An extension of two-Higgs-doublet model and the excesses of 750 GeV diphoton, muon g-2 and $h\to\mu\tau$

In this paper we simultaneously explain the excesses of the 750 GeV diphoton, muon g-2 and $h\to \mu\tau$ in an extension of the two-Higgs-doublet model (2HDM) with additional vector-like fermions and a CP-odd scalar singlet ($P$) which is identified as the 750 GeV resonance. This 750 GeV resonance has a mixing with the CP-odd scalar ($A$) in 2HDM, which leads to a coupling between $P$ and the SM particles as well as a coupling between $A$ and the vector-like fermions. Such a mixing and couplings are strongly constrained by $\tau\to\mu\gamma$, muon g-2 and the 750 GeV diphoton data. We scan over the parameter space and find that such an extension can simultaneously account for the observed excesses of 750 GeV diphoton, muon g-2 and $h\to \mu\tau$. The 750 GeV resonance decays in exotic modes, such as $P\to hA$, $P\to HZ$, $P\to HA$ and $P\to W^\pm H^\mp$, and its width can be dozens of GeV and is sensitive to the mixing angle.

Attempting to simultaneously explain the excesses of 750 GeV diphoton, h → µτ and muon g-2, we in this work introduce additional vector-like fermions and a CP-odd scalar singlet (P ) to the general 2HDM. The singlet P is identified as the 750 GeV resonance, which has a mixing with the CP-odd scalar (A) in the original 2HDM. Therefore, the model can lead to the P couplings to SM particles and the A couplings to vector-like fermions.
In addition to the 125 GeV Higgs and 750 GeV resonance data, the LFV Higgs decay τ → µγ can give strong constraints on the couplings and mixing. The dominant decays of the 750 GeV resonance can be some exotic modes, such as P → hA, P → HA, P → HZ and P → W ± H ∓ . Considering various relevant experimental constraints, we examine the diphoton production and decay of the 750 GeV resonance, as well as muon g-2 and h → µτ .
Our work is organized as follows. In Sec. II we introduce additional vector-like fermions and a CP-odd scalar singlet to the 2HDM. In Sec. III we perform numerical calculations and discuss the muon g-2, h → µτ and the diphoton production and decay of the 750 GeV resonance in the allowed parameter space. Finally, we give our conclusion in Sec. IV.

II. MODEL
We introduce a CP-odd scalar singlet field P 0 to the general 2HDM with the assumption that P 0 does not develop a vacuum expectation value (VEV). The Higgs potential is given by [18,19] In the Higgs basis, the Φ 1 field has a VEV v =246 GeV, and the VEV of Φ 2 field is zero.
The two complex scalar doublets with hypercharge Y = 1 can be expressed as The Nambu-Goldstone bosons G 0 and G + are eaten by the gauge bosons. The physical CP-even Higgs bosons h and H are the linear combinations of ρ 1 and ρ 2 : where tan 2α = 2λ The masses of two CP-even Higgs bosons are given as The field H + is the mass eigenstate of the charged Higgs boson, and the CP-odd Higgs field A 0 has a mixing with P 0 :  where tan 2θ = 2µv/(m 2 The masses of two CP-odd scalars are given as The 750 GeV Higgs boson P couplings to other Higgs bosons and gauge bosons as The general Yukawa interactions of the SM fermions are given by where Q T L = (u L , d L ), L T L = (ν L , l L ), Φ 1,2 = iτ 2 Φ * 1,2 , and y u , y d , y ℓ , ρ u , ρ d and ρ ℓ are 3 × 3 matrices in family space.
Also, we introduce a singlet quark with 2 3 electric charge and multiple singlet leptons. The Yukawa interactions of vector-like fermions are written as Then we obtain the Yukawa couplings of the neutral Higgs bosons: For the diagonal matrix elements of ρ u , ρ d and ρ ℓ , we take which corresponds to the aligned 2HDM [20]. We assume that ρ ℓ µτ and ρ ℓ τ µ are nonzero, and other nondiagonal matrix elements of ρ u , ρ d and ρ ℓ are zero.
The vector-like quark is introduced to make the 750 GeV Higgs singlet to be produced via the gluon-gluon fusion process. However, the vector-like quark can also enhance the cross section of gg → A, which will be constrained by the experimental data from the ATLAS and CMS searches. Therefore, we expect that the vector-like leptons play the main role in enhancing the 750 GeV diphoton production rate. The decay P → γγ can be enhanced by the vector-like leptons, and its amplitude is proportional to the couplings and the square of electric charge. Here we do not discuss the electric charge and coupling of every vector-like lepton as well as the quantity of vector-like leptons in detail; instead we focus on the total contribution of vector-like leptons, which depends on where L i denotes the i-th vector-like lepton.

A. Numerical calculations
In our calculations, we scan over the parameters in the following range and fix m h = 125.5 GeV m P = 750 GeV, m H ± = m A = 500 GeV, m Li = 400 GeV, m T = 700 GeV, y T = 2.0, During the scan, we consider the following experimental constraints and observables: (1) Precision electroweak data. According to the expressions for the oblique parameters S, T and U in the 2HDM [21], for −0.06 < s α < 0.06 and c α ≃ 1, the expressions in this model are approximately given as where with Here we require [22] (2) The 125 GeV Higgs data. For −0.06 < s α < 0.06, κ u = κ d = 0 and −50 < κ ℓ < 50, . We require 0.33 <μ τ τ < 2.21 and such a bound will give strong constrains on s α and κ ℓ for which the absolute value of the coupling of the 125 GeV Higgs andτ τ is around the SM value.
(3) Non-observation of additional Higgs bosons. For −0.06 < s α < 0.06 and κ u = κ d = 0, the cross sections of H and H ± at the collider are very small, and hence H and H ± can be hardly constrained by the current experimental data from the ATLAS and CMS searches. The pseudoscalar A can be produced via the gluon-gluon fusion process with vector-like quark loop, and the decay A → γγ, A → γZ and A → ZZ can be enhanced by the vector-like quark and leptons at one-loop level. For m A =500 GeV, we impose the following relevant bounds at the 8 TeV LHC [25][26][27][28][29] R γγ < 6 fb, R ZZ < 45 fb, R Zγ < 6.8 fb, R hZ < 60 fb, Rτ τ < 26 fb. (22) (4) The 750 GeV resonance data. The 750 GeV Higgs singlet P can be produced via the gluon-gluon fusion process with vector-like quark loop, and the decays A → γγ, A → γZ and A → ZZ can be enhanced by the vector-like quark and leptons at oneloop level. Due to the mixing with A, the 750 GeV singlet can decay into the SM particles, such as hZ andτ τ . For the 750 GeV Higgs singlet, we impose the following bounds at the 8 TeV LHC [25][26][27][28][29] R γγ < 2 fb, R ZZ < 12 fb, R Zγ < 4 fb, R hZ < 19 fb, Rτ τ < 12 fb.
At the 13 TeV LHC, we require the 750 GeV diphoton production rate as 2 fb < R γγ < 10 fb.
(5) The data of Br(h → µτ ). The branching ratio of h → µτ is given by where Γ h is the total width of 125 GeV Higgs. To explain the h → µτ excess reported by CMS within 2σ range, we require The muon g-2 data. The dominant contributions to the muon g-2 are from the one-loop diagrams with the Higgs LFV coupling [30], The muon g-2 can be also corrected by the two-loop Barr-Zee diagrams with the fermions loops, W and Goldstone loops. Using the well-known classical formulates [31], the main contributions of two-loop Barr-Zee diagrams in this model are given as where The experimental value of muon g-2 excess is [32] δa µ = (26.2 ± 8.5) × 10 −10 . (7) The data of Br(τ → µγ). The LFV coupling of the Higgs boson gives the dominant contributions to the decay τ → µγ. The branching ratio of τ → µγ is given by where A 1L0 , A 1Lc , A 1R0 and A 1Rc are from the one-loop diagrams with the Higgs bosons and tau lepton [17], and Here A 2L and A 2R are from the two-loop Barr-Zee diagrams with the third-generation fermions loops, vector-like fermions loops and W loops [17]: where T 3f denotes the isospin of the fermion, t 2 The terms in the first two lines of Eq. (38) come from the effective φγγ vertex and φZγ vertex induced by the third-generation fermion loop and vector-like fermion loop.

B. Results and discussions
In Fig. 1 The circles (green) are allowed by the muon g-2, the pluses (red) allowed by the muon g-2 and Br(τ → µγ), and the bullets (black) allowed by the muon g-2, Br(τ → µγ) and Br(h → µτ ).
and their absolute values are around the SM value. One is the SM-like Higgs coupling with the same sign as the coupling of the gauge boson, and the other is the Yukawa coupling with the opposite sign to the coupling of the gauge boson for a relative large κ ℓ .
From the upper-right panel of Fig. 1, we see that the muon g-2 favors ρ µτ to increase with m H . As shown in Eq. (27) versus s α , Y L versus Γ P and Y L versus s θ . Here 2 fb < R γγ < 4 fb for the circles (green), 4 fb < R γγ < 6 fb for the pluses (red), and 6 fb < R γγ < 10 fb for the bullets (black), with R γγ denoting the 750 GeV Higgs production rate at the 13 TeV LHC.
loops becomes sizable so that a large ρ µτ is required to enhance the muon g-2. In Fig. 2, we project the surviving samples on the planes of the total width of 750 GeV singlet (Γ P ) versus s θ , Y L versus Γ P and Y L versus s θ . From the left panel, we can find that Γ P is very sensitive to s θ since the P couplings to SM particles are relevant to s θ . The Γ P value increases with the absolute value of s θ , and reaches 35 GeV for |s θ | =0.3. No matter how large the Γ P value is, the 750 GeV diphoton production rate R γγ can vary from 2 fb to 10 fb. The middle panel shows that, with the increasing of the total width, Y L becomes large enough to enhance Br(P → γγ), and further make R γγ to be in the range of 2 fb and 10 fb. For Γ P = 35 GeV, R γγ > 4 fb requires Y L to be larger than 30. The right panel shows that a large absolute value of s θ favors a large Y L since s θ with a large absolute value will for the pluses (red) and Γ(P → W ± H ∓ ) for the bullets (black).
enhance the total width of 750 GeV Higgs sizably.
In Fig. 3, we project the surviving samples with 20 GeV< Γ P <40 GeV on the planes of the widths of the main decay modes of the 750 GeV Higgs singlet versus λ 3 , m H and λ 7 .
This figure shows that P → Ah, P → HZ, P → AH and P → W ± H ∓ are the main decay modes, and the decay P → hZ is insignificant due to the suppression of s α . The decay P → Ah is sensitive to λ 3 and increases with λ 3 . The width of P → Ah can reach 20 GeV and dominate over other decay modes for λ 3 = 4π. The decay P → HZ is sensitive to m H and decreases with the increasing of m H . The width of P → HZ can reach 10 GeV and be larger than those of P → AH and P → W ± H ∓ for m H = 200 GeV. The decay P → AH increases with λ 7 and can be larger than the width of P → W ± H ∓ for λ 7 = 10. The width of P → W ± H ∓ is in between 2 GeV and 4 GeV for m H ± = 500 GeV, and not sensitive to m H , λ 3 or λ 7 . With the increasing of m A and m H ± , the decay P → hA, P → HA and P → W ± H ∓ will be kinematically forbidden, which will reduce the width of 750 GeV Higgs sizably.

IV. CONCLUSION
To simultaneously accommodate the excesses of the 750 GeV diphoton, muon g-2 and h → µτ , we proposed an extension of 2HDM with vector-like fermions and a CP-odd scalar singlet P , which is identified as the 750 GeV resonance. There is a mixing between the 750 GeV Higgs and the CP-odd scalar A, which leads to the P coupling to SM particles and A coupling to vector-like fermions. In the 2HDM the Higgs bosons have tree-level LFV interactions with µ − τ , which can be responsible for the excess of h → µτ and also give sizable contributions to the muon g-2. The 750 GeV Higgs can decay into P → Ah, P → HZ, P → AH and P → W ± H ∓ , and its total width is sensitive to s θ and can reach 35 GeV for |s θ | = 0.3. Since the 750 GeV Higgs has a large width, the vector-like leptons are required to enhance Br(P → γγ) to obtain R γγ > 2 fb. Meanwhile, such vector-like leptons will give sizable contributions to Br(τ → µγ) due to the mixing of P and A. Therefore, the Higgs couplings toτ τ are required to be properly large to cancel the contributions of vector-like leptons to Br(τ → µγ). Considering the current constraints of the LHC data, precision electroweak data and Br(τ → µγ), we scanned over the parameter space and found that such an extension can simultaneously explain the excesses of the 750 GeV diphoton, muon g-2 and h → µτ .