After a brief discussion of how chiral dynamics has evolved from the
``universal V-A theory of weak interactions'', we present some evidence that
symmetry breaking for the vector meson multiplet is not simpler than but rather
analogous to that for the pseudoscalar multiplet. This provides a motivation
for speculating on how to extend in a systematic way the chiral perturbation
theory program to include vectors.Comment: Contribution to Marshak Memorial Volume, edited by E.C.G. Sudarshan,
  to be published by World Scientific, Singapore 1994, PREPRINT SU-4240-57

Schechter, J.

English

arXiv

After a brief discussion of how chiral dynamics has evolved from the ``universal V-A theory of weak interactions\u27\u27, we present some evidence that symmetry breaking for the vector meson multiplet is not simpler than but rather analogous to that for the pseudoscalar multiplet. This provides a motivation for speculating on how to extend in a systematic way the chiral perturbation theory program to include vectors

Schechter, Joseph

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 4-13-1994 Possible Extension of the Chiral Perturbation Theory Program Joseph Schechter Department of Physics, Syracuse University, Syracuse, NY Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Schechter, Joseph, "Possible Extension of the Chiral Perturbation Theory Program" (1994). Physics. 316. https://surface.syr.edu/phy/316 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:hep-ph/9404265v1  13 Apr 1994POSSIBLE EXTENSION OF THE CHIRALPERTURBATION THEORY PROGRAMJ. SCHECHTERDepartment of Physics, Syracuse University,Syracuse, NY 13244-1130, USAABSTRACTAfter a brief discussion of how chiral dynamics has evolved from the “universal V-A theory of weak interactions”, we present some evidence that symmetry breakingfor the vector meson multiplet is not simpler than but rather analogous to thatfor the pseudoscalar multiplet. This provides a motivation for speculating on howto extend in a systematic way the chiral perturbation theory program to includevectors.1. IntroductionI would like to dedicate this paper to the memory of Professor Robert Mar-shak. As an ex-graduate student at Rochester I am grateful to him for establishingan intellectually stimulating and supportive Particle Physics group there. His en-thusiasm for research and down-to-earth attitude were much appreciated by all ofus.Out of a career filled with many achievements in physics, Professor Marshak’schief one was the deduction, together with E.C.G. Sudarshan, of the “V-A” formof the weak interaction.1 On the one hand, this theory provided a basis for under-standing a wealth of experimental data. Equally important, it indicated that therelevant degrees of freedom of the observed “material” particles in nature were notthe Dirac spinor fields but rather their left and right chiral projections. The leftprojections appeared in the weak interactions while both left and right were neededfor the strong and electromagnetic interactions.The most evident application of this idea to the strong interactions of the lowlying - i.e. pseudoscalar meson - states requires us to increase the size of the “flavor”symmetry multiplet. Instead of treating the 0− mesons as belonging to a 3×3 matrixΦ which transforms as Φ → UΦU−1, U †U = 1 under the approximate symmetrygroup of the light quarks, we associate a 3× 3 matrix S of scalar mesons with Φ toform an object M = S + iΦ which transforms as M → ULMU†R under separate leftprojected and right projected unitary groups.Some consequences of this “linear” chiral symmetry approach were discussedby Prof. Marshak and collaborators in Ref. 2. However it turns out that anotherimportant physical ingredient is required. The vacuum of the strong interactiontheory is not (even in the limit of massless light quarks) invariant under the separateUL and UR transformations, although the Lagrangian is invariant. As Nambu3explained to the world, this implies that the pseudoscalar and scalar masses aredrastically split from each other, the pseudoscalar masses being forced to becomezero in the massless quark limit. This situation can be neatly handled by making thepolar decomposition M = HU , H = H†, U † = U−1 and “ freezing out” the scalarfield part H by setting H = (Fπ/2)1. The freezing out corresponds to sendingthe scalar masses to infinity while Fπ ≃ 0.132 GeV is the pion decay constant.The name of this basic constant of strong interaction physics betrays its origin inthe weak interactions. Note that U → ULUU†R under chiral transformations. Uis a function of the ordinary pseudoscalar field matrix φ (which is simply relatedto Φ and S by a “point transformation”) and one may conveniently set4 U =exp(2iφ/Fπ). φ evidently behaves non-linearly under chiral transformations. Havingisolated the proper degrees of freedom it remains only to note that the simplestinvariant Lagrangian density formed with U ,L = −F 2π8Tr (∂µU∂µU†) + ... , (1.1)provides an accurate representation of QCD at very low energies (where pertur-bation theory fails). Indeed, many of the “current algebra” theorems, carefullydiscussed in the treatise “Theory of Weak Interactions in Particle Physics”5 can beobtained using (1.1) in a simple way.It seems remarkable that, although many years have passed since the intro-duction of the “V-A” theory, chiral symmetry is still an extremely active field ofresearch. In the following, I shall very briefly describe some of the progress inthe field and shall present some speculations related to further improvement of theapproach.2. Going Beyond Very Low EnergiesModifying the simple model (1.1) in an attempt to describe QCD over thefull traditional low energy range (say up to about 1 GeV) has been the task of ageneration. And it has still not been definitively accomplished. It is, of course,necessary to recognize that the chiral symmetry is broken by the light quark massterms6 in the fundamental QCD Lagrangian,Lmass = −m̂qMq, (2.1)where q is the column vector of up, down and strange quark fields, m̂ = (mu+md)/2and M is a dimensionless, diagonal matrix which can be expanded as follows:M = yλ3 + T + xS, (2.2)with λ3 = diag (1,−1, 0), T = diag(1, 1, 0) and S = diag(0, 0, 1). x and y are thequark mass ratios:x =msm̂, y = −12(md − mum̂). (2.3)The minimal term at the effective Lagrangian level which can mock up (2.1) is:LSB = δ′ Tr [M(U + U† − 2)], (2.4)where δ′ is a numerical constant. Fitting (1.1) + (2.4) to the experimental pseu-doscalar mass spectrum yields the standard determination of the quark mass ratiosx and y.Now, there are two (in principle) straightforward ways in which one might ex-tend up in energy the description of low energy physics obtained by using (1.1) +(2.4) at tree level:i. Including loop diagrams7First compute the one loop corrections using (1.1) + (2.4) and keep terms quarticin the momenta. To eliminate the divergences add chiral invariant “counter-terms”quartic in derivatives. For this purpose count each power of M as two deriva-tives (The terms involving M will be chiral covariant rather than invariant). Thisscheme can be continued to higher orders in momenta. However since the startingLagrangian is non-renormalizable there will be an infinite number of counterterms.This is not a worry in practice as we only expect to use the method up to a lowfinite order.ii. Including additional physical particlesEven without considering any detailed models it is obvious that if we want tohave an effective theory valid up to some energy we should include the physical par-ticles whose masses lie in that energy range. How else would we get the right polesin the S-matrix at tree level? This point of view is buttressed by the Venezianomodel8 which shows how to get good high energy behavior by adding pole contri-butions from a realistic looking (infinite) set of particles. It also is supported bythe leading large Nc approximation9 to QCD in which one should keep only thetree diagrams involving all the physical mesonic states. The baryonic states can beobtained from this effective meson Lagrangian as Skyrme solitons.10Which of these two approaches is superior, or is that even the right question toask? The first approach, which has been systematized by Gasser and Leutwyler7 isknown as “chiral perturbation theory” (CPT). In practice it essentially amounts tomaking a complete list of chiral invariant and covariant counterterms, each with anunknown coupling constant. There are about ten of these at the quartic derivativeorder. With a convenient choice of the renormalization point, the contributionsfrom the loop diagrams themselves (the “chiral logs”) are typically negligible. Itseems that the CPT approach is both solid and useful for improving the descriptionof (1.1) + (2.4) at very low energies, say up to about 500 MeV in π − π scattering.But going beyond this region forces us to face the enormous peak representingthe ρ meson. It is hard to avoid including it ( and all its SU(3) partners) if wewant a realistic description. In fact it has been found11 that many of the valuesof the counterterms can be numerically understood just with vector meson poledominance. These arguments strongly suggest the suitability of a model of typeii. Should we then give up the CPT program? Here, we would like to argue for amodel combining the two approaches.Immediately, there may be a number of objections. First it seems to be discour-aging to have to include every particle multiplet with the same kind of microscopicdetail that has been applied to the pseudoscalar multiplet. In response, we maynote that a natural continuation of the present CPT program would be simply toinclude at first just the vector meson multiplet. This provides a “clean break” in thesense of retaining just the low lying S-wave quark anti-quark states in the model.It would provide coverage of the region up to around 1 GeV. The lessons learnedin such a generalization may show us how to economically include the still moremassive states.Another possible objection is related to the general feeling that, since the low ly-ing pseudoscalars are approximate Nambu-Goldstone bosons, they should be treateddifferently from the other multiplets. This apparent objection would appear to bestrengthened by the folk wisdom that while one can explain the properties of a“normal” multiplet like the vectors with a minimal deviation from SU(3) symme-try, much more elaboration, via the inclusion of many arcane symmetry breakingterms on the CPT list of counterterms, is required for the pseudoscalar multiplet.Here, we would like to point out that this folk wisdom does not seem to hold. Ina recent paper,12 which should be consulted for more explanations and references,it is shown that exactly analogous symmetry breaking terms are required for boththe vector and pseudoscalar multiplets at the Okubo-Zweig-Iizuka (OZI) rule con-serving level. This suggests that all multiplets be treated in the same way. The(approximate) spontaneous breakdown of chiral symmetry is certainly a crucial fea-ture but it would appear to affect every multiplet, presumably via the “pion cloud”intrinsic to each particle. Of course, the vector and higher multiplets have non-zeromasses in the chiral limit. One might imagine that the appropriate “large” scalewith which to compare the effects of the perturbation (m̂M) is (α′)−1/2 ≃ 1.06 GeV,α′ being the universal Regge slope parameter. The best choice of renormalizationpoint for the loop diagrams requires investigation.In Ref. 12 the vector meson nonet field ρµ(x) is introduced, for convenience, interms of auxiliary, linearly transforming “gauge fields” ALµ and ARµ by13:ALµ = ξρµξ† +ig̃ξ∂µξ†, ARµ = ξ†ρµξ +ig̃ξ†∂µξ, (2.5)where ξ = U1/2 and g̃ is related to the ρφφ coupling constant. ρµ transformsnon-linearly in this description, which corresponds to eliminating the axial vectormesons in analogy to the elimination of the scalar mesons which led to the nonlinearly transforming pseudoscalar multiplet inside U . Note that both the scalarsand axials are P -wave qq bound states so this truncation is conceptually consistent.Now we can play the CPT game, constructing all chiral invariants and covariantsup to a certain order in derivatives. Both vector and pseudoscalar fields wouldbe included. This should necessitate readjusting the coefficients of those termscontaining only pseudoscalars which were dominated by vector meson exchange.To start to explore this rather complicated scheme we will, first of all, specializeto the symmetry breaking terms and try to fit all the mass differences, includingthose which are isospin violating. We will also fit the meson decay constants andV → φφ decay widths. As a physical approximation we shall demand that (withone significant exception) the symmetry breaking terms be single traces in flavorspace and that all field matrices represent nonets. This is Okubo’s form14 of theOZI rule and appears to be respected by the existing CPT fit for the pseudoscalaronly symmetry breakers. The final approximation is to neglect the “chiral logs”.This also works in the existing CPT fits. Of course, these approximations can berelaxed in the future.Then the symmetry breaking terms which conserve the OZI rule are taken tobe (up to quartic order in derivatives):LSB = Tr {M[δ′(U + U† − 2) + α′(ALµUARµ + ARµU†ALµ)+β ′(∂µU∂µU†U + U †∂µU∂µU†) + γ′(F LµνUFRµν + FRµνU†F Lµν)]+λ′2[MU †MU † + MUMU − 2M2] + µ′(ALµMARµM)} (2.6)where F L,Rµν = ∂µAL,Rν − ∂νAL,Rµ − ig̃[AL,Rµ , AL,Rν ] while α′, β ′, γ′, δ′, λ′2, µ′ are con-stants to be determined. Notice that there are three analogous vector terms andpseudoscalar terms. Physically, each multiplet has a non-derivative and a derivativetype symmetry breaker proportional to M as well as a non-derivative term propor-tional to M2. It turns out that they are all required to fit the pseudoscalar andvector particle properties mentioned above.As explained in section III of Ref. 12 it is convenient to determine a suitablenumber of the physical quantities while holding the quark mass ratio x constant.Then there are three predictictions for each value of x. A best fit is obtained forthe quark mass ratios:x = 37 , y = −0.36. (2.7)If the γ′ term for the vectors were not present it would be very difficult to getreasonable predictions for the non-electromagnetic part of m(K∗0) − m(K∗+) andfor the width ratio Γ(K∗)/Γ(ρ). The effect of this derivative type symmetry breakingterm is to introduce non-trivial wave function renormalizations for the K∗ and φparticles. Similarly, the µ′ term improves the predictions of the mass and width ofthe φ meson in the present framework.For the pseudoscalars, it is well known that we can not restrict ourselves to justthe OZI rule conserving terms. The needed extra terms are discussed in sectionsII(c) and IV of Ref. 12. There it is shown that the minimal Lagrangian which cansolve the U(1) problem (with the aid of an auxiliary glueball field) can be modifiedby the addition of suitable symmetry breaking terms to give a reasonable descriptionof the η − η′ system. In particular, the old problem15 of too small η mass, whichhas more recently attracted some attention,16 was solved. A consequence of thisdiscussion is the presence of the OZI rule violating term ∝{Tr [M(U − U†)]}2. (2.8)On the other hand, there is no special reason to include a sizeable term of thetype{Tr [M(U + U†)]}2. (2.9)The inclusion of (2.8) but not (2.9), which is very natural in the approach men-tioned above, amounts to a practical resolution of the Kaplan-Manohar ambiguity.17Some discussion is given in section VII(c) of Ref. 12. We also have no special reasonto include the pseudoscalar OZI rule violating term of the type ∝Tr (∂µU∂µU†) Tr [M(U + U†)]. (2.10)This term is found to be very small in the usual CPT fit with pseudoscalarsonly. We have now accounted for all the terms depending upon M on the CPT listfor pseudoscalars. The only important OZI rule violating one was, as expected, theone associated with the U(1) problem and the η−η′ system. Thus our identificationof the most important symmetry breaking terms seems justified for our purpose ofmaking an appraisal of the validity of this model.Incidentally, we remark that the quark mass ratios in (2.7) are somewhat differ-ent from the usual18 ones x = 25.0± 2.5, y = −0.28± 0.03. With our determinationand a choice ms(1 GeV)=0.175 GeV we would have mu=3.2 MeV and md=7.9 MeV.To sum up, we have presented evidence that, apart from the special termsneeded to solve the U(1) problem, the treatment of symmetry breaking in the pseu-doscalar and vector multiplets involves exactly analogous terms. The implication isthat chiral perturbation theory might be extensible, as outlined, to the vectors tooin order to model low energy QCD up to around 1 GeV. Certainly a large numberof processes, including loop contributions, must be examined to fully test this idea.But the first step is to make a preliminary calculation by picking out the terms ex-pected to be most important and working at tree level. The treatment of symmetrybreaking discussed here in that manner provides an optimistic sign.3. AcknowledgmentsI would like to thank Anand Subbaraman and Herbert Weigel for a pleasantcollaboration.4. References1. E.C.G. Sudarshan and R.E. Marshak, Proceedings of the Padua conferenceon mesons and recently discovered particles, p. V-14 (1957). See also R.P.Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958) and J.J. Sakurai,Nuovo Cimento 7, 649 (1958).2. R.E. Marshak, N. Mukunda and S. Okubo, Phys. Rev 137 B698 (1965).3. Y. Nambu in Group Theoretical Concepts and Methods in Elementary ParticlePhysics, edited by F. Gürsey (Gordon and Breach, New York, 1964).4. K. Nishijima, Nuovo Cimento, 11 698 (1959); F. Gürsey, Ann. Phys. (N.Y.)12 91 (1961); J. Cronin, Phys. Rev. 161, 1483 (1967).5. R.E. Marshak, Riazuddin and C.P. Ryan, Theory of Weak Interactions inParticle Physics, Wiley-Interscience (1969).6. M. Gell-Mann, Phys. Rev. 125, 1067 (1962).7. L.-F. Li and H. Pagels, Phys. Rev. Lett. 26, 1204 (1971); S. Weinberg, Physica96A, 327 (1979); J. Gasser and L. Leutwyler, Nucl. Phys. 250, 465 (1985).8. G. Veneziano, Nuovo Cimento 57A, 190 (1968); See also L. Van Hove, Phys.Lett. 24B, 183 (1967).9. G. ’t Hooft, Nucl. Phys. B72, 461 (1974); E. Witten ibid, B160, 57 (1979).10. T. Skyrme, Proc. Roy. Soc. A262, 237 (1961).11. J. Donoghue, C. Ramirez and G. Valencia, Phys. Rev. D39, 1947 (1989); G.Ecker, J. Gasser, A.Pich and E. de Rafael, Nucl. Phys. B221, 311 (1989).12. J. Schechter, A. Subbaraman and H. Weigel, Phys. Rev. D48, 339 (1993).13. Ö. Kaymakcalan and J. Schechter, Phys. Rev. D31, 1109 (1985). An al-ternate equivalent approach is given by T. Fujiwara, T. Kugo, H. Terao, S.Uehara and K. Yamawaki, Prog. Theor. Phys. 73, 926 (1985). The equiva-lence is demonstrated in P. Jain, R. Johnson, U. Meissner, N.W. Park and J.Schechter, Phys. Rev. D37, 3252 (1988).14. S. Okubo, Phys. Lett. 5, 165 (1963).15. See Fig. 2 of V. Mirelli and J. Schechter, Phys. Rev. D15 1361 (1977).16. H. Georgi, Phys. Rev. D49, 1666 (1994); S. Peris , CERN preprint TH7109/93.17. D. Kaplan and A. Manohar, Phys. Rev. Lett. 56, 2004 (1986).18. J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982).

Possible Extension of the Chiral Perturbation Theory Program

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