Towards a new approximation for pair-production and associated-production of the Higgs boson

Abstract

We propose that loop integrals with internal heavy particles can be evaluated by expanding in the limit of small external masses. This provides a systematically improvable approximation to the integrals in the entire phase space, and works particularly well for the high energy tails of kinematic distributions (where the usual $1/M$ expansions cease to be valid). We demonstrate our method using Higgs boson pair production as an example. We find that at both one-loop and two-loop, our method provides good approximations to the integrals appearing in the scattering amplitudes. Comparing to existing expansion methods, our method are not restricted to a special phase space region. Combining our efficient method to compute the two-loop amplitude with an infrared subtraction method for the real emission corrections, we expect to have a fast and reliable tool to calculate the differential cross sections for Higgs boson pair production. This will be useful for phenomenological studies and for the extraction of the Higgs self-coupling from future experimental data. Our method can also be applied to other processes, such as the associated production of the Higgs boson with a jet or a $Z$ boson.

Figures (9)

• Figure 1: Typical Feynman diagrams for (a) Higgs boson pair production and (b) Higgs boson production associated with a jet in the gluon-fusion channel at the leading order.
• Figure 2: The real part (left plots) and the imaginary part (right plots) of the order $\epsilon^0$ coefficient of the one-loop integral $I_{1,1,1,1}$. The upper plots fix $\sqrt{-t}=\unit{200}{\GeV}$ and show the integral as a function of $\sqrt{s}$, while the lower plots fix $\sqrt{s}=\unit{1000}{\GeV}$ and show the integral as a function of $\sqrt{-t}$. In the lower panels of each plot, we show the relative errors of the approximate results against the exact result (see also the text for definition). The integral has been multiplied by $m_t^4$ to make it dimensionless.
• Figure 3: Left side: the partonic total cross section as a function of $\sqrt{s}$. Right side: the transverse momentum distribution of the Higgs boson at the parton level with $\sqrt{s}=\unit{1000}{\GeV}$.
• Figure 4: Topologies relevant to the NLO QCD corrections to Higgs boson pair production after expansion in the small $m_h$ limit. The thick lines represent massive propagators (top quarks), while the thin lines represent massless propagators (gluons). The external legs (dashed lines) are all light-like.
• Figure 5: Pre-canonical master integrals in topology E. The thick lines represent massive propagators (from top quarks) and the thin lines denote massless propagators (from gluons). The labels $s$, $t$ and $u$ on the external lines represent the (squared) momenta flowing through those legs. The external lines without labels have light-like momenta.