Nucleon Electric Dipole Moments and the Isovector Parity- and Time-Reversal-Odd Pion-Nucleon Coupling

The isovector time-reversal- and parity-violating pion-nucleon coupling $\bar{g}_{\pi}^{(1)}$ is uniquely sensitive to dimension six interactions between right-handed light quarks and the Standard Model Higgs doublet that naturally arises in left-right symmetric models. Recent work has used the $\bar{g}_{\pi}^{(1)}$-induced one-loop contribution to the neutron electric dipole moment $d_n$, together with the present experimental $d_n$ bound, to constrain the CP-violating parameters of the left-right symmetric model. We show that this and related analyses are based on an earlier meson theory $d_n$ computation that is not consistent with the power-counting appropriate for an effective field theory. We repeat the one-loop calculation using heavy baryon chiral perturbation theory and find that the resulting $d_n$ sensitivity to $\bar{g}_{\pi}^{(1)}$ is suppressed, implying more relaxed constraints on the parameter space of the left-right symmetric model. Assuming no cancellations between this loop contribution and other contributions, such as the leading order EDM low-energy constant, the present limit on $d_n$ implies $|\bar{g}_{\pi}^{(1)}| \lesssim 1.1\times 10^{-10}$.

(Dated: January 22, 2014) The isovector time-reversal-and parity-violating pion-nucleon couplingḡ (1) π is uniquely sensitive to dimension six interactions between right-handed light quarks and the Standard Model Higgs doublet that naturally arises in left-right symmetric models. Recent work has used theḡ (1) π -induced one-loop contribution to the neutron electric dipole moment dn, together with the present experimental dn bound, to constrain the CP-violating parameters of the left-right symmetric model. We show that this and related analyses are based on an earlier meson theory dn computation that is not consistent with the power-counting appropriate for an effective field theory. We repeat the one-loop calculation using heavy baryon chiral perturbation theory and find that the resulting dn sensitivity toḡ (1) π is suppressed, implying more relaxed constraints on the parameter space of the left-right symmetric model. Assuming no cancellations between this loop contribution and other contributions, such as the leading order EDM low-energy constant, the present limit on dn implies |ḡ (1) π | < ∼ 1.1 × 10 −10 .

I. INTRODUCTION
Electric dipole moments (EDMs) of neutral atoms, molecules, and the neutron provide sensitive probes of new sources of time-reversal (T) and parity (P) violation. Current upper limits on the EDMs of the 199 Hg atom [1], d A ( 199 Hg), and neutron [2], d n , place tight constraints on the QCD vacuum angle within the Standard Model (SM) as well as on possible sources of sources of CP-violation (CPV) arising from physics beyond the SM (BSM) 1 . The existence of BSM CPV is needed in order to explain the cosmic baryon asymmetry (for a recent review, see Ref. [3]). If the asymmetry had been generated at temperatures of order the electroweak (EW) scale, then d n provides a particularly sensitive probe.
At energies below the scale of BSM interactions Λ but above the EW scale, one may characterize the effects of BSM CPV interactions in terms of an effective theory involving only SM fields: where L CKM and Lθ denote the SM Cabibbo-Kobayashi-Maskawa (CKM) [4] and QCD vacuum angle [5][6][7] CPV Lagrangians, respectively, and gives the set of non-renormalizable CPV effective operators at the weak scale v = 246 GeV generated by BSM physics at a scale Λ > v. For brevity, we have indicated only those entering at dimension (d) six, while the + · · · indicate those of higher dimension 2 . Among the more widely considered d = 6 CPV operators are the elementary fermion EDMs, the quark chromo-EDMs, and the Weinberg three-gluon operator.
In this study, we focus on one particular d = 6 operator that naturally arises in left-right symmetric model (LRSM) extensions of the SM, that gives rise to EDMs of nucleons, nuclei, and diamagnetic atoms, and that has received considerably less attention than the other operators that arise at this order in the effective theory: where ϕ is the Higgs doublet,φ = iτ 2 ϕ * , D µ is the SU(2) L ×U(1) Y covariant derivative and u R (d R ) is the righthanded up-quark (down-quark) field. In LRSMs, the corresponding Wilson coefficient C ϕud is generated by mixing between the left-and right-handed W bosons in the presence of either spontaenous CPV and/or explicit CPV in the right-handed quark CKM matrix. After electroweak symmetry-breaking wherein ϕ T → (0, v/ √ 2), the exchange of the W ± contained in the covariant derivative with a left-handed quark current leads to an effective four quark interaction 3 The interaction in Eq. (4) breaks isospin symmetry, thereby giving rise to, among other interactions, the isovector TVPV πN N interaction: where N and π 0 are the nucleon and neutral pion fields, respectively. This interaction leads to long-range contributions to the nuclear Schiff moment that induces d A ( 199 Hg) as well as long-range contributions to d n that can be computed in chiral perturbation theory. The present limits on these EDMs, thus, imply constraints on the mass M WR of the right-handed W -boson and associated CPV phases in the LRSM. Following this line of reasoning, the authors of Refs. [11,12] have derived constraints on M WR and the strength of spontaneous CPV in the LRSM from the limits on d n and the corresponding implications of CPV in the neutral kaon sector. The results imply that M WR > 10 TeV. In related work, the authors of Ref. [13] observed that O ϕud will also induce a semi-leptonic CPV operator that contributes to neutron decay. Even without specifying to the LRSM, the d n limits on C ϕud then constrain the magnitude of possible effects in T-odd neutron decay correlations. In both cases, the d n constraints relied on an earlier pion-loop calculation performed by the authors of Ref. [14] using a relativistic meson-nucleon field theory approach. The results indicate that the leading term in d n resulting from the interaction (5) is proportional to the neutron anomalous magnetic moment κ n and is independent of the pion-to-nucleon mass ratio, m π /m N . From the standpoint of effective field theory (EFT), this result is surprising, as the anomalous magnetic moment vertex brings in an inverse power of the nucleon mass while consistent power counting in chiral perturbation theory requires that loops only bring in momenta of order of the pion mass. The absence of any m π /m N suppression in the computation of Ref. [14] is not consistent with this expectation.
In what follows, we repeat the pion loop computation associated with (5) using heavy baryon chiral perturbation theory (HBChPT) [15] and show that the result proportional toḡ (1) π κ n is suppressed by (m π /m N ) 2 ∼ 0.02. HBChPT implements the power counting required by an EFT by expanding about both the static nucleon (m N → ∞) and chiral (m π → 0) limits. Our results imply considerably weaker constraints on C ϕud from the long-range contribution to d n than obtained in the studies of Refs. [11][12][13]. Presently uncalculable short-distance contributions associated with loop momenta of order one GeV that reside in the nucleon EDM counterterm may imply stronger constraints as suggested by naïve dimensional analysis (NDA). In this context, one may view the relativistic meson theory computation of Ref. [14] as providing a model estimate of the short-distance contributions. Generally speaking, however, both the sign and magnitude of NDA and/or model estimates for the short distance contributions are subject to uncertainty, so the most conservative implications will be inferred from the calculable long-distance terms.
In this respect, we note that the diamagnetic EDMs provide an in principle more robust benchmark than d n , as the nuclear Schiff moment arises from tree-level pion exchange between two nucleons and is relatively free from the uncertainties associated with short-distance contributions. In practice, the many-body nuclear theory uncertainty associated with the interaction (5) are substantial [9], with the situation for 199 Hg being particularly unsettled. Looking to the future, storage-ring searches for EDMs of light nuclei such as the deuteron or 3 He nucleus [16] would provide theoretically cleaner probes of O ϕud since the short-distance contributions to such EDMs are relatively minor and since the few-body nuclear theory is well under control [17]. In the immediate term, however, the long-range contribution to d n appears to be the most trustworthy avenue for accessing O ϕud .
In the remainder of this paper, we discuss the details of our calculation. In Section II we summarize the HBChPT framework as it applies to the computation of TVPV observables and give the details of our nucleon EDM computation. In Section III we compare our results with those of Ref. [14]. We discuss the implications and summarize in Section IV.

II. HBCHPT CALCULATION
Loop computations performed with a relativistic meson-nucleon field theory and dimensional regularization include explicit contributions from loop momenta of order m N , thereby eliminating the possibility of a consistent power counting 4 . In HBChPT [15], one removes these contributions by first redefining the nucleon degrees of freedom in terms of heavy fields having fixed velocity v where with k being a residual momentum. We henceforth omit the "v" subscript. Derivatives acting on the heavy fields give the small residual momenta, and the propagator of a heavy-nucleon field no longer contains the nucleon mass. The results of loop integrals involving the N fields then scale with powers of Q/m N and Q/Λ χ , where Q is of order m π or the external momentum (assumed to be well below one GeV), Λ χ = 2πF π is the scale of chiral symmetry breaking, and F π = 186 MeV is the pion decay constant 5 . One, thus, obtains a consistent EFT power counting. The HBChPT interactions are constructed from the heavy nucleon and pion fields, the velocity v µ , and the spin S µ with S = ( σ/2, 0) in the nucleon rest frame v = ( 0, 1). It is also useful to project vectors in their components parallel and orthogonal to the velocity. We use a subscript ⊥ to denote the perpendicular components. For example, the perpendicular covariant derivative is We note that HBChPT operators are manifestly invariant under rotations and translations. Under boosts, the HBChPT operators are only invariant order by order in the Q/m N expansion, leading to non-trivial constraints on the interactions in the effective Lagrangian and on their coefficients [15,19]. The resulting, leading order (LO) P -and T -conserving Lagrangian for f fermion fields is (see Ref. [20] for more details) where g A = 1.27 is the pion-nucleon axial-vector coupling. At LO, electromagnetism can be included by making the derivatives in Eq. (9) covariant under U em (1), where e is the proton charge, e > 0 and D = 1 + π 2 /F 2 π . For brevity, in the following we omit the label "em". The anomalous magnetic moment that is of particular interest to this work first appears at next-to-leading order (NLO) in Q/m N , together with the nucleon kinetic energy, a recoil correction to the axial pion-nucleon coupling, the nucleon sigma term, and the proton-neutron mass difference Here, κ 1 and κ 0 are the isovector and isoscalar anomalous magnetic moments (AMMs), κ 1 = 3.7, κ 0 = −0.12, and ∆m N and δm N the nucleon sigma term and nucleon mass splitting originating in the quark masses. A lattice calculation found δm N = 2.26 MeV [21], while evaluations of ∆m N range between 45 and 60 MeV [22]. The "+ · · · " in Eq. (11) denote other operators with chiral index ∆ = 1, like the pion mass splitting term, which are not relevant for our discussion. In Eq. (11), the nucleon AMM couples to the magnetic field, and this interaction does not contribute directly to the nucleon EDM. However, the AMM does couple to the electric field at next-to-next-to-leading order (NNLO) through the spin-orbit interaction that appears in ∆ = 2 Lagrangian: The first term in Eq. (12) is a relativistic correction to g A , while the second term is the spin-orbit nucleon-photon coupling. Reparametrization invariance fixes the coupling constant of the interactions in Eq. (12) in terms of the couplings in Eqs. (9) and (11) [15,19]. Next we list the TVPV interactions originating in the four-quark operator in Eq. (4). The construction of these interactions has been discussed in detail in Ref. [23], and here we only summarize the results. As already mentioned, the isovector TVPV pion-nucleon interaction in Eq. (5) with couplingḡ (1) π is induced at leading order. Additionally, the quantum numbers of the neutral pion allow the existence of a pion tadpole term. This tadpole can be removed from the Lagrangian via a field redefinition of the pion and nucleon fields [24]. These field redefinitions induce an additional LO contribution to the isovector TVPV pion-nucleon that can be absorbed intoḡ (1) π . The tadpole removal also induces a contribution to the isoscalar TVPV pion-nucleon interaction Though formally LO, this contribution vanishes in the isospin limit m u = m d and is suppressed by the ratioḡ We find that there exist no NLO corrections proportional toḡ (1) π andḡ (0) π , because they depend on structures like (1 ± v /)(1 ∓ v /) that vanish. The first non-vanishing corrections proportional toḡ (1) π appear at NNLO and are given by For brevity, we do not list the remaining NLO and NNLO TVPV interactions [23] that are not relevant for the present discussion. One-loop diagrams contributing to wave function renormalization. The notation is as in Fig. 1.
Finally, the combination of Eq. (4) and electromagnetic interactions gives rise to the EDM interaction whered 0 andd 1 are, respectively, the isoscalar and isovector short-range contributions to the nucleon EDM and the "+ · · · " stand for terms with one or more pions or relativistic corrections not relevant for the discussion here.
We now compute the contributions to the nucleon EDM generated by the interactions in Eqs. (5,9,(11)(12)(13)(14)(15). In the HBChPT framework, the EDM enters as a TVPV contribution to the nucleon EM current: where q µ is the four-momentum of the outgoing photon and The dots in Eq. (16) denote relativistic corrections to the leading order current. The neutron (proton) EDM is given by d 0 − d 1 (d 0 + d 1 ). For virtual photons, the nucleon EDMs become the electric dipole form factors (see, e.g., Refs. [25][26][27]). For purposes of our calculation, however, we focus on the static limits.
Contributions to the nucleon EDM arise from tree-level diagrams involving the short-range interactions in Eq. (15) and from one-loop diagrams involving the TVPV pion-nucleon vertices. The largest one-loop diagrams involve the Pand T -conserving vertices from Eq. (9) only, while corrections appear due to insertions of vertices from Eq. (11) or (12).
The contributions to d 0,1 proportional toḡ (1) π andḡ (0) π up to order O(Q/m N ) have been computed previously in Refs. [27,28], and they are independent of κ 0,1 . The reason being that O(Q/m N ) electromagnetic interactions given in Eq. (11) do not contribute to the nucleon EDM since the interaction with the external field is purely magnetic in the rest frame of the nucleon. The contributions proportional to the AMMs therefore enter at O(Q/m N ) 2 and are generated by the vertex correction diagrams in Fig. 1 along with the corresponding wave function renormalization graphs in Fig. 2 Figs. 1(a-c) vanish individually. We also find that the sum of contributions proportional toḡ (1) π that arise from the O(Q/m N ) 2 wave function renormalization vanish. The only surviving, non-vanishing contribution is generated by the topology of Fig. 1(d).

. Becauseḡ
At O(Q/m N ) 2 , an additional set of contributions to the proton EDM proportional toḡ (1) π are generated by the graphs in Fig. 3. The latter contain only the coupling to the nucleon charge and various combinations of (a) O(Q/m N ) corrections to the πN N interactions and nucleon propagator as well as (b) O(Q/m N ) 2 corrections to one of the couplings or propagator and LO interactions for the remaining components of the diagram. Finally, at this order in principle contributions appear due to two-loop diagrams involvingḡ (1) π and LO P -and T -conserving vertices. However, the sum of these diagrams vanishes.
Before giving our result, we first quote the LO and NLO results, including the contributions involving bothḡ (1) π and independent of the anomalous magnetic moment coupling. The notation is as in Fig. 1. the CPV source in Eq. (4). The LO contribution proportional toḡ (0) π has been first calculated in Ref. [28], while the NLO corrections have been calculated in Refs. [27]. Together they give The contribution ofḡ (0) π to the isovector EDM is UV divergent, with divergence encoded in the factor in terms of number of space-time dimensions d and the Euler-Mascheroni constant γ E . The divergence is absorbed by the low-energy constantd 1 , which also contains an a priori unknown non-vanishing finite contribution. It is conventional to absorb L entirely intod 1 , thereby isolating the terms non-analytic in quark mass that are unique to the loops. Notice that, at this order, theḡ (1) π terms do not contribute to the neutron EDM. For the interaction C ϕud in Eq. (4), this implies that the largest non-analytic contributions only affect the proton EDM becauseḡ At NNLO,ḡ (1) π contributes to the nucleon EDM via the diagrams in Figs. 1 and 3. At this order in the Q/m N expansion, we neglect smallḡ (0) π m 2 π /m 2 N corrections, and only give terms proportional toḡ (1) π . We find that the contributions are divergent: Again, we can absorb L and other finite pieces analytic in the quark mass into the LECsd 0,1 . It is also instructive to express results through O(Q/m N ) 2 for the neutron and proton EDM's separately: where we have also absorbed analytic terms intod p =d 0 +d 1 (d n =d 0 −d 1 ) and evaluated the result at a renormalization scale µ = m N . We observe again that unlike d p , d n contains no terms proportional toḡ (1) π linear in m π and that its non-analytic quark mass dependence first appears at NNLO. Thus, compared to the estimates based on Ref. [14], the d n contributions proportional to κ 0,1 are suppressed by two powers of m π /m N , leading to a factor of ∼ 50 suppression.
In principle, the magnitude of the finite parts of thed 0,1 , corresponding to "short range" dynamics, may be larger than those arising from the loops. In practice, the relative importance of the short-range and loop contributions depends on the properties of the four-quark operator in consideration. To illustrate, we consider the interaction of Eq. (4). Using general chiral symmetry arguments and NDA [29], we expect [9,23] where Λ χ = 2πF π ∼ m N . Substituting the estimate (23) into Eqs. (18) we have that The result in Eq. (25) is suppressed by one power of m π /m N with respect to the NDA estimate (24). The non-analytic NNLO loop contributions are suppressed by an additional factor of m π /m N with respect to NLO contributions to d 0,1 , and of m 2 π /m 2 N with respect to the short-distance low-energy constantsd 0,1 . For the isoscalar EDM, the suppression is somewhat mitigated by the enhancement due to the large value of κ 1 .
To obtain a sense of the relative importance of various terms in Eqs. (21) and (22), we assumeḡ π = 0.02 as one expects for the source in Eq. (4). In this case, the formally LO contribution to d n , proportional toḡ (0) π , and the O(m 2 π /m 2 N ) correction proportional toḡ (1) π × κ n are numerically comparable. For the proton EDM, the NLO contribution proportional toḡ (1) π m π /m N is about twice as large as the NNLO term. The two contribution enter with opposite sign, and partially cancel, so that net contributions of the two couplingsḡ (0) π andḡ (1) π to d p are also of similar magnitude. The comparison with the short-range contributions is more uncertain. If we use the NDA estimates of d n ,d p , andḡ (1) π in Eqs. (23) and (24) we find that the pion loops enter at the 10% to 20% level with respect to the short-range terms. However, deviations from the NDA expectations can affect this conclusion. More robust estimates require a first principles determination ofd n ,d p ,ḡ (0) π andḡ (1) π in terms of the coefficient C ϕud , which, at the moment, is not available (model estimates are summarized in Ref. [9]).
Finally our result for d n in Eq. (21) can be used to obtain a limit onḡ (1) π . Using the experimental upper bound |d n | < 2.9 · 10 −26 e cm [2] and assuming no cancellations with other contributions, we obtain

III. COMPARISON WITH EARLIER WORK
It is instructive to compare the HBChPT results with those of Ref. [14]. The latter compute the contribution from Fig. 4 using the TVPV interactions in Eqs. (5) and (13), together with the T-and P-conserving pseudoscalar pion-nucleon coupling the relativistic version of the nucleon propagator, and the nucleon magnetic moment Lagrangian We have repeated the calculation using these ingredients and obtain where, when expanded for small x, The result obtained in Ref. [14] is similar apart from the first term in Eq. (30) that is 1 in their work instead of 4/3 as we find. The discrepancy can be traced back to a term containing the product of Dirac matrices γ α σ µν γ α . This combination is proportional to d − 4, but comes in front of a divergent integral. In the limit d → 4, the 1/(d − 4) from the loop integral cancels the d − 4 in the product of Dirac matrices, leaving a constant piece.
Comparing with the HBChPT results in Eq. (20), we see that the relativistic calculation with pseudoscalar coupling yields contributions proportional toḡ (1) π κ 0,1 that arise at lower order in m π /m N . The zeroth order term in Eq. (29) clearly conflicts with the power counting expected in a well-behaved EFT since there appears no m π /m N suppression. Using Eq. (23) in Eq. (29) would conflict with the chiral/NDA expectations in Eq. (25), overestimating the latter by a factor of m N /m π ∼ 10.
The term non-analytic in the quark mass (m π ∼ √ m u,d ) proportional to (m π /m N )κ 0,1 in Eq. (29) respects, in principle, the chiral/NDA estimate, but in the HBChPT calculation this term is absent. We can trace this discrepancy back to the use of the pseudoscalar coupling in Eq. (27) which breaks chiral symmetry. Repeating the calculation with the relativistic axial-vector pion-nucleon coupling that conserves chiral symmetry, we find that the (m π /m N )κ 0,1 term does not appear. The first non-analytic terms that appear are then proportional to (m 2 π /m 2 N )κ 0,1 , as they should be. We refrain from giving all details, but have confirmed that by using appropriate matching conditions the non-analytic terms appearing in the HBChPT and relativistic calculation (with axial-vector pion-nucleon coupling) are the same 6 . Of course, in the relativistic calculation with pseudovector coupling there still appear analytic terms that are lower order in m π /m N that may overestimate the dependence of the nucleon EDM onḡ

IV. IMPLICATIONS AND CONCLUSIONS
From the foregoing discussion, it is evident that any phenomenological constraints obtained from experimental limits on d n and the results in Eq. (29) are likely to be at least an order of magnitude too stringent. In HBChPT, the loop contributions proportional to κ 0,1 arise at NNLO in the Q/m N expansion, while κ-independent contributions start off at NLO. For the isoscalar EDM, the NNLO contribution is numerically comparable to the NLO term due to the large magnitude of κ 1 , whereas for the neutron the NLO contribution is absent and the NNLO suppression is more severe. In terms of the "left-right symmetric" four-quark interaction (4), the NLO loop contributions and the chiral/NDA estimates for the short distance LEC are numerically comparable although the loops can be expected to be somewhat smaller due to to a m π /m N suppression. Consequently, we see no reasonable way around the m π /m N suppression that should be applied to the analysis of the LRSM. For other scenarios that lead to a different subset of the d = 6 CPV operators, such as the Weinberg operator or chiral invariant four-quark operators, the loop contributions associated withḡ (1) π will be further suppressed compared to the short distance contributions. On the other hand, a scenario in which the isovector quark chromo-EDM is the dominant d = 6 CPV source will have a similar phenomenology as the LRSM 7 .
For the LRSM with spontaneous CPV, the relaxed constraints may allow for consistency between EDMs, the kaon CPV parameter ǫ, and a right-handed W -boson (W R ) with mass of a few TeV rather than the lower bound of 10 TeV quoted in Ref. [11] (see the right panel of Fig. 8 for an illustration of the impact of relaxed EDM constraints). In this case, discovery of the W R at the Large Hadron Collider would still be possible, though it would likely imply that a second neutral CP-even Higgs boson would be too heavy to be observed.
Similar conclusions apply to the analysis of Ref. [13] that considered the implications of d n limits for the Pconserving, T-violating neutron decay correlation J · p e × p ν with a coefficient denoted by D. Separating out the final state interactions (D f ) that can mimic bona fide T-violation (D t ), the authors of Ref. [13] use the work of Ref. [12] that, in turn, relied one the relativistic meson theory calculation of Ref. [14], to conclude that |D t /κ| < 10 −7 , where κ denotes the combination of Fermi and Gamow-Teller matrix elements, g A , and the vector coupling g V that enters the correlation. This bound lies well below the computed final state interaction contribution |D f κ| ∼ 10 −5 with an uncertainty at the 10 −7 level. In this case, it would not be possible to disentangle final state contributions from possible LRSM contributions for experiment performed with 10 −7 level sensitivity. However, applying the relaxed EDM constraint we obtain here would lead to |D t /κ| < 10 −6 , opening a window for a future D-correlation experiment to observe a non-vanishing LRSM effect.