Benchmarks for Double Higgs Production in the Singlet Extended Standard Model at the LHC

The simplest extension of the Standard Model is to add a gauge singlet scalar, $S$: the singlet extended Standard Model. In the absence of a $Z_2$ symmetry $S\rightarrow -S$ and if the new scalar is sufficiently heavy, this model can lead to resonant double Higgs production, significantly increasing the production rate over the Standard Model prediction. While searches for this signal are being performed, it is important to have benchmark points and models with which to compare the experimental results. In this paper we determine these benchmarks by maximizing the double Higgs production rate at the LHC in the singlet extended Standard Model. We find that, within current constraints, the branching ratio of the new scalar into two Standard Model-like Higgs bosons can be upwards of $0.76$, and the double Higgs rate can be increased upwards of 30 times the Standard Model prediction.


INTRODUCTION
One of the main objectives of the Large Hadron Collider (LHC) is to further our understanding of electroweak (EW) physics at the EW scale. Of particular interest are the interactions of the observed Higgs boson [1,2]. In fact, measurements of the Higgs production and decay rates are at the level of ∼ 20% precision [3]. Although these measurements help us determine if the observed Higgs boson is related to the source of fundamental masses within the Standard Model (SM), there are still many unanswered questions. One of the most pressing is the mechanism of EW symmetry breaking (EWSB). In the SM the source of EWSB is the scalar potential. Hence, it is interesting to study extensions of the SM that change the potential and their signatures at the LHC. In particular, simple extensions allow us to investigate phenomenology that is generic to more complete models.

II. THE SINGLET EXTENDED STANDARD MODEL
In this section we give an overview of the singlet extended SM, following the notation of Ref. [66]. The results of Ref. [66] are important for establishing our benchmark points.
Hence, we summarize the results of this paper regarding global minimization of the potential, vacuum stability, and perturbative unitarity. In the remaining part of the paper we will extend upon this work, thoroughly investigating the relationship of these theoretical constraints and maximization of double Higgs production.
The neutral scalar component of H is denoted as φ 0 = (h + v)/ √ 2 with the vacuum expec- We similarly write S = s + x, where the vev of S is denoted as x.
We require that EWSB occurs at an extremum of the potential, so that v = v EW = 246 GeV. Shifting the field S → S + δS does not introduce any new terms to the potential, and is only a meaningless change in parameters. Using this freedom, we can additionally choose that the EWSB minimum satisfies x = 0. Requiring that (v, x) = (v EW , 0) be an extremum 1 A similar study has been done in the case of a broken Z 2 symmetry S → −S [70]. Here we work in the singlet extended SM with no Z 2 . This model has more free parameters allowing for different benchmark rates.
of the potential gives After symmetry breaking, there are two mass eigenstates denoted as h 1 and h 2 with masses m 1 and m 2 , respectively. The new fields are related to the gauge eigenstate fields by where θ is the mixing angle. The masses, m 1 and m 2 , and the mixing angle, θ, are related to the scalar potential parameters We set the mass m 1 = 125 GeV to reproduce the discovered Higgs. The free parameter space is then We are interested in the scenario with m 2 ≥ 2m 1 , where h 2 can decay on-shell to two SMlike Higgs bosons, h 1 . After symmetry breaking, the trilinear scalar terms in the potential which are relevant to double Higgs production are The trilinear coupling λ 211 allows for the tree level decay of , the trilinear couplings are given by [66] λ 211 = 2 sin 2 θ cos θ b 3 + a 1 2 cos θ(cos 2 θ − 2 sin 2 θ) + (2 cos 2 θ − sin 2 θ) sin θ v EW a 2 −6λ sin θ cos 2 θ v EW .

A. Global Minimization of the Potential
The scalar potential, Eq (1), allows for many extrema (v, x). There are two classes that need to be considered: v = 0 and v = 0. The v = 0 extrema are given by For all of these three solutions to be real, there are constraints ∆ > 0 and v 2 ± > 0. The v = 0 extrema are given by solutions of the following cubic equation: Only real solutions for x are of interest. Manifestly real solutions for non-degenerate cubics are presented in the Appendix.
As can be seen, there is only one extremum with v = v EW . Since the scalar S is a gauge singlet, it does not contribute to the gauge boson or SM fermion masses. Hence, to reproduce the correct EWSB pattern, we require that (v EW , 0) is the global minimum.

B. Vacuum Stability
To avoid instability of the vacuum from runaway negative energy solutions, the scalar potential should be bounded from below at large field values. Vacuum stability of the potential then requires that It is clear that bounding the potential from below along the axes s = 0 and φ 0 =0 requires If a 2 > 0 as well, then the potential is always positive definite for large field values. However, a 2 < 0 is also allowed. Eq. (10) can be rewritten as The first term in Eq. (12) is always positive definite. Requiring the second term to be nonnegative for a 2 < 0 gives the bound [66] − 2 λb 4 ≤ a 2 . (13)

C. Perturbative Unitarity
Perturbative unitarity of the partial wave expansion for the scattering also constrains quartic scalar couplings, where P j (cos θ) are Legendre polynomials. Looking at the process h 2 h 2 → h 2 h 2 for large energies, the first term in the partial wave expansion at leading order is The perturbative unitarity requirement |a 0 | ≤ 0.5 gives the constraint b 4 4.2. When this bound is saturated, a minimum higher order correction of 41% is needed to restore the unitarity of the amplitude [76].
There are also perturbative unitarity constraints on the other quartic couplings: λ 4.2 and a 2 25. However, for all parameter points we consider, these constraints on λ and a 2 are automatically satisfied when all other constraints are applied.

III. EXPERIMENTAL CONSTRAINTS
The singlet model predicts that the couplings of h 1 to other SM fermions and gauge bosons are suppressed from the SM predictions by cos θ. Hence, the single Higgs production cross section is suppressed by cos 2 θ, where σ SM (pp → h 1 ) is the SM cross section for Higgs production at m 1 = 125 GeV. Since all couplings between h 1 and SM fermions and gauge bosons are universally suppressed, the branching ratios for h 1 decay agree with SM branching ratios, where X SM is any allowed SM final state. Using these properties, the most stringent constraint from observed Higgs signal strengths is from ATLAS: sin 2 θ ≤ 0.12 at 95% C. L. [31].

IV. PRODUCTION AND DECAY RATES
The contributions to double Higgs production in the singlet model are shown in Fig. 1. Figures 1(a) and 1(b) are present in the SM double Higgs production, while the s-channel h 2 contribution in Fig. 1(c) is responsible for the resonant h 1 h 1 production. The s-channel h 1 (h 2 ) contribution in Fig. 1(b) (Fig. 1(c)) depends on the scalar trilinear couplings λ 111 (λ 211 ) in Eq. 7. Hence, this process is clearly sensitive to the shape of the scalar potential.
It is expected that the resonant h 2 contribution dominates the double Higgs production cross section. We then use the narrow width approximation as follows: Although interference effects between the different contributions in Fig. 1 can be significant [68], our purpose here is to maximize the double Higgs rate in this model. Hence, for simplicity we focus on maximizing the cross section in Eq. (18). This is sufficient to attain our goal.
Due to mixing with the Higgs boson, h 2 has couplings to SM fermions and gauge bosons proportional to sin θ. The cross section for production of h 2 is then with σ SM (pp → h 2 ) being the SM Higgs production cross section evaluated at a Higgs mass of m 2 . Since the couplings to fermions and gauge bosons are proportional to the SM values, the intuition about the dominant SM Higgs production channels is valid for the production of h 2 . Hence, gluon fusion gg → h 2 is the dominant channel, as illustrated in Fig. 1(c).
The heavy scalar h 2 can decay to SM gauge bosons and fermions with partial widths of where Γ SM (h 2 → X SM ) is the SM decay width for a Higgs boson into SM final states X SM = h 1 h 1 evaluated at a mass of m 2 . The tree level decay for h 2 → h 1 h 1 has a partial width given by The branching ratio for where is the total width of h 2 . 700 GeV, we also have Γ(h 2 )/m 2 0.05. As sin θ decreases below its upper bound, the total width of h 2 will decrease as well. The value of b 4 has no effect on the partial widths of h 2 into SM fermions or gauge bosons. However, as b 4 decreases, the partial width of Γ(h 2 → h 1 h 1 ) decreases as shown in Fig. 2. Hence, the upper bound on Γ(h 2 ) in Fig. 3 is the upper bound throughout the allowed parameter regions, and h 2 is sufficiently narrow to justify the narrow width approximation in Eq. (18).

V. RESULTS
We maximize the production rate in Eq. (18) by fixing m 2 and θ, then scanning over the remaining parameters For all numerical results, the SM production cross sections and widths for a Higgs boson in Eqs. (16), (17), (19), and (20) were obtained from Ref. [82].   then the largest possible branching ratio becomes smaller. This is due to the shrinking of the allowed range for the parameters a 2 and b 3 , as shown in Fig. 2. Even for small values of b 4 , the branching ratio can still be quite substantial.
The maximum possible value of sin 2 θ is expected to decrease as more data is taken at the LHC and the measurements of the observed Higgs couplings become more precise. can be seen, the branching ratio can be larger for smaller sin θ. Hence, maximization of BR(h 2 → h 1 h 1 ) occurs at small sin θ. However, double Higgs production is not maximized with this condition.
Now we turn our attention to maximizing the double Higgs production rate. Figure 6 shows double Higgs production cross section at 13 TeV of 33.53 +5.3% −6.8% fb [82], calculated at NNLL matched to NNLO in QCD with NLO top quark mass dependence [83]. As mentioned earlier, the maximum rates occur when b 4 is at the unitarity bound b 4 = 4.2. For sin θ, although the maximum BR(h 2 → h 1 h 1 ) increases as sin θ decreases, this increase is not enough to compensate for the sin 2 θ suppression of the production cross section σ(pp → h 2 ) in Eq. (19). Hence, the maximum double Higgs production cross section occurs at the experimental bound sin 2 θ = 0.12. In the best case, the resonant double Higgs production is roughly 30 times the SM double Higgs cross section.
Finally, we provide our benchmark points in Tables I and II. We provide the parameter points that maximize the h 1 h 1 production in the singlet extended SM, as well as the corresponding BR(h 2 → h 1 h 1 ) and h 1 h 1 production cross section at a lab frame energy of √ S H = 13 TeV. As discussed before, the maximum BR(h 2 → h 1 h 1 ) occurs for b 4 = 4.2 at the unitarity bound. Hence, we fix b 4 = 4.2 for all benchmark points. Also, the maximum h 1 h 1 production cross section occurs for sin 2 θ = 0.12 at the current limit [31]. Table I contains the benchmark points for sin 2 θ = 0.12. However, as mentioned earlier, as the LHC continues to gather data it is expected that the precision Higgs measurements will further limit sin θ. The uncertainties in Higgs coupling measurements are projected to be ∼ 5% with 3000 fb −1 of integrated luminosity at the LHC [84]. This corresponds to a bound of sin 2 θ 0.05 due to the overall cos 2 θ suppression of the h 1 rate of production. Hence, we   also provide benchmark points for sin 2 θ = 0.05 in Table II.

VI. CONCLUSION
The simplest possible extension of the SM is the addition of a real gauge singlet scalar.
Although simple, this model is theoretically well-motivated and has interesting phenomenology. In particular, if the new scalar h 2 is sufficiently heavy m 2 ≥ 2 m 1 , this model can give rise to resonant double Higgs production at the LHC. We have investigated this signature.
We determined benchmark parameter points that maximize the double Higgs production rate in this model at the x 3 + Ax 2 + Bx + C = 0 (A.2) We define the intermediate variables Q and R as The discriminant D can be either positive, negative, or zero. If the discriminant is zero, the cubic has degenerate solutions. The parameter space where D = 0 has zero volume, so it is unlikely to occur. The degenerate solutions are not important to consider for our purposes.
If D < 0, the cubic has three distinct real roots. If D > 0, the cubic has a real root, and a pair of complex conjugate roots.
For the case D < 0, we define an angle θ as follows: Note that if D < 0, then we also must have Q < 0. The three real solutions to Eq. (A.1) are then given by For the case D > 0, we must look at two sub-cases. If Q < 0, we then define a hyperbolic angle η as follows: The single real solution to Eq. (A.1) is then given by For the case D > 0 and Q > 0, we also define a hyperbolic angle η: The single real solution to Eq. (A.1) is then given by