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CRITICAL ANALYSIS OF BARYON MASSES AND SIGMA-TERMS

arXiv:hep-ph/9303311v1 29 Mar 1993CRITICAL ANALYSIS OF BARYON MASSES AND SIGMA-TERMSIN HEAVY BARYON CHIRAL PERTURBATION THEORY *V´eronique BernardCentre de Recherches Nucl´eaires et Universit´e Louis Pasteur de StrasbourgPhysique Th´eorique, Bat. 40A, BP 20, 67037 Strasbourg Cedex 2, FranceNorbert KaiserPhysik Department T30, Technische Universit¨at M¨unchenJames Franck Straße, W-8046 Garching, GermanyUlf-G. Meißner†Universit¨at Bern, Institut f¨ur Theoretische PhysikSidlerstr.

5, CH–3012 Bern, SwitzerlandABSTRACTWe present an analysis of the octet baryon masses and the πN and KN σ–termsin the framework of heavy baryon chiral perturbation theory. At next-to-leading order,O(q3), knowledge of the baryon masses and σπN(0) allows to determine the three corre-sponding finite low–energy constants and to predict the the two KN σ–terms σ(1,2)KN (0).We also include the spin-3/2 decuplet in the effective theory.

The presence of the non–vanishing energy scale due to the octet–decuplet splitting shifts the average octet baryonmass by an infinite amount and leads to infinite renormalizations of the low–energy con-stants.The first observable effect of the decuplet intermediate states to the baryonmasses starts out at order q4. We argue that it is not sufficient to retain only these butno other higher order terms to achieve a consistent description of the three–flavor scalarsector of baryon CHPT.

In addition, we critically discuss an SU(2) result which allowsto explain the large shift of σπN(2M 2π) −σπN(0) via intermediate ∆(1232) states.BUTP–93/05March 1993CRN–93–06* Work supported in part by Deutsche Forschungsgemeinschaft and by SchweizerischerNationalfonds.† Heisenberg Fellow.0

I.INTRODUCTIONAt low energies, chiral symmetry governs the interaction of the low-lying hadrons.To a good first approximation the current masses of the three light quarks can be set tozero and one can expand the QCD Green functions in powers of external momenta andquark masses around the so-called chiral limit. Assuming that the order parameter ofchiral symmetry breaking, B0 = −< 0|¯uu|0 > /F 2π, is of order 1 GeV, an unambiguousscheme emerges in the meson sector [1].

In the baryon sector matters are more compli-cated. The low-lying baryons have a non-vanishing mass in the chiral limit, of the orderof the chiral symmetry breaking scale B0, which complicates the low-energy structurein the meson-baryon system considerable [2].

Making use of methods borrowed fromheavy quark effective field theories, Jenkins and Manohar [3] proposed to consider thebaryons as very heavy static sources (see also Gasser and Leutwyler [4] and Weinberg[5]). This allows to define velocity eigen-fields and to leading order the troublesomebaryon mass term can be eliminated from the Dirac lagrangian.

This procedure is com-pletely equivalent to the well-known Foldy-Wouthuysen transformation used in QED toorder operators by inverse powers of the mass of the Dirac field*. A further complicationin the baryon sector is the closeness of the first resonance multiplet.

While in the mesonsector the vector mesons only appear above 770 MeV, the mass of the ρ-meson, thespin-3/2 decuplet is only separated by 230 MeV (in average) from the spin-1/2 groundstate octet. This raises the immediate question whether these spin-3/2 fields should beincluded in the effective field theory from the very beginning.

Also ample phenomeno-logical evidence from the nucleon sector exists which points towards the importanceof the spin-3/2 fields. In ref.

[7] the effective field theory was enlarged to include thedecuplet. Furthermore, numerous calculations have been performed concerning baryonmasses, hyperon non-leptonic decays [9], nucleon polarizabilities [10] and so on withinthis framework.

All these calculations are done under the assumption that only theterms non-analytic in the quark masses are important and therefore carry a spuriousdependence on the scale introduced in dimensional regularization. In addition, kaon andeta loop contributions can be large, as first pointed out by Bijnens, Sonoda and Wise[11].

However, it is argued that the decuplet contributions to a large extent cancel theselarge kaon and eta loop terms, leaving one with a fairly well behaved chiral expansioneven in the three flavor case.While one might content oneself with these rather positive results, a closer look athow they are obtained makes one feel uneasy about them. First, chiral perturbationtheory is a systematic expansion meaning that to a given order one has to take intoaccount all contributions, may they be from loops (and thus eventually non-analyticin the quark masses) or higher order contact terms.

These are multiplied by a prioriunknown coefficients, the so-called low-energy constants.Only the sum of loop andhigher order contact terms is naturally scale independent. Second, many of the resultsbased on the approach of taking only calculable loop diagrams with octet and decuplet* For a systematic analysis in flavor SU(2), see ref.

[6].1

intermediate states only give consistent results if one uses the rather small D and Faxial vector coupling proposed in ref.[7]. These values, to our opinion, are simply anartefact of the calculational procedure and do not reflect the results of a consistent chiralperturbation calculation.To put the finger on the problem, we reconsider here the classical problem of thebaryon masses and σ-terms, namely the πN σ-term and two KN σ-terms.This isessentially the scalar sector of baryon chiral perturbation theory.

The first systematicanalysis was performed by Gasser [12] and the long-standing problem with the πN σ-term has recently been resolved by Gasser, Leutwyler and Sainio [13]. They combineda dispersion-theoretical approach with chiral symmetry constraints to show that theσ-term shift σπN(2M 2π) −σπN(0) is as large as 15 MeV.

Therefore the empirical valueσπN(2M 2π) ≃60 MeV determined from πN scattering can be reconciled if the strangequark admixture in the proton is y ≃0.2, leading to a mass shift of ms < p|¯ss|p >≃130MeV, considerably lower than expected from first order quark mass perturbation theory[12,14] and anticipated in some models of dynamical chiral symmetry breaking [15].Jenkins and Manohar have argued that the heavy fermion formulation with decupletfields is indeed consistent with these values [16].This, in fact, is the statement wewish to elaborate on. First we perform a complete calculation up to order q3, whichonly involves intermediate octet states.

At this order (one-loop approximation) one hasthree counterterms with a priori unknown but finite coefficients. These can be fixedfrom the octet masses (mN, mΛ, mΣ, mΞ) and the value σπN(0) since one of the counterterms appears in the baryon mass formulae in such a way that it always can be lumpedtogether with the average octet mass in the chiral limit.

This allows us to predict thetwo KN σ-terms, σ(1)KN(0) and σ(2)KN(0) as well as the σ-term shifts to the respectiveCheng-Dashen points and the matrix element ms < p|¯ss|p >. We then proceed andadd the low-lying decuplet fields.

Here we leave the consistent calculation since the firstvisible effect of the decuplet appears at order q4. So, in principle we should account fora host of other terms.

Our aim is, however, more modest. We simply want to checkwhether the assertions made in refs.

[7-10,16] can be considered sound. The first, andrather obvious, observation to be made is that the inclusion of the decuplet fields spoilsthe consistent chiral power counting.This can be traced back to the residual massdependence which can not be removed from the lagrangian involving velocity-dependentfields.

Stated differently, in the chiral limit the average decuplet and octet masses differby a nonzero amount of a few hundred MeV. This leads to complications similar tothe ones in the relativistic formulation of baryon chiral perturbation theory (related tothe finite value of the nucleon mass in the chiral limit) discussed in ref.[2].

Also, inthe baryon mass spectrum one finds an infinite renormalization of the previously finitecounter terms of chiral power two. These problems have not yet been spelled out inthe literature.

For the numerical results we refer the reader to later sections, whereit will become clear that one has to work harder to get a consistent picture of heavybaryon CHPT beyond q3. As a nice by-product, we find an SU(2) result which allowsto explain the large shift σπN(2M 2π) −σπN(0) in terms of the ∆(1232) states (modulo2

the many other unknown effects appearing at order q4 or higher). It is important todifferentiate between the pure SU(2) results (where the kaons and etas only contributevery little) and the SU(3) results, which are afflicted by potentially large cancellationsbetween K and η loops on one side and decuplet contributions on the other side.

It isconceivable that one should first address these genuine SU(2) results and understandthe chiral expansion for them before one turns to the more complicated three flavorsector. A first step was done in ref.

[6] and we will come back to this problem in a futurepublication. To close the introduction, let us point out that a review on baryon CHPTis available [17], another one on the heavy fermion formulation [18] and related aspectsare discussed in the more recent CHPT review [19].II.FORMALISMII.1.

Baryon masses and σ–terms at next–to–leading orderIn this section, we will give the formalism necessary to discuss the baryon massesand σ-terms. Our starting point is the effective chiral lagrangian of the pseudoscalarGoldstone bosons coupled to octet baryons (we do not exhibit the standard mesonlagrangian)L(1)φB = Tr ( ¯B iv · D B) + D Tr ( ¯BSµ{uµ, B}) + F Tr ( ¯BSµ[uµ, B])(2.1)where the Goldstone fields φ are collected in the SU(3) matrixU(x) = exp[iφ(x)/Fp],u(x) =pU(x),uµ = iu†∇µUu†(2.2)and Fp is the pseudoscalar decay constant in the chiral limit.

B is the standard SU(3)matrix representation of the low-lying spin-1/2 baryons (p, n, Λ, Σ0, Σ±, Ξ0, Ξ−) and Sµis a covariant spin operator satisfying v · S = 0 and S2 = (1 −d)/4 in d space-timedimensions. The two axial coupling constants D and F are subject to the constraintD + F = gA = 1.26.

We work in the heavy mass formalism, which means that baryonsare considered as static sources and equivalently their momenta decompose aspµ = m0 vµ + lµ(2.3)with m0 the average octet mass in the chiral limit, vµ the baryon four-velocity (v2 = 1)and lµ a small off-shell momentum. In this extreme non-relativistic limit one can definevelocity dependent fields such that the troublesome baryon mass term disappears fromthe original Dirac lagrangian for the baryons.

In the absence of the baryon mass term aconsistent low-energy expansion can be derived. The baryon propagator reads i/(v·l+iǫ).The low energy expansion resulting from loops goes along with an expansion in inversepowers of the baryon mass m0.

As indicated by the superscript (1) in eq. (2.1), treelevel diagrams calculated from the lowest order effective lagrangian are of order q, withq denoting a genuine small momentum.

At next-to-leading order one has to include the3

one-loop graphs using solely the vertices given by L(1)φB and additional chirally symmetriccounterterms of order q2 and q3, since the one-loop graphs all have chiral power q3. Thesecontact terms are accompanied by a priori unknown coupling constants and have to befixed phenomenologically.

In the present context only counter terms of chiral power q2contribute which account for quark mass insertions,L(2)φB = bD Tr ( ¯B{χ+, B}) + bF Tr ( ¯B[χ+, B]) + b0 Tr ( ¯BB) Tr (χ+)(2.4)with χ+ = u†χu† + uχ†u and χ = 2B0(M + S) where S denotes the nonet of externalscalar sources. As we will see later on, the constants bD, bF and b0 can be fixed fromthe knowledge of the baryon masses and the πN σ-term (or one of the KN σ-terms).The constant b0 can not be determined from the baryon mass spectrum alone since itcontributes to all octet members in the same way.

To this order in the chiral expansion,any baryon mass takes the formmB = m0 −124πF 2pαπBM 3π + αKB M 3K + αηBM 3η+ γDB bD + γFBbF −2b0(M 2π + 2M 2K) (2.5)The first term on the right hand side of eq. (2.5) is the average octet mass in the chirallimit, the second one comprises the Goldstone boson loop contributions and the thirdterm stems from the counter terms eq.(2.4).

Notice that the loop contribution is ultra-violet finite and non-analytic in the quark masses since M 3φ ∼m3/2q. The constants bD,bF and bs are therefore finite.

The numerical factors readαπN = 94(D + F)2,αKN = 12(5D2 −6DF + 9F 2),αηN = 14(D −3F)2;απΣ = D2 + 6F 2,αKΣ = 3(D2 + F 2),αηΣ = D2;απΛ = 3D2,αKΛ = D2 + 9F 2,αηΛ = D2;απΞ = 94(D −F)2,αKΞ = 12(5D2 + 6DF + 9F 2),αηΞ = 14(D + 3F)2;γDN = −4M 2K,γFN = 4M 2K −4M 2π;γDΣ = −4M 2π,γFΣ = 0;γDΛ = −163 M 2K + 43M 2π,γFΛ = 0;γDΞ = −4M 2K,γFΞ = −4M 2K + 4M 2π. (2.6)At this order, the deviation from the Gell-Mann-Okubo formula reads143mΛ + mΣ −2mN −2mΞ= 3F 2 −D296πF 2pM 3π −4M 3K + 3M 3η= 3F 2 −D296πF 2pM 3π −4M 3K + 1√3(M 2K −M 2π)3/2(2.7)4

where in the second line we have used the GMO relation for the η-meson mass, whichis legitimate if one works at next-to-leading order.Further information on the scalar sector is given by the scalar form factors or σ-terms which measure the strength of the various matrix-elements mq ¯qq in the proton.One defines:σπN(t) = ˆm < p′|¯uu + ¯dd|p >σ(1)KN(t) = 12( ˆm + ms) < p′|¯uu + ¯ss|p >σ(2)KN(t) = 12( ˆm + ms) < p′| −¯uu + 2 ¯dd + ¯ss|p >(2.8)with t = (p′ −p)2 the invariant momentum transfer squared and ˆm = (mu + md)/2 theaverage light quark mass. At zero momentum transfer, the strange quark contributionto the nucleon mass is given byms < p|¯ss|p >=12 −M 2π4M 2K3σ(1)KN(0) + σ(2)KN(0)+12 −M 2KM 2πσπN(0)(2.9)making use of the leading order meson mass formulae M 2π = 2 ˆmB0 and M 2K = ( ˆm +ms)B0 which are sufficiently accurate to the order we are working.

The chiral expansionat next-to-leading order for the σ-terms readsσπN(0) =M 2π64πF 2p−4απNMπ −2αKNMK −43αηNMη−2M 2π(bD + bF + 2b0)(2.10a)σ(j)KN(0) = M 2K64πF 2p−2απNMπ −3ξ(j)K MK −103 αηNMη−2ξ(j)πη απηNM 2π + MπMη + M 2ηMπ + Mη+ 4M 2K(ξ(j)D bD + ξ(j)F bF −b0)(2.10b)for j = 1, 2 with coefficientsξ(1)K = 73D2 −2DF + 5F 2,ξ(2)K = 3(D −F)2,ξ(1)πη = 1,ξ(2)πη = −3,ξ(1)D = −1,ξ(2)D = 0,ξ(1)F= 0,ξ(2)F= 1;απηN = 13(D + F)(3F −D). (2.10c)This completely determines the scalar sector at next-to-leading order.

Note that the πNσ-term is given as σπN(0) = ˆm (∂mN/∂ˆm) according to the Feynman-Hellman theorem.The shifts of the σ-terms from t = 0 to the respective Cheng-Dashen points do notinvolve any contact terms,σπN(2M 2π) −σπN(0) = M 2π64πF 2p43απN Mπ+ 23αKNM 2π −M 2K√2Mπln√2MK + Mπ√2MK −Mπ+ MK+ 49αηNM 2π −M 2η√2Mπln√2Mη −Mπ√2Mη −Mπ+ Mη(2.11a)5

σ(j)KN(2M 2K) −σ(j)KN(0) =M 2K128πF 2p43απNM 2K −M 2π√2MKln MK +√2MπMK −√2Mπ+ iπ+ Mπ+ 209 αηNM 2K −M 2η√2MKln√2Mη + MK√2Mη −MK+ Mη+ 2ξ(j)K MK+ ξ(j)πη απηN2M 2K −M 2π −M 2η√2MKln√2MK + Mπ + MηMπ + Mη −√2MK+ 2M 2π + M 2ηMπ + Mη(2.11b)Notice that the shifts of the two KN σ-terms acquire an imaginary part since the pionloop has a branch cut starting at t = 4M 2π which is below the kaon Cheng -Dashenpoint t = 2M 2K*. In the limit of large kaon and eta mass the result eq.

(2.11a) agrees,evidently, with the ancient calculation of Pagels and Pardee [20] once one accounts forthe numerical error of a factor 2 in that paper.Clearly, the σ-term shifts are non-analytic in the quark masses since they scale with the third power of the pseudoscalarmeson masses. Our strategy will be the following: We use the empirically known baryonmasses and the recently determined value of σπN(0) [13] to fix the unknown parametersm0, bD, bF and b0.

This allows us to predict the two KN σ-terms σ(j)KN(0). The shiftsof the σ-terms are independent of this fit.

Before presenting results, let us discuss theinclusion of the low-lying spin-3/2 decuplet in the effective theory.II.2. Inclusion of the decuplet-fieldsThe low-lying decuplet is only separated by ∆= 231 MeV from the octet baryons,which is just 53Mπ and considerably smaller than the kaon or eta mass.

One thereforeexpects the excitations of these resonances to play an important role even at low energies.This is also backed by phenomenological models of the nucleon in which the ∆(1232)excitations play an important role. In the meson sector, the first resonances are thevector mesons ρ and ω at about 800 MeV, i.e.

they are considerably heavier than theGoldstone bosons.It was therefore argued by Jenkins and Manohar [7] to includethe spin-3/2 decuplet in the effective theory from the start. Denote by T µ a Rarita-Schwinger fields in the heavy mass formulation satisfying v · T = 0.The effectivelagrangian of the spin-3/2 fields at lowest order readsLφBT = −i ¯T µ v · D Tµ + ∆¯T µTµ + C2 ( ¯T µuµB + ¯BuµT µ) .

(2.12)where we have suppressed the flavor SU(3) indices. Notice that there is a remainingmass dependence which comes from the average decuplet-octet splitting ∆which doesnot vanish in the chiral limit.

The constant C is fixed from the decay ∆→Nπ or the* Since we choose the GMO value for the η mass, Mπ +Mη >√2MK, the πη loop doesnot contribute to the imaginary part in eq.(2.11b). For the physical value of the η massthis contribution is tiny compared to the pion loop.6

average of some strong decuplet decays. The decuplet propagator carries the informationabout the mass splitting ∆and readsiPµνv · l −∆+ iǫ(2.13)withPµν = vµvν −gµν −4d −3d −1SµSν(2.13a)in d dimensions.

The projector obviously satisfies the constraints vµPµν = Pµνvν = 0and P µµ = −2. The appearance of the mass splitting ∆spoils the exact one-to onecorrespondence between the loop and low-energy expansion.Two scales Fp and ∆which are both non-vanishing in the chiral limit enter the loop calculations and theycan combine in the form (∆/Fp)2.

The breakdown of the consistent chiral countingin the presence of the decuplet is seen in the loop contribution to the baryon mass.The loop diagrams with intermediate decuplets states which naively count as order q4renormalize the average octet baryon mass even in the chiral limit by an infinite amount.Therefore one has to add a counter term of chiral power q0 to keep the value m0 fixedδL(0)φB = −δm0 Tr ( ¯BB)δm0 = 103C2∆3F 2pL +116π2ln 2∆λ −56L = λd−416π21d −4 + 12(γE −ln 4π −1)(2.14)with λ the scale introduced in dimensional regularization and γE = 0.577215... the Euler-Mascheroni constant. This mass shift is similar to the one in the relativistic version ofpion-nucleon CHPT, where the non-vanishing nucleon mass in the chiral limit leads tosimilar complications.Let us now turn to the baryon masses.

The inclusion of the decuplet fields has twoeffects on the mass formulae eq.(2.5). First there is an infinite loop contribution withdecuplet intermediate states and, second, an infinite renormalization of the order q2 ofthe low-energy constants bD, bF and b0.

Indeed this divergent mass shift due to decupletloops has not been treated consistently before. To account for it, we give the followingrenormalization prescription for bD, bF and b0bD = brD(λ) −∆C22F 2pLbF = brF (λ) + 5∆C212F 2pLb0 = br0(λ) + 7∆C26F 2pL(2.15)7

where the finite pieces brD,F,0(λ) will be determined by our fitting procedure (see below).Therefore the decuplet contributions to the octet masses can be written in the formδmB =C224π2F 2pβπBH(Mπ) + βKB H(MK) + βηBH(Mη)(2.16)with coefficientsβπN = 4,βKN = 1,βηN = 0;βπΣ = 23,βKΣ = 103 ,βηΣ = 1;βπΛ = 3,βKΛ = 2,βηΛ = 0;βπΞ = 1,βKΞ = 3,βηΞ = 1. (2.16a)andH(Mφ) = ∆3 ln 2∆Mφ+ ∆M 2φ32 ln Mφλ−1−(∆2 −M 2φ)3/2 ln ∆Mφ+s∆2M 2φ−1; Mφ < ∆H(Mφ) = ∆3 ln 2∆Mφ+ ∆M 2φ32 ln Mφλ−1−(M 2φ −∆2)3/2 arccos ∆Mφ;Mφ > ∆.

(2.16b)It is instructive to expand H(Mφ) for small MφH(Mφ) = 34∆M 2φ2 ln 2∆λ −1+3M 4φ32∆4 ln Mφ2∆−3+ . .

. (2.17)This shows that the leading contribution of the diagrams with intermediate decupletstates is of order M 2φ, which means linear and therefore analytic in the quark masses.This again demonstrates the problems with the chiral power counting in the presence ofa second non-vanishing scale ∆.

However, this contribution has no physical effect sinceit can be absorbed in the renormalized values of brD,F,0(λ). So the first non-trivial effectof the decuplet states on the baryon masses appears at order q4, which means beyondnext-to-leading order.

This is in agreement with the decoupling theorem [21]. Clearly,at this order there are many other contributions.

We will come back to this point lateron. The decuplet contribution to the GMO deviation reads143δmΛ + δmΣ −2δmN −2δmΞ=C2288π2F 2p−H(Mπ) + 4H(MK) −3H(Mη)(2.18)Notice that despite the appearence of the renormalization scale λ in the various H(Mφ),the right hand side of eq.

(2.18) is indeed scale independent due to the GMO relation forthe η-mass.8

Similarly, the decuplet contributes to the σ-terms at t = 0 are given byδσπN(0) = M 2πC264π2F 2p8 ˜H(Mπ) + ˜H(MK)δσ(1)KN(0) = M 2KC264π2F 2p4 ˜H(Mπ) + 43˜H(MK)δσ(2)KN(0) = M 2KC264π2F 2p4 ˜H(Mπ) + 2 ˜H(MK)(2.19)with˜H(Mφ) = ∆2 ln Mφλ −1+ 2q∆2 −M 2φ ln ∆Mφ+s∆2M 2φ−1;Mφ < ∆˜H(Mφ) = ∆2 ln Mφλ −1−2qM 2φ −∆2 arccos ∆Mφ;Mφ > ∆. (2.19a)The contribution of the decuplet to the σ-term shifts can be most economically repre-sented as a dispersion integral, the appropriate imaginary parts are collected in appendixA.

However, for the later discussion, let us consider σπN(2M 2π) −σπN(0) in SU(2), i.e.retaining only N and ∆(1232) intermediate states. One finds in this caseσπN(2M 2π) −σπN(0) = 3g2AM 4π16π2F 2πZ ∞4M2πdtt3/2(t −2M 2π)π4 (t −2M 2π) −∆pt −4M 2π+ (t −2M 2π + 2∆2) arctanpt −4M 2π2∆(2.20)where we used C = 32gA, coming from the SU(4) relation between the πNN and πN∆coupling constants.

This completes the necessary formalism. It is obvious from thediscussion so far that the inclusion of the decuplet in baryon CHPT is an incompleteattempt since there are many other terms of order q4 and higher.

For example, Jenkinsand Manohar [8,16] have included tadpole diagrams with new vertices from L(2)φB. Theseare of order q4 and can give rise to non-analytic pieces like m2q ln mq.

In what follows,we will not consider such diagrams but rather assume that anything at order q4 ismodelled by the inclusion of the low-lying spin-3/2 baryons. This can, of course, notsubstitute a full scale q4 calculation including all terms at this order, but allows us tocritically examine the role of the decuplet fields since their contribution is unique.

Weare motivated by the many papers making use of the Jenkins-Manohar proposal andwant to see to what extent such an approximation is a good thing to do.9

III. RESULTSIn this section, we will first present results for the complete q3 calculation outlinedin section II and then proceed to add the decuplet.III.1.

Results at order O(q3)First, we must fix parameters. Throughout, we us Mπ = 138 MeV, MK = 495MeV and M 2η = (4M 2K −M 2π)/3 = (566 MeV)2 as given by the GMO relation forthe pseudoscalar mesons.

This is a consistent procedure since the differences to thephysical η mass only shows up at higher order. For the pseudoscalar decay constant,we can either use Fπ = 92.6 MeV or FK = 112 MeV.

Mostly, we use an average valueFp = (Fπ + FK)/2 ≃100 MeV. Since in all terms F 2p appears, we will vary Fp from Fπto FK to find out how sensitive the results are to this higher order effect (the differenceof Fπ and FK in the meson sector is of order q4).

Furthermore, we use F = 0.5 andD = 0.75, which leads to gA = 1.25. Two other sets of D and F values, the one ofBourquin et al.

[22], F = 0.477 and D = 0.756, and the central value of Jaffe andManohar [23], F = 0.47 and D = 0.81. The four unknowns, which are the three low-energy constants bD, bF and b0 and the average octet mass (in the chiral limit) m0 areobtained from a least square fit to the physical baryon masses (N, Σ, Λ, Ξ) and the valueof σπN(0) ≃45 MeV.

This allows to predict σ(1)KN(0) and σ(2)KN(0) and the much discussedmatrix element ms < p|¯ss|p >, i.e. the contribution of the strange quarks to the nucleonmass.

We also give the value of the GMO deviation (3mΛ +mΣ −2mN −2mΞ)/4, whichexperimentally is 6.5 MeV.The results of this complete O(q3) calculation are shown in table 1. The dependenceon the values of the D and F axial vector constants is rather weak, only in the case ofthe central values of ref.

[23] an accidental cancellation occurs (D2 ≃3F 2) which makesthe GMO deviation very small. In most of the other cases one gets roughly half of theempirical value.

However, notice that it is a very small number on the typical baryonmass scale of 1 GeV and can therefore not expected to be predicted accurately. Thestrangeness matrix element in most cases is negative and of reasonable magnitude ofabout 200 MeV.

Within the accuracy of the calculation, the KN σ-terms turn out tobeσ(1)KN(0) ≃200 ± 50 MeVσ(2)KN(0) ≃140 ± 40 MeV(3.1)which is comparable to the first order perturbation theory analysis having no strangequarks, σ(1)KN(0) = 205 MeV and σ(2)KN(0) = 63 MeV [24]. Clearly, if one varies the valueof σπN(0) by ±10 MeV, the results are rather different.

This shows up in a value of b0which changes from −0.62 to −0.88 GeV−1 which has quite a dramatic impact on theKN σ-terms and the value of ms < p|¯ss|p >. For our analysis, however, we take thecentral value of σπN(0) = 45 MeV [13] as given.

Clearly a more accurate determinationof this fundamental quantity would be very much needed. We also have performed acalculation with Mη = 549 MeV, the results are very close to the ones for Mη given by10

the GMO relation (for the same D, F and Fp) with the exception of the GMO deviationfor the baryon masses.The σ-term shifts are given by σπN(2M 2π) −σπN(0) = 7.4 MeV [6,20], which ishalf of the empirical value found in ref.[13]. We will come back to this point later on.Furthermore, one findsσ(1)KN(2M 2K) −σ(1)KN(0) = (271 + i 303) MeVσ(2)KN(2M 2K) −σ(2)KN(0) = (21 + i 303) MeV(3.2)whose real part can be estimated simply Re(σ(1)KN(2M 2K) −σ(1)KN(0)) ≃[σπN(2M 2π) −σπN(0)](MK/Mπ)3 = 7.4 · 42.2 MeV = 340 MeV.

The rather small real part in ∆σ(2)KNstems from the large negative contribution of the πη–loop which leads to strong cancel-lations. Notice the large imaginary parts in σ(j)KN(2M 2K) −σ(j)KN(0) due to the two–pioncut.III.2.

Results with inclusion of the decupletIn the case of adding the decuplet, we will first keep the value of the pseudoscalardecay constant Fp = 100 MeV fixed. For the mass splitting we use either ∆= 231 MeVor ∆= 293 MeV (from the N∆splitting) and the value of C is given to be 1.8 fromthe strong decay ∆→Nπ and C = 1.5 from an overall fit to the decuplet decays [18].The scale of dimensional regularization is chosen at λ = 1 GeV [6].

It plays of courseno role in the physical results, but it should be kept in mind that the scale dependentvalues of bD,F,0(λ) are given at this scale. For the values of D and F, we will use ourcentral ones (F = 0.5, D = 0.75) and also show results with the small values of ref.

[7],D = 0.56, F = 2D/3 together with C = 2D = 1.12 [16].In table 2, we show the results for the full decuplet contribution according toeqs. (2.16) and (2.19) for the baryon masses and σ-terms, respectively.

For compari-son, table 3 gives the results accounting only for the leading term at order q4 arisingfrom the decuplet intermediate states, making use of eq. (2.17) and the expanded formof eq.(2.19).

First, one notices that for physical values of the F and D constants, aninconsistent picture emerges. While the analysis of the πN σ-term leads one to believethat the strange quark contributes of the order of 15% to the proton mass, this is com-pletely different when the decuplet is included.

Comparison of table 2 and 3 shows thatthe bulk of this effect comes from the terms of order q4. The conclusion drawn from thisexercise is that simply taking the decuplet fields at order q4 is meaningless.

It still mightbe possible as argued in ref. [16] that despite a large contribution at order M 4K ∼q4, thechiral series might converge with the small parameter (MK/4πFπ)2 = 0.16.

However,it is clear that only a complete calculation at order q4 (and beyond) can give a definiteanswer to this question. Using the small values of F, D and C [7,16], we essentiallyrecover the results of Jenkins and Manohar.

The differences stem from the fact thatwe did not account for the tadpole diagrams with one insertion of L(2)φB, eq.(2.4). In11

the spirit of the previous remarks, this is consistent. It is interesting to note that theresults for σ(j)KN(0) are quite similar to the ones of the full q3 analysis if one uses thesmall values of F, D and C.Let us now consider the πN σ-term shift σπN(2M 2π) −σπN(0).

For the preferredchoice F = 0.5, D = 0.76 and C = 32gA based on the coupling constant relation gπN∆=3gπN/√2, we findσπN(2M 2π) −σπN(0) = 15 MeV(3.3)which agrees nicely with the empirical result of ref. [13].It is interesting to discussthis result.

While the leading non-analytic piece proportional to M 3π gives 7.4 MeV,the same amount comes from the the analogous diagram with a ∆intermediate state.The kaon and η loops add a meager 1.1 MeV, i.e. they are essentially negligible.

Thisresult agrees with the phenomenological analysis of Jameson et al.. We should stressthat the spectral distribution ImσπN(t)/t2 is much less pronounced around√t = 600MeV than in ref. [13] but has a longer tail, so the total result remains the same.

The∆-contribution mocks up the higher loop corrections of the dispersive analysis of Gasseret al. [13].

A similar phenomenon is also observed in the calculation of the nuclear forcesto order q4 in heavy baryon CHPT [26]. There, the intermediate range attraction comesfrom the uncorrelated two-pion exchange and some four-nucleon contact terms, whereasthe phenomenological wisdom is that correlated two-pion exchange (also with diagramsinvolving intermediate ∆’s) is at the origin of this effect.

For a more detailed discussion,see ref.[27]. Clearly, the result eq.

(3.3) should be considered a curious accident since thecorrection to the leading term and the latter are of the same magnitude. It remains to beseen how other q4 effects and higher order corrections not yet accounted for will modifyeq.(3.3).

It is, however, important to note the essential difference to the calculationof the baryon masses and of σ(j)KN(0). In σπN(2M 2π) −σπN(0), the heavy meson loopsare irrelevant, i.e.

it is an SU(2) statement. It therefore has a better chance to survivehigher order loop corrections since the expansion parameter is (Mπ/4πFπ)2 = 0.014.

Ofcourse, there are also extra contact terms which will have to be evaluated. Finally, letus notice that the KN σ-terms shifts are large and that for the small values of F, D andC [16] σπN(2M 2π) −σπN(0) is only 6.8 MeV.IV.SUMMARY AND OUTLOOKWe have investigated the scalar sector of three-flavor baryon chiral perturbationtheory.

The baryons were treated as very massive fields, which allows to eliminate thetroublesome mass term from the lowest order effective meson-baryon Lagrangian. Ourfindings can be summarized as follows:• At next-to-leading order, i.e.

order O(q3) in the chiral expansion, one has threefinite counterterms which amount to quark mass insertions. The respective low-energy constants are denoted bD, bF and b0.

Their values can be determined froma least square-fit to the baryon masses and the pion-nucleon σ-term at t = 0.12

This allows to predict the two kaon-nucleon σ-terms, σ(1,2)KN (0). The values given ineq.

(3.1) are not very different from the lowest order analysis. The shifts to the kaonCheng-Dashen point are complex with a large real and large imaginary part, thelatter being due to the two-pion cut.

These numbers are considerably larger thanthe ones estimated by Gensini [28] a decade ago. * At this order, there is a one-to-onecorrespondence between the meson loop and small momentum expansion.• We have then proceeded and added the low-lying spin-3/2 decuplet to the effec-tive theory.

We show that the new mass scale, which is the average octet-decupletsplitting, is non-vanishing in the chiral limit and thus induces an infinite renormal-ization of the baryon self-energies. This is analogous to the infinite mass shift in therelativistic formulation of baryon CHPT as spelled out by Gasser et al.[2].

The con-sistent power counting scheme is therefore not present any longer. Similarly, thereis also an infinite renormalization of the three low-energy constant from the O(q2)effective meson-baryon Lagrangian.

Dissecting the contributions from the diagramswith intermediate decuplet states, one finds to leading order self-energy contribu-tions which are proportional to the quark masses. However, these can be absordedentirely in the finite values of the renormalized low-energy constants brD,F,0.

Thefirst non-trivial effect of the decuplet states on the baryon masses appears at orderO(q4).• The numerical evaluation of the decuplet contributions to the baryon masses and σ-terms shows a strong dependence on the values of F, D and C, the latter one beingrelated to the strong decuplet-octet-meson couplings. The decuplet contributionsare large and for the physical values of F and D, one does not have a consistentpicture of the scalar sector of baryon CHPT.

We disagree with the conclusion ofrefs. [8,16,18] that small values of F and D lead to a consistent picture at this order.First, these values stem from an incomplete calculation and, second, at order O(q4)there are many other diagrams which we (and other authors) did not take intoaccount.

The diagrams with intermediate decuplet states contribute to all orders inq2, but dominantly at O(q4) as comparison of tables 2 and 3 reveals. We concludethat it is not sufficient to include the decuplet to get an accurate machinery forbaryon CHPT in the three flavor sector.• As an interesting by-product, we have found that intermediate ∆(1232) states give acontribution to the πN σ-term shift as large as the leading order result of 7.5 MeV,so in total one has 15 MeV in agreement with the result of ref.[13].

This is an SU(2)number not affected by large kaon and eta loop contributions. However, at orderO(q4) there are other contributions not considered here which might invalidate thisresult.

From this we conclude that it is mandatory to first understand in better* For an update on the various extractions of the KN σ–terms and a discussion of theseresults see ref. [29].13

detail the two-flavor sector of baryon CHPT before one can hope to have a well-controlled chiral expansion including also the strange quark.APPENDIX: IMAGINARY PARTS OF SCALAR FORM FACTORSHere, we give explicit formulae for the imaginary parts to one loop of the threeproton scalar form factors defined in eq. (2.8):ImσπN(t) =M 2π128F 2p√t3(D + F)2(t −2M 2π)θ(t −4M 2π)+ (53D2 −2DF + 3F 2)(t −2M 2K)θ(t −4M 2K) + (D3 −F)2(t −2M 2η)θ(t −4M 2η)+ 16C23π−∆pt −4M 2π + (t −2M 2π + 2∆2) arctanpt −4M 2π2∆θ(t −4M 2π)+ 2C23π−∆qt −4M 2K + (t −2M 2K + 2∆2) arctanpt −4M 2K2∆θ(t −4M 2K)(A.1)Imσ(j)N (t) =M 2K128F 2p√t3(D + F)2( t2 −M 2π)θ(t −4M 2π) + 5(D3 −F)2( t2 −M 2η)θ(t −4M 2η)+ ξ(j)πη (D + F)(F −D3 )(t −M 2π −M 2η)θ(t −(Mπ + Mη)2) + ξ(j)K (t −2M 2K)θ(t −4M 2K)+ 8C23π−∆pt −4M 2π + (t −2M 2π + 2∆2) arctanpt −4M 2π2∆θ(t −4M 2π)+ (2 + δ2j)4C29π−∆qt −4M 2K + (t −2M 2K + 2∆2) arctanpt −4M 2K2∆θ(t −4M 2K)(A.2)The shifts of the σ-terms from t = 0 to the respective Cheng-Dashen points are mosteconomically represented in the form of a once-subtracted dispersion relation.σπN(2M 2π) −σπN(0) = 2M 2ππZ ∞4M2πdt ImσπN(t)t(t −2M 2π)Reσ(j)KN(2M 2K) −σ(j)KN(0)= 2M 2KπPZ ∞4M2πdt Imσ(j)KN(t)t(t −2M 2K)(A.3)REFERENCES1.

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DFFpbDbFb0m0σ(1)KN(0) σ(2)KN(0) SME GMO[MeV] [GeV−1] [GeV−1] [GeV−1] [GeV] [MeV][MeV] [MeV] [MeV]0.750.501000.016-0.553-0.7500.965195.3143.9-2063.80.756 0.4771000.037-0.540-0.7530.958204.3146.0-1922.30.810.471000.065-0.558-0.7890.981189.0129.8-2220.10.750.5092.60.008-0.610-0.7881.014154.0117.4-2784.50.750.501120.027-0.483-0.7030.904245.8176.4-1173.0Table 1:Results of the complete O(q3) calculation. The values of D, F andFp are input.

GMO denotes the combination (3mΛ + mΣ −2mN −2mΞ)/4 ofthe octet baryon masses. SME stands for the matrix element ms < p|¯ss|p >.DF∆CbDbFb0m0σ(1)KN(0) σ(2)KN(0) SME GMO[MeV][GeV−1] [GeV−1] [GeV−1] [GeV] [MeV][MeV] [MeV] [MeV]0.75 0.502931.80.623-1.057-1.7441.330-39.4-111.5-66711.50.75 0.502931.50.438-0.903-1.4411.21832.3-33.4-5269.20.75 0.502311.80.642-1.072-1.7521.360-58.2-131.6-70412.00.75 0.502311.50.451-0.914-1.4461.23919.3-47.4-5519.50.56 2D/32932D0.273-0.596-1.0330.975214.3115.6-1925.10.56 2D/32312D0.281-0.602-1.0360.986207.0107.8-2065.3Table 2:Results of the calculation including the full decuplet intermediatestates.

The values of D, F, ∆and C are input.16

DF∆CbDbFb0m0σ(1)KN(0) σ(2)KN(0) SME GMO[MeV][GeV−1] [GeV−1] [GeV−1] [GeV] [MeV][MeV] [MeV] [MeV]0.75 0.502931.80.547-0.995-1.6961.22238.2-23.3-5137.00.75 0.502931.50.385-0.860-1.4071.14386.227.8-4196.00.75 0.502311.80.518-0.971-1.6731.18567.010.3-4554.80.75 0.502311.50.365-0.843-1.3911.117106.151.1-3794.50.56 2D/32932D0.244-0.572-1.0140.933244.3149.7-1323.40.56 2D/32312D0.233-0.563-1.0050.919255.4162.8-1102.5Table 3:Same as in table 2 but with the decuplet contributions expanded upto and including O(q4).17

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