Polarised amplitudes and soft-virtual cross sections for b$$ \overline{b} $$→ Z H at NNLO in QCD

Abstract

Production of the Higgs boson, H in association with a massive vector boson, V , i.e., the V H process, plays an important role in the explorations of Higgs physics at the Large Hadron Collider, both for a precise study of Higgs' Standard Model couplings and for probing New Physics. In this publication we present the two-loop corrections in massless quantum chromodynamics (QCD) to the amplitude of the Higgs production associated with a Z boson via the bottom quark-antiquark annihilation channel with a non-vanishing bottom-quark Yukawa coupling, which is a necessary ingredient of the full next-to-nextto-leading-order QCD corrections to the V H process in the five-flavour scheme. The computation is performed by projecting the D-dimensional scattering amplitude directly onto an appropriate set of Lorentz structures related to the linear polarisation states of the Z boson. We provide analytic expressions of the complete set of renormalised polarised amplitudes in terms of polylogarithms of maximum weight four. To give an estimation of the size of contributions from amplitudes considered in this work, we compute numerically the resulting cross sections under the soft-virtual approximation. We also take the opportunity to make a dedicated discussion regarding an interesting subtlety appearing in the conventional form factor decomposition of amplitudes involving axial currents regularised in D dimensions.

Two-Loop QCD Helicity Amplitudes for Higgs Production Associated with a Vector Boson through Bottom Quark Annihilation

“…By substituting the obtained results of the polarisation tensor and the definition of γ 5 according to (1.3), and then performing the simplification using 4-dimensional Lorentz and spinor algebra, we choose a set of 6 projectors, P µ i to be used in D-dimensions. The explicit form of the projectors can be found in [12]. We emphasise that since by construction the momentum basis representations of polarisation state vectors fulfil all the defining physical constraints, hence the index contraction between the projectors and the amplitude (2.1) is simply done with the space-time metric tensor g µν instead of physical polarisation sum.…”

“…The normalisation factors are explicitly given in ref. [12]. Each helicity amplitude can be decomposed into vector and axial parts.…”

“…The aforementioned basis of Lorentz structures is complete in D-dimensions even without demanding the transversality of the Z boson's physical polarisation states, since we prefer to use the simple metric tensor g µν instead of the physical polarisation sum rule for index contraction in projections. By computing the inverse of the Gram matrix of the aforementioned basis, we obtain the explicit form of the projectors for the above form factor decomposition which can be found from [12]. By applying the derived projectors onto the Feynman diagrams, we obtain the vector part of the amplitudes.…”

“…which is constructed under the condition of even powers in elementary Dirac matrices combined with the P-odd constraint from the axial coupling. By substituting the D-dimensional definition of γ 5 , we calculate the D-dimensional projectors through computing the inverse of the Gram matrix whose explicit forms can be found from [12]. Now we address the question whether the vector and axial parts of the amplitudes, M vec and M axi , live in a space spanned by the aforementioned Lorentz structures depicted through (2.4) and (2.5) in D-dimensions.…”

“…The hat (ˆ) represents the bare quantity. The expressions for the aforementioned renormalisation constants can be found from [12]. The UV renormalised non-singlet current does not exhibit the Ward identity.…”