Gluon-Fusion Production Cross Sections

Using techniques as described above, Higgs boson production cross sections in various phase-space regions can be computed. Among the various Higgs boson production modes, gluon fusion stands out because at the LHC it has the largest expected cross section and for that reason needs to be modeled with particular care. QCD corrections to the LO cross section prediction are sizable. Large efforts have been made to take these into account, resulting in achieving next-to-next-to-next-to-leading order (N³LO) accuracy for the inclusive gluon fusion cross section as of today [19]. Differential gluon fusion cross sections are currently available at next-to-next-to-leading order (NNLO), as described in Section 2.6.4.

In Figure 2.3 (a) the leading-order diagram for the gluon fusion process is depicted: a pair of gluons leads to a loop of a heavy quark, which in turn produces a Higgs boson. The amplitude of this type of diagram is proportional to the Yukawa coupling of the Higgs boson to the quark in the loop. Because the top quark is the heaviest quark, the gluon fusion diagram with at-quark in the loop gives the dominant contribution to the gluon fusion cross section. However, the gluon fusion diagrams with ab-quark orc-quark in the loop contribute to the cross section as well, in particular their interference with the gluon fusion diagram with at-quark in the loop.

Inclusive Cross Section

In the calculation of the inclusive gluon fusion cross section, terms up to N³LO are included [13, 19]. Results of NLO QCD corrections are described in References [23–25]. NNLO QCD corrections have been discussed in References [26–32]. References [33–36] present electroweak corrections to the gluon fusion cross section. Finally, the N³LO QCD corrections are described in References [20–22].

Ideally, the exact quark masses enter the calculations at each order of perturbation theory. At low scales, however, the dominant t-quark-loop contribution can be safely treated as a point-like interaction that effectively couples two gluons to a Higgs boson, i.e. thet-quark loop can be integrated out by taking m_t→ ∞, which greatly simplifies the cross section computation.

However, progress in the computation of matrix elements has allowed incorporating effects of a finitet-quark mass up to NNLO. The exact dependence onm_tis taken into account up to NLO [24, 25], whereas at NNLO an expansion in 1/m_t is used to correct for finite-m_t effects [29–32].

Similarly, at LO and NLO, the complexity of the calculation is sufficiently small to include also the exact effects from other quarks such asb, andc-quarks [24, 25, 81–86]. The contributions of b-quark or c-quark gluon fusion diagrams beyond NLO, including their interference with thet-quark loop, are not known. An estimate of the effect on the gluon fusion cross section of omitting these corrections has been computed in Reference [19] based on NLO results and is of the order of 1 %.

Differential Cross Sections

Differential cross sections are collections of cross sections in different phase-space regions as defined by a kinematic variable such as the Higgs boson transverse momentum p^H_T. See Section 3.2.2 for an introduction of some basic kinematic variables, including the transverse momentum. The perturbative expansion for non-inclusive cross sections tends to be less accurate than in the inclusive case. At the same time, the restriction of the phase space to the boundaries of the individual exclusive region as defined by the variable in consideration (bin) can make a resummation in some regions of phase space necessary.

In Chapter 6, an analysis of the p^H_T distribution aiming at extracting limits on quark Yukawa couplings is shown, which makes it worthwhile to discuss predictions for the p_T^Hdistribution in more detail. State-of-the-art predictions for the differential gluon fusion cross section in Higgs boson transverse momentum are of perturbative order NNLO in α_s and include logarithmic terms up to next-to-next-to-leading logarithms (NNLL). While the t-quark-loop contribution to the gluon fusion cross section and its dependence on thet-quark mass is a well understood problem, the calculation of the contribution of theb-quark loop to the gluon fusion differential cross sections is more challenging. At transverse momenta below thet-quark mass, thet-quark loop can be integrated out, which simplifies the computation by removing one loop from the considered diagram. In this approximation, NNLO accuracy was achieved. A similar approach for theb-quark loop is not possible except for transverse momenta below theb-quark mass, which is approximately 4 GeV at the relevant scales [87]. For transverse momenta above theb-quark mass, such approximation is invalid, and one has to consider the resolvedb-quark loop. The resulting complexity of the calculation currently restricts the accuracy to NLO. Even though the b-quark Yukawa coupling is considerably smaller than thet-quark Yukawa coupling, theb-quark contributions to the transverse momentum distribution should not be neglected, as they can be enhanced by large logarithmic factors of the form ln(m²_H/m²_b) and ln(p²_T/m²_b) [87]. Therefore, a resummation to all orders of these terms could be indicated; however, this has not been achieved so far. Similarly, a resummation of equivalent terms including the mass of thec-quark would be advantageous, although less important due to the smaller contribution to the gluon fusion cross section.

Below, the predictions for distributions in several kinematic variables that are presented in

comparison with the measured differential cross sections in Chapter 5 are outlined. The different predictions differ in matters such as resummation procedure and in some cases are specialized to describe particular distributions well. Several predictions of differential cross sections are obtained by means of asoft-collinear effective theory(SCET) [88, 89], which is an effective theory of QCD and can be used to resum logarithmic terms associated with emissions of soft and collinear gluons.

PowhegNNLOPS

The gluon fusion MC simulation that has been used in the measurement of inclusive and differential cross sections in Section 5 is PowhegNNLOPS [90], interfaced with the Pythia8 parton showering program [76, 77]. Gluon fusion production cross sections with different jet multiplicities in the final state are combined using the Minlomergingscheme [91]. Events are reweighted according to the Higgs boson rapidity such that resulting inclusive distributions are accurate to NNLO [92, 93]. Afterwards, a comparison of the resulting distribution in p_T^Hwith a prediction at NNLO accuracy of the p^H_T distribution as obtained by HRes[94, 95] is made, ensuring that both distributions agree within uncertainties. The masses oft- andb-quark are considered at LO and NLO [96]. The central renormalization and factorization scales have been chosen to bemH/2.

SCETlib+MCFM8

The MC simulation program MCFM8 [97] enables an NNLO calculation of the differential cross section in the rapidity of the Higgs boson. In this calculation, a regularization of the divergences of soft and collinear radiation is necessary. Both virtual loop corrections and radiative corrections involving soft or collinear partons lead to divergences in the computation if considered separately. In this prediction, a measure of the soft emission intensity is given by the observableτ₀, which quantifies how much an event appears like an event without jets [98].

Logarithmic terms of the form ln²(−1) = −π² are introduced to the gluon form factors³[99, 100], which are sufficiently large to impede the convergence of the perturbative expansion. The coefficients of those logarithmic terms are related to infrared singularities and exist at all orders inα_s. By resumming those logarithmic terms with NNLL⁰_φaccuracy, the resulting cross section

3Form factors are terms that encapsulate the effect of a non-point-like charge distribution in scattering objects at a given four-momentum transfer.

is more accurate and the perturbative uncertainties are reduced. The prime in NNLL⁰_φ means that also some parts of N³LL resummation contributions are taken into account, while the subscriptφ signifies that the resummation is performed in the gluon form factor.

SCETlib(STWZ)

The SCETlib(STWZ) prediction is a NNLO prediction of the differential distribution in the momentum of the leading jet, p_T^j1 [101]. The SCET framework is used [100], and the corre-sponding resummation of logarithms of the form ln(p_T^j1/mH) takes into account logarithms up to next-to-next-to-leading logarithm and someα²_s corrections that are not part of the resummation (NNLL⁰).

NNLOjet+SCET

The NNLOjet+SCET prediction [102] for the differential cross section in p_T^H consists of an NNLO QCD computation of the gluon fusion process in the limit of an infinitet-quark mass.

The large logarithmic terms log(p_T^H/m_H) at low transverse momentum of the Higgs boson are resummed to all orders, using SCET and includes sub-leading logarithmic terms up to the third level (N³LL) [103].

RadISH

The RadISH (short forRadiation offInitial State Hadrons) gluon fusion prediction [104] for the differential cross section in p_T^H as used in this thesis has NLO accuracy in QCD perturbation theory. It is obtained through differentiation of the resummed NNLL+NNLO result in the total phase space. Details on the treatment of quark masses are described in Reference [105].