Discovery Prospects for Electron and Neutron Electric Dipole Moments in the General Two Higgs Doublet Model

Baryon asymmetry of the Universe offers one of the strongest hints for physics Beyond the Standard Model (BSM). Remarkably, in the general two Higgs Doublet Model (g2HDM) that possesses a second set of Yukawa matrices, one can have electroweak baryogenesis (EWBG) while the electron electric dipole moment (eEDM) is evaded by a natural flavor tuning that echoes SM. We show that eEDM may first emerge around $10^{-30}\,e$ cm or so, followed by neutron EDM (nEDM) down to $10^{-27}\,e$ cm. We illustrate a cancellation mechanism for nEDM itself, which in turn can be probed when a facility capable of pushing down to $10^{-28}\,e$ cm becomes available.

Introduction.-Withno BSM physics emerging at the Large Hadron Collider (LHC), particle physics is in a state of exasperation.It is not clear whether one can address lofty issues such as the Baryon Asymmetry of the Universe, arguably one of the strongest hints for BSM physics that calls for the existence of large CP violating (CPV) phase(s) beyond the Kobayashi-Maskawa phase [1] of SM.The current frontier is the experimental race to measure electron EDM, where the bound held by the ACME experiment [2] has recently been surpassed at JILA [3], giving d e < 0.41 × 10 −29 e cm at 90% C.L.This is several orders of magnitude stronger than the current nEDM bound of d n < 1.8 × 10 −26 e cm by the nEDM experiment at PSI [4].However, by using ultra cold neutrons (UCN), nEDM measurement is poised to improve by two orders of magnitude within two decades [5], with many experiments joining the fray.
In fact, the EDM experiments, much smaller than the behemoth LHC and its associated experiments, pose a general challenge: since BAU demands extremely large BSM CPV, can one survive the EDM bounds, especially eEDM?We explore this theme and promote the general two Higgs doublet model (g2HDM), where dropping the usual Z 2 symmetry one can have enough CPV for BAU, but the observed flavor (fermion mass and mixing) hierarchies -a mystery in itself -allows for an exquisite natural flavor cancellation mechanism to work for eEDM.We project that eEDM and nEDM could well emerge in the next decade or two, and extend the parameter range beyond previous considerations.
With one Higgs doublet observed, the two Higgs doublet model [6] should be a no-brainer.A Z 2 symmetry is usually imposed to enforce the natural flavor conservation (NFC) condition posited by Glashow and Weinberg [7] to forbid extra Yukawa matrices of charged fermions.But as first illustrated by Cheng and Sher [8], the flavor hierarchies may help alleviate Glashow's worries about flavor changing neutral couplings (FCNCs).It was pointed [9] out, even before the top discovery, that the process to watch, then, is t → ch.The bound at the LHC, however, has reached the stringent B(t → ch) < 0.00073 [10].But as stressed in 2013 [11] after the obser-vation of h(125), as the ρ tc coupling is associated more with the exotic H and A bosons, the tch coupling should be ρ tc c γ , where c γ ≡ cos γ is the h-H mixing angle between the two CP -even scalars.Who would have guessed that Nature would throw in, circa 2015, the alignment (small c γ ) phenomenon from the purely Higgs sector, to protect t → ch decay.
Having introduced the ρ tc element of the up-type extra Yukawa matrix, it was subsequently shown [12] that λ t Im ρ tt can robustly drive EWBG [13], with top Yukawa λ t ∼ = 1 recently measured [1], and with first order phase transition arising from O(1) [14] Higgs quartic couplings, where there are a total of 7 in absence of Z 2 .It was further inferred with emergent alignment that the exotic scalars are likely sub-TeV [15] in mass and populate 300-600 GeV, openning up a search program at the LHC [16][17][18][19], where Ref. [19] is from ATLAS.
The large Im ρ tt at O(λ t ) ∼ 1 that drives EWBG brings up our theme of how to survive eEDM.A typical two-loop Barr-Zee diagram [20] for eEDM is given in Fig. 1.To cancel the leading effect due to ρ tt and ρ ee , specifically the ϕγγ * insertion, one finds [21] |ρ ee /ρ tt | = r|λ e /λ t |, arg(ρ ee ρ tt ) = 0, with r ≃ 0.7, where the first relation follows from a phase-lock between ρ ee and ρ tt for ϕ = A. Eq. ( 1) is remarkable in that the ρ matrices seem to "know" the quark mass and mixing hierarchies in SM.
The purpose of this Letter is to show that the combined eEDM and nEDM effort provides the cutting edge probe of ρ tt -driven EWBG in g2HDM: as the experimental competition heats up, we may first observe eEDM in the 10 −30 -10 −31 e cm range, followed by confirmation at n2EDM at PSI for d n ∼ 10 −26 -10 −27 e cm in about a decade.But as we will illustrate a general cancellation mechanism for nEDM itself, a more advanced nEDM experiment may confirm down to 10 −28 e cm in two decades.To unravel the underlying dynamics, the "decadal mission" [24] with direct exotic scalar search at the LHC, flavor physics explorations with LHCb and Belle II, plus µ and τ studies, would be needed.g2HDM and EDMs.-For simplicity, we assume CPconserving [15,23] Higgs potential of g2HDM, removing it as a CPV source without discussing it any further here, so CPV is relegated to extra Yukawa couplings.As already stated, O(1) Higgs quartics supply [12,14] the prerequisite first order EW phase transition for BAU, which is a bonus in g2HDM.
To clarify the flavor and EWBG discussion in the Introduction, without any Z 2 symmetry, there are extra Yukawa matrices ρ f for charged fermions f = u, d, ℓ [18,23], which are complex and nondiagonal, with generation indices i, j summed over, L, R = 1 ∓ γ 5 , and s γ ≡ sin γ.The A, H + couplings are c γ -independent, while in the alignment limit (c γ → 0, s γ → −1), h couples diagonally and H couples via extra Yukawa couplings −ρ f ij , which can drive BAU.Thus, besides massmixing hierarchy protection [9] of FCNCs, alignment provides [15] further safeguard, such as for t → ch, without the need of NFC.Furthermore, the µ 2 12 Φ † Φ ′ term in the Higgs potential is eliminated after symmetry breaking by minimization, leaving a unique h-H mixing parameter, η 6 , which can be O(1) [15] for small c γ , with H, A, H + likely in the 300-600 GeV mass range.
Considering how effective g2HDM evades stringent flavor constraints, and to address the question "What makes g2HDM so well hidden so far?", we guessed a "rule of thumb" [22] for flavor control: with j ̸ = 1.This allows ρ tt = O(1) but ρ bb ≃ 0.02.However, ρ d ij seems to be an order of magnitude weaker by flavor constraints.
With complications of transport equations for EWBG [12], the simplified case with H, A, H + degenerate at 500 GeV was studied.The ACME experiment [2] taught us the lesson to keep the weakest ρ ee coupling in the Barr-Zee diagrams of Fig. 1, where the exquisite cancellation mechanism of Eq. (1) was uncovered [21].The prowess of ACME, however, led one to illustrate with the timid |ρ tt | ≃ 0.1, which we seek to extend here.
Results: Interplay of eEDM and nEDM.-Inour numerical illustration, we shall keep the degeneracy at 500 GeV, but explore a broader range of and follow the numeric ansatz [21] for f = u, c; d, s, b, where r ≃ 0.71 is a combination of loop functions that is insensitive [21] to exotic Higgs spectrum.In Fig. 2 we illustrate the natural "flavor tuning" [21] of Eq. ( 1), for ρ tt values in Eq. ( 4) and numeric ansatz of Eq. ( 5), where both bounds of ACME [2] and JILA [3] are shown.We take some liberty in the visual effect of the light purple band, with left side taken from the reddashed d ϕγ e curve [21], and right side from the red-solid d e curve.This is in part because, though the cancellation point (black-solid curve sitting in the middle, with final shift from C S effect [21]) is insensitive to the spectrum [21], there should be some spread in exotic scalar masses, which we refrain from exploring.
From left to right in Fig. 2, as ρ tt strength rises, the "funnel" is raised, but at 10 −30 e cm, the openning of the funnel is still decent, suggesting a still robust discovery likelihood, although by 10 −31 e cm, it approaches a pinpoint and may no longer seem plausible.In any case, these plots are for numeric illustration.
Turning to nEDM, besides effects of ρ uu and ρ dd through Barr-Zee type diagrams, there are also chromomoments and the Weinberg operator, with progressively larger theory uncertainties.While the classic review of Pospelov and Ritz [25] continue to be widely cited, it is a bit dated.We use the more recent formula [26], where we evaluate chromo-moments du,d and the Weinberg operator C W term [27] by following Refs.[28] and [29], respectively.A recent discussion on uncertainties can be found in Ref. [27].
We give in Fig. 3 the scan plot for r ∈ [0.6, 0.8] for same range of ρ tt and exotic Higgs masses as in Fig. 2, showing both the JILA bound [3] on eEDM, and PSI bound [4] on nEDM.One survives the PSI bound even for |ρ tt | ≃ 0.3 √ 2, while r ≃ 0.7 nicely illustrates the natural flavor cancellation of eEDM.The follow-up experiment to nEDM at PSI, i.e. n2EDM [30], plans to reach down to 10 −27 e cm sensitivity within a decade, and should be able to cover the range illustrated in Fig. 3.
But we should admit that Eq. ( 5) is nothing but an ansatz [21] for sake of numeric illustration.The fact is, we have little knowledge of the actual strength of extra Yukawa couplings such as ρ uu .Our "rule of thumb" of Eq. ( 3) is our guess of the "flavor protection" in g2HDM, which echos the remarkable cancellation mechanism of Eq. ( 1) for eEDM.Taking Eq. ( 3) literally, it states that |ρ uu | = O(λ u ), with phase unknown.Thus, taking the usual sense of "an order of magnitude", we vary while keeping other ρ f f s according to Eq. ( 5).This explores the impact of ρ uu strength and phase on nEDM.Since ρ tt is in the 3rd quadrant in Eq. ( 4), in the convention of Eq. ( 7), arg ρ tt = −3π/4.A scan plot of the variation of Eq. ( 7) is given in Fig. 4 for illustration.For negative arg ρ uu , nEDM is closer to the PSI bound (red and yellow scan points), and for the largest |ρ tt | = 0.3 √ 2 (right plot), the bound cuts a little bit into the scan space.But interestingly, for positive arg ρ uu , i.e. opposite the sign of arg ρ tt , the blue scan points extend below 10 −27 e cm, which can evade n2EDM of PSI.Therefore, the scan in Fig. 4 illustrates a general cancellation mechanism that may well be operative in Nature for neutron EDM.It can be probed, however, at more advanced nEDM facilities, such as the nEDM experiment under construction at the Spallation Neutron Source [31] at Oak Ridge National Lab (ORNL), which utilizes UCN and can probe down to 10 −28 e cm.Although this may go beyond the next decade, the possibility appears to be covered fully, as the blue scan points tend to run out by 10 −28 e cm.
Thus, if g2HDM is the source of EWBG, the combined effort of eEDM and nEDM experiments seem poised for major discoveries in the coming decade or two.
Discussion and Summary.-Thiswork was actually stimulated by the ability at the LHC to probe top CPV, i.e. top chromo-moments [32].As this is a new beginning, top chromo-moment bounds are still rather weak.We realized instead that prospects for electron and neutron EDMs are rather good in g2HDM.
We have kept H, A and H + degenerate at 500 GeV and have not revisited EWBG, but we have checked that features at 300 GeV are quite similar, where baryogenesis should be more efficient.The actual parameter space should therefore be considerably larger.For example, breaking the degeneracy, one would need to face precision electroweak constraints [1], where either one keeps m A = m H + (custodial symmetry), or take the twistedcustodial [33] case of m H = m H + .
We have emphasized as our theme that it is nontrivial that g2HDM can provide electroweak baryogenesis while surviving the eEDM constraint, a remarkable feat rooted in the flavor structure as revealed by the SM sector.With exotic H, A and H + bosons sub-TeV in mass, search programs at the LHC [19] have started, while there are also some good flavor probes [22].Any BSM theory of EWBG would need to face the litmus test of surviving the eEDM bound [3].
We may sound optimistic in the discovery prospect for eEDM at 10 −30 e cm.Note that both the JILA

FIG. 1 .
FIG. 1.A two-loop Barr-Zee diagram for electron EDM with extra Yukawa coupling ρee on electron line, and top (hence ρtt) and W run in the gray blob for neutral scalar ϕ = h, H, A. Neutron EDM has many more contributions, including u-and d-quark chromo-moments and the Weinberg operator.

FIG. 3 .
FIG.3.Combined scan result for r [0.6, 0.8] for electron and neutron EDM for same range of ρtt and exotic Higgs masses as in Fig.2, with ρ f f fixed according to Eq. (5).