Schiff Theorem and the Electric Dipole Moments of Hydrogen-Like Atoms

0.04 MB·English

by  C. -P. Liu

Schiff Theorem and the Electric Dipole Moments
of Hydrogen-Like Atoms
C.-P. Liu †, W. C. Haxton , M. J. Ramsey-Musolf‡, R. G. E.
∗ ∗∗
Timmermans and A. E. L. Dieperink
∗ ∗
6
KernfysischVersnellerInstituut,Zernikelaan25,9747AAGroningen,TheNetherlands
0 ∗
0 †TheoreticalDivision,LosAlamosNationalLaboratory,LosAlamos,NM87545,USA
2 ∗∗INTandDept.ofPhysics,Univ.ofWashington,Box351550,Seattle,WA98195-1550,USA
‡KelloggRadiationLaboratory,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
n
a
J
Abstract. The Schiff theorem is revisited in this work and the residual P- and T-odd electron–

9 nucleusinteraction,aftertheshieldingtakeseffect,iscompletelyspecified.Anapplicationismade
totheelectricdipolemomentsofhydrogen-likeatoms,whosequalitativefeaturesandsystematics
1
haveimportantimplicationforrealisticparamagneticatoms.
v
5
2
0 INTRODUCTION
1
0
6 The permanent electric dipole moment (EDM) of a physical system is an indication of
0
time-reversal(T)violationwhich,byCPT invariance,isequivalenttoCPviolation,one
/
h of the most profound puzzles in elementary particle physics. Although a neutral atom
t
- is ideal for such a precision measurement, much of its EDM evades detection because
l
c of the re-arrangement of its constituentsin order to screen the applied electric field and
u
keep thewholesystemstationary.
n
: As the shielding is not exact, many experiments have been performed over the years
v
tomeasurethesetiny,residualEDMswithgraduallyimprovedtechniquesandaccuracy.
i
X
Since an atom contains both electrons and nucleons, its EDM receives contributions
r
a from all possibleP- and T-odd (P/T/) dynamics in leptonic, semi-leptonic,and hadronic
sectors. The first part of this work is to completely specify, after incorporating the
shielding effect, the residual P/T/ electron–nucleus (eN) interaction, H . The second
eN
part concerns an application to the EDMs of hydrogen-like(H-like) atoms where some
e
general features and systematics of the contributions from different P/T/ sources are
extracted.
SCHIFF THEOREM REVISITED
The so-called Schiff theorem states the following: for a nonrelativistic system made
up of point, charged particles which interact electrostatically with each other and with
an arbitrary external field, the shielding is complete [1]. Applying this theorem to
atoms, the assumptionsof this theorem are not exactly satisfied because: (1) the atomic
electrons can be quite relativistic, (2) the atomic nucleus has a finite structure, and
(3) the electromagnetic (EM) interaction between ee or eN has magnetic components.
Therefore, a measured atomic EDM, d , or any upper bound on it, is a combined
A
manifestation of these effects. This can be best summarized by H , through which
eN
d can beexpressedas
A
e
(cid:229) 1
d ddd = 0 exxx n n H 0 +c.c. , (1)
A ≡h Ai∼ E E h | | ih | eN| i
n 0 n (cid:16) (cid:17)
−
e
where 0 and n represent theatomicgroundand excitedstates, respectively.
| i | i
Whiletheimplementationoftheshieldingeffect,whichcanbecarriedoutbyvarious
ways [see, e.g., 1, 2, 3, 4, 5, 6, 7] is too tedious to be shown here, a simplified example
below can be used to illustrate the basic points. The key relationship is to re-write
the internal EDM interaction Hd as a commutator involving the unperturbed atomic
eN
Hamiltonian,H , plussomeremainingterms, ifany,
0
e
Hd =[ddd (cid:209)(cid:209)(cid:209) ,H ] +.... (2)
eN 0
·
Substituting the commutator ineEq. (1), a closure sum can be performed and leads to
the result: ddd . Therefore, if there is no term left beyond the commutator, Hd then
eN
−h i
contributesto thetotalEDM exactlyoppositeto ddd :thisisthecompleteshielding.
h i e
ItshouldbeemphasizedthattheSchifftheoremisaquantum-mechanicaldescription
of the shielding effect, and this implies that Eq. (2) should be realized at the operator
level, i.e., every quantity is operator. While we obtained the similar expression for the
residualinteraction duetotheelectron EDM,d , as Refs. [2, 8], theresidualinteraction
e
due to the nuclear EDM, d , differs from existing literature. These differences can be
N
summarizedin theSchiffmoment,S,wefound
e 5 1 4√p
S SSS = r2rrr r2(1 Y (rˆ)) rrr , (3)
2
≡h i 10 (cid:18)h i−3 Z (cid:28)(cid:20) − 5 ⊗ (cid:21) (cid:29)(cid:19)
1
incontrastto theusual definition[see, e.g., 4, 5,6, 7].
Besides the additional quadrupole term we include in the derivation (it was usually
ignored), the main difference is a matrix element of the composite operator r2 rrr in
the former versus a product of the matrix elements r2 rrr in the latter, whhe⊗re "i "
denotes the recoupling of angular momenta. This sthemis f⊗rohmithe way we treat ddd (cid:181)⊗rrr
N
as an operator in Eq. (2), and the previous definition already takes the matrix element
ddd (cid:181) rrr before the derivation. As the nucleons in the nucleus do not necessarily
N
h i h i
respondinacoherentmanneraftertheshieldingsetsin,ourdefinitionaccountsforthese
additional dynamics which is left out in the previous one. A calculation for deuteron
shows quite some difference: in S, the three terms contribute as 1 : 5/3 : 4/3; in
the traditional definition, supposed the quadrupole term is included as− r2Y − rrr ,
h 2i⊗h i 1
the ratio becomes 1 : 0.59 : 0.07. While the huge difference in d(cid:2)euteron can (cid:3)be
− −
attributedto its loosebinding,other nuclearSchiff momentsshouldberevised, because
theyreceivemostcontributionsfromthesurfaceregion,wherethebindingisusuallynot
asstrongas in thecore.
Ourderivationalsotreated thefinite-sizeeffect morecarefully,followingthesugges-
tionofRef. [7], and includedall themagneticeN interactions intoaccount. As a result,
thefullform of H contains all possiblenuclearmoments,eitherlong-ranged orlocal,
eN
chargeormagnetic.Thedetailswillbepresentedin alaterpublication[9].
e
ELECTRIC DIPOLE MOMENTS OF HYDROGEN-LIKE ATOMS
The parity admixture of an atomic ground state, 0 (1s for H-like atoms) can be
1/2
| i
solvedfromtheSternheimerequation[10].ForthecaseswherethePauliapproximation
f
is valid, the results can be expressed analytically. Separating the contributions from
(1) d , (2) C0 , a representative case in the semi-leptonic eN interaction, which is
e PS,S
isoscalarand nuclearspinindependent,(3)S,and (4)Smag,themagneticequivalenceof
S, which contains, e.g., the magnetic quadrupole moment, they roughly grow with the
atomic number Z as Z2, ZA, ZS, and ZSmag. The common factor Z1 comes from the
atomic structure calculation of 0 , and the remaining growth factor indicates how the
corresponding P/T/ interaction sc|ailes. As S and Smag both involve r2-weighted nuclear
f
moments,theyroughlyscale withA2/3.
The main reason that heavy paramagnetic atoms are suitable for constraining d is
e
usuallyjustifiedfrom aZ3 enhancementfactor[see, e.g., 11,12]. Based on thesystem-
atics found in H-like atoms, actually part of the enhancement, Z2, comes with 0 , and
| i
can be crudely understood from the fact that it takes less energy for a p-state excitation
f
from a ns state with n > 1 than from 1s . As this Z2 enhancement also applies to
1/2 1/2
other P/T/ sources, the competition between the four contributors mentioned above only
goes with Z, A, S, and Smag. Suppose the semi-leptonicor hadronic P/T/ intreactions are
large, i.e., largeC or S and Smag, then the dominanceof the d contributionis question-
e
able.Inthissense,itisbettertohaveaseriesofEDMmeasurementsandusetheirresults
toconstraintheseP/T/ sourcessimultaneously,oratleast,oneshouldsemi-quantitatively
determinetheconditionsunderwhichd isclearlythewinner.Ineithercase,thecalcula-
e
tionsofsemi-leptonicandhadroniccontributionsareindispensableforamorethorough
studyofEDMsofparamagneticatoms.
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