Charged Lepton EDM with Extra Yukawa Couplings

In a two Higgs doublet model with extra Yukawa couplings, we assess the new physics contributions to charged lepton electric dipole moment. We focus especially on muon (and tau) EDM where in the coming decade several experiments $-$ Muon g-2, J-PARC and PSI (and Belle II) $-$ will push sensitivities down by several orders of magnitude. With the working assumption that extra Yukawa couplings are analogous to SM ones in strength and taking exotic scalar masses in sub-TeV range, we find that $\mu$ and $\tau$ EDM can be enhanced to values larger than new physics scenarios that scale with lepton mass. The main effect comes from the flavor-conserving extra top coupling $\rho_{tt}$ via well-known two-loop diagrams. Deviating from our working assumption, if the muon $g-2$ anomaly arises from the one-loop diagram, driven by singly enhanced lepton flavor violating $\rho_{\tau\mu}$ coupling, it can also induce rather large muon EDM, accessible at upcoming experiments.


I. INTRODUCTION
All charged-parity violation (CPV) observed in the laboratories so far are in the quark sector. The Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] that govern the weak interactions of the Standard Model (SM) has been impressively successful in accounting for all such effects. But, as is known, the KM phase does not provide enough CPV to satisfy the Sakharov condition [3] for the disappearance of antimatter in the Universe. Various new physics (NP) models attempt to make up for this by supplying new complex phases to hopefully explain the baryon asymmetry of the Universe (BAU).
The general two Higgs doublet model (g2HDM) [4,5], also known as 2HDM Type-III [6], is attractive in this context. In g2HDM, one does not introduce a Z 2 symmetry to enforce Natural Flavor Conservation [7]. Thus, the second Higgs doublet brings in extra Yukawa coupling matrices, which are in general complex. The new phases, especially those associated with extra top quark couplings, ρ tt and ρ tc , which are likely the stronger in strength, can drive electroweak baryogengesis (EWBG) and explain BAU [8,9]. Complex extra Yukawa couplings can also contribute to CPV observables accessible in the laboratory, such as charged lepton electric dipole moment ( EDM), which we denote as d .
Lepton EDMs are powerful probes of NP CPV. The SM predictions for EDM are extremely tiny (e.g. |d e | ∼ 10 −38 e cm [10][11][12]) because the sole complex phase, the KM phase, survives only at the four-loop level. With essentially no SM background, detection of EDM will establish the presence of NP of CPV nature.
Recently, there has been very good progress in the search for eEDM. The ACME experiment exploits the large internal electric field of ThO molecules to measure eEDM. Already in 2013, ACME provided the best upper limit on eEDM, |d e | < 8.7×10 −28 e cm [13]. Impressively, the bound was further improved to [14] |d e | < 1.1 × 10 −29 e cm, (ACME 2018) (1) which is the best upper limit on any EDM. The bound of Eq. (1) puts stringent constraint on any BSM model that seek to address the BAU problem. For g2HDM, consistency with Eq. (1) has helped shed light on the structure of extra Yukawa couplings. As noted in Ref. [9], one can accommodate EWBG in g2HDM with flavor conserving coupling |ρ tt | 0.1 (see Ref. [8] for flavor violating coupling ρ tc which is immune to eEDM constraint), but parameter space with single coupling scenario runs into trouble with the ACME 2018 bound. This can be avoided by introducing extra electron Yukawa coupling ρ ee , which helps salvage parameter space favorable for EWBG. Interestingly, the organization of these couplings follow the pattern [9], suggesting that the extra ρ couplings of g2HDM have similar hierarchical structure as Yukawa couplings in SM. Eq.
(2) provides not only an elegant way of complying with the ACME 2018 bound, it connects EWBG with tiny CPV effects in the laboratories, such as eEDM.
Seeing that eEDM is highly sensitive to extra Yukawa couplings, in this paper we wish to explore the detailed contributions of extra Yukawa couplings to µ and τ EDM. Such an investigation is well-timed for the muon, where the Fermilab Muon g-2 experiment [15] has affirmed recently the significant deviation from SM expectation in a related observable -the anomalous magnetic moment of the muon. Any NP in muon g − 2 certainly can have important implications for µEDM, as we will show later.
The current bound on µEDM is by the Muon g-2 experiment at Brookhaven National Laboratory (BNL) [16], while an indirect limit, extracted from eEDM bound of ACME 2018, is claimed [17] at ∼ 2 × 10 −20 e cm. In the coming years, several experiments will search for µEDM and improve Eq. (3). The Fermilab Muon g-2 [18] and the J-PARC muon g-2/EDM [19] experiments project sensitivities at ∼ 10 −21 e cm. Another experiment being planned at PSI [20] aims at improving by at least another order of magnitude, eventually reaching a sensitivity of 6 × 10 −23 e cm. Thus, prospects for µEDM in the coming decade appears promising. Compared to eEDM and µEDM, τ EDM measurement is hampered by the short τ lifetime. The present limits are derived from spin-momentum correlations of final state decay products in e + e − → γ * → τ + τ − , where τ EDM enters the matrix element of the γτ τ vertex. An old limit of the Belle experiment [21] with 29.5 fb −1 was very recently updated [22] to full dataset of 833 fb −1 . These limits on the real and imaginary parts of d τ , at ∼ 10 −17 e cm, can be read off Table I. An indirect bound extracted from eEDM [17] seems much better, but real and imaginary parts were not separately given.
Belle II [23] should improve the bound by more than an order of magnitude. Furthermore, some improvement is possible by studying "optimal" observables [24].
If one assumes that the NP contribution scales as m , such as models with [25] minimal flavor violation (MFV), one expects |d µ /d e | m µ /m e . Taking the current limit on eEDM of Eq. (1), d µ is expected at only ∼ 2 × 10 −27 e cm. A glance at Table I shows that future sensitivities fall short by more than four orders of magnitude, and it is even farther away for d τ . But this m scaling may not hold in NP models where the underling flavor dynamics is quite different [26,27], and the situation need not be as pessimistic. The g2HDM provides a simple example where extra Yukawa couplings can give much larger d µ and d τ . We will show that, in a scenario favored by the muon g − 2 anomaly, very large d µ that can be probed in the coming years is possible. This paper is organized as follows. In the next section, we give the extra Yukawa interactions in g2HDM and our working assumptions for their strength. We then discuss µEDM in Sec. III and present results for the oneand two-loop contributions, while in Sec. IV we analyze τ EDM in a similar fashion. In Sec. V, we discuss extra Yukawa couplings that violate our working assumptions, as possibly hinted by muon g−2, and discuss implications for EDM of all three charged leptons. Finally, we offer our conclusion in Sec. VI.

II. PRELIMINARIES
Before we proceed to the specifics, let us lay out the framework of the model. The Yukawa interactions in g2HDM are given by [5,28], where L (R) = (1 ∓ γ 5 )/2 are projection operators, Q f is the charge of fermion f in units of e, and V is the CKM matrix. The shorthand c γ ≡ cos γ and s γ ≡ sin γ represent mixing between two CP even scalars, h and H, which is known to be very small. Although ρ ij couplings are generic and do not possess information about SM Yukawa couplings λ i , it is intriguing that data suggest the ρ matrices imitate closely the hierarchical structure of SM Yukawa matrices (see Eq. (2)), as already mentioned in the Introduction. The ∆F = 2 processes such as B s(d) −B s(d) and K 0 −K 0 mixing constrain off-diagonal entries of down-type ρ d to very small values [29], which indicates that ρ d is nearly diagonal. On the other hand, the extra top Yukawa coupling ρ tt , important for our study, is allowed to be significant in strength [30]. For the lepton sector, the ρ coupling related to third and second generation are tightly constrained from LFV decays h → τ µ and τ → µγ, but with ρ tt ∼ 1, values of ρ τ µ = ρ µτ O(λ τ ) are easily allowed even for c γ ∼ 0.1 [31].
With these arguments in mind, we assume the following strengths for ρ ij related to u-and -type matrices [32], where the first two conditions can also be applied to elements of the ρ d matrix. Down-type couplings, however, do not play much role in our discussion, so Eq. (5) will be our working assumption while estimating the typical size of EDM in g2HDM. But we note that deviation from Eq. (5) is possible. For example, NP in muon g − 2 may call for large deviation from the last condition in Eq. (5) for interpretation in 2HDM [38]. We will discuss in Sec. V the implications of such a scenario for EDM and assess the impact on our results.
The h → τ τ process provides an avenue for the study of CP properties of the h boson at the LHC (see e.g. Refs. [35,36]). Very recently, CMS [37] has carried out a study of the CP structure of the hτ τ coupling with 137 fb −1 data, reporting the phase of the τ Yukawa coupling, φ τ τ , at 4 • ±17 • (±36 • ) at the 68% (95%) C.L., with rather large errors. In the notation of Ref.

III. µEDM
The EDM, d , of lepton is defined through the effective Lagrangian, where σ αβ = i[γ α , γ β ]/2, and F αβ is the electromagnetic field strength tensor. In g2HDM, the first finite contribution to EDM appears at one-loop. A sample one-loop diagram for µEDM in g2HDM is given in Fig. 1. The dipole operator in Eq. (8) is chirality violating, so an additional mass insertion is required on the fermion line to obtain the correct chiral structure. This means that analogous diagrams with lighter leptons in the loop are chirally suppressed, and we neglect them. Fig. 1 gives [39], where x ij = m 2 i /m 2 j . We note that the contribution of the h boson is negligible in Eq. (9) because c 2 γ 1, while the contributions of CP even and odd scalars have a relative sign between them, indicating that in the H-A degeneracy limit, one-loop contributions will be highly suppressed, which is indeed the case. For example, taking (5), we obtain |d µ | 4 × 10 −26 e cm at one-loop, which is three orders of magnitude below PSI sensitivity. With m A and m H sufficiently apart, the one-loop contribution increases, but it remains outside of experimental reach. For example, with m H = 300 GeV and m A = 500 GeV and keeping other values as before, we obtain |d µ | 7 × 10 −25 e cm. Thus, we find that our working assumption of ρ in Eq. (5) renders the oneloop contribution too small. Next, we consider two-loop diagrams -also known as Barr-Zee or Bjorken-Weinberg diagrams [40,41].
It is well known that, despite having additional loop suppression, certain two-loop diagrams can dominate over one-loop diagrams. This can be understood qualitatively by noting that Fig 1 contains three chirality flips while only one flip in Fig. 2, which can let the latter compensate for the additional loop factor (∼ α/π). Barr and Zee [40] calculated neutral scalar contributions with top quark and gauge bosons in the loop, which was later extended [42][43][44][45][46] to include other two-loop diagrams.
Following the convention used in Ref. [9], we categorize two-loop contributions to µEDM into three parts, related to effective φγγ, φγZ and H + γW + vertices, respectively, as in is written as the sum of two terms [9], where d φγ where , and loop functions f (x) and g(x) are given in Appendix A. Note that for x ∼ 1, f (x), g(x) ∼ x, and for small x, f (x), g(x) ∼ x(log x) 2 /2, which further exemplifies why the top quark diagram is dominant. Eq. (12) is due to mixing of SM Yukawa couplings with extra Yukawa couplings, and therefore has a linear dependence on Im ρ ii with no dependance on m A . On the other hand, Eq. (13) contains contributions from extra Yukawa couplings only. For very small c γ , the "mix" term becomes negligible, and the top loop contribution is governed by Eq. (13). One also notes that, for ρ µµ and ρ tt both purely real or purely imaginary, Eq. (13) does not contribute, which yields (d φγ µ ) t (d φγ µ ) mix t . The diagram with W + in the loop induces contribution to d φγ µ for finite c γ . Defining the φW + W − coupling as i gm W C φW W [47,48], where C hW W = s γ , C HW W = c γ , and C AW W = 0 , the diagram gives, with loop function I γ (a) given in Appendix A. This diagram has sign opposite the top-loop, hence large cancellation can occur in some parameter space for finite c γ . The cancellation is, however, sensitive to complex phases of extra Yukawa couplings. Adding Eqs. (11) and (14) and taking c γ = 0.1, m H = m A = 300 GeV, we illustrate d φγ µ as solid lines in Fig. 3 vs ρ tt coupling. We take |ρ µµ | = λ µ in accordance of Eq. (5) and assume it to be purely imaginary. The red (black) lines correspond to purely real (imaginary) ρ tt . For φ tt = 0, as ρ tt increases, top-loop diagram starts to compete with W -loop, resulting in cancellation between the two, seen as the dip in the plot. For larger values of ρ tt , the top-loop overwhelms the W -loop, thereby bringing d φγ µ to ∼ 10 −24 e cm. For φ tt = π/2, (d φγ µ ) t is given by Eq. (12) only, where the ρ tt term is suppressed further by m µ , hence (d φγ µ ) t is feebly sensitive to ρ tt and never overpowers W -loop even for large |ρ tt | ∼ 1.
For d φZ µ , one simply replaces the inner photon propagator (one connecting loop with muon line) with the Z propagator. By parity, only the vector part of Zµµ coupling contributes. Its smallness, together with 1/m 2 Z from the Z propagator, the d φZ µ effect is typically suppressed with respect to d φγ µ . The corresponding formulae are relegated to Appendix A. In Fig. 3, d φZ µ are given as dot-dashed lines, with color coding as before. For both red and black lines, |d φZ µ | always lie far lower than |d φγ µ |, except near cancellation regions of |d φγ µ |. On the other hand, complex extra Yukawa couplings of charged Higgs boson in Eq. (4) contribute significantly to d H + W + µ . The main effect comes from top-bottom in the loop, which gives [45], where φ stands for H + only, and functions F q (a, b) are given in Appendix A. Diagrams with W + and scalars in the loop were calculated in Ref. [46]. Adapting to our case, we find, where I 4 (a, b) is given in Appendix A. Combining Eqs. (15) and (16), we give d H + W + µ in Fig. 3 as dashed lines, taking the same values of scalar masses and c γ as before. For φ tt = 0 (red), d H + W + µ is basically governed by Eq. (15) and slightly lower than d φγ µ , but significantly larger than d φW µ . At ρ tt = 0, the origin, d H + W + µ is given by Eq. (16), which lies between 10 −27 -10 −26 e cm. For extremely tiny ρ tt , the effects from Eqs. (15) and (16) tend to cancel. This behavior is similar to the case of d φγ µ discussed previously. For φ tt = π/2 (black), there is no dependence on ρ tt and the total contribution is given by Eq. (16), which is represented as a horizontal black dashed line.
We see from Fig. 3 that, our working assumption of Eq. (5) suggests µEDM in g2HDM can be enhanced to ∼ 10 −24 e cm. This, however, lies out of reach of muon g-2 experiments at FNAL and J-PARC by three orders of magnitude. Even with improved sensitivity at PSI, shown as cyan band in Fig. 3, it cannot probe extra Yukawa couplings with strength as given in Eq. (5).
Some comments are in order. There are additional subdominant diagrams related to charged Higgs loops which depend on trilinear Higgs couplings. The expressions of these are given in Appendix A. We ignore them since our scope is to cover effects from Yukawa interactions of Eq. (4). These contribution can be important in models with CPV scalar potentials, or 2HDM models with softly broken Z 2 symmetry, where large contributions such as in Eq. (15) are negligible (see e.g. Refs. [46,[49][50][51][52]). For the former case, we should mention that there are additional so-called "kite diagrams", calculated recently in Ref. [51], which are required to obtain a gauge-invariant result.

IV. τ EDM
The diagrams contributing to τ EDM is similar to µEDM, except a more significant coupling ρ τ τ ∼ λ τ is involved, hence one would expect further enhancement.
One obtains the one-loop effect for τ EDM by replacing "µ" → "τ " in Fig. 1. Diagrams with internal µ and e are again neglected because of chiral suppression. Since the flavor conserving φτ τ vertex is involved, the SM Yukawa λ τ also contributes via interference with ρ τ τ . Using the results of Ref. [29], the one-loop contribution is In the small c γ limit, λ τ effects decouple. Taking c γ = 0.1, m H = m A = 300 GeV, and Re ρ τ τ = Im ρ τ τ ≤ λ τ / √ 2 for τ coupling following Eq. (5), we find |d τ | 6 × 10 −25 e cm, which is about five orders below Belle II projections. For m H = 300 GeV and m A = 500 GeV, the one-loop contribution increases to |d τ | 1.3×10 −24 e cm, but remains out of Belle II reach.
The discussion of two-loop contribution to d τ is analogous to d µ in Sec. III. Therefore, similar to Eq. (10), we write for d τ at two-loop, where expressions for each term can be obtained from Eqs. (11) and (14)-(16) by change of label "µ" → "τ ".
In Fig. 4 we give results for each term in Eq. (18) as functions of ρ tt . The same benchmarks are used as in previous section: m H = m A = m H ± = 300 GeV, c γ = 0.1, and we take ρ τ τ = iλ τ . Red (black) lines represent φ tt = 0 (π/2). The characteristics of each type of contribution are the same as the µEDM case in Sec. III, except that each contribution is now larger by ρ τ τ /ρ µµ . From Fig. 4 and for our benchmarks, d τ can reach ∼ 10 −23 e cm, but again falls short of Belle II sensitivity.

V. DISCUSSION
With our working assumption for ρ ij given in Eq. (5) and sub-TeV exotic scalars, we find that µ and τ EDM can be enhanced in g2HDM, but are not large enough to be probed in the coming years. However, recent progress in muon g − 2 may suggest good prospects for µEDM.
Small ρ tt and c γ will suppress the two-loop contributions to EDM: the top-loop diagrams by small ρ tt , and W -loop diagrams by c γ → 0. But large ρ τ µ (ρ µτ ) values can give rather large µEDM via the one-loop diagram.
Defining ρ τ µ ρ µτ = |ρ τ µ ρ µτ | exp(φ CP ), we illustrate in Fig. 5 the range of d µ accessible in the |ρ τ µ ρ τ µ |-φ CP plane while simultaneously accounting for muon g −2, Eq. (19). The left plot is for m H = 300 GeV and m A = 340 GeV, which are not far from degenerate, while the right plot is for the same m H but a much heavier m A = 500 GeV. We see from both plots that, given the complexity of ρ τ µ ρ µτ , the solution space for muon g − 2 projects large d µ values that are well within PSI sensitivity! In the more optimistic scenario where m H and m A are farther apart, the parameter space in the right plot may even appear to be within reach of Fermilab and J-PARC muon g-2 experiments, namely ∼ 10 −21 e cm. We caution, however, that this would demand very large |ρ τ µ | ∼ |ρ µτ | values.
On the other hand, one does not expect such extraordinary enhancements for d τ . In fact, if muon g − 2 arises from g2HDM as described, then d τ will be much smaller than our estimates in Sec. IV. This is because the twoloop contribution to d τ in Eq. (18) becomes suppressed by small ρ tt and c γ , as argued previously. Furthermore, one-loop enhancement similar to µEDM is not possible because simultaneously large ρ τ τ and ρ τ µ (ρ µτ ) would make τ → µγ too large at one-loop and violate the Belle bound, as we showed recently in Ref. [57].
What about eEDM? The leading one-loop results (τ in the loop) for eEDM are given by Eq. (9), replac-ing "µ" → "e". Therefore, at one loop, d e is proportional to Im(ρ τ e ρ eτ ). The µ → eγ process constrains ρ τ e (ρ τ e ) severely. For values of ρ τ µ consistent with muon g − 2 solution, the current MEG [58] bound of B(µ → eγ) < 4.2×10 −13 gives |ρ τ e | = |ρ eτ | O(λ e ) [57], which is in line with our working assumption in Eq. (5). Taking | Im(ρ τ e ρ eτ )| λ 2 e , c γ = 0, and m H = 300 GeV and m A = 500 GeV, we find |d e | 5 × 10 −32 e cm, which is more than two orders of magnitude below the current ACME bound, |d e | < 1.1 × 10 −29 e cm. For m H closer to m A , d e will be further suppressed. The two-loop results for d e can be obtained by replacing "µ" → "e" in the corresponding expressions for µEDM in Sec. III. If one assumes ρ ee ∼ 0, then two-loop diagrams for eEDM is essentially given by the second term of Eq. (12) (after obvious change of lepton indices), which depends on |s 2γ Im ρ tt | and m H . For m H = 300 GeV, we find |d e | 2 × 10 −27 |s 2γ Im ρ tt | e cm. With both parameters constrained to be small, c γ < 0.005 and |ρ tt | 0.1, it is rather easy to satisfy the current ACME bound, without resort to the cancellation mechanism [9] to evade the experimental bound. It remains to be seen whether parameter space exists where simultaneously small ρ tt and c γ can still provide solutions to BAU in g2HDM.

VI. CONCLUSION
We study one-and two-loop mechanisms for charged lepton EDM in g2HDM, the two Higgs doublet model that possesses extra Yukawa couplings. We find both d µ and d τ can be enhanced to considerably larger valuesthanks to the extra top Yukawa coupling ρ tt ∼ O(1).
The question for g2HDM is how it has managed to hide so well from scrutiny, if the exotics Higgs bosons are sub-TeV in mass. This we have elucidated in a previous work [32], where we developed Eq. (5) as our working assumption for the strength of extra ρ f ij couplings, namely ρ ii O(λ i ), ρ 1i O(λ 1 ), and ρ 3j O(λ 3 ) for j = 1. It reflects the mass-mixing hierarchies observed in the SM sector [6], and finds further support from the cancellation mechanism [9], or interplay between ρ ee and ρ tt to satisfy the recent ACME 2018 bound on d e , while ρ tt still drives baryogenesis. Flavor changing couplings of h(125) are further suppressed by the recent emergent phenomenon of alignment, i.e. small h-H mixing.
With our working assumption as above and taking central values, we showed that d µ lies in the ballpark range of 10 −24 e cm, while d τ is in the 10 −23 e cm range. These numbers could go up by another order of magnitude as allowed by Eq. (5). But though they are much larger than typical expectations in MFV-like scenarios, with future sensitivities as stated in Table 1, it is unlikely that d µ and d τ would be observed in the near future. Thus, Eq. (5) continues to "protect" g2HDM from revealing itself, and we have advocated elsewhere for LHC direct search [30], as well as several flavor probes [32].
The recent experimental affirmation of the muon g − 2 "anomaly", however, brings in some hope, but at a cost.
In g2HDM, the one-loop mechanism can quite simply account for the discrepancy, if |ρ τ µ | |ρ µτ | = O(20λ τ ) ∼ 0.2. This would grossly violate our working assumption of Eq. (5); but Nature herself is the judge and the jury, and maybe this is what she has been telling us through the "anomaly". The point is, the same one-loop diagram can generate d µ through the complexity of ρ τ µ ρ µτ , which can be tested with projected experimental sensitivities. But this single-enhanced coupling scenario would then imply quite small ρ tt 0.1, while alignment must be near-perfect, which by themselves are rather peculiar. These in turn suggest that d τ will be rather suppressed, and d e can be much below ACME bound without resort to cancellation mechanism.