A $\mu$-$\tau$-philic Higgs doublet confronted with the muon g-2, $\tau$ decays and LHC data

We propose a two-Higgs-doublet model in which one Higgs doublet has the same interactions with fermions as the SM, and another Higgs doublet only has the $\mu$-$\tau$ LFV interactions. Assuming that the Yukawa matrices are real and symmetric, we impose various relevant theoretical and experimental constraints, and find that the excesses of muon $g-2$ and lepton flavour universality in the $\tau$ decays can be simultaneously explained in the region of small mass splittings between the heavy CP-even Higgs and the CP-odd Higgs ($m_A>m_H$). The multi-lepton event searches at the LHC can sizably reduce the mass ranges of extra Higgses, and $m_H$ is required to be larger than 560 GeV.


I. INTRODUCTION
The muon anomalous magnetic moment g − 2 has been a long-standing puzzle since the announcement by the E821 experiment in 2001 [1]. There is an almost 3.7σ discrepancy between the experimental value and the prediction of the SM [2] ∆a µ = a exp µ − a SM µ = (274 ± 73) × 10 −11 .
In this work we propose a 2HDM in which one Higgs doublet has the same interactions with fermions as the SM, and another Higgs doublet only has the µ-τ LFV interactions, namely µ-τ -philic Higgs doublet. After imposing the joint constraints from the theory, the precision electroweak data, and the LFU in the Z decays, we examine the parameter space explaining the excesses of muon g − 2 and LUF in τ decays. Next, we apply the ATLAS and CMS direct searches at the LHC to constrain the parameter space.
Our work is organized as follows. In Sec. II we recapitulate the model. In Sec. III we discuss the muon g − 2, LUF in τ decays, and other relevant constraints, and then use the direct search limits at the LHC to constrain the model. Finally, we give our conclusion in Sec. IV.
II. THE 2HDM WITH µ-τ -PHILIC HIGGS DOUBLET An inert Higgs doublet Φ 2 is introduced to the SM under a discrete Z 2 symmetry, while all the SM particles are unchanged. The scalar potential of Φ 2 and Φ 1 is given as We focus on the CP-conserving case, and all λ i are real. The two complex scalar doublets can be written as The Φ 1 field has the vacuum expectation value (VEV) v=246 GeV, and the VEV of Φ 2 field is zero. We determine Y 1 by requiring the scalar potential minimization condition.
The G 0 and G + are the Nambu-Goldstone bosons which are eaten by the gauge bosons.
The H + and A are the mass eigenstates of the charged Higgs boson and CP-odd Higgs boson. Their masses are given as The two CP-even Higgses h and H are mass eigenstates, and there is no mixing between them. In this paper, the light CP-even Higgs h is taken as the SM-like Higgs. Their masses are given as The masses of fermions are obtained from the Yukawa interactions with Φ 1 where Q T L = (u L , d L ), L T L = (ν L , l L ), Φ 1 = iτ 2 Φ * 1 , and y u , y d and y ℓ are 3 × 3 matrices in family space. To obtain µ-τ -philic Higgs doublet, we introduce the Z 2 symmetry-breaking From Eq. (9), we can obtain µ-τ LFV coupling of extra Higgses (H, A, and H ± ). We assume that the Yukawa matrices of Φ 2 are CP-conserving, that is ρ µτ and ρ τ µ are real and At the tree-level, the light CP-even Higgs h has the same couplings to fermions and gauge boson as the SM, and the µ-τ LFV coupling of h is absent. In our calculations, we take λ 2 , λ 3 , m h , m H , m A and m H ± as the input parameters, which can determine the values of λ 1 , λ 5 and λ 4 from Eqs. (6,7). λ 2 controls the quartic couplings of extra Higgses, and does not affect the observables considered in our paper. Therefore, we simply take λ 2 = λ 1 . λ 3 is adjusted to satisfy the theoretical constraints. We fix m h = 125 GeV, and scan over several key parameters in the following ranges: At the tree-level, the SM-like Higgs has the same couplings to the SM particles as the SM, and no exotic decay mode for such Higgs mass spectrum. The masses of extra Higgses are beyond the exclusion range of the searches for the neutral and charged Higgs at the LEP.
Because the extra Higgses have no couplings to quarks, the bounds from meson observables can be safely neglected.
In our calculation, we consider the following observables and constraints: (1) Theoretical constraints and precision electroweak data. The 2HDMC [23] is employed to implement the theoretical constraints from the vacuum stability, unitarity and coupling-constant perturbativity, and calculated the oblique parameters (S, T , U).
Adopting the recent fit results in Ref. [24], we use the following values of S, T , U, The correlation coefficients are given by The oblique parameters favor that one of H and A has a small mass splitting from H ± , and therefore we simply take m A = m H ± in this paper.
(2) Muon g − 2. The model contributes to the muon g − 2 through the one-loop diagrams involving the µ-τ LFV coupling of H and A [17], From Eq. (13), the model can give a positive contribution to the muon g − 2 for m A > m H . This is reason why we scan over the parameter space of m A > m H .
(3) Lepton universality in the τ decays. The HFAG collaboration reported three ratios from pure leptonic processes, and two ratios from semi-hadronic processes, τ → π/Kν and π/K → µν [3]: with HereΓ denotes the partial width normalized to its SM value. The correlation matrix for the above five observables is  In this model,Γ Where δ tree can give a positive correction to τ → µνν, and is from the tree-level diagram mediated by the charged Higgs, δ τ loop and δ µ loop denote the corrections to vertices Wν τ τ and Wν µ µ, respectively, which are from the one-loop diagrams involving H, A, and H ± . Since we take ρ µτ = ρ τ µ ≡ ρ, and therefore δ τ loop = δ µ loop . Following results of [10,12,22], where In the model, We perform χ 2 τ calculation for the five observables. The covariance matrix constructed from the data of Eq. (14) and Eq. (16) has a vanishing eigenvalue, and the corresponding degree is removed in our calculation.
(4) Lepton universality in the Z decays. The measured values of the ratios of the leptonic Z decay branching fractions are given as [25]: where the SM value g e L = −0.27 and g e R = 0.23. δg loop L and δg loop R are from the one-loop corrections, which are given as where C Z (r) = ∆ ǫ 2 + 1 2 − r 1 + log(r) + r 2 log(r) log(1 + r −1 ) Due to ρ µτ = ρ τ µ ≡ ρ, we can obtain (5) The exclusions from the ATLAS and CMS searches at the LHC. The extra Higgs bosons are dominantly produced at the LHC via the following electroweak processes: pp →Z * → HA, For small mass splitting among H, A, and H ± , the dominant decay modes of these Higgses are When m A and m H ± are much larger than m H , the following exotic decay modes will open with m A = m H ± , In order to restrict the productions of the above processes at the LHC for our model, we perform simulations for the samples using MG5 aMC-2.4.3 [26] with PYTHIA6 [27] and Delphes-3.2.0 [28], and adopt the constraints from all the analysis for the 13 TeV LHC in version CheckMATE 2.0.26 [29]. Besides, the latest multi-lepton searches for electroweakino [30][31][32][33][34] implemented in Ref. [35] and the ATLAS search for direct stau production with 139 fb −1 13 TeV events [36] are also taken into consideration.

B. Results and discussions
We find that the constraints from theory, oblique parameters and Z decays can be easily satisfied in the parameter space taken in this paper. The allowed ranges of m H , m A , m H ± , and ρ are not reduced by the those constraints. Therefore, we won't show their results in the following discussions. After imposing the constraints of the theory, the oblique parameters, and Z decays, in The lower panels of Fig. 1 show that the experimental data of ∆a µ and τ decays favor ρ to increase with m H . The lower-right panel shows that the excesses of ∆a µ and τ decays can be simultaneously explained in the range of 300 GeV < m H < 800 GeV, and the corresponding ρ is imposed upper and lower bounds. Taking m H = 500 GeV for an example, the excesses of ∆a µ and LUF in the τ decays can be simultaneously explained for 0.6 < ρ < 0.9.
For m H = 500 GeV and ρ < 0.6, ∆a µ can be explained, but the τ decays can not be accommodated. For m H = 500 GeV and ρ > 0.9, ∆a µ and the τ decays can be respectively explained. However, the former favors a small ∆m and the latter favors a large ∆m, which leads that the two anomalies can not be simultaneously explained for m H = 500 GeV and ρ > 0.9.
After imposing the constraints of the direct searches at the LHC, those samples of Fig.   1 are projected on the planes of m H versus ρ and m H versus ∆m, as shown in Fig. 2. Since the excesses of ∆a µ and τ decays require the mass splitting between m A (m H ± ) and m H to be smaller than 50 GeV, H, A and H ± will dominantly decay into τ µ, τ ν µ , and µν τ .
The direct searches at the LHC exclude region of m H < 560 GeV, and the corresponding ρ is required to be larger than 0.68. Since ∆m is such small, it hardly affects the excluded region. We also adopt the ATLAS search for direct stau production with 139 fb −1 integrated luminosity data at 13 TeV LHC [36]. Although the integrated luminosity is much higher than the multilepton search [32], the stau search can not constrain any sample with m H > 300 GeV. We can see from the left panel of Fig. 3 that the largest R-value, the ratio of event yields in signal region to the corresponding 95% experimental limit, is 0.24 for our samples. It is because that both the signal regions of the stau search [36], SR-lowMass and face the similar issues, such as requiring multiple jets [39][40][41], hard jet [42,43], b-tagged jet [44], or exactly two light flavor leptons [42,45].
Given the fact that hundred of integrated luminosity 13 TeV events have been recorded at LHC, we further estimate the exclusion power of LHC with higher luminosity by normalizing signal and background event yields in the signal regions SR-A44 and SR-C18 of [32]. We compute the significance as S = 2(n s + n b )ln(1 + n s /n b ) − 2n s , where n s and n b are the normalized signal and background event yields, respectively. We show the result in the right panel of Fig. 3. The samples with m H < 645 (700) GeV will be excluded at 2σ confidence level with 139 (300) fb −1 integrated luminosity data. If the signal regions of [32] are optimized for the production and decay modes in Eq. (30)(31)(32)(33)(34), the detection ability of LHC for this model could be further enhanced.

IV. CONCLUSION
In this paper, we proposed a 2HDM in which one Higgs doublet has the same interactions with fermions as the SM, and another Higgs doublet only has the µ-τ LFV interactions.
Assuming the Yukawa matrices to be real and symmetric, we considered various relevant theoretical and experimental constraints, and found that the excesses of muon g − 2 and LUF in the τ decays can be simultaneously explained in many parameter spaces with 300 GeV < m H < 800 GeV, ∆m < 50 GeV, and 0.36 < ρ < 1. The parameter spaces are sizable reduced by the direct search limits from the LHC, and m H is required to be larger than 560 GeV.
Note added: When this manuscript is being prepared, a similar paper appeared in the arXiv [46]. Here we discussed different scenario, and obtain different conclusions.