Enhanced Electron EDM with Minimal Flavor Violation

The latest data from the ACME experiment have led to the most stringent limit to date on the electric dipole moment de of the electron. Nevertheless, the standard model (SM) prediction for de is many orders of magnitude below the new result, making this observable a powerful probe for physics beyond the SM. We perform a model-independent study of de in the SM plus right handed neutrinos and its extension with the seesaw mechanism under the framework of minimal flavor violation (MFV). We find that de crucially depends on whether neutrinos are Dirac or Majorana fermions. In the Majorana case, de can reach its measured bound, which therefore constrains the scale of MFV to be above a few hundred GeV. We also consider extra CP-violating sources in the Yukawa couplings of the right-handed neutrinos. Such new sources can have important effects on de.


Introduction
Electric dipole moments (EDMs) constitute highly sensitive indicators for the presence of new sources of the violation of charge parity (CP ) and time reversal (T ) symmetries beyond the standard model (SM) of particle physics [1,2,3,4]. Recently the ACME experiment [5], which searched for the electron EDM, d e , utilizing the polar molecule thorium monoxide, has reported a fresh result of d e = (−2.1±3.7 stat ±2.5 syst )×10 −29 e cm, which corresponds to an upper bound of |d e | < 8.7 × 10 −29 e cm at 90% confidence level. This is more stringent than the previous best limit by about an order of magnitude, but still way above the SM expectation for d e , which is at the level of 10 −44 e cm [6]. Hence there is plenty of room between the current bound on d e and its SM value where potential new physics may be detected in future measurements.
Extra ingredients beyond the SM can boost d e considerably with respect to its SM prediction, even up to its experimental limit. Such tremendous enhancement of d e may hail from various origins depending on the specifics of the new physics models. Therefore it is desirable to carry out an analysis of d e beyond the SM which deals with some general features of the physics in a model-independent fashion. This turns out to be feasible under the framework of the so-called minimal flavor violation (MFV) which presupposes that the sources of all flavor-changing neutral currents (FCNC) and CP violation reside in renormalizable Yukawa couplings [7,8]. This offers a systematic method to organize and study possible SM-related flavor-and CP -violating new interactions.
Here we discuss an MFV treatment of d e recently performed in Refs. [9,10] within the SM with three right-handed neutrinos and its extension with the neutrino seesaw mechanism. The relevant contributions arise from effective dipole operators. As will be shown below, the predicted size of d e depends significantly on whether light neutrinos are Dirac or Majorana in nature. In the Majorana case, d e can be as large as its experimental bound, which therefore limits the scale of MFV to be above a few hundred GeV or higher [9,10].

Leptonic MFV
In the SM slightly expanded with the addition of three right-handed neutrinos, the renormalizable Lagrangian for lepton masses is given by where k, l = 1, 2, 3 are summed over, L k,L represents left-handed lepton doublets, ν l,R (E l,R ) denotes right-handed neutrinos (charged leptons), Y ν,e are matrices for the Yukawa couplings, H is the Higgs doublet,H = iτ 2 H * , and M ν is the Majorana mass matrix for ν l,R . The M ν part is essential for the seesaw mechanism to generate light neutrino masses [11].
Since it is still unknown whether light neutrinos are Dirac or Majorana particles, we deal with the two possibilities separately. If neutrinos are of Dirac nature, the M ν terms in Eq. (1) are absent, and according to the MFV hypothesis for leptons [8] the Lagrangian is formally invariant under the global group U This entails that the three generations of L k,L , ν k,R , and E k,R transform as fundamental representations of SU(3) L,ν,E , respectively, whereas the Yukawa couplings are spurions transforming according to Taking advantage of the invariance under G , we work in the basis where with v 246 GeV being the vacuum expectation value of H, and the fields ν k,L , ν k,R , E k,L , and E k,R refer to the mass eigenstates. We can then express L k,L and Y ν in terms of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix U PMNS as where m 1,2,3 are the light neutrino eigenmasses and in the standard parametrization [12] U PMNS = with δ being the CP violation phase, c kl = cos θ kl , and s kl = sin θ kl . If neutrinos are of Majorana nature, the M ν part in Eq. (1) is allowed. As a consequence, for M ν M D = vY ν / √ 2 the seesaw mechanism [11] becomes operational involving the 6×6 neutrino mass matrix in the (U * PMNS ν c L , ν R ) T basis. The resulting mass-matrix for the light neutrinos is where now U PMNS contains the diagonal matrix P = diag(e iα 1 /2 , e iα 2 /2 , 1) multiplied from the right, α 1,2 being the Majorana phases. It follows that Y ν in Eq. (5) is no longer valid, and one can instead take Y ν to be [13] where O is a matrix satisfying OO T = 1l and M ν = diag(M 1 , M 2 , M 3 ). As we see later, O can give rise to new CP -violation effects besides those from U PMNS .
To arrange nontrivial two-lepton FCNC and CP -violating interactions satisfying the MFV principle, one assembles an arbitrary number of the Yukawa coupling matrices Y ν ∼ (3,3, 1) and Y e ∼ (3, 1,3) as well as their Hermitian conjugates to devise the G representations The MFV hypothesis implies that the coefficients ξ ijk··· have to be real because otherwise they would introduce new CP -violating sources beyond what is already contained in Y e,ν . Using the Cayley-Hamilton identity X 3 = X 2 TrX +X[TrX 2 −(TrX) 2 ]/2+1l DetX for a 3×3 invertible matrix X, this infinite series can be resummed into a finite number of terms [14]: where 1l is a 3×3 unit matrix. Although ξ ijk··· are real, the reduction of the infinite series into the 17 terms can make the coefficients ξ r in Eq. (10) complex due to imaginary parts among the traces of the matrix products A i B j A k · · ·. Such imaginary contributions turn out to be proportional to the Jarlskog invariant Im Tr( 1 [15]. The implication is that, with all ξ ijk··· being expected to be at most of O(1), the impact of ξ r on d e is suppressed by a factor of m 2 µ m 2 τ /v 4 compared to the contribution from ABA 2 which has the smallest number of suppressive factor Y e among the products in Eq. (10) that contribute to d e . Therefore, hereafter we ignore Im ξ r .

Lepton EDMs with MFV
The EDM d l of a lepton l is described by L d = −(id l /2)lσ κω γ 5 lF κω , where F κω is the photon field strength tensor. In the MFV framework, this arises at lowest order from the operators in the effective Lagrangian [8] where mass scale Λ characterizes the heavy new physics underlying these flavor-violating interactions, W and B denote the SM gauge fields with coupling constants g and g , respectively, τ j are Pauli matrices, and ∆ 1,2 are of the form in Eq. (10) with generally different ξ r 's. Expanding Eq. (11), we find that the charged-lepton EDMs are proportional to Im(Y † e ∆ 1,2 ) kk , but that in ∆ 1,2 only the ABA 2 and AB 2 A 2 terms are relevant. Thus for the electron where ξ r = ξ r . If neutrinos are Dirac particles, we obtain from Eqs. (5) and (12) d invariant for U PMNS . In the Majorana neutrino case, if the right-handed neutrinos ν k,R are degenerate, M ν = M1l, and O is a real orthogonal matrix, from Eq. (9) we have and consequently neglecting the ξ 16 term. Since m k M, obviously d D e is highly suppressed relative to d M e . In the preceding paragraph, d e depends on the CP -violating Dirac phase δ in U PMNS , and the Majorana phases α 1,2 therein do not enter. However, if ν k,R are not degenerate, nonzero α 1,2 can lead to an extra effect on d e even with a real O = 1l. If O is complex, its phases may induce an additional contribution to d e , whether or not ν k,R are degenerate. We explore these scenarios numerically later.
To evaluate d e , we need the values of the various pertinent quantities, such as the elements of U PMNS as well as the masses of neutrinos and charged leptons. In Table 1, we have listed sin 2 θ kl and δ from a recent fit to global neutrino data [16]. Most of these numbers depend on whether neutrino masses fall into a normal hierarchy (NH) or an inverted one (IH). Since the Majorana phases α 1,2 remain undetermined, we will select specific values for them in our illustrations.
For Dirac neutrinos, we scan the empirical ranges of the parameters from Table 1 to maximize d D e in Eq. (13). The result in the NH or IH case is d D e = 1.3 × 10 −99 ξ 12 (GeV 2 /Λ 2 ) e cm. This is negligible compared to the latest experimental upper bound [5]. Table 1. Results of a recent fit to the global data on neutrino oscillations [16]. The neutrino mass hierarchy may be normal (m 1 < m 2 < m 3 ) or inverted (m 3 < m 1 < m 2 ). If neutrinos are Majorana fermions, in contrast, d e can be sizable. We start with the simplest possibility that ν k,R are degenerate, M ν = M1l, and the O matrix in Eq. (9) is real. Thus d e is already given in Eq. (15). Scanning again the empirical parameter ranges in Table 1 where M is specified below. Hence |d e | exp < 8.7 × 10 −29 e cm [5] translates intô Since d M e in Eq. (16) is proportional to M 3 , one might naively think that d M e can easily reach its measured bound, which would therefore constrainΛ to an arbitrarily high level with a very large M. However, the size of M is capped based on the condition that the series in Eq. (10) which supposedly includes arbitrarily high powers of A and B must converge [9,15]. Therefore, in this work, we require that the biggest eigenvalue of A not exceed 1. Accordingly, we arrive at M = 6.16 (6. In general OO † = e 2iR , where R is a real antisymmetric matrix with nonzero elements denoted by r 1 = R 12 = −R 21 , r 2 = R 13 = −R 31 , and r 3 = R 23 = −R 32 . Since OO † is not diagonal, the Majorana phases in U PMNS can also enter A if α 1,2 = 0. We concentrate first on the CP -violating effect of O by setting α 1,2 = 0. For illustrations, we pick two possible sets of r 1,2,3 , namely, (i) r 1 = −r 2 = r 3 = −ρ and (ii) r 1 = 2r 2 = 3r 3 = ρ, and employ the central values of the data in Table 1. In Figure 1 we present the resulting d M eΛ 2 versus ρ for the NH (IH) of light neutrino masses with m 1(3) = 0. Since δ is not yet well-determined, we also depict the variations of d M e over the one-sigma ranges of δ quoted in Table 1 with the lighter blue and red bands. Within these bands, the blue and red solid curves belong, respectively, to the NH and IH central values in the table. We also graph the (dashed) curves for δ = 0 to reveal the CP -violating role of O alone.
With α 1,2 = 0, the CP -violating effect of O can still occur even if it is real, provided that ν k,R are nondegenerate, in which case from Eq.    evident from these graphs that the Majorana phases produce additional important CP -violating impact on d e beyond δ.
For the EDMs of the muon and tau lepton, one could do a similar analysis. However, their experimental limits are still much weaker than |d e | exp . Consequently, they lead to bounds onΛ which are not competitive to that in the electron case [9,10].

Conclusions
We have investigated the electron EDM, d e , under the MFV framework and found that d e can reach its experimental limit if neutrinos are Majorana in nature. Moreover, from the latest data on d e , we infer that the MFV scale has to be a few hundred GeV or higher. We demonstrate that d e has the potential to probe not only the Dirac phase in the lepton mixing matrix, but also the Majorana phases therein, as well as extra CP -violation sources in the Yukawa couplings of the right-handed neutrinos.