Muon g$-$2 in two-Higgs-doublet models

Updating various theoretical and experimental constraints on the four different types of two-Higgs-doublet models (2HDMs), we find that only the ``lepton-specific"(or ``type X") 2HDM can explain the present muon g$-$2 anomaly in the parameter region of large $\tan\beta$, a light CP-odd boson, and heavier CP-even and charged bosons which are almost degenerate. The severe constraints on the models come mainly from the consideration of vacuum stability and perturbativity, the electroweak precision data, the $b$-quark observables like $B_S \to \mu\mu$, the precision measurements of the lepton universality as well as the 125 GeV boson property observed at the LHC.

Since the first measurement of the muon anomalous magnetic moment a µ = (g − 2) µ /2 by the E821 experiment at BNL in 2001 [1], much progress has been made in both experimental and theoretical sides to reduce the uncertainties by a factor of two or so establishing a consistent 3σ discrepancy ∆a µ ≡ a EXP µ − a SM µ = +262 (85) × 10 −11 (1) which is in a good agreement with the different group's determinations. Followed by the 2001 announcement, there have been quite a few studies in the context of 2HDMs [2][3][4], however, restricted mainly to the type I and II models out of four different types of 2HDMs ensuring natural flavour conservation. Considering the recent experimental development confirming more precisely the Standard Model (SM) predictions, including the discovery of the 125 GeV Brout-Egnlert-Higgs boson, it would be timely to revisit the issue of the muon g-2 in 2HDMs. An additional contribution to th muon g-2 from an extra boson in 2HDMs, shown in Fig. 1, may be the origin of the positive excess in the ∆a µ . This can happen in the type II or X (lepton-specific) 2HDM which allows a light boson having large Yukawa couplings enhanced by tan β. While the type II option is completely ruled out by now, the type X model [5] remains an unique option to explain the a µ anomaly evading all the recent experimental constraints [6][7][8][9]. The previous studies on the muon g-2 in the type II 2HDM (2HDM-II) and various experimental constraints were nicely summarised in Ref. [3]: • The one-loop correction mediated by a light CP-even (CP-odd and charged) boson gives a positive (negative) contribution to ∆a µ , and thus the current 3σ deviation can be explained by a CP-even boson lighter than about 5 GeV. However, such a light boson was already in contradiction to non-observation of radiative Υ decays Υ → γ + X.
• Contrary to the one-loop correction, the major two-loop contribution from the Barr-Zee diagram [10] mediated by a light CP odd (even) boson can give a sizable positive (negative) contribution to ∆a µ . Thus, a light CP odd boson A with large tan β can account for the muon g-2 deviation. However, most of the muon g-2 favoured region in the lower and upper M A part are excluded by the LEP and TEVATRON search on Z → bbA(bb), respectively, except a small gap of M A ≈ 25 − 70 GeV with tan β 20.
• However, let us note that such a light A gives a huge contribution to B s → µ + µ − as its rate is proportional to tan 4 β/M 4 A [11], and thus the above gap is completely closed now by the LHC measurement which is consistent with the SM prediction [12]. Now the situation can be quite different in the type X 2HDM (2HDM-X) where all the extra boson couplings to quarks (leptons) are proportional to cot β (tan β). Due to this, the 2HDM-X becomes hadrophobic in the large tan β limit invalidating most hadron-related constraints applied to the type II model. On the other hand, its leptophilic property brings sever constraints from the precision leptonic observables. The key features in confronting the type X model with the muon g-2 anomaly can be summarized as follows [6][7][8][9].
• As in the 2HDM-II, the one-loop correction with a light CP even boson H can account for the muon g-2 excess.
While the Upsilon decay suppressed by 1/ tan 2 β cannot provide a meaningful constraint, the Belle and LHCb searches for B → Kµµ shut down the muon g-2 favoured region except tiny gaps around M H ≈ 3 and 4 GeV [13]. In any case, such a light Higgs boson is excluded by the current measurement of • The Barr-Zee diagram with the tau lepton in the loop can account for the muon g-2 anomaly again in a parameter region of small M A and large tan β evading all the constraints from the hardron colliders and the b-quark observables [6] except the process B s → µ + µ − which rules out M A 10 GeV [7].
• However, the lepton universality test by HFAG [14] combined with the Z → τ τ decay turns out to limit severely the muon g-2 favoured region of the type X model [8] allowing (only at 2σ) a small region below M A ≈ 80 GeV and tan β ≈ 60 [9].
• With such a light A, the exotic decay of the 125 GeV boson h → AA or AA * (τ τ ) becomes generically too large unless a certain cancelation is arranged to suppress the hAA coupling λ hAA which turns out to be possible only in the wrong-sign limit of the lepton Yukawa coupling [7].

II. FOUR TYPES OF 2HDMS
Non-observation of flavour changing neutral currents restricts 2HDMs to four different classes which differ by how the Higgs doublets couple to fermions [16]. They are organized by a discrete symmetry Z 2 under which different Higgs doublets and fermions carry different parities. These models are labeled as type I, II, "lepton-specific" (or X) and "flipped" (or Y). Having two Higgs doublets Φ 1,2 , the most general Z 2 symmetric scalar potential takes the form: where a (soft) Z 2 breaking term m 2 12 is introduced. Minimization of the scalar potential determines the vacuum expectation values Φ 0 1,2 ≡ v 1,2 / √ 2 around which the Higgs doublet fields are expanded as The model contains the five physical fields in mass eigenstates denoted by H ± , A, H and h. Assuming negligible CP violation, H ± and A are given by where the angle β is determined from t β ≡ tan β = v 2 /v 1 , and their orthogonal combinations are the corresponding Goldstone modes G ±,0 . The neutral CP-even bosons are diagonalized as where h (H) denotes the lighter (heavier) state. The gauge couplings of h and H are given schematically by where V = W ± or Z. Taking h as the 125 GeV boson of the SM, the SM limit corresponds to s β−α → 1. Indeed, LHC finds, c β−α 1 in all the 2HDMs confirming the SM-like property of the 125 GeV boson [18]. Normalizing the Yukawa couplings of the neutral bosons to a fermion f by GeV, we have the following Yukawa terms: where the normalized Yukawa couplings y h,H,A f are summarized in Table I for each of these four types of 2HDMs. Let us now recall that the tau Yukawa coupling y τ ≡ y h l in Type X (also y b ≡ y h d in Type II) can be expressed as which allows us to have the wrong-sign limit y τ ∼ −1 compatible with the LHC data [15] if c β−α ∼ 2/t β for large t β ≡ tan β favoured by the muon g−2. Later we will see that a cancellation in λ hAA can be arranged only for y h τ < −1 to suppress the h → AA decay.

III. THE MUON g−2 FROM A LIGHT CP-ODD BOSON
Considering all the updated SM calculations of the muon g−2, we obtain a SM µ = 116591829 (57) × 10 −11 (9) comparing it with the experimental value a EXP µ = 116592091 (63) × 10 −11 , one finds a deviation at 3.1σ: ∆a µ ≡ a EXP µ − a SM µ = +262 (85) × 10 −11 . In the 2HDM, the one-loop contributions to a µ of the neutral and charged bosons are These formula show that the one-loop contributions to a µ are positive for the neutral scalars h and H, and negative for the pseudo-scalar and charged bosons A and H ± (for M H ± > m µ ). In the limit r 1, f A (r) = + ln r + 11/6 + O(r), showing that in this limit f H ± (r) is suppressed with respect to f h,H,A (r). Now the two-loop Barr-Zee type diagrams with effective hγγ, Hγγ or Aγγ vertices generated by the exchange of heavy fermions gives Note the enhancement factor m 2 f /m 2 µ of the two-loop formula in Eq. (17) relative to the one-loop contribution in Eq. (10), which can overcome the additional loop suppression factor α/π, and makes the two-loop contributions may become larger than the one-loop ones. Moreover, the signs of the two-loop functions g h,H (negative) and Allowing such a light CP-odd boson, there could be a strong limit on the extra boson masses coming from the electroweak precision test. To see this, we compare the theoretical 2HDMs predictions for M W and sin 2 θ lept eff with their present experimental values via a combined χ 2 analysis. These quantities can be computed perturbatively by means of the following relations where sin 2 θ W = 1 − M 2 W /M 2 Z , and k l (q 2 ) = 1 + ∆k l (q 2 ) is the real part of the vertex form factor Z → ll evaluated at q 2 = M 2 Z . We than use the following experimental values: The results of our analysis are displayed in Fig. 3 confirming a custodial symmetry limit [17] Taking λ 1 as a free parameter, one can have the following expressions for the other couplings in the large t β limit: where we have used the relation (8) neglecting the terms of O(1/t 2 β ). In the right-sign (RS) limit of the lepton (tau, in particular), y τ s β−α → +1, one finds a strong upper limit of [6] On the other hand, in the wrong-sing (WS) limit, y τ s β−α → −1, the heavy boson masses up to the perturbativity limit, can be obtained.
Let us finally remark that the hAA coupling is generically order one and thus can leads to a sizable non-standard decay of h → AA or AA * (τ τ ) if allowed kinematically. Then, one needs to have |λ hAA /v| 1 to avoid an exotic decay of the SM boson. Noting that where we have put s 2 β−α = 1. In the RS or SM limit, the condition λ hAA ≈ 0 can be met for a rather light H with M 2 H ≈ 1 2 M 2 h + M 2 A which is disfavoured in the explanation of the muon g-2. On the other hand, one can arrange a cancellation for λ hAA ≈ 0 in the wrong-sign limit for arbitrary value of M H if the tau Yukawa coupling satisfies

VI. LEPTON UNIVERSALITY TESTS
In the limit of large tan β, the charged boson can generate significant corrections to τ decays at the tree level and furthermore the extra Higg boson contribution to one-loop corrections can also be significant [19]. The recent study [8] showed that a stringent bound on the charged boson contributions can be obtained from the lepton universality condition obtained by the HFAG collaboration [14]. Given the precision at the level of 0.1 %, the lepton universality data put the strongest bound on the type X 2HDM parameter space in favor of the muon g-2. Thus, let us now make a proper analysis of the HFAG data.
From the measurements of the pure leptonic processes, τ → µνν, τ → eνν and µ → eνν, HFAG obtained the constraints on the three coupling ratios, (g τ /g µ ) = Γ(τ → eνν)/Γ(µ → eνν), etc. Defining δ ll ≡ (g l /g l ) − 1, let us rewrite the data from the leptonic processes: δ l τ µ = 0.0011 ± 0.0015, δ l τ e = 0.0029 ± 0.0015, δ l µe = 0.0018 ± 0.0014 (33) In addition, combing the semi-hadronic processes π/K → µν, HFAG also provided the averaged constraint on (g τ /g µ ) which is translated into It is important to notice that only two ratios out of the three leptonic measurements are independent and thus the three data (33) are strongly correlated. For a consistent treatment of the data, one combination out of the three has to be projected out. One can indeed check that the direction δ l τ µ − δ l τ e + δ l µe has the zero best-fit value and the zero eigenvalue of the covariance matrix, and thus corresponds to the unphysical direction. Furthermore, two orthogonal directions δ l τ µ + δ l τ e and −δ l τ µ + δ l τ e + 2δ l µe are found to be uncorrelated in a good approximation. As a result, the 2HDM contribution to δ ll are calculated to be Here δ tree and δ loop are given by [19]: , and x φ = m 2 φ /m 2 H ± . From Eqs. (33), (34) and (35), one obtains the following three independent bounds: δ loop = 0.0001 ± 0.0014.
We will use these constraints to put a strong limit on the (g − 2) µ favoured region in the M A -tan β plane in the next section. Let us recall that the Z → τ τ data, although less strong than the HFAG data, provides an independent bound [8] which further cuts out some corner of parameter space.

VII. PINNING DOWN THE WHOLE 2HDM-X PARAMETER SPACE
It is an interesting task to narrow down the allowed region of the type X 2HDM parameter space collecting all the relevant experimental data including those outlined in Section I and the 125 GeV boson data from LHC, in particular. The scan ranges of all the 2HDM-X input parameters are listed in Table II. For our scan, we adopt the convention −π/2 < α − β < π/2 and 0 < β < π/2, and use the parameter λ 1 as an input parameter instead of m 2 12 .

2HDM parameter Range
Scalar boson mass (GeV) 125 < mH < 400 Pseudoscalar boson mass (GeV) 10 < mA < 400 Charged boson mass (GeV) 94 < m H ± < 400  Fig. 4 shows the allowed region in the m A -tan β plane from the profile-likelihood study taking all the other 2HDM-X parameters as nuisance parameters. To see the impact of the lepton universality data by HFAG, we overlay the contour lines of the lepton universality likelihood at the 99%, 95% and 90% confidence level based on the constraints (37). The allowed region opened up for tan β > 140 needs a comment. Note that the δ loop is always negative while δ tree becomes positive for larger tan β/m H ± . Thus, there appears a fine-tuned region around tan β/m H ± ∼ 1 GeV where the positive δ tree and the negative δ loop cancel each other to give a good fit. However, such regions are excluded by the Z → τ τ data [8] and thus we are left with the tightly limited region of M A ≈ 10 − 80 GeV and tan β ≈ 25 − 60 at the 95% confidence level. The region allowed in Fig. 4 can be either in the right-sign (y τ ≡ ξ l h > 0) or wrong-sign (y τ ≡ ξ l h < 0) domain as shown in the left and middle panels of Fig. 5. One can see that the rigt-sign limit is tightly constrained to a small region of m A ≈ 60 − 80 GeV while the wrong-sign limit is favoured in a wider range of parameter space. The right panel shows the sizes of the coupling λ hAA restricted by the LHC data on the exotic decay of the 125 GeV boson, putting a generous bound of Br(h → AA ( * ) ) < 40%. As explained before, the suppressed value of λ hAA for m A m h /2 is shown to appear only in the wrong-sign domain.
Given the possible existence of a light CP-odd boson explaining the muon g-2 in the type X 2HDM, it would be important to look for its trails at the LHC. Fig. 6 shows the allowed mass ranges of all the extra bosons. Region A following the pattern of m A m H ≈ m H ± is favoured while Region B with m A ≈ m H ± m H is already excluded as discussed before.

VIII. TAU-RICH SIGANTURES AT THE LHC
The bulk parameter space with m A m H ∼ m H ± is a clear prediction of the type X 2HDM as the origin of the muon g-2 anomaly. Since the extra bosons are mainly from the "leptonic" Higgs doublet with large tan β, all the three members are expected to dominantly decay into the τ −flavor, leading to τ −rich signatures at the LHC via the following production and ensuing cascade decay chains: To probe Region A, we select six benchmark points with different combinations of m A and m H presented in Table III.
For each point, we take a simple parametrization of tan β = 1.25(m A /GeV) + 25 and m H ± = m H + 15GeV. Note that we included the points with m A > 80 GeV for the sake of the LHC study although they are forbidden by the lepton universality tests. In Table III, we show the production cross-section, the selection cuts and the significance for each benchmark expected for the integrated luminosity of 25/fb at the 14 TeV LHC.  In Fig. 7, we present the exclusion region coming mainly from the chargino-neutralino search at the LHC8, and the expected discovery reaches at LHC14 with the integrated luminosity of 25/fb. A heavy CP-even boson with m H > 200 GeV and a light CP-odd boson with m A < 50 GeV are still allowed, and the LHC14 can explore some of the regions. The sensitivities are weaker for larger m H just because of smaller cross sections, and for smaller m A because τ s from lighter A become softer and thus the acceptance quickly decreases. Moreover, the H/H ± → AZ/W ± decay modes also start open to decrease the number of hard τ s from direct H/H ± decays. In such a region, a light A from heavy H + /H decay will be boosted, resulting in a collimated τ −pair which becomes difficult to be tagged as two separated τ -jets. It is one of the reasons to have less acceptance for this parameter region. We can estimate the separation R τ τ of the τ leptons from A decay: Since the jets are usually defined with R = 0.5, the τ −pair starts overlapping. We indicated the region with the overlapping τ problem in red lines in the right panel of Fig. 7. Further studies on how to capture the kinematic features of the boosted A → τ + τ − are required to probe such a small m A region.

IX. SUMMARY
The type X 2HDM is still a viable option for the explanation of the muon g-2 in the parameter region with large tan β and a light CP-odd boson A. Being "hadrophobic and leptophilic" in the large tan β limit, it can be easily free from all the hadron-related constraints, particularly, coming from the decay B s → µµ which puts only a mild bound of m A 10 GeV. However, such a region is tightly limited by the lepton universality tests from the HFAG and Z → τ τ data. Combining all the current bounds, we find allowed at the 95% confidence level a limited region of tan β ≈ 15 − 60 and m A ≈ 10 − 80 GeV with m H ≈ m H ± m A . It will be an interesting task to search for such a light CP-odd boson A and the extra heavy bosons H, H ± in the next run of the LHC mainly through pp → AH, AH ± followed by the decays H ± → τ ± ν and A, H → τ + τ − which requires further studies to improve the (boosted) tau identification.