Abstract
A recently proposed dispersive approach to hadronic light-by-light is described. In this talk I have presented a dispersive approach to hadronic light-by-light (HLbL) which has been recently proposed in [1]. This approach aims to take into account only the cuts in the hadronic tensor which are due to singleor double-pion intermediate states – this approximation is justified by the fact that in explicit calculations higher-lying singularities (like the one due to two kaons) give small contributions [2]. Further, we split the hadronic tensor as follows: Πμνλσ = Π π0−pole μνλσ + Π FsQED μνλσ + Πμνλσ + · · · , (1) where the first term takes into account the one-pion pole, the second one two-pion intermediate states with simultaneous cuts in the s and t channel (and all possible cyclic permutations including u), and the third one is the one for which we write down a dispersion relation. We briefly discuss the three contributions. 1 Pion pole The dominant contribution to HLbL scattering at low energy is given by the π0-poles. Their residues are determined by the on-shell, doubly-virtual pion transition form factor Fπ0γ∗γ∗ (q1, q2), which is defined as the current matrix element i ∫ d4x eiq·x 〈 0 ∣∣∣T { jμ(x) jν(0)}∣∣∣π0(p)〉 = μναβqαpβFπ0γ∗γ∗(q2, (p − q)2). (2) In these conventions, the π0-pole HLbL amplitude reads Π π0-pole μνλσ = Fπ0γ∗γ∗(q21, q22)Fπ0γ∗γ∗(q23, 0) s − M2 π0 μναβqα1q β 2 λσγδq γ 3k δ + Fπ0γ∗γ∗(q21, q23)Fπ0γ∗γ∗(q22, 0) t − M2 π0 μλαβqα1q β 3 νσγδq γ 2k δ + Fπ0γ∗γ∗(q22, q23)Fπ0γ∗γ∗(q21, 0) u − M2 π0 νλαβqα2q β 3 μσγδq γ 1k . (3) DOI: 10.1051/ C © Owned by the authors, published by EDP Sciences, 2014 /2014 0005 , 000 (2014) Conferences EP Web of J 80 80 ep conf j 56 6 This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20148000056 Its contribution to aμ can be expressed as [3] a 0-pole μ = −e6 ∫ dq1 (2π)4 ∫ dq2 (2π)4 1 q1q 2 2s ( (p + q1) − m2)((p − q2) − m2) (4) × {Fπ0γ∗γ∗(q21, q22)Fπ0γ∗γ∗(s, 0) s − M2 π0 T1(q1, q2; p) + Fπ0γ∗γ∗(s, q22)Fπ0γ∗γ∗(q21, 0) q1 − M2 π0 T2(q1, q2; p) } ,
Highlights
Where the first term takes into account the one-pion pole, the second one two-pion intermediate states with simultaneous cuts in the s and t channel, and the third one is the one for which we write down a dispersion relation
One first takes the two-pion cut in the s-channel, which gives the discontinuity as the product of two γ∗γ∗ → ππ amplitudes, and selects the Born term in each of the two amplitudes. The singularity of this diagram is given by four π+π−γ∗ vertices with on-shell pions—which implies that these vertices are nothing but the full pion vector form factors
The singularity structure of this contribution is identical to that of a Feynman box diagram with four pion propagators: since the four vertices depend only on the momentum squared of the external photons and on none of the internal momenta, this contribution is given by the box-diagram multiplied by three pion vector form factors
Summary
The dominant contribution to HLbL scattering at low energy is given by the π0-poles Their residues are determined by the on-shell, doubly-virtual pion transition form factor Fπ0γ∗γ∗ (q21, q22), which is defined as the current matrix element i d4 x eiq·x 0 T jμ(x) jν(0) π0(p) = μναβqα pβFπ0γ∗γ∗ q2, (p − q). Their residues are determined by the on-shell, doubly-virtual pion transition form factor Fπ0γ∗γ∗ (q21, q22), which is defined as the current matrix element i d4 x eiq·x 0 T jμ(x) jν(0) π0(p) = μναβqα pβFπ0γ∗γ∗ q2, (p − q)2 Due to the q1 ↔ −q2 symmetry of the integrand, the t- and u-channel terms give the same contribution
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