We show, at leading-order in the multipole expansion of the electron-nucleus
interaction, that nucleon electric-dipole moments are completely shielded by
electrons so that they contribute nothing to atomic electric-dipole moments,
even when relativity in the nucleus is taken into account. It is well known
that relativistic electron motion, by contrast, leads to dipole moments that
are not screened; we discuss the reasons for the difference.Comment: 4 pages, typeset by REVTeX, submitted to PR

Engel, J.

Liu, C. -P.

English

arXiv

We show, at leading-order in the multipole expansion of the electron-nucleus interaction, that nucleon electric-dipole moments are completely shielded by electrons so that they contribute nothing to atomic electric-dipole moments, even when relativity in the nucleus is taken into account. It is well known that relativistic electron motion, by contrast, leads to dipole moments that are not screened; we discuss the reasons for the difference

Liu, C. P.

Carolina Digital Repository

arXiv:0705.1981v1  [nucl-th]  14 May 2007LA-UR-07-2423Schiff screening of relativistic nucleon electric-dipole moments byelectronsC.-P. Liu∗T-16, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA 87545J. Engel†Department of Physics and Astronomy,University of North Carolina, Chapel Hill, NC, USA 27599-3255AbstractWe show, at leading-order in the multipole expansion of the electron–nucleus interaction, thatnucleon electric-dipole moments are completely shielded by electrons so that they contribute noth-ing to atomic electric-dipole moments, even when relativity in the nucleus is taken into account. Itis well known that relativistic electron motion, by contrast, leads to dipole moments that are notscreened; we discuss the reasons for the difference.∗Electronic address: cpliu@lanl.gov†Electronic address: engelj@physics.unc.edu1Some years ago, Schiff [1] showed that in the limit of a pointlike nucleus and nonrelativisticelectrons, any electric-dipole moments (EDMs) carried by the electrons and nucleus arecompletely screened by atomic polarization, so that the EDM of the atom vanishes. Shortlythereafter, Sandars [2] pointed out that in heavy polarizable atoms, relativistic correctionsto the electron-EDM operator not only survive screening but can enhance atomic EDMs.For nucleons in a nucleus, v/c is about 0.1, and one may wonder whether relativity allowsnucleon EDMs to evade screening at some level. Ref. [3] claims that it does, at a levelmuch larger than that resulting from the finite size of the nucleus. The authors use theresult to argue that the limit on the neutron EDM from experiments in 199Hg should be|dn| <∼ 2.5 × 10−26e cm, a value nearly an order of magnitude tighter than the generallyaccepted limit, |dn|<∼ 4× 10−25e cm [4], and comparable to the current best limit from directmeasurement, |dn| < 2.9 × 10−26e cm [5]. But nucleons are different from electrons in thatthey are confined to a much smaller volume. This confinement, as we show below, impliesthat relativistic contributions to nucleon EDMs are just as screened as their nonrelativisticcounterparts.To see this, let us divide the parity-and-time-reversal-conserving part of the system’sinternal Hamiltonian H0 into a piece that involves electrons only — the electron kineticenergy and electron-electron interactions — a corresponding part that involves nucleonsonly, and a part that contains the electron-nucleon interaction:H0 = Hinte +Hintnuc +He–nuc , (1)where “int” stands for “internal”. We expand electron–nucleus part of H0 (which is thesame Hamiltonian as the H0 in Eq. (2.3) of Ref [3]) in multipoles:He–nuc =Z∑i=1Z e2ri+ . . . , (2)where ri ≡ |ri| is the coordinate of the ith electron. We shall neglect in this discussionthe sub-leading terms “. . .”, which involve nuclear moments (static and local) beyond thelowest-order monopole. These higher moments, corresponding to what are often called“finite-size effects”, do in fact allow Schiff screening to be evaded, but at a low level that issystematically discussed in Ref. [6].We can now calculate corrections to the ground-state energy induced by the parity- andtime-reversal-violating interaction of the nucleon dipole moments with an external electricfield Eext. The first order shift is∆E(1) = −〈g.s.|A∑j=1djN γ0j Σj |g.s.〉 ·Eext , (3)where γ0j =(1 00 −1)is the Dirac matrix for the jth nucleon, Σj =(σj 00 σj)is the spinoperator for that nucleon, djN is the magnitude of the EDM for the same nucleon, and |g.s.〉is the ground state of the unperturbed Hamiltonian H0. Often the relativistic operator inEq. (3) is divided into two pieces by writing γ0 as 1 + (γ0 − 1); the second piece is thenpurely relativistic (see, e.g., Ref. [2]). We do not make this split because for nucleons bothpieces will be shielded.2To see that no portion of the dipole operator is left unscreened in the point-like (lowest-order monopole) limit, we now calculate the second-order addition to Eq. (3) from polariza-tion of the electrons. With the nucleus kept in its ground state (see below for justification),this contribution is∆E(2) =−∑n1Eg.s. − En〈g.s.|(A∑j=1djN γ0j Σj)·(Z∑i=1∇iA0)|n〉× 〈n|eZ∑k=1rk ·Eext|g.s.〉+H.c. , (4)where the label n on the states refers to electronic configurations, and A0 denotes the electricpotential at the origin (where the nucleus is located) generated by the electrons:A0 =Z∑i=1eri= −1Z eHe–nuc . (5)The gradient ∇i A0 in Eq. (4) is then just the electric field at the origin produced by theith electron. In writing these expressions we have assumed only that the full wave functionfactors into products of atomic and nuclear wave functions to good approximation.The truncation of the multipole expansion at leading order — the point-like approxima-tion — allows us to restrict attention to the electric field at the origin and write He–nuc interms of electron operators only in Eq. (5). As a consequence, the nuclear state is not per-turbed; excited nuclear states do not contribute to Eq. (4), even if the coupling of nucleonsto Eext is included (nuclear polarization). For there to be a contribution, the excitationswould have to be created by a nuclear operator with positive parity (i.e., Σ) and destroyedby one with negative parity (i.e., r). That can’t happen, however, because H intnuc is symmetricunder reflection so that all unperturbed nuclear states have good parity.These facts allow us evaluate sum in ∆E(2). From Eq. (5) it follows that(A∑j=1djN γ0j Σj)·(Z∑i=1∇iA0)= −1Z e[A∑j=1Z∑i=1djN γ0j Σj ·∇i , He–nuc]= −1Z e[A∑j=1Z∑i=1djN γ0j Σj ·∇i , H0 −Hinte −Hintnuc]. (6)Now, noting that1. the electron-electron interaction is pair-wise, so thatZ∑i=1[djN γ0j Σj ·∇i , Hinte ] =Z∑i=1djN γ0j Σj · [∇i , Hinte ] = 0 , (7)and2. the nucleus is not excited so that the commutator of any nuclear operator with Hnucyields a vanishing expectation value:nuc 〈g.s.|[djN γ0j Σj ·∇i , Hintnuc]|g.s.〉nuc = nuc 〈g.s.|[djN γ0j Σj , Hintnuc]|g.s.〉nuc ·∇i = 0 ,(8)3we can eliminate the energy denominators in ∆E(2) and perform the sum over intermediatestate in closure. The result is∆E(2) =1Z〈g.s|[(A∑j=1Z∑i=1djN γ0j Σj ·∇i),Z∑k=1rk ·Eext]|g.s.〉=〈g.s.|A∑j=1djN γ0j Σj |g.s.〉 ·Eext = −∆E(1) , (9)so that the first- and second-order contributions cancel each other exactly. Thus, atomicpolarization screens the nucleon EDMs even in relativistic quantum theory. The finite sizeof the nucleus, leading to a difference between the monopole and dipole charge densities,is still the dominant nuclear contribution to the atomic EDM. Relativistic corrections tonuclear wave functions affect the result only a little because their contributions to densitiesare of O(v2/c2) ≈ 1%.Why then do relativistic corrections to electron EDM operators evade screening? Thereason is that the γ0 in the relativistic operator doesn’t commute with the relativistic free-electron Hamiltonian γ0(me + p · γ), so that the first commutator in Eq. (7) doesn’t vanishif the Dirac matrices act on electrons (see Refs. [2, 6, 7] for details). By contrast, Eq. (8)vanishes even if one uses a relativistic form for the nuclear Hamiltonian because of theexpectation value. And as mentioned above, off-diagonal nuclear matrix elements contributenothing because of the parity symmetry of H intnuc.In summary, at leading-order in the multipole expansion for He–nuc, where the electronssee the nucleus as a point particle, even fully relativistic nucleon EDMs are screened byelectron polarization. The effects of relativity in the nucleus will only add small correctionsto the usual finite-size effects encoded in the nuclear “Schiff moment”.AcknowledgmentsThis work was supported in part by the U.S. Department of Energy under ContractsDE-AC52-06NA25396 (CPL) and DE-FG02-97ER41019 (JE).[1] L. I. Schiff, Phys. Rev. 132, 2194 (1963).[2] P. G. H. Sandars, J. Phys. B 1, 511 (1968).[3] Sachiko Oshima, Takehisa Fujita, and Tomoko Asaga, Phys. Rev. C 75, 035501 (2007).[4] V. F. Dmitriev and R. A. Sen’kov, Phys. Rev. Lett. 91, 212303 (2003).[5] C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006).[6] C.-P. Liu, M. J. Ramsey-Musolf, W. C. Haxton, R. G. E. Timmermans, and A. E. L. Dieperink,arXiv:0705.1681 [nucl-th].[7] E. Lindroth, B. W. Lynn, and P. G. H. Sandars, J. Phys. B 22, 559 (1989).4

Schiff Screening of Relativistic Nucleon Electric-Dipole Moments by Electrons

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