The present status of low energy test of chiral invariance via chiral
perturbation theory is reviewed, both in the meson and baryon sectors, and
future prospects are discussed.Comment: Invited talk given at PANIC 96, Standard 8 page Latex fil

Holstein, Barry R.

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The present status of low energy test of chiral invariance via chiral perturbation theory is reviewed, both in the meson and baryon sectors, and future prospects are discussed

Holstein, BR

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University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics1996Low Energy Tests of Chiral Symmetry1BR Holsteinholstein@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, "Low Energy Tests of Chiral Symmetry1" (1996). Physics Department Faculty Publication Series. 526.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/526arXiv:hep-ph/9606228v1  4 Jun 1996Low Energy Tests of Chiral Symmetry1Barry R. HolsteinDepartment of Physics and AstronomyUniversity of MassachusettsAmherst, MA 01003AbstractThe present status of low energy tests of chiral invariance via chiral per-turbation theory is reviewed, both in the meson and baryon sectors, andfuture prospects are discussed.1 IntroductionWhen I was a student back in the 1960’s, the goal of particle and nuclear physicistswas to seek the fundamental laws of nature, and one of the requirements of sucha “fundamental” law was that it be renormalizable. Well now it’s 1996 and we’velearned a few things in these three decades. One is that nonrenormalizable effectivefield theories can be just as if not more useful than their renormalizable siblings incertain situations. One of these is the case of QCD, where we have what we feel isa correct model of nature. However, it is written in terms of the “wrong” degrees offreedom (quarks and gluons rather than hadrons) and is impossible to solve becauseof its strong coupling and inherent nonlinearity. Much more useful in the arena of lowenergy physics is an effective Lagrangian, which is written in terms of experimentaldegrees of freedom—mesons and baryons—and which encodes the symmetries of theunderlying QCD interaction—specifically for our purposes chiral symmetry, whichexists in the limit in which the quark mass can be taken as vanishing. This is aprogram which was begun in the 60’s with the effective two-derivative Lagrangian[1]L(2) = F2π4TrDµUDµU † +F 2π4Tr2B0m(U + U†) (1)which describes the interaction of the Goldstone fields φi, i = 1..8. Here Fπ = 92.4MeV is the pion decay constant, m is the quark mass matrix, B0 is a phenomenologicalconstant andU = exp(iFπ8∑j=1λjφj) (2)is a nonlinear function of the fields which transforms as LUR† under chiral rotations.When used at tree level this interaction is rather successful in predicting low energy1Invited talk given at PANIC 96, Williamsburg, VA1Coefficient Value OriginL1 0.65± 0.28 ππ scatteringL2 1.89± 0.26 andL3 −3.06± 0.02 Kℓ4 decayL5 2.3± 0.2 FK/FπL9 7.1± 0.3 < r2π >L10 −5.6± 0.3 π → eνγTable 1: Gasser-Leutwyler coefficients Li and the means of determination.interactions.[1] For example, expanding to fourth order in the fields one finds thewell-known Weinberg predictions for ππ scattering lengths[2]a00 =7mπ32πF 2πa20 = −mπ16πF 2π(3)which are confirmed by experiment. However, in order to go further and include loopeffects one must include additional four-derivative pieces into the effective Lagrangian,with ten phenomenological constants Li, i = 1...10 which can be determined experi-mentally as shown by Gasser and Leutwyler,[3] yielding values for these parametersas given in Table 1.2 MesonsAs mentioned above, the one loop chiral expansion has been carried out in the cae ofGoldstone boson interactions by many investigators. As emphasized by Weinberg,[4]this is basically an expansion in energy-momentum with a scale parameter Λχ ∼ 1GeV, so that one is entitled to quit at one loop provided that energies are smallcompared to this scale. It is this for this reason that this is called chiral perturbationtheory (χpt). Although ten seems at first like a large number of parameters, χpt isvery predictive and the extent to which these predictions are valid is at some levela probe of the validity of QCD itself. This subject has been extensively reviewedin many places[5] and there is in general very good agreement between predictedand measured quantities as shown in Table 2 The one area here where there is apossible problem has to do with the required relationship between the charged pionpolarizability απE and the axial structure function hA in radiative pion decay[8]απE =αhA√2Fπmπ(4)The chirally required value for the polarizability—2.8×10−4 fm3—is at variance withthe value found at Serpukov via radiative pion scattering[6] but not with that found2Reaction Quantity Theory Expt.γ → π+π− < r2π > (fm2) 0.44 (input) 0.44± 0.02γ → K+K− < r2K > (fm2) 0.44 0.34± 0.05π+ → e+νeγ hV (m−1π ) 0.027 0.029± 0.017hA/hV 0.46 (input) 0.46± 0.08K+ → e+νeγ (hA + hV )(m−1π ) 0.038 0.043± 0.003π+ → e+νee+e− rA/hV 2.6 2.6± 0.6γπ+ → γπ+ (αE + βM)(10−4) fm3 0 1.4± 3.1αE(10−4fm3) 2.8 6.8± 1.4[6]2.2± 1.1[7]K → πe+νe ξ = f−(0)/f+(0) -0.13 −0.20± 0.08λ+ (fm2) 0.067 0.065± 0.005λ0 (fm2) 0.040 0.050± 0.012Table 2: Comparison between chiral predictions and experimental values for param-eters in the Goldstone sector.at SLAC in γγ → π+π−.[7] This is clearly an area which deserves further study andwork in this regard is presently underway at DAΦNE, Fermilab and Mainz.3 BaryonsIn the case of pion-nucleon intereactions, chiral perturbative calculations can also beperformed. However, things are much less clean for reasons which will become clear.One begins as before with the simplest πN Lagrangian having chiral symmetryLπN = N¯(i 6D −mN + gA26uγ5)N (5)where Dµ is a covariant derivative, gA is a coupling constant to be determined, anduµ = iu†∇µUu† with u2 = U . Expanding to lowest order we findLπN = N¯(i 6∂ −mN − igA~τ · 6~Aγ5 − gAFπ~τ · 6∇~φ+ . . .)N (6)so that gA is to be identified with the axial coupling in neutron beta decay. Also wesee that chiral symmetry requires the Goldberger-Treiman relation between gA andthe πNN coupling constant[9]FπgπNN = mNgA (7)Using the best present values we have1201MeV = 92.4MeV× 13.0 vs. 939MeV× 1.26 = 1183MeV (8)3πN → πN Scattering lengths, σπNπN → ππN LET’s, ππ Scattering lengthsγN → γN Polarizabilities, DHG sum ruleγN → πN LET’sγ∗N → πN LET’s, gA(q2)γN → ππN Chiral loop effects in π0π0Table 3: Examples of nucleon reactions which have been examined via χpt.and the agreement to better than two percent strongs confirms the validity of chiralinvariance in the nucleon sector. A second probe in this arena arises from the fea-ture that the nucleon matrix element of the axial current also includes a pion polecontribution, leading to a prediction that in the muon capture process one requires[10]rP =gP (q2 = −0.9m2µ)gA(q2 = −0.9m2µ)=2mNmµm2π + 0.9m2µ= 7.0 (9)which can be experimentally checked. Present results arerP =7.4± 2.0[11] µ− capture 3He6.5± 2.4[12] µ− captureH10± 1[13] radiativeµ− capture H(10)Obviously the discrepancy in the case of the radiative capture needs to be furtherexplored.In order to go further one requres loop corrections, just as in the mesonic case. Aproblem arises here that for the nuclons one has an additional dimensionful paramter—mN—which is the same size as the chiral scale, which makes the entire renormalizationprocedure doubtful. This problem can be gotten around, however, by using so-calledheavy baryon methods, which are equivalent to the use of a Foldy-Wouthuysen trans-formation and which make a consistent power counting scheme possible. Of course,renormalization introduces new low energy constants (six, e.g. at O(p2)) but never-theless this program has been carried out, predominantly by the group of Bernard,Kaiser, and Meissner,[14] and applications are reported in many systems: One areaof particular interest here is that of pion photoproduction. In this case for chargedproduction the feature that the pion derivative in Eq. 6 is covariant leades to theKroll-Ruderman predictions for the E0+ (electric dipole) multipole at thresholdEth0+ ={ √2D(1− 32µ+O(µ2)) π+n−√2D(1− 12µ+O(µ2)) π−p (11)where D = egπNN/8πmN = 23.9 (×10−3/mπ) and µ ≡ mπ/mN . In this case theoryand experiment are in good agreement, as shown below However, the experimental4theory expt.Eth0+(π+n) 26.3 27.9± 0.5[15]28.8± 0.7[16]Eth0+(π−p) -31.3 −31.4± 1.3[15]−31.2± 1.2[17]Table 4: Threshold values of E0+ for charged pion photoproduction (×10−3/mπ).theory expt.E0+(π0p)(×10−3/mπ) -1.2 −1.31± 0.08[20]−1.3± 0.5± 0.6[21]P1/|~q|(π0p)(×GeV−2) 0.480 0.47± 0.01[20]0.41± 0.03[21]Table 5: Threshold parameters for neutral pion photoproduction.numbers quoted are from old emulsion measurements and resutls from the recent SALexperiment are eagerly awaited.In the case of neutral pion photoproduction, things are more interesting. In thiscase the venerable Low Energy Theorems (LET) for the threshold E0+ multipoleEth0+ ={ −D(µ− 12µ2(3 + κp) +O(µ3)) = −2.3 π0p−D(12µ2κn +O(µ3)) = +0.5 π0n (12)were shown in 1991 to be incorrect due inappropriate assymptions concerning analyticity.[18]New loop contributions at O(µ2)∆Eth0+ = −Dµ2(mN4Fπ)2 (13)are large and destroyed the agreement which appeared to exist between the originalLET predictions and preliminary π0p experiments at both Mainz and at Saclay. How-ever, there has been a good deal of recent activity. On the theoretical side, Bernardet al. have performed an analysis at O(p4).[18] In doing so they require the valuesof two counterterms. Estimating these via ∆, ρ, ω dominance, they predict a valuefor E0+ in good agreement with new experiments performed at both MAMI and SALThis may be somewhat accidental, as the convergence of the series appears slow—E0+ = C(1 − 1.26 + 0.59 + . . .). However, it has been pointed out that the P-wavecalculations do not suffer from this slow convergence, yielding what should be a verysolid prediction1|~q|P1 ≡ (M1+ −M1− + 3E1+)5theory expt.αpE 10.5 11.6± 0.6± 0.6[25]βpM 3.5 2.6∓ 0.6∓ 0.6[25]αnE 13.4 12.6± 1.5± 2.0[26]βnM 7.8 3.2∓ 1.5∓ 2.0[26]Table 6: Experimental values of electric and magnetic polarizabilities compared tochiral predictions at O(p4). (All ×10−4 fm3)=DMN[1 + κp − µ(1 + 12κp − g2πNN(10− 3π)48π) + . . .] = 0.480GeV−2 (14)There may be a remaining problem in the size of the E1+ multipole,[21] but thisinvolves a significant theoretical cancellation.A second arena of activity is that of Compton scattering. In this case recent precisemeasurements of both the proton and neutron polarizabilities have been performedyielding values as shown below. As can be observed, the electric polarizability ofthe neutron is comparable to but slightly larger than its proton counterpart. This isinteresting since a valence quark model cannot produce such a result, yielding insteada prediction[22]αpE − αnE =α3MN(< r2p > − < r2n >) ≈ 4.6(×10−4 fm3) (15)On the other hand a chiral expansion of the polarizability does not have this problem,starting off from equal values for the proton and neutron[23]αpE = αnE = 10βpM = 10βnM =e2g2A192π3F 2πMN(5π2µ+ . . .) = 12.4(×10−4 fm3) (16)A full O(p4) calculation of both electric and magnetic polarizabilities, with coun-terterms evaluated via resonance dominance yields very satisfactory results as shownTable 6, although again the convergence of the series is in doubt.[24] Similarly agood deal of work has been done in the case of polarized Compton scattering, bothexperimentally and theoretically, but we do not have the space to discuss this here.4 ConclusionsObviously in a short report such as this the discussion above can provide only a briefintroduction to the multitude of work which is presently underway. Additional areasof activity include6i) γp → π0π0p for which the significant near threshold cross section observed viathe TAPS group at Mainz is being confronted with the pion loop correctedamplitude. The loop correction is large but it is too early to tell whether itagrees with the experimental findings.ii) γ∗p → π0p for which k2 ≃ −0.1 GeV2 data from both NIKHEF and MAMIseems to contradict at least some of the chiral perturbative predictions. How-ever, this value of momentum transfer is probably above the range where onecan expect agreement, and we await the lower k2 data to be taken at Mainz.iii) πN → ππN for which previous analysis in order to extract the ππ scatteringlengths has utilized the Olsson-Turner parameterization, which is inconsistentwith a modern chiral analysis.iv) Kℓ4 measurements at DAΦNE and elsewhere should be able to resolve the ques-tion concerning the use of standard vs. generalized chiral perturbation theory.v) ~γ~p → γp measurements at CEBAF and elsewhere combined with precise reso-nance photoproduction data should shed light on the validity of the DHG sumrule.vi) on the theoretical side it is important to include effects of the ∆(1240) as abaryonic degree of freedom and not just as a heavy state which contributes toa counterterm. Work to this end is underway and should appear soon.Overall then I hope that I have been able to convey the sense that the area of lowenergy tests of the standard model via chiral perturbation theory is an active andexciting one, with plenty of work remaining to be done on both the theoretical andexperimental sides.Acknowledgement: This research is supported in part by the National ScienceFoundation. It is a pleasure to acknowledge the warm hospitality of the OberlinCollege physics department where this work was performed.References[1] See, e.g. S. Gasiorowicz and D.A. Geffen, Rev. Mod. Phys. 41, 531 (1969).[2] S. Weinberg, Phys. Rev. Lett. 17, 616 (1966).[3] J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158, 142 (1984); Nucl. Phys.B250, 465 (1985).[4] S. Weinberg, Physica A96, 327 (1979).[5] See, it e.g. B.R. Holstein, Int. J. Mod. Phys. A7, 7873 (1992).7[6] Yu. M. Antipov et al., Phys Lett. B121, 445 (1983).[7] D. Babusci et al., Phys. Lett. B277, 158 (1992).[8] J.F. Donoghue and B.R. Holstein, Phys. Rev. D48, 137 (1993).[9] M.L. Goldberger and S.B. Treiman, Phys. Rev. 110, 1478 (1958).[10] For a modern treatment see V. Bernard, N. Kaiser, and U.G. Meissner, Phys.Rev. D50, 6899 (1994).[11] L.B. Auerbach et al., Phys. Rev. 138, B127 (1965); D.R. Clay et al., Phys. Rev.140, B586 (1965).[12] G. Bardin et al., Phys. Lett. B104, 320 (1981).[13] Preliminary TRIUMF result, M. Blecher (private communciation).[14] For a useful summary, see V. Bernard, N. Kaiser and U.G. Meissner, Int. J. Mod.Phys. E4, 193 (1995).[15] J.P. Burg, Ann. De Phys. (Paris) 10, 363 (1965.[16] M.J. Adamovitch et al., Sov. J. Nucl. Phys. 2, 95 (1966).[17] E.L. Goldwasser et al., Proc. XII Int. conf. n High Energy Physics, Dubna, 1964,ed. Ya.-A.Smorodinsky, Atomizdat, Moscow (1966).[18] V. Bernard, J. Gasser, N. Kaiser and U.G. Meissner, Phys. Lett. B268, 291(1991).[19] V. Bernard, N. Kaiser, and U.G. Meissner, to be published in Zeit. fur Physik.[20] M. Fuchs et al., Phys. Lett. B368, 20 (1996).[21] J.C. Bergstrom et al., Phys. Rev. C53, R1052 (1996).[22] B.R. Holstein, Comm. Nucl. Part. Phys. 20, 301 (1992).[23] V. Bernard, N. Kaiser and L.G. Meissner, Phys. Rev. Lett. 67, 1515 (1991);Nucl. Phys. B373, 364 (1992).[24] V. Bernard, N. Kaiser, A. Schmidt, and U.G. Meissner, Phys. Lett. B3119, 269(1993).[25] F.J. Federspiel et al., Phys. Rev. Lett. 67, 1511 (1991); E.L. Hallin et al., Phys.Rev. C48, 1497 (1993); A. Zeiger et al., Phys. Lett. B278, 34 (1992).[26] J. Schmiedmyer et al., Phys. Rev. Lett. 66, 1015 (1991).8

Low Energy Tests of Chiral Symmetry1

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Low Energy Tests of Chiral Symmetry

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