5.2 Higgs Pair Production and Anomalous Couplings

At proton-proton colliders, the main production mechanism for a Higgs pair is gluon fusion. The gluon fusion interaction at leading order includes a fermion loop as depicted in the top diagrams of Figure 5.1, which is largely dominated by the contribution from top and bottom quarks, and thus explaining the low expected production rate listed in Equation 5.1. The most common production mode, labelled as (b) in Figure 5.1, features a triangular fermion loop followed by the production of an off-self Higgs boson, that in turn decays on two on-shell Higgs bosons via a triple Higgs boson interaction vertex. In addition, within the SM is also possible to produce a pair of Higgs bosons at leading order through a fermion box loop, as shown in diagram (a) of Figure 5.1, which evidently does not depend on the Higgs self-coupling. Both box and triangle loop contributions interfere destructively in the SM amplitude to give rise to the total HH production.

Figure 5.1: Set of HH production Feynman diagrams, representing all gluon-induced processes at leading order. The interactions depicted by (a) and (b) represent processes that are expected within the SM, while the contact interactions between the Higgs bosons and gluons (c) and (d), as well the contact interaction of two Higgs bosons with top quarks (e), are effective diagrams of BSM interactions. Figure adapted from [148].

New physics at higher energy scales can affect processes and observables at the electroweak scale, such as Higgs pair production. As reviewed in Section 1.2.2, the effective field theory (EFT) approach is a way to calculate observables of possible extensions of the SM without being tied to a certain class of BSM model, by adding non-renormalisable local interactions. In the context of Higgs pair production, the effect of new operators can be parametrised by the following effective Lagrangian: \[ \begin{aligned} \mathcal{L}_\textrm{H} = \frac{1}{2} \partial_{\mu}\, \textrm{H} \, \partial^{\mu} \textrm{H} - \frac{1}{2} m_\textrm{H}^2 \textrm{H}^2 - {\color{red} \kappa_\lambda} \, \lambda_\textrm{SM} v\, \textrm{H}^3 \\ - \frac{m_\textrm{t}}{v}(v+ {\color{red}\kappa_\textrm{t}} \, \textrm{H} + \frac{{\color{red}c_2}}{v} \textrm{HH} ) \,( \bar{\textrm{t}}_\textrm{L} \textrm{t}_\textrm{R} + \textrm{h.c.}) \\ + \frac{1}{4} \frac{\alpha_\textrm{S}}{3 \pi v} ( {\color{red}c_\textrm{g}} \, \textrm{H} - \frac{ {\color{red}c_\textrm{2g}}}{2 v} \, \textrm{HH}) \, G^{\mu \nu}G_{\mu\nu}\, \end{aligned} \qquad(5.2)\]

where \(v=246\ \textrm{GeV}\) is the vacuum expectation value of the Higgs field. After neglecting the enhanced coupling of the Higgs boson with bottom quarks due its experimental constraints and the presence of new light particles, a total of five EFT parameters remain, which are highlighted by using red colour in Equation 5.2. The factors \(\kappa_\lambda = \lambda_\textrm{HHH}/\lambda_\textrm{SM}\) and \(\kappa_\textrm{t}= y_\textrm{t}/y_\textrm{SM}\) account for possible deviations from the SM of the Higgs boson trilinear coupling and the top quark Yukawa coupling, thus effectively modifying the relative weight of the SM Feynman diagrams described at the beginning of the section. The absolute parameters \(c_g\), \(c_{2g}\) and \(c_2\) instead lead to new contact interactions not expected within the SM, represented in the (c), (d) and (e) Feynman diagrams of Figure 5.1, and which could arise by mediation of heavy particles beyond the electroweak scale. The previous parametrisation is commonly referred to as dimension-six non-linear or anomalous couplings EFT, however alternative approaches exist, such as the so-called linear EFT [173] which is more appropriate to model smaller BSM effects.

A theoretical prediction for the differential and total cross section for each point in the mentioned five-dimensional EFT parameter space \((\kappa_\lambda, \kappa_\textrm{t}, c_2, c_\textrm{g}, c_\textrm{2g})\) can be computed as outlined in Section 1.3. The distribution of the final state kinematical variables, i.e. the relative angles and momenta of the Higgs pair, can depend substantially on the value of some of these couplings. A naive grid or random scan of the full five-dimensional space would require simulated samples of observations at too many EFT points and hence it is not feasible. While this signal modelling issue could be tackled by means of event re-weighting, as described in Section 3.1.2.3, it is useful to consider a different methodology to represent the main properties of the anomalous couplings parameter space where only a reduced number of EFT points are considered.

For the analysis presented in this work, a total of twelve EFT points referred to as benchmarks are considered, which have been chosen via a agglomerative clustering procedure so they represent the main kinematical topologies in the parameter space. The details of the clustering methodology are detailed in [174], but they amount to the construction of a distance between the main kinematic distributions at generator level of each pair of EFT points. The parameters corresponding to each of the benchmarks, as well as those corresponding to the SM model and the case where Higgs boson self coupling is zero, are included in Table 5.1.

Table 5.1: Effective field theory parameters for the anomalous couplings benchmarks considered in this analysis, as defined in [174], as well as the modified couplings corresponding to the Standard Model.