Dispersive Approach to Hadronic Light-by-Light

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 + Π̄μνλσ + · · · , (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 μναβqα1 q β 2 λσγδq γ 3k δ + Fπ0γ∗γ∗(q21, q23)Fπ0γ∗γ∗(q22, 0) t − M2 π0 μλαβqα1 q β 3 νσγδq γ 2k δ + Fπ0γ∗γ∗(q22, q23)Fπ0γ∗γ∗(q21, 0) u − M2 π0 νλαβqα2 q 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) } ,

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 single-or 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: 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.

Pion pole
The dominant contribution to HLbL scattering at low energy is given by the π 0 -poles.Their residues are determined by the on-shell, doubly-virtual pion transition form factor F π 0 γ * γ * (q 2 1 , q 2 2 ), which is defined as the current matrix element ( In these conventions, the π 0 -pole HLbL amplitude reads Its contribution to a μ can be expressed as [3] a T 2 (q 1 , q 2 ; p) , with Due to the q 1 ↔ −q 2 symmetry of the integrand, the t-and u-channel terms give the same contribution.

FsQED contribution
The precise meaning of the FsQED contribution can be explained as follows: Π FsQED μνλσ includes the contribution due to simultaneous two-pion cuts in two of the channels (by crossing symmetry it contains three contributions with simultaneous singularities in the (s, t), (s, u), and (t, u) channels, respectively).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 then selects the Born term (the pure pole term) in each of the two amplitudes.The singularity of this diagram is therefore 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. On the other hand, 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 (since one of the photons is on-shell).In sQED the box diagram is not gauge invariant on its own, however.The photon-scalar-scalar vertex comes together with the seagull term (two-photon-two-scalar vertex), with couplings strictly related to each other: in any amplitude with two or more photons both vertices have to be taken into account to form a subset of gauge-invariant diagrams.Therefore, in sQED the box diagram has to be accompanied by a triangle and a bulb diagram in order to respect gauge invariance.We do the same here and define our gauge-invariant box diagram as the charged pion loop calculated within sQED multiplied by the pion vector form factors.
We find the representation where F V π (s) is the pion vector form factor and the expression for I s,u and J 1,2 can be found in [1].

Two-pion cuts
A central result of our analysis is that after separating Π π 0 −pole μνλσ and Π FsQED μνλσ from the rest, we have been able to derive explicit unitarity relations for the remainder Πμνλσ and relate the imaginary parts to the helicity amplitudes for γ * γ * → ππ.

00056-p.2
In general, the HLbL tensor with one of the four photons on-shell contains 29 independent scalar amplitudes.We have explicitly constructed 29 independent gauge-invariant Lorentz tensors, but doing so in a way that makes crossing symmetry manifest, or even easy to express, is more difficult.For our purposes we find it more convenient to use a redundant basis, in which however crossing symmetry is evident.Therefore, we exploit the crucial property of the A μνλσ i,s that if we project the s-channel HLbL tensor on helicity amplitudes, only a single function Π s i ≡ Π i (s, t, u) contributes for each helicity amplitude, and write The 45 tensors in (7) form a complete, though redundant, basis.In fact, already the 30 tensors A μνλσ i,s and A μνλσ i,t are sufficient to saturate the number of linearly independent structures.The construction of dispersion relations for the Π i functions becomes greatly simplified if we consider that here we are only interested in the HLbL contribution to a μ .This involves the derivative of the HLbL tensor with respect to k evaluated at k = 0. We therefore construct dispersion relations only for this very special limit and omit from the start any contribution to the HLbL tensor of O(k 2 ).The dispersive representation of the Π s i amplitudes which we have provided has the following properties 1.For each Π s i we only take into account the discontinuity due to the lowest partial wave.
2. We fix the discontinuity to what unitarity prescribes.
3. The Π s i amplitudes have the required soft-photon zeros.
4. The exact form of the soft-photon zeros follows from γ * γ * → ππ by means of factorization.
5. The number of subtractions is chosen according to what the implementation of the soft-photon zeros naturally generates (which is sufficient to ensure convergence).
These arguments uniquely lead to the following dispersive integrals for the Π s i amplitudes1