A light pseudoscalar of 2HDM confronted with muon g-2 and experimental constraints

A light pseudoscalar of the lepton-specific 2HDM can enhance the muon g-2, but suffer from various constraints easily, such as the 125.5 GeV Higgs signals, non-observation of additional Higgs at the collider and even $B_s\to \mu^+\mu^-$. In this paper, we take the light CP-even Higgs as the 125.5 GeV Higgs, and examine the implications of those observables on a pseudoscalar with the mass below the half of 125.5 GeV. Also the other relevant theoretical and experimental constraints are considered. We find that the pseudoscalar can be allowed to be as low as 10 GeV, but the corresponding $\tan\beta$, $\sin(\beta-\alpha)$ and the mass of charged Higgs are strongly constrained. In addition, the surviving samples favor the wrong-sign Yukawa coupling region, namely that the 125.5 GeV Higgs couplings to leptons have opposite sign to the couplings to gauge bosons and quarks.


I. INTRODUCTION
The ATLAS and CMS Collaborations found a 125.5 GeV Higgs boson at the LHC [1,2]. The latest experimental data show that the properties of this particle agree with the Standard Model (SM) predictions. Especially the diphoton signal strength is changed from 1.6 ± 0.4 to 1.17 ± 0.27 for ATLAS [3] and from 0.78 +0. 28 −0.16 to 1.12 +0.37 −0.32 for CMS [4], which are well consistent with the SM prediction within 1σ range. Thus, the 125.5 GeV Higgs signal data can give the strong constraints on the effects of new physics.
The two-Higgs-doublet model (2HDM) has very rich Higgs phenomenology, including two neutral CP-even Higgs bosons h and H, one neutral pseudoscalar A, and two charged Higgs H ± . The recent Higgs data have been used to constrain the 2HDM, see some recent examples [5]. In addition, a light pseudoscalar with a large tan β can account for the 3.1σ deviation between the SM predicted and measured values of the muon anomalous magnetic moment [6][7][8][9]. Due to the experimental constraints, the type-II 2HDM [10] is very difficult to explain the muon g-2 anomaly [7,9], but the lepton-specific 2HDM (L2HDM) [11] can still give a valid explanation [8,9]. Compared to the recent study [9], we focus on a light pseudoscalar for which a relative small tan β is required to account for the muon g-2 anomaly.
For a light pseudoscalar, the 125.5 GeV Higgs decay into the pseudoscalars is open, and the rare decay B s → µ + µ − can obtain the additional important contributions from the very light pseudoscalar exchange diagrams. Therefore, the 125.5 GeV Higgs signal data and even B s → µ + µ − can give the important constraints on the very light pseudoscalar. Also we consider the theoretical constraints, electroweak precision data, the non-observation of additional Higgs at collider, and the flavor observables B → X s γ, ∆m Bs and ∆m B d .
Our work is organized as follows. In Sec. II we recapitulate the L2HDM. In Sec. III we introduce the numerical calculations. In Sec. IV, we show the implications of muon g-2 and experimental data on the L2HDM. Finally, we give our conclusion in Sec. V.

II. L2HDM
The general Higgs potential is written as [12] 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 h and H, one neutral pseudoscalar A, and two charged scalar H ± . We can rotate this basis to the Higgs basis by a mixing angle β, where the VEV of Φ 2 field is zero.
In the Higgs basis, the mass eigenstates are obtained from The right fields of the equations denote the interaction eigenstates in the Higgs basis. The corresponding masses and couplings of Eq. (1) are changed in the Higgs basis [13]. For example, both λ 6 and λ 7 are taken as zero in the physics basis, but the rotation into the Higgs basis can generate non-zero values for λ 6 and λ 7 .
In the Higgs basis, the general Yukawa interactions with no tree-level FCNC are give [14] L whereΦ i (x) = iτ 2 Φ * i (x) and M ′ d,u,ℓ are the Yukawa matrices. For the L2HDM, The couplings of neutral Higgs bosons with respect to the SM Higgs boson are give by Where V denotes Z and W , and f denotes u, d and ℓ. The charged Higgs couplings are give as where M f are the diagonal fermion mass matrices.

III. NUMERICAL CALCULATIONS
The in-house code is used to calculate the muon g-2, χ 2 fit to 125.5 GeV Higgs signal, 15] is employed to implement the theoretical constraints from the vacuum stability, unitarity and coupling-constant perturbativity, and calculate the oblique parameters (S, T , U) and δρ. SuperIso-3.4 [16] is used to implement the constraints from B → X s γ. HiggsBounds-4.1.3 [17] is employed to implement the exclusion constraints from the neutral and charged Higgses searches at the LEP, Tevatron and LHC at 95% confidence level. Now we introduce the calculations of some constraints, which are the main motivations of this paper: Muon g-2: The recent measurement on the muon anomalous magnetic moment is a exp µ = (116592091±63)×10 −11 [18], which has approximately 3.1σ deviation from the SM prediction [19,20], ∆a µ = a exp µ − a SM µ = (262 ± 85) × 10 −11 [9].
In the L2HDM, a µ gets the additional 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 [21], where For two-loop contributions, where i = h, H, A. m f , Q f and N c f are the mass, electric charge and number of color degrees of freedom of the fermion f in the loop. The functions g i (r) are [6,7] 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 may be larger than one-loop ones. In the L2HDM, since the CP-odd Higgs coupling to the tau lepton is proportional to tan β, the L2HDM can enhance sizably the muon g-2 for a light CP-odd Higgs with a large tan β.
Global fit to the 125.5 GeV Higgs signal data: We take the light CP-even Higgs as the 125.5 GeV Higgs. Using the method taken in [22], we perform a global fit to the 125.5 GeV Higgs data of 29 channels after ICHEP 2014, which are summarized in the Tables I-V of [23]. A number of new measurements or updates of existing ones were presented by ATLAS and CMS Collaborations [3,4,[24][25][26][27][28][29][30][31]. The signal strength for the i channel is defined as Where R j = (σ×BR) j (σ×BR) SM j with j denoting the partonic processes ggH, V BF, V H, and ttH. ǫ i j denotes the assumed signal composition of the partonic process j, which are given in Tables   I-V of [23]. For an uncorrelated observable i, where µ exp i and σ i denote the experimental central value and uncertainty for the i channel.
We retain the uncertainty asymmetry in our calculations. For the two correlated observables, we take where ρ is the correlation coefficient. We sum over χ 2 of 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 regions in any two dimensional plane of the model parameters when explaining the Higgs data (corresponding to be within 2σ range).
Following the method taken in Ref. [36], we define where the CKM matrix elements and hadronic factors cancel out. Combining the SM prediction with the experimental result,R sµ = 0.79 ± 0.2 is required.
The 2HDM 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 M 2 Bs /M 2 W . 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 [37]. 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 m A is very small. Such contributions are also sensitive to m H ± and small tan β. For the large tan β, the terms proportional to cot β and the higher order terms can be neglected.
Using the formulas in [36], we calculate the parameter P and S in the L2HDM. Note that the mixing of two CP-even Higgses in this paper is different from [36], therefore some corresponding couplings need be replaced.
In our calculations, m h = 125.5 GeV is fixed, and the input parameters are sin(β − α), tan β, the physical Higgs masses (m H , m A , m H ± ) and the coupling of hAA (λ hAA ), where λ hAA is used to replace the soft-breaking parameter m 2 12 . We focus on 5 GeV < m A < 62.75 GeV, and such light CP-odd Higgs will be more strongly constrained, especially for the 125.5 GeV Higgs signal. Assuming that the Higgs couplings to fermions and gauge bosons are the same as the SM, Br(h → AA) is larger than 40% for |λ hAA | > 20 GeV and m A < 62.5 GeV. Therefore, we scan λ hAA in the range of -20 GeV ∼ 20 GeV. In addition to that the theoretical constraints are satisfied, we require the L2HDM to explain the experimental data within the 2σ range, and fit the current Higgs signal data at the 2σ level. The experimental values of electroweak precision data, B → X s γ, ∆m Bs and ∆m B d are taken from [38].

IV. RESULTS AND DISCUSSIONS
Without the constraint of muon g-2, we find a surviving sample with a minimal value of Our numerical results show that for the surviving samples within the 2σ range of χ 2 , Br(h → AA) is required to be smaller than 24%.
In Fig. 1, we project the surviving samples on the plane of sin(β − α) versus tan β.
Without the constraint of muon g-2, the surviving samples lie in two different regions. In one region, the 125.5 GeV Higgs couplings are near the SM values, called the SM-like region.
In the other region, the Higgs couplings to leptons have opposite sign to the corresponding couplings to V V , called the wrong-sign Yukawa coupling region. In the SM-like region, the absolute value of sin(β − α) is required to be larger than 0.986, while sin(β − α) is allowed to have more sizable deviation from 1.0 in the wrong-sign Yukawa coupling region, sin(β − α) > 0.89. This can be understandable from the Higgs couplings to leptons. For the wrong-sign Yukawa coupling and SM-like Yukawa coupling, the Higgs couplings to lepton are respectively Where ε is larger than zero, but much smaller than 1.0. For sin(β − α) approaches to 1.0, cos(β − α) of the former is much larger than that of the latter for the same tan β.   versus m H ± , respectively. The left panel shows that there is a small mass difference between m H and m H ± for the surviving samples, especially for including the constraint of muon g-2.

Allowed by muon g-2 and
Such small mass difference is mainly required by the electroweak parameter ρ to produce a pseudoscalar with mass in the range of 5 GeV ∼ 62.75 GeV.
The right panel shows that tan β is required to be larger than 2.0 for m H ± < 230 GeV, and the main constraints are from the flavor observables ∆m Bs and ∆m B d . In addition, for m A < 16 GeV, there is strong correlation between m A and m H ± due to the constraint of B s → µ + µ − . In particular, m H ± is required to be larger than 450 GeV for tan β < 2 and m A < 10 GeV, and m H ± is allowed to be sharply decreased with the increasing of m A . For m A > 20 GeV, the contributions from the exchange of A diagrams to the coefficient C P are difficult to overcome the suppression factor M 2 Bs /M 2 W , therefore B s → µ + µ − is not sensitive to m A . Also the constraint of B s → µ + µ − on m A and m H ± can be relaxed by a modest large tan β, but not sensitive to the enough large tan β. Including the constraint of muon g-2, m A is allowed to be as low as 10 GeV, but the corresponding m H ± is required to be larger than 250 GeV.
In Fig. 4, the surviving samples are projected on the plane of m A versus tan β. For m A < 26 GeV, the upper bound of tan β is strongly constrained by the exclusion experiments of Higgs at the collider, and some intermediate values are excluded by the 2σ constraint of χ 2 fit to the Higgs signal. Including the constraint of muon g-2, the range of tan β is sizably narrowed with m A approaching to 10 GeV.

V. CONCLUSION
In the L2HDM, a light pseudoscalar with a large tan β can account for the muon g-2 anomaly. Assuming that the light CP-even Higgs is the 125.5 GeV Higgs, we study the implications of the relevant theoretical and experimental constraints on a pseudoscalar with the mass below the half of 125.5 GeV, especially for the muon g-2 anomaly, 125.5 GeV Higgs signal and B s → µ + µ − . We find that the pseudoscalar can be allowed to be as low as 10 GeV, and tan β is required to be larger than 34. As the increasing of tan β, sin(β − α) is very closed to 1.0. For m A approaches to 10 GeV, the range of tan β is sizably narrowed, and m H ± is required to be larger than 250 GeV. In addition, the 125.5 GeV Higgs couplings to leptons are favored to have opposite sign to the couplings to gauge bosons and quarks, and over 30% deviation from the SM values.