Interpreting The 750 GeV Diphoton Excess Within Topflavor Seesaw Model

We propose to interpret the 750 GeV diphoton excess in a typical topflavor seesaw model. The new resonance X can be identified as a CP-even scalar emerging from a certain bi-doublet Higgs field. Such a scalar can couple to charged scalars, fermions as well as heavy gauge bosons predicted by the model, and consequently all of the particles contribute to the diphoton decay mode of the X. Numerical analysis indicates that the model can predict the central value of the diphoton excess without contradicting any constraints from 8 TeV LHC, and among the constraints, the tightest one comes from the Z \gamma channel, \sigma_{8 {\rm TeV}}^{Z \gamma} \lesssim 3.6 {\rm fb}, which requires \sigma_{13 {\rm TeV}}^{\gamma \gamma} \lesssim 6 {\rm fb} in most of the favored parameter space.


Introduction
Recently in the searches for new physics at the LHC Run-II with √ s = 13TeV and 3 fb −1 integrated data, both the ATLAS and CMS collaborations reported a diphoton excess with an invariant mass around 750 GeV [1,2]. Combined with the 8 TeV data, the favored rate of the excess is given by σ 750GeV γγ = (4.4 ± 1.1) fb . (1.1) in the narrow width approximation [3]. Although the local significance is not very high, which is only 3.9σ for ATLAS data and 2.6σ for CMS data, this excess was widely regarded as a possible hint of new physics beyond the Standard Model (SM). So far about one hundred theoretical papers have appeared to interpret the excess in various models [3,4,5,6,7,8,9,10,11,12,13], and most of them employed the process gg → X → γγ with X denoting a 750 GeV scalar particle to fit the data. From these studies, one can infer two essential properties of the X. One is that its interactions with the SM particles other than gluons and photons should be significantly weaker than those of the SM Higgs boson. In this case, the rates of the X-mediated processes pp → X → ZZ, W W * , hh, ff are suppressed so that no excess on these channels was observed at the LHC Run-I [13]. The other is that the X must interact with new charged and colored particles to induce the effective Xγγ and Xgg couplings through their loop effects. In order to explain the excess in a good way, the new particles should be lighter than about 1 TeV, and meanwhile their interactions with the X must be rather strong.
Among the new physics models employed to interpret the excess, the minimal theoretical framework is the extension of the SM by one gauge singlet scalar field and vector-like fermions [4]. This model was extensively discussed since it provides a very simple but meanwhile feasible explanation of the excess. However, as pointed out in [14], in order to explain the excess the Yukawa couplings of the fermions are usually so large that the vacuum state of the scalar potential becomes unstable at a certain energy scale, which implies that other new physics must exist. This motivates us to speculate what is the fundamental theory behind the minimal model. We note that for the interaction λ F XF F with F denoting a vector-like fermion, its contribution to the Xγγ coupling is determined by the ratio λ F M F under the condition 4M 2 F (750GeV) 2 . In a usual theory, λ F M F ∼ 1 v with v ∼ 1 TeV denoting the typical size for both the fermion mass and other new particle masses in the theory. On the other hand, if the fermion acquires its mass in a complicated way and consequently λ F and M F are less correlated, the ratio can be much larger than 1/v with v M F denoting the mass scale of the other new particles. For such a situation, the effective theory at the TeV scale contains only the scalar X and the fermions, which is similar to the minimal model, but it usually has a more complicated scalar sector than the minimal model. Building such a model and using it to explain the diphoton excess are the aims of this work. Obviously, such a study can improve our understanding on the minimal framework.
To actualize the idea, we are motivated by the top-specific theories, such as the top condensation models [15,16], the top seesaw model [17,18] and the topflavor model [19], which attempts to interpret the relatively large top quark mass in comparison with the other SM particles. Roughly speaking, we assume that the third generation fermions undergo a different SU (2) weak interaction from the first two generation fermions [20]. At the same time, we introduce new vector-like fermions and split their mass spectrum by seesaw mechanism [21]. In this way, the 750GeV resonance is identified as a CP-even scalar emerging from a bi-doublet Higgs, which triggers the breaking of the two SU(2) gauge symmetry into SU (2) L , and its interactions with photons are induced by relevant scalars, fermions as well as gauge bosons. Due to these features, our model has more freedom to explain the diphoton excess than the minimal model. We remind that our model is somewhat similar to the topflavor seesaw model proposed in [21], so we dub it hereafter the typical topflavor seesaw model. This paper is organized as follows. In section 2, we introduce the structure of the typical top flavor seesaw model, and list its particle spectrum. In section 3, we choose benchmark scenarios to study the diphoton excess. Subsequently we draw conclusions in section 4. We also present more details of our model in the Appendix.

The framework of the topflavor seesaw model
In this section, we recapitulate the structure of the typical topflavor seesaw model. This model is based on the gauge symmetry group SU ( where the third generation of fermions transform non-trivially under the SU (2) 2 group, while the first two generation of fermions transform by the SU (2) 1 group. The symmetry breaking of the gauge group into the electromagnetic group U (1) Q is a two-stage mechanism: first, 1) Q at the electorweak scale. This breakdown can be accomplished by introducing two Higgs doublets and one bi-doublet Higgs with the following SU ( In this model, we also introduce following vector-like quarks and leptons to couple with the bi-doublet and write their mass terms as follows where the dimensionful coefficient M H may have a dynamical origin or just be imposed by hand.
With the above field settings, the remaining Lagrangian is where V (Φ) represents the potential of the field Φ, V (H 1 , H 2 ) corresponds to the potential of a general two Higgs doublet model, y α with α = t, b, · · · and y β with β = T, B, N, E are Yukawa coupling coefficients, and m V,Q and m V,L are dimensionful parameters. The general expression of V (Φ) is given by [22] V About the Lagrangian in Eq. (2.4), two points should be noted. One is that the field Φ may have a different dynamical origin from that of the fields H 1 and H 2 . So in writing down the scalar potential, we neglect the couplings between Φ and H 1 , H 2 . The other is that, the third generation fermions, which are specially treated in our model, can in principle mix with the vector-like quarks. If any of y β , y β and m V is large, the flavor mixing of the SM fermions may differ significantly from its SM prediction [21]. Although this situation may still be allowed by the precise measurements in flavor physics, we require all the coefficients to be sufficiently small to suppress the decay of the 750GeV resonance into the state tt. We will turn to this issue later.
In the following, we list the spectrum of the particles that we are interested in.

Scalar sector
In the topflavor seesaw model, the bi-doublet Higgs contains 8 real freedoms, and it can parameterized by where ρ 1 and ρ 2 are CP-even fields with √ 2v being their vacuum expectation value (vev), η 1 and η 2 are CP-odd fields and V + 1 and V − 2 denote charged fields. The non-zero v triggers the breaking of the group SU (2) 1 × SU (2) 2 into the diagonal SU (2) L group. In such a process, the field combinations η 1 − η 2 and V + 1 − (V − 2 ) * act as Goldstone modes, and are absorbed by the gauge bosons corresponding to the broken SU (2) H group. Their orthogonal combinations correspond to the charged and CP-odd scalars respectively, which are given by

7)
As for the CP-even fields ρ 1 and ρ 2 , they mix to form mass eigenstates h 0 and H 0 in following way With the physical states, the field Φ can be reexpressed as This form is useful to understand our expansion result of the V (Φ). From Eq.(2.5), one can get the minimization condition of the potential, 4κv 2 = µ 2 1 +2µ 2 2 with κ = λ 1 + λ 2 + 2λ 3 + 2λ 4 , and the vacuum stability condition κ > 0. One can also get the spectrum of the scalars as follows In a similar way, the two Higgs doublets H 1 and H 2 can be written as In this process, the alignment of the fields H 0 1 and H 0 2 forms a lightest CP-even scalar, which corresponds to the 125 GeV higgs boson discovered by the ATLAS and CMS collaborations at the LHC [23,24].
Throughout this work, we identify the state h 0 from Φ as the 750 GeV resonance, which is responsible for the diphoton excess by the parton process gg → h 0 → γγ. Since we have neglected the mixing between H i and Φ, we do not consider the decay of h 0 into the SM-like Higgs boson pair. In fact, if we switch on the mixing, the upper bound on the di-Higgs signal at the LHC Run I has required the mixing to be small [13].

Gauge bosons
In our theory, the covariant derivative that appears in the kinetic term of Φ is given by where T a 1 and T b 2 with a, b = 1, 2, 3 are the SU (2) generators, Y is the hypercharge generator, and h 1 , h 2 and g Y are gauge coupling coefficients. After the first step symmetry breaking, the SU (2) L coupling coefficient g 2 is related with h 1 and h 2 by which implies h 1 = g 2 / cos θ, h 2 = g 2 / sin θ with tan θ ≡ h 1 /h 2 , and the gauge fields corresponding to the broken SU (2) H group (usually called flavoron and denoted by F i µ hereafter) and the SU (2) where i = ±, 3. At this stage, the fields F ± µ and F 3 µ are massive with a common squared mass of (h 2 1 + h 2 2 )v 2 = 4g 2 2 v 2 (csc 2 2θ), and by contrast all the fields W i µ keep massless. After the second step symmetry breaking, the masses of the fields F ± µ keep unchanged, but the field F 3 µ mixes with the other neutral gauge fields to form mass eigenstates. In the basis (F 3 µ , W 3 µ , B µ ), the squared mass matrix is given by where h = h 2 1 + h 2 2 , s θ ≡ sin θ and c θ ≡ cos θ. This matrix can be diagonalized by a rotation U to get mass eigenstates (Z , Z, γ).

Heavy Fermions
After the first step gauge symmetry breaking, the mass matrix of the vector-like fermions V and V are given by The corresponding eigenstates are given by the combinations In the basis (t, T 1 , T 2 ), the mass matrix of the heavy up-type quarks at the weak scale is given by This matrix can be diagonalized to get the mass eigenstates (t 1 , t 2 , t 3 ). Since we are interested in the case that We emphasize that the mixings between t and T i can induce the h 0t 1 t 1 , H 0t 1 t 1 and H +t 1 b interactions with t 1 identified as the top quark measured in experiments. In our discussion about the diphoton excess, we assume the mixings to be sufficiently small so that Br(h 0 → gg) Br(h 0 → t 1t1 ), and consequently we neglect the contribution of h 0 → t 1t1 to the total width of h 0 . The same mixings can also induce the decay t 2 , t 3 → W b, t 1 Z, and the LHC searches for vector-like fermions have required m t 2,3 800GeV [25].
Note that similar discussions can be applied to the down-type quarks and leptons.

The diphoton excess
If the diphoton excess observed by both ATLAS and CMS collaborations is initiated by gluon fusion, its production rate can be written as [13,26] where Γ h 0 →gg is the width for the decay h 0 → gg, Γ SM H→gg = 6.22 × 10 −2 GeV denotes the width of the SM Higgs H decay into gg with m H = 750GeV and σ SM √ s=13TeV (H) = 735 fb is the NNLO production rate of the H at the 13 TeV LHC [27]. As pointed out in [3], after combining the diphoton data at the 13 TeV LHC with those at the 8 TeV LHC, the preferred rate for the excess at the 13 TeV LHC is This rate can be transferred to the requirement where Γ tot denotes the total width of h 0 , and in the topflavor seesaw model it is given by

Useful formulae for calculation
In this part, we list the formulae for the partial widths to calculate the Γ tot .
• The widths of h 0 → γγ, gg are given by where I h 0 γγ and I h 0 gg parameterize the h 0 γγ and h 0 gg interactions, and their general expressions are 1 In above expressions, the coefficient g h 0 XX with X = V, F, S represents the coupling of the h 0 X * X interaction, m X , N c,X and Q X are the mass, color number and electric charge of the particle X respectively, and τ X = 4m 2 X /m 2 h 0 . The involved loop functions are defined by [28] Obviously, the three terms in I h 0 γγ correspond to the contributions from vector bosons, fermions and scalars, respectively. In our model, they are given by: gauge bosons V = F + µ , F − µ , fermions F = t 2 , t 3 , b 2 , b 3 , τ 2 , τ 3 and scalars S = H + , H − . In the appendix, we present all couplings used in our calculation, including the expressions of g h 0 V V , g h 0 F F and g h 0 SS .
• The width for h 0 → Zγ can be obtained in a way quite similar to that for h 0 → γγ, and it is given by [13] where with g ZXX (X = V, F, S) standing for the coefficient of the ZX * X interaction. Note that in getting this expression, we have neglected the Z boson mass appeared in the loop functions since m 2 h 0 , m 2 X m 2 Z , and consequently the involved loop functions can be greatly simplified.
• In the topflavor seesaw model, the decay h 0 → ZZ, W W * are also induced by loop effects. Their width expressions are slightly complex, but can be still obtained in a way similar to that of h 0 → γγ if one neglects the vector boson mass appeared in the relevant loop functions. Explicitly speaking, we have [13] where the possible particles in the loops areṼ ,Ṽ = F + µ , F 3 µ , F, F = t 2 , t 3 , b 2 , b 3 , τ 2 , τ 3 , ν τ 2 , ν τ 3 and S, S = H + , H 0 respectively.
About above formulae, it should be noted that, if the W and Z mass appeared in the squared amplitudes and phase spaces are also neglected, their expressions can be greatly simplified, and consequently they take similar forms. In this case, their expressions in our model are approximated by where we have defined  (3.13) In the following, we will use this condition to find the solutions to the diphoton excess in the topflavor seesaw model. Eq. (3.12) also indicates that the branching ratios for the decays h 0 → W W * , ZZ, Zγ are at least several times larger than that of h → γγ. As a result, the h 0 production can generate sizable W W * , ZZ and Zγ signals. Our model should be compatible with the LHC Run I constraints that are given by (3.14)

Discussion and numerical results
From Eq. (3.12), one can learn that the involved parameters for the diphoton signal are • the parameters in the scalar sector, which are m h 0 = 750GeV, v, x and m H + = m H 0 .
• the parameter tan θ in the gauge sector, which determines the flavoron mass.
• the parameters in the fermion sector, which are λ V and m H used to determine the fermion masses and their Yukawa couplings.
In order to illustrate our explanation of the excess in a concise way, we assume that all new particles other than h 0 are significantly heavier than h 0 so that τ X = 4m 2 X /m 2 h 0 Eq.(3.13) can be reexpressed by

In this case, since
This equation reveals the following information • The vector boson contribution interferes destructively with the fermion contribution. While for the scalar contribution, it may interfere either constructively (if x > 0) or destructively (if x < 0) with the fermion contribution.
• If m H 0, the contribution from each fermion is usually significantly smaller than the vector boson contribution, but the total fermion contribution in our model can cancel strongly with the vector contribution regardless the value of λ V . On the other hand, if m H is sufficiently large so that 1/v, the fermion contribution may be dominant. This guides us to get the solution for the diphoton excess.
• For x ∼ 1, the scalar contribution is very small in comparison with the other contributions. However, if |x| 1, which is somewhat unnatural but still possible by tuning µ 2 1 and µ 2 2 to get m H + in Eq.(2.14), the scalar contribution can be important.
• For a large v, contributions from the vector boson and the scalar decrease quickly since they are proportional to 1/v 2 . In contrast, if one keeps the lighter vectorlike fermions at TeV scale by requiring (λ V v − m H ) ∼ 1TeV, the vector fermion contribution can still be sizable even for a very large v. In this case, the effective theory of our model at TeV scale is similar to the minimal model mentioned in the Sec 1.
• For v = 10TeV, λ V − m H = 1TeV and x = 0, we can make an estimation with Eq.(3.15) that λ V 6.3 ± 0.9 can explain the diphoton excess at 1σ level. The corresponding Yukawa coefficient for the h 0t 2 t 2 interaction is 3±0.4, which is about 3 times the top quark Yukawa coupling, but still significantly below the non-perturbative bound 4π/ N c,F .
In our numerical study, we fix tan θ = 1, m h 0 = 750GeV, m H + = m H 0 = v, and λ V v −m H = 1TeV. We vary λ V and v to get the favored parameter region for the diphoton excess with x = 0, 20, 30, 40 at each time. The contours for σ Zγ 8TeV = 3.6 fb, σ ZZ 8TeV = 12 fb and σ W W * 8TeV = 37 fb in the λ v − v plane are also obtained. The corresponding results are shown in Fig.1, where the upper panels are for the results with x = 0, 20 and the lower panels correspond to the results with x = 30, 40. From this figure, one can get following conclusions • The topflavor seesaw model can explain the diphoton excess without conflicting with the constraints from the data at the LHC Run I, and the central value of the excess can be obtained even for v ∼ 10TeV.
• Given a sufficiently large v, e.g. v 6TeV, λ V 6 is usually needed to predict the central value of the excess.
• For v 1TeV and x = 40, which corresponds to a tuning of 1/x in getting the mass of H + , λ V 4 is required to predict the central value of the excess. Moreover, to explain the excess at 2σ level, the coupling can be as small as λ V 2.5.
• The LHC data at Run I has imposed rather tight constraints on our model. For the field configuration in our theory, the strongest constraint comes from the upper bound on the Zγ channel σ Zγ 8TeV 3.6fb, and it has required σ γγ 13TeV 6fb. We remind that if we use σ Zγ 8TeV 6fb adopted in [3] as the constraint, the upper bound of σ γγ 13TeV becomes about 10fb.

Conclusion
We propose to interpret the 750 GeV diphoton excess in a typical topflavor seesaw model. The new resonance X can be identified as a CP-even scalar emerging from a certain bidoublet higgs field. Such a scalar can couple to charged scalars, fermions as well as heavy gauge bosons predicted by the model, and as a result all of these particles contribute to the diphoton decay mode of the X. Numerical analysis indicates that the model can predict the central value of the diphoton excess without contradicting any constraints from 8 TeV LHC, and among the constraints, the tightest one comes from the Zγ channel, σ Zγ 8TeV 3.6fb, which requires σ γγ 13TeV 6fb in most of the favored parameter space. From theoretical point of view, our model has advantages in comparison with minimal frameworks. As we mentioned in Sec 1, the key factor λ F /M F for the diphoton rate in the minimal model scales like 1/v for a large v. However, in our model the factor λ F /M F is equal to c v /v with c v 1. This is possible because the simply relation M F = λ F v is unleashed and an effective negative contribution to the vector-like fermions is generated to decrease M F . Note that in the minimal model imposing such a negative contribution by hand to spoil the relation M F = λ F v is very unnatural. Moreover, in minimal seesaw model with vector-like heavy top quarks [21], the seesaw mechanism is fully responsible for the top quark mass. As a result, there is undesirable strong correlation between the diphoton decay rate and the h 0 →tt decay rate, and the constraint from the h 0 mediated tt resonance at LHC Run I can falsify the model even though the mixing between the top quark and heavy top T is very tiny. In our model, however, the seesaw mechanism contributes only a small portion of the top quark mass, and consequently there is no such a correlation.

A. The couplings needed in our calculation
In the section, we enumerate the couplings needed in our calculation. These interactions comes from the kinetic term 1) and consequently, we have • The couplings of h 0 to vector-like quarks.
These couplings are given by Note that there is an additional factor 1 2 for the coupling coefficient. Also note that the vector-like leptons have same Yukawa couplings as the quarks.
• The couplings of h 0 to heavy scalars These coupling originates from the Φ potential presented in Eq. (2.5). After tedious expansion of the V (Φ), we find that they take following forms where in the last step we introduce a dimensionless quantity x to parameterize the interaction. From Eq.(2.14), one can learn that x = 1 if 2µ 2 2 = µ 2 1 , and x > 1 (x < 1) if 2µ 2 2 < µ 2 1 (2µ 2 2 > µ 2 1 ).

A.2 The couplings of W and Z bosons to the heavy scalars
These couplings originate from the kinematic term in Eq.(A.1), and the terms we will use are given by The corresponding Feynman rules are • H(p 1 ) − H − (p 2 ) − W + µ (p 3 ) : ig 2 (p 1 − p 2 ) µ , • H + − H − − Z µ − Z ν : 2ig 2 2 cos 2 θ W g µν , In getting the first four rules, we have defined the direction of the momentum as that pointing to the vertex.

A.3 The couplings of W and Z bosons to the heavy fermions
Denoting F to be any of the fermion fields t 2 , t 3 , b 2 , b 3 , τ 2 , τ 3 , ν τ 2 , ν τ 3 , we have following Feynman rules for W and Z bosons Moreover, we also find that the coupling of the F + F − Z interaction is same as that of the W + W − Z interaction in the SM, and the coupling of the F + W − F 3 interaction differs from that of the W + W − Z interaction by a factor of 1/ cos θ W .