Muon $g-2$ in a $U(1)$-symmetric Two-Higgs-Doublet Model

We show in this paper that, in a $U(1)$-symmetric two-Higgs-doublet model (2HDM), the two additional neutral Higgs bosons would become nearly degenerate in the large $\tan\beta$ regime, under the combined constraints from both theoretical arguments and experimental measurements. As a consequence, the excess observed in the anomalous magnetic moment of the muon could not be addressed in the considered framework, following the usual argument where these two neutral scalars are required to manifest a large mass hierarchy. On the other hand, we find that, with an $\mathcal{O}(1)$ top-Yukawa coupling and a relatively light charged Higgs boson, large contributions from the two-loop Barr-Zee type diagrams can account for the muon $g-2$ anomaly at the 1$\sigma$ level, in spite of a large cancellation between the scalar and pseudoscalar contributions. Furthermore, the same scenario can survive the tight constraints from the $B$-physics observables.


I. INTRODUCTION
There is a long-standing deviation between the standard model (SM) prediction and the experimental measurement for a µ ≡ (g − 2)/2, the anomalous magnetic moment of the muon [1][2][3][4][5]. The precision measurement of a µ has been conducted by the E821 experiment at Brookhaven National Laboratory [6], with the current world-averaged result given by [7] a exp µ = 116592091 (54) where the first error is statistical and the second systematic. The two next generation muon g − 2 experiments at Fermilab [8] and at J-PARC [9] have also been designed to reach a four times better precision. If we adopt the SM prediction from the Particle Data Group [7]: where the errors are due to the electroweak, lowest-order hadronic, and higher-order hadronic contributions, respectively, the difference between experiment and theory ∆a µ = a exp µ − a SM µ = (288 ± 80) × 10 −11 (3) would show a 3.6σ discrepancy, hinting at tantalizing new physics (NP) beyond the SM. There exist various NP scenarios to explain the muon g − 2 excess; for recent and thorough reviews, see e.g. Refs. [1][2][3][4][5]. In this paper, we will consider the two-Higgs-doublet model (2HDM) [10,11], a simple extension of the SM Higgs sector, as a prospective solution to the muon g − 2 anomaly. It has been pointed out that the anomaly can be hardly addressed in the type-II 2HDM with a small pseudoscalar Higgs boson mass around 40 GeV [12,13]. This is mainly because the charged Higgs boson mass is now restricted to be larger than 580 GeV by the branching ratio of the inclusivē B → X s γ decay [14], and the resulting mass splitting between the pseudoscalar and the charged Higgs boson would violate the electroweak precision measurements as well as the flavor physics data [15][16][17][18][19][20][21][22]. For the leptonspecific 2HDM, while it can accommodate ∆a µ at the 2σ level with the parameter space 10 M A 30 GeV,200 M H,H ± 350 GeV and 30 tan β 50, the 1σ-level fitted region [13,23] is already excluded by the lepton universality data in Z and τ decays [24,25]. In a general 2HDM with tree-level flavor-changing neutral current (FCNC) at the charged-lepton sector, the muon g − 2 anomaly can be accounted for with a sizeable µ − τ violating coupling [26][27][28]. A reasonable solution to the anomaly can also be provided in the aligned 2HDM [29][30][31][32]. Generically, however, all these scenarios mentioned require a significant mass splitting between the scalar and pseudoscalar Higgs bosons.
In our previous work [33], a general 2HDM endowed with electroweak-scale right-handed neutrinos was considered to explain the B-physics anomalies R D ( * ) [34][35][36][37][38][39][40][41][42] and R K ( * ) [43,44], as well as the neutrino mass problem. We proposed a global U (1) symmetry to induce the sub-eV neutrinos indicated by the neutrino oscillation experiments [45][46][47]. In addition, a large value of tan β, the ratio of the two Higgs vacuum expectation values, is favored in light of the R D ( * ) fits. As will be shown in Sec. II, in such a large tan β regime, the scalar and pseudoscalar Higgs bosons would become nearly degenerate, as is required by the combined constraints from both theoretical arguments and experimental measurements. As a consequence, an explanation of the muon g − 2 anomaly in our scenario cannot be realized in the usual way. However, with the particular up-quark FCNC texture specified in Refs. [33,48], we will illustrate in this paper that, in spite of the large cancellation between the scalar and pseudoscalar contributions, the 1σ range of ∆a µ can be accommodated through the two-loop Barr-Zee type diagrams [30,31,[49][50][51][52][53][54][55][56][57], in which a sizeable top-quark Yukawa coupling and a relatively light charged Higgs boson are simultaneously presented.
The paper is organized as follows. In Sec. II, we outline the framework of U (1)-symmetric 2HDM and analyze the Higgs mass spectrum. In Sec. III, we discuss the dominant contributions to the muon g − 2 from the two-loop Barr-Zee type diagrams, and then in Sec. IV the constraints from the B-physics observables are investigated. Our conclusions are finally made in Sec. V. Details of the relevant formulas are relegated to the Appendix.
In the 2HDM, the CP-conserving scalar potential with a softly broken Z 2 symmetry can be written as [11,58] where Φ 1 and Φ 2 are the two Higgs doublets, and m 2 12 and λ 5 are real parameters. We will consider the SMlike limit [58][59][60][61][62] in which the CP-even neutral scalar h mimics the SM Higgs and its couplings to gauge bosons and fermions attain the corresponding SM values, namely β − α = π/2, where α denotes the mixing angle of the two neutral scalars, and In such a limit, the parameters λ 1−5 can be expressed in terms of the physical scalar masses, tan β and m 2 12 as [58,63] which are valid for tan β 1. Here, M h,H and M A are the masses of scalar and pseudoscalar Higgs bosons, respectively, while M H ± is the charged Higgs boson mass.
As in our previous work [33], we introduce a U (1) symmetry with the charge assignment L(Φ 1 ) = 0 and L(Φ 2 ) = −1, leading to λ 5 = 0, while m 2 12 is the only soft symmetry breaking source in the Higgs sector. In this case, one can see immediately from Eq. (5) that, in the large tan β 1 regime, a large mass splitting between M H and M A would prompt a too large λ 1 , spoiling therefore the validity of perturbativity. Moreover, the perturbativity criterion of λ 3,4 constrains the mass splitting between M A and M H ± . On the other hand, the bounded-from-below conditions (see e.g. Ref. [11]) on the scalar potential require λ 1,2 0, λ 3 − √ λ 1 λ 2 and λ 3 + λ 4 − |λ 5 | − √ λ 1 λ 2 . In addition, the perturbative unitarity of the S-matrix for scalar and longitudinal vector boson scattering puts tight bounds on the Higgs boson masses [64][65][66][67][68]. Finally, the mass spectrum can also have a significant impact on the oblique parameters S, T and U [69,70].
Taking all these constraints into account, we will adopt the perturbativity criteria |λ i | π/2 [13], the perturbative unitarity bounds |a 0 i,± | 1/2, with a 0 i,± being the eigenvalues of the s-wave amplitude matrix for the elastic scattering of two-body bosonic states [24], as well as the 1σ ranges of S, T and U parameters [7], with the corresponding formulas collected in the Appendix. In Fig. 1, we show the mass regions allowed in the (M A , M A − M H ) plane by these constraints with tan β = 50, where the blue, red and black boundaries correspond to M H ± = 200, 300 and 400 GeV, respectively. One can see clearly from the figure the nearly degenerate pattern in the masses of scalar and pseudoscalar Higgs bosons. Furthermore, the mass of the charged Higgs boson is restricted by the difference |M H ± − M A,H | 100 GeV. More thorough analyses could also be found e.g. in Refs. [21,71,72].
We should emphasize that, except for the custodial symmetry that protects M A M H ± [70,73] or the twisted custodial symmetry that protects M H M H ± [70,74], there exists no any symmetry protecting M H M A . Then, for large tree-level M H and M A , the radiative self-energy corrections may generate dangerous mass splitting between M H and M A , and hence violate the constraints under discussion. To this end, we require the (pseudo)scalar masses to be relatively small, e.g., M H,A 100 GeV and M H ± 200 GeV. Such a relatively light charged Higgs boson is also motivated by a feasible explanation for the muon g − 2 excess under the tight constraints from the B-physics observables, as will be discussed in the subsequent sections.

III. MUON ANOMALOUS MAGNETIC MOMENT IN THE DEGENERATE REGIME
The U (1) symmetry was also used to constrain the Yukawa coupling textures [33]. The scalar-fermion interaction Lagrangian in our scenario is specified as whereΦ i = iτ 2 Φ * i with τ 2 being the Pauli matrix; Q L , E L , u R , d R and e R denote the left-handed quark and lepton doublets, the right-handed up-quark, down-quark, and charged-lepton singlets, respectively. Note that the right-handed neutrino singlet N R was introduced to account for the small neutrino mass and the R K ( * ) anomaly, details of which can be found in Ref. [33].
In this paper, we generalize the previously studied texture of the up-quark Yukawa coupling to where V u L,R denote the basis transformation matrices in the up-quark sector. The Yukawa texture in Eq. (7) was studied in the general 2HDM, namely the type-III 2HDM, with or without the Cheng-Sher ansatz [28,33,48,[75][76][77][78][79][80][81]. It has been pointed out that a nonzero tc can improve the discrepancy observed in R D ( * ) [33,48,77], while O(1) tt is required to explain the R K ( * ) anomaly [33]. It will be shown in this section that O(1) tt is also responsible for resolving the muon g − 2 anomaly. On the other hand, the large contributions from tt toB → X s γ and B s −B s mixing can be cancelled to a large extent, if a nonzero ct is presented at the same time [28,[78][79][80], as will be discussed in Sec. IV.
In our scenario, the two crucial elements for addressing the muon g − 2 excess are: (i) a sizeable tt and (ii) a relatively light charged Higgs boson. They provide the dominant contributions to the muon g − 2 via the typical two-loop Barr-Zee type diagrams shown in Fig. 2(a) [30,31,[49][50][51][52][53][54][55][56][57] as well as the new sets of two-loop Barr-Zee type diagrams shown in Fig. 2(b) [30]. For Fig. 2(a), the fermion loops come from the top quark and the τ lepton, while the bottom-quark loop has a suppression factor 1/ tan β in the large tan β regime and there is no neutrino loop. Note that the amplitude with the photon propagator replaced by that of the Z boson is suppressed by a factor −1/4 + sin 2 θ W ≈ −0.02 [51]. As for the new sets of two-loop Barr-Zee type diagrams discussed in Ref. [30], we would like to make the following remarks. Due to the assumption of CP conservation in the scalar potential, there are neither AH ± H ∓ nor AW ± W ∓ couplings. Then, contributions from all the diagrams involving these vertices shown in Fig. 2 would vanish. Meanwhile, the contribution involving the HH ± H ∓ vertex is suppressed in the large tan β regime, because the HH ± H ∓ coupling is proportional to 1/ tan β (we have confirmed this point with the code SARAH [82]). Finally, due to the absence of the HW ± W ∓ coupling in the SM-like limit, there exist no contributions from the diagrams involving this vertex either. Therefore, in our scenario, the diagrams shown in Fig. 2(b) with the topbottom (bottom-top) loop should give the dominant contributions among all these two-loop Barr-Zee type diagrams. We should also mention that, although the heavy neutrino with mass around the electroweak scale is introduced in our framework, we need not consider its contribution coming from Fig. 2(b) with the light neutrino propagator replaced by that of the heavy neutrino N , because the O(1) H ± N µ ∓ coupling considered in Ref. [33] is compensated by the small W ± N µ ∓ one [83][84][85][86].
Based on these observations, we will only consider the two-loop Barr-Zee type diagrams shown in Fig. 2. The corresponding amplitudes as well as the one-loop contributions generated by the H, A and H ± propagators are collected in the Appendix. In the numerical analysis, we fix tan β = 50 and M H ± = M H,A + 100 GeV which is allowed by the mass spectrum shown in Fig. 1. In addition, the heavy neutrino mass is fixed at 200 GeV and an O(1) neutrino Yukawa coupling is adopted [33] (see the Appendix). The total NP contributions to the muon g−2 are shown in Fig. 3, where the dependence of ∆a µ on the  It is worth mentioning that the recent lattice QCD calculations [87,88] of the leading-order hadronic vacuum polarization contribution to the muon g −2 obtained consistent results that are larger than the previous ones (see e.g. Ref. [89] and references therein). If these results are confirmed, the discrepancy between the SM prediction and the experimental measurement will be diluted. This in turn indicates that, if NP really exists as the one considered here, it will be unnecessary to invoke large tt and small M H ± to account for the muon g − 2 excess, making therefore the scenario considered much easier to avoid the tight constraints from the B-physics observables.

IV. CONSTRAINTS FROM B-PHYSICS OBSERVABLES
In this section, we analyze the tight constraints from the B-physics observables, concentrating on the branching ratio of the inclusiveB → X s γ decay and the mass difference in the B s −B s mixing, as the sizeable top-Yukawa coupling tt and the relatively small charged Higgs boson mass give large contributions to these observables [28,[78][79][80].

A.B → Xsγ
The low-energy effective Hamiltonian for the inclusivē B → X s γ decay is given by where the current-current operators O 1,2 and the QCDpenguin operators O 3−6 could be found e.g. in Ref. [90], while the dipole operators O 7γ and O 8g are defined, respectively, as Note that the primed operators O 7γ,8g that are obtained from Eq. (9) by replacing P R with P L need not be included in Eq. (8), because the Wilson coefficients of these operators are suppressed by m s /m b relative to those coming from O 7γ,8g in the SM, and are zero in our scenario due to the absence of FCNCs in the down-quark sector.
Following Ref. [91], the branching ratio of the inclusivē B → X s γ decay can be expressed as where R is an overall factor and determined to be R = 2.47 × 10 −3 [92,93], while N (E γ ) denotes the non-perturbative correction, with N (E γ ) = (3.6 ± 0.6) × 10 −3 [92] for a photon-energy cut off E γ > 1.6 GeV in theB-meson rest frame. The Wilson coefficient C 7γ (µ b ) can be decomposed into the sum of the SM and the NP contributions: For the SM part, we adopt the result at the next-tonext-to leading order in QCD which gives B SM sγ = (3.36 ± 0.23) × 10 −4 for E γ > 1.6 GeV [94,95], while for the NP contribution, we use the leading order result at the scale where the Wilson coefficients C NP 7γ (µ H ) and C NP 8g (µ H ) at the initial scale µ H = O(M H ± ) are collected in the Appendix. For the numerical study, we take the magic numbers κ 7 = 0.524 and κ 8 = 0.118 evaluated at µ H = 200 GeV and µ b = 2.5 GeV [91].

B. Bs −Bs mixing
Adopting the overall normalization of the SM contribution, the low-energy effective Hamiltonian for B s −B s mixing can be written as where the four-quark operators could be found e.g. in Refs. [96,97]. In our scenario, only the Wilson coefficient C 1 of the operator receives a significant NP contribution, while the NP contributions to the other Wilson coefficients are either absent as no FCNCs occur in the down-quark sector or suppressed by 1/ tan β in the large tan β regime. This can also be understood e.g. from Refs. [79,80]. A prominent constraint from the B s −B s mixing is due to the mass difference between the two mass eigenstates of the neutral B s mesons which is given as ∆M s = 2|M s 12 |, where M s 12 is the off-diagonal element in the B smeson mass matrix, with the known SM result given by [98][99][100] M s,SM where M Bs and f Bs are the B s -meson mass and decay constant, respectively. The factorη B encodes the shortdistance QCD corrections [101], while the bag parameter B [102], together with the decay constant f Bs , parameterizes all the long-distance QCD effect contained in the hadronic matrix element B s |O 1 |B s . Note that a different convention for the QCD correction and the bag parameter is also used in the literature through the relationη B B ≡ η BB (see e.g. Ref. [103] and the relevant discussions in Refs. [100,104]). The Inami-Lim function [105] is calculated from the W -boson box diagrams with two internal top-quark exchanges, the value of which is obtained with the central value of the top-quark MS mass m t (m t ) [106].
Normalizing to the SM contribution, we can express ∆M s in the presence of NP contributions as where ∆M SM s = (18.3 ± 2.7) ps −1 [99] and ∆ NP It should be mentioned that, in obtaining Eq. (16), we have assumed that the renormalization group running effect on the NP contributions [96,97] from the initial scale µ H down to the low scale µ b is identical to that in the SM part (which is encoded in the factorη B ). This approximation is quite reasonable, because the charged Higgs boson mass is considered to be around the electroweak scale, being not far from the top-quark mass.

C. Combined constraints from B(B → Xsγ) and ∆Ms
We now proceed to analyze the combined constraints from the branching ratio B(B → X s γ) and the mass difference ∆M s . To this end, we confront our theoretical predictions with the 1σ ranges of the world-averaged results compiled by the Heavy Flavor Averaging Group for these two observables [107]: B exp sγ = (3.32 ± 0.15) × 10 −4 for E γ > 1.6 GeV and ∆M exp s = (17.757 ± 0.021) ps −1 . In Fig. 4, by fixing tc = 0.1 in light of the combined fits for R D ( * ) and B − c → τ −ν [33,77], we show the allowed parameter space in the ( tt , ct ) plane, with M H ± = 200, 300 and 400 GeV, corresponding to the blue, red and black boundaries, respectively. We can see clearly from this figure that it is possible to allow tt = −0.75 and M H ± = 200 GeV even with a tightly constrained ct : −0.04 ct −0.06. It is therefore concluded that the parameter region with large tt and small M H ± required by the muon g − 2 excess can still be reached under the tight constraints from these two observables.  different global U (1) charges, the two additional neutral Higgs bosons would become nearly degenerate in the large tan β regime, due to the constraints from both theoretical arguments and experimental measurements. As a result, it is impossible to address the muon g −2 anomaly in this framework, because a significant mass splitting between the scalar and pseudoscalar Higgs bosons, as is generally required in the usual 2HDMs, cannot be realized. However, with an O(1) top-Yukawa coupling and a relatively light charged Higgs boson mass ( 200 GeV), we found that there exist large contributions to the muon g−2 arising from the two-loop Barr-Zee type diagrams, in spite of a large cancellation between the scalar and pseudoscalar sectors, providing therefore an explanation for the muon g − 2 excess at the 1σ level. At the same time, the required top-Yukawa coupling and charged Higgs boson mass can also survive the tight constraints from the branching ratio of the inclusiveB → X s γ decay and the mass difference in the B s −B s mixing.

APPENDIX
In this Appendix, we collect the relevant expressions for the oblique parameters S, T and U , the well-known one-loop and the two-loop Barr-Zee type contributions to the muon g −2, as well as the relevant Wilson coefficients in the inclusiveB → X s γ decay and the B s −B s mixing, in the framework of U (1)-symmetric 2HDM.

A. Oblique parameters
Following Refs. [69,70], the oblique parameters S, T and U in the SM-like limit are given, respectively, by with where B. Formulas for muon g − 2 The amplitude for the typical two-loop Barr-Zee type diagrams shown in Fig. 2(a) is given by [30,31,[49][50][51][52][53][54][55][56][57] ∆a (a) µ = with where G F is the Fermi constant and α the fine-structure constant; r i f = m 2 f /M 2 i with i = H, A, and m f , Q f and N f c are the mass, the electric charge (in unit of the elementary charge), and the number of colors for fermion f = t, τ , respectively; N H (x) = 2x(1 − x) − 1 and N A (x) = 1. In our scenario, the fermion couplings of neutral Higgs bosons are given by y H µ,τ = −y A µ,τ = tan β and y H t = y A t = v m t tt . The amplitude for the new sets of two-loop Barr-Zee type diagrams shown in Fig. 2(b) is given by [30,48] from the one-loop penguin diagram with charged Higgs boson exchange. They are given as [28,[78][79][80] C NP 7γ,8g (µ H ) = where y t,c ≡ m 2 t,c /M 2 H ± , and the scalar functions E 7,8 are defined by E 7 (y) = y 18y 2 − 12y ln y − 8y 3 + 3y 2 + 12y − 7 72(y − 1) 4 , E 8 (y) = y −6y ln y − y 3 + 6y 2 − 3y − 2 24(y − 1) 4 , which can be found e.g. in Refs. [28,[78][79][80]90]. For the B s −B s mixing, in our scenario, the dominant NP contribution to the mass difference ∆M s comes from the one-loop box diagrams involving the charged Higgs boson exchange, and only the Wilson coefficient C 1 is significantly affected (see the text). The total NP contributions to C 1 at the initial scale µ H = O(M H ± ) can be written as The part of C NP 1 (µ H ) from the pure charged Higgs boson box diagrams can be written as