Abstract
The Standard Model prediction for μ-e scattering at Next-to-Next-to-Leading Order (NNLO) contains non-perturbative QCD contributions given by diagrams with a hadronic vacuum polarization insertion in the photon propagator. By taking advantage of the hyperspherical integration method, we show that the subset of hadronic NNLO corrections where the vacuum polarization appears inside a loop, the irreducible diagrams, can be calculated employing the hadronic vacuum polarization in the space-like region, without making use of the R ratio and time-like data. We present the analytic expressions of the kernels necessary to evaluate numerically the two types of irreducible diagrams: the two-loop vertex and box corrections. As a cross check, we evaluate these corrections numerically and we compare them with the results given by the traditional dispersive approach and with analytic two-loop vertex results in QED.
Highlights
JHEP02(2019)027 in the space-like region (q2 < 0) in μ-e scattering as a function of the squared momentum transfer t [22, 23]
The Standard Model prediction for μ-e scattering at Next-to-Next-to-Leading Order (NNLO) contains non-perturbative QCD contributions given by diagrams with a hadronic vacuum polarization insertion in the photon propagator
By taking advantage of the hyperspherical integration method, we show that the subset of hadronic NNLO corrections where the vacuum polarization appears inside a loop, the irreducible diagrams, can be calculated employing the hadronic vacuum polarization in the space-like region, without making use of the R ratio and time-like data
Summary
We will give a short review of the hyperspherical integration method. In the hyperspherical approach, we cannot shift the loop momentum and we are left at the denominator with the product of propagators of the form 1/[(q − p)2 − m2], that has two poles in the q0 complex plane at q0± = p0 ± (q − p )2 + m2 ∓ iε. The general solution of an integral with three denominators in D = 4 was given long time ago by Laporta in a not very-well-known article [47] This is the main reason why our calculation is carried out in D = 4 and not in dimensional regularization, even if the hyperspherical method can be applied to D dimensions as well It is even possible to avoid an explicit calculation of IR divergent integrals containing Πhad(q2) by observing that in the soft limit the box diagrams are proportional to the “tree-level” amplitude, i.e. the Born amplitude with a dressed photon propagator in figure 1a. The integral Iμ contains, through the scalar product ei · Q, terms like Kj · Q/Dj or Kk · Q/Dk, which can be simplified as before, leading to an integrand without Q in the numerator
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