You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: '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.'
author:
- Amine Ahriche
- Abdesslam Arhrib
- Salah Nasri
title: 'Triple Higgs Coupling as a Probe of the Twin-Peak Scenario'
---

On July 2012, ATLAS and CMS collaborations [@ATLAS; @CMS] have shown the existence of a Higgs-like resonance around 125 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; $m_{h}=125.3\pm0.4(\text{stat}.)\pm0.5(\text{syst}.)$ for CMS [@CMS2G] and $m_{h}=125.0\pm0.5$ for ATLAS [@ATLAS2G] from the diphoton channel and $m_{h}=125.5\pm0.37(\text{stat}%
.)\pm0.18(\text{syst}.)$ from combined channels. Despite this relatively large uncertainty, a scenario of two degenerate scalars around 125.5 resonance is neither excluded nor confirmed [@TP].

In the twin peak scenario (); it is assumed that there are two scalars $h_{1,2}$ with almost degenerate masses around 125 . To our knowledge, there is no indication from experimental data which disfavor this scenario. The couplings of the twin peak Higgs to SM particles $g_{h_{i}XX}$ are simply scaled with respect to SM rate by $\cos\theta$ (for $h_{1}$) and $\sin\theta$ (for $h_{2}$), where $\theta$ is a mixing angle, such that we have the following approximate sum rule:$$g_{h_{1}f\bar{f}}^{2}+g_{h_{2}f\bar{f}}^{2}\simeq g_{h_{SM}f\bar{f}}%
^{2},~g_{h_{1}VV}^{2}+g_{h_{2}VV}^{2}\simeq g_{h_{SM}VV}^{2}, \label{sr}%$$ where $f$ can be any of the SM fermions and $V=W,Z$ vector boson. In fact, the branching ratios of the Higgs to SM particles are SM-like only if the Higgs invisible is very suppressed or kinematically forbidden as will be considered in our example. Consequently, the single Higgs production such as gluon-gluon fusion at LHC, Higgs-strahlung, Vector Boson Fusions, and $t\bar{t}H$ 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 $\cos^{2}\theta +\sin^{2}\theta=1$. However, for processes with di-Higgs final states ($pp(e^{-}e^{+})\rightarrow hh+X$), 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 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 $pp\rightarrow gg\rightarrow hh$ [@double1]. Such measurement is rather challenging at the LHC, and for this purpose several parton level analysis have been devoted to this process. It turns out that $hh\rightarrow
b\bar{b}\gamma\gamma$ [@double3], $hh\rightarrow b\bar{b}\tau^{+}\tau^{-}$ [@double3; @double0] and $hh\rightarrow b\bar{b}W^{+}W^{-}$ [@double0; @ww] final states are very promising for High luminosity. Recently, CMS report a preliminary result on the search for resonant di-Higgs production in $b\bar{b}\gamma\gamma$ channel [@CMS-hh]. 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 [@ILC]. Using recoil technique for the Higgs-strahlung process, the Higgs mass can be measured with an accuracy of about 40 [@ILC]. We note that at LHC with high luminosity we can measure the Higgs mass with about 100 uncertainty which is quite comparable to $e^{+}e^{-}$ colliders. The triple Higgs coupling can be extracted from $e^{+}%
e^{-}\rightarrow Zh^{\ast}\rightarrow Zhh$ at 500 and even better from $e^{+}e^{-}\rightarrow\nu\overline{\nu}h^{\ast}\rightarrow\nu
\overline{\nu}hh$ at $\sqrt{s}>800$ . In this regards, the LHC and $e^{+}e^{-}$ LC measurements are complementary [@Weiglein:2004hn].

In Ref. [@Gunion], the authors have provided a tool to distinguish the two-degenerate states scenario from the single Higgs one. The approach of [@Gunion] applies only to models which enjoy modifications of $h\rightarrow\gamma\gamma$ 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 [@CMS2G; @ATLAS2G]. 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 [@TSM], where the SM is extended with two real scalar fields $S_{0}$ and $\chi_{1}$; each one is odd under a discrete symmetry $\mathbb{Z}_{2}^{(0)}$ and $\mathbb{Z}_{2}^{(1)}$ respectively. The field $\chi_{1}$ has a non vanishing vacuum expectation value, which breaks $\mathbb{Z}_{2}^{(1)}$ spontaneously, whereas, $\left\langle S_{0}\right\rangle =0$; and hence, $S_{0}$ is a dark matter candidate. Both fields are SM gauge singlets and hence can interact with the ’visible’ particles only via the Higgs doublet $H$. The spontaneous breaking of the electroweak and the $\mathbb{Z}_{2}^{(1)} $ symmetries introduces the two vacuum expectation values $\upsilon$ and $\upsilon_{1}$ respectively. The physical Higgs $h_{1}$ and $h_{2}$, with masses $m_{1}$ and $m_{2}\gtrsim
m_{1}$, are related to the excitations of the neutral component of the SM Higgs doublet field, Re$(H^{(0)})$, and the field $\chi_{1}$ through rotation with a mixing angle $\theta$ and, with a specific choice in the parameter space, could give rise to two degenerate scalars around 125 . In what follows, we denote by $c=\cos\theta$ and $s=\sin\theta$. The quartic and triple couplings of the physical fields $h_{i}$ are given in the appendices in [@Arh]. In our analysis we require that [^1]: *(i)* all the dimensionless quartic couplings to be $\ll4\pi$ for the theory to remain perturbative, *(ii)* the two scalar eigenmasses should be in agreement with recent measurements [@CMS2G; @ATLAS2G]: we have checked that for the Two-Singlets model, the splitting between $m_{1}$ and $m_{2}$ could be of the order of 40 . *(iii)* the ground state stability to be ensured; and *(iv)* we allow the DM mass $m_{0}$ to be as large as 1 . 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, $\sigma^{LHC}(hh)$ and $\sigma^{LHC}(hh+t\bar{t})$ at 14 ; $\sigma^{LC}(hh+Z)$ at 500  and $\sigma^{LC}(hh+E_{miss})$ at 1 . All these processes include, at least, one Feynman diagram with triple Higgs coupling. For the , the total cross section gets contributions from the final states $h_{1}h_{1}$, $h_{1}h_{2}$ and $h_{2}%
h_{2}$. Therefore the quantity to be compared with the standard scenario can be expressed as: $$\sigma^{\mathrm{TPS}}\left( hh+X\right) =\sigma\left(
h_{1}h_{1}+X\right) +2\sigma\left( h_{1}h_{2}+X\right) +\sigma\left(
h_{2}h_{2}+X\right) ,
\label{TPS}%$$ which can be parameterized as: $$\sigma^{\mathrm{TPS}}=\sigma_{aa}r_{1}+\sigma_{ab}r_{2}+\sigma_{bb},$$ with $\sigma_{aa}+\sigma_{ab}+\sigma_{bb}=\sigma^{SM}\left(
hh+X\right) $ and $\sigma_{aa}$, $\sigma_{bb}$ and $\sigma_{ab}$ 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 $\theta$ and the Higgs triple couplings $\lambda_{ijk}^{(3)} $.

In the , 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 $c\lambda_{111}^{(3)}/\lambda_{hhh}^{SM}$ and $s\lambda
_{112}^{(3)}/\lambda_{hhh}^{SM}$, respectively. We denote the total amplitude of these two contributions by $\mathcal{M}_{(a)}$. (2) The ones with no triple Higgs couplings. Their amplitude, denoted by $\mathcal{M}_{(b)}$, is given by the one of the SM scaled by a factor of $c^{2}$. Therefore, the amplitudes $\mathcal{M}_{(a,b)}$ (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: $$%
\begin{tabular}
[c]{lll}\hline\hline $h_{1}h_{1}:$ & &
$\mathcal{M}_{(a)}=[(c\lambda_{111}^{(3)}+s\lambda
_{112}^{(3)})/\lambda_{hhh}^{SM}]\mathcal{M}_{(a)}^{SM},$\\
& & $\mathcal{M}_{(b)}=c^{2}\mathcal{M}_{(b)}^{SM},$\\\hline
$h_{2}h_{2}:$ & & $\mathcal{M}_{(a)}=[(c\lambda_{122}^{(3)}+s\lambda
_{222}^{(3)})/\lambda_{hhh}^{SM}]\mathcal{M}_{(a)}^{SM},$\\
& & $\mathcal{M}_{(b)}=s^{2}\mathcal{M}_{(b)}^{SM},$\\\hline
$h_{1}h_{2}:$ & & $\mathcal{M}_{(a)}=[(c\lambda_{112}^{(3)}+s\lambda
_{122}^{(3)})/\lambda_{hhh}^{SM}]\mathcal{M}_{(a)}^{SM},$\\
& & $\mathcal{M}_{(b)}=cs\mathcal{M}_{(b)}^{SM},$\\\hline\hline
\end{tabular}
\ \$$ where $\lambda_{hhh}^{SM}$ is the SM triple Higgs coupling calculated at one-loop. Then the parameters $r_{i}$ are given by: $$\begin{gathered}
r_{1}=\left\{
c^{2}[\lambda_{111}^{(3)2}+\lambda_{122}^{(3)2}+2\lambda
_{112}^{(3)2}]+s^{2}[\lambda_{112}^{(3)2}+\lambda_{222}^{(3)2}+2\lambda
_{122}^{(3)2}]+2cs[\lambda_{111}^{(3)}\lambda_{112}^{(3)}+2\lambda_{112}%
^{(3)}\lambda_{122}^{(3)}+\lambda_{122}^{(3)}\lambda_{222}^{(3)}]\right\}
/\left( \lambda_{hhh}^{SM}\right)^{2},\nonumber\\
r_{2}=\{c^{3}\lambda_{111}^{(3)}+3c^{2}s\lambda_{112}^{(3)}+3cs^{2}%
\lambda_{122}^{(3)}+s^{3}\lambda_{222}^{(3)}\}/\lambda_{hhh}^{SM}. \label{ri}%\end{gathered}$$ Thus, the values of $r_{i}$ quantify by how much each di-Higgs process deviates from the SM case. In Fig. \[rri\], we show the parameters $r_{i}$ as a function of $\sin\theta$ for about 600 chosen sets of the model parameters within the condition (\[sr\]).

![*Numerical values of the parameters $r_{i}$ in (\[ri\]) for 600 benchmarks that fulfill the above mentioned requirements.*[]{data-label="rri"}](ri.eps){width="7cm" height="5.5cm"}

We see that for very small mixing angle $r_{i}$’s are approximately equal to unity, while for $\sin\theta>0.8$ and $\sin\theta
<-0.2$, the parameter $r_{1} $ becomes larger than unity and $r_{2}$ acquires negative values. This behavior could lead to an enhancement/reduction to the cross section depending on the sign of the interference contribution, $\sigma_{ab}$, to the total cross section. This means that the measurement of the following ratio: $$\xi\left( hh+X\right) =\frac{\sigma^{\mathrm{TPS}}\left( pp(e^{-}%
e^{+})\rightarrow hh+X\right) }{\sigma^{SM}\left(
pp(e^{-}e^{+})\rightarrow
hh+X\right) }, \label{xi}%$$ 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 $\xi\left( hh+X\right) $ 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, $r_{1}=\left( 1+\Delta\right)
^{2}$ and $r_{2}=1+\Delta$, where $\Delta$ represents the relative enhancement of the triple Higgs coupling due to SM+MP. As we will show later, our considered scenario for small or large mixing could be distinguished from the case of SM+MP by combining the ratio (\[xi\]) for different processes.

In Table \[T2\], we give the values of $\sigma_{aa}$, $\sigma_{ab}$ and $\sigma_{bb}$ for the corresponding di-Higgs production processes. We note that their contributions to the LHC process $pp\rightarrow hh$ and to the LC one $e^{+}e^{-}\rightarrow Zhh$ seem to be uncorrelated, which makes the Higgs triple coupling useful to probe this scenario and distinguish it from (SM+MP).

[max width=0.8]{}

  -----------------------------------------------------------------------------------------
        $\sigma_{aa}~(fb)$       $\sigma_{ab}~(fb)$        $\sigma      $\sigma^{SM}~(fb)$
                                                         _{bb}~(fb)$   
  -- ------------------------ ------------------------- -------------- --------------------
              $9.66$                   $-49.9$              $70.1$           $29.86$

      $3.3164\times 10^{-2}$          $0.13952$               $%              $1.02$
                                                           0.84731$    

      $9.0206\times 10^{-3}$   $4.6999\times 10^{-2} $   $9.005\times       $0.14607$
                                                           10^{-2}$    

      $5.1631\times 10^{-2}$         $-0.20867$               $%            $0.14004$
                                                           0.29708$    
  -----------------------------------------------------------------------------------------

  : *Different contributions to the considered processes cross sections. Numbers for LHC are taken from [@Spira:1995mt] at NLO.* []{data-label="T2"}

For the benchmarks considered previously in Fig. \[rri\], we illustrate in Fig. \[cs\] the production cross section of di-Higgs at $e^{+}e^{-}$ LC and LHC and in Fig. \[rat\] the ratio $\xi$. As it can be seen, in the TPS, the cross section of the processes $pp\rightarrow hh$, $pp\rightarrow hh+t\bar{t}$ and $e^{-}e^{+}\rightarrow hh+E_{miss}$ are mostly enhanced, while for $e^{-}e^{+}\rightarrow hh+Z$ it is enhanced just for the mixing values $0.5<\sin\theta<0.8$.

![*The cross section values (\[TPS\]) for the di-Higgs production processes for the 600 benchmarks used previously. The solid lines correspond to the SM cross sections.*[]{data-label="cs"}](pphh.eps "fig:"){width="7cm" height="5.5cm"} ![*The cross section values (\[TPS\]) for the di-Higgs production processes for the 600 benchmarks used previously. The solid lines correspond to the SM cross sections.*[]{data-label="cs"}](eehh.eps "fig:"){width="7cm" height="5.5cm"}

![*The ratios $\xi$ given in (\[xi\]) for the di-Higgs production processes for the 600 benchmark used previously. The green benchmarks correspond to the large mixing case where $0.35<cos^{2}\theta
<0.65$, and the blue point represents the SM; and the solid curve represents the case of a SM extension, where the new physics affects the triple Higgs coupling as $\lambda_{hhh}=\lambda_{hhh}^{SM}(1+\Delta)$; and the value of the relative enhancement $\Delta$ can be read from the palette.*[]{data-label="rat"}](LHC.eps "fig:"){width="7.8cm" height="5.5cm"} ![*The ratios $\xi$ given in (\[xi\]) for the di-Higgs production processes for the 600 benchmark used previously. The green benchmarks correspond to the large mixing case where $0.35<cos^{2}\theta
<0.65$, and the blue point represents the SM; and the solid curve represents the case of a SM extension, where the new physics affects the triple Higgs coupling as $\lambda_{hhh}=\lambda_{hhh}^{SM}(1+\Delta)$; and the value of the relative enhancement $\Delta$ can be read from the palette.*[]{data-label="rat"}](ILC.eps "fig:"){width="7.8cm" height="5.5cm"}

Now let us discuss the possibility of disentangling the from the SM+MP. It is clear from Fig. \[rat\] that for both LHC and LC processes with large mixing, $0.35<\cos^{2}\theta<0.65$, the TPS may coincide with SM+MP. However, for non-maximal mixing values the TPS is clearly different than SM+MP where all benchmarks have the following feature $$\xi_{1}^{TPS}+\xi_{2}^{TPS}>\xi_{1}^{SM+MP}\left( \Delta\right)
+\xi
_{2}^{SM+MP}\left( \Delta\right) , \label{crit}%$$ where $\xi_{i}^{TPS}$ the ratio in (\[xi\]) for any LHC or LC processes and $\xi_{i}^{SM+MP}\left( \Delta\right) $ is the same ratio due the existence of massive particles. Therefore, when measuring the quantities (\[xi\]) for both the LHC and $e^{+}e^{-}$ LC processes, and one finds that the criterion (\[crit\]) is not fulfilled, then it is a certain exclusion exclusion for this scenario. In case where the criterion (\[crit\]) is fulfilled, detailed analysis is required for in order to identify the mixing angle, the parameters $\mathbf{r}_{\mathbf{i}}$ and therefore the Higgs triple couplings. 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 $e^{-}e^{+}\rightarrow hh+E_{miss}$ [@Ahriche:2014xra], new colored scalar singlets contribution to $pp\rightarrow hh$ (or $hh+t\bar{t}$) [@Kribs:2012kz], or the presence of a heavy resonant Higgs [@resonant].

In order to show whether this scenario can be tested at colliders, we consider three benchmarks that may be distinguished from SM+MP (i.e., three red points from Fig. \[rat\]), 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 $\xi_{i}$ are given in Table \[BB\], and in Table \[en\], 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 $pp\rightarrow2b2\tau$ at the LHC and $e^{-}%
e^{+}\rightarrow4b+E_{miss}$ at LC’s, while it is reduced for the processes $pp\rightarrow4b+t\bar{t}$ and $e^{-}e^{+}\rightarrow4b+Z$. For benchmark $B_{1}$, the events number of the processes $pp\rightarrow2b2\tau$ and $e^{-}e^{+}\rightarrow4b+E_{miss}$ is SM-like but it is reduced for the processes $pp\rightarrow4b+t\bar{t}$ and $e^{-}e^{+}\rightarrow4b+Z$. For benchmark $B_{3}$, the events number is reduced for the considered processes.

[max width=0.8]{}

                                       $B_{1}$      $B_{2}$      $B_{3}$
  ---------------------------------- ------------ ------------ ------------
            $\sin \theta $            $0.53555$    $0.90126$    $-0.39802$
               $r_{1}$                $2.95386$    $,2.88466$   $5.62286$
               $r_{2}$                $1.31634$    $0.28189$    $-1.26011$
       $\xi \left( hh\right) $        $1.10345$    $2.80975$    $6.27248$
   $\xi \left( hh+t\bar{t}\right) $   $2.69728$    $2.51821$    $4.66603$
      $\xi \left( hh+Z\right) $       $ 1.22243$   $0.88532$    $0.55827$
   $\xi \left( hh+E_{miss}\right) $   $1.24900$    $2.76488$    $6.07213$

  : *Different values of the ratios (\[ri\]) and (\[xi\]) for the three chosen benchmarks.*[]{data-label="BB"}

[max width=0.8]{}

\[c\][cccccc]{}$Events~number$ & $channel$ & $SM$ & $B_{1}$ & $B_{2}$ & $B_{3}$\
$pp\rightarrow hh$ & $4b$ & $966.75$ & $1066.8$ & $2716.3$ & $6063.9$\
& $2b2\tau$ & $106.70$ & $117.74$ & $299.8$ & $669.27$\
& $2b2\gamma$ & $3.89$ & $4.29$ & $10.93$ & $24.4$\
$pp\rightarrow hh+t\bar{t}$ & $4b$ & $33.02$ & $89.06$ & $83.15$ & $154.07$\
$e^{-}e^{+}\rightarrow hh+Z$ & $4b$ & $23.65$ & $28.91$ & $20.94$ & $13.2$\
$e^{-}e^{+}\rightarrow
hh+E_{miss}$ & $4b$ & $45.34$ & $56.63$ & $125.36$ & $275.31$\

In Fig. \[Mhh\], we illustrate the di-Higgs invariant mass distribution ($M_{h,h}$) for the process $e^{-}e^{+}\rightarrow hh+E_{miss}$. Clearly, the can be easily distinguished from the SM, especially in the case of non-maximal mixing. However, the full confirmation of the requires the enlargement of the investigation by taking into account other di-Higgs production channels such as $hhjj$, $hhW^{\pm}$, $hhZ$ and $hhtj$ at the LHC [@Frederix:2014hta] and the $e^{+}e^{-}$ LC [@ILC].

![*Normalized di-Higgs invariant mass distribution for the process $e^{-}e^{+}\rightarrow hh+E_{miss}$ for the background (BG) and the considered benchmarks in Table \[BB\].*[]{data-label="Mhh"}](MhhILC.eps){width="8cm" height="5.5cm"}

In conclusion, we have investigated the case of twin-peak at the 125 GeV observed scalar resonance by considering different di-Higgs production processes at both LHC and $e^{+}e^{-}$ LC. We have introduced a criterion whose violation excludes the TPS scenario, otherwise this scenario can be surely distinguished from the SM and SM extended by massive fields in case of non-maximal mixing.

Last but not least, we should note that this scenario could be realized within SM +(real/complex) singlet scalar, or any larger scalar field content. This includes neutral or charged scalars that are members any multiplets, where two degenerate scalar eigenstates $h_{1,2}$ at 125 , do couple to the SM gauge fields and fermions by more than $\sim$90%, i.e., the sum rule (\[sr\]) is fulfilled [^2]. 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 scenario.

If the measurement of the couplings $hf\bar{f}$ and $hVV$ become much more precise from the future experiment data, it may be possible that one could be sensitive to the radiative corrections effect to these couplings. Such radiative corrections to $hf\bar{f}$ and $hVV$ couplings in a variety of extended Higgs sector have been evaluated in [@Kanemura:2014dja; @Aoki:2012jj; @Arhrib:2003ph]. These one-loop effects are of the order of 2-10% and even more in some special cases. The present LHC measurements are not yet sensitive to such effects.

Acknowledgments {#acknowledgments .unnumbered}
---------------

We would like to thank A. Djouadi and R. Santos for the valuable comments; and E. Vryonidou for sharing with us her code and for many useful discussions. A. Ahriche is supported by the Algerian Ministry of Higher Education and Scientific Research under the CNEPRU Project No. D01720130042; and partially by DAAD and ICTP. A. Arhrib is supported in part by the Moroccan Ministry of Higher Education and Scientific Research: “projet des domaines prioritaires de la recherche scientifique et du developpement technologique”.

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[^1]: Actually, we considered that all quartic couplings to be of order unity; and the singlet vev $\upsilon_{1}=\left\langle \chi_{1}\right\rangle =20\sim2000$ $\mathrm{GeV.}$

[^2]: In the 2HDM, twin pick scenario has been studied in [@Ferreira:2012nv], but the study concentrated only on the diphoton channel. According to this study [@Ferreira:2012nv], this scenario is not ruled out.