Strong CP violation in nuclear physics

Electric dipole moments of nuclei, diamagnetic atoms, and certain molecules are induced by CP-violating nuclear forces. Naive dimensional analysis predicts these forces to be dominated by long-range one-pion-exchange processes, with short-range forces entering only at next-to-next-to-leading order in the chiral expansion. Based on renormalization arguments we argue that a consistent picture of CP-violating nuclear forces requires a leading-order short-distance operator contributing to ${}^1S_0$-${}^3P_0$ transitions, due to the attractive and singular nature of the strong tensor force in the ${}^3P_0$ channel. The short-distance operator leads to $\mathcal O(1)$ corrections to static and oscillating, relevant for axion searches, electric dipole moments. We discuss strategies how the finite part of the associated low-energy constant can be determined in the case of CP violation from the QCD theta term by the connection to charge-symmetry violation in nuclear systems.

Introduction. Electric dipole moments (EDMs) of nuclei, atoms, and molecules are excellent probes of new sources of CP violation [1]. CP violation in the quark and lepton mixing matrices of the standard model (SM) leads to immeasurably small values for EDMs [2,3], implying that any nonzero measurement is either due to the so-far undiscovered QCDθ term or from beyond-the-SM (SM) sources of CP violation. Current experimental EDM limits [4][5][6] set strong constraints on BSM models with additional CP -violating phases such as supersymmetry, leptoquarks, multi-Higgs, or left-right symmetric models, and various scenarios of electroweak baryogenesis [7]. In the framework of the SM effective field theory (SMEFT), EDM limits constrain a large set of CPodd dimension-six operators at the multi-TeV scale, well above limits from high-energy collider experiments [8].
The interpretation of EDM experiments requires care. It is a non-trivial task to connect EDMs of complex objects such as nuclei or molecules to the underlying CPviolating source at the quark level. Recent years have seen significant theoretical improvements towards modelindependent first-principle calculations of EDMs from a combination of lattice QCD [9][10][11], chiral EFT (χEFT) [12][13][14], and nuclear calculations [15][16][17][18][19]. The chain of logic is roughly as follows: the SMEFT framework allows for the derivation of a general set of dimension-four (the QCDθ term) and -six CP -violating operators involving light quarks, gluons, and photons. χEFT, the low-energy EFT of QCD, is used to construct the corresponding CPviolating interactions among the relevant low-energy degree of freedoms: pions, nucleons, and photons. Each interaction in the chiral Lagrangian comes with a lowenergy constant (LEC) that encodes the nonperturbative QCD dynamics that is ideally calculated from lattice QCD (LQCD). EDMs can be then be calculated in terms of the LECs in the CP-odd chiral Lagrangian.
The derivation of the CP -odd NN potential of Refs. [19,26] is based on Weinberg's power-counting scheme [27]. In this scheme, the CP -odd potential arises from one-pion-exchange (OPE) diagrams, whose LECs can in principle be fixed from processes involving just nucleons and pions (only in principle as πN scattering experiments are not sufficiently accurate). Chiral symmetry does not forbid purely nuclear short-distance interactions with LECs that can only be fixed in nuclear systems. Indeed, in the CP -conserving potential the leading-order (LO) potential consists of OPE diagrams and two non-derivative contact interactions in 1 S 0 and 3 S 1 waves. In the CP -violating case, NN interactions require at least one space-time derivative and Weinberg's power-counting scheme predicts short-distance operators to enter at N 2 LO in the chiral expansion. This is welcome news, as it implies that nuclear EDMs can be calculated in terms of only a few LECs and ratios of EDMs can be used to pinpoint the underlying CP -violating source [28].
Weinberg's power counting scheme is based on naive dimensional analysis (NDA) of the NN LECs [29] which is not always reliable for nuclear physics. NDA does not in all cases lead to order-by-order renormalized nuclear amplitudes [30,31], as required in a consistent EFT. This is most clear in partial waves where OPE is attractive and non-perturbative, such as the 3 P 0 channel, where phase shifts show oscillatory limit-cycle-like cut-off dependence [32] that can not be renormalized at LO in Weinberg's scheme. The same problem affects external currents inserted in NN scattering states in perturbation theory [33,34]. In this work, we investigate longdistance CP -violating OPE potentials and demonstrate that renormalization requires a LO short-distance operator for 1 S 0 -3 P 0 transitions. This has direct consequences for the interpretation of EDM experiments in terms of the QCDθ term or higher-dimensional operators, and axion searches via oscillating nuclear EDM experiments [35,36].
Setup of the calculation. We first consider the case of strong CP violation from the QCDθ term. The relevant Lagrangian is given by [37,38] where q = (u d) T denotes the quark field, D µ is the color and electromagnetic covariant derivative, M = diag(m u , m d ) the quark mass matrix, m = m u m d /(m u + m d ), and the QCD angleθ. The relevant chiral Lagrangian can be constructed with well-known methods [39], and the leading CP -even and CP -odd pionnucleon interactions are given by in terms of the non-relativistic nucleon doublet N = (p n) T and the pion triplet π, g A 1.27 is the nucleon axial coupling, andḡ 0 = O m θ /F π a CP -odd LEC. The dots denote interactions involving more pions. The QCDθ term is related by a chiral rotation to the isospinbreaking component of the quark masses [22], giving a precise determination ofḡ 0 [40] where δm str N is the quark-mass induced part of the protonneutron mass splitting that has been calculated with LQCD [41] and ε = (m u − m d )/(m u + m d ). The value of g 0 agrees with a LQCD extraction [11].
From the interactions in Eq. (2) we calculate the OPE NN potentials where q = p − p is the momentum transfer between inand outgoing nucleon pairs with relative momenta p and p respectively (|p| = p and |p | = p ), and m π denotes the pion mass. In addition, we consider CP -even NN interactions in the 1 S 0 , 3 S 1 , and 3 P 0 waves where P s,t,p project respectively on the 1 S 0 , 3 S 1 , and 3 P 0 waves. In Weinberg's power counting the S-wave contact terms appear at LO while the P -wave counter term enters at N 2 LO. To obtain the strong NN scattering wave functions we solve a Lippmann-Schwinger (LS) equation in terms of a momentum space cut-off Λ. The LS equation is solved numerically for a wide range of Λ to ensure that observables are cut-off independent. We briefly discuss results for waves with total angular momentum j = 0, 1 and give explicit results in the Appendix. Solving the LS equation for just the strong OPE potential leads to 1 S 0 and 3 S 1 -3 D 1 phase shifts and mixing angles that are cut-off dependent. In the 3 P 1 and 1 P 1 waves, the strong OPE potential lead to cut-off independent phase shifts that at low energies agree well with experimental data. In the 3 P 0 channel, however, the phase shifts arising from OPE are strongly cut-off dependent and undergo a dramatic limit-cycle like behavior, see Fig. 4. In Weinberg's power counting, the regulator dependence of the S-wave phase shifts can be absorbed into the LO counter terms C s and C t but there is no counter term for the 3 P 0 channel. Following Ref. [32], we promote C P to LO and fit C s,t,p to the phase shifts at a center-of-mass energy E CM = 5 MeV. The resulting phase shifts are Λ independent for a wide range of energies demonstrating that the strong wave functions are properly renormalized. The LECs C s,t,P , of course, show significant Λ dependence, but this is of no concern as they are not observable. All results are in agreement with Refs. [32,42].
Having obtained renormalized scattering states, we insert the CP -odd potential Vḡ 0 which causes 1 S 0 -3 P 0 and 3 S 1 -1 P 1 transitions. We can treat Vḡ 0 to very good accuracy in perturbation theory and write (9) where m N is the nucleon mass. In the j = 0 channel we parametrize the S matrix by where 0 SP ∼θ denotes the small 1 S 0 -3 P 0 mixing angle. The j = 1 channel is more complicated due to strong 3 S 1 -3 D 1 mixing, and for simplicity we expand in the small S-D mixing angle . Up to O( 3 ) corrections we write in terms of two CP -odd mixing angles 1 SP and DP . S is antisymmetric in the S-P and P -D elements due to time-reversal violation. The CP -odd mixing angles 0,1 SP and DP are in principle observable in, for example, spin rotation of polarized ultracold neutrons on a polarized hydrogen target [43], but it is unlikely that these experiments can reach a sensitivity that is competitive with EDM experiments, although neutron transmission experiments using heavy target nuclei might be up to the task [44,45]. Nuclear EDMs can be written as linear combinations of the mixing angles in addition to contributions from CP-odd electromagnetic currents such as constituent nucleon EDMs.
The CP-odd mixing angles are observable and should be independent of the value of Λ. We find that this is the case for 1 SP and DP which quickly converge as shown in the top panel of Fig. 1. However, 0 SP shows an oscillatory behavior and even changes sign as function of Λ. There is no sign of convergence whatsoever. We have checked that no regulator dependence appears for any j = 2 transition after renormalizing the strong j = 2 scattering states. The difference between the behavior of the 1 S 0 -3 P 0 and 3 {S, D} 1 -1 P 1 arises from the absence of a strong counter term in the 1 P 1 channel. The observed regulator dependence arises from divergences in diagrams contributing to Tḡ 0 with topology of the left diagram in Fig. 2, where Vḡ 0 is dressed on both sides by a strong short-distance interaction (an infinite number of related LO diagrams are generated by adding additional strong interactions on either side). At LO this only occurs for 1 S 0 -3 P 0 transitions. In χEFT calculations using Weinberg's power counting, P -wave counter terms appear at N 2 LO, but are iterated to all orders in the solution of the LS equation [46]. Divergent diagrams with the topology of Fig. 2 reappear and the CP -odd transitions become regulator dependent. In practice, this might be hard to see numerically as regulators are only varied in a tiny window around Λ = 450 MeV [17,19].
The need for a counter term. The observation that 0 SP is cut-off dependent implies that CP -odd nuclear observables that depend on 1 S 0 -3 P 0 mixing cannot be directly calculated fromḡ 0 , and thusθ via Eq. (3). An observable that shows regulator dependence in an EFT calcula- tion indicates there must be an associated counter term that encapsulates missing short-distance physics and absorbs the divergence. In the present context, such counter terms are provided by short-range CP -odd NN interactions, see the right diagram of Fig. 2, of the form [13,14] which projects on 1 S 0 -3 P 0 .C 0 is a LEC that depends on Λ in such a way to make 0 SP Λ-independent. NDA sug-gestsC 0 = O(m θ /(F 2 π Λ 2 χ )) and a N 2 LO contribution, but renormalization enhancesC 0 by (4π) 2 making it LO instead.
We now show that promotingC 0 to LO indeed renormalizes the 1 S 0 -3 P 0 transition. We fitC 0 at a specific kinematical point to a fictitious measurement of 0 SP , picking 0 SP,fit = 0.01ḡ 0 at E CM = 5 MeV for concreteness. The regulator dependence ofC 0 is shown in the bottom panel of Fig. 1 and shows a limit-cycle-like behaviour driven by C P . The resulting 0 SP is regulator in-FIG. 2: Left: A particular diagram contributing to the regulator dependence of 0 SP . Solid (dashed) lines denote nucleons (pions). The square denotes an insertion ofḡ 0 while the circles denote the g A or C s vertices. The circled circle denotes an insertion of C P . Right: short-distance contribution proportional toC 0 . dependent for a wide range of energies as depicted by the dashed lines in the top panel of Fig. 1. While this method accounts for the regulator-dependent part of the shortdistance contributions and renormalizes the CP -odd amplitude, it cannot account for possible finite contributions fromC 0 . That is, the results in Fig. 1 can shift up or down (they remain flat) if we were to pick different values for 0 SP,fit . The best way to obtain the total shortdistance contribution is by fitting to a measurement of 0 SP . This is at present not possible, and even if there was data it would not be satisfactory. We would like to use such data to extract a value ofθ.
Fixing the value of the short-distance LEC. We discuss two potential methods to obtain a value forC 0 in the absence of data. The first one is to perform a LQCD calculation of NN → NN scattering in the presence of a nonzeroθ background. There have been significant recent developments in calculations of the nucleon EDM arising from the QCDθ term by applications of the gradient flow [11,47], and the same techniques could be used to study four-point functions in aθ vacuum. A major challenge will be to control the signal-to-noise. Already for CP -conserving NN → NN scattering, signal-to-noise considerations demand pion masses well above the physical point [48]. Going to smaller pion masses is even more daunting in case of CP violation from theθ term, as the signal scales as ∼θm 2 π . If such a LQCD calculation is possible, we can obtainC 0 from a matching calculation of χEFT to the lattice data after taking the appropriate continuum and infinite-volume limits.
On a shorter time-scale a more promising approach is to apply chiral-symmetry relations between theθ term and the quark masses similar to the relation betweenḡ 0 and δm str N in Eq. (3). Using SU (2) L × SU (2) R χEFT, the operators in Eq. (17) arise from the structures where χ − = u † χu † − uχ † u, u = exp(i τ · π/(2F π )), χ = 2B(M + im θ ), and B = − qq /F 2 π related to the chiral condensate. Expanding out the trace gives C 0 = (Bm θ )C 0 and a relation to the CP -conserving but isospin-breaking NN π operators [26] These operators contribute to charge-symmetry-breaking (CSB) in NN → NN π processes [49][50][51][52]. One of the LO contributions to this CSB process arises from the N ππ vertex related to δm str N by chiral symmetry The contact operator in Eq. (14) contributes at N 2 LO in Weinberg's counting (in agreement with Ref. [51] that relegate counter terms to N 4 LO in an expansion in p/Λ χ ). At the pion threshold, where final-state πN interactions can be neglected, the transition operator for the process 1 S 0 -3 P 0 + π due to Eq. (15) is of the same form as Vḡ 0 . As such, the regulator dependence seen in Fig. 1 appears and C 0 must be promoted to LO for renormalization. Unfortunately the simplest process where CSB data is available, pn → dπ 0 , is not sensitive to C 0 due to the isosinglet nature of the deuteron. This motivates an investigation of dd → απ 0 using renormalized χEFT to fit C 0 to CSB data [53], which would provide a determination ofC 0 = (Bm θ )C 0 .
Consequences for other sources of CP or P violation. At the dimension-six level in the SMEFT there appear other CP -odd sources involving light quarks. The most relevant operators for the present discussion are quark chromo-EDMs and chiral-breaking four-quark operators, which are induced in a wide range of BSM models [28,54]. In addition to the isoscalarḡ 0 term in Eq. (2), the LO CP -odd chiral Lagrangian contains an isovector term A potential isotensor term is subleading for all dimensionsix operators [13]. In combination with the strong g A vertex, an OPE involvingḡ 1 causes 1 S 0 -3 P 0 and 3 S 1 -3 P 1 transitions. Strong 3 P 1 interactions arise solely from long-distance OPE such that the divergent diagrams in Fig. 2 do not appear and we expect no regulator dependence for 3 S 1 -3 P 1 transitions. This is confirmed by explicit calculations. The j = 0 transition, up to an isospin factor, shows the same regulator dependence as theḡ 0 case and thus a LO isospin-breaking counter term is needed. The associated operator takes the form which projects unto 1 S 0 -3 P 0 , but only for the neutronneutron and proton-proton case. The simplest EDM that depends onḡ 1 is the deuteron EDM [55], which is targeted in storage-ring experiments [56]. Due to the isosinglet nature of the deuteron, its EDM only depends on 3 S 1 -3 P 1 transitions which do not require a counter term for renormalization. There is no such selection rule for more complex EDMs such as 3 He,199 Hg, or 225 Ra [16-19, 57, 58], andC 1 must be included at LO.
Finally, the finiteness of 3 S 1 -3 P 1 transitions is relevant for the field of hadronic parity (P ) violation [59]. The LO P -odd, but CP -even, chiral Lagrangian induced by P -odd four-quark operators contains a single πN term [60], usually parametrized as (h π / √ 2)N ( π × τ ) 3 N that in combination with g A leads to 3 S 1 -3 P 1 transitions [61,62]. We have checked explicitly that no regulator dependence appears and no counter terms are needed. The value of h π that has been recently determined from P -violating asymmetries in np → dγ [63], can thus be directly applied in calculations of other P -odd observables.
Conclusion. We have argued the need for a leadingorder short-range CP -violating counter term in 1 S 0 -3 P 0 transitions that affects calculations of EDMs and CP violation in neutron-nucleus scattering at the O(1) level. This directly affects the interpretation of experimental limits, and hopefully future signals, in terms of the QCD θ term and other CP -odd sources, and the interpretation of axion dark matter searches via oscillating EDMs. For CP violation from theθ term, we have proposed strategies to obtain the value of the associated low-energy constant,C 0 , from existing data on charge-symmetrybreaking in pion production in few-body systems. We hope our results stimulate determinations ofC 1 using lattice QCD and analyses of CSB data, and calculations of the impact of the short-range operator on observables of experimental interest such as (oscillating) EDMs and time-reversal-odd scattering observables.
We would like to thank Emanuele Mereghetti and Bira van Kolck for valuable discussions. We thank Nodoka Yamanaka for discussions in the initial stage of this work. JdV is supported by the RHIC Physics Fellow Program of the RIKEN BNL Research Center.

Renormalization of the strong scattering states
We consider NN scattering in the center-of-mass frame with energy E CM . The momenta of the incoming (outgoing) nucleons and their quantum numbers are denoted by p (p ) and α = α((ls)jm j , tm t ) (α ), respectively. l, s, j, m j , t, m t denote, respectively, orbital angular momentum, spin, total angular momentum, third component of total angular momentum, total isospin, and third component of isospin. We focus on the LO CP -even potential V str as given by Weinberg's power counting The LS equation is given by which we solve numerically after introducing the regulator function f Λ (p, p ) in Eq. (7). The phase shifts and mixing angles calculated using just the OPE potential are cut-off dependent in the 1 S 0 and 3 S 1 -3 D 1 channels, see the top panel of Fig. 3. This is resolved by including the short-distance counter terms C s and C t acting in the 1 S 0 and 3 S 1 waves. We fit the LECs is the result after promoting C P to LO. Bottom: Low-energy constants C s , C t , and C P as a function of Λ.
to reproduce the strong phases shifts at E CM = 5 MeV. The phase shifts then become cut-off independent for a wide range of energies, as exemplified for E CM = 50 MeV in the bottom panel of Fig. 3. The regulator dependence of C s and C t is given in the bottom panel of Fig. 4.
Using just the strong OPE potential leads to cut-off independent phase shifts in the 1 P 1 and 3 P 1 channels, see the top panel of Fig. 4. In the 3 P 0 wave, however, the strong tensor force is attractive leading to phase shifts that are very sensitive to short-distance physics and the phase shifts show a limit-cycle behaviour as a function of Λ. Unlike for the 1 S 0 and 3 S 1 channels there does not appears a counter term that can absorb this regulator dependence in Weinberg's power counting. We therefore promote the 3 P 0 counter term with LEC C P in Eq. (5) to LO and fit C P to the 3 P 0 phase-shift at E CM = 5 MeV. With this modified power counting the phase-shifts becomes cut-off independent, see top panel of Fig. 4. The regulator dependence of C P is given in the bottom panel of Fig. 4.