Abstract
Precision studies of the Higgs boson at future e+e− colliders can help to shed light on fundamental questions related to electroweak symmetry breaking, baryogenesis, the hierarchy problem, and dark matter. The main production process, e+e−→ HZ, will need to be controlled with sub-percent precision, which requires the inclusion of next-to-next-to-leading order (NNLO) electroweak corrections. The most challenging class of diagrams are planar and non-planar double-box topologies with multiple massive propagators in the loops. This article proposes a technique for computing these diagrams numerically, by transforming one of the sub-loops through the use of Feynman parameters and a dispersion relation, while standard one-loop formulae can be used for the other sub-loop. This approach can be extended to deal with tensor integrals. The resulting numerical integrals can be evaluated in minutes on a single CPU core, to achieve about 0.1% relative precision.
Highlights
These ISR effects are enhanced by logarithmic terms of the form log(s/m2e)
This article proposes a technique for computing these diagrams numerically, by transforming one of the sub-loops through the use of Feynman parameters and a dispersion relation, while standard one-loop formulae can be used for the other sub-loop
The derivation of the numerical integral representations for the planar and non-planar two-loop box diagrams is discussed in detail
Summary
Let us initially consider a basic scalar planar integral, which contains the propagators of the diagram in figure 1 (top-left) but 1 in the numerator: Iplan =. The σ integral blows up at the lower boundary, and the term in [ ] is divergent for σ → σ0, whereas only the sum of the two is finite To circumvent this problem, one can modify the integrand according to dx dy. The q1 loop will in general contain terms with different powers of q1 in the numerator, some of which originate from eq (2.11). These lead to Passarino-Veltman functions D1, D2, D3, D00, etc. For the double-box diagrams in figure 1 tensors up to rank 3 in at least one of the two loops are encountered
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