Based on analyticity, unitarity, and Lorentz invariance the contribution from
hadronic vacuum polarization to the anomalous magnetic moment of the muon is
directly related to the cross section of e^+e^- --> hadrons. We review the main
difficulties that impede such an approach for light-by-light scattering and
identify the required ingredients from experiment. Amongst those, the most
critical one is the scattering of two virtual photons into meson pairs. We
analyze the analytic structure of the process gamma^* gamma^* --> pi pi and
show that the usual Muskhelishvili-Omnes representation can be amended in such
a way as to remain valid even in the presence of anomalous thresholds.Comment: 5 pages, 3 figures, Proceedings for the International Workshop on
  e^+e^- collisions from phi to psi 2013, Rome, Italy, September 9-12, 201

Colangelo, Gilberto

Hoferichter, Martin

Procura, Massimiliano

Stoffer, Peter

English

arXiv

Based on analyticity, unitarity, and Lorentz invariance the contribution from hadronic vacuum polarization to the anomalous magnetic moment of the muon is directly related to the cross section of e+e− → hadrons. We review the main difficulties that impede such an approach for light-by-light scattering and identify the required ingredients from experiment. Amongst those, the most critical one is the scattering of two virtual photons into meson pairs. We analyze the analytic structure of the process γ*γ* → ππ and show that the usual Muskhelishvili–Omnès representation can be amended in such a way as to remain valid even in the presence of anomalous thresholds

Bern Open Repository and Information System (BORIS)

November 18, 2014 16:35 WSPC/CRC 9.75 x 6.5 1460400e+e− Collisions from Phi to Psi 2013 (PHIPSI2013)International Journal of Modern Physics: Conference SeriesVol. 35 (2014) 1460400 (6 pages)c© The AuthorsDOI: 10.1142/S2010194514604001VIRTUAL PHOTON–PHOTON SCATTERINGMARTIN HOFERICHTER∗, GILBERTO COLANGELO,MASSIMILIANO PROCURA and PETER STOFFERAlbert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,University of Bern, Sidlerstrasse 5, CH–3012 Bern, Switzerland∗hoferichter@itp.unibe.chPublished 18 December 2014Based on analyticity, unitarity, and Lorentz invariance the contribution from hadronicvacuum polarization to the anomalous magnetic moment of the muon is directly relatedto the cross section of e+e− → hadrons. We review the main diﬃculties that impedesuch an approach for light-by-light scattering and identify the required ingredients fromexperiment. Amongst those, the most critical one is the scattering of two virtual photonsinto meson pairs. We analyze the analytic structure of the process γ∗γ∗ → ππ and showthat the usual Muskhelishvili–Omne`s representation can be amended in such a way asto remain valid even in the presence of anomalous thresholds.Keywords: Dispersion relations; anomalous magnetic moment of the muon; Comptonscattering; meson–meson interactions.PACS Numbers: 11.55.Fv, 13.40.Em, 13.60.Fz, 13.75.Lb1. Hadronic Vacuum PolarizationThe leading contribution of strong interactions to the anomalous magnetic momentof the muon g − 2 originates from hadronic intermediate states in the polarizationtensor of the photon.1 By means of gauge invariance, the polarization tensor maybe expressed in terms of one single-variable scalar function Π(k2)k, µ k, ν= −i(k2gµν − kµkν)Π(k2). (1)Due to analyticity, the renormalized self energy satisﬁes a subtracted dispersionrelationΠren = Π(k2)−Π(0) = k2π∫ ∞4M2πdsImΠ(s)s(s− k2) . (2)Unitarity relates the imaginary part to the e+e− hadronic cross sectionThis is an Open Access article published by World Scientiﬁc Publishing Company. It is distributedunder the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distributionof this work is permitted, provided the original work is properly cited.1460400-1Int. J. Mod. Phys. Conf. Ser. 2014.35. Downloaded from www.worldscientific.comby UNIVERSITY OF BERN on 01/15/15. For personal use only.source: http://boris.unibe.ch/62060/ | downloaded: 13.3.2017November 18, 2014 16:35 WSPC/CRC 9.75 x 6.5 1460400M. Hoferichter et al.Fig. 1. Classes of unitarity diagrams in light-by-light scattering. The grey blobs denote (transi-tion) form factors, the blobs with vertical line a polynomial contribution in the crossed channel.Short-dashed lines refer to pions, wiggly lines to photons, and the long-dashed lines indicate cutpropagators. Crossed diagrams are not shown.ImΠ(s) =s4πασtot(e+e− → hadrons). (3)In this way, general principles obeyed by the polarization tensor provide a directlink between its contribution to g − 2 and observables.2. Light-by-Light Scattering2.1. Structure of the Light-by-Light TensorNo such immediate relation to experiment is known for the light-by-light tensorΠµνλσ , describing the scattering processγ∗(q1, µ)γ∗(q2, ν)→ γ∗(−q3, λ)γ(k, σ). (4)In contrast to vacuum polarization, there are 29 independent Lorentz structures,cf. Ref. 2, and 5 independent kinematic variables (2 Mandelstam variables and 3virtualities), so that the full amplitude should be expanded in a suitable set of basisfunctionsaΠµνλσ(q1, q2, q3)=29∑i=1Aµνλσi(q1, q2, q3)Πi(s, t, q21 , q22 , q23). (5)In order to write down dispersion relations for the scalar coeﬃcients Πi, the basisfunctions Aµνλσi need to be chosen in such a way that the Πi are free of kinematicsingularities and that crossing symmetry, e.g. invariance under (q1, µ)↔ (q2, ν), ismaintained.The complicated structure of the light-by-light tensor prohibits a comprehensiveanalysis of all intermediate states allowed by unitarity. However, the most importantstates (besides the pseudoscalar meson poles) in the low/intermediate energy regionare two-meson reducible. They can be classiﬁed according to the analytic structurein the crossed channel as shown in Fig. 1. There are classes of box, triangle, and bulbunitarity diagrams, depending on whether the crossed-channel amplitude involvesnon-polynomial terms. Such non-polynomial contributions are given by the pionpole and multi-pion exchange, whereas the polynomials for instance include eﬀectsaAs shown in Ref. 3, gauge invariance for the on-shell photon implies that only the derivative withrespect to kρ at k = 0 is needed for the application in g − 2.1460400-2Int. J. Mod. Phys. Conf. Ser. 2014.35. Downloaded from www.worldscientific.comby UNIVERSITY OF BERN on 01/15/15. For personal use only.November 18, 2014 16:35 WSPC/CRC 9.75 x 6.5 1460400Virtual photon–photon scatteringdue to ππ rescattering. In practice, the multi-pion diagrams may be approximatedby resonance exchange, i.e. ρ and ω/φ for 2 and 3 pions, respectively. While forω and φ a narrow-width approximation is certainly viable, the eﬀect of the ﬁnitewidth of the ρ is captured through a spectral-function approach that relies on theamplitude for γ∗π → ππ as input.2.2. Input from ExperimentThe experimental ingredients necessary for this program follow from Fig. 1. Dia-grams with a pion pole require the pion vector form factor, those with resonanceexchange the corresponding transition form factors and the γ∗π → ππ amplitude.This input for the multi-pion diagrams can again be checked for consistency withina framework respecting analyticity and unitarity.4–6 The most critical input con-cerns the polynomial pieces, since they involve the pole-subtracted partial wavesfor the process γ∗γ∗ → ππ. Absent direct experimental information for arbitraryvirtualities, e.g. from e+e− → ππ+−, these partial waves are again reconstructeddispersively, see Refs. 7, 8 for two on-shell photons and Ref. 9 for one photon withnon-vanishing virtuality. Finally, the dispersion relations for the Πi will involve acontribution of the pion-pole diagram, with a residue determined by the (on-shell)pion transition form factor Fπγ∗γ∗(M2π , q21 , q22). In order to eliminate the model-dependence as far as possible, also input for this form factor should fulﬁll analyticityand unitarity requirements and be backed by data wherever available.10–123. Analytic Structure of γ∗γ∗ → ππIn principle, the partial waves for γ∗γ∗ → ππ are constrained by a similar setof dispersion relations as derived in Refs. 7, 8, 9. Within a simpliﬁed scalar toyexample, where the left-hand cut is approximated by the pion pole, one thus obtainsthe following Muskhelishvili–Omne`s representation for the pole-subtracted S-wavef0(s; q21 , q22)=Ω0(s)π∫ ∞4M2πds′N0(s′; q21 , q22)sin δ0(s′)(s′ − s)|Ω0(s′)| , (6)with the projection of the pole termN0(s; q21 , q22)=2Lσs√λ(s, q21 , q22) , L = log s− q21 − q22 + σs√λ(s, q21 , q22)s− q21 − q22 − σs√λ(s, q21 , q22),σs =√1− 4M2πs, λ(x, y, z) = x2 + y2 + z2 − 2(xy + xz + yz), (7)1460400-3Int. J. Mod. Phys. Conf. Ser. 2014.35. Downloaded from www.worldscientific.comby UNIVERSITY OF BERN on 01/15/15. For personal use only.November 18, 2014 16:35 WSPC/CRC 9.75 x 6.5 1460400M. Hoferichter et al.the Omne`s function Ω0(s), and ππ S-wave t0(s)Ω0(s) = exp{sπ∫ ∞4M2πds′δ0(s′)s′(s′ − s)}, t0(s) =1σseiδ0(s) sin δ0(s). (8)The analytic continuation of this solution in the virtualities q2i in the case that bothphotons are oﬀ-shell is complicated by the occurrence of anomalous thresholds,13i.e. the singularities of the logarithm in Eq. (7) located ats± = q21 + q22 −q21q222M2π± 12M2π√q21(q21 − 4M2π)q22(q22 − 4M2π). (9)In this way, left- and right-hand cut become intertwined, which invalidates the directderivation of Eq. (6) for large virtualities.In order to elucidate the role of these anomalous thresholds we ﬁrst consider thescalar triangle loop functionC0(s) =1iπ2∫d4k(k2 −M2π)((k + q1)2 −M2π)((k − q2)2 −M2π) . (10)If q21 + q22 ≥ 4M2π, its dispersive representation involves an additional, anomalouspiece that emerges because the anomalous branch point’s moving onto the ﬁrst sheetdistorts the integration contour, see Fig. 2 and Ref. 14. The numerical results inFig. 3 show that the dispersive reconstruction of C0(s) indeed works for arbitraryvirtualities as long as the anomalous contribution is taken into account (upperpanel), but that substantial deviations occur in the region of large virtualities if theanomalous piece is ignored (lower panel).In fact, this procedure to perform the analytic continuation in the q2i for C0(s)transfers immediately to f0(s; q21 , q22), the crucial observation being that the inte-grand of Eq. (6) coincides with the discontinuity of C0(s),q22→ −∞q22→ ∞q22= 0q22= 4M2π− q21q22= 4M2πFig. 2. Trajectory of the anomalous branch point s+ as a function of q22 for 0 ≤ q21 ≤ 4M2π. Forq22 → −∞, s+ lies on the second sheet, then migrates onto the ﬁrst sheet through the unitaritycut, and there requires a deformation of the integration contour.1460400-4Int. J. Mod. Phys. Conf. Ser. 2014.35. Downloaded from www.worldscientific.comby UNIVERSITY OF BERN on 01/15/15. For personal use only.November 18, 2014 16:35 WSPC/CRC 9.75 x 6.5 1460400Virtual photon–photon scatteringnumericalanalyticdispersive-10-5 0 5 10-10-5 0 5 10-2-1 0 1 2 3q21q22−ReC0(s)numericalanalyticdispersive-10-5 0 5 10-10-5 0 5 10 0 1 2 3 4 5q21q22−ImC0(s)numericalanalyticdispersive-10-5 0 5 10-10-5 0 5 10-2-1 0 1 2 3q21q22−ReC0(s)numericalanalyticdispersive-10-5 0 5 10-10-5 0 5 10 0 1 2 3 4 5q21q22−ImC0(s)Fig. 3. C0(s) for s = 5 and Mπ = 1 calculated numerically, analytically, and dispersively. Thelower panel shows the eﬀect of switching oﬀ the anomalous contribution in the dispersive formula.N0(s; q21 , q22)sin δ0(s)|Ω0(s)| = −discC0(s)πiσssin δ0(s)|Ω0(s)| = −discC0(s)πit0(s)Ω0(s), (11)up to a factor t0(s)/Ω(s), which is independent of q2i and well-deﬁned in the wholecomplex s-plane. Therefore, the full result for f0(s; q21 , q22) becomes merely amendedby an additional term that takes care of the anomalous thresholdsf0(s; q21 , q22)∣∣∣anom= θ(q21 + q22 − 4M2π)Ω0(s)2πi∫ 10dx∂sx∂xdiscan f0(sx; q21 , q22)sx − s ,discan f0(s; q21 , q22)= − 8π√λ(s, q21 , q22) t0(s)Ω0(s) , sx = 4M2π x + (1− x)s+.AcknowledgmentsWe would like to thank Bastian Kubis, Bachir Moussallam, and Sebastian Schnei-der for numerous useful discussions. Financial support by the Swiss NationalScience Foundation is gratefully acknowledged. The AEC is supported by the“Innovations- und Kooperationsprojekt C-13” of the “Schweizerische Univer-sita¨tskonferenz SUK/CRUS.”References1. F. Jegerlehner and A. Nyﬀeler, Phys. Rept. 477, 1 (2009).1460400-5Int. J. Mod. Phys. Conf. Ser. 2014.35. Downloaded from www.worldscientific.comby UNIVERSITY OF BERN on 01/15/15. For personal use only.November 18, 2014 16:35 WSPC/CRC 9.75 x 6.5 1460400M. Hoferichter et al.2. J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. B 474, 379 (1996).3. J. Aldins, T. Kinoshita, S. J. Brodsky and A. J. Dufner, Phys. Rev. D 1, 2378 (1970).4. M. Hoferichter, B. Kubis and D. Sakkas, Phys. Rev. D 86, 116009 (2012).5. F. Niecknig, B. Kubis and S. P. Schneider, Eur. Phys. J. C 72, 2014 (2012).6. S. P. Schneider, B. Kubis and F. Niecknig, Phys. Rev. D 86, 054013 (2012).7. R. Garc´ıa-Mart´ın and B. Moussallam, Eur. Phys. J. C 70, 155 (2010).8. M. Hoferichter, D. R. Phillips and C. Schat, Eur. Phys. J. C 71, 1743 (2011).9. B. Moussallam, Eur. Phys. J. C 73, 2539 (2013).10. E. Czerwin´ski et al., arXiv:1207.6556 [hep-ph].11. M. J. Amaryan et al., arXiv:1308.2575 [hep-ph].12. M. Hoferichter, B. Kubis, S. Leupold, F. Niecknig and S. P. Schneider, in preparation.13. S. Mandelstam, Phys. Rev. Lett. 4, 84 (1960).14. W. Lucha, D. Melikhov and S. Simula, Phys. Rev. D 75, 016001 (2007).1460400-6Int. J. Mod. Phys. Conf. Ser. 2014.35. Downloaded from www.worldscientific.comby UNIVERSITY OF BERN on 01/15/15. For personal use only.

Virtual photon-photon scattering

 Based on analyticity, unitarity, and Lorentz invariance the contribution from hadronic vacuum polarization to the anomalous magnetic moment of the muon is directly related to the cross section of e+e− → hadrons . We review the main difficulties that impede such an approach for light-by-light scattering and identify the required ingredients from experiment. Amongst those, the most critical one is the scattering of two virtual photons into meson pairs. We analyze the analytic structure of the process γ*γ* → ππ and show that the usual Muskhelishvili–Omnès representation can be amended in such a way as to remain valid even in the presence of anomalous thresholds. </jats:p

MARTIN HOFERICHTER

GILBERTO COLANGELO

MASSIMILIANO PROCURA

PETER STOFFER

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International Journal of Modern Physics Conference Series

VIRTUAL PHOTON–PHOTON SCATTERING

https://boris.unibe.ch/62060/1/S2010194514604001.pdf

Virtual photon-photon scattering

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