Heavy Higgs signal–background interference in $gg\rightarrow VV$ in the Standard Model plus real singlet

Abstract

For the Standard Model extended with a real scalar singlet field, the modification of the heavy Higgs signal due to interference with the continuum background and the off-shell light Higgs contribution is studied for $gg\rightarrow ZZ, WW \rightarrow 4$ lepton processes at the Large Hadron Collider. Interference effects can range from $\mathcal{O}(10\,\%)$ to $\mathcal{O}(1)$ effects for integrated cross sections. Despite a strong cancellation between the heavy Higgs–continuum and the heavy Higgs–light Higgs interference, the full interference is clearly non-negligible and modifies the heavy Higgs line shape. A $\left| M_{VV} - M_{h2}\right| < \Gamma _{h2}$ cut mitigates interference effects to $\mathcal{O}(10\,\%)$ or less. A public program that allows one to simulate the full interference is presented.

1 Introduction

In 2012, the ATLAS and CMS experiments at the CERN Large Hadron Collider (LHC) announced the discovery of a new scalar resonance with a mass of approximately 125 GeV [1, 2]. The discovered particle is so far consistent with the Higgs boson predicted by the Standard Model (SM) Higgs mechanism [3–7], but many extensions to the SM preserve the minimal assumptions of an $\mathrm{SU}(2)$ doublet which acquires a vacuum expectation value thus inducing a physical Higgs boson that couples to fermions and vector bosons in proportion to their mass, while also allowing for an expanded Higgs sector with additional, heavier (or lighter) Higgs-like scalar particles. The search for high-mass Higgs-like particles in the $gg\rightarrow H\rightarrow ZZ$ and $gg\rightarrow H\rightarrow WW$ channels at the LHC is ongoing [8–17].

With inclusive NNLO signal uncertainties of $\mathcal {O}(10\,\%)$ in gluon-fusion Higgs production at the LHC, which can be further reduced by experimental selection cuts, it is important to study signal–background interference in the $H\rightarrow VV$ decay modes ($V=Z,W$), because it can be of similar size or larger for Higgs invariant masses above the weak-boson pair threshold. For Higgs invariant masses much larger than $2M_V$, the occurring sizeable Higgs–continuum interference is linked to the preservation of unitarity. In the SM, interference between the Higgs signal and continuum background in $gg (\rightarrow H)\rightarrow VV$ and including fully leptonic decays has been studied in Refs. [18–32].¹ Higgs–continuum interference results for a heavy SM Higgs boson with a $\Gamma _H/M_{H}$ ratio of $\mathcal {O}(10\,\%)$ or more have been presented in Refs. [21–23, 25, 26, 28, 30]. We note that all Higgs–continuum interference calculations are at leading order (LO), except for Refs. [25, 28, 32], where approximate higher-order corrections have been calculated.

Since a Higgs boson with $M_{H}\approx 125$ GeV has been discovered, a theoretically consistent search for an additional Higgs boson has to be based on a model that is beyond the SM. The simplest extension of the Higgs sector of the SM introduces an additional real scalar singlet field which is neutral under the SM gauge groups. The remaining viable parameter space of this 1-Higgs-singlet extension of the SM (1HSM) after LHC Run 1 has been studied in Refs. [37, 38].² Here, we focus on the case where the additional Higgs boson is heavier than the discovered Higgs boson. In this case, the heavy Higgs signal is affected not only by sizeable interference with the continuum background, but also by a non-negligible interference with the off-shell tail of the light Higgs boson [24]. A calculation including full interference effects in a Higgs portal model has been carried out in Ref. [42]. But the occurring interference effects (which are discernible in the distributions shown in Fig. 8 of Ref. [42]) have not been analysed quantitatively there.³ A dedicated study of heavy Higgs-light Higgs interference in the 1HSM with an additional $Z_2$ symmetry was presented in Ref. [43].⁴

In this paper, we extend the analysis of Ref. [43] by taking into account the full signal–background interference which includes the heavy Higgs–continuum interference.⁵ Furthermore, in addition to $gg\rightarrow h_2\rightarrow ZZ\rightarrow 4$ leptons, where $h_2$ is the heavy Higgs boson, we also calculate results for $gg\rightarrow h_2\rightarrow WW\rightarrow 4$ leptons. Our calculations are carried out with a new version of the parton-level integrator and event generator gg2VV, which we have made publicly available [46].

In Sect. 2 we discuss the 1HSM and specify the used benchmark points. Calculational details are discussed in Sect. 3. Integrated cross sections and differential distributions in $M_{VV}$ for the heavy Higgs signal and its interference with the continuum background and off-shell light Higgs contribution are presented in Sect. 4 for $g g \rightarrow h_2 \rightarrow Z Z \rightarrow \ell \bar{\ell }\ell ' \bar{\ell }'$ and $g g \rightarrow h_2 \rightarrow W^- W^+ \rightarrow \ell \bar{\nu }\bar{\ell }' \nu '$. Conclusions are given in Sect. 5.

2 Model

As minimal theoretically consistent model with two physical Higgs bosons, we consider the SM with an added real singlet field which is neutral under all SM gauge groups.

The 1HSM has been extensively explored in the literature [47–67]. Higgs-singlet models with an additional $Z_2$ symmetry have generated some interest recently because of the possibility of the additional Higgs boson being a dark matter candidate, but here we consider the most general extension. In the following, we give a brief summary of the model. A more detailed description can be found in Refs. [67, 68].

The SM Higgs sector is extended by the addition of a new real scalar field, which is a singlet under all the gauge groups of the SM and which also gets a vacuum expectation value (VEV) under electroweak symmetry breaking. The most general gauge-invariant potential can be written as [48, 50]

$$\begin{aligned} V= & {} \lambda \left( \Phi ^\dagger \Phi - \frac{v^2}{2}\right) ^2 + \frac{1}{2} M^2 s^2 + \lambda _1 s^4 +\lambda _2 s^2 \nonumber \\&\times \left( \Phi ^\dagger \Phi - \frac{v^2}{2}\right) +\mu _1 s^3 + \mu _2 s \left( \Phi ^\dagger \Phi - \frac{v^2}{2}\right) , \end{aligned}$$

(2.1)

where s is the real singlet scalar which is allowed to mix with the SM $\mathrm{SU}(2)$ Higgs doublet, which in the unitary gauge can be written as

$$\begin{aligned} \Phi = \begin{pmatrix} 0 \\ (\phi + v)/\sqrt{2} \end{pmatrix} \end{aligned}$$

(2.2)

with VEV $v\simeq 246$ GeV. Here it has already been exploited that (without the $Z_2$ symmetry) shifting the singlet field simply corresponds to a redefinition of the parameter coefficients and due to this freedom one can take the VEV of the singlet field to zero, which implies $M^2>0$. To avoid vacuum instability the quartic couplings must satisfy

$$\begin{aligned} \lambda >0,\quad \lambda _1 > 0,\quad \lambda _2 > -2 \sqrt{\lambda \lambda _1}. \end{aligned}$$

(2.3)

The trilinear couplings $\mu _1$ and $\mu _2$ can have positive or negative sign. Substituting Eq. (2.2) into Eq. (2.1), one obtains the potential

$$\begin{aligned} V= & {} \frac{\lambda }{4}\phi ^4 + \lambda v^2 \phi ^2 + \lambda v \phi ^3 + \frac{1}{2}M^2s^2 + \lambda _1 s^4 \nonumber \\&+ \frac{\lambda _2}{2}\phi ^2 s^2 + \lambda _2 v \phi s^2 + \mu _1 s^3 + \frac{\mu _2}{2}\phi ^2 s + \mu _2 v \phi s. \end{aligned}$$

(2.4)

The mass eigenstates can be parametrised in terms of a mixing angle $\theta$ as

$$\begin{aligned} h_1&= \phi \cos \theta - s \sin \theta , \end{aligned}$$

(2.5)

$$\begin{aligned} h_2&= \phi \sin \theta + s \cos \theta , \end{aligned}$$

(2.6)

where $h_1$ is assumed to be the lighter Higgs boson with a mass of approximately 125 GeV, and

$$\begin{aligned} \tan 2\theta = \frac{- \mu _2 v}{\lambda v^2 - \frac{1}{2} M^2} \end{aligned}$$

(2.7)

with

$$\begin{aligned} -\frac{\pi }{4} < \theta < \frac{\pi }{4} \end{aligned}$$

(2.8)

under the condition $M^2>2\lambda v^2$.

The model has six independent parameters, which we choose to be $M_{h1}, M_{h2}, \theta , \mu _1, \lambda _1$ and $\lambda _2$. The dependent model parameters are:

$$\begin{aligned} \lambda&=\frac{\cos \left( 2\theta \right) (M_{h1}^2 - M_{h2}^2) + M_{h1}^2 + M_{h2}^2 }{4v^2}, \end{aligned}$$

(2.9)

$$\begin{aligned} M^2&= \frac{M_{h2}^2 - M_{h1}^2 + \sec \left( 2\theta \right) (M_{h1}^2 + M_{h2}^2)}{2\sec \left( 2\theta \right) }, \end{aligned}$$

(2.10)

$$\begin{aligned} \mu _2&= -\tan \left( 2\theta \right) \ \frac{\lambda v^2 - \frac{1}{2}M^2}{v}. \end{aligned}$$

(2.11)

We set $M_{h1}$ to 125 GeV in accordance with the mass of the observed resonance and study three values for the mass of the heavy Higgs resonance: $M_{h2}=300$ GeV, $M_{h2}=600$ GeV and $M_{h2}=900$ GeV. We choose the mixing angle $\theta$ so as not to alter the predicted light Higgs cross section too much. To illustrate how interference effects change with the mixing angle, we study the two values $\theta =\pi /15$ and $\theta =\pi /8$, which is consistent with current limits on the Higgs signal strength and does not appear to be in conflict with limits given in Ref. [38], but strictly speaking these apply to the model with the additional $Z_2$ symmetry and are not directly applicable here. Furthermore, we consider model benchmark points with vanishing coupling parameters $\mu _1, \lambda _1$ and $\lambda _2$. ($\lambda _1>0$ is treated as approximately zero.) We emphasise that this does not imply that the $h_2\rightarrow h_1 h_1$ decay width is zero. For instance, for the mixing angle $\theta =\pi /8$ and $M_{h2}=$ 300 (600) [900] GeV the branching ratio $\Gamma (h_2\rightarrow h_1 h_1)/\Gamma _{h2}$ is 28 % (20 %) [19 %]. The $h_2\rightarrow h_1 h_1$ decay mode is therefore not suppressed in our study. Furthermore, the implementation in gg2VV is not restricted to benchmark points with vanishing $\mu _1, \lambda _1$ and $\lambda _2$. Nonzero values of $\mu _1, \lambda _1$ and $\lambda _2$ affect the calculation of the signal–background interference only via a change of the heavy Higgs width. In combination with FeynRules, our implementation in gg2VV therefore allows one to calculate full signal–background interference effects for arbitrary benchmark points of the general 1HSM. See Sect. 3 for further details.

Reference [67] gives bounds on the $\lambda _1$ and $\mu _1$ parameters for $M_{h2} \lesssim 500$ GeV and a similar $\theta$, which are in agreement with our choice of zero for these parameters. Our choice for the coupling parameters is also in agreement with upper limits on the combination of these parameters from experimental searches [69, 70].

3 Calculational details

Fig. 1

Representative Feynman graphs for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ,WW \rightarrow 4$ leptons. The heavy Higgs ($h_2$) graphs define the signal process, which interferes with the light Higgs ($h_1$) graphs (a, b). They also interfere with the gluon-induced continuum background graphs (c, d)

In Sect. 4 we present results calculated with a new version of gg2VV [22, 24, 26], which is publicly available [46]. Representative Feynman graphs for the light and heavy Higgs and interfering continuum background processes are shown in Fig. 1. The heavy Higgs ($h_2$) graphs define the signal process. They interfere with the light Higgs ($h_1$) graphs and with the gluon-induced continuum background graphs.

The amplitudes are calculated using a modified (for compatibility only) output of FeynArts/FormCalc [71, 72], using a custom coded UFO [73] model file generated by FeynRules [74]. The Higgs boson widths are calculated using FeynRules for consistency. The used width values are given in Table 1.

The PDF set MSTW2008LO [75] with default $\alpha _s$ is used and the CKM matrix is approximated by the unit matrix, which causes a negligible error [22]. As input parameters, we use the specification of the LHC Higgs Cross Section Working Group in App. A of Ref. [76] with $G_\mu$ scheme and LO weak-boson widths for consistency. More specifically, $M_W = 80.398$ GeV, $M_Z = 91.1876$ GeV, $\Gamma _W = 2.141$ GeV, $\Gamma _Z = 2.4952$ GeV, $M_t = 172.5$ GeV, $M_b = 4.75$ GeV, $G_F = 1.16637\times 10^{-5}$ GeV$^{-2}$ are used. Finite top and bottom quark mass effects are included. Lepton masses are neglected. A fixed-width Breit–Wigner propagator is employed for the weak bosons and the Higgs boson. The width parameter of the complex pole of the Higgs propagator is defined in Eq. (16) of Ref. [77]. The box graphs shown in Fig. 1c, d are affected by numerical instabilities when Gram determinants approach zero. In these critical phase space regions the amplitude is evaluated in quadruple precision. Residual instabilities are eliminated by requiring that $p_{T,W}$ and $p_{T,Z}$ are larger than 1 GeV. This criterion is also applied to the Higgs amplitudes, which are not affected by numerical instabilities, to obtain consistent cross section-level results. The numerical effect of this technical cut has been shown to be small [22, 27]. Furthermore, minimal selection cuts are applied: $M_{\ell \bar{\ell }} > 4$ GeV and $M_{\ell '\bar{\ell }'} > 4$ GeV cuts are applied for the $gg\rightarrow Z(\gamma ^*)Z(\gamma ^*)\rightarrow \ell \bar{\ell }\ell '\bar{\ell }'$ process to eliminate the soft photon singularities. The renormalisation and factorisation scales are set to $M_{VV}/2$ and the pp collision energy is $\sqrt{s}=8$ TeV.

The phase space integration is carried out using the multi-channel Monte Carlo integration technique [78], in which every kinematic structure has its own mapping from random variables to the phase space configuration such that singularities or peaks in the amplitude are compensated, and the inverse Jacobi determinants of all mappings are summed to give the inverse weight at each phase space point. This approach has the advantage of a straightforward systematic extension from the SM to two-Higgs models: an extra channel with a mapping for the heavy Higgs resonance is added. The multi-channel technique has been implemented in the new version of gg2VV, and has been tested thoroughly. Each mapping was phase space integrated individually to check that the result matches the analytically known phase space volume for massless final state particles. Cross sections for the continuum background and $h_1$ only contributions⁶ to the processes considered here were found to be in agreement with the results of Ref. [26], which were calculated using a previous version of gg2VV with a different phase space implementation based on a decomposition into sections. Furthermore, results for similar processes calculated using the same code show excellent agreement with a fully independent implementation [79].

Table 1 Widths of the physical Higgs bosons $h_1$ and $h_2$ in the 1HSM with mixing angles $\theta =\pi /15$ and $\theta =\pi /8$ as well as $\mu _1= \lambda _1=\lambda _2=0$

4 Results

In this section we present integrated and differential cross section-level results for the $h_2$ signal (S) and its interference (I) at the LHC for the processes

$$\begin{aligned} g g (\rightarrow \{h_1,h_2\}) \rightarrow Z(\gamma ^*)Z(\gamma ^*) \rightarrow \ell \bar{\ell }\ell '\bar{\ell }' \end{aligned}$$

(4.1)

and

$$\begin{aligned} g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu }\bar{\ell }'\nu ' \end{aligned}$$

(4.2)

with input parameters, settings and cuts as described in Sect. 3.

The following notation is used:

$$\begin{aligned} S\sim & {} \left| \mathcal{M}_{h2}\right| ^2\end{aligned}$$

(4.3)

$$\begin{aligned} I_{h1}\sim & {} 2\,\mathrm {Re}(\mathcal{M}_{h2}^*\,\mathcal{M}_{h1})\end{aligned}$$

(4.4)

$$\begin{aligned} I_\mathrm{bkg}\sim & {} 2\,\mathrm {Re}(\mathcal{M}_{h2}^*\,\mathcal{M}_\mathrm{bkg})\end{aligned}$$

(4.5)

$$\begin{aligned} I_\mathrm{full}= & {} I_{h1}+I_\mathrm{bkg}\end{aligned}$$

(4.6)

$$\begin{aligned} R_i= & {} \frac{S+I_i}{S}. \end{aligned}$$

(4.7)

The interference of the heavy Higgs signal with the light Higgs and continuum background is given separately. We also give the combined interference to illustrate the overall effect. The ratios $R_{h1}$, $R_\mathrm{bkg}$ and $R_\mathrm{full}$ illustrate the relative change of the heavy Higgs signal due to interference with the light Higgs and continuum background amplitude contributions.

Integrated results for processes 4.1 and 4.2 are shown in Tables 2, 3, 4 and 5. As illustrated by the differential distributions shown below, a $|M_{VV}-M_{h2}| < \Gamma _{h2}$ window cut is an effective means to eliminate or mitigate the interference.⁷ Therefore, integrated results with window cut are presented in Tables 6, 7, 8 and 9.

Table 2 Cross sections for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions at $\sqrt{s}=8$ TeV at loop-induced leading order in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 300, 600, 900$ GeV and mixing angle $\theta =\pi /15$. Results for the heavy Higgs ($h_2$) signal (S) and its interference with the light Higgs ($I_{h1}$) and the continuum background ($I_\mathrm{bkg}$) and the full interference ($I_\mathrm{full}$) are given. The ratio $R_i = (S+I_i)/S$ illustrates the relative change of the heavy Higgs signal due to interference with the light Higgs and continuum background amplitude contributions. Minimal cuts are applied (see main text). Cross sections are given for a single lepton flavour combination. The integration error is displayed in brackets

Table 3 Cross sections for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Table 2

Table 4 Cross sections for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions at $\sqrt{s}=8$ TeV in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 300, 600, 900$ GeV and mixing angle $\theta =\pi /15$. Other details as in Table 2

Table 5 Cross sections for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Table 4

Table 6 Cross sections for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions at $\sqrt{s}=8$ TeV in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 300, 600, 900$ GeV and mixing angle $\theta =\pi /15$. An additional window cut $\left| M_{ZZ} - M_{h2}\right| < \Gamma _{h2}$ is applied. Other details as in Table 2

Table 7 Cross sections for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Table 6

Table 8 Cross sections for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions at $\sqrt{s}=8$ TeV in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 300, 600, 900$ GeV and mixing angle $\theta =\pi /15$. An additional window cut $\left| M_{WW} - M_{h2}\right| < \Gamma _{h2}$ is applied. Other details as in Table 2

Table 9 Cross sections for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Table 8

Corresponding $M_{VV}$ distributions for processes 4.1 and 4.2 and $M_{h2} = 300, 600, 900$ GeV are shown in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15. Results for the heavy Higgs signal and including interference with the light Higgs and the continuum background are displayed. Where appropriate, vertical dashed lines at $M_{VV}=M_{h2} \pm \Gamma _{h2}$ are used to visualise the effect of a $\left| M_{VV} - M_{h2}\right| < \Gamma _{h2}$ window cut. For invariant VV masses with negative signal plus interference, the distributions are shown in Figs. 6 and 13.

As seen in the tables and figures, interference effects increase significantly with increasing heavy Higgs mass. They can range from $\mathcal{O}(10\,\%)$ to $\mathcal{O}(1)$ effects for integrated cross sections. With window cut we find that interference effects are mitigated to $\mathcal{O}(10\,\%)$ or less. We note that the heavy Higgs–continuum background interference is negative above $M_{h2}$ and positive below $M_{h2}$, while the heavy Higgs–light Higgs interference has the opposite behaviour. Consequently, in the heavy Higgs resonance region a strong cancellation occurs when both interference contributions are added. It is therefore essential to take both contributions into account in phenomenological and experimental studies. Despite the occurring cancellation, the full interference is clearly non-negligible and modifies the heavy Higgs line shape. We find overall $\mathcal{O}(10\,\%)$ effects for integrated cross sections, even if a window cut is applied. The results for $\theta =\pi /15$ and $\theta =\pi /8$ are in qualitative agreement. Relative interference effects show a mild quantitative dependence on the mixing angle.

Fig. 2

Invariant ZZ mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions at $\sqrt{s}=8$ TeV at loop-induced leading order in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 300$ GeV and mixing angle $\theta =\pi /15$. Results for the heavy Higgs ($h_2$) signal (S) and including interference with the light Higgs ($S+I_{h1}$) and the continuum background ($S+I_{h1}+I_\mathrm{bkg}$) are shown. Minimal cuts are applied (see main text). Other details as in Table 2

Fig. 3

Invariant ZZ mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Vertical dashed lines are shown at $M_{VV}=M_{h2} \pm \Gamma _{h2}$. Other details as in Fig. 2

Fig. 4

Invariant ZZ mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions at $\sqrt{s}=8$ TeV in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 600$ GeV and mixing angle $\theta =\pi /15$. Other details as in Fig. 2

Fig. 5

Invariant ZZ mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Fig. 4

Fig. 6

Invariant ZZ mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. As Fig. 5, but with linear $\mathrm{d}\sigma /\mathrm{d}M_{ZZ}$ scale, to illustrate negative $S+I_{h1}$ and $S+I_{h1}+I_\mathrm{bkg}$

Fig. 7

Invariant ZZ mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions at $\sqrt{s}=8$ TeV in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 900$ GeV and mixing angle $\theta =\pi /15$. Other details as in Fig. 2

Fig. 8

Invariant ZZ mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow ZZ \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Fig. 7

Fig. 9

Invariant WW mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions at $\sqrt{s}=8$ TeV in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 300$ GeV and mixing angle $\theta =\pi /15$. Other details as in Fig. 2

Fig. 10

Invariant WW mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Fig. 9

Fig. 11

Invariant WW mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions at $\sqrt{s}=8$ TeV in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 600$ GeV and mixing angle $\theta =\pi /15$. Other details as in Fig. 2

Fig. 12

Invariant WW mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Fig. 11

Fig. 13

Invariant WW mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. As Fig. 12, but with linear $\mathrm{d}\sigma /\mathrm{d}M_{WW}$ scale, to illustrate negative $S+I_{h1}$

Fig. 14

Invariant WW mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions at $\sqrt{s}=8$ TeV in the 1HSM with $M_{h1} = 125$ GeV, $M_{h2} = 900$ GeV and mixing angle $\theta =\pi /15$. Other details as in Fig. 2

We note that our results for heavy Higgs–light Higgs interference are qualitatively in agreement with those given in Ref. [43], where this interference is considered for $g g\rightarrow \{h_1,h_2\} \rightarrow ZZ \rightarrow 4\ell$, but in the 1HSM model with an extra $Z_2$ symmetry.

Fig. 15

Invariant WW mass distributions for $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ in pp collisions in the 1HSM with mixing angle $\theta =\pi /8$. Other details as in Fig. 14.

5 Conclusions

In the 1HSM, the modification of the heavy Higgs ($h_2$) signal due to interference with the continuum background and the off-shell light Higgs ($h_1$) contribution has been studied for the $g g (\rightarrow \{h_1,h_2\}) \rightarrow Z(\gamma ^*)Z(\gamma ^*) \rightarrow \ell \bar{\ell } \ell ' \bar{\ell }'$ and $g g (\rightarrow \{h_1,h_2\}) \rightarrow W^-W^+ \rightarrow \ell \bar{\nu } \bar{\ell }' \nu '$ processes at the LHC. Interference effects increase significantly with increasing heavy Higgs mass. They can range from $\mathcal{O}(10\,\%)$ to $\mathcal{O}(1)$ effects for integrated cross sections. With a $\left| M_{VV} - M_{h2}\right| < \Gamma _{h2}$ window cut, we find that interference effects are mitigated to $\mathcal{O}(10\,\%)$ or less. We find that the heavy Higgs–continuum background interference is negative above $M_{h2}$ and positive below $M_{h2}$, while the heavy Higgs–light Higgs interference has the opposite behaviour. Consequently, in the heavy Higgs resonance region a strong cancellation occurs when both interference contributions are added. It is therefore essential to take both contributions into account in phenomenological and experimental studies. Despite the occurring cancellation, the full interference is clearly non-negligible and modifies the heavy Higgs line shape. We find overall $\mathcal{O}(10\,\%)$ effects for integrated cross sections, even if a window cut is applied to mitigate the interference effects. Our calculations have been carried out with a parton-level integrator and event generator, which we have made publicly available.