The breaking of the $SU(2)_L\times U(1)_Y$ symmetry: The 750 GeV resonance at the LHC and perturbative unitarity

If the di-photon excess at 750 GeV hinted by the 2015 data at the LHC is explained in terms of a scalar resonance participating in the breaking of the electro-weak symmetry, this resonance must be accompanied by other scalar states for perturbative unitarity in vector boson scattering to be preserved. The simplest set-up consistent with perturbative unitarity and with the data of the di-photon excess is the Georgi-Machacek model.

If the di-photon excess at 750 GeV hinted by the 2015 data at the LHC is explained in terms of a scalar resonance participating in the breaking of the electro-weak symmetry, this resonance must be accompanied by other scalar states for perturbative unitarity in vector boson scattering to be preserved. The simplest set-up consistent with perturbative unitarity and with the data of the di-photon excess is the Georgi-Machacek model.

I. MOTIVATIONS
Irrespective of whether it will stay or not-the recent excess in the 2015 LHC data with two photons in the final state at invariant mass of about 750 GeV [1] reminds us that even after the discovery of the Higgs boson we may still not know all the details of the breaking of the electro-weak (EW) symmetry.
Let us interpret the LHC di-photon excess as a new scalar resonance.
The simplest (although perhaps least interesting) possibility is that this resonance takes no part in the breaking of the EW symmetry. In this case, it is possible to reproduce the di-photon excess by coupling the resonance-in a generic fashion-to extra scalar or fermionic degrees of freedom (see, for instance, [2] and [3]). If this is the case, the rationale of such new physics is bound to remain rather mysterious and we might be justified in thinking that it would be for the best if the di-photon excess were to disappear from the new data in 2016.
On the other hand, if this resonance takes part in the EW symmetry breaking, its existence would tell us something new about such a mechanism, in particular that it is not realised by the vacuum expectation value (VEV) of the Higgs boson alone. Moreover-and more importantly for the present work-the presence of such a state necessarily affects the high-energy behavior of the theory: to the extent that the perturbative unitarity of vector boson scattering is to be preserved, such a resonance cannot come by itself or with arbitrary couplings [4].
Let us classify states after symmetry breaking according to their properties under custodial SU (2) C and take the new resonance to be a singlet. There are two possibilities. This custodial singlet either • comes from one or more doublets (this choice leads to the two Higgs doublet model (2HDM) [5] and related constructions) and its coupling to the gauge bosons is fixed by gauge invariance to combine with * marco@sissa.it † alfredo.leonardo.urbano@cern.ch that of the Higgs boson to cancel the unitarity violating growth with the center-of-mass (CM) energy; or • its coupling to the gauge bosons does not combine with that of the Higgs boson as to cancel the unitarity violations, and we must also include a quintuplet of custodial SU (2) C -the only scalar with a contribution in the high-energy amplitudes of the opposite sign with respect to that of the Higgs boson and other singlets [6]-in order for unitarity to be preserved.
The inclusion of a custodial singlet resonance arbitrary coupled to the gauge bosons therefore leads naturally to the Georgi-Machacek (GM) model [7]-the simplest model to contain a custodial quintuplet and in which symmetry breaking is achieved by three scalar fields: one doublet (with hypercharge 1/2) and two triplets (with hypercharges 1 and 0).
If neither of the above is the case, perturbative unitarity cannot be preserved and the singlet resonance must belong to a non-perturbative regime. This would imply the exciting discovery of a new interaction that is strong at the EW scale. A fit of the di-photon excess in terms of a non-perturbative resonance is possible and has been already discussed in the literature (for instance, see [3]).
In this paper we expand on the reasoning above. We discuss to what extent a singlet resonance can take part in the EW symmetry breaking and still belong to a perturbative regime in which reliably computations can be performed. The GM model seems to emerge as the simplest model satisfying these requirements that also explains the di-photon excess at the LHC for a realistic choice of its parameters.

A. Perturbative unitarity
Perturbative unitarity limits the possible models in which the leading orders of perturbation theory are expected to be a reliable guide to physics [6]. If perturbative unitarity is satisfied, EW interactions are described by a renormalisable gauge theory and the strength of the arXiv:1601.02447v3 [hep-ph] 28 Apr 2016 interactions among the particle remains weak at all energies. If this is not the case, unitarity is recovered by the inclusion of higher order terms; these, however, cannot be small and a non-perturbative regime is entered.
The requirement of perturbative unitarity is stated in terms of partial-wave amplitudes a J (s) where the amplitude of vector boson scattering is and s and t are the Mandelstam variables. Unitarity requires that In general, the partial-wave amplitude in vector boson scattering is given by with terms growing as the fourth power and the square of the CM energy, and a constant, respectively. A vanishes by gauge invariance that implies g 4V = g 2 3V . B vanishes in the standard model (SM) because of the Higgs boson h contribution and the relationship among the couplings (with self-explanatory notation). The constant terms in C sets a limit on the Higgs boson mass in the SM and on the masses of other states in its extensions. If there are more singlets, for instance two: H 1 and H 1 , their couplings must satisfy in order for the coefficient B in eq. (3) to vanish. This is realised in the 2HDM and variations of the same. The other possibility is to have a negative contribution: this can only come from a quintuplet (see [6] and [8]) of custodial SU (2) C . In fact, for interactions and between the longitudinal components of the vector boson fields Σ = exp [−i/v σ a π a ] and the singlet in eq. (6) and quintuplet in eq. (51), the amplitudes for singlet scalars are always with the same sign as the Higgs boson, while gives a (repulsive) negative contribution.
Considering the limit s m 2 W , m H1 , m H1 m H5 -and having the Higgs boson contribution already cancel the contribution from the vector bosons to the coefficient B in eq. (3)-an exact cancellation between eq. (8) and eq. (9) requires As shown below, such a cancellation, and the unitarity of the theory, are automatically implemented in the GM model.

II. THE FIRST POSSIBILITY: THE 2HDM
The first possibility considered in the introduction section is the simplest: perturbative unitarity is maintained by having the scalar resonance coupling at a special value fixed by gauge invariance (see eq. (5)).
This would be the first choice in trying to incorporate the resonance within a model. Unfortunately, the parameters of the 2HDM model must be pushed to rather unrealistic values in order to accomodate the di-photon data [9]. These values are particularly worrisome in the light of the required size of the the Yukawa couplings, the renormalized values of which bring the theory into a non-perturbative regime [10].
We therefore consider the other case discussed in section I.

III. THE GM MODEL
The GM model contains a complex SU (2) L doublet field φ (Y = 1), a real triplet field ξ (Y = 0), and a complex SU (2) L triplet field χ (Y = 2). The scalar content of the theory can be organised in terms of the SU (2) L ⊗ SU (2) R symmetry, and we define the following multiplets whose VEVs are The VEVs of the two triplets must be the same in order to preserve custodial SU (2) C .
The doublet and the two triplet states can be written in components: The most general potential that conserves SU (2) C is given by where τ and t are the SU (2) generators in the doublet and triplet representation respectively, and U a matrix that rotates ∆ into the Cartesian basis. From the (canonically normalised) kinetic terms we can read the interactions with the EW gauge bosons.
Considering the neutral components of the scalar fields in eq. (11), a direct computation gives The imaginary part of φ and χ does not interact with the EW gauge bosons as a consequence of CP invariance. The gauge boson masses are given by Under SU (2) C we have the group representations (2, 2) ∼ 1 ⊕ 3, and (3, 3) ∼ 1 ⊕ 3 ⊕ 5. One of the two triplets is unphysical, since it represents the Goldstone bosons eaten by the EW gauge bosons. Accordingly, the GM model has ten physical degrees of freedom: two SU (2) C singlets H 0 1 , H 0 1 (the Higgs and the additional scalar resonance), one SU (2) . If compared with the setup envisaged in section I, the spectrum of the GM model has one additional scalar triplet. However, the triplet H 3 does not interact with the EW gauge bosons.
The mass eigenstates in terms of gauge eigenstates are From the Lagrangian in eq. (19) we find the physical couplings where the doublet-triplet mixing angle is given by As far as the charged interactions are concerned, we find, in the g → 0 limit, From the interactions in eqs. (22)-(24) we have for the singlets, and for the quintuplet. The cancellation of the coefficient B in the vector boson scattering amplitude follows from A. Mass spectra and couplings After EW symmetry breaking, a mixing between the neutral singlet scalar states H 0 1 and H 0 1 is generated. The corresponding mass matrix is with The mass matrix can be easily diagonalized by introducing the physical states where α is a mixing angle and we used the short-hand notation c α ≡ cos α, s α ≡ sin α from which α = ± sin −1 [(1 − c 2α )/2]. The mass eigenvalues are The mixing angle is defined by The masses of the custodial triplet and quintuplet are given by Neglecting loop-induced mass splitting, the mass is degenerate within the same custodial multiplet. As a consequence of the rotation in eq. (30) and the ratio of VEVs in eq. (23) the Higgs couplings with gauge bosons and fermions are modified with respect to the corresponding SM values. One finds with g hW + W − = c 2 W g hZZ .

IV. FITTING THE 750 GEV DI-PHOTON EXCESS
There exists a number of constraints that the parameters of the GM model must satisfy in order to reproduce the observed di-photon excess while, at the same time, not be in violation of other known observables.
First of all, for the model to be consistent, its parameters must • Satisfy perturbative unitarity. Perturbative unitarity on the 2 → 2 scalar field scattering amplitudes provides a set of stringent constraints on the parameters of the scalar potential [11]: In addition, we also have • Have a potential bounded from below. This requirement restricts λ 3,4 in the following interval In addition, we must verify that, for each choice of parameters, known experimental constraints are satisfied. These are: • Modification of the SM Higgs couplings. Higgs coupling measurements [12] strongly constrained the allowed values of v ∆ and α.
• Electroweak precision tests. The presence of additional scalar states, charged under the EW symmetry, generates a non-zero contribution to the S parameter [14].
In order to explore the model, we perform a parameter scan by proceeding as follows: 3. The remaining parameters v ∆ and M 2 are randomly generated within the intervals v ∆ ∈ (0, 50) GeV, |M 2 | ∈ (1, 10 4 ) GeV. The VEV v φ is given by v φ = v 2 − 8v 2 ∆ ; 4. For each sample of values the mass matrix in eq. (28)-and hence the mixing angle α-and the mass eigenstates in eq. (33) can be computed;

5.
As a final step in our Monte-Carlo generation, we check that the values of v ∆ and α are consistent with the Higgs coupling measurements at the 2-σ level. Following [13], we perform a two-parameter χ 2 fit of the most recent ATLAS and CMS measurements [12]. We show in fig. 1 the corresponding 1-and 2-σ confidence level contours in the plane (α, v ∆ ).
We also check that the correction to the S parameter is within 3-σ of the LEP-I and LEP-II fit of the EW precision observables. In fig. 3 we show the constraint from the EW parameter S on the scan of the parameters v ∆ and α of the GM model.
Having set the scope and range of the parameter scan, we are now in the position to discuss the fit of the diphoton excess.

A. Production cross section
The mixing with the Higgs boson in eq. (30) and the presence of a non-zero VEV v ∆ automatically allows for H production via both Vector Boson Fusion (VBF) and gluon fusion (ggF). The former is triggered by tree-level H couplings with the EW gauge bosons, the latter at one loop by H coupling to SM fermions, with the top quark providing the most sizable contribution.
The relevant couplings are The H production cross section can be straightforwardly obtained by rescaling the production cross section of a SM Higgs with m h = 750 GeV. At √ s = 13 TeV we have σ(VBF → h) m h =750 GeV 0.1307 pb and σ(ggF → h) m h =750 GeV 0.736 pb [15], and the rescaling is simply given by The rescaled cross sections crucially depend on the values of v ∆ and α. In fig. 1 we show contours of constant VBF (left panel, blue lines) and ggF (right panel, green lines) H production compared with the reference values of the SM Higgs with m h = 750 GeV (red lines). As clear from the plot, in the allowed region of the (α, v ∆ ) plane we always observe a reduction if compared with the SM case.
In addition to VBF and ggF, we also includefollowing [16]-production via photon fusion (γγF) for inelastic, partially elastic and elastic collisions.

B. Total decay width and di-photon decay
The di-photon signal strength at √ s = 13 TeV is given by where the last line accounts for production via γγF [16]. Given the preliminary status of the experimental analysis, we do not perform any complicated fit. On the contrary, the purpose of this section is to check whether the GM model can account for a di-photon signal strength of the order of few fb, that is the order of magnitude suggested by present data. As discussed in section II-a positive answer is anything but trivial in weakly coupled theories (in particular without invoking the presence of extra vector-like fermions with either large multiplicities, electric charge or Yukawa couplings) and would be a remarkable result if achieved in the GM model.
In order to evaluate eq. (48) we need to compute the total decay width of the singlet, Γ H , and the di-photon decay width.
At the tree level, H predominantly decays-as far as the SM final states are concerned-into W + W − , ZZ, tt and hh. The corresponding decay widths can be computed rescaling those of the SM Higgs boson. We find where The trilinear scalar coupling is [11] The singlet H can also decay into the custodial triplet and quintuplet if the corresponding channels are kinematically allowed. If m H > m H5 /2 (m H > m H3 /2), the new decay channels are Γ ). The decay widths can be computed as in eq. (49), and the relevant couplings are [11]: with g HH 0 , and g HH 0 with g HH 0 Finally, H can decay into a vector boson plus a custodial triplet scalar. If m H > m W + m H3 and m H > m Z + m H3 the corresponding decay channels are Γ where the kinematic function λ is λ(x, y) = (1 − x − y) 2 − 4xy. The relevant couplings are The sum of the tree-level decay widths reconstruct the total width Γ H .
The loop-induced di-photon decay width for the scalar singlet H = h, H is therefore where the loop functions are known and can be find, for instance, in [17]. The last term in eq. (56) represents the contribution of the electrically charged scalar states, and we have β s ≡ g HHsH * s v/2m 2 s . The electrically charged scalars affect the di-photon decay of both the new scalar resonance H and the Higgs h (the scalar couplings in eqs. (51,52) for the Higgs boson can be found in [11]). The challenge is to explain the di-photon signal strength observed by ATLAS and CMS without introducing big deviation in the di-photon Higgs decay. Result of the parameter scan in terms of mixing angle α versus triplet VEV v∆. We superimpose the analysed points to the region allowed by Higgs coupling measurements. The constraint from the EW parameter S on the scan is shown in green (1-, 2-and 3-σ confidence level regions correspond to lighter shades).
dominating the loop in Γ H γγ . In fig. 3 we recast the parameter scan in the plane (α, v ∆ ). The yellow contours agree with [18]. We see that points where µ H ∼ O(1) fb correspond to small and negative mixing angle, α ∼ −3 • and triplet VEV v ∆ 20 GeV. In this region the dominant contribution to the production cross section is given by γγF. Production by means of ggF and VBF contributes up to 20%. As a consequence, the tension between the diphoton excess observed at √ s = 13 TeV and the absence of such signal in the dataset at √ s = 8 TeV is alleviated. The production cross section via ggF-going from √ s = 13 TeV to √ s = 8 TeV-is reduced by the factor σ(ggF → H) 13 TeV /σ(ggF → H) 8 TeV = 4.693 while the production cross section via γγF is reduced by a factor of 2. These scaling factors make the di-photon excess at √ s = 13 TeV consistent with the bound extracted from the √ s = 8 TeV dataset. In fig. 4 we recast the parameter scan in the plane (m H5 , m H3 ). Points where µ H ∼ O(1) fb correspond to m H5 ∼ 400 − 600 GeV, m H3 ∼ 650 − 700 GeV. This feature is expected because for these values the corresponding loop in the di-photon decay amplitude of H is maximised.
The explanation of the di-photon excess in the context of the GM model predicts the presence of additional light scalar degrees of freedom, including the doubly-charged state H ++ 5 . Notice that tree-level decays of H into triplet or quintuplet scalar states are not kinematically allowed at the red points of the scan. The characteristic phenomenology [18,19] of these scalar states represents a signature of the model.
We checked that the model, for the chosen choice of parameter values, is consistent with other searches for resonant production of a pair of SM particles which constrain the tree-level decay modes of H [2].
As it can be seen in fig. 3-there is a moderate tension with the EW parameter S for which the fit of the diphoton excess (the red dots) only agrees at the 3-σ level. This is to be expected given the presence of the additional charged new states.
The Higgs scaling factor κ γ is defined as the ratio between the loop-induced h → γγ coupling in the GM model with respect to that of the SM. At the red points in fig. 2, we find 0.8 κ γ 1.2. The presence of such deviation is consistent with the present experimental bound [12].
We find that the other two neutral scalars, H 0 3 and H 0 5 give a negligible contribution to the di-photon cross section.
Concerning the total decay width Γ H , points where µ H ∼ O(1) fb correspond to Γ H ∼ 1 GeV. The value of the total decay width suggested by data represents at the moment the most controversial aspect of the diphoton excess. Since the typical di-photon invariant mass resolution at 750 GeV is estimated to be around 10 GeV, it is natural to expect a large total decay width, Γ H 40 GeV. At this stage of the experimental analysis no conclusive statements can be made, and the value Γ H ∼ 1 GeV is perfectly consistent with the data. However, if large values of Γ H are confirmed by future analysis, an explanation of the di-photon excess in terms of weakly coupled theories will be disfavored.

D. Perturbative reliability
The result above is qualitatively different with respect to both the case in which the resonance is not taking part in the EW symmetry breaking (and one is forced to introduce additional electrically charged vector-like fermions to boost both production cross section and di-photon decay) and the 2HDM (in which the condition µ H ∼ O(1) fb requires unrealistically large Yukawa couplings). In our scan, all the dimensionless couplings of the GM model are kept within the perturbative regime.
This point is better understood in terms of the overall size of the di-photon decay induced by the loop of scalar particles. In full generality, we can consider the effective Lagrangian