Bounds to the Higgs Sector Masses in Minimal Supersymmetry from LHC Data

We update our analysis of the Higgs sector masses in Minimal Supersymmetry and in the Two-Higgs Doublet Model to the final ATLAS and CMS data from the 2011-2012 LHC run.

Introduction. The search for positive indications of low energy Supersymmetry is one of the challenges of present LHC experiments. Realistic models with broken Supersymmetry, however, admit many free parameters so as to make the search for the supersymmetric companions of the known particles a rather complex affair.
One univocal prediction of Supersymmetry is the need for an extended Higgs field structure, with respect to the Standard Theory: at least two SU (2) ⊗ U (1) doublets in its simplest version, the Minimal Supersymmetric Standard Model (MSSM).
In a previous paper [1] we stressed that, with the newly discovered boson h(125) [2,3] identified with the lightest MSSM Higgs boson, the physics of the Higgs sector has a very mild dependence upon the parameters of the full supersymmetric theory. The leading radiative corrections may be fixed by the h mass so that the Higgs sector physics is described by two parameters only, the ratio of the two vacuum expectation values, conventionally denoted as tan β, and the mass of one of the two other neutral Higgs bosons, H (J P = 0 + ) or A (J P = 0 − ), for a recent discussion see [6].
In a second paper [4], a comparison was made of the Higgs sector MSSM description with the first data on the production and decay modes of h(125) made available by the ATLAS and CMS collaborations at CERN [2,3] and by the CDF and D0 collaborations at the Tevatron [5]. A consistent fit was obtained and lower bounds to the mass of H were derived, of 320 (2σ) and 360 GeV (1σ), with the masses of A and H ± essentially degenerate with H in this mass range. The theoretical decay branching fractions of H and A were reported as functions of M H and tan β. A picture of partial decoupling of the additional Higgs sector with tan β = 1 − 10 seems viable. The partial decoupling scenario goes well with the picture of a relatively light scalar top envisaged in [7] and is not inconsistent with the lack of observed deviations, so far, in flavour changing neutral currents from the predictions of the Standard Theory (ST), see e.g. [8,9].
In the present paper we update our analysis to the final ATLAS and CMS data from the 2011-2012 LHC run [10,11], also extending the MSSM analysis to the more general Two-Higgs Doublet Model (THDM). The extended Higgs sector of the Next to Minimal Supersymmetric Model (NMSSM) has been compared to present data in [12,13].
Couplings. We denote, as usual, by H 0 d and H 0 u the 0 + neutral scalar fields and define 0 ≤ β ≤ π/2 according to: For conserved CP symmetry, the physical 0 + particles are orthogonal combinations of these two fields, determined by a second angle α. The two angles determine the couplings of the Higgs scalars to quarks, leptons and gauge bosons. In formulae: The ratios of the couplings of h and H to the ST couplings for the different channels are summarised in Tab. I. We may require the coupling to vector bosons to be positive and α is restricted to the range: The line α = β in the (β, α) plane maps into the ST point c t = c b = c V = 1 in coupling constant space. The above equations suggest to fit the cross section and decay rates data of h(125) with three independent couplings [4], c t , c b , c V for the up quarks and leptons, down quarks and leptons and for the vector boson, respectively, rather than with a single normalization factor (denoted by µ in Refs. [2,3]) or with one fermion and one vector coupling [10,11] [14,15].
The MSSM mass matrix. The structure of the scalar mass matrix in MSSM has been studied extensively (see e.g. Ref. [16]). We start from the expression: Radiative corrections due to b and other lighter quarks are neglected, which is justified for the relatively small values of tan β we shall consider. The corrections embodied by δ 1 and δ arise from t quark and t scalar quarks loop and are given approximately by [17]: where M 2 SUSY = Mt 1 Mt 2 fixes the scale of supersymmetry breaking andX t is a mixing parameter [17] with: We note that δ 1 is a subdominant correction. In fact, δ 1 = O(g 2 y 2 t ) while δ = O(y 4 t ), with g and y t the electroweak and the t-quark Yukawa couplings, see Eqs. (6) and (7). Numerically, we find |δ 1 /δ| ≤ 0.25 in the range 0.5 ≤ M SUSY ≤ 10 TeV and 0 ≤X t ≤ 6. Within this precision 1 , we shall neglect δ 1 .
One obtains δ as function of M A and tan β from the secular equation for M h , without any assumption on the mixing parameterX t .
One finds, explicitly: with the large M A limit: Eq. (10) has a singularity for: 1 alternatively, one may assume a given value ofXt, express both δ 1 and δ as functions of t, Eq. (8), and use the secular equation of M h to eliminate t. In this case all results depend upon the assumed value ofXt. In the range (9), we find the results to be insensitive to the value ofXt and numerically indistinguishable from the results obtained by neglecting δ 1 .
which implies a lower bound for M 2 A , since, below this value, the second eigenvalue, M 2 H , becomes negative. Within the validity of Eq. (5) with δ 1 neglected, we find, numerically: Substituting Eq. (10) into (5) Three-couplings fit to the h(125) data. It is straightforward to express the ST decay widths of h in terms of the parameters (c t , c b , c V ): we refer to [1,4] for details. We compute the ST widths with the code HDECAY [18]. The t quark couplings appears in the h → γγ loop amplitude, together with c V , so that the decay width is sensitive to their relative sign. In MSSM there are also contributions from the scalarst 1 andt 2 which renormalize the t quark coupling: c t →c t , making it different from the coupling of the up-like quarks. For a mass of the lighter scalar t quark around 0.5 TeV, the correction amounts [4] to about 3% and, in any case, the contribution from u and c quarks to the decay width are entirely negligible, so we may for the moment disregard this effect.
The total width, necessary to obtain the branching fractions, is computed by adding individual widths in the ST channels. There may be invisible channels in h decay, e.g. due to light SUSY neutralinos [19]. This would appear as a reduction of the couplings obtained from the fit by a common factor f < 1.
The h production cross section receives two contributions. A dominant one due to gluon-gluon fusion, which scales with c t and the subdominant Vector Boson Fusion, which scales with c V 2 . We have computed the ST cross sections for p + p → H + 2 jets using Alpgen [20] and applied the experimental cuts on jet energy and on the diphoton rapidity, for the p + p → h → γγ case, spelled out by the CMS collaboration [3]. Similar cuts apply to the ATLAS case, see [4] for details.
With the new data in [10,11] and Tevatron data [5] we find one absolute minimum with positive c t : c t = 0.99c b = 1.23c V = 1.11 (χ 2 = 11.7, χ 2 /dof = 1.46) (18) and a slightly unfavored local minimum: We illustrate in Fig. 1 Positions of the two minima of the three-dimensional fit, the 1σ and 2σ regions, and the ST point, which lies well inside the 1σ region of the absolute minimum, corresponding to c t > 0. For comparison with other work, we have made a fit with c t = c b = c F . In this case, the difference of χ 2 between the absolute minimum (c t > 0, χ 2 = 12.6) and the local minimum (c t < 0, χ 2 = 14.6) is more pronounced.
We have performed also a χ 2 fit to the two parameters, α, β of the Two Higgs Doublets Model (THDM). The distribution of the points with χ 2 < 15 is reported in Fig. 2. Points cluster along the ST line α = β, confirming the results in [21], see also [22,23].
The situation in coupling constants space is illustrated in Fig. 3. and: These values are considerably larger than the bounds in Eq. (13) and, more important, they are consistent with and slightly larger than the experimental lower bounds derived by ATLAS and CMS [24,25]. Unfortunately, there are no upper bounds to M H since, for M H increasing, the SUSY point is driven towards the ST point, which is well within 1σ from the best fit.
Conclusions. The intermediate decoupling region, with tan β = 1 − 10, M H in the few hundreds GeV region and a scalar top below 1 GeV, is not excluded by the data on h(125) and by Flavor Changing Neutral Currents data. The three couplings fit to the data is very effective for mapping the allowed region of the heavier scalars predicted by MSSM. The lower bounds: M H ≤ 340(360) GeV are derived at 2σ (1σ) confidence level, with A and H ± essentially degenerate. These values are consistent with recent experimental bounds by ATLAS and CMS [24,25]. The fit is sensitive to the experimental cuts. A fit produced by the LHC collaborations with  the precise experimental cuts they have adopted would be extremely useful.