Abstract
We compute the two-loop QCD helicity amplitudes for the production of a Higgs boson in association with a bottom quark pair at a hadron collider. We take the approximations of leading colour and work in the five flavour scheme, where the bottom quarks are massless while the Yukawa coupling is non-zero. We extract analytic expressions from multiple numerical evaluations over finite fields and present the results in terms of an independent set of special functions that can be reliably evaluated over the full phase space.
Highlights
In this article we consider the two-loop amplitudes relevant for the production of a Higgs boson in association with a bottom-quark pair, i.e. pp → bbH, in the leading colour approximation. bbH production at the LHC has been a subject of great phenomenological interest due to its potential in directly measuring the bottom-quark Yukawa coupling
In the Standard Model (SM), the coupling strengths of the Higgs boson to the fermions and vector bosons are proportional to their mass, causing the rate of the bbH production to be suppressed with respect to, for example, Higgs production in gluon fusion or vector boson fusion, associated production with a vector boson, and associated production with a top-quark pair
We provide Mathematica scripts which illustrate how to evaluate the finite remainders interfered with the tree-level amplitudes for all the partonic channels contributing to the process pp → bbH, which we label as gg : g(−p3) + g(−p4) → ̄b(p1) + b(p2) + H(p5), qq : q(−p3) + q(−p4) → ̄b(p1) + b(p2) + H(p5), qq : q(−p3) + q(−p4) → ̄b(p1) + b(p2) + H(p5), bb : b(−p3) + ̄b(−p4) → ̄b(p1) + b(p2) + H(p5), bb : ̄b(−p3) + b(−p4) → ̄b(p1) + b(p2) + H(p5), bb : b(−p3) + b(−p4) → b(p1) + b(p2) + H(p5), bb : ̄b(−p3) + ̄b(−p4) → ̄b(p1) + ̄b(p2) + H(p5)
Summary
The tree-level amplitudes can be obtained using the BCFW recursion relations [93, 94] within the spinor helicity formalism. We ensure that the shifted brackets [ˆi , |ˆj] do not belong to particles of helicities i−, j+. For thebbggH channel we obtain the following non-vanishing tree-level partial amplitudes, A(0)(1 ̄+b , 2+b , 3+g , 4+g , 5H ) =. For thebbqqH channel the “all-plus” and MHV configurations vanish, and we are left with. In both cases, due to the colour decomposition of the full amplitudes given by eq (2.7), the A(0)(1+, 2+, 3−, 4+, 5H ) partial amplitude is related to A(0)(1+, 2+, 3+, 4−, 5H ) by swapping the particles 1 ↔ 2, 3 ↔ 4 , and flipping the overall sign for the subprocessbbggH. The remaining non-vanishing helicity configurations can be obtained by parity transformations, that is by swapping the brackets ↔ [ ]
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