Measurement of the $\gamma 3\pi$ process has revealed a possible conflict
with what should be a solid prediction generated by the chiral anomaly. We show
that inclusion of appropirate energy-momentum dependence in the matrix element
reduces the discrepancy.Comment: 8 page standard Latex fil

A. Bramon

Barry R. Holstein

E. Witten

J. Bijnens

J. S. Bell

J. Wess

K. Kawarabayashi

L. Brown

L. Montanet

M. A. B. Beg

M. V. Terent'ev

R. Aviv

R. Omnes

Riazuddin

S. L. Adler

S. Rudaz

S. Weinberg

T. D. Cohen

T. N. Truong

Yu. M. Antipov

English

arXiv

Crossref

Chiral anomaly and γ3π

Measurement of the γ3π process has revealed a possible conflict with what should be a solid prediction generated by the chiral anomaly. We show that inclusion of appropriate energy-momentum dependence in the matrix element reduces the discrepancy

Holstein, BR

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics1996Chiral anomaly and gamma 3 piBR Holsteinholstein@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics 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, "Chiral anomaly and gamma 3 pi" (1996). PHYSICAL REVIEW D. 348.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/348arXiv:hep-ph/9512338v1  18 Dec 1995Chiral Anomaly and γ3π∗Barry R. HolsteinDepartment of Physics and AstronomyUniversity of MassachusettsAmherst, MA 01003February 1, 2008AbstractMeasurement of the γ3π process has revealed a possible conflictwith what should be a solid prediction generated by the chiral anomaly.We show that inclusion of appropriate energy-momentum dependencein the matrix element reduces the discrepancy.∗ Research supported in part by the National Science Foundation.11 IntroductionThe chiral anomaly is a well-known and fascinating aspect of QCD. Firstidentified in the context of the “triangle diagram” contribution to π0 → 2γ,[1]it has been shown to have much more general consequences which can becharacterized in terms of an effective Lagrangian1[2]LWZW =Nc48π2ǫµναβ [eAµTr(Q(RνRαRβ + LνLαLβ))−ie2FµνAαTr(Q2(Lβ +Rβ) +12(QU †QURβ +QUQU†Lβ))] (1)where U = exp(i∑λiφi/Fpi) is the usual nonlinear matrix describing thepseudoscalar Goldstone fields, Rµ ≡ (∂µU†)U, Lµ ≡ U(∂U†) are right,left-handed currents respectively andQ = e3(2,−1,−1)diag is the quark chargematrix. One immediately identifies the theoretical prediction for π0 → γγwhich arises from the second line of Eq. 1Amppi→γγ = −iAγγǫµναβǫ∗µkνǫ′∗αk′βwith Aγγ =αNc3πFpiNc=3−→ 0.025GeV−1 (2)which is in excellent agreement with the experimental value[3]Aγγ = 0.025± 0.001GeV−1 (3)In a corresponding fashion one can read off from Eq. 1 the prediction forthe γπππ vertexAmpγpi+pi−pi0 = −iA3pi(0)ǫµναβǫµp1νp2αp0βwith A3pi(0) =eNc12π2F 3piNc=3−→ 9.7GeV−3 (4)In this case, agreement with the value quoted experimentally[4]Aexp3pi = 12.9± 0.9± 0.5GeV−3 (5)is not particularly convincing and could even be said to favor the valueNc = 4! However, since such a violation would have severe consequences1We include here only the component relevant to electromagnetic interactions.2about the very foundations of QCD it warrants a more careful look, which isthe purpose of the present note. Since the prediction of the anomaly strictlyspeaking hold only at zero four-momentum, while the experimental data isobtained over a range of energies above threshold, it is essential to under-stand the energy dependence of the γ3π amplitude generated by O(p6) andhigher contributions, and this is done in section II. Then in section III weuse these results to confront existing experimental information and commenton implications for future experiments such as that approved at CEBAF.[5]2 Finite Energy CorrectionsThe issue of finite energy correction to predictions of the anomaly has beenaddressed by a number of authors and is now reasonably well understood.The first such consideration was that of Terent’ev who, on phenomenologicalgrounds, suggested the form[6]A3pi(s, t, u) = A3pi(0)[1 + Cρeiδ(sm2ρ − s+tm2ρ − t+um2ρ − u)] (6)where s = (p1 + p2)2, t = (p1 + p0)2, u = (p2 + p0)2, δ is an phenomenologicalphase factor, andCρ =2gρpipigpiργm3ρA3pi(0)= 0.478 (7)represents the pure vector dominance contribution. The next step was takenby Rudaz who, noting that the amplitude for π0 → γγ could be generatedentirely via the vector dominance diagram π0 → ωρ → γγ, cf. Figure 1a,proposed the same for the γ3π process, cf. Figure 1b, yielding[7]A3pi(s, t, u) =13A3pi(0)[m2ρm2ρ − s+m2ρm2ρ − t+m2ρm2ρ − u] (8)However, it was soon realized that this expression conflicted both withthe KSRF relation[8] as well as with the anomalous Ward identities of Avivand Zee[9] and that the correct form was[10]A3pi(s, t, u) = −12A3pi(0)[1− (m2ρm2ρ − s+m2ρm2ρ − t+m2ρm2ρ − u)] (9)3Figure 1: Vector dominance contributions to the reactions π0 → γγ (a) andγ → 3π (b).which contains both a vector dominance piece and a contact term.In recent years, the problem has also been addressed via a one loop ex-pansion in chiral perturbation theory, yielding the form, correct to O(p6) inthe derivative expansion[11]2A3pi(s, t, u) = A3pi(0)[1+3m2pi2m2ρ+m2pi24π2F 2pi(34lnm2ρm2pi+F (s)+F (t)+F (u))] (10)whereF (s) =(1− s4m2pi)√s−4m2pisln1+√s−4m2pis−1+√s−4m2pis− 2 s > 4m2pi2(1− s4m2pi)√4m2pi−sstan−1√s4m2pi−s− 2 s ≤ 4m2pi(11)The vector dominance form—Eq. 9—may be made consistent with itschiral counterpart—Eq. 10—provided we include the effects of final statep-wave pi-pi scattering. We begin by noting that the N/D formt1(s) = tCA1 (s)/D1(s), (12)2Here we use the mass shell condition s+ t+ u = 3m2piand determine the coefficient ofthe term linear in s,t,u (a free parameter in strict chiral perturbation theory) by demandingagreement with expansion of the vector dominance form Eq. 9.4withtCA1 (s) =s− 4m2pi96πF 2pi(13)being the familiar p-wave Weinberg or current algebra prediction[12] andD1(s) = 1−sm2ρ−s96π2F 2pilnm2ρm2pi−m2pi24π2F 2piF (s) (14)providing an analytic approximation to the Omnes function,3[13] provides arather successful representation for the ℓ = 1 pi-pi scattering amplitude[14]t1(s) =√ss− 4m2pieiδ1(s) sin δ1(s). (16)Likewise, a reasonable approximation to the electromagnetic form factor ofthe charged pion is[15]Gpi(s) = 1/D1(s) ≈m2ρm2ρ − s− imρΓρ(s)(17)whereΓρ(s) = θ(s4m2pi− 1)g2ρpipis48πmρ(1−4m2pis) 32(18)is an energy dependent quantity which reduces to the rho width when s = m2ρ.Here we have noted thatm2pi24π2F 2piImF (s) =1mρΓρ(s) (19)and have utilized the KSRF relation g2ρpipi = m2ρ/2F2pi .[8] We observe that Eqs.9 and 10 can be made to agree to low order in s, t, u provided we use the3One could also use the experimental p-wave phase shifts and the definitionD1(s) = exp(−sπ∫∞4m2pids′δ1(s′)s′(s′ − s− iǫ))(15)but the result is similar.5formA3pi(s, t, u) = −12A3pi(0)[1− (m2ρm2ρ − s+m2ρm2ρ − t+m2ρm2ρ − u)]×1− sm2ρD1(s)1− tm2ρD1(t)1− um2ρD1(u) (20)which is suggested by the feature that rescattering occurs in each of the threepi-pi channels simultaneously. It should also be noted that Eq. 20 satisfiesthe requirements of the Fermi-Watson theorem (i.e. unitarity) for the processγπ → ππ and provides the preferred form to use in future analysis.3 Comparison with ExperimentAs mentioned in the introduction, it is often asserted that the experimentaland theoretical values for A3pi(0) are in significant disagreement. However,a more careful look at the paper of Antipov et al.[4] reveals that this isnot the case. In fact, the experimental value quoted in Eq. 5 obtains onlyunder the assumption that the matrix element A3pi(s, t, u) is independent ofmomentum. On the other hand, averaging the from given by Terent’ev overthe experimental spectrum yields (in units of GeV−3)[4]A23pi(0) + 1.9 cos δA3pi(0) + 1 = 166± 23± 13 (21)Since the spectral shape given by Terent’ev—Eq. 6—is basically in agreementwith the form given by anomaly considerations— Eq. 9—provided cos δ = 1,and since the experiment of Antipov et al. was primarily at low values ofthe energy where unitarity corrections given by Eq. 19 are small we find thesolutionA3pi(0) = 11.9± 0.9± 0.5GeV−3 (22)Thus the disagreement with the number required by the chiral anomaly is atthe 1.6σ level rather than the 2.3σ level generally quoted. Nevertheless, theexperimental value is still on the high side and should certainly be subjectedto additional experimental scrutiny, as will take place in the approved CLASexperiment at CEBAF.[5] When such data are analyzed they should useforms such as Eq. 20 which both satisfy chiral and unitarity restrictions6Table 1: Spectral modifications to the process γπ → ππ generated via Eqs.10,9,20 respectively. All values of s,t are in units of m2pi and the numbersquoted in the table represent percentage deviations from the anomaly pre-diction.s,—t— 0.5 1.0 2.0 3.0 4.04 6.0 6.0 6.0 6.3 6.65.9 5.9 5.9 6.0 7.26.9 6.9 6.9 8.5 105 6.6 6.6 6.6 6.7 6.86.0 6.0 6.0 7.2 7.38.5 8.5 8.5 10 1110 8.7 8.7 8.7 8.6 8.514 14 14 14 1421 21 21 21 2115 11 11 11 11 1132 31 30 30 3045 45 44 44 4420 14 14 14 14 1470 70 70 68 6896 95 95 93 93as well as the phenomenological requirements of vector dominance. Thatuse of such a form can make a significant difference can be seen in Table 1,where we compare the modifications of the lowest order anomaly predictionas generated by Eqs. 10,9,20.4 In the region 4m2pi < s < 13m2pi; 0.5m2pi < |t| <3.5m2pi explored by the Antipov et al experiment the differences between thevarious forms are moderate, but in CEBAF proposal much larger values ofenergy and momentum transfer are involved—4m2pi < s, |t| < 50m2pi and theuse of a properly unitarized form for the decay amplitude is essential in orderto extract the value of the anomaly.4The top line of each row is equivalent to the results quoted previously by Bijnens,Bramon and Cornet, ref 9.7References[1] S.L. Adler, Phys. Rev. 177, 2426 (1969); J.S. Bell and R. Jackiw, NuovoCim. 60, 147 (1969).[2] J. Wess and B. Zumino, Phys. Lett. B37, 95 (1971); E. Witten, Nucl.Phys. B223, 422 (1983).[3] Part. Data Group, Phys. Rev. D50, 1449 (1994).[4] Yu. M. Antipov et al., Z. Phys. C27, 21 (1985); Phys. Rev. D36, 21(1987).[5] CEBAF PR-94-015, R. Miskimen, spokesman.[6] M.V. Terent’ev, Phys. Lett. B38, 419 (1972); Sov. Phys. Usp. 17, 20(1974).[7] S. Rudaz, Phys. Rev. D10, 3857 (1974).[8] K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16, 233 (1966);Riazuddin and Fiazuddin, Phys. Rev. 147, 1071 (1966).[9] R. Aviv and A. Zee, PLhys. Rev. D5, 2372 (1972).[10] S. Rudaz, Phys. Lett. B145, 281 (1984); T.D. Cohen, Phys. Lett. B233,467 (1989).[11] J. Bijnens, A. Bramon and F. Cornet, Phys. Lett. B237, 488 (1990); A.Bramon, A. Grau, E. Pallante, G. Pancheri and R. Petronzio, DaΦneHandbook, INFN, Frascati 305 (1992).[12] S. Weinberg, Phys. Rev. Lett. 17, 616 (1966).[13] R. Omnes, Nuovo Cim. 8, 1244 (1958).[14] L. Brown and R. Goble, Phys. Rev. Lett. 20, 346 (1968); M.A.B. Begand A. Zepeda, Phys. Rev. D6, 2912 (1972).[15] T.N. Truong, Phys. Rev. Lett. 61, 2526 (1988).8This figure "fig1-1.png" is available in "png" format from:http://arXiv.org/ps/hep-ph/9512338v1