Revisiting lepton-specific 2HDM in light of muon g-2 anomaly

We examine the lepton-specific 2HDM as a solution of muon $g-2$ anomaly under various theoretical and experimental constraints, especially the direct search limits from the LHC and the requirement of a strong first-order phase transition in the early universe. We find that the muon g-2 anomaly can be explained in the region of 32 $<\tan\beta<$ 80, 10 GeV $<m_A<$ 65 GeV, 260 GeV $<m_H<$ 620 GeV and 180 GeV $<m_{H^\pm}<$ 620 GeV after imposing the joint constraints from the theory, the precision electroweak data, the 125 GeV Higgs data, the leptonic/semi-hadronic $\tau$ decays, the leptonic $Z$ decays and Br$(B_s \to \mu^+ \mu^-)$. The direct searches from the $h\to AA$ channels can impose stringent upper limits on Br$(h\to AA)$ and the multi-lepton event searches can sizably reduce the allowed region of $m_A$ and $\tan\beta$ (10 GeV $<m_A<$ 44 GeV and 32 $<\tan\beta<$ 60). Finally, we find that the model can produce a strong first-order phase transition in the region of 14 GeV $<m_A<$ 25 GeV, 310 GeV $<m_H<$ 355 GeV and 250 GeV $<m_{H^\pm}<$ 295 GeV, allowed by the explanation of the muon $g-2$ anomaly.


I. INTRODUCTION
The muon anomalous magnetic moment (g − 2) is a very precisely measured observable.
The muon g−2 anomaly has been a long-standing puzzle since the announcement by the E821 experiment in 2001 [1,2]. The experimental value has an approximate 3σ discrepancy from the SM prediction [3][4][5]. As a popular extension of the SM, the two Higgs doublet models (2HDM) have been applied to explain the muon g − 2 anomaly in the literature .
Among these extensions, the lepton-specific 2HDM (L2HDM) provides a simple explanation for the muon g − 2 anomaly [11,[14][15][16][17]22]. This model includes two neutral CP-even Higgs bosons h and H, one neutral pseudoscalar A, and a pair of charged Higgs bosons H ± . The lepton Yukawa couplings can be sizably enhanced by a large tan β. The pseudoscalar can give positive contributions to muon g − 2 via the two-loop Barr-Zee diagrams, and the muon g − 2 excess favors a light pseudoscalar with a large coupling to lepton.
After the discovery of the SM-like Higgs boson at the LHC, it was found [15] that the muon g − 2 explanation favors the lepton Yukawa couplings of the SM-like Higgs to have an opposite sign with respect to the SM couplings. The observation of Br(B s → µ + µ − ) gives a new constraint on the parameter space of L2HDM [15]. Further, it was found [16] that the leptonic Z decays and leptonic/semi-hadronic τ decays can also give strong constraints on the parameter space of L2HDM, and a more precise calculation is performed in [22]. The L2HDM can lead to τ −rich signatures at the LHC in the parameter region favored by muon g−2. The study in [17] derived the constraints on the model by using the chargino/neutralino search at the 8 Tev LHC and analyzed the prospects at the 14 TeV LHC.
In this work we examine the parameter space of L2HDM by considering the joint constraints from the theory, the precision electroweak data, the 125 GeV Higgs signal data, the muon g − 2 anomaly, the lepton universality in the τ and Z decays, the measurement of Br(B s → µ + µ − ), as well as the direct search limits from the LHC (the data of some channels analyzed by the ATLAS and CMS are corresponding to an integrated luminosity up to about 36 f b −1 recorded in proton-proton collisions at √ s = 13 TeV). On the other hand, it is known that the 2HDM can trigger a strong first-order phase transition (SFOPT) in the early universe [32,33], which is required by a successful explanation of the observed baryon asymmetry of the universe (BAU) [34] and can produce primordial gravitational-wave (GW) signals [35] potentially detectable by future space-based laser interferometer detectors like eLISA [36]. Due to the importance of SFOPT in cosmology, we will also analyze whether a SFOPT is achievable in the parameter space in favor of the muon g − 2 explanation.
Our work is organized as follows. In Sec. II we recapitulate the L2HDM. In Sec. III we discuss the muon g − 2 anomaly and other relevant constraints. In Sec. IV, we constrain the model using the direct search limits from the LHC. In Sec. V, we discuss some benchmark scenarios leading to a SFOPT. Finally, we give our conclusion in Sec. VI.

II. THE LEPTON-SPECIFIC 2HDM
The general Higgs potential is given as [37] In this paper we focus on the CP-conserving case where all λ i and m 2 12 are real. In the L2HDM, a discrete Z 2 symmetry is introduced to make λ 6 = λ 7 = 0, and allow for a soft-breaking term with m 2 12 = 0. The two complex scalar doublets have the hypercharge Y = 1, where the vacuum expectation values (VEVs) v 2 = v 2 1 + v 2 2 = (246 GeV) 2 , and the ratio of the two VEVs is defined as usual to be tan β = v 2 /v 1 . There are five mass eigenstates: two neutral CP-even states h and H, one neutral pseudoscalar A, and two charged scalars H ± .
The Yukawa couplings of the neutral Higgs bosons normalized to the SM are given by where V denotes Z or W , κ ℓ ≡ − tan β, κ d = κ u ≡ 1/ tan β and α is the mixing angle of the two CP-even Higgs bosons.
III. MUON g − 2 ANOMALY AND RELEVANT CONSTRAINTS

A. Numerical calculations
In this paper, we take the light CP-even Higgs h as the SM-like Higgs, m h = 125 GeV.
Since the muon g − 2 anomaly favors a light pseudoscalar with a large coupling to lepton, we scan over m A and tan β in the following ranges: 10 GeV < m A < 120 GeV, 20 < tan β < 120.
In our calculation, we consider the following observables and constraints: (1) Theoretical constraints and precision electroweak data. The 2HDMC [40] is employed 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).
(2) The signal data of the 125 GeV Higgs. Since the 125 GeV Higgs couplings with the SM particles in this model can deviate from the SM ones, the SM-like decay modes will be modified. Besides, for m A is smaller than 62.5 GeV, the invisible decay h → AA is kinematically allowed, which will be strongly constrained by the experimental data of the 125 GeV Higgs. We perform χ 2 h calculation for the signal strengths of the 125 GeV Higgs in the µ ggF +tth (Y ) and µ V BF +V h (Y ) with Y denoting the decay mode γγ, ZZ, W W , τ + τ − and bb, where µ ggH+ttH (Y ) and µ V BF +V H (Y ) are the data best-fit values and a Y , b Y and c Y are the parameters of the ellipse, which are given by the combined ATLAS and CMS experiments [41].
(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 [42]: with g τ g µ 2 ≡Γ(τ → eνν)/Γ(µ → eνν), HereΓ denoting the partial width normalized to its SM value. The correlation matrix for the above five observables is  In the L2HDM we have the ratios g τ g µ ≈ 1 + δ loop , g τ g e ≈ 1 + δ tree + δ loop , g µ g e ≈ 1 + δ tree , where δ tree and δ loop are respectively corrections from the tree-level diagrams and the one-loop diagrams mediated by the charged Higgs. They are given as [16,22] where and We perform χ 2 τ calculation for the five observables. The covariance matrix constructed from the data of Eq. (7) and Eq. (9) 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 [43]: with a correlation of +0. 63 where the SM value g e L = −0.27 and g e R = 0.23. δg 2HDM L and δg 2HDM R are from the 6 one-loop corrections of L2HDM, which are given as where The recent measurement is a exp µ = (116592091±63)×10 −11 [44], which has approximately 3.1σ deviation from the SM prediction [45], ∆a µ = a exp µ − a SM µ = (262 ± 85) × 10 −11 . In the L2HDM, the muon g − 2 obtains contributions from the one-loop diagrams induced by the Higgs bosons and also from the two-loop Barr-Zee diagrams mediated by A, h and H. For the one-loop contributions [6] we have The two-loop contributions are given by where i = h, H, A, and m f , Q f and N c f are the mass, electric charge and the number of color degrees of freedom of the fermion f in the loop. The functions g i (r) are The contributions of the CP-even (CP-odd) Higgses to a µ 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 can be larger than one-loop ones. In the L2HDM, since the CP-odd Higgs coupling to the lepton is proportional to tan β, the L2HDM can sizably enhance the muon g-2 for a light CP-odd Higgs with a large tan β.
where the CKM matrix elements and hadronic factors cancel out, and The L2HDM can give the additional contributions to coefficient C 10 by the Z-penguin diagrams with the charged Higgs loop. Unless there are large enhancements for C P and C S , their contributions can be neglected due to the suppression of the factor For example, the C P and C S of type-II 2HDM can be dominant due to the enhancement of the large tan 2 β terms [47]. Although such large tan 2 β terms are absent in the L2HDM, C P can obtain the important contributions from the CP-odd Higgs exchange diagrams for a very small m A . The experimental data of Br(B s → µµ) is given as [48] Br   The 125 GeV Higgs signal data and the lepton universality data from τ decays include a large number of observales. We perform a global fit to the 125 GeV Higgs signal data and the lepton universality data from τ decays, and define χ 2 as χ 2 = χ 2 h + χ 2 τ . We 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 be within the 2σ range in any two-dimension plane of the model parameters when explaining the signal data of the 125 GeV Higgs and the data of the lepton universality from τ decays.

B. Results and discussions
In Fig. 1, we project the surviving samples within 1σ, 2σ, and 3σ ranges of ∆χ 2 on the planes of tan β versus m A , tan β versus m H ± , and m A versus m H ± after imposing the constraints from theory, the oblique parameters, the exclusion limits from searches for Higgs at LEP, the signal data of the 125 GeV Higgs, and the lepton universality in τ decays. We  shows that the value of χ 2 is favored to increase with tan β and with a decrease of m H ± . This is because the lepton universality in τ decays is significantly corrected by the tree-level  processes: pp →Z * /γ * → HA, In our scenario, the important decay modes of the Higgs bosons are H →τ + τ − , ZA, · · · · · · , H ± →τ ± ν, W ± A, · · · · · · .
Here the light pseudo-scalar A indeed decays into τ τ essentially at 100% due to the enhanced lepton Yukawa couplings by large tan β. The other decay branch ratios and mass spectrum for the samples satisfying constraints described in Sec. III are presented in Fig. 4  Br(H ± → W ± A) decrease, and Br(H → τ + τ − ) and Br(H ± → τ ± v τ ) increase. In conclusion, the dominated finial states generated at LHC of our samples are 3 or 4 τ s with or without gauge boson from pp →Z * /γ * → HA → 4τ or 4τ + Z.
In order to restrict the productions of the above processes at the LHC for our model, we perform simulations for the samples in Fig. 4 using MG5 aMC-2.4.3 [53] with PYTHIA6 [54] and Delphes-3.2.0 [55], and adopt the constraints from all the analysis for the 13 TeV LHC in version CheckMATE 2.0.7 [56]. Besides, the latest multi-lepton searches for electroweakino [57][58][59][60][61] implemented in Ref. [62] are also taken into consideration because of the dominated multi-τ final states in our model.
The results from CheckMATE are presented in Fig. 5   CheckMATE. The orange stars and green dots stand for the samples excluded and allowed by the LHC Run-2 data at 95% confidence level, respectively.
experimentally measured 95% confidence limit on signal event in signal region i. Obviously, R > 1 means that the corresponding point is excluded at 95% confidence level by at least one search channel. We can see that the constraints from current LHC 13 TeV data shrink m A from [10,65] GeV to [10,44] GeV and tan β from [32,80] to [32,60]. For the samples excluded by current 13 TeV LHC data, the strongest constraint comes from the search for electroweak production of charginos and neutralinos in multilepton final states [58]. In this analysis, 7 categories of signal region are designed for event with τ in final state, SR-C to SR-F and SR-I to SR-K. The most sensitive signal region is SR-K for most of the parameter space. It requires at least two light-flavor leptons and two τ jets without b-tagged jet.
The signal region is subdivided by missing energy E / T to three bins, SR-K01, SR-K02, and SR-K03. The main contributions of our samples to the bins are from processes in Eq. (38) and Eq. (39) with at least two of the τ s decaying hadronically.
In Fig. 5 Specially designed signal regions are needed to detect this region.

V. THE STRONG FIRST-ORDER PHASE TRANSITION
Originally the 2HDM was proposed to spontaneously break CP-conservation [63]. Thus by combining with baryon number violation and departure from the thermal equilibrium, it is possible to explain BAU [34]. Baryon number conservation is broken by electroweak sphaleron process [64], and the departure from the thermal equilibrium can be realized by a SFOPT. The SM electroweak sector cannot trigger a SFOPT, because SFOPT in the SM requires a Higgs mass of around 70 GeV ∼ 80 GeV [65]. This problem can be solved by the extended Higgs sector in 2HDM which has more degrees of freedom [32]. In this section we study the possibility to obtain a parameter space in L2HDM that can trigger a SFOPT and explain muon g − 2 anomaly at the same time.
A. Thermal effective potential In order to know the strength of phase transition in our scenario, we need to study the effective potential with thermal correction included. The thermal effective potential V (φ 1 , φ 2 , T ) at temperature T is composed of four parts: where V 0 is the tree-level potential, V CW is the Coleman-Weinberg potential, V CT is the counter term and V T is the thermal correction.
The tree-level potential V 0 (φ 1 , φ 2 ) can be obtained by replacing scalar fields Φ 1 (x) and : CP violation is not introduced here because its impact on phase transition is usually not significant [66].
The one-loop Coleman-Weinberg potential V CW (φ 1 , φ 2 ) [67] under MS renormalization scheme is where the index i runs over all the massive particles, and n i is the number of degrees of freedom, equal to -12, -4, 6, 3, 2, 1, 2, and 1 for quark, lepton, W ± , Z, H ± , G 0 , G ± , and neutral scalars, respectively. Here we include the contribution from goldstone G 0 and G ± because their mass can be non-zero for (φ 1 , φ 2 ) = (v 1 , v 2 ). c i is equal to 5 6 for gauge bosons, and is equal to 3 2 for other particles. m 2 i (φ 1 , φ 2 ) is the field value dependent mass square of different particles. Renormalization scale µ is set to zero temperature VEV v.
The tree-level spectrum and mixing angles will be shifted by the Coleman-Weinberg correction. In order to offset the shift, a counter term V CT (Φ 1 , Φ 2 ) needs to be added to Lagrangian. For a CP-conserving 2HDM, V CT (Φ 1 , Φ 2 ) can be expressed as where the δ's are determined by the "on-shell" renormalization conditions: with ψ i denoting all the component scalar fields of Φ 1 and Φ 2 . These conditions are evaluated at the minimum of scalar potential at zero temperature, where Thus our tree-level input parameters can be preserved after including loop correction. And the corresponding potential V CT (φ 1 , φ 2 ) needs to be added to V (φ 1 , φ 2 , T ).
The thermal correction including daisy resummation [68,69] is where the index i denotes all gauge bosons and scalars, j denotes leptons and quarks, and k denotes scalars and longitudinal component of gauge bosons. The integral functions J B,F are given by Andm 2 k (φ 1 , φ 2 , T ) is thermal Debye mass. Expression of thermal Debye mass can be found in literature [70].

B. Numerical results
The condition for a SFOPT is usually taken to be [71]: where T c is the critical temperature at which a second minimum of V (φ 1 , φ 2 , T ) with non-zero VEV appears, and v c = φ 2 1 + φ 2 2 is the corresponding VEV at T c . Due to the complicated form of V (φ 1 , φ 2 , T ), a numerical calculation is always required to analyze the geometry evolution of V (φ 1 , φ 2 , T ). In this work we use package BSMPT [72] to do the analysis. In GeV < m A < 25 GeV, 310 GeV < m H < 355 GeV, and 250 GeV < m H ± < 295 GeV. As pointed out in [70], m A < 100 GeV is not favored by SFOPT in 2HDM. Thus a certain level of fine-tuning is required if one wants to explain the muon g − 2 anomaly, which needs a light m A , and SFOPT in the meantime. For the narrow parameter space which can achieve SFOPT and accomodate the muon g − 2 anomaly, we list several benchmark points in Table   II. A thorough study of its observability at the LHC or future colliders will be performed elsewhere. and satisfy the relevant theoretical and experimental constraints.

VI. CONCLUSION
The L2HDM can provide a simple explanation for the muon g −2 anomaly. We performed a scan over the parameter space of L2HDM to identify the ranges in favor of the muon g − 2 explanation after imposing various relevant theoretical and experimental constraints, especially the direct search limits from LHC and a SFOPT in the early universe. We found that the muon g-2 anomaly can be accommodated in the region of 32 < tan β < 80, 10 GeV < m A < 65 GeV, 260 GeV < m H < 620 GeV and 180 GeV < m H ± < 620 GeV  after imposing the joint constraints from the theory, the precision electroweak data, the 125 GeV Higgs signal data, the lepton universality in τ and Z decays, and the measurement of Br(B s → µ + µ − ). The direct search limits from the LHC can give stringent constraints on m A and tan β for small m H and m H ± : 10 GeV < m A < 44 GeV and 32 < tan β < 60.
The direct search limits from the h → AA channels at the LHC can impose stringent upper limits on Br(h → AA). Finally, we found that a SFOPT can be achievable in the region of 14 GeV < m A < 25 GeV, 310 GeV < m H < 355 GeV, and 250 GeV < m H ± < 295 GeV while the muon g − 2 anomaly is accommodated.