Gauged Two Higgs Doublet Model confronts the LHC 750 GeV di-photon anomaly

In light of the recent 750 GeV diphoton anomaly observed at the LHC, we study the possibility of accommodating the deviation from the standard model prediction based on the recently proposed Gauged Two Higgs Doublet Model. The model embeds two Higgs doublets into a doublet of a non-abelian gauge group $SU(2)_H$, while the standard model $SU(2)_L$ right-handed fermion singlets are paired up with new heavy fermions to form $SU(2)_H$ doublets, and $SU(2)_L$ left-handed fermion doublets are singlets under $SU(2)_H$. An $SU(2)_H$ scalar doublet, which provides masses to the new heavy fermions as well as the $SU(2)_H$ gauge bosons, can be produced via gluon fusion and subsequently decays into two photons with the new fermions circulating the triangle loops to account for the deviation from the standard model prediction.


I. INTRODUCTION
Recent results from LHC [1][2][3] exhibit an intriguing anomaly on the diphoton channel at the scale around 750 GeV. Numerous attempts  have been put forward to explain the excess, while Refs. [14,43,57] are based on two Higgs doublet models, similar to this work.
In Ref. [78], a combined result from run I and II gives a cross section σ(pp → X → γγ) ∼ O(6) fb for a scalar particle X with mass around 750 GeV. In this paper, we will show that the newly proposed Gauged Two Higgs Doublet Model [82] (G2HDM) is able to provide a cross section with such magnitude. Moreover, the vev of the triplet induces the SM Higgs vev, breaking SU (2) L ×U (1) Y down to U (1) Q , while H 2 does not obtain any vev and the neutral component of H 2 could be a dark matter (DM) candidate, whose stability is protected by the SU (2) H gauge symmetry and Lorentz invariance, without resorting to an ad-hoc Z 2 symmetry. In order to write down SU (2) H × U (1) X invariant Yukawa couplings, we introduce heavy SU (2) L singlet Dirac fermions, the right-handed component of which is paired up with the SM right-handed fermions to comprise SU (2) H doublets. In this setup, the model is anomaly-free regarding all gauge groups involved.
In this work, we focus on the role of φ 2 which is a physical component in Φ H and whose vev φ 2 = v Φ gives masses to the new heavy fermions. Since it couples to new colored fermions, it can be produced radiatively via gluon fusion and also decay radiatively into a pair of photons with the heavy charged fermions in loops. We will demonstrate that φ 2 can be a good candidate if LHC eventually could confirm the diphoton anomaly. Moreover, the observed width of the bump can be simply obtained from φ 2 decay into the additional fermions with O(1) Yukawa couplings.
The paper is organized as follows. First, we briefly discuss the G2HDM in Section II restraining ourselves only to those aspects most relevant to γγ mode. Next, in Section III we compute the diphoton cross section through φ 2 exchange and the partial decay width of φ 2 into the new heavy fermions. In Section IV, we briefly comment on implications of such the new heavy fermions in terms of collider searches, electron and muon magnetic dipole moment measurements, and the electroweak precision test data. Finally, we conclude in Section V.

II. G2HDM SET UP
In this Section, we review the G2HDM (cf. Ref [82]) with the particle content summarized in Table I Besides the the doublet H, we also introduce SU (2) H triplet and doublet, ∆ H and Φ H , which are singlets under SU (2) L .
The Higgs potential invariant under both SU (2) L × U (1) Y and SU (2) H × U (1) X can be written down easily as with Nonzero vevs v, v Φ and v ∆ will induce the mixing among the scalars, leading to two mass matrices. In this work, the relevant mass matrix in the basis of {h, δ 3 , φ 2 } is given by To simplify the diphoton excess analysis below, we focus on the simplest but representative scenario where all off-diagonal terms vanish by choosing and the scalar masses become where the value of λ H is exactly the same as in the SM. In this scenario, there is no mixing among h, δ 3 , φ 2 1 and the scalar φ 2 is responsible for the diphoton excess as we shall see below.
Next, the fermion sector together with new Yukawa couplings will be discussed.
where the subscript denotes hypercharge. As a consequence, we have Yukawa couplings where "·" refers to SU (2) H multiplication 2 and transforms as 2 under SU (2) H . After the electroweak symmetry breaking H 1 = 0, u and d obtain their masses but u H and d H remain massless since H 2 does not get a vev.
To provide masses to the additional species, we make use of the SU (2) H scalar doublet The lepton sector mimics the quark sector as in which E T R = (e H R e R ) −1 and N T R = (ν R ν H R ) 0 where ν R and ν H R correspond to the righthanded neutrino and the SU (2) H partner of it respectively, while χ e and χ ν are SU (2) L,H singlets with Y (χ e ) = −1 and Y (χ ν ) = 0. Similarly all SM leptons and their heavy counterparts will obtain masses from H 1 and Φ 2 .
As mentioned above, because φ 2 (a member of Φ H ) couples to the new heavy fermions, it can be radiatively produced via loops of the new colored particles and radiatively decays into the diphoton final state via loops of the new charged particles to accommodate the observed bump. On the other hand, although φ 2 is a singlet under the SM gauge group, it does couple to SM fermions and gauge bosons at tree level via the h − φ 2 mixing. That is the reason why we work in the zero mixing limit to evade direct search bounds from, for instance, dijet or dilepton channels. Note that there are no excesses in the ZZ, dijet or dilepton channels near the invariant mass of 750 GeV.

III. DIPHOTON ANOMALY
Equipped with the basics of G2HDM, we are now in a position to calculate the diphoton cross section via φ 2 exchange. The cross section at the φ 2 -resonance can be well approximated by [84] σ with the center of mass energy √ s = 13 TeV and the integral of the parton (gluon in this case) distribution function product evaluated at the scale µ = m φ 2 , using MSTW2008NNLO [85] and the value is consistent with Ref. [15]. The partial decay width of φ 2 into a heavy fermion and antifermion in the presence of a Yukawa term, y f φ 2f f / √ 2, that also gives a mass m f to the heavy fermion where N c = 3 for heavy colored particles while N c = 1 for heavy leptons.
The partial decay width of φ 2 into diphoton mediated by heavy fermions is [86][87][88] and the function f (τ ) is defined as On the other hand, the partial decay width of φ 2 into 2 gluons mediated by colored heavy fermions is [86][87][88] In our model, there are 6 heavy colored Dirac fermions, including 3 generations of uptype and down-type heavy quarks (with electric charge of 2/3 and 1/3, respectively) which contribute in Γ (φ 2 → gg) while for Γ (φ 2 → γγ) there are additional 3 heavy charged leptons with one unit of electric charge in addition to the heavy quarks. From the CMS run I and CMS+ATLAS run II diphoton data combined, the best fit value for the diphoton cross section is 6.2 ± 1.0 femtobarn [78]. It implies in units of i.e, with √ s = 13 TeV and Γ φ 2 /m φ 2 0.06 [1].
In the Fig. 1, we color the 1σ region in purple on the m f − v Φ plane to accommodate the γγ anomaly where all heavy fermions involved are assumed to have the same mass m f for simplicity. The green shaded region corresponds to the total decay width of φ 2 , obtained from Eq. (17) by including all neutral and charged heavy fermions (u H , d H , e H , ν H ), at the range of 0.05 < Γ φ 2 /m φ 2 < 0.07 that is consistent with the observed resonance width [1]. By contrast, the blue shaded region denotes the total decay width of φ 2 with 0.05 < Γ φ 2 /m φ 2 < 0.07, including heavy charged particles only (u H , d H , e H ). The red solid line corresponds to the perturbativity limit since m 2 φ 2 = 2λ Φ v 2 Φ in the limit of zero mixing among h, φ 2 and δ 3 . In order to have the diphoton excess, one can see that the new fermion masses have to be around 360 GeV with the vev v Φ at 250 GeV. However, we can also relax our assumption to allow for non-degenerate heavy fermion masses. In this case, one can still achieve the diphoton excess and the desired total decay width of φ 2 , while the heavy charged fermion masses are not longer constrained to be around 360 GeV.
We conclude this Section by commenting on impacts of having v Φ around 250 GeV. As discussed in Ref. [82], v Φ is restrained to be of order TeV to avoid various constraints. Small v Φ will induce a large mixing between the SM Z and SU (2) H Z , which can be avoided if the SU (2) H gauge coupling g H is small. To be more clear, the mixing angle, in the limit of where g and g are the SM SU (2) L and U (1) Y gauge coupling constants, respectively. One can in principle make g H small to have a very small mixing, resulting in very light SU (2) H gauge bosons. On the other hand, the DM matter candidate in this case could be the new neutral lepton (ν H R or χ ν ), the SU (2) H W or the neutral Higgs H 0 2 , depending on the parameter space. The DM stability is protected by the SU (2) H gauge symmetry and the Lorentz invariance as demonstrated in Ref. [82].

IV. IMPLICATIONS OF A FEW HUNDRED GEV HEAVY FERMIONS
In this Section, we briefly comment on some of consequences of SU (2) H heavy fermions with masses of order 360 GeV, required to realize the diphoton excess. A detailed study is, however, beyond the scope of this paper and deserves a separate work.
A. Muon and Electron magnetic dipole moment g − 2 At one-loop level, the charged leptons (electron and muon) anomalous magnetic moment (g − 2) receive three additional radiative contributions 3 involving loops of W with H , H 2 with H and Z with H , out of which the H 2 contribution can be neglected because it is highly suppressed by the corresponding small SM electron and muon Yukawa couplings and H 2 are assumed to be heavy. Taking into account the fact W and Z only couple to the 3 To simplify the analysis, we treat U (1) X as a global symmetry by setting g X = 0. right-handed SM fermions, the gauge boson contributions to the anomaly a ≡ (g − 2)/2 are [89,90] and where In addition, the Z − Z mixing with the angle given in Eq. (24) also induces an extra contribution to a l , obtained by multiplying Eq. (26) by (sin θ ZZ ) 2 and replacing g H by g/(cos θ w ), where θ w is the Weinberg angle. In contrast, due to the quantum number assignment, W is electrically neutral and will not mix with the SM W boson, unlike Z . Thus, Eq. (25) is the total contribution from W . Moreover, the W and Z boson masses are We present our results in Fig. 2 [94][95][96][97]. For g H 10 −3 , the electron anomaly ∆a e scales as g 2 H m 2 e /m 2 (W ,Z ) , which is simply m 2 e /v 2 (∆,Φ) since m 2 (W ,Z ) ∼ g 2 H v 2 (∆,Φ) . This implies independence of ∆a e on g H . However, for g H 10 −2 it is proportional to g 2 H , since for m H ∼ m W m , a W e ∼ g H m m H from Eq. (25).

B. Collider Searches
In previous subsection, we showed that in order to accommodate the diphoton excess without contradicting the electron and muon g −2 measurement, the SU (2) H gauge coupling g H is confined to be less than 10 −2 . Thus, at the LHC the heavy fermions will be mainly produced via the 750 GeV φ 2 decay due to large Yukawa couplings of O(1) instead of being generated through W -and Z -exchange processes. By virtue of the SU (2) H gauge symmetry, the decay of these heavy fermions must be accompanied by the DM particle in the final state as well.
For illustration, we use τ H as an example. It has three different decay channels, corresponding to three possible DM candidates ν H , H 0 2 and W in G2HDM, respectively: where in the first channel one could have multiple leptons or jets in addition to missing energy depends on whether ν R decays into ν L and H 1 within the detector or not, while the last two channels feature one lepton plus missing transverse energy.
The energy of SM fermion τ in the final state depends on the mass difference between τ H and the DM. If the mass splitting is too small, this may lead to very soft τ which fails to pass the event selection. The process gg → φ 2 → τ H τ H → null (DM + soft τ s), which will be largely excluded by the DM mono-jet searches as pointed out in Ref. [98]. On the other hand, if the mass splitting is large enough, the final state τ is visible and the situation will require delicate study, see Ref. [98] for more details. The heavy fermions, as SU (2) L singlets, will not contribute to electroweak corrections described by the oblique parameters, ∆S, ∆T and ∆U , as can be easily seen from the definition of the parameters [99]. Moreover, as demonstrated above the Z − Z mixing is constrained by the electron g − 2 bound to be less than 10 −2 or so, implying contributions to the oblique parameters at the order of 10 −4 or smaller. Finally as long as the mass splitting between H ± 2 and H 0 2 is small, corrections to ∆S, ∆T and ∆U will be suppressed [100]. All in all, this model can survive from the electroweak precision test.

V. CONCLUSION
In this work, we address a possible solution to the diphoton anomaly observed at the LHC based on the recent G2HDM model proposed by us. In the G2HDM, the two Higgs doublets The new heavy fermions receive masses from the vev of the SU (2) H scalar doublet, that also provides masses to the additional gauge bosons. A physical component φ 2 inside the doublet can be produced radiatively via gluon fusion with the additional heavy colored fermions in loops and in turn radiatively decays into two photons with the heavy charged fermions involved. We have shown that in the limit of the universal fermion mass, in order to reproduce the anomaly, the vev of φ 2 ranges from 180 to 300 GeV with the new fermion mass of few hundred GeV. The desired total decay width of Γ φ 2 0.06m φ 2 , by having φ 2 decay into the new fermions, can be realized with m f ∼ 360 GeV and v Φ ∼ 250 GeV. The favorable region could be further extended if the additional neutral fermions are allowed to have arbitrary masses.
The existence of SU (2) H gauge bosons can also explain the anomalous muon magnetic dipole moment. There are three radiative corrections to muon g − 2: W with µ H , Z with µ R and the correction induced by the Z − Z mixing. We have found out with m µ H = 360 GeV and g H ∼ 7 × 10 −3 , resulting in GeV or sub-GeV W and Z depending on the vevs of Φ H and ∆ H , the muon anomaly ∆a µ of order 10 −9 can be realized while the corresponding contributions to electron anomaly ∆a e is highly suppressed by the very small electron mass.
We conclude by pointing out that except for the diphoton anomaly, the LHC run-II data do not feature any significant deviation from the SM prediction. Our model can avoid overproducing other SM model particles through the same φ 2 exchange process since φ 2 couples only to the extra fermions at tree level in the limit of the vanishing h − φ 2 mixing.
The heavy fermions from φ 2 decays, however, subsequently decay into SM particles plus the DM particles, that manifest as missing transverse energy. The resulting SM particle energy spectra depend on the mass difference between the new heavy fermions and DM, and the spectra could be very soft if the mass difference is small just like the compressed spectra in various supersymmetry models. Finally, for the zero h − φ 2 mixing, one can expect the Zγ and ZZ signals with a similar order of magnitude as in the γγ anomaly through the same φ 2 exchange process.