T violation in radiative $\beta$ decay and electric dipole moments

In radiative $\beta$ decay, $T$ violation can be studied through a spin-independent $T$-odd correlation. We consider contributions to this correlation by beyond the standard model (BSM) sources of $T$-violation, arising above the electroweak scale. At the same time such sources, parametrized by dimension-6 operators, can induce electric dipole moments (EDMs). As a consequence, the manifestations of the $T$-odd BSM physics in radiative $\beta$ decay and EDMs are not independent. Here we exploit this connection to show that current EDM bounds already strongly constrain the spin-independent $T$-odd correlation in radiative $\beta$ decay.


I. INTRODUCTION
The Standard Model of particle physics (SM) cannot account for the baryon asymmetry of the universe [1][2][3], and additional sources of CP violation might be expected to arise beyond the SM (BSM). Searches for additional time-reversal (T ) violation, and equivalently CP violation, are, therefore, promising probes of BSM physics. Especially interesting are observables with a very low SM background, such as the electric dipole moments (EDMs) of hadrons, nuclei, atoms, and molecules.
In β decay, T violation is probed by the triple-correlation coefficients D and R [4]. However, these observables are not independent from EDM measurements. In fact, the stringent neutron EDM limit bounds D more than an order of magnitude better than current β-decay experiments [5][6][7][8]. Molecular and atomic EDMs constrain scalar and tensor electron-nucleon couplings, which leads to strong constraints on the R coefficient. These constraints are several order of magnitude better than the current best β-decay bounds [9][10][11][12].
In radiative decays it is possible to study spin-independent T -odd triple-correlations [13][14][15], which are not present in β decay. In this paper, we consider such a correlation in radiative β decay generated by high-energy BSM sources of CP violation. As in β decay, we find that this T -odd correlation and EDMs are connected, which allows EDM bounds to strongly constrain the spin-independent T -odd correlation.
We work in an effective field theory (EFT) framework in which dimension-6 operators parametrize the new sources of CP violation. We first discuss these operators. Then we consider their contributions to radiative β decay in Section III, while discussing the contribution of these operators to the EDM in section IV. Finally, we give the current EDM bounds on these operators and end with a brief discussion.

II. FORMALISM
We consider the effects of new T -violating physics on the correlation K p ν · ( p e × k), where k is the photon momentum, and neglect the small T -violating SM contributions generated by the CP -odd phase of the CKM matrix and the QCD θ-term [16]. Besides these true SM T -odd sources, there are also electromagnetic final-state interactions (FSI) that mimic Tviolation and that also contribute to the triple-correlation (similar FSI contribute to D and R). These FSI have been studied for the neutron, 19 Ne, and 35 Ar [14,15,17], and contribute to the T -odd asymmetry at O(10 −3 ) − O(10 −5 ), depending on the detectable photon energy and the used isotope.
The effects of new T -violating physics, arising at a high scale Λ, can be studied in an EFT framework. At low energies, the new physics is effectively described by higherdimensional operators. We consider dimension-six operators, for which the complete set of gauge-invariant operators has been derived in Ref. [18]. We divide the operators relevant for radiative β decay into two groups.
(i) The first group consists of four-fermion operators that also contribute to β decay [18,19]. The relevant part of the effective β-decay Lagrangian is [20] where we have set V ud = 1 for convenience. G F is the Fermi coupling constant and we sum over the chirality (L, R) of the final states. These four-fermion operators modify the V − A coupling of the SM, by generating scalar/pseudoscalar (A) and tensor (α) couplings [4,21].
Besides contributing to β decay, the operators in Eq. (1) also contribute to radiative β decay after being dressed with bremsstrahlung photons.
(ii) The second group of T -violating operators is given in Table I 1 . At the scale of new physics, Λ, the relevant terms for radiative β decay, are where v ≈ 246 GeV is the vacuum expectation value of the Higgs field ϕ = 1 √ 2 v, the photon field is denoted by A µ and s w = sin θ w is the sine of the Weinberg angle (c w = cos θ w ). The where q f is the charge of the fermion. C X is the coupling constant associated with the operator Q X defined in Table I. Fig. 1 shows how these operators contribute to radiative β decay. At low energies, µ ≈ the field strengths of the SU (2) and U (1) Y gauge fields respectively. Finally, the duals of the field strengths areX µν = ε µναβ X αβ , where ε 0123 = +1.
1 GeV, after integrating out the W ± boson, we obtain where Γ µν = g µν D 2 − D ν D µ − igs w F µν , whose leading contribution to K arises from of the Q qW operators [22,23]. The dots represent terms which are necessary to maintain gauge invariance, but that do not contribute to K at leading recoil order.
The first term in Eq. (3) is similar to the interaction studied in Eq. (2) of Ref. [15].
Although we find that such a term is not T -violating when it arises from a pseudo-Chern-Simons term (i.e. Eq. (1) in Ref. [15]), it is clear that it can be generated by BSM physics such as Q ϕW B .

III. T -VIOLATING RADIATIVE β DECAY
The new sources of T violation contribute to the radiative β decay rate where M n,p are the neutron and proton masses and dΓ 0 contains the integral over the phase space. The dots represent higher-order recoil terms as well as T -even terms that are present in the SM [24]. The K coefficient can be inferred from the asymmetry [14,15] where Γ + corresponds to the φ ν range [0, π] and Γ − to the φ ν range [π, 2π] if p e is in theẑ direction, such that k and p e fix theẑ −x plane [14,15]. The asymmetry depends on the Qvalue of the interaction and on the threshold energy of the photon detector ω min , typically in the range of MeV. The asymmetry grows with increasing ω min , following Ref. [15] we evaluate the integrals at ω min = 0.3MeV. We discuss the form of K and the contribution to A for the two groups of operators.

MeV, the asymmetry is
which is in part small due to the nucleon mass suppression in Eq. (6). The asymmetry only contains quadratic couplings, which also appear in the R correlation. The constraints on these couplings from EDM measurements and the R coefficient range from O(10 −3 )−O(10 −6 ) [11,12]. Improving these bounds in radiative β decay would require a measurement of the asymmetry to better than 10 −11 . (ii) At leading order, the interactions in Eq. (3) give For ω min = 0.3 MeV, the asymmetry for neutron radiative β decay is in terms of the couplings at Λ = 1 TeV. We used g A = 1.27, s 2 w = 0.23, g = 0.64, g T ∼ 1 and M W = 80.4 GeV. For clarity we have redefined C ′ ≡ v 2 C, such that the couplings C ′ are dimensionless. Clearly, the contribution of these operators to the asymmetry is rather small.
However, the sensitivity of the asymmetry to the T -odd BSM sources can be improved by choosing isotopes with larger Q-values. For 37 K [25], we find, for example, that it is 20 times more sensitive than neutron decay. In the next Section we discuss the stringent limit from EDMs on these couplings.

IV. CONSTRAINTS FROM EDMS
The second class of operators also contribute to the neutron EDM (nEDM) and electron (eEDM). At the scale M W , these operators induce  Table II.
Q ϕud Table II shows that, at leading order C ϕud , only contributes to Ξ 1 . This interaction is generated after integrating out the W ± boson through a tree-level diagram. The relevant interaction is similar to Fig. 1b without the photon, and where the W ± should now be coupled to quarks instead of leptons [5]. As Q ϕud does not evolve under QCD renormalization, the relation in Table II is independent of Λ.

Q ϕW B
In contrast, Q ϕW B does not contribute to the interactions in Eq. (10) at the tree-level, but it induces quark EDMs at the one-loop-level. This operator also contains Higgs − γγ and Higgs − Zγ interactions. These interactions and the W W γ interaction in Eq. (2) contribute to the quark EDMs through the diagrams in Fig. 2 [26,27]. As a consequence, the operator Q ϕW B mixes with the quark EDMs when it is evolved from the scale of new physics, Λ, down to M W . The electron EDM (eEDM) is induced through the same mechanism. The results in Table II take into account both the mixing between C ϕW B and d q , and the running of d q 2 [22,23].
Q uW and Q dW Finally, the Q qW operators contribute to d u,d directly, and we have, The estimate for d dq n is in agreement with QCD sum-rule results [32,33], while the estimate of d Ξ n agrees with the results of Refs. [34][35][36]. Combining Table II, Eq. (12) and (13) with the upper limit on the nEDM, |d n | ≤ 2.9 × 10 −26 e cm [37], and eEDM, |d exp e | ≤ 8.7 × 10 −29 e cm [38], we finally obtain the bounds shown in Table III.

V. CONCLUSION
Radiative β decay offers the possibility to study a spin-independent T -violating triplecorrelation coefficient K. We have considered T -violating BSM physics arising above the electroweak scale that contributes to this correlation. The dimension-6 operators that contribute to K also contribute to the spin-dependent EDMs. The EDM limits therefore stringently constrain these operators. In fact, comparing the EDM bounds in Table II to Eq. (9), we find that improving the EDM bounds would require a measurement of the neutron asymmetry better than 10 −16 . This accuracy cannot be reached in present experiments.
In conclusion, the T -odd correlation K is not "EDM-safe" when considering CP -violating dimension-6 operators above the electroweak scale.