Muon $g-2$ in two Higgs doublet models with vectorlike leptons

We calculate contributions to the anomalous magnetic moment of the muon from heavy neutral and charged Higgs bosons and new leptons in two Higgs doublet models extended by vectorlike leptons. We present detailed predictions of two models with type-II couplings to standard model fermions, motivated by a $Z_2$ symmetry and supersymmetry. In addition, we compare the results with the standard model extended by vectorlike leptons. We find that the model motivated by a $Z_{2}$ symmetry can generate much larger contributions to the magnetic moment compared to the standard model, even by two orders of magnitude due to $\tan^{2}\beta$ enhancement, while satisfying current constraints. As a consequence, the standard model explanation of the anomaly requires much larger corrections to muon couplings making this model easier to probe at future precision machines. Additionally, we find that the model with couplings motivated by supersymmetry typically leads to much smaller contributions to the anomaly as a result of cancellations. However, we identify interesting scenarios where contributions from the charged Higgs boson can fully explain the anomaly.


I. INTRODUCTION
The Standard Model (SM) provides a spectacular description of nature, surviving stringent tests at both the current energy and precision frontiers. Indeed the absence of any direct signal for new particles at the LHC implies strong bounds for many kinds of new particles up to several TeV. Further, the discovery of the Higgs boson and the subsequent measurements of the Higgs couplings to gauge bosons and fermions indicate that the SM is the appropriate effective theory of electroweak (EW) symmetry breaking.
Despite the lack of any direct clue for new particles, some discrepancies with SM predictions still persist evoking a variety of models for new physics whose low-energy effects could be probed indirectly. In particular, the measurement of the magnetic moment of the muon deviates from the SM prediction by more than four standard deviations [1][2][3]. Examples of models which may lead to an explanation of this discrepancy with particles at or slightly above EW include possible new fermions, scalars, gauge bosons, or combinations of new particles, e.g. in the MSSM. For detailed reviews see [4][5][6][7][8] and references therein. Naively, new particles which can account for the anomalous magnetic moment cannot be too far above the EW scale, since the typical contribution from new particles can be parameterized by ∆a µ g 2 N P m 2 µ /16π 2 m 2 N P , where g N P and m N P are the coupling and mass of new particles. In some cases, certain enhancements can allow for heavier particles. For instance in the MSSM, the contribution can be enhanced by tan β [9]. Alternative explanations involve very light particles which, to avoid a variety of constraints, must be singlets under the SM [10][11][12][13][14][15][16].
In models with new fermions which have the same quantum numbers as SM leptons, the contributions to (g − 2) µ associated to new physics are proportional to the mixing parameter, m LE µ , which simultaneously contributes to the muon mass. The contribution to (g − 2) µ can be estimated by ∆a µ m µ λ 3 v/16π 2 m 2 N P m µ m LE µ /16π 2 v 2 [17,18]. In this case chirality flipping operators lead to a chiral enhancement, λv/m µ , compared to the typical contribution. Chiral enhancement effects related to (g − 2) µ are additionally motivated by connections with recent B anomalies [19][20][21][22][23], the Cabibbo angle anomaly [24,25], and dark matter [26][27][28].
In this paper, we focus mainly on type-II 2HDM models with vectorlike leptons as an explanation for the anomalous measurement of (g − 2) µ . We study in detail a type-II 2HDM motivated by a Z 2 symmetry (2HDM-II-Z 2 ), highlights of which were presented previously in [29]. A striking feature of this scenario is found in a tan 2 β enhancement in the contributions of heavy Higgses to (g − 2) µ compared to those of W, Z, and h in addition to the chiral enhancement expected in models with VL. In contrast, the same couplings which generate a large correction to (g − 2) µ also lead to corrections of W, Z, and h couplings to the muon resulting from mixing that are tan 2 β suppressed. This would allow for a contribution to ∆a µ even two orders of magnitude larger than the measured value while simultaneously satisfying low-energy observables, or an explanation of the measured value, ∆a exp µ , with tiny corrections to SM couplings, or even an explanation of ∆a exp µ from new leptons with masses of tens of TeV. Interestingly, future precision measurements can fully explore scenarios with heavy new leptons indirectly [29]. In addition, a muon collider would be perfectly suited to explore heavy lepton masses directly [30][31][32].
We also discuss a version of the 2HDM motivated by supersymmetry (2HDM-II-S).
In particular, we call attention to the fact that in either model, the couplings of the Higgs doublets to SM leptons are indistinguishable. However, when the models are extended with VL each symmetry dictates a different structure of Yukawa couplings leading to drastically different results. In this version of the model we find that the contributions to ∆a µ from vectorlike leptons and heavy Higgses with comparable masses tend to cancel with those of W, Z, and h. Viable explanations of ∆a exp µ can be achieved either by decoupled heavy Higgses or from the charged Higgs contribution if vectorlike neutral singlets are included. Furthermore, we extend previous studies of the SM extended with vectorlike leptons [17,18]. In particular, we include couplings to vectorlike neutral singlets (also considered previously in [33]), and extend the range of possible couplings and masses that can explain ∆a exp µ . In addition, we impose updated experimental constraints emphasizing the impact of recent measurements of the SM Higgs coupling to the muon [34]. It has been noted that the correlation of the Higgs coupling to the muon with other observables can often give complementary information on models for new physics [17,18,29,35].
Interestingly, we find that this constraint limits the possible contribution to ∆a µ in the SM with vectorlike leptons close to the current central value, while in the 2HDM-II-Z 2 it allows for even two orders of magnitude larger contribution to ∆a µ than the measured value. However, the 1σ range of ∆a exp µ can be explained with a similar range of heavy lepton masses as in the 2HDM-II-Z 2 . To illustrate the impact of future precision measurements, we study possible modifications of W, Z, and h couplings to the muon.
In our discussion we focus on scenarios where vectorlike leptons share analogous quantum numbers to SM leptons. This allows for straightforward extensions of the SM by complete vectorlike families in the context of simple unified models. The extension of the SM with vectorlike familes provides a possible explanation for the observed hierarchy of gauge couplings [36,37], while the MSSM with a complete vectorlike family can explain the structure of the seven largest couplings in the SM at the EW scale when all new particles are in the multi-TeV range [38][39][40]. Vectorlike quarks around the same scales can also lead to more natural EW symmetry breaking [41,42]. For other examples of explanations of ∆a exp µ with vectorlike leptons either with the same or different quantum numbers, see also Refs. [19,20,23,24,26,[43][44][45][46][47][48][49]. 1 Related studies of (g − 2) µ in the MSSM with vectorlike leptons (not including one-loop contributions from heavy Higgses) were presented in [50][51][52]. For previous studies of supersymmetric models with vectorlike leptons, see also [53,54]. Related discussions of collider searches for heavy new leptons can be found in [55][56][57][58][59][60] and similar studies with vectorlike quarks 1 In particular, similar 2HDM variants with VL have been explored recently in [47,48]. We find disagreement with the results in [48] in connection with (g−2) e , where the neutral Higgs contributions are incomplete. Further, the authors claim that the charged Higgs contribution does not have any chiral enhancement, which we do not find to be correct. In [47] the authors do not consider chirally-enhanced one-loop contributions and rather solely consider two-loop Barr-Zee contributions to (g − 2) µ . However, we find that these contributions are negligible compared to chirally-enhanced one-loop contributions by several orders of magnitude.
This paper is organized as follows. In section II, we describe the 2HDM-II-Z 2 , 2HDM-II-S, and SM extended with vectorlike leptons which mix with the muon at tree level. In section III, we present formulae for contributions to (g − 2) µ in models with extended Higgs and lepton sectors that can be applied to any model. We discuss details of our analysis and a variety of constraints relevant to heavy leptons and Higgs bosons in section IV. We present detailed results and discussion for all three models in section V and conclude in section VI. In Appendix A, we provide general formulas for couplings of the muon to Z, W , and Higgs bosons in the 2HDM-II-Z 2 and provide an explicit derivation of the Goldstone boson equivalence theorem for couplings of the Z and W boson. In addition we list useful approximate formulas which aid in understanding of the results. We provide details of the 2HDM-II-S in Appendix B. We comment on the relative size of possible Barr-Zee contributions in Appendix C.

II. MODELS
We consider a two Higgs doublet model extended with vector-like leptons (VL) in which both SU (2) doublet, L L,R , and singlet representations, E L,R and N L,R , are included. We assume that the left-handed new doublet, L L , transforms under the same representations as the left-handed SM leptons. Likewise, the right-handed charged singlet, E R , has the same quantum numbers as the right-handed SM leptons. Further, we assume couplings of SM leptons to the Higgs doublets as in type-II models where H d couples exclusively to the down-sector leptons and H u to the up-sector. This can be achieved by assigning appropriate charges under a Z 2 symmetry. Alternatively, the supersymmetric extension of the SM automatically leads to couplings of SM fermions of type-II [64]. However, when VL are included the Z 2 symmetry and supersymmetry enforce different structures of their Yukawa couplings to the Higgs doublets, and thus we distinguish the two models. We will also compare these models with the SM extended with VL. In all cases, the leading contributions of the model to (g − 2) µ originate from After electroweak symmetry breaking, the electric charge is given by possible mixing of VL leptons to the 2nd generation SM leptons. Thus, for simplicity we will consider only Yukawa couplings leading to mixing of VL leptons to the muon and muon neutrino.
A. 2HDM-II-Z 2 with vectorlike leptons For the main focus of this paper, we consider the type-II two Higgs doublet model motivated by Z 2 symmetry. The quantum numbers of SM leptons, Higgs doublets, and vector-like fields are summarized in Table I. A similar model with vector-like quark doublets and singlets was considered in [61]. While the phenomenology related to vector-like quarks will not be pertinent in this paper, generalizing the model to a 2HDM with a complete VL family is straightforward.
In the basis where the SM lepton Yukawa couplings are diagonal, the most general lagrangian of Yukawa couplings and VL masses under these assumptions is given by where the doublet components are labeled as In the process of electroweak symmetry breaking the neutral components of the Higgs GeV, and we define tan β = v u /v d . Additionally, the charged lepton mass matrix becomes Similarly, for the neutral leptons we obtain where for convenience we have inserted ν R = 0 to present the mass matrix in 3 × 3 form. The mass matrices can be diagonalized by bi-unitary transformations to obtain lepton mass eigenstates. We label new charged leptons as e 4 and e 5 , and neutral leptons as ν 4 and ν 5 . The mixing of VL to the 2nd generation will induce modifications of the muon couplings to gauge and Higgs bosons, leading in particular to flavor non-diagonal lepton couplings. Details of all couplings in the mass eigenstate basis, as well as approximate formulas for individual couplings in the limit of heavy VL masses are given in the Appendix A.

B. 2HDM-II-S with vectorlike leptons
Another well motivated 2HDM-type scenario is the MSSM extended with vectorlike leptons. We do not consider contributions from superpartners which depend on further assumptions about the SUSY-breaking sector. These could be simply added to the contributions from heavy Higgses and VL. Alternatively, our results are complete in the limit of heavy superpartners such that the relevant low-energy particle content of the model is the same as the 2HDM-II-Z 2 . Despite the same particle content, slight differences in the structure of Yukawa couplings will lead to very different results in this case. In the supersymmetric version of the model (2HDM-II-S) the requirement that the superpotential be holomorphic forbids the termsλH † dĒ L L R andκH † uN L L R . However, similar terms are generated through couplings with H u and H d respectively.
We defer to Appendix B for detailed discussion of the model.
The resulting structure of mixing matrices and couplings follows similarly as in the 2HDM-II-Z 2 case with the exception thatλv d →λv u andκv u →κv d in Eq. 5 and 6.
This results in replacement ofλ →λ tan β andκ →κ/ tan β in the couplings of gauge bosons and the light SM higgs, while the couplings for the heavy CP-even, CP-odd, and charged Higgses are found with the replacementλ → −λ/ tan β andκ → −κ tan β. In later sections, we will see that this will result in dramatic differences in the predictions for (g − 2) µ compared to the 2HDM-Z 2 version.

C. SM with vectorlike leptons
The SM extended with VL and the corresponding contributions to (g − 2) µ have been studied in detail in [17,18]. In section V, we will briefly elaborate on these results, in particular updating the viable parameter space with respect to recent improved measurement of h → µ + µ − . In this case there is essentially no difference in the structure of Yukawa couplings or mixing matrices compared to the 2HDM-II-Z 2 version of the model with the caveat that the vevs in Eq. 5 and 6 should be replaced by v d → v and v u → v (for couplings of the light Higgs this also translates to cos β → 1 in Eq. A25 and related approximate formulas). The 1-loop contributions to (g − 2) µ from new particles induced by mixing with the muon in two Higgs doublet models are shown in Fig. 1. In this section, we present analytical formulas for these contributions in a general two Higgs doublet model. Contributions from SM bosons were previously calculated in [17,18].
Defining the couplings of lepton mass eigenstates to the W -boson by the corresponding contribution to (g − 2) µ is where x a W = m 2 νa /M 2 W , and the loop functions, F W (x) and G W (x), are given by Similarly, we define couplings of charged or neutral leptons, generically denoted by f a , to the Z-boson by The Z-boson contribution to (g − 2) µ is then given by where the sum is over charged leptons, e 4 and e 5 , and x a Z = m 2 ea /M 2 Z . The associated loop functions are given by The contributions from neutral Higgses to (g − 2) µ involving new charged leptons are where x a φ = m 2 ea /m 2 φ and Finally, couplings of charged and neutral leptons to the Higgs in the mass eigenstate basis can be defined by The contribution to (g − 2) µ from loops with the charged Higgs is then given by where x a H ± = m 2 νa /m 2 H ± and We emphasize that the formulas given in this section are not specific to any particular 2HDM strucutre (type-I, type-II, type-X, etc.) and can be used in any model We require M L > 800 GeV, M E > 200 GeV, and M N > 100 GeV in order to generically satisfy constraints from searches for new leptons [65][66][67][68]. However, it should be noted that the limits vary significantly with the assumed pattern of branching ratios of new leptons to W , Z and h [69] and, in the model we consider an arbitrary pattern of branching ratios can occur [57] (for a more detailed discussion of branching ratios and approximate formulas for relevant couplings of vectorlike quarks which are completely analogous to leptons, see also Ref. [61]). General pattern of branching ratios can allow significantly lighter new leptons than we consider here, especially SU (2) singlets.
For dimensionless parameters we will typically explore values of Yukawa couplings up to ±0.5 or ±1. Values up to ±1 are motivated by perturbativity limits at very large energy scales, possibly the GUT scale (depending on other details of the model).
Occasionally, we will extend the range of couplings up to ± √ 4π which is motivated by perturbativity limits of couplings at the scale of new physics. Note that the signs of three Yukawa couplings are not physical and can be absorbed into a redefinition of three vectorlike lepton fields. For example, λ L , λ E and κ N can be chosen to be positive.
We impose constraints from precision EW data related to the muon and muon neutrino that include Z-pole observables, the W partial width, and the muon lifetime.
We also impose constraints from oblique corrections [70,71]. These are obtained from data summarized in ref. [3].
Precision EW measurements constrain possible modification of couplings of the muon to the Z and W bosons at ∼ 0.1% level which, in the limit of small mixing, translates into 95% C.L. bounds on λ E and λ L couplings [17]: assuming only the Yukawa couplings in the charged sector. In the neutral lepton sector the strongest limits are obtained from the muon lifetime. These were discussed in ref. [56] together with constraints from the invisible widths of the Z boson. The constraint on the W − ν − µ coupling translates into an approximate 95% C.L. upper bound on the size of κ N and λ E couplings: which is slightly lower compared to the one quoted in ref. [56] due to lower uncertainty in the W mass [3].
For simplicity, for the 2HDM-II-Z 2 we assume degenerate heavy Higgs masses m A = m H = m H ± and for 2HDM-II-S we assume the standard tree-level relations between masses of heavy Higgs bosons. Thus, we only impose ATLAS limits on H(A) → τ + τ − [73] and on H + → tb [81] which are currently the strongest at large and small tan β respectively. These assumptions are also sufficient to satisfy constraints from flavor observables [82].
In addition to constraints on heavy Higgs masses, there are relevant constraints on the SM Higgs coupling to the muon through its modified relation to the muon mass.
In the present case, the physical muon mass originates from its coupling to H d as well as mixing with heavy leptons where we have defined that would give the muon mass in the absence of y µ as can be seen from the determinant of Eq. 5. Thus, for a given set of parameters that fix m LE µ , y µ can be iteratively determined so that Eq. 25 leads to the measured value of the muon mass. However, the sign of the muon mass determined by Eq. 25 is not physical and thus there are two solutions, y ± µ , leading to ±m µ , either of which is acceptable in principle. The wrongsign of the mass can always be rotated away by proper field redefinition of eigenstates.
Due to the arbitrary overall sign of m LE µ it is always possible to restrict to y + µ solutions.
From the Higgs coupling to the muon [34] by far exclude this possibility. Thus, in our numerical analysis we restrict to regions of parameters where m LE µ < m µ , and thus y + µ > 0. We will explore the impact of h → µ + µ − constraints in this region further in the following section.
We note that similar loops as in Fig. 1 will also generate a correction to the muon Yukawa coupling. This could lead to large corrections to y µ compared to the value needed to reproduce the muon mass. As a simple example, we will see in the following sections that regions of parameters which achieve ∆a exp µ within 1σ in the SM also require that the tree-level Higgs coupling to the muon is typically y µ 2m µ /v. Loop corrections to the muon Yukawa coupling in our model scale as ∆y µ λ L λ Eλ /8π 2 and reach this value for couplings ∼ 0.5. For couplings of order 1, motivated by perturbitivity in the UV, a tuning of only about 10% between tree-and loop-level contributions to y µ is expected in these scenarios. However, it could be argued that scenarios with larger couplings suffer from a fine-tuning problem with respect to the physical muon mass.
See also [32] for a related discussion.

V. RESULTS
The current measurement of the muon anomalous magnetic moment sits at more than four standard deviations from the predicted value in the SM [1, 2] Contributions to ∆a µ from charged and neutral vectorlike leptons with mixing to the muon are given by loops with h, Z and W bosons as well as those with heavy Higgses, A, H and H ± . The contributions involving vectorlike leptons and SM bosons were calculated previously in [17,18]. The complementarity of contributions from charged vectorlike leptons to ∆a µ and other precision observables in a 2HDM-II-Z 2 was presented in [29]. In this paper, we extend the calculation to include mixing in the neutral lepton sector. In the following subsections, we provide a detailed study of the 2HDM-II-Z 2 followed by a discussion of the corresponding predictions for ∆a µ in the 2HDM-II-S.
We also compare these results to the current status of the SM with VL.
A. 2HDM-II-Z 2 with vectorlike leptons Contributions to (g − 2) µ can be calculated following the analytic formulas in section III and Appendix A. In the following, it will prove useful to have approximations on hand to estimate the impact of individual particles to ∆a µ in terms of lagrangian parameters. In Tables II and III we summarize individual contributions from doublet-and singlet-like new leptons to ∆a µ normalized by m µ /16π 2 , in the limit of VL masses well above the EW scale (note the comments after Eq. A64 for the appropriate approximations used). We also assume that the masses of heavy Higgs bosons are comparable  to that of new leptons. 2 The derivation of each contribution is straightforward from 2 Our approximations are accurate to within 10% in the range 1 Though, in our numerical results we do not use any approximations. For heavier lepton masses, in Tables II and III for charged . We note that the latter expansion for approximate couplings listed in Appendix A 4. The total approximate contributions assuming heavy lepton masses can be found by summing the corresponding rows in the tables. We find for gauge bosons and SM-like Higgs. For the contributions from heavy Higgses with masses comparable to new leptons we find , where k W = 1, . We have additionally ignored terms ∝ λ in the CP-even and CPodd Higgs contributions as these terms would cancel in the total contribution when We see that the leading contribution from SM bosons is ∝ −m LE µ and likewise for CP-even and CP-odd heavy Higgses up to terms proportional to λ. The terms in neutral Higgs loops is numerically good to within a factor of 2 up to x a 0.1. give an order of magnitude larger contribution than those from SM bosons over most of the parameter space. For instance, for comparable heavy lepton and Higgs masses ∆a H µ 0.6 tan 2 β × ∆a h µ . Note that since CP-odd and charged Higgs contributions tend to cancel the enhancement is largely driven by the CP-even Higgs contribution for most of the parameter space. This can be seen when comparing to the right panel of Fig. 3 where we show the total contribution to (g − 2) µ for tan β = 1, 5, 50. In both panels we show the regions of m LE µ /m µ that are excluded by h → µ + µ − . Note that both m LE µ /m µ = 0 and -1 lead to the same prediction of h → µ + µ − as in the SM which can be seen from Eq. (27).
In both panels, the dark and light shaded green bands represent the 1 and 2σ levels of ∆a exp µ , respectively. We see that for couplings up to 0.5 (1), the correction to the magnetic moment spans a range about 4 (10) times the measured central value. As a curiosity, we mention that allowing couplings up to the perturbativity limit, ∼ √ 4π, the possible contribution to ∆a µ ∼ 200 × 10 −9 can be achieved while still satisfying all relevant constraints.
Regarding contributions from up-type couplings, it is clear from Tables II and III that corrections to (g − 2) µ from charged currents are the only relevant pieces. Mixing with the neutral component of the doublets L 0 L,R further dictates that additionally λ L should be non-zero to have any non-vanishing effects from κ's at leading order. Further, it is expected that any effect from loops in involving the W -boson are small, see Eq. 30, since the leading order contributions from SU (2) doublets tend to cancel those from singlets. In fact, we find that the sub-leading contribution from the W -loop can be found by further expanding x a G W (x a ) at the next order in x a In Fig. 4 we show the size of corrections to (g − 2) µ (color shading) in the limit that only λ L , κ,κ, and κ N are non-zero. We have fixed λ L and κ N to their maximum values allowed by precision EW constraints. In the left panel we fix κ = −κ = 0.5, while in  [29]. In previous sections we highlighted the fact that contributions from heavy Higgs bosons can dominate the total correction to the magnetic moment in most of the parameter space largely due to the tan 2 β enhancement. In Fig. 6, we show contributions to ∆a µ for couplings up to 0.5 (left) and 1 (right) from heavy Higgses relative to the total contribution with respect to m H and tan β when ∆a exp µ is achieved within 1σ. Lightly shaded crosses correspond to scenarios where heavy Higgses contribute less than 50% to the total correction. We see that the heavy Higgs corrections are generically the In Fig. 7, we show the corresponding range of masses in the same plane as Fig. 6.
Here the range of viable vectorlike lepton masses to at least 3 (8.5) TeV assuming couplings not exceeding 0.5 (1) are explicit. While these upper ranges may be out of reach for future colliders, similar comments as made in [29] also apply here, where comple-

B. 2HDM-II-S with vectorlike leptons
We remarked in section II that the supersymmetric version of the model in the limit of heavy superpartners has similar structure up toλ andκ couplings. In Appendix B we provide the corresponding approximate formulas for individual contributions to (g−2) µ .
The heavy Higgs contributions in the 2HDM-II-S contain both tan β enhanced and suppressed pieces as before. However, the tan β enhanced pieces of these contributions tend to cancel in the leading approximation for M L , M E , M N m A . Further, the total contribution from Z, W , and h loops tends to cancel that from heavy Higgses in this limit.
In Fig. 8, we show the total contributions to ∆a µ in the 2HDM-II-S with respect to m A and min(M L , M E , M N ) (left) and m A and tan β (right) for couplings up to 1.
As a result of the cancellation mentioned above the total contribution is smaller than that of the 2HDM-II-Z 2 over most of the plane. However, it is worth noting that the performance of the model improves as heavy Higgses are decoupled and the total con- Such scenarios can be seen in the left corner of either panel in Fig. 8 with m A 10 TeV where contributions from heavy Higgses make up more than 50% of the total contribution indicated by points with filled circles. To summarize, the 2HDM-II-S performs less favorably than the 2HDM-II-Z 2 version with respect to (g − 2) µ considering the loops in Fig. 1 largely due to the cancellation of tan β enhanced contributions. However, it should be stressed that the contributions presented in Fig. 8 can be considered in addition to the usual contributions from superpartners, e.g. through chargino/sneutrino or neutralino/slepton loops [9,[50][51][52].

C. SM with vectorlike leptons
The standard model with vectorlike leptons was previously studied in [17,18] as an explanation for ∆a exp µ . Here we extend the region of parameters considered in the model and show the impact of recent measurements of h → µ + µ − . We also explore the correlation of the contribution to ∆a µ with modifications of gauge and Yukawa couplings.
In Fig. 9, we show individual contributions to ∆a µ with respect to m LE µ /m µ . All Notably, these ranges are similar to the study without up-type couplings [29].
However, if the typically dominant contributions from down-type couplings are not present the contribution from the charged Higgs itself can still explain ∆a exp µ within 1σ due to the presence of couplings to H u . Apart from the main results, we also emphasize that the model can generate ∆a µ one (two) orders of magnitude larger than the central measured value with couplings up to 1 ( √ 4π) while satisfying all current precision constraints. While it is expected that even the LHC running at 14 TeV with 3ab −1 luminosity can only exclude (doublet) VL masses up to 1250 GeV [59] (depending on the decay modes), the high range of masses we present here can be probed indirectly at future precision machines [29].
In addition to our study of the 2HDM-II-Z 2 , we emphasize that while Yukawa couplings of SM leptons in this model are indistinguishable to those in the MSSM, couplings of VL are necessarily different due to the requirement that the superpotential be holomorphic. This leads to drastically different results in the contributions to ∆a µ from the same particle content. We find that the 2HDM-II-S can typically generate ∆a µ within 1σ from the central measured value in the limit that heavy Higgses are decoupled, For completeness we have extended previous studies of the SM with VL [17,18] by including couplings to heavy leptons which are SM singlets. Interestingly, we find that while the reach of lepton masses which can lead to ∆a µ within 1σ from the central We consider a complete generation of VL's which can mix with the 2nd generation leptons of the SM. In section III, we present one-loop formulas giving contributions to (g−2) µ in a generic 2HDM. In the following appendices we derive general expressions for all relevant couplings in the 2HDM-II-Z 2 we consider and present useful approximations for individual couplings in the limit of heavy lepton masses.

Couplings to Z and W bosons
Expressions for couplings of charged and neutral leptons to Z and W bosons have been given previously in the SM extended with vectorlike leptons, and in the 2HDM-II-Z 2 [18,56]. We summarize these expressions for completeness. 3 In the following it will be convenient to define the 3-component vectors e L,Ra ≡ (µ L,R , L − L,R , E L,R ) T , and ν L,Ra ≡ ((ν µ ) L,R , L 0 L,R , N L,R ) T in the gauge eigenstate basis. We denote 3-vectors of mass eigenstates by e L,R = U e L,Rê L,R and ν L,R = U ν L,Rν L,R , where U e L,R and U ν L,R are the diagonalization matrices given by Eqs. (5) and (6). We label the components of mass eigenstate vectors by a = 2, 4, 5.
The couplings to the Z bosons follow from the kinetic terms of leptons: where the covariant derivative is given by Defining the couplings of the Z boson to leptons f a and f b as the couplings of left-and right-handed fields immediately follow from Eq. A2 where a, b = 2, 4, 5. Since we only introduce mixing to the muon and muon neutrino, couplings of the first and third generation leptons in the SM are not modified.
The couplings of the W boson to charged and neutral leptons arise from the kinetic Defining the couplings of the W boson to mass eigenstatesν a andê a as we find

Couplings to Higgs bosons
Here we provide our conventions for the Higgs sector and couplings of VL leptons to physical Higgs and Goldstone bosons.
In the basis where the Yukawa couplings of SM leptons are diagonal, the Yukawa couplings of the neutral Higgs components to the muon and VL leptons are given by To write these interactions in terms of mass eigenstates, we additionally rotate the Higgs fields to the basis where physical and Goldstone degrees of freedom are apparent.

This basis is defined by
and for the charged sector Inverting these relations, the mass eigenstates of the neutral Higgs and Goldstone bosons are given by where h and H, A, and G are the CP-even, CP-odd, and neutral Goldstone bosons, respectively. By requiring a light Higgs with couplings to gauge bosons that are identical to those in the SM we have α = β − π/2, and the mass eigenstates for h and H are Thus, in term of mass eigenstates the Yukawa couplings of charged and neutral leptons to h and H are where Y E and Y N are given by The lagrangian for Yukawa couplings to CP-even Higgses can be written as where The couplings for the CP-odd Higgs can be derived in a similar way. The langrangian for the Yukawa couplings to A reads where Y A E and Y A N are given by Writing the lagrangian as we have The couplings for the neutral Goldstone boson, G, follow similarly. Defining we get For couplings to the charged Higgs bosons we first define Then, reading off the interactions from the lagrangian we get The charged Higgs mass eigenstates H ± and G ± are related to the gauge eigenstates Thus, the Yukawa couplings to charged Higgs bosons, in terms of mass eigenstates, are given by where Finally, writing the lagrangian for charged Higgs Yukawa couplings as we have The couplings for the charged Goldstone bosons, G ± , follow similarly. Defining we get where

Goldstone boson equivalence theorem
The Goldstone boson equivalence theorem (GBET) gives a relation between Smatrix elements of massive vector bosons and unphysical goldstone bosons at high energies through the requirement of tree unitarity [83][84][85][86].
which, in terms of lagrangian parameters, can be written as We introduce the following matrices and note that Inserting this relation into the vertex factor and applying the unitary relations of U e L and U e R results in where we have identified and Similar calculations lead to the following relation between the charged Goldstone and W boson couplings where we have identified

Approximate couplings
In this Appendix, we list various approximate formulas for couplings which enter the contributions to (g − 2) µ and relevant constraints on the model from mixing of the muon to VL's. Contributions coming from the SU (2) doublet VL are labeled with index L, whereas contributions coming from SU (2) charged and neutral VL singlets are labeled with index E or N , respectively, regardless of the hierarchy of masses. We assume that all mixing parameters are of similar order. In this case, the mixing matrices in Eqs. 5 and 6 can be written as an expansion in the dimensionless parameters and The above formulas are valid assuming that the mass eigenstates e 4 and ν 4 are mostly doublet-like, while e 5 and ν 5 are mostly singlet-like. This is equivalent to m e 4 M L , For couplings of the Z boson in this approximation we find and where in some formulas we indicate leading order terms in E,N .
For the corresponding couplings of the W boson to charged and neutral leptons we find and For the light SM-like Higgs boson, h, we obtain The couplings for the CP-even heavy Higgs H can be obtained simply by replacing one factor of cos β by sin β in the couplings for h.
The couplings of the CP-odd heavy Higgs, A, are given by Finally, the couplings for the charged Higgs boson are given by where the doublet components are labeled as and SU (2) doublets are contracted using antisymmetric , e.g. H d l = H 0 d l 2 − H − d l 1 = ab H da l b where 12 = − 12 = +1. Note thatL field is related to L R introduced in Eq. 1 viaL = iσ 2 L * R , and similarlyĒ is the chiral supermultiplet which contains E † R . In addition, note that the Higgs doublets are defined with opposite hypercharges than in the 2HDM-II-Z 2 . They are related by the field redefinitions H d = iσ 2H * d and H u = −iσ 2H * u , where the tilde fields are the Higgs doublets defined in the 2HDM-II-Z 2 . Signs of couplings have been chosen so that entries in the mass matrices of charged and neutral leptons have the same sign as in the 2HDM-II-Z 2 case.
The mass eigenstates, couplings of fermions to gauge and Higgs bosons, and contributions to (g − 2) µ can be calculated in a straightforward way following the procedure detailed in the previous appendix. Note that in the conventions used here the Higgs sector (in alignment limit) is decomposed as and where and we identify H ± = (H ∓ ) * . With these definitions, the differences in the couplings of gauge and Higgs bosons appear only throughλ andκ terms, and they are summarized in the main text. Contributions to (g −2) µ can then be found from the general formulas given in Section (III).
The approximate contributions to (g − 2) µ from Z, W , and h in the limit of heavy comparable lepton masses are given by Assuming M L,E,N m A , the contributions from H, A, and H ± are given by Compared to the 2HDM-II-Z 2 version, the loops involving SM bosons are now tan β enhanced. Heavy Higgs contributions contain both tan β enhanced and suppressed terms. Though in the limit when heavy Higgs masses are equal and comparable to heavy lepton masses, the tan 2 β enhanced contributions cancel in the total contribution. The approximate formulas highly simplify when M L,E,N = m A and vanishing κ couplings.
In this case we find ignoring terms which cancel between ∆a H µ and ∆a A µ . Note that comparing contributions from Z, W , and h to those from heavy Higgses, we find that ∆a µ 0 in this approximation.

Appendix C: Comments on Barr-Zee contributions
Two-loop contributions to (g − 2) µ from Barr-Zee (BZ) diagrams can sometimes be competitive with one-loop predictions due to chiral enhancement in the closed fermion loop [87]. In the models we have discussed, the chiral enhancement is generated already at the one-loop level, ∆a 1 loop µ ∼ m µ v/M 2 , where M is the scale of new physics. The dominant contribution from BZ-type diagrams is generated through diagrams with a neutral Higgs and photon in the internal legs [88]. General formulae for this diagram are given in [88,89]. In our notation for Higgs couplings, this contribution from neutral Higgses is given by where x a φ = m 2 ea /m 2 φ and The relative size of the BZ contribution from CP-even Higgses (noting that we work in a CP-conserving 2HDM) compared to the one-loop contribution, Eq. (16), is estimated by where φ = h, H. In the limit of heavy lepton masses, we have [90] f (x)