Triple Higgs coupling in the most general 2HDM at SM-like scenario

We consider the triple Higgs coupling for $h(125)$ Higgs boson within the most general 2HDM. At moderate values of parameters of model, allowing by modern data, noticeable deviation of this coupling from its SM value is improbable. This deviation can be sizable only if some measurable parameters of the model are exotic.


Introduction
The recent discovery of a Higgs boson with M ≈ 125 GeV at the LHC [1]- [3] suggests that the spontaneous electroweak symmetry breaking is most probably brought up by the Higgs mechanism. The simplest realization of the Higgs mechanism introduces a single scalar isodoublet φ with the Higgs potential V H = −m 2 (φ † φ)/2 + λ(φ † φ) 2 /2. This model is usually called "the Standard Model" (SM). The mentioned data do not rule out the possibility of realization of beyond SM models (BSM) which include both neutral Higgs scalars h a (generally without definite CP parity) and charged Higgs scalars H ± b with masses M a and M b± respectively.
In the discussion that follows we use the relative couplings for each neutral Higgs boson h a (for the case with single charged Higgs boson H ± ):

SM
[P = V (W, Z), q = t, b, ..., ℓ = τ, ...] , The quantities χ P a are the ratios of the couplings of h a with the fundamental particles P to the corresponding couplings for the would be SM Higgs boson with M h = M a . The other relative couplings describe interaction of h a with charged Higgs boson. Couplings χ V a and χ ± a are real due to Hermiticity of Lagrangian, while other couplings are generally complex.
We omit the adjective "relative" below.

SM-like scenario
Current data allow us to suggest that Nature realizes the SM-like scenario 1 : The observed particle with mass M ≈ 125 GeV is a Higgs boson, we call it h 1 . It interacts with the gauge bosons and t-quarks with coupling strengths that are close to those predicted by the SM within experimental accuracy (see e.g. [4]- [6]). In particular, for coupling with the gauge bosons In estimates we will have in mind ε V 0.1.

Two Higgs doublet Model (2HDM)
The 2HDM presents the simplest extension of the standard Higgs model [9]. It offers a number of phenomenological scenarios with different physical content in different regions of the model parameter space, such as a natural mechanism for spontaneous CP violation, etc.
In the most general 2HDM the couplings (1) obey the following sum rules [14]- [16]: We have constructed in [16] the minimal complete set of measurable quantities ("observables") which determine all parameters of the 2HDM. This set contains v.e.v. of Higgs field v = 246 GeV, masses of Higgs bosons M a , M ± (a = 1, 2, 3), two out of three couplings χ V a , In the most general 2HDM, these observables are independent of each other. In some particular variants of 2HDM, additional relations between these parameters may appear (for example, in the CP conserving case we have χ V 3 = 0, χ ± 3 = 0).

Limitations for parameters
The values of parameters λ a of 2HDM (and -therefore -mentioned basic parameters) obey two groups of constraints (see e.g. [10], [11]). Positivity constraints are conditions for the stability of Higgs potential at large quasi-classical values of fields. They do not restrict parameters from above.
Perturbativity (and unitarity) constraints make it possible to use the first non-vanishing approximation of perturbation theory for description of physical phenomena with reasonable accuracy -perturbative description. (This is a tree approximation for most of phenomena and a one-loop approximation for the phenomena which are absent at tree level, e.g. decays h → γγ , h → Zγ, h → gg). The starting point in obtaining of these constraints is the observation that the effective parameter of perturbative expansion is not λ i (i = 1, 2, ...7) but λ i /∆ with ∆ = 8π or 4π. The perturbativity condition is written usually in the form |λ i | < ∆.
At |λ i | ≈ ∆ perturbative description of physical phenomena is incorrect even at low energies. In particular, the equations, expressing masses and couplings via parameters of Lagrangian, become invalid. Good example provide the one-loop radiative corrections (RC) to the triple Higgs coupling [17]- [21]. In the SM-like scenario these RC reach 150 ÷ 200% at |λ i | ≈ ∆. (The ref. [21] presents example with clear details. Authors consider Inert Doublet Model, i.e. 2HDM with exact Z 2 symmetry in the SM-like case, at λ 4 = λ 5 = 0 and λ 1 = λ SM . The one-loop corrections to the g(h 1 h 1 h 1 ) are described by single parameter λ 3 , they reach 180% at |λ 3 | ≈ ∆.) The first non-vanishing approximation of perturbation theory describes physical phenomena with relative inaccuracy k only at In particular, in the region of parameters, provided accuracy of standard description in 30% one should have k = 0.3. In this region of parameters the value of RC, discussed in [17]- [21], does not exceed 20%. Below we will have in mind this very limitation with k ≈ 0.3.
The realization of the SM-like scenario imposes additional restrictions on the parameters. Because of sum rules (3) In the SM-like scenario the perturbativity constraints lead to additional restrictions. In particular, according to Eq. (23) from Ref. [16], the perturbativity constraint (5) imposes the limitation on the coupling of h 1 to charged Higgs bosons: It means that the heavy charged Higgs boson gives only small contribution to the two-photon width of the observed Higgs boson h 1 . Next, we consider heavy neutral Higgs bosons h a (a = 2, 3) in the SMlike scenario. The couplings χ V a are small (see (6)), while Eq. (23) from Ref. [16] allows to have big values of χ ± a ( 1/ √ ε V ). Therefore, the twophoton width of the boson h a is strongly different from the similar width calculated for the would-be SM Higgs boson with the mass M a .

Triple Higgs vertex
The observation of hh production and the extraction of the triple Higgs vertex g(hhh) from the future experiments is scheduled at the LHC and other colliders. This is a necessary step in the verification of the Higgs mechanism. Hopefully, these observations will allow us to see the effects of BSM 3 . The studies of triple Higgs coupling have long history, for recent reviews see e.g. [22]. There are two major parts. The first one is whether it is possible or not to observe hh production, caused by hhh vertex. The second one is whether it is possible to use these observations for extraction of New Physics effect beyond SM.
The accuracy in the extraction of a triple Higgs vertex g(hhh) from the future data cannot be high, since in each case corresponding experiments deal with interference of two channels -an independent production of two Higgses and production of Higgses via hhh vertex. This interference is mainly destructive [23]. For example, for 100 TeV hadron collider with total luminosity 3/ab one can hope to reach accuracy of 40% in the extraction of this vertex from future data [24]; at ILC the accuracy in the extraction of g(hhh) will be better than 80% only after 10 years of operation [25]. Therefore, the effects of New Physics will be distinguishable in the data of g(hhh) in the realistic future only if the deviation of this coupling from its SM value is high enough, |χ 111 − 1| 1.
One of the approaches in the description of the SM violations is to add in the SM Lagrangian terms with anomalous interactions of Higgs boson. It was found for many reasonable benchmark points that these anomalous interactions are difficult for observation [26].
The other approach is to consider some special form of BSM. The review of the whole variety of possible BSM models is beyond our scope. We limit ourself considering 2HDM in its most general form.
The potential of such g(h 1 h 1 h 1 ) observation was studied for some benchmark points of parameters of 2HDM mainly in the case of CP conservation and with moderate values of parameters [27], for the MSSM with CPconservation [28] or with violated CP [29] mainly beyond SM-like scenario. The case of SM-like scenario with similar limitations was considered in [30].

Triple Higgs coupling via observables
The transition from neutral components of basic fields φ 1,2 to the neutral Higgs bosons h a is described by some mixing matrix. The equation for triple Higgs coupling in the most general 2HDM via parameters of Lagrangian and elements of this mixing matrix is obtained simply (see e.g. Eq. (25) of Ref. [16]). The expression of this coupling in terms of the introduced observables (4) was obtained in Eq. (36) of Ref. [16]. We transform it to the following form:

Triple Higgs coupling in SM-like scenario
With estimates (6), (7) we have We see that at moderate values of parameters, the relative coupling χ 111 is close to 1, and it is difficult to expect a sizable effect 4 .
Nevertheless, it is interesting to consider the special exotic values of model parameters that provide sizable deviations of triple Higgs coupling from its SM value, i.e. |χ 111 − 1| 1. We consider the effect of different terms R i , entering R.
The term R 1 can give |χ 111 − 1| 1 if the charged Higgs boson H ± is heavy enough and the coupling χ ± 1 of the observed Higgs boson with H ± deviates substantially from the value χ ± 1 ≈ 1. In view of Eq. (8), it can happen if this coupling is either very small or negative.
The term can give |χ 111 − 1| 1 only if at least one of other Higgs bosons h 2,3 is heavy enough, M 2,3 > 1 TeV. Direct discovery of such Higgs bosons seems to be a difficult task. Therefore the value of g(h 1 h 1 h 1 ) might become an important source of knowledge about such heavy neutrals for a long time.
The term R 3 contains small factors χ V 2,3 , χ H + W − 1 and factors χ ± 2,3 which can be large (up to 1/ √ ε V ). The term R 3 can be not small if H ± is heavy.
Certainly, the real range of possible values of discussed parameters is restricted by other observations. Better estimates are possible only after measuring of ε V with reasonable accuracy. In particular, at ε V ≪ 0.1 we cannot expect sizable effects in the triple Higgs vertex.

Summary
• Measuring hh production at various colliders is a necessary step in the verification of Higgs mechanism of EWSB. Within the SM-like scenario in the 2HDM, these measurements can give information about the New Physics beyond SM only at exotic values of the parameters listed above. The enlargement of the field of parameters of 2HDM at the transition from CP conserved softly Z 2 broken potential to the most general case gives no new essential opportunities in the deviation of triple Higgs coupling from its SM value.
In our conclusions we limit ourself perturbative limitations in the form (5) with k ∼ 0.3. These limitations guarantee us applicability of first orders of perturbation theory for description of model (including the expressions of masses and couplings via parameters of Lagrangian) and small value of quantum (loop) corrections.
• In other models deviation of the triple Higgs coupling from its SM value can be stronger than that in 2HDM at moderate values of parameters, see [33]. In the particular case of the nMSSM (2HDM +Higgs singlet) values χ 111 can range from -5 to 20 [28], [29].
• If the mass M 2 of the heavier Higgs boson h 2 lies within the interval (250 ÷ 400) GeV and |χ t 2 | > 1 (in the SM-like scenario for h 1 ), the following interesting phenomenon takes place. The boson h 2 becomes relatively narrow and the cross section of gluon fusion gg → h 2 can be larger than that for the would-be SM Higgs boson with mass M 2 . The process gg → h 2 → h 1 h 1 can be seen as a resonant production of the h 1 h 1 pair. In principle, it allows us to discover the mentioned h 2 at LHC (see examples in [27], [34], [35] for special sets of parameters).