A new implementation of the NMSSM Higgs boson decays

Supersymmetric theories provide elegant extensions of the Standard Model. Among these the NMSSM is a model which in addition to the content of the MSSM includes a new singlet to solve the μ problem. We present here the basic features of the NMSSM Higgs sector and a new Fortran implementation of the decays of the Higgs bosons in the NMSSM, both with real and complex parameters.


Introduction
Now that a new boson has been discovered by the AT-LAS and CMS Collaborations [1,2], an important question remains to be solved: is it the Standard Model (SM) Higgs boson or some exotic beyond the SM (BSM) Higgs particle? Among the various BSM models, supersymmetric (SUSY) extensions are the primary candidates to solve various problems of the SM, as for example the hierarchy problem. The Next-to-Minimal Supersymmetric SM (NMSSM) is a SUSY extension where in addition to two Higgs doublets H u and H d a Higgs singlet field S is included to dynamically generate the µ-term, thereby solving the so-called " µ-problem" of the MSSM. Indeed, a SUSY invariant term µH u · H d can be found in the MSSM Lagrangian. The coupling µ, being a parameter with mass dimension not linked to any SUSY-breaking scale, is naturally of the order of the largest scale available in the theory (or equal to zero), see Ref. [3]. This is in contradiction to phenomenological requirements that µ is of the order of the electroweak scale, unless a high level of fine-tuning is involved. By dynamically generating the µ-term in the NMSSM through the singlet Higgs vacuum expectation value v s in the trilinear coupling λ SH u ·H d , µ eff = λv s / √ 2, this fine-tuning problem is avoided. The Higgs sector is then less restricted: the tree-level bound on the lightest Higgs boson mass is higher than in the MSSM, meaning that smaller quantum corrections are required to get M h ∼ 126 GeV for the SM-like Higgs boson. A review of the NMSSM can be found in Ref. [4,5].
Using complex parameters gives rise to CP violation in the model as required by the baryon asymmetry in the Universe, by its implementation already at tree-level in the NMSSM Higgs sector for example. In addition, the phenomenology of the Higgs bosons is substantially modified as all the Higgs states can mix and there are no a Speaker; e-mail: baglio@particle.uni-karlsruhe.de pure CP-even or CP-odd states anymore. In order to compare with recent experimental results it is mandatory to assess, besides the spectrum, the decay pattern of the NMSSM Higgs bosons. In the following, we will present our implementation of the NMSSM Higgs boson decays either with real (CP-conserving R-NMSSM) or complex (CP-violating C-NMSSM) parameters. In particular the state-of-the-art higher-order corrections will be included as well as the renormalization group evolution of the parameters.

Parametrization
In this section we concentrate on the (complex) Higgs sector and its parametrization. The NMSSM superpotential gives rise to the following scalar Higgs potential, with complex λ, κ, A λ and A κ . We then introduce the vacuum expectation values for the Higgs fields and use the following parametrization with the phases φ u , φ s , In the Higgs sector there are only six CP-violating phases that are needed in the scalar potential: φ u , φ s , φ λ , φ κ , φ A λ and φ A κ . In other sectors of the model further CP-violating phases appear such as those of M 1 , M 2 , etc. Among the six phases described above, only one single combination is actually relevant at tree-level, the other possible combinations being fixed by tadpole conditions [4,5].

Higgs boson masses at one-loop level
Higher order corrections to the masses of the Higgs bosons are available at one-loop in the R-NMSSM [6] and in the C-NMSSM [7], and even up to two-loop order in the R-NMSSM [8]. In Refs. [6,7] a mixed renormalization scheme was used in which on-shell renormalization was used for the physical parameters while the DR scheme was used to renormalize the parameters tan β, v s , |λ|, |κ|, |A κ | and the phases in case of the C-NMSSM. It has been shown that all the counter-terms for the phases, with the exception of the phases linked to the two tadpole conditions t a and t a s , vanish [7]. The effects of quantum corrections are sizeable. We display two cases taken from Ref. [7], first with tree-level CP violation where Ψ 0, more precisely φ κ 0, φ λ = φ u = φ s = 0, then without tree-level CP violation where Ψ = 0 but with φ λ = φ κ and φ κ 0.
One-loop corrections to the masses are of utmost importance to conduct realistic studies, as can be seen in  GeV. The dashed area shows the region not compatible with this requirement at one-loop, the grey area shows the region not compatible with exclusion limits from the LEP, Tevatron and 7 TeV LHC data. In the case with no CP violation at tree-level but still non vanishing phases φ λ , φ κ , the impact of the latter is negligible for the H 1 and H 2 states. The stronger the coupling to the stops is, the stronger the one-loop corrections to the Higgs boson mass are influenced, the stop mass being a ffected byφ λ . As in this scenario φ λ = φ κ , the heavier H 3 mass increases with increasing φ κ (= φ λ ) as can be seen in Fig. 2 while the H 1 and H 2 masses remain nearly constant, H 3 having a large h u component strongly coupled to the stops. Note also that again the quantum corrections are not negligible as they increase M H 3 by 3 − 6% as shown in Ref. [7].

Description of the Fortran implementation
Our new implementation of the decays of the NMSSM Higgs bosons uses the one-loop masses decribed in section 2.2 as input masses. We have built a Fortran code for the decays of the NMSSM Higgs bosons based on the program HDECAY [9,10]. The user can choose to use either real (CP-conserving) or complex (CP-violating) parameters with CP-violating phases in the Higgs sector. Higher order corrections are included in the relevant decays as well as the renormalization group equations (RGEs) at one loop in the R-NMSSM [11,12]. The implementation of the RGEs in the C-NMSSM is still work in progress.

Comparison between the R-NMSSM and the C-NMSSM
In order to show the potential of the code, we have started preliminary analyses. We start from real NMSSM benchmark points [12] and tune the phases with two requirements: • the SM-like Higgs state should reproduce the observed excess within a 10 GeV range, 120 GeV ≤ M H SM ≤ 130 GeV;   • the phases φ A λ and φ A κ are fixed by tadpole conditions, φ λ = φ κ = φ s = 0 and φ u 0. We start with a fixed value φ u = 0.1.
The set of parameters is given in Table 1. All the parameters are set at the SUSY scale Q = 540 GeV in the DR scheme.
We display in Fig. 3 the decay branching fractions for the H 1 state as a function of the Higgs mass M H 1 obtained when varying |A λ |. On the left-hand side the branching ratios are depicted, on the right-hand side the ratio between the branching ratios and the SM predictions is displayed. We show a comparison between the SM, the C-NMSSM and the R-NMSSM. In the latter model we use the same set of parameters described above but with the phases turned to zero. The NMSSM branching fractions are higher than those of the SM except for H 1 → bb and H 1 → τ + τ − so that the sum of the branching fractions is equal to 1. The shapes are similar in the real and complex NMSSM as shown in Fig. 3, with the notable exception in the region near M H 1 = 130 GeV, where the C-NMSSM predictions are much closer to the SM predictions than in the real NMSSM case.
We display in Figs. 4 and 5 the branching fractions in two specific channels to comment in more detail on the decay pattern. Starting with the decay H 1 → bb, Fig. 4 shows that the NMSSM predictions are smaller than the SM predictions. Nevertheless, the C-NMSSM branching fraction is always closer to the SM prediction than the R-NMSSM prediction. There is even a cross-over at M H 1 ≃ 128 GeV (the same cross-over is found in the H → WW * branching fraction) where the C-NMSSM prediction matches the SM branching fraction. The same comment can also be made for the decay H 1 → τ + τ − .
It can be noted that in this particular case turning on the phases in the Higgs sector helps to reproduce the observed value of the signal strength in the diphoton channel σ obs /σ SM ≃ 1.8 at M H = 125 GeV as can be seen in    5. In this decay the situation is inverted compared to H → bb decay and the SM prediction is the lowest. This is due to the reduced decay width into bb, which is the dominant decay, hence it induces a smaller total decay width and therefore an enhanced diphoton branching ratio. This means that although reducing the deviations from the SM, the C-NMSSM scenario can nevertheless account for the excess of the data in the diphoton channel 1 .

Phase variation in the C-NMSSM
We now modifiy the set-up of section 3.2. The phase φ u is not fixed anymore and varies in the range 0 ≤ φ u ≤ 0.35, starting from an R-NMSSM scenario (φ u = 0). Above the upper limit of φ u = 0.35 the tadpole conditions cannot be fulfilled and the model is not valid anymore. The goal is to demonstrate the sensitivity of the Higgs boson decays to the complex phases in the Higgs sector. In our particular example the variation of the phase φ u has a significant impact on the branching fractions. Note that the chosen scenario for this particular example is not required to fulfill collider constraints when the phase φ u is different from φ u = 0.1.
We display in Fig. 6 the masses of the two lightest Higgs states H 1 and H 2 as a function of the phase φ u . The lightest mass is strongly affected by the variation of φ u and M H 1 ≃ 125 GeV is obtained for φ u = 0.2. The heavier state H 2 , however, has a nearly constant mass all over the range, M H 2 ≃ 167 GeV. Interestingly enough, while the mass of the state H 1 is strongly affected by the phase φ u , the branching fractions are hardly modified when increasing φ u (except for φ u ≥ 0.30), as depicted in Fig. 7. The decomposition of the lightest Higgs state H 1 in terms of the electroweak eigenstates is shown in Fig. 8, where |S 1 j | 2 is the square of the absolute value of the mixing term between H 1 and the electroweak eigenstate Φ j : • 0.1 < φ u ≤ 0.2: H 1 is still mostly a CP-even doublet state but the singlet a s component increases; • φ u > 0.2: H 1 is a CP-mixed state with singlet admixture, the a s component being more important than the h d component. 1 This assumes that the production cross sections are not strongly affected by the phases. This has to be confirmed, though. Even though H 1 is a specific C-NMSSM state and even though its mass is strongly affected by the phase φ u , its decay branching fractions are hardly affected in this scenario.
The case of the heavier state H 2 is very different. The decomposition into electroweak eigenstates again shows three distinct regions, see Fig. 9: • φ u ≤ 0.03: the H 2 state is a pure singlet state nearly CP-even; • 0.03 < φ u ≤ 0.27: H 2 is still mostly a singlet state but there is a strong CP mixing between the h s and a s states. There is a cross-over at φ ≃ 0.27 where a s becomes the dominant singlet component; • φ u > 0.27: H 2 is a CP-mixed state with a strong mixing between the singlet and the doublet components. There is a cross-over at φ ≃ 0.33 where the CP-even h d state dominates over the CP-odd h s state.
Nevertheless, although M H 2 is nearly constant when varying φ u , the decay pattern is strongly affected by the variation of the phase φ u . Indeed, as can be seen in Fig. 10 (upper), the branching fractions of the decays H 2 → bb, τ + τ − are strongly enhanced, by up to a factor of 75 − 85 at φ u ≃ 0.065. A zoom on the smaller decay branching fractions in Fig. 10 (lower) shows that this strong enhancement at φ u ≃ 0.065 corresponds to a suppression of the decays into weak bosons. The decay H 2 → γγ is also enhanced with a branching fraction five times higher than in the SM.

Conclusion
We have presented the characteristic features of the Higgs sector in the NMSSM using complex parameters in order to take into account CP-violating effects in the Higgs sector already at tree-level. We have implemented the decays of the NMSSM Higgs bosons within the real and the complex NMSSM in a Fortran code based on HDECAY. The implementation includes the available higher-order corrections as well as the renormalization group evolutions, in the R-NMSSM, of the input parameters at one-loop order. As shown in section 2.2, it is necessary to use the input Higgs masses at higher order which has been done in our implementation. Preliminary results have been discussed in section 3 and show that in some cases the deviation in the branching fractions with respect to the SM can be reduced when turning the complex phases on. The decay pattern can also be strongly affected by the complex phases with a large increase in some decay channels. A public version of the code will soon be available [13]. Updates of the code, as e.g. the implementation of the RGEs in the case of the C-NMSSM, will follow.