Higgs boson decay into 2 photons in the type~II Seesaw Model

We study the two photon decay channel of the Standard Model-like component of the CP-even Higgs bosons present in the type II Seesaw Model. The corresponding cross-section is found to be significantly enhanced in parts of the parameter space, due to the (doubly-)charged Higgs bosons' $(H^{\pm \pm})H^\pm$ virtual contributions, while all the other Higgs decay channels remain Standard Model(SM)-like. In other parts of the parameter space $H^{\pm \pm}$ (and $H^{\pm}$) interfere destructively, reducing the two photon branching ratio tremendously below the SM prediction. Such properties allow to account for any excess such as the one reported by ATLAS/CMS at $\approx 125$ GeV if confirmed by future data; if not, for the fact that a SM-like Higgs exclusion in the diphoton channel around 114-115 GeV as reported by ATLAS, does not contradict a SM-like Higgs at LEP(!), and at any rate, for the fact that ATLAS/CMS exclusion limits put stringent lower bounds on the $H^{\pm \pm}$ mass, particularly in the parameter space regions where the direct limits from same-sign leptonic decays of $H^{\pm \pm}$ do not apply.


JHEP04(2012)136
similar lines but will not be discussed in the present paper.] Section 4 is devoted to the theoretical and experimental constraints as well as to the numerical analysis for the physical observables. We conclude in section 5.

The DTHM model
In [20] we have performed a detailed study of DTHM potential, derived the most general set of dynamical constraints on the parameters of the model at leading order and outlined the salient features of Higgs boson phenomenology at the colliders. These constraints delineate precisely the theoretically allowed parameter space domain that one should take into account in Higgs phenomenological analyses. We have also shown that in most of the parameter space the DTHM is similar to the SM except in the small µ regime where the doublet and triplet component of the Higgs could have a maximal mixing.
The scalar sector of the DTHM model consists of the standard Higgs doublet H and a colorless Higgs triplet ∆ with hypercharge Y H = 1 and Y ∆ = 2 respectively. Their matrix representation are given by: The most general SU(2) L × U(1) Y gauge invariant renormalizable Lagrangian in the scalar sector is [12,20]: L Y ukawa contains all the SM Yukawa sector plus one extra term that provides, after spontaneous electroweak symmetry breaking (EWSB), a Majorana mass to neutrinos. Once EWSB takes place, the Higgs doublet and triplet acquire vacuum expectation values inducing the Z and W masses (2.5) with v 2 = (v 2 d + 4v 2 t ) ≈ (246 GeV) 2 . The DTHM is fully specified by seven independent parameters which we will take: λ, λ i=1... 4 , µ and v t . These parameters respect a set of dynamical constraints originating from -3 -

JHEP04(2012)136
the potential , particularly perturbative unitarity and boundedness from below constraints . The model spectrum contains seven physical Higgs states: a pair of CP even states (h 0 , H 0 ), one CP odd Higgs boson A, one simply charged Higgs H ± and one doubly charged state H ±± . The squared masses of the neutral CP-even states and of the charged and doubly charged states are given in terms of the VEV's and the parameters of the potential as follows, For a recent and comprehensive study of the DTHM, in particular concerning the distinctive properties of the mixing angle between the neutral components of the doublet and triplet Higgs fields, we refer the reader to [20]. We close this section by stressing an important point which is seldom clearly stated in the literature. Recall first that the general rational justifying the name 'type II seesaw' assumes µ ∼ M ∆ ∼ M GUT (or any other scale much larger than the electroweak scale). One then obtains naturally v t v d , as a consequence of the electroweak symmetry breaking conditions, and thus naturally very small neutrino masses for Yukawa couplings of order 1. But then one has also µ v t and consequently a very heavy Higgs sector, largely out of the reach of the LHC, apart from the lightest state h 0 , as can be seen from the above mass expressions; this leaves us at the electroweak scale with simply a SM Higgs sector. Put differently, a search for the DTHM Higgs states at the LHC entails small µ(∼ O(v t )) and thus implicitly questions the validity of the seesaw mechanism. Since we are interested in new physics visible at the LHC, we will take up the latter assumption of small µ in our phenomenological study, which can also have some theoretical justification related to spontaneous soft lepton-number violation.

H → γγ
The low SM Higgs mass region, [110,140] GeV, is the most challenging for LHC searches. In this mass regime, the main search channel through the rare decay into a pair of photons can be complemented by the decay into τ + τ − and potentially the bb channel (particularly  for the lower edge of the mass range and/or for supersymmetric Higgs searches), while the W W * , ZZ * channels are already competitive in the upper edge (130-140 GeV) of this mass range [1] the Higgs being produced mainly via gluon fusion [21,22].
The theoretical predictions for the loop induced decays H → γγ (and H → γZ) have been initiated since many years [23][24][25]. Several more recent studies have been carried out looking for large loop effects. Such large effects can be found in various extensions of the SM, such as the Minimal Supersymmetric Standard Model (MSSM) [26][27][28][29], the two Higgs Doublet Model [30][31][32][33][34][35][36], the Next-to-MSSM [37][38][39], the little Higgs models [40,41], in models with a real triplet [42] and in Randall-Sundrum models [43]. To the best of our knowledge there is no H → γγ study in the context of a triplet field with hypercharge Y = 2, that is comprising charged and doubly-charged Higgs states.
We turn here to the study of the latter case explaining how these charged and doubly charged Higgs states of the DTHM could enhance or suppress the 2 photons decay rate. Furthermore, since one or the other of the two CP-even neutral Higgs bosons h 0 , H 0 present in the DTHM can behave as a purely SM-like Higgs depending on the regime under consideration (see [20]), we will refer to the SM-like state generically as H in the following. It should be kept in mind, however, that when H = h 0 all the other DTHM Higgs states are heavier than H while when H = H 0 they are all generically lighter than H, thereby leading possibly to a different phenomenological interpretation of the present experimental exclusion limits for H → γγ channel.
The decay H → γγ is mediated at 1-loop level by the virtual exchange of the SM fermions, the SM gauge bosons and the new charged Higgs states. Using the general results for spin-1/2, spin-1 and spin-0 contributions, [25] (see also [44][45][46]), one includes readily the extra contributions to the partial width which takes the following form, where the first two terms in the squared amplitude are the known SM contributions up to the difference in the couplings of H to up and down quarks and W ± in the DTHM, when H is not purely SM-like. The relevant reduced couplings (relative to the SM ones) are summarized in table. 1. In eq. instance [46]). The last two terms correspond to the H ± and H ±± contributions whose Feynman diagrams are depicted in Fig 1. The structure of the H ± and H ±± contributions is the same except for the fact that the H ±± contribution is enhanced by a relative factor four in the amplitude since H ±± has an electric charge of ±2 units. The scalar function for spin-0 A H 0 is defined as and the function f (τ ) is given by while the reduced DTHM trilinear couplings of H to H ± and H ±± are given bỹ The latter can be read off from the couplings of h 0 , in the limit where h 0 becomes a pure SM Higgs, i.e. when s α → 0, or from the couplings of H 0 , obtained simply from the above couplings by the substitutions in the limit where H 0 becomes a pure SM Higgs, i.e. when c α → 0, taking also into account that [In the above equations α and β stand for the mixing angles in the CP-even and charged Higgs sectors with the shorthand notations s x , c x for cos x, sin x; In eqs. (3.6), (3.7) we have denoted by¯ the sign of s α in the convention where c α is always positive, which is defined as¯ [20].] Obviously, in the limit where one of the two CP-even Higgs states is JHEP04(2012)136 Table 1.
The CP-even neutral Higgs couplings to fermions and gauge bosons in the DTHM relative to the SM Higgs couplings, α and β denote the mixing angles respectively in the CP-even and charged Higgs sectors, e is the electron charge, m W the W gauge boson mass and s W the weak mixing angle.
SM-like, the other state behaves as a pure triplet ∆ 0 with suppressed couplings to H ± and H ±± given by Due to the smallness of v t /v d the states h 0 , H 0 are mutually essentially SM-like or essentially triplet, apart from a very tiny and fine-tuned region where they carry significant components of both. (see [20] for more details). We can thus safely consider that any experimental limit on the SM Higgs decay in two photons can be applied exclusively either to h 0 or to H 0 , depending on whether we assume H to be the lightest or the heaviest among all the neutral and charged Higgs states of the DTHM. 1 As a cross-check on our tools, an independent calculation using the FeynArts and FormCalc [47,48] packages for which we provided a DTHM model file was also carried out and we found perfect agreement with eq. (3.1). Clearly the contribution of the H ±± and H ± loops depends on the details of the scalar potential. The phase space function A 0 involves the scalar masses m H ± and m H ±± , while g HH + H − and g HH ++ H −− are functions of several Higgs potential parameters. It is clear from eqs. (3.6), (3.7) that those couplings are not suppressed in the small v t and/or sin α limit but have a contribution which is proportional to the vacuum expectation value of the doublet field and hence can be a source of large enhancement of H → γγ (and H → γZ).
As well known, the decay width of H → γγ in the SM is dominated by the W loops which can also interfere destructively with the subdominant top contribution. In the DTHM, the signs of the couplings g HH + H − and g HH ++ H −− , and thus those of the H ± and H ±± contributions to Γ(H → γγ), are fixed respectively by the signs of 2λ 1 + λ 4 and λ 1 , eqs. (3.4), (3.5), (3.6), (3.7). However, the combined perturbative unitarity and potential boundedness from below (BFB) constraints derived in [20] confine λ 1 , λ 4 to small regions. For instance, in the case of vanishing λ 2,3 , λ 1 is forced to be positive while λ 4 can have either signs but still with bounded values of |λ 4 | and |2λ 1 + λ 4 |. Moreover, since we are considering scenarios where µ ∼ O(v t ), negative values of λ 4 can be favored by the experimental bounds on the (doubly)charged Higgs masses, eqs. (2.8), (2.9). For definiteness we stick in the following to λ 1 > 0, although the sign of λ 1 can be relaxed if λ 2,3 are non-vanishing. Also in the considered mass range for H, H ± and H ±± the function A H 0 (τ ) is real-valued and takes positive values in the range 0.3 − 1. An increasing value of λ 1 will JHEP04(2012)136 thus lead to contributions of H ± and H ±± that are constructive among each other but destructive with respect to the sum of W boson and top quark contributions. [Recall that ReA H 1 (τ ) takes negative values in the range −12 to −7.] As we will see in the next section, this can either reduce tremendously the branching ratio into diphotons, or increase it by an amount that can be already constrained by the present ATLAS/CMS results.

Theoretical and experimental constraints, numerics and discussions
In order to infer limits on the parameters of our model from the experimental searches, we consider the ratio σ γγ /σ γγ SM that has been constrained by the recent ATLAS and CMS limits and compare it to the quantity Note that the experimental limits on σ γγ /σ γγ SM assume all Higgs production channels, as well as SM-like Higgs decay branching ratios, taking into account the known QCD and EW corrections and uncertainties in the proton-proton collision, [49], some of which cancel out in the ratio. In contrast, the ratio R γγ obviously concerns only the leading parton level gluon fusion Higgs production contribution, and assumes the narrow width approximation. We wish to comment here briefly on the approximations involved when identifying these two ratios, keeping in mind that in our present exploratory study we do not aim at a high precision analysis that would require a more quantitative estimate of the effects of these approximations. Strictly speaking, in the narrow width approximation the cross-section σ(gg → H → γγ) is related to Br(H → gg) × Br(H → γγ) up to a phase space factor, [50]. The ratio r = σ(gg → H → γγ) DT HM /σ(gg → H → γγ) SM of the DTHM to SM values of this cross-section reduces then to r = Γ(H → all) SM /Γ(H → all) DT HM × R γγ (H). Taking into account the fact that in the SM-like Higgs regime of DTHM, the branching ratios of all Higgs decay channels are the same as in the SM, except for H → γγ (and H → γZ) where they can significantly differ but remain very small compared to the other decay channels, one expects Γ(H → all) SM /Γ(H → all) DT HM ≈ 1. R γγ (H) is thus a very good estimate of r in this regime.
Nonetheless, using the experimental limits on σ γγ /σ γγ SM to constrain R γγ (H) entails in principle subtracting from the former the subleading channels; mainly the W -boson fusion process which is of order 5-8% of the gluon fusion for m H in the range 115-130GeV, [49], and taking into account possible differences in the experimental analysis efficiency between these two channels. In the pure SM-like regime, there can be some instances where all details of the production channels cancel out in the ratio σ γγ /σ γγ SM due to a complete factorization of the Higgs branching ratio into two photons that contains all the new DTHM effects. In this case σ γγ /σ γγ SM becomes insensitive to the production channels and one retrieves essentially R γγ (H)(≈ Br(H → γγ) DT HM /Br(H → γγ) SM ). However, this cancellation can be model-dependent and does not necessarily occur when the W -boson fusion process is included. Indeed in this case the relevant parton-level process, qq → qqH → qqγγ, with the requirement of two hard jets in the final state, does not proceed via the exchange of a -8 -JHEP04(2012)136 single intermediate unstable particle, due to the W W intermediate states. The validity of the factorization of Br(H → γγ) through the narrow width approximation requires here, and in contrast with the gg → H → γγ case, that the variation of Γ(H → γγ) as a function of m H should remain much larger than the Higgs total width itself, [50]. We find that the latter requirement is far from being satisfied in our case. In particular, we checked that in the experimentally interesting Higgs mass range around 125GeV, Γ(H → γγ) becomes a very flat function of m H leading to a variation at least an order of magnitude smaller than Γ(H → all). This invalidates the above mentioned factorization so that the ratio σ γγ /σ γγ SM remains sensitive to the initial state and does not reduce trivially to R γγ . One should thus keep in mind the corresonding approximations when using R γγ instead of the experimentally constrained quantity. Of course the PDF uncertainties as well as the initial state leading QCD corrections drop out in the ratios r and R γγ , even in the regime where H is not purely SM-like. However, QCD corrections to the fermion loop contribution to H → γγ should be in principle included as they can somewhat affect the interference pattern between the standard model and the (doubly)charged Higgs states contributions to be discussed in section 4.3.

DTHM parameter scans and theoretical constraints
All the Higgs mass spectrum of the model is fixed in terms of λ, λ 1,2,3,4 , v t and µ which we will take as input parameters, [20]. As one can see from eq. (2.3) λ 2 and λ 3 enter only the purely triplet sector. Since we focus here on the SM-like (doublet) component, their contributions will always be suppressed by the triplet VEV value and can be safely neglected as compared to the contributions of λ 1 and λ 4 which enter the game associated with the doublet VEV, eqs. (3.8)- (3.11). Taking into account the previous comments, λ 2,3 will be fixed and we perform a scan over the other parameter as follows: v t = 1 GeV, λ = 0.45 ∼ 1, 0 < λ 1 < 10, The chosen range for λ values ensures a light SM-like Higgs state and the scanned domain of the λ i 's is consistent with the perturbative unitarity and BFB bounds mentioned earlier.

Experimental constraints
Here we will discuss the experimental constraints on the triplet vev as well as on the scalar particles of the DTHM. In the above scan, the triplet vev has been taken equal to 1 GeV in order to satisfy the constraint from the ρ parameter [51] for which the tree-level extra contribution δρ should not exceed the current limits from precision measurements: |δρ| 0.001. Nowadays, the doubly charged Higgs boson is subject to many experimental searches. Due to its spectacular signature from H ±± → l ± l ± , the doubly charged Higgs has been searched by many experiments such as LEP, Tevatron and LHC. At the Tevatron, D∅ [52,53] and CDF [54, 55] excluded a doubly charged Higgs with a mass in the range 100 → 150 GeV. Recently, CMS also performed with 1 fb −1 luminosity a search for doubly charged -9 -

JHEP04(2012)136
Higgs decaying to a pair of leptons, setting a lower mass limit of 313 GeV from H ±± → µ ± µ ± , e ± e ± , µ ± e ± [56]. The limit is lower if we consider the other decay channel with one electron or more [56,57].
We stress that all those bounds assume a 100% branching ratio for H ±± → l ± l ± decay, while in realistic cases one can easily find scenarios where this decay channel is suppressed whith respect to H ±± → W ± W ±( * ) [11,12,58,59] which could invalidate partially the CDF, D∅, CMS and ATLAS limits. In our scenario with v t 1 GeV we estimated that the decay channel H ±± → W ± W ± * can still overwhelm the two-lepton channel for m H ±± down to ≈ 110 GeV . It has been shown in [60] that (cascade) channels such as H ±± → H ± W ± * → H 0 W ± * W ± * can also compete strongly with H ±± → l ± l ± , lowering the m H ±± mass bound to 100 GeV. We will take m H ±± ≈ 110 GeV as a nominal lower bound in our numerical analysis.
In the case of charged Higgs boson, if it decays dominantly to leptons or to light quarks cs (for small v t ) we can apply the LEP mass lower bounds that are of the order of 80 GeV [61,62]. For large v t , i.e. much larger than the neutrino masses but still well below the electroweak scale, the dominant decay is either H + → tb or one of the bosonic decays H + → W + Z, H + → W + h 0 /W + A 0 . For the first two decay modes there has been no explicit search neither at LEP nor at the Tevatron, while for the H + → W + A 0 decay (and possibly for H + → W + h 0 / if h 0 decays similarly to A 0 ), one can use the LEPII search performed in the framework of two Higgs doublet models. In this case the charged Higgs mass limit is again of the order of 80 GeV [62].

Numerical results
In the subsequent numerical discussion we use the following input parameters: G F = 1.166 × 10 −5 GeV −2 , α −1 ≈ 128, m Z = 91.1875G eV, m W = 80.45 GeV and m t = 173 GeV. We also compute the total width of the Higgs boson taking into account leading order QCD corrections as given in [63] as well as the off-shell decays H → W W * and H → ZZ * [64,65].
We show in figure 2 the branching ratio for the CP-even Higgs bosons decays into two photons as a function of λ 1 , illustrated for several values of λ 4 and λ = 0.45, v t = 1 GeV. In the left panel we take µ = 1 GeV, implying that the lightest CP-even state h 0 carries 99% of the SM-like Higgs component, with an essentially fixed mass m h 0 ≈ 114-115 GeV over the full range of values considered for λ 1 and λ 4 . In the right panel, where µ = 0.3 GeV, the heaviest CP-even state H 0 carries most of the SM-like Higgs component [∼ 90% for λ 1 3] with a mass more sensitive to the λ 1 and λ 4 couplings, m H 0 ≈ 115-123 GeV. 2 As can be seen from the plots, Br(H → γγ) is very close to the SM prediction [≈ 2 × 10 −3 ] for small values of λ 1 , irrespective of the values of λ 4 . Indeed in this region the diphoton decay is dominated by the SM contributions, the H ±± contribution being shutdown for vanishing λ 1 , cf. eq. (3.6), while the sensitivity to λ 4 in the H ± contribution, eqs. (3.5), (3.7), is suppressed by a large m H ± mass, m H ± ≈ 164-237 GeV for −1 < λ 4 < 1. Increasing λ 1 (for fixed λ 4 ) enhances the g HH ± H ∓ and g HH ±± H ∓∓ couplings. The  destructive interference, already noted in section 3, between the SM loop contributions and those of H ± and H ±± becomes then more and more pronounced. The leading DTHM effect is mainly from the H ±± contribution, the latter being enhanced with respect to H ± by a factor 4 due to the doubled electric charge, but also due to a smaller mass than the latter in some parts of the parameter space, m H ±± ≈ 110-266 GeV. It is easy to see from eqs. (3.1), (3.4)-(3.7) that the amplitude for H → γγ is essentially linear in λ 1 , since m H ± and m H ±± , eqs. (2.8), (2.9), do not depend on λ 1 while the dependence on this coupling through m H is screened by the mild behavior of the scalar functions A H 0,1/2,1 . Furthermore, the latter functions remain real-valued in the considered domain of Higgs masses. There exit thus necessarily values of λ 1 where the effect of the destructive interference is maximized leading to a tremendous reduction of Γ(H → γγ). Since all the other decay channels remain SM-like, the same reduction occurs for Br(H → γγ). The different dips seen in figure 2 are due to such a severe cancellation between SM loops and H ± and H ±± loops, and they occur for λ 1 values within the allowed unitarity & BFB regions. Increasing λ 1 beyond the dip values, the contributions of H ±± and H ± become bigger than the SM contributions and eventually come to largely dominate for sufficiently large λ 1 . There is however another interesting effect when λ 4 increases. Of course the locations of the dips depend also on the values of λ 4 , moving them to lower values of λ 1 for larger λ 4 . Thus, for larger λ 4 , there is place, within the considered range of λ 1 , for a significant increase of Br(H → γγ) by even more than one order of magnitude with respect to the SM prediction. This spectacular enhancement is due to the fact that larger λ 4 leads to smaller H ±± and H ± which can efficiently boost the reduced couplings that scale like the inverse second power of these masses. For instance varying λ 4 between −1 and 1 in the left panel case, decreases H ±± from 266 to 110 GeV, while varying it from −1 to 0 in the right panel case decreases H ±± In figures 4, 5 we illustrate the effects directly in terms of the ratio R γγ ≈ σ γγ /σ γγ SM defined in eq. (4.1), for benchmark Higgs masses. We also show on the plots the present experimental exclusion limits corresponding to these masses, taken from [5]. As can be seen from figure 4, one can easily accommodate, for m H ≈ 125GeV, a SM cross-section, R γγ (m H = 125GeV) = 1, or a cross-section in excess of the SM, e.g. R γγ (m H = 125GeV) ∼ 3-4, for values of λ 1 , λ 4 within the theoretically allowed region, fulfilling as well the present experimental bound m H ± 80 GeV and the moderate bound m H ±± 110 GeV as discussed previously. The excess reported by ATLAS and CMS in the diphoton channel can be readily interpreted in this context. However, one should keep in mind that all other channels remain SM-like, so that the milder excess observed in W W * and ZZ * should disappear with higher statistics in this scenario. This holds independently of which of the two states, h 0 or H 0 , is playing the role of the SM-like Higgs.
We comment now on another scenario, in case the reported excess around m H ≈ 125 GeV would not stand the future accumulated statistics. Figure  . The sensitivity to the coupling λ 1 can be seen by comparing figures 6 and 7. For low values of λ 1 as in figure 6, the ratio R γγ remains below 1 even for increasing H ±± and H ± masses. The reason is that these masses become large when λ 4 is large (and negative) for which the loop contribution of H ± does not vanish, as can be easily seen from eqs. (2.8), (3.5), (3.7). In contrast, we see that for the parameter set of figure 7, R γγ can take SM-like values for m H ±± of order 180 GeV, while an excess of 2 to 6 can be achieved for m H ±± ≈ 130-160 GeV, and a deficit in R γγ , down to 2 orders of magnitude, for m H ±± between 200 and 300 GeV. Increasing m H ±± (and m H ± ) further, increases R γγ again, but rather very slowly towards the SM expectation as can be seen from the upper green region of the plots.

Conclusions
The very recent ATLAS and CMS exclusion limits for the search for the Higgs boson, clearly indicate that if such a light SM-like state exists, it should be somewhere in the region between 114.4 (LEP) and 130 GeV. The diphoton channel is thus expected to play a crucial role in the near future data analyses, eventually confirming the not yet statistically significant excess around 125 GeV. In this paper we have shown that the diphoton channel can be interpreted in a peculiar way in the context of the Type II Seesaw model, provided that the full Higgs sector of the model lies below the TeV scale. While there is always in this model a neutral Higgs state coupling essentially like the SM Higgs, the diphoton channel can be drastically enhanced or reduced by several factors with respect to the SM prediction, as a result of the loop effects of the doubly-charged (and charged) Higgs states, while all the other relevant decay (as well as production) channels remain at their SM level.

JHEP04(2012)136
Theoretically consistent domains of the parameter space in the small µ regime can thus account either for an excess or for a deficit or even for a SM value of the diphoton crosssection, making the model hard to rule out on the basis of the neutral Higgs observables alone. In particular, the exclusion of a mass region through the diphoton channel does not exclude SM-like Higgs events in the other channels (including the LEP ZH channel) for the same mass region. Rather, it can be re-interpreted in terms of bounds on the masses of the doubly-charged (and charged) Higgs states of the model. The experimental search for such light doubly-charged states through their decay into (off-shell) W bosons is a crucial test of the model while the present bounds based on same-sign di-lepton decays do not necessarily apply in our scenario.