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

TL;DR

This paper establishes benchmark scenarios for resonant double Higgs production in the singlet extended Standard Model at the LHC, highlighting potential enhancements over the Standard Model predictions.

Contribution

It identifies optimal benchmark points that maximize double Higgs production rates within current experimental constraints.

Findings

Double Higgs production rate can be increased up to 30 times the Standard Model prediction.

Branching ratio of the new scalar into two Higgs bosons can reach 0.76.

Provides benchmarks for experimental searches at the LHC.

Abstract

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.

Tables

Table 1. Table 1: Benchmark points that maximize BR ​ ( h 2 → h 1 ​ h 1 ) BR → subscript ℎ 2 subscript ℎ 1 subscript ℎ 1 {\rm BR}(h_{2}\rightarrow h_{1}h_{1}) with b 4 = 4.2 subscript 𝑏 4 4.2 b_{4}=4.2 and sin 2 ⁡ θ = 0.12 superscript 2 𝜃 0.12 \sin^{2}\theta=0.12 . The cross sections are evaluated at a lab frame energy of S H = 13 subscript 𝑆 𝐻 13 \sqrt{S_{H}}=13  TeV.

$m_2$	$a_2$	$b_3/v_{EW}$	$\mathrm{BR}\left(h_2\to h_1{h}_2\right)$	$\sigma \left(pp\to h_2\right)\mathrm{BR}\left(h_2\to h_1{h}_1\right)$
$300$ GeV	$-0.79$	$-2.7$	$0.76$	$0.89$ pb
$400$ GeV	$-0.40$	$-3.9$	$0.60$	$0.68$ pb
$500$ GeV	$0.059$	$-5.4$	$0.48$	$0.26$ pb
$600$ GeV	$0.56$	$-7.1$	$0.42$	$0.10$ pb
$700$ GeV	$1.0$	$-8.7$	$0.37$	$0.042$ pb
$800$ GeV	$1.6$	$-11$	$0.35$	$0.019$ pb

Table 2. Table 2: Benchmark points that maximize BR ​ ( h 2 → h 1 ​ h 1 ) BR → subscript ℎ 2 subscript ℎ 1 subscript ℎ 1 {\rm BR}(h_{2}\rightarrow h_{1}h_{1}) with b 4 = 4.2 subscript 𝑏 4 4.2 b_{4}=4.2 and sin 2 ⁡ θ = 0.05 superscript 2 𝜃 0.05 \sin^{2}\theta=0.05 . The cross sections are evaluated at a lab frame energy of S H = 13 subscript 𝑆 𝐻 13 \sqrt{S_{H}}=13  TeV.

$m_2$	$a_2$	$b_3/v_{EW}$	$\mathrm{BR}\left(h_2\to h_1{h}_2\right)$	$\sigma \left(pp\to h_2\right)\mathrm{BR}\left(h_2\to h_1{h}_1\right)$
$300$ GeV	$-1.2$	$-1.6$	$0.76$	$0.37$ pb
$400$ GeV	$-1.0$	$-2.7$	$0.60$	$0.29$ pb
$500$ GeV	$-0.78$	$-3.9$	$0.48$	$0.11$ pb
$600$ GeV	$-0.59$	$-5.0$	$0.42$	$0.042$ pb
$700$ GeV	$-0.31$	$-6.5$	$0.38$	$0.017$ pb
$800$ GeV	$-0.015$	$-8.1$	$0.35$	$0.0079$ pb

Equations

$$V(H,S)=-{\mu }^2{H}^{†}H+\lambda (H^{†}H{)}^2+\frac{a_1}{2}{H}^{†}HS+\frac{a_2}{2}{H}^{†}H{S}^2+b_{1}S+\frac{b_2}{2}{S}^2+\frac{b_3}{3}{S}^3+\frac{b_4}{4}{S}^4.$$

$${\mu }^2$$

$$b_1$$

$$\left(\begin{array}{c}{h}_1\\ h_2\end{array}\right)=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}h\\ s\end{array}\right).$$

$$a_1$$

$$b_2+\frac{a_2}{2}{v}_{EW}^2$$

$$\lambda$$

$$m_2\text{, }\theta \text{, }{a}_2\text{, }{b}_3\text{, and }{b}_4.$$

$$V(h_1,h_2)\supset \frac{{\lambda }_{111}}{3!}{h}_1^3+\frac{{\lambda }_{211}}{2!}{h}_2{h}_1^2.$$

$${\lambda }_{111}$$

$${\lambda }_{211}$$

$$x_{\pm }$$

$$v_{\pm }^2$$

$$\Delta$$

$$b_1+b_{2}x+b_3{x}^2+b_4{x}^3=0.$$

$$4\lambda {\varphi }_0^4+2a_2{\varphi }_0^2{s}^2+b_4{s}^2>0.$$

$$\lambda >0\mathrm{and}{b}_4>0.$$

$$\lambda (2{\varphi }_0^2+\frac{a_2}{2\lambda }{s}^2{)}^2+(b_4-\frac{a_2^2}{4\lambda })s^4>0.$$

$$-2\sqrt{\lambda b_4}\le a_2.$$

$$\mathcal{M}=16\pi \sum _{j=0}^{\infty }(2j+1)a_j{P}_j(\cos \theta ),$$

$$a_0(h_2{h}_2\to h_2{h}_2)=\frac{3b_4}{8\pi }.$$

$$\sigma (pp\to h_1)={\cos }^2\theta {\sigma }_{SM}(pp\to h_1)$$

$$\text{BR}(h_1\to X_{SM})={\text{BR}}_{SM}(h_1\to X_{SM})$$

$$\sigma (pp\to h_2\to h_1{h}_1)\approx \sigma (pp\to h_2)\text{BR}(h_2\to h_1{h}_1)\text{.}$$

$$\sigma (pp\to h_2)={\sin }^2\theta {\sigma }_{SM}(pp\to h_2)$$

$$\Gamma (h_2\to X_{SM})={\sin }^2\theta {\Gamma }_{SM}(h_2\to X_{SM})$$

$$\Gamma (h_2\to h_1{h}_1)=\frac{{\lambda }_{211}^2}{32\pi m_2}\sqrt{1-\frac{4m_1^2}{m_2^2}}$$

$$\text{BR}(h_2\to h_1{h}_1)=\frac{\Gamma (h_2\to h_1{h}_1)}{\Gamma (h_2)},$$

$$\Gamma (h_2)=\Gamma (h_2\to h_1{h}_1)+{\sin }^2\theta {\Gamma }_{SM}(h_2\to X_{SM})$$

$$a_2\text{, }{b}_3\text{, and }{b}_4.$$

$$b_1+b_{2}x+b_3{x}^2+b_4{x}^3=0$$

$$x^3+A{x}^2+Bx+C=0$$

$$A=\frac{b_3}{b_4}$$

$$B=\frac{b_2}{b_4}$$

Full text

I 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 Chatrchyan:2012xdj ; Aad:2012tfa . In fact, measurements of the Higgs production and decay rates are at the level of $\sim 20\%$ precision Khachatryan:2016vau . 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.

The simplest extension of the SM is the addition of a gauge singlet real scalar, $S$: the singlet extended SM. At the renormalizable level, the only allowed interactions between $S$ and the SM are with the Higgs field. Hence, this model is a useful laboratory to investigate deviations from the SM Higgs potential. Although this is the simplest possible extension, it is well-motivated. This scenario arises in Higgs portal models Binoth:1996au ; Davoudiasl:2004be ; Schabinger:2005ei ; Patt:2006fw ; Bowen:2007ia ; Bock:2010nz ; Djouadi:2011aa ; Englert:2011aa ; Englert:2011yb ; Alanne:2014bra . In these models, the scalar singlet couples to a dark matter sector. Through its interactions with the Higgs field, the new scalar provides couplings between the dark sector and the SM. Additionally, scalar singlets can help provide the strong first order EW phase transition necessary for EW baryogenesis Ham:2004cf ; Profumo:2007wc ; Ashoorioon:2009nf ; Bodeker:2009qy ; Espinosa:2011ax ; No:2013wsa ; Alanne:2014bra ; Curtin:2014jma ; Profumo:2014opa ; Huang:2015bta ; Damgaard:2015con ; Kozaczuk:2015owa ; Huang:2015tdv ; Xiao:2015tja ; Huang:2016cjm ; Kotwal:2016tex ; Curtin:2016urg ; Vaskonen:2016yiu .

If there is no $Z_{2}$ symmetry, $S\rightarrow-S$, after EWSB the new scalar will mix with the SM Higgs boson. This mixing induces couplings between the new scalar and the rest of the SM particles. Hence, the new scalar can be produced and searched for at the LHC, as well as affecting precision Higgs measurements. The simplicity of the singlet extended SM allows for easy interpretation of precision Higgs measurements Aad:2015pla ; Khachatryan:2016vau and resonant searches for heavy scalars Aad:2015agg ; Khachatryan:2015cwa ; Aad:2015kna ; Aad:2015xja ; Khachatryan:2015yea ; Khachatryan:2016sey ; CMS:2016knm ; CMS:2016tlj ; ATLAS:2016ixk ; ATLASbbgamgam ; CMS:2016ilx ; CMS:2016jpd ; ATLAS:2016oum ; ATLAS:2016kjy ; Aaboud:2016okv ; CMS:2016zte ; Khachatryan:2015sma ; Aad:2015fna .

There have been many phenomenological studies of the singlet extended SM at the LHC Schabinger:2005ei ; OConnell:2006rsp ; Bowen:2007ia ; Profumo:2007wc ; Barger:2008jx ; Dawson:2009yx ; Englert:2011yb ; Bertolini:2012gu ; Pruna:2013bma ; Caillol:2013gqa ; Curtin:2014jma ; Lopez-Val:2014jva ; Profumo:2014opa ; Buttazzo:2015bka ; Robens:2015gla ; Falkowski:2015iwa ; Bojarski:2015kra ; Costa:2015llh ; Fischer:2016rsh ; Fichet:2016xpw . Of particular interest to us is if the new scalar is sufficiently heavy, it can decay on-shell into two SM-like Higgs bosons, mediating resonant double Higgs production at the LHC Dolan:2012ac ; Cooper:2013kia ; No:2013wsa ; Chen:2014ask ; Martin-Lozano:2015dja ; Dawson:2015haa ; Godunov:2015nea ; Robens:2016xkb ; Nakamura:2017irk ; Huang:2017jws ; Chang:2017niy ; Lu:2015qqa ; Ren:2017jbg . This can greatly enhance the double Higgs rate over the SM prediction. We will provide benchmark points that maximize double Higgs production in the singlet extended SM. These benchmark points are needed to help determine when the experimental searches for resonant double Higgs production ATLAS:2016ixk ; ATLASbbgamgam ; CMS:2016tlj ; CMS:2016knm ; Khachatryan:2015yea ; Khachatryan:2016sey ; Aad:2015xja are probing interesting regions of parameter space.111A similar study has been done in the case of a broken $Z_{2}$ symmetry $S\rightarrow-S$ Robens:2016xkb . Here we work in the singlet extended SM with no $Z_{2}$. This model has more free parameters allowing for different benchmark rates.

In Section II we provide an overview of the model, including the theoretical constraints on the model. Experimental constraints are discussed in Section III. Resonant double Higgs production is discussed in Section IV. In Section V we discuss the maximization of the double Higgs rate and provide the benchmark points. We conclude with Section VI.

II The Singlet Extended Standard Model

In this section we give an overview of the singlet extended SM, following the notation of Ref. Chen:2014ask . The results of Ref. Chen:2014ask 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 model contains the SM Higgs doublet, $H$, and a new real gauge singlet scalar, $S$. The new singlet does not directly couple to SM particles except for the Higgs doublet. Allowing for all renormalizable terms, the most general scalar potential is

[TABLE]

The neutral scalar component of $H$ is denoted as $\phi_{0}=(h+v)/\sqrt{2}$ with the vacuum expectation value (vev) being $\langle\phi_{0}\rangle=\frac{v}{\sqrt{2}}$. 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 $S\rightarrow S+\delta S$ 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 of the potential gives

[TABLE]

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

[TABLE]

where $\theta$ is the mixing angle. The masses, $m_{1}$ and $m_{2}$, and the mixing angle, $\theta$, are related to the scalar potential parameters

[TABLE]

We set the mass $m_{1}=125$ GeV to reproduce the discovered Higgs. The free parameter space is then

[TABLE]

We are interested in the scenario with $m_{2}\geq 2m_{1}$, where $h_{2}$ can decay on-shell to two SM-like Higgs bosons, $h_{1}$. After symmetry breaking, the trilinear scalar terms in the potential which are relevant to double Higgs production are

[TABLE]

The trilinear coupling $\lambda_{211}$ allows for the tree level decay of $h_{2}\rightarrow h_{1}h_{1}$. At the EWSB minimum $(v,x)=(v_{EW},0)$, the trilinear couplings are given by Chen:2014ask

[TABLE]

II.1 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: $v\neq 0$ and $v=0$. The $v\neq 0$ extrema are given by $(v,x)=(v_{EW},0)$ and $(v,x)=(v_{\pm},x_{\pm})$ where Chen:2014ask

[TABLE]

For all of these three solutions to be real, there are constraints $\Delta>0$ and $v_{\pm}^{2}>0$.

The $v=0$ extrema are given by solutions of the following cubic equation:

[TABLE]

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.

II.2 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

[TABLE]

It is clear that bounding the potential from below along the axes $s=0$ and $\phi_{0}$=0 requires

[TABLE]

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

[TABLE]

The first term in Eq. (12) is always positive definite. Requiring the second term to be nonnegative for $a_{2}<0$ gives the bound Chen:2014ask

[TABLE]

II.3 Perturbative Unitarity

Perturbative unitarity of the partial wave expansion for the scattering also constrains quartic scalar couplings,

[TABLE]

where $P_{j}(\cos\theta)$ are Legendre polynomials. Looking at the process $h_{2}h_{2}\rightarrow h_{2}h_{2}$ for large energies, the first term in the partial wave expansion at leading order is

[TABLE]

The perturbative unitarity requirement $|a_{0}|\leq 0.5$ gives the constraint $b_{4}\lesssim 4.2$. When this bound is saturated, a minimum higher order correction of $41\%$ is needed to restore the unitarity of the amplitude Schuessler:2007av .

There are also perturbative unitarity constraints on the other quartic couplings: $\lambda\lesssim 4.2$ and $a_{2}\lesssim 25$. However, for all parameter points we consider, these constraints on $\lambda$ 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 $\cos\theta$. Hence, the single Higgs production cross section is suppressed by $\cos^{2}\theta$,

[TABLE]

where $\sigma_{SM}(pp\rightarrow h_{1})$ 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,

[TABLE]

where $X_{SM}$ is any allowed SM final state. Using these properties, the most stringent constraint from observed Higgs signal strengths is from ATLAS: $\sin^{2}{\theta}\leq 0.12$ at $95\%$ C. L. Aad:2015pla .

As mentioned earlier, there are also direct constraints from searches for heavy scalar particles Khachatryan:2015cwa ; Aad:2015kna ; Aad:2015agg ; Aad:2015xja ; Khachatryan:2016sey ; Khachatryan:2015yea ; CMS:2016knm ; CMS:2016tlj ; ATLAS:2016ixk ; ATLASbbgamgam ; CMS:2016ilx ; CMS:2016jpd ; ATLAS:2016oum ; ATLAS:2016kjy ; Aaboud:2016okv ; CMS:2016zte ; Khachatryan:2015sma ; Aad:2015fna . For the mass range $250\textrm{ GeV}\leq m_{2}\leq 1000\textrm{ GeV}$ considered here, the direct constraints on $\sin\theta$ are weaker than those from the Higgs signal strengths Robens:2016xkb . Nevertheless, independently and using HiggsBounds arXiv:0811.4169 ; arXiv:1102.1898 ; arXiv:1301.2345 ; arXiv:1311.0055 ; arXiv:1507.06706 , we verify that our benchmark points satisfy all experimental constraints.

IV Production and Decay Rates

The contributions to double Higgs production in the singlet model are shown in Fig. 1. Figures 1 and 1 are present in the SM double Higgs production, while the $s$-channel $h_{2}$ contribution in Fig. 1 is responsible for the resonant $h_{1}h_{1}$ production. The $s$-channel $h_{1}$ ($h_{2}$) contribution in Fig. 1 (Fig. 1) depends on the scalar trilinear couplings $\lambda_{111}$ ($\lambda_{211}$) 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:

[TABLE]

Although interference effects between the different contributions in Fig. 1 can be significant Dawson:2015haa , 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 $\sin{\theta}$. The cross section for production of $h_{2}$ is then

[TABLE]

with $\sigma_{SM}(pp\rightarrow h_{2})$ 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 $gg\rightarrow h_{2}$ is the dominant channel, as illustrated in Fig. 1.

The heavy scalar $h_{2}$ can decay to SM gauge bosons and fermions with partial widths of

[TABLE]

where $\Gamma_{SM}(h_{2}\rightarrow X_{SM})$ is the SM decay width for a Higgs boson into SM final states $X_{SM}\neq h_{1}h_{1}$ evaluated at a mass of $m_{2}$. The tree level decay for $h_{2}\rightarrow h_{1}h_{1}$ has a partial width given by

[TABLE]

The branching ratio for $h_{2}\rightarrow h_{1}h_{1}$ is

[TABLE]

where

[TABLE]

is the total width of $h_{2}$.

The parameter $b_{4}$ does not explicitly affect $\textrm{BR}(h_{2}\rightarrow h_{1}h_{1})$. However, through the constraints of vacuum stability and $(v,x)=(v_{EW},0)$ being the global minimum of the scalar potential [Sec. II.1], $b_{4}$ affects the allowed ranges for the other parameters $a_{2}$ and $b_{3}$. These parameters appear in the trilinear coupling $\lambda_{211}$ in Eq. (7), which is relevant for $\Gamma(h_{2}\rightarrow h_{1}h_{1})$. Figure 2 shows the allowed parameter region satisfying these constraints for (a) $b_{4}=4.2$ and (b) $b_{4}=0.2$ with $m_{2}=260$ GeV and $\sin^{2}\theta=0.12$. It is clear from the figures that a lower value of $b_{4}$ shrinks the allowed region. The shading in the figures indicates the value of $\textrm{BR}(h_{2}\rightarrow h_{1}h_{1})$, where the values of $\Gamma_{SM}(h_{2}\rightarrow X_{SM})$ were obtained from Ref. deFlorian:2016spz . It was found that the maximum ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ always occurs with $b_{4}=4.2$ at the unitarity bound.

In Fig. 3 we show allowed ranges of $\Gamma(h_{2})/m_{2}$ as a function of the mass of $m_{2}$ for $b_{4}=4.2$ and $\sin^{2}\theta=0.12$. The total width is always bounded by $\Gamma(h_{2})/m_{2}\lesssim 0.09$. For $m_{2}\lesssim 700\textrm{ GeV}$, we also have $\Gamma(h_{2})/m_{2}\lesssim 0.05$. As $\sin\theta$ 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 $\Gamma(h_{2}\rightarrow h_{1}h_{1})$ decreases as shown in Fig. 2. Hence, the upper bound on $\Gamma(h_{2})$ 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 $\theta$, then scanning over the remaining parameters

[TABLE]

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. deFlorian:2016spz .

The maximum and minimum ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ for different values of $m_{2}$ are shown in Fig. 4. We set $b_{4}=4.2$ at the perturbative unitarity bound and $\sin^{2}\theta=0.12$ at the experimental bound Aad:2015pla . The largest possible branching ratio occurs at around $280\textrm{ GeV}$ with $\textrm{BR}(h_{2}\rightarrow h_{1}h_{1})=0.76$. Even up to masses of $1000\textrm{ GeV}$ the branching ratio to double Higgs can be larger than $0.3$. Additionally, for $m_{2}\gtrsim 600$ GeV there is a minimum on ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$.

Figure 5 shows the dependence of the maximum branching ratio ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ on the parameter $b_{4}$. As can be seen, if the parameter $b_{4}$ is less than the unitarity bound of $4.2$ 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 $\sin^{2}{\theta}$ is expected to decrease as more data is taken at the LHC and the measurements of the observed Higgs couplings become more precise. Figure 5 shows the maximum possible ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ for several values of $\sin^{2}{\theta}$. As can be seen, the branching ratio can be larger for smaller $\sin\theta$. Hence, maximization of ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ occurs at small $\sin\theta$. 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 the maximum $\sigma(pp\rightarrow h_{2})\textrm{BR}(h_{2}\rightarrow h_{1}h_{1})$ at an LHC energy of $\sqrt{S_{H}}=13\textrm{ TeV}$ for various (a) $b_{4}$ and (b) $\sin\theta$ values as a function of mass $m_{2}$. The values are scaled by the SM double Higgs production cross section at $13\textrm{ TeV}$ of $33.53^{+5.3\%}_{-6.8\%}\textrm{ fb}$ deFlorian:2016spz , calculated at NNLL matched to NNLO in QCD with NLO top quark mass dependence Borowka:2016ypz . As mentioned earlier, the maximum rates occur when $b_{4}$ is at the unitarity bound $b_{4}=4.2$. For $\sin\theta$, although the maximum ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ increases as $\sin\theta$ decreases, this increase is not enough to compensate for the $\sin^{2}\theta$ suppression of the production cross section $\sigma(pp\rightarrow h_{2})$ in Eq. (19). Hence, the maximum double Higgs production cross section occurs at the experimental bound $\sin^{2}\theta=0.12$. 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 1 and 2. We provide the parameter points that maximize the $h_{1}h_{1}$ production in the singlet extended SM, as well as the corresponding ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ and $h_{1}h_{1}$ production cross section at a lab frame energy of $\sqrt{S_{H}}=13$ TeV. As discussed before, the maximum ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ 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 $\sin^{2}\theta=0.12$ at the current limit Aad:2015pla . Table 1 contains the benchmark points for $\sin^{2}\theta=0.12$. However, as mentioned earlier, as the LHC continues to gather data it is expected that the precision Higgs measurements will further limit $\sin\theta$. The uncertainties in Higgs coupling measurements are projected to be $\sim 5\%$ with $3000\textrm{ fb}^{-1}$ of integrated luminosity at the LHC Dawson:2013bba . This corresponds to a bound of $\sin^{2}\theta\lesssim 0.05$ due to the overall $\cos^{2}\theta$ suppression of the $h_{1}$ rate of production. Hence, we also provide benchmark points for $\sin^{2}\theta=0.05$ in Table 2.

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 $m_{2}\geq 2\,m_{1}$, 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 $\sqrt{S_{H}}=13$ TeV LHC. These benchmark points are important for gauging when the ongoing experimental searches for resonant double Higgs production are probing interesting regions of parameter space of well-motivated models. We have found that ${\rm BR}(h_{2}\rightarrow h_{1}h_{1})$ as high as $0.76$ and $h_{1}h_{1}$ production rates up to 30 times the SM rate are still possible.

Acknowledgements

We thank S. Dawson for valuable discussions about double Higgs production in the singlet extended SM. This investigation was supported in part by the University of Kansas General Research Fund allocation 2302091.

Appendix A $v=0$ Extrema

In this appendix we give solutions to Eq. (9) where $v=0$. We update the results from Ref. Chen:2014ask , presenting these solutions in a manifestly real form.

Solutions to the extrema conditions were solutions to Eq. (9), repeated below.

[TABLE]

All the coefficients in Eq. (25) are real. We divide Eq. (25) by $b_{4}$ to normalize the cubic term:

[TABLE]

We define the intermediate variables $Q$ and $R$ as

[TABLE]

The polynomial discriminant of Eq. (26) is then given by

[TABLE]

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 $\theta$ as follows:

[TABLE]

Note that if $D<0$, then we also must have $Q<0$. The three real solutions to Eq. (25) are then given by

[TABLE]

For the case $D>0$, we must look at two sub-cases. If $Q<0$, we then define a hyperbolic angle $\eta$ as follows:

[TABLE]

The single real solution to Eq. (25) is then given by

[TABLE]

For the case $D>0$ and $Q>0$, we also define a hyperbolic angle $\eta$:

[TABLE]

The single real solution to Eq. (25) is then given by

[TABLE]