018.8.

J. Gasser, M.E. Sainio and A.

ˇSvarc, Nucl. Phys.

B 307 (1988) 779.9. E. Jenkins and A.V.

Manohar, Phys. Lett.

B255 (1991) 558.10. V. Bernard, N. Kaiser, J. Kambor and Ulf-G. Meißner, Nucl.

Phys. B388 (1992)315.11.

E. Jenkins and A.V. Manohar, Phys.

Lett. B259 (1991) 353.12.

H. Meyer, University of Regensburg preprint, TPR–92–40.13. B.L.

Ioffe, V.A. Khoze and L.N.

Lipatov, ”Hard processes”, Vol.1, North Holland,1984.14. V. Bernard, N. Kaiser, and Ulf-G. Meißner, Nucl.

Phys. B373 (1992) 364.15.

F.E. Low, Phys.

Rev. 96 (1954) 1428;M. Gell-Mann and M.L.

Goldberger, Phys. Rev.

96 (1954) 1433.16. T. Ericson and W. Weise, ”Pions and Nuclei”, Clarendon Press, Oxford, 1988.FIGURE CAPTIONSFig.1.

a) One loop diagrams contributing to the spin-dependent Compton tensor in theheavy mass formulation of CHPT. Dashed lines denote pions.b) One loop Compton graphs including the ∆(1232) resonance in the heavy massapproach (denoted by a thick line).Fig.2.

The momentum dependence of the extended DHG sum rule ˜Ip(k2). The solid linegives the one–loop result in the heavy mass limit of baryon CHPT.

The dashed lineis obtained from one–loop graphs involving nucleons as well as ∆(1232) resonances.The dashed–dotted line gives the result of the relativistic version of baryon CHPTto one loop.11

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