Triple Higgs Coupling as a Probe of the Twin-Peak Scenario

In this letter, we investigate the case of a twin peak around the observed 125 GeV scalar resonance, using di-Higgs production processes at both LHC and $e^{+}e^{-}$ Linear Colliders. We have shown that both at LHC and Linear Collider the triple Higgs couplings play an important role to identify this scenario; and also that this scenario can be distinguishable from any Standard Model extension by extra massive particles which might modify the triple Higgs coupling. We also introduce a criterion that can be used to ruled out the twin peak scenario.

In this letter, we investigate the case of a twin peak around the observed 125 GeV scalar resonance, using di-Higgs production processes at both LHC and e + e − Linear Colliders. We show that the triple Higgs couplings play an important role to identify this scenario; and also that this scenario is surely distinguishable from any Standard Model extension by extra massive particles which might modify the triple Higgs coupling.
On July 2012, ATLAS and CMS collaborations [1,2] have shown the existence of a Higgs-like resonance around 125 GeV confirming the cornerstone of the Higgs mechanism that predicted such particle long times ago. All Higgs couplings measured so far seem to be consistent, to some extent, with the Standard Model (SM) predictions. Moreover, in order to establish the Higgs mechanism as responsible for the phenomena of electroweak symmetry breaking one still needs to measure the self couplings of the Higgs and therefore to reconstruct its scalar potential.
Recent measurements at the LHC show that there is still uncertainty on the Higgs mass; m h = 125.3 ± 0.4(stat.) ± 0.5(syst.) GeV for CMS [3] and m h = 125.0 ± 0.5 GeV for ATLAS [4] from the diphoton channel and m h = 125.5 ± 0.37(stat.) ± 0.18(syst.) GeV from combined channels. Despite this relatively large uncertainty, a scenario of two degenerate scalars around 125.5 GeV resonance is neither excluded nor confirmed [5].
In the twin peak scenario (TPS); it is assumed that there are two scalars h 1,2 with almost degenerate masses around 125 GeV. The couplings of the twin peak Higgs to SM particles g hiXX are simply scaled with respect to SM rate by cos θ (for h 1 ) and sin θ (for h 2 ), where θ is a mixing angle, such that we have the following approximate sum rule: where X can be any of the SM fermions or vector bosons. Consequently, the single Higgs production such as gluongluon fusion at LHC, Higgs-strahlung, Vector Boson fusions, and ttH at LHC and e + e − Linear Colliders (LC) will obey the same sum rule. The summation of event numbers (both for production and decay) of the two possible cases will be identical to SM case since cos 2 θ + sin 2 θ = 1. However, for processes with di-Higgs final states (pp(e − e + ) → hh + X), the triple Higgs couplings may play an important role, and therefore these processes can be useful to distinguish between the cases of one scalar or two degenerate ones around the observed 125 GeV resonance.
It is well known that the triple Higgs couplings can be, in principle, measured directly at the LHC with high luminosity option through double Higgs production pp → gg → hh [6]. Such measurement is rather challenging at LHC, and for this purpose several parton level analysis have been devoted to this process. It turns out that hh → bbγγ [7], hh → bbτ + τ − [7,8] and hh → bbW + W − [8,9] final states are very promising for High luminosity. Recently, CMS report a preliminary result on the search for resonant di-Higgs production in bbγγ channel [10].
The LC has also the capability of measuring with better precision: the Higgs mass and some of the Higgs couplings together with the self coupling of the Higgs [11]. Using recoil technique for the Higgs-strahlung process, the Higgs mass can be measured with an accuracy of about 40 MeV [11]. We note that at LHC with high luminosity we can measure the Higgs mass with about 100 MeV uncertainty which is quite comparable to e + e − colliders. The triple coupling can be extracted from e + e − → Zh * → Zhh at 500 GeV and even better from e + e − → ννh * → ννhh at √ s > 800 GeV. In this regards, the LHC and e + e − LC measurements are complementary [12].
In Ref. [13], the authors have provided a tool to distinguish the two-degenerate states scenario from the single Higgs one. The approach of [13] applies only to models which enjoy modifications of h → γγ rate with respect to the SM. However, according to the latest experimental results, both for ATLAS and CMS the di-photon channel seem to be rather consistent with the SM [3,4]. In this work we propose a new approach to distinguish the TPS. This approach is based on the di-Higgs production which is sensitive to the triple Higgs coupling, that is modified in the majority of SM extensions.
Here, as an example, we consider, the Two-Singlets Model proposed in [14], where the SM is extended with two real scalar fields S 0 and χ 1 ; each one is odd under a discrete symmetry Z In what follows, we denote by c = cos θ and s = sin θ. The quartic and triple couplings of the physical fields h i are given in the appendices in [15].
In our analysis we require that 1 : (i) all the dimensionless quartic couplings to be ≪ 4π for the theory to remain perturbative, (ii) the two scalar eigenmasses should be in agreement with recent measurements [3,4]: we have checked that for the Two-Singlets model, the splitting between m 1 and m 2 could be of the order of 40 MeV. (iii) the ground state stability to be ensured; and (iv) we allow the DM mass m 0 to be as large as 1 TeV.
In our work, we consider di-Higgs production processes at the LHC and e + e − LC, whose values of the cross section could be significant, namely, σ LHC (hh) and σ LHC (hh + tt) at 14 TeV; σ LC (hh + Z) at 500 GeV and σ LC (hh + E miss ) at 1 TeV. All these processes include, at least, one Feynman diagram with triple Higgs coupling. For the TPS, the total cross section get contributions from the final states h 1 h 1 , h 1 h 2 and h 2 h 2 . However, each contribution should be weighted by the h 1,2 modified couplings since the Higgs is detected through its SM final states decay. Therefore the quantity to be compared with the standard scenario can be expressed as: which can be parameterized as: with σ aa + σ ab + σ bb = σ SM (hh + X) and σ aa , σ bb and σ ab correspond to the cross section contributions coming from triple Higgs diagrams (a), non-triple Higgs diagrams (b) and the interference term in the amplitude, respectively. The coefficients r i are dimensionless parameters, that receive contributions from the final states h i h j , which depend on the mixing angle θ and the Higgs triple couplings λ (3) ijk . The SM case can be obtained by taking s = 0 and r i = 1 2 . In the TPS, the amplitudes for di-Higgs production processes have SM Feynman diagrams where the the Higgs field h is replaced by h i . To compute the parameters r i , we first estimate how does each amplitude get modified with respect to the corresponding SM one for each case h i h j . For example, in the case of h 1 h 1 production, there are two types of diagrams: (1) The ones that involve triple scalar interactions h 1 h 1 h 1 and h 2 h 1 h 1 , with couplings equal to the one of a SM times a factor of cλ (3) 111 /λ SM hhh and sλ 112 /λ SM hhh , respectively. We denote the total amplitude of these two contributions by M (a) . (2) The ones with no triple Higgs couplings. Their amplitude, denoted by M (b) , is given by the one of the SM scaled by a factor of c 2 . Therefore, the amplitudes M (a,b) (where a (b) stand for triple Higgs (non-triple Higgs) Feynman diagrams) for the di-Higgs production can be written in terms of their corresponding SM values as: where λ SM hhh is the SM triple Higgs coupling calculated at one-loop. Then the parameters r i are given by: 112 ] 2 + s 4 [cλ (3) 122 + sλ Thus, the values of r i quantify by how much each di-Higgs process deviates from the SM case. In Fig. 1 mixing angle r i 's are approximately equal to unity, where as for |s| > 0.8, the parameter r 1 (r 2 ) becomes larger than unity (negative). This behavior could lead to an enhancement/reduction to the cross section depending on the sign of the interference contribution, σ ab , to the total cross section. This means that the measurement of the ratio: could be very useful to confirm or exclude this scenario based on the deviation of any of the parameters r i from unity.
For instance, the ratio ξ (hh + X) can deviate from unity if the SM is extended with massive particles (SM+MP) that couple to the Higgs doublet and contribute to the triple Higgs coupling as well the Higgs mass. In this case, r 1 = (1 + ∆) 2 , r 2 = 1 + ∆ and r 3 = 1, where ∆ represents the relative enhancement of the triple Higgs coupling due to SM+MP. As we will show later, our discussed scenario will be surely distinguished from the case of SM+MP by combining the ratio (5) for different processes.
In Table I, we give the values of σ aa , σ ab and σ bb for the corresponding di-Higgs production processes. We note that their contributions to the LHC process pp → hh and to the LC one e + e − → Zhh seem to be uncorrelated, which makes the Higgs triple coupling useful to probe this scenario and distinguish it from (SM+MP). For the benchmarks considered previously in Fig. 1, we illustrate in Fig. 2 the production cross section of di-Higgs at e + e − LC and LHC and in Fig. 3 the ratio ξ. As it can be seen, in the TPS, the cross section of the processes pp → hh + tt and e − e + → hh + Z are always reduced, while for pp → hh and e − e + → hh + E miss it could be enhanced or reduced depending on the mixing angle. Now let us discuss the possibility of disentangling the TPS from the SM+MP. According to the ratios ξ (hh) and ξ (hh + tt) (Fig. 3-left), the TPS coincides with the SM+MP in two tight regions of the triple Higgs coupling relative enhancement ∆ ∼ −0.5, −0.7 and ∆ ∼ −1.7. While for the ratios ξ (hh + Z) and ξ (hh + E miss ) (Fig. 3-right), the TPS coincides with the SM+MP only for ∆ ∼ −2.2. Therefore, by measuring these quantities at both the LHC and e + e − LC it possible to confirm/exclude the TPS since a coincidence between the TPS and the SM+MP can not takes place in both measurements. Moreover, if the observed 125 GeV scalar resonance is a twin-peak, then one needs to measure (5) for three well chosen di-Higgs production processes (either at the LHC, e + e − LC or both of them) in order to deduce the values of the three parameters r i , while any other remaining di-Higgs production processes (at both LHC & e + e − LC) could be used to confirm/exclude this scenario. In fact, by studying all the di-Higgs production channels at both LHC and e + e − LC one not only confirm/exclude this scenario, but also distinguished it from models where only one type of processes gets modified by new physics such as: it manifests as new sources of missing energy in e − e + → hh + E miss [17], new colored scalar singlets contribution to pp → hh (or hh + tt) [18], or the presence of a heavy resonant Higgs [19].  In order to show whether this scenario can be tested at colliders, we consider three benchmarks and compare the di-Higgs distribution (of the di-Higgs invariant mass as an example) with the SM one. The corresponding values of ratios r i and ξ i are given in Table II, and in Table III, we present the expected number of events at both the LHC and LC. We see that for benchmark B 2 , the events number is significantly larger than the SM for the channels pp → 2b2τ at the LHC and e − e + → 4b + E miss at LC's, while it is reduced for the processes pp → 4b + tt and e − e + → 4b + Z. For benchmark B 1 , the events number of the processes pp → 2b2τ and e − e + → 4b + E miss is SM-like but it is reduced for the processes pp → 4b + tt and e − e + → 4b + Z. For benchmark B 3 , the events number is reduced for the considered   In Fig. 4, we illustrate the di-Higgs invariant mass distribution (M h,h ) for the process e − e + → hh + E miss . Clearly, the TPS can be easily distinguished from the SM, especially in the case where |sin θ| > 0.2, i.e far from the decoupling limit. However, the full confirmation of the TPS requires the enlargement of the investigation by taking into account other di-Higgs production channels such as hhjj, hhW ± , hhZ and hhtj at the LHC [20] and the e + e − LC [11].  Table II.
In conclusion, we have investigated the case of twin-peak at the 125 GeV observed scalar resonance, where we have shown that by considering different di-Higgs production processes at both LHC and e + e − LC, this scenario that can be surely distinguished from the SM and SM extended by massive fields. It has been shown also that in the case where the mixing between singlet and doublet is slightly small, the di-Higgs production processes would mimic SM predictions and therefore not distinguishable from SM.
Last but not least, we should note that this scenario could be realized within SM +(real/complex) singlet scalar, or any larger scalar content model where two degenerate scalar eigenstates h 1,2 at 125 GeV and couple together to the SM gauge fields and fermions by more than ∼90%, i.e., the sum rule (1) is fulfilled. If the measurement of di-Higgs processes at LHC and/or e + e − LC turn out to be consistent with SM predictions, then it will be very challenging to distinguish the TPS scenario.