Maximal Electric Dipole Moments of Nuclei with Enhanced Schiff Moments

The electric dipole moments (EDMs) of heavy nuclei, such as 199Hg, 225Ra and 211Rn, can be enhanced by the Schiff moments induced by the presence of nearby parity-doublet states. Working within the framework of the maximally CP-violating and minimally flavour-violating (MCPMFV) version of the MSSM, we discuss the maximal values that such EDMs might attain, given the existing experimental constraints on the Thallium, neutron and Mercury EDMs. The maximal EDM values of the heavy nuclei are obtained with the help of a differential-geometrical approach proposed recently that enables the maxima of new CP-violating observables to be calculated exactly in the linear approximation. In the case of 225Ra, we find that its EDM may be as large as 6 to 50 x 10^{-27} e.cm.


Introduction
Electric dipole moments (EDMs) are among the most promising potential signatures for CPviolating physics beyond the Standard Model (SM), and one of the most promising options for extending the SM is provided by supersymmetry (SUSY) [1]. The minimal SUSY extension of the SM (the MSSM) already contains many possible CP-violating phases, even in its minimally flavour-violating (MFV) version. The maximally CP-violating MFV version of the MSSM, the MCPMFV model [2,3], has six CP-violating phases, to which may be added the QCD vacuum phase θ QCD . These phases are tightly constrained by the present experimental upper limits on the EDMs of 205 Tl, the neutron and 199 Hg. Nevertheless, one could in principle envisage (accidental) cancellations [4] between the contributions to the measured EDMs of the six (seven) phases of the MCPMFV model (including θ QCD ), which might leave open the possibility of large net contributions to other EDMs. However, for any fixed values of the CP-conserving MCPMFV parameters, the compact ranges of the CP-violating parameters imply that the value of any other CP-violating observable, e.g., an EDM, is necessarily bounded. The question then arises whether the prospective sensitivity of any proposed experiment reaches below the maximum value attainable in any given theoretical framework, such as the MCPMFV model (with or without the possibility that θ QCD = 0). Clearly, any experiment that has insufficient sensitivity to search below the maximum value is not interesting for testing the MCPMFV model, whereas any experiment capable of reaching below the maximum value may either make a measurement or exclude part of the MCPMFV parameter space.
In a recent paper [5] we proposed a novel analytical technique, based on a differentialgeometrical construction (see also [6]), for finding the maximal values of CP-violating observables subject to the existing EDM constraints, which is exact in the linear approximation. We applied this technique to find maximal values of the EDMs of the Deuteron and muon, the CP-violating asymmetry in b → sγ decay, A CP , and the B s mixing phase [5]. We found that, whereas the EDM of the Deuteron in the MCPMFV model (allowing also for θ QCD = 0) might be one (two) orders of magnitude larger than the prospective experimental sensitivity, and A CP might also be detectable, the EDM of the muon and the contribution to the B s mixing phase in the MCPMFV model are likely to be too small to be observable in the near future.
In this paper we extend the applications of our analytical method [5,6] to calculate the maximal values in the MCPMFV of the EDMs of some nuclei that are enhanced by the Schiff moment contributions associated with nearby parity-doublet states [7]. An experimental campaign is now being considered for HIE-ISOLDE that could search for EDMs of radium isotopes [8], accompanied by measurements of octupole collectivity in radium isotopes [9] that would be needed to interpret EDM measurements in terms of time-reversal violating interactions.
We find that values of the 225 Ra EDM that are considerably larger than 10 −27 e · cm (well within the estimated sensitivity of the proposed HIE-ISOLDE experiment) are possible in the MCPMFV model. How much larger depends quite sensitively on the implementation of the experimental constraint on the EDM of 199 Hg. As discussed below, several theoretical calculations of the EDM are available, and most of them give very similar allowed ranges for the EDM of 225 Ra, in the range 6 to 10 × 10 −27 e · cm. However, one theoretical calculation of the 199 Hg EDM yields much weaker constraints on the CP-violating parameters of the MCPMFV, and hence yields a much larger maximal value of the 225 Ra EDM, namely 50 × 10 −27 e · cm.

Schiff Moments for Selected Nuclei
A CP-violating atomic EDM may arise from intrinsic EDMs of the constituent nucleons and atomic electrons, CP-odd electron-nucleon interactions, and the CP-odd nuclear moment known as the Schiff moment. The Schiff moments of several atoms have been calculated in the literature, and expressed in terms of sums of products of CP-even and CP-odd pionnucleon-nucleon (πNN) couplings. We recall that the CP-even strong πNN interaction is given by [10] L strong where g πN N = 13.45. On the other hand, the CP-odd (T-odd) πNN interactions are given by [10,11] πN N N τ a Nπ a +ḡ (1) πN N N Nπ 0 +ḡ (2) πN N N τ a Nπ a − 3Nτ 3 Nπ 0 =ḡ in terms of the isoscalarḡ πN N , and isotensorḡ (2) πN N T-violating pion-nucleon couplings.
The Schiff moment is linear in the CP-odd πNN couplingsḡ (i) πN N , and may be written as [12] S = (a 0 + b) g πN Nḡ Here the coefficients a i specify the dependence of the Schiff moment on the CP-odd interactions and the coefficient b specifies its dependence on the nucleon dipole moments. The coefficients a i with i = 0, 1, 2 and b depend on the type of atom of interest, and some theoretical estimates for 199 Hg, 225 Ra and 211 Rn, in units of e · fm are collected in Table 1.
In the case of the 199 Hg Schiff moment, the results [13] adopted in [11,14,15] are significantly different from the more recent calculations of [12,16]. In particular, the coefficient a 0 may be enhanced by a factor of ∼ 30, whilst the coefficient a 1 may be reduced by a factor of ∼ 10 and could even take the opposite sign. After considering the different numbers from the most recent calculations [12] in Table 1, we have used the SIII (HF) calculation for our numerical illustration. As motivation, we note that the first SLy4 and the SV calculations do not yield the right quantum numbers for the ground state, that the Table 1: The coefficients a i , i = 0, 1, 2 and b of the Schiff moments of 199 Hg, 225 Ra, and 211 Rn, expressed in units of e · fm 3 . The labels HB and HFB stand for calculations in the Hartree-Fock and Hartree-Fock-Bogoliubov approximations, respectively: see Ref. [12] for details. We have changed the signs of the coefficients a 0 , a 1 , and b to follow the conventions of Ref. [11]. The CP-odd couplingsḡ (i) πN N may be generated by chromoelectric dipole moments (CEDMs) of the quarks and/or dimension-six four-fermion interactions. In Ref. [20], one may find full expressions for the contributions to the CP-odd isoscalarḡ (0) πN N and isovector g (1) πN N couplings from the CEDMs of the light quarks * : (2.4) We note that a 'best value' is not available forḡ πN N /ḡ (1) , as follows from assuming that the couplingsḡ (0) πN N andḡ (1) πN N are proportional to the matrix elements ofūu −dd andūu +dd in the nucleon state, respectively. It is known that the matrix element with the minus (−) sign is smaller than the one with the plus (+) sign by a a factor ∼ 5 to 10 [21].
We have also included the contribution of the dimension-six four-fermion interactions to the isovector coupling which can be enhanced for a large value of tan β [22,23]: where the couplings appearing in (2.5) are defined via the interaction Lagrangian and κ ≡ N|m ss s|N /220 MeV ≃ 0.50 ± 0.25. The contribution of the four-fermion interactions to the isoscalar couplingḡ πN N is ignored and the isotensor couplingḡ (2) πN N , which changes the isospin by two units, is neglected in this work (as in Ref. [20]), since it can be generated only at the expense of an additional m u − m d suppression.
In the following, we concentrate on the EDMs of 199 Hg and 225 Ra atoms, because there is already a stringent experimental upper limit on the former, and there is a proposal to measure the latter at HIE-ISOLDE [8]. Moreover, we note that the estimates of the contributions to the Schiff moment of 211 Rn shown in Table 1  with the coefficient taking the values C S Hg = −4 [26,27], −2.8 [28] and +5.07 [29]. We note that we have changed the sign of the most recent calculation based on relativistic coupledcluster theory [29], so as to match the conventions in the previous calculations [26][27][28] based on coupled perturbed Hartree-Fock calculations, and that it has a different sign. For our numerical calcluatons we take the estimates of the coefficient C S Hg that have negative signs. To be specific, we have used the following four calculations of the Mercury EDM: • From Ref. [11], πN N /GeV , (2.9) • Taking C S Hg = −2.8 [28] and the coefficients from Ref. [13], πN N /GeV , (2.10) • Taking C S Hg = −2.8 [28] and the average coeffcients from Ref. [16], • Taking C S Hg = −2.8 [28] the SIII(HF) coeffcients from Ref. [12], On the other hand, the EDM of 225 Ra is related to its Schiff moment by [28] d (2.13) Taking the coefficients given in Ref. [18], see Table 1, we obtain We note that theḡ (1) πN N contribution the EDM of 225 Ra is about 200 times larger than to the Mercury EDM d I Hg [S], an enhancement due to the existence of a nearby parity-doublet states [17].

Differential-Geometrical Optimization Method
We briefly review our powerful analytical approach for finding the optimal choice of CPodd phases which maximize the size of a given CP-violating observable O, while remaining compatible with the present EDM constraints [6]. We have applied this approach previously to estimate maximal values of the Deuteron and muon EDMs, the CP asymmetry in b → sγ, and the phase in B s mixing [5], and it may be applied similarly to the case where the observable O is the EDM of 225 Ra.
We consider a theory such as the MCPMFV SUSY model with six CP-odd phases, Φ, represented by a 6D phase vector, subject to three EDM constraints denoted by E a,b,c = 0, corresponding to the non-observation of the Thallium, neutron and Mercury EDMs. For any given value of the CP-conserving parameters in the MCPMFV model, we may expand these EDMs and the observable O in the small-phase approximation, defining the four 6D vectors E a,b,c = ∇E a,b,c and O = ∇O, and we assume that the four vectors E a,b,c and O are linearly independent.
We then introduce the triple exterior product where the Greek indices label the components of the vectors in the 6D space, i.e., α, β, γ = 1, 2, . . . , 6. The square brackets on the RHS of (3.15) indicate that the tensor A αβγ is obtained by fully antisymmetrizing the vectors E a α , E b β and E c γ in the indices α, β, γ, i.e., A αβγ = −A βαγ = −A αγβ , etc. Borrowing a term from the calculus of differential forms, A αβγ is a 3-form. We also introduce the 2-form where summation over repeated indices is implied and ε µνλρστ is the usual Levi-Civita tensor generalized to 6D. In the language of differential forms, B µν is, up to an irrelevant overall factor, the Hodge-dual product between the 1-form O λ , representing the CP-violating observable, and the 3-form A αβγ .
The components Φ * α of the optimal EDM-free direction maximizing O can now be obtained from the Hodge-dual product of the 3-form A βγδ and the 2-form B µν . Explicitly, where we have included an unknown overall normalization factor N . By construction, the 6D phase vector Φ * is orthogonal to the three vectors E a,b,c , and therefore satisfies the desired EDM constraints, E a = E b = E c = 0, in the small-phase approximation. We observe that the magnitude φ * ≡ |Φ * |, and hence the overall normalization factor N , can only be determined by a numerical analysis of the actual experimental limits on the three EDMs. As in the 3D example, the maximum allowed value of the CP-violating observable O is given in the small-phase approximation by where the caret denotes the components of a unit-norm vector. As discussed in [5], quadratic and higher-order derivative terms with respect to the CP-odd phases will generically prefer a particular sign for the optimal value of O.
We can also allow for the possible presence of a non-zero strong CP phase θ QCD in the theory, in which case the corresponding CP-odd phase vector Φ becomes seven-dimensional (7D) in the MCPMFV SUSY model. The generalization of the above construction of the optimal value of the observable O is discussed in Section 5 of [5].

Results
In this Section we use the above the differential-geometrical approach to analyze the maximal value of the 225 Ra EDM obtainable in CP-violating variants of the following representative CMSSM benchmark scenario which predicts the mass spectrum of SUSY particles in the sub-TeV region: at the GUT scale, introducing non-zero CP-violating phases and varying tan β (M SUSY ). We adopt the convention that Φ µ = 0 • , and we vary independently the following six MCPMFV phases at the GUT scale: In the 7D case, in addition to the 6 CP phases, we consider a non-zero strong CP phase θ QCD . The scenario (4.19) considered in this work is similar to one with |M 1,2,3 | = 250 GeV that we considered previously [2,5,14]. The somewhat larger value |M 1,2,3 | = 350 GeV is chosen here for consistency with the recent SUSY search results reported by the CMS Collaboration at the LHC [30]. When tan β = 10, Φ 1,2,3 = 0 • , and Φ Au,A d ,Ae = 180 • , the previous scenario with |M 1,2,3 | = 250 GeV became the well-known SPS1a point [31] (also known as benchmark B). The scenario (4.19) is more similar to benchmarks C, G and I of [31] when tan β = 10, 20 or 35, respectively. Our calculations of the EDMs are based on Refs. [5,14], which include the two-loop diagrams mediated by the γ-H ± -W ∓ and γ-W ± -W ∓ couplings, and we take into account the effects of the different computations of the Schiff moment of the Mercury nucleus as explained in Section 2.

The MCPMFV SUSY model with 6 CP phases
In order to analyze the scenario (4.19), we first make Taylor expansions of the following four EDMs in terms of the MCPMFV CP-violating phases: where we choose the following normalization factors: which correspond to the current experimental limits on the EDMs of Thallium [32], the neutron [33], and Mercury [24,25]. The normalization factor for the 225 Ra EDM, namely 10 −27 e · cm, is typical of the estimated experimental sensitivity.
In Fig. 1, we show the absolute values of the components of the three 6D MCPMFV vectors characterizing the existing EDM constraints and the 6D vector representing the d Ra observable, for the scenario (4.19) varying tan β in a small-phase expansion around the CP-conserving point The solid lines represent the CP-violating phases of the gaugino mass parameters, and the dashed lines the trilinear A parameters, respectively. The components corresponding to the CP-violating gaugino phases dominate in all cases, increasingly as tan β grows, with the exception of the Φ 3 component of E d Tl . For example, when tan β = 40, |d Tl | and |d n | are larger than the current experimental limits by factors ∼30 and ∼5, respectively, even when Φ 2 = 1 • , whereas |d I Hg | is larger than the current limit by a factor ∼15 (∼8) when Φ 3 = 1 • (Φ 2 = 1 • ) ‡ . We see that |d Ra | can be as large as ∼ 100 × 10 −27 e cm (∼ 40 × 10 −27 e cm) if  The relative contributions of the different CP-violating MCPMFV phases to d Hg vary according to the choice of theoretical calculation, as shown in Fig. 2 § . Specifically we observe the Φ 3 contribution exhibits strong variations, and that it is much suppressed if the |d IV Hg | calculation is used. In Fig. 3 we show the cosines of the angles between the 6D MCPMFV vector repre- § We again display (as magenta lines) the corresponding components for possible 7th components corresponding to the QCD phase θ, which we discuss later.   senting the observable O d Ra and the EDM vectors, which are defined by    Having the vectors representing the EDM constraints and the observable d Ra in hand, we now combine them to construct the optimal directions in the 6D space of CP-violating MCPMFV phases, using (3.17), so as to maximize d Ra in the linear approximation. For comparison, we also consider two reference directions, which have Φ 1 = Φ Ae = 0 and Φ 2 = Φ 3 = 0, respectively. These two reference directions can be constructed by defining where, for each direction, the two null directions N for the direction Φ 2 = Φ 3 = 0.
We display in Fig. 4 the absolute values of the six components of the normalized optimal vectors in the direction along which d Ra is maximized, as obtained using d I Hg (upper left), d II Hg (upper right), d III Hg (lower right), and d IV Hg (lower left) for the Mercury EDM. We first observe that the Φ 1,2,3 components (solid lines) are relatively small, and decrease as tan β increases. Hence, all the optimal directions are mostly given by some combination of Φ Au (black dashed line) and Φ A d (red dashed line) directions implying, for tan β = 40, that (Φ Au ,A d ) max ∼ φ * whereas (Φ 1,2,3 ) max ∼ φ * × 10 −2 , as will be shown in the following.
In Fig. 5, we consider the products Φ * · O of the 6D vectors in the normalized optimal directions for d Ra , and the Radium EDM, taking account of the uncertainty of the Mercury EDM calculation. The products determine the sizes of d Ra along its optimal direction through the relations given in (3.18) when φ * = 1 • . As shown below, φ * could be as large as ∼ 100 • before the small-phase approximation breaks down and one of the three EDM constraints is violated. We observe that the direction constructed using the geometric prescription given in Section 3 indeed gives the larger values of d Ra than do the two reference directions with Φ 1 ,Ae = 0 and Φ 2 ,3 = 0. We note in particular that, in the case of d IV Hg , the Radium EDM may become about an order of magnitude larger than is possible with another choices for the Mercury EDM calculation.  respectively.
In Fig. 7, we show the maximal values of d Ra attainable in the 6D case after imposing the three EDM constraints. We see the that small-phase approximations for d I ,II ,III Hg break down for φ * ∼ 40 • , as seen by comparing with the calculations along the two reference directions, and that d Ra can be as large as ∼ 6 × 10 −27 e · cm or more. On the other hand, in the case of d IV Hg , d Ra can be as large as ∼ 50 × 10 −27 e · cm. Finally, in Fig. 8 we show the 6 CP-violating phases at the GUT scale (top and middle) and the 3 CP-violating phases of the third-generation A parameters at the SUSY scale (bottom). We observe that the CP phases of the gaugino mass parameters Φ 1 , Φ 2 and Φ 3 can only be as large as 2 • , 0.4 • and 0.4 • , respectively, whereas Φ Au (Φ A d ) at the GUT scale can be as large as ∼ 100 • (60 • ), as previously seen in Fig. 4. These CP-violating phases are suppressed at the SUSY scale by RG running from the GUT scale [2], but sizeable non-trivial CP-violating phases are still allowed at the SUSY scale:

The 7D Case of non-zero θ QCD
Hitherto, we have implicitly assumed that the CP-violating QCD θ-term: is negligible, whereG µν,a ≡ ǫ µνρσ G a ρσ /2 and the parameterθ is given by the sum of the QCD θ QCD and the strong chiral phase for the quark mass matrix: In the weak basis where Arg Det M q = 0, we haveθ = θ QCD .
The dimension-four operator (4.26) would in general contribute to the neutron, Mercury and Radium EDMs, e.g., through the CP-odd pion-nucleon-nucleon interactions (2.2). Explicitly, for the neutron EDM, we use the estimate [5,11] d n (θ) ≃ 2.5 × 10 −16θ e · cm. (4.28) For theθ-induced Mercury EDM, we neglect the contribution from theḡ where C I Hg = 1.8, C II Hg = 1.0, C III Hg = 1.4 and C IV Hg = 9.5 × 10 −2 , with [5,34] g (1) πN N (θ) ≃ 1.1 × 10 −3θ . (4.30) Finally, for the Radium EDM, we use We have analyzed the possible maximal values of d Ra in this 7D case including θ following a procedure similar to that we used in the 6D case. Looking again at Fig. 1, we note the horizontal magenta lines representing the 7th components of the vectors representing the present EDM constraints on d Tl (upper left), d n (upper right), and d Hg (lower left), and of the vector representing the EDM of 225 Ra (lower right). We observe that the 7th component is missing in the d Tl case, because this observable has no contribution from the θ term in our approach. On the other hand, the θ component is close to unity for d n (cf. the discussion at the end of the previous paragraph), ∼ 10 −1 for d I Hg , and somewhat less than unity for d Ra . This implies that a measurement of d Ra at the level of 10 −27 e cm would already be a competitive measurement of θ, even in the absence of the other MCPMFV phases.
As seen in Fig. 2, we have made similar analyses using the d II ,III ,IV Hg calculations, finding similar θ components in the first two cases, but a value about an order of magnitude smaller in the d IV Hg case. We have also analyzed (not shown) the cosines of the angles between the observable vector O d Ra and the EDM-constraint vectors E d Tl , E dn and E d I, II, III, IV Hg in the 7D model, as functions of tan β. We find results that are very similar to the 6D case shown in Fig. 3, the most significant difference being quite small and limited to tan β < 10 in the d n case.
We turn now to Fig. 9, which is the analogue of Fig. 4, but including the QCD θ term (4.26). We see significant differences at intermediate and large tan β, where we note that the optimal vectors in all four models for d Hg exhibit not only relatively large θ components, but also larger components for the gaugino mass phases than in the 6D case. We have also analyzed (not shown) the products Φ * · O for the optimal d Ra direction in the 7D space, as obtained using the calculations d I, II, III, IV Hg for the Mercury EDM, finding results that are generally very similar to those shown in Fig. 5 with a little rise around tan β = 40. Figure 10 displays the values of the Thallium, neutron and Mercury EDMs along the directions optimized for d Ra , analogously to Fig. 6 Fig. 6, we see that the magnitudes of the Thallium and Mercury EDMs increase quickly as φ * deviates from 0 and it could be as large as only about 50 • . Turning to the resulting 7D estimates of d Ra shown in Fig. 11, we see that they have somewhat smaller magnitudes than the 6D estimates shown in Fig. 7 in the linear approximation. However, larger values of the CP-violating gaugino phases are allowed in the 7D case than in the 6D case, as seen by comparing Figs. 8 and 12. Specifically, we observe that Φ 1 , Φ 2 and Φ 3 could be as large as ∼ 15 • , ∼ 4 • and ∼ 1 • , respectively. We also see in Fig. 13 that sizeable θ could be much larger than the upper limit of ∼ 10 −10 usually quoted, with values as large as θ ∼ 2.5 × 10 −9 becoming possible in the presence of non-zero MCPMFV phases.

Conclusions
In this paper we have extended our previous analyses of the MCPMFV model, with its 6 CP-violating phases to determine (in the linear approximation) the largest value of d Ra that is allowed by the present constraints on the neutron, Thallium and Mercury EDMs, using the differential-geometric approach developed in [5]. Numerically, we obtain rather similar results whether we include the CP-violating QCD vacuum phase θ in the analysis, or not.
The results are much more sensitive to the theoretical treatment of the Mercury EDM constraint, and we compare the results obtained with four different calculations of d Hg . Three of them yield quite similar results for d Ra , but one calculations indicates a smaller dependence of d Hg on the CP-violating phases of the MCPMFV model, and hence allows larger numerical values of these phase and, in general, larger values of d Ra become possible. The maximal values we find for d Ra using three of the d Hg calculations are typically ∼ 6 × 10 −27 e · cm or more, whereas the fourth calculation allows d Ra ∼ 50 × 10 −27 e · cm.
For comparison, we recall that there is a proposal to measure d Ra with an sensitivity approaching ∼ 10 −27 e · cm in one day of data-taking. This experiment would clearly have interesting potential to probe regions of the MCPMFV parameter space that have not been  This potential surely extends to many other models with several sources of CP violation, which could also be analyzed using the differential-geometric approach [5] exploited here. As long as a limited number n of EDMs have been bounded (or measured) by experiments, any model with N > n CP-violating parameters will be underconstrained, and (partial) cancellations [4] are possible that would allow large values for other CP-violating observables. The MCPMFV model is one such example, in which N = 6 (or 7 if θ is included) and n = 3, so far. As we have shown in this paper using this differential-geometric approach, constraining or measuring d Ra at the level of ∼ 10 −27 e · cm or better would be a valuable addition to the existing arsenal of experimental probes of CP violation.
However, it would still not complete the set of constraints needed for a model with N > 4, such as the MCPMFV. For this reason, other measurements, e.g., of the CPviolating asymmetry in b → sγ decay. We have discussed elsewhere the maximal value that this observable might take in the MCPMFV, and it would be an interesting and  We conclude by drawing reader's attention to potential caveats in the physics of CP violation. The baryon asymmetry of the Universe is one of the strongest pieces of evidence for physics beyond the Standard Model, as it cannot be generated successfully within the standard Kobayashi-Maskawa model of CP violation. There must be new sources of CP violation beyond the Kobayashi-Maskawa phase, and it behooves experiments to chase down all those within reach. Some CP-violating phases may manifest themselves at the TeV scale and be accessible to contemporary collider experiments, e.g. at the LHC. However, baryogenesis could equally well be achieved via CP-violating phases appearing at higher energy scales, and EDMs have the potential to probe beyond the TeV scale, in particular because the Standard Model Kobayashi-Maskawa predictions for EDMs are quite small. As our analysis exemplifies, new EDM observables probe complementary region of parameter space including the strong CP phase θ QCD . Therefore, constraining or even measuring d Ra is an interesting experimental objective.