$h\to\mu\tau$ and muon g-2 in the alignment limit of two-Higgs-doublet model

We examine the $h\to \mu\tau$ and muon g-2 in the exact alignment limit of two-Higgs-doublet model. In this case, the couplings of the SM-like Higgs to the SM particles are the same as the Higgs couplings in the SM at the tree level, and the tree-level lepton-flavor-violating coupling $h\mu\tau$ is absent. We assume the lepton-flavor-violating $\mu\tau$ excess observed by CMS to be respectively from the other neutral Higgses, $H$ and $A$, which almost degenerates with the SM-like Higgs at the 125 GeV. After imposing the relevant theoretical constraints and experimental constraints from the precision electroweak data, $B$-meson decays, $\tau$ decays and Higgs searches, we find that the muon g-2 anomaly and $\mu\tau$ excess favor the small lepton Yukawa coupling and top Yukawa coupling of the non-SM-like Higgs around 125 GeV, and the lepton-flavor-violating coupling is sensitive to another heavy neutral Higgs mass. In addition, if the $\mu\tau$ excess is from $H$ around 125 GeV, the experimental data of the heavy Higgs decaying into $\mu\tau$ favor $m_A>230$ GeV for a relatively large $H\bar{t}t$ coupling.


I. INTRODUCTION
The ATLAS and CMS collaborations have probed the lepton-flavor-violating (LFV) Higgs decay h → µτ around 125 GeV at the LHC run-I [1][2][3] and early run-II [4]. By the analysis of data sample corresponding to an integrated luminosity of 20.3 fb −1 at the √ s = 8 TeV LHC, the ATLAS Collaboration found a mild deviation of 1σ significance in the h → µτ channel and set an upper limit of Br(h → µτ ) < 1.43% at 95% confidence level with a best fit Br(h → µτ ) = (0.53 ± 0.51)% [2]. Based on the data sample corresponding to an integrated luminosity of 19.7 fb −1 at the √ s = 8 TeV LHC, the CMS collaboration imposed an upper limit of Br(h → µτ ) < 1.51% at 95% confidence level, while the best fit value is Br(h → µτ ) = (0.84 +0.39 −0.37 )% with a small excess of 2.4σ [3]. At the √ s = 13 TeV LHC run-II with an integrated luminosity of 2.3 fb −1 , the CMS collaboration did not observe the excess and imposed an upper limit of Br(h → µτ ) < 1.2% [4]. However, the CMS search result at the early LHC run-II can not definitely kill the excess of h → µτ due to the low integrated luminosity.
If the h → µτ excess is not a statistical fluctuation, the new physics with the LFV interactions can give a simple explanation for the excess. On the other hand, the longstanding anomaly of the muon anomalous magnetic moment (muon g-2) implies that the new physics is connected to muons. The two excesses can be simultaneously explained by the LFV Higgs interactions, such as the general two-Higgs-doublet model (2HDM) with the LFV Higgs interactions. There have been many studies on the h → µτ excess in the framework of 2HDM [5][6][7] and some other new physics models [8].
In this paper, we discuss the excesses of h → µτ and muon g-2 in the exact alignment limit of the general 2HDM where one of the neutral Higgs mass eigenstates is aligned with the direction of the scalar field vacuum expectation value (VEV) [9]. In the interesting scenario, the SM-like Higgs couplings to the SM particles are the same as the Higgs couplings in the SM at the tree level, and the tree-level LFV coupling hµτ is absent. We assume the µτ excess observed by CMS to be respectively from the other neutral Higgses, H and A, which almost degenerates with the SM-like Higgs at the 125 GeV. In our discussions, we impose the relevant theoretical constraints from the vacuum stability, unitarity and perturbativity as well as the experimental constraints from the precision electroweak data, B-meson decays, τ decays and Higgs searches.
Our work is organized as follows. In Sec. II we recapitulate the alignment limit of 2HDM.
In Sec. III we perform the numerical calculations and discuss the muon g-2 anomaly and the µτ excess around 125 GeV after imposing the relevant theoretical and experimental constraints. Finally, we give our conclusion in Sec. IV.

II. TWO-HIGGS-DOUBLET MODEL AND THE ALIGNMENT LIMIT
The alignment limit of 2HDM is defined as the limit in which one of the two neutral CP-even Higgs mass eigenstates aligns with the direction of the scalar field VEV [9]. The alignment limit can be easily realized in the decoupling limit [10], namely that all the non-SM-like Higgses are very heavy. The possibility of alignment without decoupling limit was first noted in [10], "re-invented" in [11][12][13] and further studied in [9,[14][15][16][17]. The alignment limit is basis-independent, and clearly exhibited in the Higgs basis. The alignment limit also exists in the Minimal Supersymmetric Standard Model which is a constrained incarnation of the general 2HDM. There are some detailed discussions in [18,19] and a very recent study in [20].
A. Two-Higgs-doublet model in the Higgs basis The general Higgs potential is written as [21] All µ i and k i are real in the CP-conserving case. In the Higgs basis, the H 1 field has a VEV v =246 GeV, and the VEV of H 2 field is zero. The two complex scalar doublets have the hypercharge Y = 1, The general Higgs potential is written as [21] The parameters m ij and λ i are the linear combinations of the parameters in the Higgs basis: The general Yukawa interaction can be given as 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 u1,2 , Y d1,2 and Y ℓ1,2 are 3 × 3 matrices in family space.
To avoid the tree-level FCNC couplings of the quarks, we take where this choice, the interaction corresponds to the aligned 2HDM [23,24].
For the Yukawa coupling matrix of the lepton, we take Where X = V L Y ℓ2 V † R , and V L (V R ) is the unitary matrix which transforms the interaction eigenstates to the mass eigenstates of the left-handed (right-handed) lepton fields. The other nondiagonal matrix elements of X are zero.
The Yukawa couplings of the neutral Higgs bosons are given as Where The κ ℓ is a free input parameter, which is used to parameterize the matrix element of the lepton Yukawa coupling, as shown in Eq. (12). In other words, the matrix elements of the lepton Yukawa coupling are taken as the Eq. (12) in order to obtain the Yukawa couplings of lepton in Eq. (13).
The neutral Higgs bosons couplings to the gauge bosons normalized to the SM Higgs boson are given by where V denotes Z and W .
In the exact alignment limit, namely cos(β − α) = 0, the Eq. (13) and Eq. (14) show that the 125 GeV Higgs (h) has the same couplings to the fermions and gauge bosons as the SM values, and the tree-level LFV couplings are absent. The heavy CP-even Higgs (H) has no coupling to the gauge bosons, and there are the tree-level LFV couplings for the A and H.

III. NUMERICAL CALCULATIONS AND DISCUSSIONS
A. Numerical calculations In the exact alignment limit, the SM-like Higgs has no tree-level LFV coupling. In order to explain the h → µτ excess reported by CMS, we assume the signal to be respectively from H and A, which almost degenerates with the SM-like Higgs at the 125 GeV. Here we take two scenarios simply: (i) m A =126 GeV and (ii) m H =126 GeV.
In our calculations, the other involved parameters are randomly scanned in the following ranges: In order to relax the constraints from the observables of down-type quarks, we take For the cases of m A = 126 GeV and m H =126 GeV, we respectively take ρ µτ = −ρ τ µ and ρ µτ = ρ τ µ to produce a large positive contribution to the muon g-2. The pseudoscalar A can give the positive contributions to the muon g-2 via the two-loop Barr-Zee diagrams with the lepton-flavor-conserving (LFC) coupling. Therefore, we take | κ ℓ |< 150 to examine the possibility of explaining the muon g-2. In the exact alignment limit, the hττ coupling is independent on κ ℓ and equals the SM value. However, the Aττ and Hττ couplings can reach 1.08 and be slightly larger than 1 for | κ ℓ |= 150, which can not lead to the problem on the perturbativity due to the suppression of the loop factor. In addition, for such large κ ℓ the Br(A → ττ ) and Br(H → ττ ) can reach 1. Due to κ d = 0 and cos(β − α) = 0, the cross sections of A and H are equal to zero in the bb associated production mode and vector boson fusion production mode. However, the searches for gg → A/H → ττ can give the constraints on κ u . We will discuss the constraints in the following item (5).
During the scan, we consider the following experimental constraints and observables: (1) Theoretical constraints and precision electroweak data. We use 2HDMC-1.6.5 [25] to implement the theoretical constraints from the vacuum stability, unitarity and coupling-constant perturbativity, as well as the constraints from the oblique parameters (S, T , U) and δρ.
(2) B-meson decays and R b . Although the tree-level FCNCs in the quark sector are absent, they will appear at the one-loop level in this model. We consider the constraints of B-meson decays from ∆m Bs , ∆m B d , B → X s γ, and B s → µ + µ − , which are respectively calculated using the formulas in [26][27][28]. In addition, we consider the R b constraints, which is calculated following the formulas in [29]. In fact, in this paper we take κ d = 0 and 0 ≤ κ u ≤ 1.2, which will relax the constraints from the bottom-quark observables sizably.
The two µ − in final states can be exchanged. In the exact alignment limit H 0 k denotes A and H.
(3) τ decays. In this model, the non-SM-like Higgses have the tree-level LFV couplings to τ lepton, and the LFC couplings to lepton can be sizably enhanced for −150 ≤ κ ℓ ≤ 150.
Therefore, some τ decay processes can give very strong constraints on the model.
(ii) τ → µγ. The main Feynman diagrams of τ → µγ in the model are shown in Fig. 2.
In the exact alignment limit, the SM-like Higgs has no tree-level LFV coupling, and the heavy CP-even Higgs couplings to the gauge bosons are equal to zero. Therefore, the SM-like Higgs does not contribute to the τ → µγ, and the τ → µγ can not be where A 1L0 , A 1Lc , A 1R0 and A 1Rc are from the one-loop diagrams with the Higgs boson and τ lepton [6], The A 2L and A 2R are from the two-loop Barr-Zee diagrams with the third-generation fermion loop [6], where T 3f denotes the isospin of the fermion, and The two terms of A 2L come from the effective φγγ vertex and φZγ vertex induced by the third-generation fermion loop. The current experimental data give an upper bound of Br(τ → µγ) [33], (iii) τ → µπ 0 . The τ can decay into a lepton and a pseudoscalar meson at the tree level via the CP-odd Higgs with the LFV couplings, such as τ → µπ 0 . The corresponding Feynman diagrams are shown in Fig. 3. The width of τ → µπ 0 is given as [34], The current upper bound of Br(τ → µπ 0 ) is [35], (4) muon g-2.
The dominant contributions to the muon g-2 are from the one-loop diagrams with the Higgs LFV coupling [36], and the corresponding Feynman diagrams can be obtained by replacing the initial states τ with µ in Fig. 2 (a) and Fig. 2 (b). In the exact alignment limit, At the one-loop level, the diagrams with the Higgs LFC coupling can also give the contributions to the muon g-2, especially for a large lepton Yukawa coupling [37]. The corresponding Feynman diagrams can be obtained by replacing τ in the initial state and loop with µ in Fig. 2 (a) as well as replacing the initial state τ with µ and ν τ in the loop with ν µ in Fig. 2 (b). The contributions from the one-loop diagrams with the Higgs LFC coupling are given as where r φµ = m 2 µ /m 2 φ and y H ± µµ = y Aµµ . For r φµ ≪ 1, The muon g-2 can be corrected by the two-loop Barr-Zee diagrams with the fermions loops by replacing the initial state τ with µ in Fig. 2 (c). Further replacing the fermion loop with W loop, we obtain the two-loop Barr-Zee diagrams with W loop which can contribute to muon g-2 for the SM-like Higgs h as the mediator in the exact alignment limit. Using the well-known classical formulates in [38], the main contributions of two-loop Barr-Zee diagrams in the exact alignment limit are given as The experimental value of muon g-2 excess is [39] δa µ = (26.2 ± 8.5) × 10 −10 .
Where R j = (σ × BR) j /(σ × BR) SM j with j denoting the partonic process ggĤ, V BFĤ, VĤ, or ttĤ. ǫ i j denotes the assumed signal composition of the partonic process j. If A (H) almost degenerates with the SM-like Higgs, φ denotes A (H). For an uncorrelated observable i, where µ exp i and σ i denote the experimental central value and uncertainty for the ichannel. We retain the uncertainty asymmetry in the calculation. For the two correlated observables, we take where ρ is the correlation coefficient. We sum over χ 2 in the 29 channels, and pay particular attention to the surviving samples with χ 2 −χ 2 min ≤ 6.18, where χ 2 min denotes the minimum of χ 2 . These samples correspond to the 95.4% confidence level region in any two-dimension plane of the model parameters when explaining the Higgs data (corresponding to the 2σ range).
(iii) The Higgs decays into τ µ. In the exact alignment limit, the µτ excess around 125 GeV is from A (H) → τ µ where A (H) almost degenerates with the SM-like Higgs.

The width of A (H) → µτ is given by
We take the best fit value of Br(h → µτ ) = (0.84 +0.39 −0.37 )% based on the CMS search for the h → µτ at the LHC run-I. Since the µτ excess is assumed to be from the A (H), we require the production rates of pp → A (H) → µτ to vary from σ(pp → h) × 0.1% to σ(pp → h) × 1.62%.
In addition, the CMS collaboration did not publish the bound on the heavy Higgs decaying into µτ . Ref.
[7] gave the bound on the production rate of pp → φ → µτ by recasting results from the original h → µτ analysis of CMS.

B. Results and discussions
In Fig. 4, we project the surviving samples on the planes of ρ τ µ versus κ ℓ and κ u versus ρ τ µ . The lower panels show the κ u is required to be smaller than 1 due to the constraints of B-meson decays and R b . The upper panels show that there is a strong correlation between ρ τ µ and κ ℓ , which is mainly due to the constraints of Br(τ → 3µ) on the product |ρ τ µ × κ ℓ |, and obviously affected by the constraints of Br(τ → µγ). For example, in the case of m A = 126 GeV, |ρ τ µ | is required to be smaller than 0.06 for κ ℓ = −10.
In the case of m A = 126 GeV, there are two different regions where the muon g-2 anomaly can be explained. (i) ρ τ µ = 0 and |κ ℓ | > 100: The Higgs LFV couplings are absent due to ρ τ µ = 0, and the muon g-2 can be only corrected via the diagrams with the Higgs LFC couplings. Without the contributions of top quark loops, the contributions of the CP-even (CP-odd) Higgs to muon g-2 are negative (positive) at the two-loop level and positive (negative) at one-loop level. As m 2 f /m 2 µ could easily overcome the loop suppression factor α/π, the two-loop contributions may be larger than one-loop ones. Therefore, the muon g-2 can obtain the positive contributions from A loop and negative contributions from H loop. For the enough mass splitting of H and A, the muon g-2 can be sizably enhanced by the diagrams with the large Higgs LFC couplings. The corresponding κ u is required to be smaller than 0.2 due to the constraints of the search for gg → A → ττ at the LHC, see the pluses (red) with ρ τ µ = 0 shown in the lower-left panel of Fig. 4. (ii) 0.04 < |ρ τ µ | < 0.18 and −9 < κ ℓ < 3: The muon g-2 can be corrected by the diagrams Note that there is the κ ℓ asymmetry in the regions of 0.04 < |ρ τ µ | < 0.18, −9 < κ ℓ < 3 and 0.02 < κ u < 0.1 for m A = 126 GeV where muon g-2 can be explained. The main reason is from the constraints of τ → µγ. In the above regions, the top quark can give sizable contributions to τ → µγ via the "A 2L " and "A 2R " terms as shown in Eq. (24), which have destructive (constructive) interferences with the "A 1L0 " of Eq. (20) and "A 1R0 " of Eq. (22) induced by the one-loop contributions of τ for κ ℓ < 0 (κ ℓ > 0). Therefore, | κ ℓ | for κ ℓ < 0 is allowed to be much larger than that for κ ℓ > 0. Similar reason is for the κ ℓ asymmetry in the case of m H = 126 GeV but the destructive (constructive) interferences for κ ℓ > 0 (κ ℓ < 0).
In Fig. 6, we project the surviving samples on the planes of ρ τ µ versus m H and ρ τ µ versus m A in the cases of m A = 126 GeV and m H = 126 GeV, respectively. We find that ρ τ µ is sensitive to the mass of heavy Higgs, and the absolute value decreases with increasing of the mass of heavy Higgs in order to explain the muon g-2 anomaly and the µτ excess around 125 GeV. As we discussed above, there is an opposite sign between the contributions of the H loops and A loops to the muon g-2. Therefore, with the decreasing of the mass splitting of H and A, the cancelation between the contributions of H and A loops becomes sizable so that a large absolute value of ρ µτ is required to enhance the muon g-2.
In Fig. 7 GeV (m H = 126 GeV), which is due to the constraints of the oblique parameters and δρ. In this paper we focus on the exact alignment limit. If the alignment limit is approximately realized, the µτ excess can be from the SM-like Higgs (h) in addition to H or A around the 125 GeV. Therefore, the upper limits of κ u become more stringent. When the µτ excess is mainly from h, the lower limit of κ u will disappear since the htt coupling hardly changes with κ u , and the A(H)tt coupling is (nearly) proportional to κ u . In addition, the upper limit of ρ µτ can become more strong for the proper deviation from the alignment limit. For example, for sin(β − α) = 0.996, Br(h → µτ ) < 1.62% will give an upper limit of | ρ µτ |< 0.0408, which is much smaller than that in the exact alignment limit. In the exact alignment limit, the widths of H → hh, W W ( * ) , ZZ ( * ) and A → hZ are equal to zero, and increase with decreasing of | sin(β−α) |. Therefore, the searches for H → hh, W W ( * ) , ZZ ( * ) and A → hZ can be used to probe the deviation from the alignment limit. These signatures refer to the H or A whose mass is not near 125 GeV. Otherwise, its signal would be indistinguishable from that coming from the SM-like light Higgs, and even H → hh (A → hZ) is absent for H (A) near 125 GeV. Some similar studies have been done in the singlet extension of the SM [42].
In the previous studies, the µτ excess is assumed to be from the SM-like Higgs h. In this paper we discuss another interesting scenario where the µτ excess is from either H or A near the observed Higgs signal. There is no AV V coupling due to the CP-conserving. The HV V coupling is absent and the hV V coupling is the same as the SM value in the exact alignment limit. Therefore, the two scenarios can be distinguished by observing the µτ signal via the vector boson fusion production process at the LHC with high integrated luminosity. In other words, if the µτ signal excess is observed in the gluon fusion process and not observed in the vector boson fusion process, the scenario in this paper will be strongly favored. Even when sin(β − α) deviates from the alignment limit sizably, the production rates of µτ signal via the gluon fusion and vector boson fusion can still have different correlations in the two different scenarios.

IV. CONCLUSION
In this paper we examine the muon g-2 anomaly and the µτ excess around 125 GeV in the exact alignment limit of 2HDM. In the scenario, the SM-like Higgs couplings to the SM particles are the same as the Higgs couplings in the SM at the tree level, and the treelevel LFV coupling hµτ is absent. We assume the µτ signal excess observed by CMS to be respectively from the H and A, which almost degenerates with the SM-like Higgs at the 125 GeV. After imposing various relevant theoretical constraints and experimental constraints from precision electroweak data, B-meson decays, τ decays and Higgs searches, we obtain the following observations: For the case of m A = 126 GeV, the muon g-2 anomaly can be explained in two different regions: (i) ρ τ µ = 0 and |κ ℓ | > 100; (ii) 0.04 < |ρ τ µ | < 0.18 (|ρ τ µ | is sensitive to m H ) and −9 < κ ℓ < 3. Further, the µτ excess around 125 GeV can be explained in the region (ii) with 0.02 < κ u < 0.1 where all the surviving samples are allowed by the experimental constraints of the heavy Higgs decaying into µτ .
However, most samples in the ranges of 0.07 < κ u < 0.15 and m A < 230 GeV are further excluded by the experimental constraints of the heavy Higgs decaying into µτ .