We argue that the difficulties associated with the convergence properties of
conventional SU(3) chiral perturbation theory can be ameliorated by use of a
cutoff, which suppresses the model-dependent short distance effects in such
calculations.Comment: Talk given at Workshop on Quark-Hadron Degrees of Freedom, CSSM,
  Adelaide, March 199

Holstein, Barry R.

English

arXiv

We argue that the difficulties associated with the convergence properties of conventional SU(3) chiral perturbation theory can be ameliorated by use of a cutoff, which suppresses the model-dependent short distance effects in such calculations

Holstein, BR

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics1998Is SU(3) Chiral Perturbation Theory an EffectiveField Theory?BR HolsteinUniversity of Massachusetts - Amherst, holstein@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physical Sciences and Mathematics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationHolstein, BR, "Is SU(3) Chiral Perturbation Theory an Effective Field Theory?" (1998). Physics Department Faculty Publication Series.529.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/529arXiv:nucl-th/9806037v1  12 Jun 1998Is SU(3) Chiral Perturbation Theory an Effective Field Theory?Barry R. HolsteinDepartment of Physics and Astronomy, University of Massachusetts, Amherst,MA 01003, USAE-mail: holstein@phast.umass.eduWe argue that the difficulties associated with the convergence properties of con-ventional SU(3) chiral perturbation theory can be ameliorated by use of a cutoff,which suppresses the model-dependent short distance effects in such calculations.1 IntroductionTony Thomas presented us with a challenge in setting up this workshop—tomake the connection between quark and hadronic degrees of freedom. I haveaccepted this challenge and will go even further by attempting to make contactbetween the physics of Tony Thomas and that of Ulf Meissner! My point ofcontact will be the study of low energy baryon phyics, studied a number of yearsago by Tony and his collaborators in the context of the cloudy bag model andexamined in recent years by Ulf and his collaborators using the technique ofchiral perturbation theory.Let’s begin with some simple baryon phenomenology which was studied inthe 1960’s via the quark model, but which has now been updated via SU(3)chiral perturbative studies. In each case the simple quark model picture isreasonably successful but the addition of chiral loops, which are supposed tobe essentially model independent, destroys experimental agreement:i) Baryon masses, wherein the Gell-Mann-Okubo relation obtains at firstorder in quark mass and is well satisfield experimentally. Chiral loops,however, make significant—O(50-100%)—corrections to individual masses.1ii) Semileptonic hyperon decay, wherein a simple SU(3) representation ofthe axial couplings in terms of F,D couplings yields an excellent fit toexperiment. Chiral loops make O(30-50%) corrections to individual cou-plings and destroy this agreement.2iii) Nonleptonic hyperon decay, wherein a basic SU(3) fit to s-wave ampli-tudes provides an excellent representation of the experimental numbersin terms of f,d couplings. Chiral loops make O(30-50%) corrections toindividual terms and destroy this agreement. (The situation is somewhat1more confused in the case of the p-waves wherein a significant cancella-tion between pole terms exists at lowest order and the validity of thechiral expansion is suspect.3)iv) Magnetic moments, wherein an SU(3) representation (especially withthe strange quark mass accounted for) provides a reasonable fit to mea-sured quantities, but where loop corrections make O(50-90%) corrections,which invalidate the goodness of fit.4Of course, the disagreement brought about by such loop modifications can beeliminated by contributions from yet higher order terms in the chiral expansion.However, this gives one concern about the convergence properties of the series.Thus I would answer the question raised in my title—yes it is an effective fieldtheory but it is not an effective field theory!Below we shall make the case that this problem can be solved by usinga cutoff regularization in order to calculate chiral corrections rather than theconventional dimensional regularization. The important point here is thata significant component of the loop correction arises from meson propagationover distances small compared to the ∼ 1 fm size of a typical baryon. This pieceis not only large but is model dependent and unphysical—the representation ofthe physics in terms of point particles misrepresents the short distance physics.We shall develop a procedure whereby only the—model independent—longdistance component is retained and will show how this resolves the problemwith respect to the convergence of the chiral expansion.In the next section we shall make a somewhat detailed analysis of theproblem of baryon mass renormalization, and then in section III we will extendour discussion to include the other aspects of low energy baryon physics referredto above. A short summary appears in a concluding section IV.2 Baryon MassAn example of the difficulties of baryon chiral perturbation theory is providedby the analysis of baryon masses. Such masses have an expansion in the massesof the quarks (mq), or equivalently in terms of the pseudoscalar meson masses(mM )MB = M0 +Σq b¯qmq +Σq c¯qm3/2q +Σq d¯qm2q + ... (1)= M0 +ΣM bMm2M +ΣM cMm3M +ΣM dMm4M + ... (2)Here M0 is a common mass and bM and dM contain adustable parametersrepresenting terms in the effective Lagrangian. However, the non-analyticm3/2q2terms come from loop diagrams, and the coefficients are not adjustable—theyare given in terms of the baryon-meson coupling constants. The leading SU(3)breaking terms involving bM go back to Gell-Mann and Okubo5, the non-analytic corrections from one-loop diagrams, represented above by cM , werefirst calculated by Langacker and Pagels6, and the complete m4M corrections(including diagrams up to two loops) were calculated by Borasoy andMeissner1.The convergence difficulties in the expansion are demonstrated by the resultingfit for the nucleon mass where, in the same sequence, the different contributionsare given, in GeV, by1MN = 0.711 + 0.202− 0.272 + 0.298 + . . . (3)or, more dramatically for the Ξ,MΞ = 0.767 + 0.844− 0.890 + 0.600 + . . . (4)The non-analytic terms appear unavoidably large and the expansion has cer-tainly not converged at this order. The final fit also violates the Gell-Mann-Okubo relation by an amount that is five times larger than the experimentallyobserved value.To one-loop order, the explicit form of the contributions to the baryonmasses is given by7MN = Mˆ0 − 4m2KbD + 4(m2K −m2pi)bF + LNMΛ = Mˆ0 −43(4m2K −m2pi)bD + LΛMΣ = Mˆ0 − 4m2pibD + LΣMΞ = Mˆ0 − 4m2KbD − 4(m2K −m2pi)bF + LΞ (5)whereMˆ0 =M0 − 2(2m2K +m2pi)b0 (6)with M0, bD, bF and b0 as free parameters. (Note that M0 and b0 do not haveseparate effects, but only enter in the combination Mˆ0.) The ingredients LBcontain the nonanalytic contributions from loop diagrams, and have the formLi = −124πF 2pi∑jκjim3j (7)where κji are given in terms of the D and F baryon axial-vector couplings inref. 1. These non-analytic terms are quite large, having valuesLN = −0.31 GeV,3LΛ = −0.66 GeV,LΣ = −0.67 GeV,LΞ = −1.02 GeV. (8)using D = 0.806, F = 0.46 and Fpi = 93MeV. In particular, the Ξ mass shift isclearly unphysically large. It is not possible to obtain a reasonably convergentfit to the masses with such large non-analytic terms to this order.2.1 Cutoff RegularizationThe mass analysis is especially simple in the heavy baryon formalism8,9,10 usingdimensional regularization—all of the mass shifts are proportional to a singleintegral∫d4k(2π)4kikj(k0 − iǫ)(k2 −m2P + iǫ)= −iδijI(m2P )24πwithI(m2P ) = m3P (9)where mP is the mass of the Goldstone boson that is involved in the loop.Here we note a peculiarity of dimsional regulariztion—although the integralappears cubically divergent by power counting arguments, the dimensionallyregularized result is finite. There us also the disturbing feature that whilethe long-distance pion exchange component is most model-independent andbelievable, Eq. 9 emphasizes the contributions from the short distance K andη exchange by a factor of nearly 30! It is this feature which is responsiblefor the large loop corrections and the consequent problems with SU(3) chiralconvergence. This problem can be solved by use of a regularization schemewhich eliminates the large and model-dependent short distance effects. Whatone needs is to multiply the relevant loop integrals by a function which isunity for long distance propagation (small loop momenta) but which vanishesfor short distance (large loop momenta). The precise form of this function isunimportant, only that it is present.a For the purposes of this paper, we shallemploy a dipole cutoff which, when included in the integration in Eq. 9, yieldsI¯(m2P ) = −8πi∫d4k(2π)4k · k(k0 − iǫ)(k2 −m2P + iǫ)(Λ2Λ2 − k2)2=[Λ4(Λ2 −m2P )2(m3P − Λ3) +32Λ5Λ2 −m2P](10)aThere exist a few subtleties concerning the maintainance of various symmetries such aschiral invariance within a cutoff regularization. However, one can deal with these problemsby the addition of appropriate noncovariant terms to the perturbation expansion.114Table 1: Given are numerical values of the integral I(m2P) (Eq. 9) in GeV3 for various valuesof the cutoff Λ given in MeV.dim. Λ = 300 Λ = 400 Λ = 500 Λ = 600I¯pi 0.0025 0.0061 0.0157 0.0320 0.0567I¯K 0.1213 0.0024 0.0077 0.0183 0.0363I¯η 0.1815 0.0020 0.0069 0.0163 0.0329Table 2: Given (in GeV) are the nonanalytic contributions to baryon masses in dimensionalregularization and for various values of the cutoff parameter Λ in MeV.dim. Λ = 300 Λ = 400 Λ = 500 Λ = 600N -0.31 -0.04 -0.11 -0.22 -0.40Σ -0.67 -0.03 -0.08 -0.18 -0.34Λ -0.66 -0.03 -0.08 -0.18 -0.34Ξ -1.02 -0.02 -0.06 -0.15 -0.29In Table 1 is shown the size of this integral for various cutoff values of orderinverse baryon size. Note that unlike the case of dimensional regularization, the(long distance) pion contribution is now emphasized, as we desire. For baryonmasses, the previous formulae are unchanged except for the substitutionm3P →I¯(m2P ), and in Table 2 we list the resulting loop corrections Obviously, thereare no longer any large SU(3) violating loop pieces and the chiral expansion isnow under control.That we have not altered the chiral invariance via this procedure can byexamining the form of the one loop contribution in the limit Λ >> mPδMi = −124πF 2pi∑j(12Λ3 −12Λm2j +m3j + . . .) (11)Obviously the term in m3j is identical to that arising in conventional dimen-sional regularization, but more interesting are the contributions proportionalto Λ3 and to Λm2P . The piece cubic in Λ has the formδMΛ3i = −Λ248πF 2pi∑jκji (12)and is independent of baryon type—it may be absorbed into a renormalizationof M0—M r0 =M0 − (5d2A + 9f2A)Λ348πF 2pi(13)5On the other hand the terms linear in ΛδMΛi =Λ48πF 2pi∑jκjim2j (14)must be able to be absorbed into renormalizations of the coefficients involvingmq, and indeed this is found to be the case—one verifies that12drm = = dm −3f2A − d2A128πF 2piΛf rm = fm −5dAfA192πF 2piΛbr0 = b0 −13d2A + 9f2A576πF 2piΛ (15)That this renormalization can occur involves a highly constrained set of con-ditions and the fact that they are satisfied is a significant verification of thechiral invariance of the cutoff procedure. Of course, once one has defined renor-malized coefficients, since they are merely phenomenological parameters whichmust be determined empirically, the procedure is identical to the results of theusual dimensionally regularized technique when the masses are smaller thanthe cutoff.3 Additional ApplicationsHaving seen how to perform the long-distance regularization in the case ofthe mass renormailization, it is now straightforward to carry through a similaranalyis in the other cases of low energy baryon phenomenology mentionedabove.13 In the case of the axial couplings and the S-wave hyperon decay therelevant heavy baryon integral is of the form∫d4k(2π)4kikj(k0 − iǫ)2(k2 −m2 + iǫ)= −iδijJ(m2)16π2(16)In dimensional regularization the integral has the valueJdim−reg(m2) = m2 lnm2µ2(17)while the cutoff version is given by∫d4k(2π)4kikj(k0 − iǫ)2(k2 −m2 + iǫ)(Λ2Λ2 − k2)2 = −iδijJΛ(m2)16π2. (18)6HereJΛ(m2) =Λ4m2(Λ2 −m2)2lnm2Λ2+Λ4Λ2 −m2(19)and has the form m2 lnm2 in dimensional regularization. However, using longdistance regularization the short distance effects are under control and the∼ (50%) corrections from loop diagrams are reduced substantially, allowingagreement with experimental findings to be reproduced without large nextorder counterterms. For both axial couplings and hyperon decay it can beverified that chiral invariance is maintained in the limit that m2p << Λ2 viathe renormalizationsdrA = dA −32dA(3d2A + 5f2A + 1)Λ216π2F 2pif rA = fA −16fA(25d2A + 63f2A + 9)Λ216π2F 2pi(20)for f, d axial couplings anddrw = dw −12[dw(1 + 13d2A + 9f2A) + 18fwdAfA]Λ216π2F 2pif rw = fw −12[fw(1 + 5d2A + 9f2A) + 10dwdAfA]Λ216π2F 2pi(21)in the case of hyperon decays. Likewise one can verify that the fit to semilep-tonic and to nonleptonic hyperon decay amplitudes is quite satisfactory andthat agreement with the Lee-Sugawara relation is restored for the latter.Finally, in the case of the magnetic moments, the relevant heavy quarkintegral is∫d4k(2π)4kikj(k0 − iǫ)(k2 −m2 + iǫ)2= −iδijK(m2)16π(22)The dimensionally regularized form is given by Kdim−reg(m2) = m. Onceagain, the integral shows no sign of its true linear divergence, and grows atlarge values of m, indicating short distance dominance at large m. The use ofthe dipole cutoff yieldsK(m2) = −13Λ41(Λ +m)3, (23)7In this case use of the long-distance regularized version of the loop integral alsorestores agreement between the experimental magnetic moments and their one-loop corrected values and chiral invariance is maintained in the casem2P << Λ2via the renormalizations of the lowest order parameters fµ, dµ viadrµ = dµ +M0Λ4πF 2pidAfAf rµ = fµ +M0Λ24πF 2pi(53d2A + 3f2A) (24)Thus in each case we are able include chiral loop corrections without destroyingeither chiral invariance or experimental agreement.4 ConclusionsWe have argued that a significant component of the poor convergence seenin previous calculations in SU(3) baryon chiral perturbation theory is due tothe inclusion of large and spurious short-distance contributions when loop pro-cesses are regularized dimensionally. The use of a momentum space cutoffkeeps only the long distance portion of the loops and leads to a greatly im-proved behavior. Indeed although we have formulated our discussion in termsof merely a different sort of regularization procedure, it is interesting to notethat our results are quite consistent with the sort of SU(3) breaking effectsfound in chiral confinement models such as the cloudy bag, when the effects ofkaon and/or eta loops is isolated.14One might ask why baryon chiral perturbation theory has this problemwhile mesonic chiral theories do not. (Most applications in mesons work per-fectly well using dimensional regularization.) At first sight one might arguethat the separation scale in baryons corresponds to lower energies because thephysical size of baryons is larger than mesons. While this is a true statement,it does not really answer the question, since the baryon problem surfaces en-tirely within the point particle theory. For some reason, given the same mesonmasses, the loop corrections are larger in the baryonic point particle theorycompared to a mesonic point particle theory. This feature can perhaps beblamed on the baryon propagator in the loop integral which, being linear in themomentum, suppresses high momentum contributions less than a correspond-ing quadratic mesonic propagator. However, the existence of the problem isbeyond doubt, given the troubles discussed in the introduction. Fortunately,we do not as a consequence have to abandon all such chiral calculations—arevised regularization scheme seems capable of resolving the dilemma.8The simplicity that underlies baryon physics is more evident when chiralloops are calculated with a long-distance regularization. In this context, wehope that baryon chiral perturbation theory will become more phenomenolog-ically useful.AcknowledgmentsIt is a pleasure to acknowledge support of the Alexander von Humboldt Foun-dation as well as the warm hospitality of the Institut fu¨r Kernphysik at Forchungszen-trum Ju¨lich. This work was also supported in part by the National ScienceFoundation.References1. B. Borasoy and U. G. Meissner, Phys. Lett. B365, 285 (1996); Ann.Phys. (NY) 254, 192 (1997).2. J. Bijnens, H. Sonoda, and M.B. Wise, Nucl. Phys. B261, 185 (1985);J.F. Donoghue and B.R. Holstein, Phys. Lett. 160B, 173 (1985); E.Jenkins and A. Manohar, Phys. Lett. 259B, 353 (1991).3. E. Jenkins and A. Manohar, ref. 4; B. Borasoy and B.R. Holstein, UMasspreprint (1998).4. U-G. Meissner and S. Steininger, Nucl. Phys. B499, 349 (1997).5. M. Gell-Mann in M. Gell-Mann and Y. Ne’eman, the Eightfold Way,Benjamin, New York (1962) and Phys. Rev. 125, 1067 (1962); S. Okubo,Prog. Theo. Phys. 27, 949 (1962).6. P. Langacker and H. Pagels, Phys. Rev. D8, 4595 (1975).7. V. Bernard, N. Kaiser, and U.-G. Meissner, Int. J. Mod. Phys. E4, 193(1995).8. H. Georgi, Phys. Lett. B240, 447 (1990).9. N. Isgur and M. Wise, Phys. Lett. B232, 113 (1989).10. E. Jenkins and A.V. Manohar, Phys. Lett. B255, 558 (1991).11. I. Gerstein, R. Jackiw, B.W. Lee, and S. Weinberg, Phys. Rev. D3, 2486(1971).12. J.F. Donoghue and B.R. Holstein, hep-ph/9803312.13. J.F. Donoghue, B.R. Holstein, and B.Borasoy, to appear.14. See, e.g. T. Yamaguchi et al., Nucl. Phys. A500, 429 (1989); K.Kubodera et al., Nucl. Phys. A439, 695 (1985); S. Theberge et al.,Phys. Rev. D22, 2838 (1980); A.W. Thomas, J. Phys G7, L283 (1981);R.E. Stuckey and M.C. Birse, J. Phys. G23, 29 (1997).9

Is SU(3) Chiral Perturbation Theory an Effective Field Theory?

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Is SU(3) Chiral Perturbation Theory an Effective Field Theory?

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