Quark Seesaw, Vectorlike Fermions and Diphoton Excess

We present a possible interpretation of the recent diphoton excess reported by the $\sqrt s=13$ TeV LHC data in quark seesaw left-right models with vectorlike fermions proposed to solve the strong $CP$ problem without the axion. The gauge singlet real scalar field responsible for the mass of the vectorlike fermions has the right production cross section and diphoton branching ratio to be identifiable with the reported excess at around 750 GeV diphoton invariant mass. Various ways to test this hypothesis as more data accumulates at the LHC are proposed.


Introduction
In the early run II data from the √ s = 13 TeV Large Hadron Collider (LHC), both CMS [1] and ATLAS [2] experiments have reported an excess of γγ events over the SM background with invariant mass around 750 GeV. The signal cross section is reported to be (6 ± 3) fb by CMS [1] and (10 ± 3) fb by ATLAS [2]. While this excess has a local statistical significance of around 2.6σ (CMS) to 3.9σ (ATLAS) and needs more data to firmly rule out the possibility of a statistical fluctuation, it has nonetheless generated a great deal of recent activity in the theory community as a possible signal of beyond the Standard Model (SM) physics and many possible interpretations have been advanced; for a non-exhaustive list of ideas and speculations, see . In this note we add another one in the context of a theory proposed many years ago as a solution to the strong CP problem without an axion [99,100].
The model is based on the assumption that there exist TeV-scale vectorlike fermions which are responsible for the seesaw masses for the quarks and charged leptons [101][102][103][104] in the context of a left-right (LR) symmetric model based on the gauge group SU The Higgs sector of the model consists of SU (2) L,R doublets χ L,R which break the left and right SU (2)'s and a real singlet S that gives mass to the vectorlike fermions. An appropriate discrete Z 2 symmetry forbids the bare mass of the vectorlike fermions. Under this Z 2 symmetry, the Higgs fields χ L and S are odd as are the right-handed (RH) chirality of the vectorlike fermions; all other fields are even. The Yukawa couplings are given in this case by the Lagrangian whereχ L,R = iσ 2 χ * L,R (σ 2 being the second Pauli matrix), and y F , f F (with F = U, D, E) are the Yukawa couplings with potential beyond SM CP violations. Once both the doublets and the singlet obtain their non-vanishing vacuum expectation values (VEVs) v L , v R , v S respectively, we get the seesaw form for the 2 × 2 mass matrix for a single quark or lepton flavor: which generates the small fermion masses in the SM (except the top-quark mass) and alleviates the hierarchy problem in the Yukawa couplings: Both the two new VEVs are assumed to be at the (multi-)TeV scale, whereas v L 246.2 GeV is the electroweak scale. Clearly, the simple seesaw mass formula in Eq. (1.4) is not a good approximation for the top quark, as it is expected that the matrix entries y F v R and f F v S are of similar magnitude, which implies a large "right-handed" mixing of the top quark and its partner through sin α t R ∼ 1 Therefore in general, one should take into consideration the whole 2 × 2 mass matrix (1.3) and diagonalize M F M † F to get the mass eigenvalues of the SM quarks and their partners.
As far as the flavor structure and quark mixing are concerned, we can have either heavy quarks with degenerate masses (respectively for the up and down type flavors) in which case the SM quark mixings are completely determined by the flavor structure of the Yukawa couplings y U, D [108], or the couplings y U, D are hierarchical but diagonal (e.g. from some discrete symmetry assignments) and the matrices f U, D are of order O(1) in which case we have flavor anarchic [109].
As a direct result of the Lagrangian in (1.2), the heavy vectorlike quarks decay dominantly to the SM gauge and Higgs bosons plus SM quarks, especially for the top and bottom partners. Due to the Goldstone equivalence theorem, the branching ratios for the decays to W , Z and Higgs are approximately 2 : 1 : 1. The current LHC constraints put a 95% confidence level (CL) lower limit on the top partner mass from 715-950 GeV and on the bottom partner mass from 575-813 GeV [110], depending on their decay branching ratios. These lower bounds are much stronger than those from the flavor changing neutral currents (FCNCs) mediated by the heavy quarks [108].

Scalar sector
The gauge and Z 2 -invariant scalar potential including the singlet is given by As in [108] we keep in the potential the mass terms µ 2 L, R different so that they break the LR symmetry softly. Among all the terms given above, the λ 2 term mixes the left and right-handed doublets and the λ 3 term couples the singlet S to the doublets. The mass square matrix for the scalars after minimization of the potential is given by The FCNC constraint on the W R boson from K-meson mixing implies a lower bound on the VEV v R (which depends also on the gauge coupling g R ) [111], and is much larger than the SM VEV v L . Then the mixing between the SM Higgs h and its heavy RH partner H is suppressed by the ratio λ 2 v L /2λ 1 v R [108]. The h − S mixing is expected to be of order λ 3 v L /λ S v S . On the contrary, as long as λ 3 ∼ λ S , under the condition of v S ∼ v R at the (multi-)TeV scale, H and S can potentially have sizable mixing. Then in this case the singlet S can decay into the SM top quark, W and Z gauge bosons and SM Higgs pairs by the large mixing to H [108]. It is also subject to some constraints from flavor observations [112,113], precision tests [114,115] and neutrinoless double beta decay [116].
To explain the 750 GeV diphoton excess, we make the simple choice that the mixing term λ 3 at tree level is very small, so that the SM Higgs observables [117][118][119] and the electroweak vacuum stability analysis [108] do not get affected much by the presence of S.

Production and decay of the singlet
The dominate production channel for the singlet S at the LHC is the gluon-gluon fusion process mediated by the TeV-scale vectorlike fermions in the triangle loop. As the heavy fermion couplings to S are proportional to their masses, it is expected that all three generations of heavy fermions contribute significantly to the gluon fusion loops, which enhances the production cross section by roughly a factor of N 2 f (with N f being the number of flavors). At √ s = 13 TeV, the cross section for gluon fusion channel for a 750 GeV scalar with SM Higgs-like couplings is σ 13 TeV 0 850 fb, taking into account the large k−factor of order 2 [120]. The corresponding production cross section of the singlet S in our model can be obtained easily by rescaling the loop factor A 1/2 and the Yukawa couplings and masses in the fermion loop: σ(gg → S) 13 TeV = σ 13 TeV 0 r, where the scaling factor is defined as with the first term on the RHS for the top quark loop (with M t 173.2 GeV) and the second term for all the heavy vectorlike quark partners. Here τ F = M 2 S /4M 2 F and the fermion loop function is given by (see e.g. [121]) where the function f (τ ) is defined as In the parameter space of interest, the top quark contribution is suppressed by the LH mixing where we have chosen v S ∼ 1.2 TeV, as is required to explain the diphoton excess (see Figure 4 below) and the Yukawa parameter y T is given by On the other hand, due to the large Yukawa coupling and v R , the RH t − T mixing is generally very large, of order one, and the top partner mass is largely from or even dominated by the v R term For a typical value of v R = 3 TeV and v S = 1 TeV, the top partner mass can reach up to 2.5 TeV. The LH and RH mixing of other SM quarks and their corresponding heavy partners are comparatively much smaller and can be safely neglected, for which M F f F v S with f F a universal Yukawa coupling for all the flavors of vectorlike quarks and leptons in the Lagrangian (1.2). 1 The production of S is then predominately from these five heavy quark loops in Eq. (3.1). For a fixed value of f F ∼ O(1), the production cross section is suppressed by the heavy fermion masses via approximately σ(gg → S) ∝ M −2 F . As far as the decay of S is concerned, due to our choice of parameters, it couples dominantly to the heavy vectorlike fermions with a coupling f F of order one. All the couplings of S to other particles are from mixing or radiative effects. The largest coupling to the SM fermions is to the top quark from t − T mixing, i.e. f T sin α t L sin α t R , which is however suppressed by the small LH mixing sin α t L . For loop-induced channels, S can decays into gg, γγ, γZ and ZZ mainly via the large number of heavy quark and lepton loops. As the SM W boson does not couple directly to the heavy fermions, the S → W W channel is dominantly induced from the top quark loop which is comparatively suppressed by the small mixing sin α t L . On the other hand, the Z boson couples to the heavy vectorlike fermions with couplings proportional to Q F s 2 w , where Q F is the electric charge of heavy fermions and s w ≡ sin θ w , θ w being the SM weak mixing angle. As M S 2M Z , it is expected that the γγ, γZ and ZZ decay channels have comparable fractional widths Γ γγ : Γ Zγ : Γ ZZ = 1 : 2 tan 2 θ w : tan 4 θ w , which is a clean signal of vectorlike fermions. Combining all these decay channels, we find that since they are suppressed either by the small mixing or by the loop factors, the width of hidden singlet S is rather small, as we will show below. The various partial decay widths of S from tree and loop-level interactions are given by (3.12) For the γZ and ZZ channels we show the formulae in the massless Z boson limit in which we can see clearly the relation given in Eq. (3.7). The exact expressions for these two channels are given in the Appendix. In Eqs.
is the next-to-leading order factor for the gluon decay channel. The strong coupling α s ≡ g 2 s /4π and the fine-structure constant α ≡ e 2 /4π are evaluated at the resonance scale of 750 GeV. The loops run over the top and heavy vectorlike fermions for which the effective Yukawa couplings f F are respectively f T sin α t L sin α t R (top quark), f T cos α t R (top partner) and f F (all other heavy fermions). The total decay width of the scalar S is then given by In the minimal LR seesaw model, only the universal Yukawa coupling f F and the VEV v S are input by hand, apart from the RH scale v R . The total decay width for some typical values of v R is shown in Figure 1 as a function of the VEV v S . For this and subsequent plots, we have set the universal Yukawa coupling f F = 1, unless otherwise specified. Note that the total width of S is at most a few GeV and decreases with higher v S . Therefore, if this model is right, S will always appear as a narrow resonance at the LHC. This prediction can be easily tested in near future. In particular, if the wide resonance behavior of the reported excess with Γ S ∼ 45 GeV as claimed by ATLAS [2] persists with more data, the minimal version of this model will be ruled out. The various branching ratios for the channels discussed above are shown in Figure 2 as a function of the VEV v S for v R = 3 and 5 TeV. Comparing the different decay channels, we find that in most of the parameter space of interest, gg is the dominate decay channel. The tt channel depends quadratically on the Yukawa coupling and increases when v S is larger, which can be easily seen from the seesaw relation (1.4). For v S ∼ 1 TeV it is even comparable with the loop-induced gg channel. The decay rates into SM gauge bosons γγ, γZ and ZZ are suppressed by both the gauge coupling α 2 /α 2 s and loop factors and are at the per mil level or even smaller. The diphoton signal cross section σ γγ ≡ σ(gg → S) × BR(S → γγ) as function of v S is shown in Fig. 3 for different values of the RH scale v R = 3, 5 and 10 TeV. The horizontal (green) shaded region shows the preferred range of the observed diphoton signal: σ obs γγ = (8 ± 5) fb [1, 2]. The vertical (orange) shaded region is the 95% CL exclusion region from direct searches for the bottom partner at the √ s = 8 TeV LHC [110]. The corresponding limits for the top partner mass are always satisfied for the choice of v R in our model. We note that the signal cross section decreases when the vectorlike fermion mass is heavier, as expected. Furthermore, the mild dependence on the RH scale v R in some region of the parameter space is mainly from the mixing of SM quarks and their heavy partners. In order to see whether the allowed parameter space shown in Fig. 3 is consistent with the upper limits on signal cross sections in other channels from the publicly available √ s = 8 TeV data, we evaluate the coresponding production cross section σ(gg → S) 8 [123][124][125][126][127][128][129], namely, σ tt < 450 fb, σ gg < 2.5 pb, σ γγ < 1.5 fb, σ γZ < 4 fb, σ ZZ < 12 fb. (3.14) In Fig. 4, we compare the parameter space in the (v S , v R ) plane as preferred by the √ s = 13 TeV diphoton excess (green shaded region) with the √ s = 8 TeV exclusion limits given by Eq. (3.14). We find that the most stringent constraint comes from the √ s = 8 TeV diphoton search, as shown by the red shaded region in Fig. 4, which disfavors part of the √ s = 13 TeV diphoton excess region. The γZ and dijet constraints also rule out the low vectorlike fermion mass region, as shown by the magenta and blue shaded regions, respectively. The tt and ZZ limits are much weaker in this model and only rule out the very low v R and M F values (not shown in this plot). The direct search limit [110] rules out the parameter space with M B < 575 GeV, as shown by the orange shaded region. In any case, we find that M F (F = T ) values between about 700-1150 GeV are still compatible with the existing constraints and can explain the observed diphoton excess in this model.
It is interesting to note that for a fixed value of the singlet VEV v S , there exists an upper limit on the RH scale v R . This is shown in Figure 5 where the different contours show the diphoton signal cross section σ γγ in fb at √ s = 13 TeV LHC. We find that for v S = 1 TeV, v R must be smaller than 65 TeV to satisfy the observed value of σ γγ . For higher values of v S , the upper limit on v R becomes stronger. This provides another potential way to distinguish this model from other explanations of the diphoton excess.
For the testability of this model in the ongoing run of the √ s = 13 TeV LHC in near future, we choose two optimum sets of parameters with the RH scale v R = 3 TeV and 5 TeV respectively, for which the total production cross section, total decay width, the dominant fractional width and branching ratios of S are listed in Table 1. Table 1. Input parameters (f F and v S ) for v R = 3 TeV and 5 TeV, the total production cross section and decay width of the scalar S, and the signal cross sections in various relevant channels at the √ s = 13 TeV LHC in the minimal LR seesaw model.

High-Scale Validity
Due to the presence of a large number of extra fermions in our model and their O(1) Yukawa couplings with the singlet scalar field S, it is expected that the singlet scalar self-coupling λ S would become negative well before the Planck scale, thus potentially destabilizing the singlet vacuum. This would mean that our TeV-scale model provides only an effective description of the 750 GeV diphoton excess and must be augmented by some other new physics for its ultraviolet (UV)-completion. In order to find this cut-off scale, we write down the one-loop renormalization group (RG) equation for the scalar self-coupling λ S , as well as for other relevant couplings affecting the evolution of λ S above the scale v S : 2 where t ≡ log(µ) is the renormalization scale, and for simplicity, we have assumed a common Yukawa coupling f F (with F = U, D, E) for each of the three generations of the vectorlike fermions. We have further assumed the coupling λ 3 in Eq. (2.1) to be very small, so that it does not affect the RG evolution of other couplings and the electroweak vacuum stability analysis, as given in Ref. [108]. For a given value of the singlet VEV v S , the initial value of For illustration, we choose v S = 1.1 TeV which is close to the maximum value allowed by the diphoton excess [cf. Figure 4]. For the vectorlike fermion couplings f F (v S ) = 1, we find a cut-off scale around 10 TeV, as shown in Figure 6 (left). For larger initial values of the Yukawa couplings f F , the cut-off scale will be even lower, as is evident from Eq.

Discussions
In general vectorlike fermion and (pseudo)scalar interpretations of the diphoton events recently observed at 750 GeV, the Yukawa couplings of vectorlike quarks to the scalar resonance tend to be very large, or the electric charges of the heavy fermions are exotically large. It is preferable in the LR seesaw model that these problems are avoided by the naturally large number of heavy partners for the seesaw generation of SM fermion masses. We emphasize here that while the main features of the diphoton explanation in this model are quite similar to earlier works which introduced a SM-singlet scalar and vectorlike fermions, the specific nature of our model constrained by the strong CP requirement, i.e. seesaw quark masses, leads to some differences. In particular, our minimal LR interpretation of the diphoton excess implies the following features which could be tested by the upcoming LHC data and future precision measurements: (i) As the gluon fusion production for S goes down like M −2 F , the observation of diphoton events with cross section at the level of few fb implies the existence of heavy fermions at the TeV scale.
(ii) In the minimal LR seesaw model, for heavy partners at the scale of 1 TeV, the RH scale lies naturally at the few TeV scale. In such scenarios we expect a heavy W R nearby which should be accessible at the √ s = 13 TeV LHC. 3 (iii) The decay of S in the LR model is suppressed either by small mixing parameters or the loop factors, and thus it would be a narrow resonance and could be easily tested by the upcoming LHC data.
(iv) A large portion of the S decay width is into top quark pairs, with a branching ratio varying from a few percent to half or even larger (subject to the top quark mixing and other parameters). Thus, the tt observations could not only test the minimal LR model but also be used to probe the t − T mixing which is essential to search for new physics in the top quark sector.
(v) In the minimal scenario of the LR seesaw model, the singlet S decouples from the doublets for gauge symmetry breaking. In such cases the stability analysis of the scalar potential is the same as in [108]. In a large parameter space the SM vacuum is stable up to the GUT scale or even the Planck scale.
(vi) The presence of the new scalar S and the vectorlike fermions has a potential impact on the precision electroweak observables [142]. However, with the small t − T mixing angle sin α t L as in our case [cf. Eq. (3.4)], these constraints are easily satisfied for the range of top-partner masses considered here. In future, a better measurement of the top-quark decay width which has currently a large uncertainty of O(1 GeV) [143,144] could be another way to test our hypothesis.