The Standard Model of particle physics violates time-reversal (T) invariance, but apparently only through a single phase in the Cabibo-Kobayashi-Maskawa matrix that mixes quark flavors. The resulting T violation in flavor-conserving observables is therefore very weak and static electric-dipole moments (EDMs) of neutrons, electrons, or atoms, all of which are nonzero if T is violated, have never been observed. Standard-Model T violation is also too weak to account for the baryon asymmetry of the universe, which must come from as yet undiscovered physics. Happily, most theories of what lies beyond the Standard Model contain enough phases that flavor-conserving T violation will be unsuppressed. Current levels of sensitivity in EDM experiments are already sufficient to rule out or pressure many extra-Standard models, and it seems quite possible that with slightly improved sensitivity, new T-violating physics will be discovered.
Some of the tightest constraints on T violation come from atomic EDM experiments. The best of these at present is an experiment  with Hg, but it has become clear recently [2,3,4] that atoms with octupole-deformed nuclei are potentially more sensitive than Hg. The primary reason is that given any T violation in the nucleon-nucleon interaction, an asymmetric nuclear shape and an associated parity doubling create a collective ``Schiff'' moment, a kind of radially weighted dipole moment (see below). Because of screening by atomic electrons, the Schiff moment, rather than the nuclear EDM, is the quantity that directly induces an atomic EDM (at least in lowest order; see Ref. ). In nuclei with symmetric shapes, a collective contribution to the Schiff moment develops only in fluctuations around that shape .
In this paper we calculate the Schiff moment of Ra, or more precisely, its dependence on any T-violating couplings, in a mean-field theory that allows us to break all possible symmetries, consider a variety of phenomenologically successful strong (Skyrme) interactions, implicitly include the RPA polarization of the even-even core by the valence neutron, treat both the direct and exchange parts of the full pion-mediated interaction responsible for creating the Schiff moment, and include short-range two-body correlations between nucleons that modify the effects of this T-violating interaction. Though further refinements are possible, they will probably have to include correlations beyond mean-field theory and/or careful and systematic work on Skyrme functionals; the results presented here will not be easy to supersede.
Simpler calculations of Schiff moments have been attempted before. Ref.  applied an independent-particle model in Hg, Xe, and other symmetrically deformed or spherical isotopes. Refs. [8,9] carried out a much more sophisticated RPA-based calculation in Hg; it's main drawback was the use of a single phenomenological interaction that made it difficult to estimate uncertainty. Ref.  made estimates in a particle-rotor model of the enhancement due to octupole deformation, and in Ref.  we applied a preliminary version of our technique to Ra, an experiment on which is in the works . That paper, however, assumed the range of the T-violating interaction to be zero, an especially bad approximation for exchange matrix elements, and was unable (obviously) to examine the effects of short-range correlations.
We briefly review some definitions and ideas. The Schiff moment is given by
The asymmetric shape of Ra implies parity doubling
(see e.g. Ref. ), i.e. the existence of a very
low-energy state, in this case 55keV  above the ground state
, that dominates the sum in Eq. (1) because of the corresponding small denominator. With the
the shape deformation is rigid,
ground state and its negative-parity partner in octupole-deformed nucleus are
projections onto good parity and angular
momentum of the same ``intrinsic state" (see Fig. 1), which represents the wave function of
the nucleus in its own body-fixed frame
with the total angular momentum aligned along the symmetry axis.
Equation (1) then reduces to 
The octupole deformation enhances , the first factor in Eq. (4), making it collective, robust, and straightforward to calculate with an error of a factor of two or less. The interaction expectation value is harder to estimate because it is sensitive to the nuclear spin distribution, which depends on delicate correlations near the Fermi surface. Our calculation allows the breaking of Kramers degeneracy in the intrinsic frame and, consequently, spin polarization.
we constructed a new version of
the code HFODD (v2.14e) [15,16]. The code
uses a triaxial harmonic-oscillator basis and Gaussian integration
to solve self-consistent mean-field equations for zero-range Skyrme interactions.
Evaluating matrix elements of the finite-range interaction (3) is much harder numerically, but
efficient techniques have already been developed  for Gaussian interactions,
which are separable in three Cartesian directions. The spatial dependence in
Eq. (3) is different, the derivative of a Yukawa function, and we
include short-range correlations between nucleons (which the mean-field does
not capture) by multiplying the interaction by the square of a correlation
function  that cuts off the two-nucleon wave functions below a
relative distance of about a fermi:
HFODD works with any Skyrme energy functional. In the context of the present study the best is SkO [19,20]. The ``time-even'' terms in this interaction, which act in even-nucleus ground states, were fit with special attention to nuclei around Pb and to spin-orbit splitting. The ``time-odd'' terms responsible for core-polarization in an odd nucleus were adjusted in Ref.  to reproduce Gamow-Teller resonances, resulting in an effective Landau parameter . (The isoscalar parameter was set to , following common practice.) For comparison, we also carry out the calculation with the older parameterizations SLy4, SIII, and SkM* (with time-odd terms that are not fixed by gauge invariance neglected, and then again, with the simplest time-odd terms modified so that the Landau parameters have the same values as in SkO) but have the most confidence in SkO. Ref.  presented predictions by these functionals for the binding energies, separation energies, intrinsic dipole moments, and spin-orbit splittings in the even Ra isotopes. SkO and SIII seemed to do the best job.
Table 1 shows the calculated value, with SkO, of the three
coefficients at several levels of approximation. The finite range reduces
the direct matrix elements of the interaction (and the corresponding )
from the zero-range limit  significantly. The exchange terms are
reduced much more, so that they are always smaller than the direct terms. The
effects of the short-range correlations, which also reduce the coefficients,
are relatively small as well but non-negligible. The coefficients produced by
SIII are similar to those in the table, while the other two forces give numbers
that are larger by factors of two or three, whether or not the Landau
parameters are adjusted, i.e., the effects of adjusting those parameters seem
to be fairly small. Finally, as pointed out in Ref. ,
relativistic effects in electron wave functions correct the effects of the Schiff moment; the
authors summarize the corrections in a quantity the call the ``local
nuclear dipole moment''. Our local
dipole moment in Ra is 82% of the Schiff moment for the
atomic transition and 86.5% for the
|zero-range (direct only)
|finite-range (direct only)
|finite-range + src (direct only)
|finite-range + src (direct + exchange)
What is the uncertainty in our numbers? The mean-field omits correlations that could have some effect on the result; those could be explored by refining the calculation through angular-momentum and parity projection, i.e., the restoration of symmetries broken by the mean field. In addition, an optimal Skyrme functional has yet to be identified. Those we tested give results that differ from the SkO numbers by factors of up to two or three, as mentioned above. But some low-order terms in the T-odd part of the Skyrme functional are never used even in SkO, because they have never been fit. Ref.  constrained some combinations of those terms but others were set to zero for lack of sufficient Gamow-Teller data in spherical nuclei. One might imagine trying to fit in deformed nuclei, or looking at spin-strength distributions with different total angular momentum and parity; the channel would be particularly useful because those are the quantum numbers of . At the same time, it would probably help to explicitly study the sensitivity of the Schiff moments to changes in the various Skyrme parameters, both in the time-odd and time-even sectors. With enough work on all these fronts we could give a firmer estimate of the uncertainty than our current guess: a factor of two or three.
If we accept our current results as reasonably accurate, we are in a
position to quantify the advantages of Ra for an EDM
measurement. A recent RPA calculation [8,9] of the for Hg gives
In conclusion, we have evaluated the Schiff moment of Ra in a completely symmetry-breaking mean-field approach, including in particular the finite-range matrix elements of the T-violating nucleon-nucleon interaction. The results indicate that EDM experiments in this system are very promising. The remaining uncertainties of a factor of two or three are related primarily to deficiencies in nuclear effective interactions, which can be removed but not easily.
Fruitful discussions with W. Nazarewicz are gratefully acknowledged. This work was supported in part by the U.S. Department of Energy under Contracts Nos. DE-FG02-96ER40963 (University of Tennessee), DE-AC05-00OR22725 with UT-Battelle, LLC (Oak Ridge National Laboratory), DE-FG05-87ER40361 (Joint Institute for Heavy Ion Research), DE-FG02-97ER41019 (University of North Carolina); by the Polish Committee for Scientific Research (KBN) under Contract No. 1 P03B 059 27; and by the Foundation for Polish Science (FNP).