ON THE RENORMALIZATION OF THE SINGLET EXTENSION OF THE STANDARD MODEL

Abstract

In June of 2012, the LHC experiment [1, 2] has finally completed the spectrum of the Stan dard Model with the discovery of the Higgs boson, predicted in the 60’s by Higgs [3, 4], Englert, Brout [5], Guralnik, Hagen and Kibble [6]. However, the structure and the physics behind the Higgs sector are not completely clear and this represents a possible gateway to the manifold conceivable extensions of the Standard Model (SM). One of the simplest renormal izable enlargement of the Higgs sector is constructed by adding to the SM Lagrangian one additional spinless real electroweak singlet, which develops its own vacuum expectation value [7, 8, 9, 10, 11, 12, 13, 14]. Beside being easy to implement, the physics of a scalar singlet has received a lot of attention in the recent years for several reasons; among them, it can help in solving the issues related to the metastability of the electroweak vacuum [15, 16] if the Higgs potential receives a correction due to new physics which modify it at large field values [17] and it could provide a door to hidden sectors [18] to which it is coupled. The singlet model has the advantage of depending on relatively few parameters and this implies a feasible experimental study at the LHC for the analysis of the new physic effects in the Higgs boson couplings, searches for heavy SM-like Higgs bosons [19, 20] and direct searches for resonant di-Higgs production [21, 22, 23]; in the absence of linear and triple self-interactions, this model possesses a Z2-symmetry and the singlet can be a viable candidate for dark matter, although for masses somehow larger than 500 GeV [24, 25] the couplings of the dark matter to the known particles occur only through the mixing of the singlet field with the SM Higgs boson. Without a Z2-symmetry a strong first order electroweak phase transition is allowed and additional sources of CP violation occur in the scalar potential. In this thesis we limit ourselves to a situation where the new singlet s 0 communicates with the SU(2)L doublet φ only via a quartic interaction of the form, κ (φ †φ)(s 0 ) 2 . This implies that the would-be Higgs boson of the SM mixes with the new singlet leading to the existence of two mass eigenstates, the lighter of which (H) is the experimentally observed Higgs boson whereas the heaviest one (S) is a new state not seen so far in any collider experiments. We call this model the Singlet Extension of the SM (SSM). Since only φ is coupled to ordinary matter, the main production mechanisms and decay channels of H and S are essentially the same as those of the usual SM Higgs particle, with couplings rescaled by quantities which depend on the scalar mixing angle, called α, whose bounds have been 1 discussed in details in [10, 11, 26, 27]. For masses larger than & 200 GeV, the most important decay channels of the heavy state S are those to a pair of vector bosons S → V V and, when kinematically allowed (mS > 2 mH), to a pair of lighter scalars and top quarks, S → HH,tt¯ . With the run II at LHC, the exploration of the scalar sector is expected to reveal more details. So, the comparison between theory and data requires precise predictions obtained through higher-order calculations. To this aim, we evaluated the radiative corrections to the main decay rates Γ(S → ZZ, W+W−, tt, HH ¯ ) and studied in details their dependence on the singlet mass mS as well as on the mixing angle α and the singlet vev w. Interestingly enough, the SSM scalar sector implies no natural way of defining the renormalized scalar mixed mass (or alternatively, the scalar mixing angle) and the non-diagonal fields through a physically motivated renormalization scheme. As a consequence, it is possible to construct different prescriptions to renormalize the non-diagonal scalar sector; nevertheless, we have to pay attention to their definitions since some of them manifest a gauge dependence in the physical observables. To compute the next-to-leading order (NLO) EW decay rates, we use the "improved on-shell" renormalization scheme which is totally gauge-invariant [29]. To give a comment on the gauge dependence effect on the renormalized decay widths we also consider a second scheme, called "minimal field", which exhibits a gauge dependence [29]. The minimal field scheme is defined by renormalization conditions which need the introduction of a renormalization scale µR. We prove that it is possible to obtain a gauge independent result by fixing this scale at µ 2 R = (m2 H + m2 S )/2 since, for this specific value, the improved on-shell and the minimal field schemes are equivalent. The main result of this thesis is that for the singlet scalar mass range 200 ≤ mS ≤ 1000 GeV the gauge independent EW corrections to the decay widths reach a maximum of O(6%) in the W+W− channel, O(5%) in the ZZ channel and O(4%) in the HH, tt¯ channels for masses lower than 450 GeV and almost independently on the mixing angle α (the HH channel is the only one to show a more pronounced mixing dependence in the mass region for which its NLO correction is maximal), whereas for larger masses (mS & 700 GeV) these corrections take negative values. Besides, we discuss the impact of the QCD corrections on the S → tt¯ channel which can be directly deduced by the SM QCD one-loop contributions to the Higgs decay into a top quark pair. For the total decay width Γ(S → all), we obtain a maximum correction of O(6%) for mS ∼ 200 GeV. Finally, we have analyzed the impact of the gauge dependence on the decay rates with respect to µR for two fixed values of mS = 400, 1000 GeV and found that it causes a variation on the NLO decay widths which is less than . |3|% in all decay channels. The structure of the thesis is as follows: in Chap.1 we recall the relevant features of the SSM and its theoretical and experimental constraints; in Chap.2, we describe and analyze the full set of the leading-order (LO) expressions of the scalar singlet decay widths; in Chap.3 we illustrate the details of our renormalization procedure that we apply in Chap.4 to discuss the structure of the Γ(S → ZZ, W+W−,tt, HH, all ¯ ) renormalized decay widths. The radiative corrections to these decay rates are numerically computed in Chap.5; the last chapter is devoted to our conclusions.