Comments on Recent Developments in Theory of Hadronic Light-by-Light

Transcription

1 INT Workshop on Hadronic Light-by-Light Contribution to the Muon Anomaly February 28 -March 4, 2011, Seattle Comments on Recent Developments in Theory of Hadronic Light-by-Light Arkady Vainshtein William Fine Theoretical Physics Institute University of Minnesota

2 With Joaquim Prades and Eduardo de Rafael we wrote in 2008 a kind of white paper on HLbL summarizing our understanding of the problem at that time. It is was very sad to learn that Ximo passed away in August. In our 08 mini-review we combined different calculations with some educated guesses about possible errors to come to: a HLbL = (105 ± 26) However the error estimates are quite subjective and further study of different exchanges is certainly needed.

3 While I do not think that there were significant changes during the last 3 years I ll try to comment on few suggestions which appeared at this period.

4 Off-shell Form Factors Nyffeler, Jeherlehner OPE constraints k 1 q 0 q 0 ( H γ γ 5 In the range where i k k 2 3 d 4 x d 4 y e iq 1x iq 2 y T {j µ1 (x),j µ2 (y)} = utations, we can q1 2 q2 2 q3 2 and t q3 2 Λ2 QCD { } h the well-known ptotic expres d 4 z e i(q 1+q 2 )z 2i ˆq 2 ɛ µ 1 µ 2 δρ ˆq δ j ρ 5 (z)+ ˆq =(q 1 q 2 )/2 q 1 q 2 ng (in the limit of large Euclide k 3

5 we get for the HLbL amplitude M = α 2 N c Tr [ ˆQ 4 ] A A= 4 q 2 3 ˆq2 {f 2 f 1 }{ ff 3 } } 4 q 2 3 ˆq4 ( {q 2 f 2 f1 ff3 q 3 } A { }{ } +{q 1 f 1 f2 ff3 q 3 }+ q2 1 + q2 2 4 ( { } ) {f 2 f1 }{ ff 3 } + where f µν i = q µ i ɛν i qν i ɛµ i are field strengths of photons. aces denote either tr Thus, the amplitude is unambiguously fixed in the range utations, we can though th q1 2 q2 2 q3 2 Λ 2 QCD, Note an absence yet of any reference h the well-knownthe OPE f to the pion pole. By quantum numbers the first line refers to pseudoscalar exchange, the second -- pseudovector.

6 µ 3 γδf = ˆq 2 ɛ µ 1 µ 2 δρˆq W w short-distance L (q 3 ) q 3 q 3 f σµ3 constraints φ+ w for a lbl L (0, T (q 0) 3 ) = q 3 f µ Compare this now with the meson exchange ;itisknownthatthis µ3 +q is not always the case ( see [3] 4π2 for af recent 2, 3µ3 q 3 f σ q 3 q 3 f σµ3 (17) + lighta=3,8,0 scattering amplitude takes the form, φ (3) π L (0, 0) near = the N c The abs discussion). The short-distance QCD constraints are most 4πrestric- tive 2 Eq.(18) F 2, di π meson pole. For the so that pion the in the pole model pseudoscalar we for the have isovector pion exchange channel. in In the a special light-bylight N cw (3) F πγ γ (q1 2,q2 2 ) A π 0 = kinematic scattering limit, amplitude where invariant takes the masses form, of two virtual o hierarchy between q3 2 and Λ2 QCD so is 2πassumed. 2 that Fπ 2 the q3 2 + {f 2 f1 }{ fore. model The continuation for the ff lations of 3 } m2 π pion to Euclidean exchange photons are much larger than the invariant mass of impact space the on mo i hts W (a) are defined! 0, a as 1 +permutations. third photon, this channel is completely saturated by neutral A π 0 = pion. N cerns cw (3) the change F πγ γ (q 2 in 1,q2 sign 2 ) for all qi 2 The 2π 2 saturation F is complete in the sense that it works for arbitrary π 2 q small3 2 invariant + {f 2 f1 }{ ff and the overa light scattering amplitude takes (18) the contributio form, in sign for the amplitude A, sinceitinvolvesth ( ) 3 } 2 m2 π Tr [λ ˆQ2 of two Levi-Cevita tensors. mass of the The third result can b a The ] πγ γ form factor F W (a) = virtual photon, +permutations. in spite of the fact that, in general, the (18) Tr [λ 2 a]tr [ ˆQ 4 ] ; (11) πγ γ (q by 1 2 comparison,q2 2 with the direct computation of OPE applies A π 0 only = box when N )isdefinedas pion pole cw (3) F πγ γ diagram, that mass for arbitrary is much larger q (q than b,presentedin 1 2,q2 the 2 ) absenc TheΛπγ QCD γ. W (3) = 1 form factor F πγ γ This happens because in the (q1 2 kinematic,q2 2 )isdefinedas 4, W(8) = 1 12, W(0) = 2 I. There2π we show that the amplitude can be accounts for projection 3. in terms the of isovector Fπ 2 q3 nineteen independent limit part 2 + convergenc F m2 πγ γ π (q1,q 2 2)= 2 φ(3) L (q2 1,q2) 2 the result. (19) described tensor struc φ (3) of utions to the light-by-light scatterp,(b) limitexchangeofneutralpionand q ing diagram to the famous anomalous triangle dia- above, the OPE five L (0, 0) happens. relates independent hadronic form-factors. light-by-light In scatter- what follows, the axial current. +permutations. F gram with one πγ γ axial (q1,q 2 2 and2)= φ(3) L (q2 1,q2) Λ2 QCD,Eq.(10)canbesimplifiedusasymptotic expressions Eq.(7) for the invariant currents. Because deal with the approximate form of the amplitud The comparison with the OPE constraint given by the but we make. (19) two vector occasional φ (3) references to Further general e relevant term in Eq.(10) leads both perturbative and non-perturbative L (0, 0) corrections to s w (a) into The OPE asymptotics of form Appendix factor I. tions on th L,T. Convoluting the tensor amplitude with s natural to expect the chiral pabe a better expansion parameter chiral the anomalous triangle are absent in the limit of exact on polarization vectors and analytically The comparison continuclidean space, we arrive at: symmetry, with the pion thepole OPE contribution constraint is unambiguous both term given by the re careful analysis indicates that relevant lim F πγ γ atin (q small Eq.(10) 2,q 2 )= 8π2 F 2 recently re π q Λleads QCD and to atc. large The q model Λ QCD s do, work differently. In particodels = 4 used to estimate a lbl µ,the momenta and provides an important constraint thereby. momenta. This observation connects the two regions of ion contribution is always much In terms lim q 2 Λ of 2 the F diagram πγ γ (q 2 in,q 2 Fig. )= 8π2 F q π {f f ˆq2 2 1 }{ ff 3 } 4 ( F πγ γ (q1,q 2 2)= 2 φ(3) L (q2 1,q 2 q 2) 2 Λ 2 N QCD c q 2, (20) their LMD q3 2 {q 2 f 2 f1 ff3 q 3 } ˆq4 Two different terms Eq.(10) 2a, Nthe constraint QCD c q 2, φ (3) can be (20) ident L (0, 0) matches the asymptotics hanced {q contribution. We present amounts to the statement that the form factor is present 1 f 1 f2 ff3 q 3 }+ q2 1 + ) of the exchanges HLbL amplitude of the pseudoscalar derived (pseudovector q2 2 above. So our {f 2 f1 }{ irally enhanced O(N4 model ff 3 } +. (12) for the functions w (a) c 0 )contribuf this paper where we argue that absent in the πγ γ L,T vertex if both photons are (q2 3). Extrapolating Eq. The correctly comparison interpolates. with the virtual OPEbut constrain it is q F if that vertex 1,2 2 contains Λ2 QCDthe to external arbitrary magnetic q2 1,2,wearriveatthe field. be ν = accidental. q µ i ɛν i qν i ɛµ i are the field strength relevant πγ γ (q1 2,q2 2 Although tensors, term )= 4π2 Fπ 2 q Let 1 2q2 2 (q2 1 + us q2 2 ) h show 2q1 2q2 2 + h 5(q that suggested off-shell changes model: do N c not Eq.(10) fit. leads (q1 2 to the pseudoscalar channel has been + the M 1 2 subject )(q2 1 + M 2 2)( s example denote either is provided traces of products by the of of themany matrices detailed studies in the past, this constraint has zation contribution to a.there, been overlooked F and, as the result, the π 0 πγ γ (q 2,q 2 )= 4π2 Fπ 2 -pole q2 1 q2 2 contribu- (q2 1 + q2 2 ) h 2 eir convolutions with vectors q i. A=A PS + A PV +permutations, 2 the literatu {f 2 f The πγ γ form factor F πγ γ (q 2 1,q2 2 )isde

7 The off-shell approach by Jegerlehner and Nyffeler implies that form factors at each vertex are functions of all three virtualities F πγ γ (q 2 1,q2 2 ; q2 3 ) including virtuality of pion q 2 3 =(q 1 + q 2 ) 2. In the vertex with the external magnetic field it becomes F πγ γ (0,q 2 3 ; q2 3 ) 2 2 The idea is that this function at large q 2 3 is a constant different from F πγ. Comparing with γ (0, 0; 0) = 1 utations, we can though th asymptotics at q1 2 q2 2 q3 2 Λ 2 QCD, we see that such h the well-knownthe OPE f deviation is not allowed.

8 An additional note: If we introduce a form factor F πγ γ (0,q 2 ; q 2 ) with the external magnetic field it will add F πγ γ (0,q 2 ; q 2 ) 1 q 2 in the vertex where the pion propagator is included. Clear that this does not contain the pion pole at q 2 =0. Moreover, it does not modified the longitudinal part, only the transverse one. Thus, it changes what we call the pseudovector exchange in the model. The longitudinal part is protected both perturbatively and nonperturbatively, it s only perturbative for the transversal part associated with the pseudovector exchanges.

9 Quark-based HLbL calculations Goecke, Fischer, and Williams suggested to use the Dyson-Schwinger approach to calculation of the HLbL quark loop and claim a considerable enhancement of the HLbL contribution. No theoretical control or independent check. Compare with ENJL approach with separation of scales. In large Nc limit the enhancement have to be transferred into enhancement of meson-gamma-gamma vertex. A possible check of the approach is to use for calculation of vacuum polarization where experimental data exist.

10 Quark loop estimates were recently discussed by Erler and Sanchez who followed Pivovarov s work of He used m u = m d = m s 180 MeV = 166 ± 1MeV. to fit the vacuum polarization in the leading order as well as in NLO with quark loop without strong interactions. Then he used these masses and the Laporta-Remiddi result a LBL µ (had) = Erler and Sanchez formulate it as an upper bound a LBL µ (had) < Strange duality but at least supported by few fits.

11 One more approach: instanton induced nonlocal quark interaction by Dorohov and collaborators. Again there is no much of theoretical control but the approach fits VP and then HLbL numbers are in to the same ballpark as others.

12 e Pseudovector Puzzle γ f Γ(f 1 (1285) γγ ) = (2.8 ± 0.8) kev γ e This is compatible with our model of pseudovector exchange. However, Γ(f 1 (1285) γρ 0 ) Γ total = (5.5 ± 1.3) 10 2 ± leads to a strong enhancement (of order of 5) for PV exchange. Could be an example of strong enhancement if would be not contradictive.

13 Conclusions Having in mind that the new g-2 experiment is in its way more efforts are needed to improve accuracy for the hadronic light-by-light contribution. In my view it should involve new measurements of hadronic two-photon production which provides a good test of theoretical models for HLbL.