Diphoton decay for a 750 GeV scalar boson in a $SU(6)\otimes U(1)_{X}$ model

We propose a new $SU(6)\otimes U(1)_{X}$ GUT model free from anomalies, with a 750 GeV scalar candidate which can decay into two photons, compatible with the recent diphoton signal reported by ATLAS and CMS collaborations. This model gives masses to all fermions and may explain the 750GeV signal through one loop decays to $\gamma\gamma$ with charged vector and charged Higgs bosons, as well as up- and electron-like exotic particles that arise naturally from the condition of cancellation of anomalies of the $SU(6)\otimes U(1)_{X}$ group. We obtain, for different width approximations, allowed mass regions from 900 GeV to 3 TeV for the exotic up-like quark, in agreement with ATLAS and CMS collaborations data.


Introduction
This paper is organized as follows. In section 2, we show the particle content of a SU (6) ⊗ U (1) X model as an anomaly free theory which contains the 331, 321 and 31 subgroups and their SSB scheme. We describe the Yukawa Lagrangian showing that four Higgs fields are sufficient to give masses to all fermions. We also show the most general Higgs potential terms compatible with the symmetries and identify the relevant quartic couplings that will induce the process pp → ξ → γγ. Section 3 is devoted to explore allowed regions consistent with the reported cross section of the 750 GeV signal. Finally, in section 4, we summarize our conclusions.
2 SU (6) ⊗ U (1) X model SU (6) ⊗ U (1) X strong-electroweak models provide us a new framework which contains 331 and SM models for one family of fermions as effective low energy field theories. In order to include the three families we consider replicas of the first family as in the SM. Below, we describe some remarkable properties of these models.
• The symmetry breaking SU (6) → SU (3) C ⊗ SU (3) L ⊗ U (1) Y gives us the following branching rules: where (n ⊗ m) Y are tensorial products of n SU (3) C multiplet with m SU (3) L multiplets and Y corresponds to the U (1) Y quantum number normalized as 2Y / √ 3, where: This gives us the following multiplets for the first family: 1 : where U is a new up-like quark, E − , E − 1 and E − 2 are new exotic charged leptons and ν e1R , N EL and N ER1 are new neutrinos. In order to obtain fermion mass hierarchies among families, discrete symmetries can be introduced to obtain suitable mass matrix ansatz. The additional sterile neutrino ν S with X S = 0 is necessary to produce see-saw mechanisms between neutrinos [26,27,28].
• The covariant derivatives for each type of multiplets are defined as follows: where Latin indices run from 1 to 6, while Greek indices run from 1 to 35. The15 generators are given by (T α • Gauge bosons are described by the 35 =35 adjoint representation which obey the branching rule where In addition, there is a new electrically neutral vector boson X µ from U (1) X symmetry. In total, the SU (6) ⊗ U (1) X group has 36 gauge bosons: eight gluons, eight electroweak bosons, eighteen leptoquark bosons and two electrically neutral bosons.
• Electric charge are constructed using all diagonal generators of SU (6) ⊗ U (1) X : where the a, b, c, d, e and f constants are fixed such that the electric charge match with each charge from the multiplets. We find where Y is the usual hypercharge operator of the SM.
• The fermions contained in the model have the charges listed in Table 1. • The scalar sector is introduced to obtain the correct SSB chain. The two last symmetry breakings are fulfilled using two Higgs fields represented by sextets6 with X H1,H2 = 1/3. The directions of their VEV, V 1 and v 2 , are selected to obtain electrically neutral vacua. In addition, V 1 is at the TeV scale while v 2 is at the electroweak scale. Two additional Higgs fields represented by15 multiplets with X H3 = −2/3 and X H S = 1/3 are introduced to give masses to down quarks and neutrinos, respectively.
The first SSB needs a Higgs field from 35 adjoint representation with the following VEV: where V GUT ∼ 10 17 GeV breaks the gauge symmetry to SU (3) C ⊗SU (3) L ⊗U (1) Y providing masses to the leptoquark bosons. For the second and third SSBs, we define the following Higgs scalar multiplets: where In this way, the SSB chain is given by Eq. (1).
• Vector boson masses: there are two electroweak SSBs in the low-energy SU (3) L ⊗ U (1) X model, the first V at TeV and the second at v GeV scale. After the TeV SSB SU ( µ and X µ mix them together into the weak hypercharge boson B µ and a new massive electrically neutral gauge boson Z Xµ trough the following mixing matrix with the mixing angle The new gauge coupling constant is the electroweak hypercharge g = g X sin θ X = − √ 3g cos θ X . The gauge bosons W ± 3µ , W 0 3µ and W 0 3µ acquire the same mass M W3 which is related to M Z X by sin θ W in the following way where In addition, the ξ gauge couplings are given by Here ξ does not couple to B µ because M B = 0. Secondly, for the GeV SSB SU (2) L ⊗ U (1) Y → U (1) Q will bring the well-known gauge boson mixing through the Weinberg angle tan θ W = g /g The new gauge coupling constant is the electromagnetic charge e = g 2 sin θ W = g Y cos θ W and the new gauge boson masses are where v 2 = v 2 2 + 2v 2 3 + 2v 2 S = (246 GeV) 2 . In addition, Z W µ and Z Xµ acquire the following masses There is an additional gauge boson mixing between the two neutral Z W and Z X through the mixing angle tan obtaining the physical gauge boson masses Z and Z ,

Yukawa Lagrangian
The Yukawa Lagrangian that describes interactions between the Higgs and the fermion sector is the following where a, · · · , f = 1, · · · , 6. The terms ψ T LaĈ h ψi Ψ ab L H ib and χ T LaĈ h χi Ψ ab L H ib give masses to up quarks and leptons at the order of H i and with Yukawa coupling constants h ψi and h χi . Terms containing couplings with singlet leptons, as for example ψ T LaĈ h ψei H a i e + L , give masses to e + L , E + L and E + 2L with coupling constants h ψei , h ψEi and h ψE2i , respectively. The term that contains couplings with the scalar fields H 3 , gives masses to down quarks with coupling constant h 3 . The last two terms in Eq. (25) induce see-saw neutrino masses [29,30,31,32], where ν SL is a Majorana neutrino of mass M S which is fixed to give a light neutrino ν e , with mass at the order of eV.
First, for up quarks, we obtain the following non-diagonal mass matrix, By diagonalizing the symmetric matrix M 0 uU T M 0 uU , we obtain the following masses Charged leptons acquire masses through the following mass matrix in the (e − , E − , E − 1 , E − 2 ) basis, while neutrinos acquire masses through the following mass matrix in the (ν e , ν c e1 , N E , N c E1 , ν S ) basis, Finally, the down quark acquire mass proportional to

Higgs potential
The interactions between the four Higgs bosons are described by the following scalar potential, From H 2 , H 3 and H S (equations (14), (15) and (16)) we obtain three electroweak Higgs doublets which breaks SU (2) L × U (1) Y into U (1) Q . They are expressed as follow, Thus, the model contains an effective three Higgs doublet model. For the charged sector, ϕ ± 2 , ϕ ± 3 and ϕ ± S rotate into the Goldstone bosons G ± associated to W ± , and the physical charged Higgs bosons H ± 1 and H ± 2 .

Couplings with ξ
As we mentioned before, from the mixing terms between the real fields ξ 1 in H 1 and ξ S in H S , we obtain two real scalar fields: our candidate for the 750 GeV signal ξ, and one ξ at the TeV scale. In particular, we are interested in the following trilinear terms coming from the quartic couplings between the weak multiplets of the Higgs potential Eq.( (31)), where λ H1 and λ H2 are linear combinations of λ 12 , λ 13 and λ 1S . The field φ ± 3 is a charged singlet Higgs-like boson coming from the 15-dimensional representation H 3 in Eq. (15). For simplicity, we choose λ H = λ H1 ∼ λ H2 ∼ λ 13 . Thus, the equation (33) becomes For the vector boson sector, for simplicity and following [33] we consider general interactions of the form In addition, from the fact that the scalar boson ξ has not electroweak isospin and hypercharge, its couplings with the A 3 µ and B µ gauge bosons are completely null, hence after the electroweak SSB at the GeV scale the ξ boson remains without interaction with A µ and Z W µ . Moreover, as the gauge bosons Z W µ and Z Xµ mix them together into the physical gauge bosons Z µ and Z µ , there could be some interaction between ξ and Z µ . However, it is strongly suppressed by the Z−mixing angle tan θ Z ∝ v 2 /V 2 .
Finally, for the fermionic sector, the flavor eigenstates of quarks are related to their mass eigenstates trough the following mixing matrix, The off-diagonal blocks mix SM and non-SM quarks. In particular, the t and U quarks are related by where R tt = R U U = cos θ tU and R U t = −R tU = sin θ tU ∝ v 2 /V 1 . Since v 2 /V 1 1 the mixing between t and U is small resulting in the suppression of the coupling between ξ and t. In the same way the off-diagonal components of the mixing matrix in Eq. (37) are proportional to v 2 /V 1 resulting in the suppression of these mixing terms splitting the up-quark sector in SM and non-SM up-quarks.

Diphoton decay
For the analysis of the diphoton decay, we take into account all the possible decay modes of the 750 GeV candidate. Firstly, the masses of charged Higgs bosons H ± 1,2 and φ ± 3 are at the TeV scale, so the decay of ξ at tree level into these charged Higgs bosons in the model is kinematically forbidden. Secondly, when the SSB SU (3) L → SU (2) L takes place, U L does not acquire a SU (2) L quantum number resulting in a SU (2) L singlet. As a consequence of that, the decay ξ → W W is forbidden too. Thirdly, the ξ → ZZ decay is negligible at tree-level as it is suppressed by the Z−mixing angle tan θ Z ∝ v 2 /V 2 . Similarly, the ξ → tt decay is negligible at tree-level as the ξ − t coupling is proportional to sin θ tU ∝ v 2 /V 1 . Finally, the decay ξ → hh is strongly constrained by ATLAS and CMS at 95%CL [34]. In this way, we obtain the following total decay width for ξ,  Figure 2: Contour plots of the production cross-section σ(pp → ξ → γγ) in femtobarns. The dashed line corresponds to the central value at 6 fb, and the shaded bands corresponds to regions at 68.3% (green), 95.5% (yellow) and 99.7% (light blue) C.L. exclusion limits from ATLAS and CMS combined data.
The experimentally reported width of the resonance ranges between 0 and 100 GeV, and can be larger ('broad') or smaller ('narrow') than the experimental resolution of about 6-10 GeV [35]. The best-fit width reported by the ATLAS Collaboration is Γ ∼ 45 GeV ∼ 0.06 m ξ . So, in view of some tension with the CMS data we use three approximations for the decay width: • A width approximation given by the experimentally reported width from the ATLAS Collaboration Γ = 45 GeV.
• A width approximation for a narrower resonance with Γ = 1 GeV.
Following [33,36] the decay rates of ξ are given by where P(x) = √ 1 − 4x 1 − 4x + 6x 2 is a factor correcting the massive final states in the decay width. Here, h 2 U = (h 2 χ1 + h 2 ψ1 )/2 , i.e, we assume the same Yukawa coupling for the three families for simplicity and with the same mass m U and we have made κ = κ , λ = λ . The functions F i are spin dependent functions for the loop factor. For τ i > 1 the function f (τ i ) is

Production cross section
The total cross section σ(pp → ξ → γγ) in the narrow width approximation is given by where is the dimensionless partonic integral computed at the scale µ = m ξ = 750 GeV and center of mass energy √ s = 13 TeV, obtaining C gg = 2137 [37]. For the analysis we have taken the combined-rescaled results for the cross section from CMS and ATLAS, σ(pp → ξ → γγ) = (2 − 8) fb equally valid for √ s = 8 TeV and C gg = 174 [34]. Fig. 1 shows all the possible one loop contributions from exotic charged Higgs bosons, gauge bosons and fermions. In the fermionic loop to γγ we take into account the multiplicity coming from the three families, i.e., three exotic quarks and nine exotic charged leptons. For this reason, the contribution coming from the charged Higgs bosons is almost negligible. We also take m W ± 3 ∼ 3 TeV according to experimental constraints obtained by ATLAS and CMS Collaboration [39]. However, for m W ± 3 ∼ 3 TeV the associated form factor F 1 reaches its asymptotic value so the cross section dependence on m W ± 3 is suppressed. So, the production cross section will depend only on the Yukawa coupling h U , the mass of the quarks m U and on the exotic charged lepton masses m E , m E1 and m E2 . From the lower bound reported by the ATLAS Collaboration searches on exotic heavy charged leptons [40] we set m E = m E1 = m E2 ∼ 600 GeV.
Taking into account all the above conditions, we display in Fig.2 contour plots of the production crosssection σ(pp → ξ → γγ) as function of the up-type quark mass m U and the Yukawa coupling normalized as h U /4π for Γ = 1 GeV and Γ = 45 GeV. The lower bound of 900 GeV for m U corresponds to the reported value in recent searches on top-and bottom-like heavy quarks from ATLAS and CMS Collaborations [41] and the upper bound of 3 TeV corresponds to the asymptotic value obtained from the fermionic from factor F 1/2 . We obtain allowed regions for both Γ = 1 GeV and Γ = 45 GeV widths for the scalar particle of 750 GeV in agreement with the ATLAS and CMS Collaborations data. In Fig.2 (a)  from 0.2 to 1.0 and an allowed mass region for the up-like quark from 900 GeV to 3.0 TeV at 99.7% CL. In Fig.2 (b) the model is excluded for h U /4π < 0.6 and m U > 1.5 TeV at 99.7% CL. Finally, for the case Γ = Γ γγ + Γ gg + Γ Zγ + Γ ZZ , we show in Fig. 3 the different contributions in Eq. (40) for the decay width of ξ. From Fig. 3 (a), the case κ = λ = η = 0 and h U = 0.5 corresponds to pure fermionic contributions into the loops. We can see that the contributions (ignoring the dominant Γ gg ) Γ γγ , Γ Zγ , Γ ZZ have branching ratios of order 64%, 25%, 11% respectively. On the other hand, the case κ = λ = η = 0.5 and h U = 0.5 in Fig. 3 (b), corresponds to both fermionic and bosonic contributions into the loop with BR γγ , BR Zγ , BR ZZ of order 43%, 56%, 1% respectively.
In this way, and taking into account current bounds on Γ Zγ /Γ γγ and Γ ZZ /Γ γγ [42], we display in Fig.4 contour plots of the production cross-section σ(pp → ξ → γγ) in the Γ Zγ /Γ γγ -Γ ZZ /Γ γγ plane. For simplicity, we have set λ ef f ≡ κ = λ = η = h U in such a way that the contour plots only depend on m U and λ ef f . In general, for low values of m U the ratio Γ Zγ /Γ γγ is of order Γ Zγ /Γ γγ ∼ 1, and for greater values of m U we have Γ Zγ /Γ γγ < 1. We also observe that the greater the ratio Γ Zγ /Γ γγ , the stronger the coupling λ ef f . However, if λ ef f > 3 the model is completely excluded by the bound Γ Zγ < 20Γ γγ for all m U .

Summary
We have presented an anomaly-free model based on the electroweak-strong unification group SU (6) ⊗ U (1) X , containing the SU (3) C ⊗ SU (3) L ⊗ U (1) X as a subgroup. We break the gauge symmetry down to SU (3) C ⊗ U (1) Q and at the same time give masses to the fermion fields in the model in a consistent way by five Higgs fields Φ, H 1 , H 2 , H 3 and H S . These Higgs fields and their VEVs set two different mass scales: υ = 256 GeV << V . From the mixing terms between the real fields ξ 1 in H 1 and ξ S in H S , we obtain two real scalar fields: our candidate for the 750 GeV signal ξ, and one ξ at the TeV scale. For the analysis of the diphoton decay, we take into account all the possible decay modes of the 750 GeV candidate considering three approximations for the decay width: Γ = 1 GeV, Γ = 45 GeV and Γ = Γ γγ + Γ gg + Γ Zγ + Γ ZZ . Then, taking various simplified assumptions on the parameter space, we show that the states U , E, E 1 , E 2 , W ± 3 , H ± 1,2 and φ ± 3 into the loop can explain the diphoton excess for each one of the width approximations according to ATLAS and CMS bounds on all the particle masses involved and on the decay widths Γ Zγ and Γ ZZ .