Relations between matrix elements of different weak interactions and interpretation of the PNC and EDM measurements in atoms and molecules

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by  V. A. Dzuba

Relations between matrix elements of different weak interactions and interpretation of
the PNC and EDM measurements in atoms and molecules
V. A. Dzuba, V. V. Flambaum, and C. Harabati
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
(Dated: January 26, 2012)
The relations between matrix elements of different P- and T-odd weak interactions are derived.
We demonstrate that similar relations hold for parity nonconserving (PNC) transition amplitudes
2 andelectronelectricdipolemoments(EDM)ofatomsandmolecules. Thisallows toexpressP-and
1 T-oddeffectsinmany-electronsystemscausedbydifferentsymmetry-breakingmechanismsviaeach
0 otherusing simple analytical formulas. Weusethese relations for theinterpretation of theanapole
2 moment measurements in cesium and thallium and for the analysis of the relative contributions
of the scalar-pseudoscalar CP-odd weak interaction and electron EDM to the EDM of Cs, Tl, Fr
n
a and other atoms and many polar molecules (YbF, PbO, ThO, etc.). Model-independent limits on
J electron EDMandtheparameterofthescalar-pseudoscalar CP-odd interaction arefoundfrom the
analysis of theEDM measurements for Tl and YbF.
5
2
PACSnumbers: 11.30.Er,31.15.A-

]
h
p I. INTRODUCTION The extraction of the nuclear anapole moment from
- the PNC measurements for thallium [15] also presents a
m
problem. The analysis of the experimental data based
The study of the parity and time invariance viola-
o on atomic calculations favors a negative value for the
tion in atoms, molecules and nuclei is a low-energy, rel-
t
a atively inexpensive alternative to the high-energy search thallium anapole moment [15, 16] while all nuclear cal-
. culations produce positive values [17].
s for new physics beyond the standard model (see, e.g. a
c review [1]). Accurate measurements of the parity non- Since the analysis based on sophisticated atomic cal-
i
s conservation(PNC)inatomsisoneofthemostpromising culations is very complicated it is important to have ad-
y waysofexploringthispath. Itculminatedinveryprecise ditional tools which may help to check the calculations
h
measurementsofthePNCincesium[2]. Interpretationof forconsistencyorevenavoidthe calculations. Inpresent
p
the measurementsbasedonaccurateatomic calculations paper we build such tools by studying the relations be-
[
led to perfect agreement with the standard model and tween matrix elements for different symmetry-breaking

3 putstrongconstrainsonanynewphysicsbeyondit[3,4] operators. We show that using these relations the result
v
(see also review [1] for more detailed discussion). First foraspecificoperatorcanbeeasilyobtainedanalytically
2
unambiguousmeasurementof the nuclearP-oddanapole ifthecalculationsforanotheroperatorareavailable. The
8
0 momentwasalsoreportedintheCsPNCexperiment[2]. relations can also be used to check different calculations
6 At present,the study ofsymmetry violationsin atoms for consistency. Additional benefit comes from the need
. andmoleculesgoesmostlyinthreemajordirections(see, to separatethe effects of different sourcesto the PNC or
9
0 e.g. [5]): (i) the PNC measurements for a chain of iso- EDM of atoms or molecules in the analysis of the exper-
1 topes;(ii)themeasurementsofnuclearanapolemoments; imental data. Recent measurement of T and P violation
1 and (iii) the measurements of the P,T-odd permanent in YbF molecule [18] combined with the Tl EDM mea-
: electric dipole moments of atoms and molecules. Inter- surement [19] would allow to obtain independent limits
v
i pretation of the anapole moment and EDM measure- onelectronEDMandCP-violatinginteraction. However,
X ments requires sophisticated atomic or molecular cal- the relative sign of these contributions is different in dif-
r culations. The calculations are difficult and sometimes ferent calculations [9, 20]. This signproblem is solvedin
a
disagree with each other. For example, the calcula- the present work by calculating the ratio of the matrix
tions of the thallium electric dipole moment (EDM) by elements. ThenindependentlimitsonelectronEDMand
Natarajet al[6]andbySahooet al[7]disagreewithear- scalar-pseudoscalarCP-odd interaction is presented.
lier calculations [8, 9], calculations of the nuclear spin- There is an active search for the CP-violating per-
dependent PNC in cesium by Mani and Angom [10] dis- manent EDM of polar molecules, such as YbF [18],
agree with earlier calculations [11–14], etc. The diffi- PbO [21, 22], ThO [23], etc. Interpretation of the
culties are mostly due to the strong inter-electron cor- measurements requires complicated molecular calcula-
relations which need to be treated to all orders of the tions. Most of calculations consider only one possible
many-body perturbation theory. Majority of publica- source of the molecular EDM: electron EDM or scalar-
tions present the effect of one particular P- or CP-odd pseudoscalarCP-oddinteraction. Weshowthattheana-
interaction for a specific atom or a molecule. If another lytical ratios of the matrix elements provide reliable link
weak interaction Hamiltonian is considered, the calcula- between molecular EDMs caused by different sources.
tions are to be done again. Therefore, if calculations exist for a particular CP-odd
2
operator,nonewcalculationsareneededtofindanEDM
TABLE I: The ratio of the matrix elements of the spin-
caused by a different operator.
dependenttospin-independentP-oddweakinteractions(R1)
The approachdeveloped in this paper is easy to apply
andscalar-pseudoscalar CP-oddinteractiontoelectron EDM
when matrix elements of the symmetry-breaking opera-
(R2) for the 6s,6p1/2 states of cesium with and without the
tor between s1/2 and p1/2 single-electron states strongly inclusion of dominating many-body effects. The ratios are
dominateoverothermatrixelements. Thisisthecasefor stable while the matrix elements change by up to two times.
thePNCamplitudesbetweenatomics-statesandforthe Units: κ/(−QW) for R1 and 10−15CPS/de a.u. for R2.
EDM of atoms and molecules caused by electron EDM
orscalar-pseudoscalarCP-oddweakinteraction. Wealso Approximation R1 R2
RHF hψHF|H |ψHFi 4.78 8.96
use this approachfor the thallium anapolesignproblem. 6s W 6p
RPA hψ6HsF|HW +δVcore|ψ6HpFi 4.88 8.94
BOa hψBO|H |ψBOi 4.78 9.03
6s W 6p
II. SINGLE-ELECTRON MATRIX ELEMENTS BO+CPb hψ6BsO|HW +δVcore|ψ6BpOi 4.85 9.01
Analytical, eq. (A10,A13,A15) 4.84 9.01
OF WEAK INTERACTION AND MANY-BODY
EFFECTS aBrueckner orbitals
bBrueckner orbitalsandcorepolarization
The PNC amplitude of an electric dipole transition
between states of the same parity a and b is equal to:
| i | i
then due to the proportionality of the wave functions
E1PNC = hb|d|nihn|HPNC|ai on short distances, the same proportionality, with the
ba (cid:20) E E samevalueofRwouldholdforanypairofsingle-electron
Xn a− n
states and for the total EDM of atom (2). Furthermore,
bHPNC n nda
+ h | | ih | | i , (1) ifthes1/2 p1/2 matrixelementsofthe weakinteraction
E E (cid:21) −
b n strongly dominate over other matrix elements then the
−
proportionalityis not affected by the many-body effects.
whered=−e iri istheelectricdipoleoperator,HPNC Indeed,ifotherweakmatrixelementsareneglectedthen
is the operatorPof a P-odd CP-even weak interaction.
any many-body expression is a sum of terms with one
The EDM of an atom in state a is given by
s p weak matrix element in each term. Since all
1/2 1/2
−
adn nHCP a of them are proportional with the same proportionality
da = 2 h | | ih | | i (2) coefficient R the proportionality would hold for the sum
E E
Xn a− n as well.
whereHCP istheoperatorofaCP-oddweakinteraction. Table I shows the ratios of the h6s1/2|HW|6p1/2i ma-
trix elements for cesium atom for different P-odd and
States a, b and n in (1) and (2) are the many-electron
atomic states. However,we can start for simplicity from CP-odd operators of weak interaction HW in the rel-
ativistic Hartree-Fock (RHF) approximation and with
an atom with one external electron above closed shells
dominating many-body effects included. The ratios are
keeping in mind cesium atom as an example. Then in
comparedwith the analyticalresult presented in the ap-
the lowest order of the perturbation theory in residual
pendix. The many-bodyeffects include corepolarization
Coulomb interaction the PNC amplitude and the EDM
(CP)andBrueckner-typecorrelations. Corepolarization
of the atom are given by (1) and (2) in which states a,
can be understood as the change of the self-consistent
b and n are single electron (e.g. Hartree-Fock) states. It
corepotentialdue tothe effectofanexternalfield(weak
is easy to see that all single-electron matrix elements of
interactionin our case). Its inclusion above the Hartree-
weak interaction are proportionalto each other. Indeed,
Fock level is often called the random-phase approxima-
only short distances contribute to the value of the ma-
tion (RPA). Brueckner-type correlations are the correla-
trix elements (see appendix for analytical estimations).
tionswhichcanbereducedtoaredefinitionofthesingle-
These distances are r < r for the PNC interactions
N
electronorbitals, replacing the Hartree-Fockones by the
and for the scalar-pseudoscalar CP-odd interaction (R
N
Bruecknerorbitals(BO)[24]. Inbothcasesofthe P-odd
isnuclearradius)andr a0/Z fortheelectronEDMop-
≤ and CP-odd weak interactions the ratios of the matrix
erator(a0 isBohrradius). Onthesedistancestheenergy
elements are stable within the 1-2% accuracy while the
ofthe single-electronstatecanbe neglectedcomparedto
the nuclearpotential Ze2/r andDiracequationsforall matrix elements change by up to two times.
−
single-electron states become identical. The only differ-
encecomesfromthenormalizationofthestates. Suppose
wehavetwodifferentoperatorsof,say, CP-oddweakin-
′ III. PARITY NON-CONSERVATION
teraction HCP and HCP. If we establish proportionality
betweenmatrixelementsforaparticularpairi,jofsingle
electron states Hamiltoniandescribingparity-nonconservingelectron-
nuclear interaction can be written as a sum of spin-
′
iHCP j =R iHCP j , (3) independent(SI) andspin-dependent(SD) parts(weuse
h | | i h | | i
3
atomic units: ~= e =m =1): It is convenient to present the amplitude in a compact
e
| |
form
HPNC = HSI+HSD
= GF QWγ5+ καI ρ(r), (4) hJf,Ff||dSI||Ji,Fii=C(S1+S2), (9)
√2(cid:16)− 2 I (cid:17)
whereC C(F ,J ,F ,J )istheangularcoefficientand
where GF ≈ 2.2225×10−14 a.u. is the Fermi constant sums S1 ≡and S2fdofnotideipend on Ff and Fi:
of the weak interaction, Q is the nuclear weak charge,
W
α = (cid:18)σ0 σ0 (cid:19) and γ5 = (cid:18) I0 −I0(cid:19) are Dirac matri- S1 = hJf||d||n,EJnihnE,Jn|HSI|Jii,
ces, I is the nuclear spin, an−d ρ(r) is the nuclear den- Xn i− n
sitynormalizedto1. Thestrengthofthespin-dependent S2 = hJf|HSI|n,Jnihn,Jn||d||Jii.
E E
PNCinteractionisproportionaltothedimensionlesscon- Xn f − n
stant κ which is to be found from the measurements.
There are three major contributions to κ arising from For the SD PNC amplitude we have
(i) electromagnetic interaction of atomic electrons with
nuclear anapole moment [25], (ii) electron-nucleus spin- hJf,Ff||dSD||Ji,Fii=
dependent weak interaction [26] , and (iii) combined ef- (I +1)(2I+1)(2F +1)(2F +1)/I
i f
fect of the spin-independent weak interaction and the q
magnetic hyperfine interaction[27] (see, also review [1]). ( 1)Jf−Ji Jn Ji 1 Jn Jf 1
Itnribthuitsiownosrtkowκedanodnoptredsiesntitntghueisrhesbuelttwseienntderiffmesreonfttcootna-l ×XJn (cid:20)d−n,J n,(cid:26)J Ib IJ Fi (cid:27)(cid:26) Ff Fi I (cid:27)
κ which is the sum of all possible contributions. h f|| || nih n|| || ii (10)
× E E
Within the standard model the weak nuclear charge n− i
QW is given by [28] +( 1)Ff−Fi Jn Jf 1 Jn Ji 1
− (cid:26) I I Ff (cid:27)(cid:26) Fi Ff I (cid:27)
Q 0.9877N+0.0716Z. (5)
W ≈− J b n,J n,J d J
f n n i
h || || ih || || i ,
Here N is the number of neutrons, Z is the number of × E E (cid:21)
n f
protons. −
The PNC amplitude of an electric dipole transition where b is the electron part of the SD weak interaction
between states of the same parity i J F M and
i i i
| i ≡ | i G
f JfFfMf is equal to: b= Fαρ(r)κ. (11)
| i≡| i √2
E1PNC = hf|d|nihn|HPNC|ii
fi (cid:20) E E Like in the spin-independent PNC amplitude (8), it
Xn i− n is convenient to present the SD amplitude in a compact
f HPNC n ndi form
+ h | | ih | | i , (6)
E E (cid:21)
f − n 4
Fwh=ereId+=J i−sethPeitroitaisl atnhgeuellaerctmriocmdeinptoulemo.perator, and hJf,Ff||dSD||Ji,Fii=Xi=1ciSi′. (12)
Applying the Wigner-Eckart theorem we can express
Here c c(F ,J ,F ) (i = 1,2,3,4) are angular coeffi-
the amplitudes via reduced matrix elements i ≡ f n i ′
cientswhichcanbe extractedfrom(10). Sums S donot
i
E1PNC = ( 1)Ff−Mf Ff 1 Fi depend on Ff and Fi:
fi − (cid:18) Mf q Mi (cid:19)
− J d J J b J
× hJfFf||dPNC||JiFii. (7) S1′ = h f|| |E| nihEn|| || ii,
Xn n− i
Detailed expressions for the reduced matrix elements of J d J′ J′ b J
the SI and SD PNC amplitudes can be found e.g. in S′ = h f|| || nih n|| || ii,
2
E E
Refs. [29] and [30]. For the SI amplitude we have Xn n− i
hJf,Ff||dSI||Ji,Fii=(−1)I+Fi+Jf+1 S3′ = hJf||b|E|JnihJEn||d||Jii,
J J 1 Xn n− f
×q(2Ff +1)(2Fi+1)(cid:26) Ffi Ffi I (cid:27) (8) S′ = hJf||b||Jn′iJn′||d||Jii.
4
hJf||d||n,Jnihn,Jn|HSI|Jii Xn En−Ef
× (cid:20) E E
Xn i− n Equation (12) has more terms than (9) due to the elec-
Jf HSI n,Jn n,Jn d Ji tron vector nature of the nuclear-spin-dependent opera-
+ h | | ih || || i .
E E (cid:21) tor.
f n
−
4
The total PNC amplitude can be presented in a form A. PNC in cesium
convenientforextractionofthevaluesofκfromthePNC
measurements (see, e.g. [31, 32]) Experimental values for the two different hfs compo-
nents of the parity non-conserving 6s 7s transition in
EPNC =E(1+R). (13) cesium are [2] −
HereE istheSIPNCamplitudegivenby(8)andRisthe
ratio of the SD to SI PNC amplitudes. In this work we EPNC(6sF=4−7sF=3)=1.6349(80)mV/cm,
are mostly interested in the values of R. Using compact EPNC(6sF=3 7sF=4)=1.5576(77)mV/cm. (19)
−
expressions (9) and (12) one can write
To extract κ we use (13) and calculate R for these
R= c1S1′ +c2S2′ +c3S3′ +c4S4′. (14) tJwo=trJan=sitJions=u1s/i2n,gQ(18=).-73.T1o9 (dsoeesEoq.w(5e))naonted tthakaet
C(S1+S2) f i n W
S1/S2 = 0.3459fromRef.[33]. Thisleadstothesystem
−
Accordingtothediscussionoftheprevioussection,ratios of equations
oftheSDandSIweakmatrixelementsdonotdependon
E(1+0.06739κ)=1.6349(80),
the principal quantum number n. Therefore, using the
ratios E(1 0.05937κ)=1.5576(77). (20)
−
r1 = Jn b Ji / Jn HSI Ji , Thesolutionforκ isκ=0.382(56). Thisresultisingood
r2 =hhJn′||||b||||Jiii/hhJn||HSI||Jiii, aagnrdeeκm=e0n.t37w5i(t5h6)thfreomvalRueefs.κ[1=2]01..39A3c(c5u6r)atfreocmalcRuelaf.ti[o1n1s]
r3 =hJf||b||Jni/hJf|HSI|Jni, (15) similartowhatisreportedinourpreviousworkforBa+,
r4 =hJf||b||Jn′i/hJf|HSI|Jni, Yb+ and Ra+ [32] lead to the value κ = 0.376 which
is in perfect agreement with the all-order calculations of
′ ′ ′ ′
we can get rid of the sums S , S , S , S , involving the
1 2 3 4 Ref. [12]. This is an illustration of the accuracy of the
SD matrix elements and replace them with the SI sums
analysis based on the ratio of the matrix elements. The
S1,S2: value κ=0.382 coming from this analysis differs by less
(c1r1+c2r2)S1+(c3r3+c4r4)S2 than 2% from the value κ=0.376 coming from the accu-
R = rate calculations. This difference is due to two simplifi-
C(S1+S2)
cations: (a) neglecting the s p matrix elements of
3/2
= (c1r1+c2r2)S1/S2+(c3r3+c4r4). (16) theSDweakinteraction,and(−b)assumingthattheratio
C(S1/S2+1) of the matrix elements is the same for all single-electron
states.
The only parameter in (14) which comes from numerical
Recent relativistic coupled-cluster calculations of the
calculations is the ratio S1/S2 of two different contri- nuclear spin-dependent PNC in Cs [10] report the val-
butions to the SI PNC amplitude (see Eqs.(8,9)). All
ues of the SD PNC matrix elements which are about
other parameters are given by analytical expressions.
30% smaller than those of the all-order calculations of
The c1,c2,c3,c4 and C parametersare just angularcoef- Ref. [12]. This is in disagreement not only with this
ficients. Theratiosr1,r2,r3,r4 canalsobeapproximated work but with all earlier calculations of the SD PNC in
by analytical expressions which will be discussed below.
cesium [11–14]. Given the proportionality of the matrix
The expression (14) can be further simplified in an
′ elements of the SI and SD weak interactions discussed
important case of an ns ns transition (e.g. the 6s-7s
− above, the results of Ref. [10] are also in disagreement
transitioninCs). Onafewpercentlevelofaccuracythe
with all most accurate calculations of the SI PNC in ce-
s p matrix elements of the SD PNC interaction can
− 3/2 sium (see, e.g. [3, 4, 33]). The latter calculations have
be neglected [32]. This means that r2 =r4 =0. We also accuracybetter than0.5%andareusedtotesttheStan-
have r1 =r3 =r and (16) is reduced to dard Model in Cs PNC experiment [2] where the accu-
c1S1/S2+c3 racy is 0.35 %.
R=r . (17)
C(S1/S2+1)
Substituting r from (A10) leads to B. Anapole moment of thallium
R=4.90(1 0.073Z2α2)c1S1/S2+c3 κ . (18) The value of the nuclear anapole moment of thallium,
− C(S1/S2+1)( QW) extracted from the measurements of the PNC in the
−
Q is the weak nuclear charge. As in (14) the only
W
parameter which comes from numerical calculations is
theratioS1/S2. Ontheotherhand,knowingthevalueof 1 Refs.[11] and [12] use different definition of κ. The conversion
R for at least two hfs components of the PNC transition
factors are (I+1/2)/(I+1) for Ref. [11] and I for Ref. [12]. I
is sufficient for extraction of κ from the measurements. isnuclearspin,I =7/2for133Cs.
5
TABLEII:Parametersξ(F,F′)ofthenuclearspin-dependent TABLE III: The ratio of the s1/2 − p1/2 matrix elements
PNC amplitude in the 6p (F) − 6p (F′) transitions in of the electron EDM operator (A15) to that of the scalar-
1/2 3/2
thallium. pseudoscalar CP-odd operator (A13). Numerical results for
′ EDMofCs,TlandYbFarealsogivenforcomparison. Units:
F F this work Ref. [16] d /(CSP10−18e cm). (Forother isotopes R′=A′R/A)
0 1 0.947 1.10 e
1 1 -0.325 -0.462 Z Atom Ratios
1 2 -0.325 -0.348 Analytical Numerical
37 85Rb 228
55 133Cs 158 163a
56 138Ba+ 152
6p 6p transition [15] is in disagreement with
1/2 − 3/2 70 173Yb+ 114 115b
the results of nuclear calculations (see, e.g. ref. [17]).
81 205Tl 89 83c
The analysis of the experimental data based on sim-
82 208Pb 88
ple single-electron approximations gives the value κa = 87 211Fr 83
0.22 0.30 [15]. The analysis based on sophisticated 90 232Th 75
m−any-b±ody calculations gives very close value of κ =
a
0.26 0.27 [16]. On the other hand, the “best value” aCsatom,Ref.[9].
−obtaine±d in nuclear calculations is κ = 0.24 [17]. To bYbFmolecule,Ref.[36].
extractthesignofκ fromtheexperimaentoneneedsthe cTlatom,Ref.[9].
a
relative sign of SI and SD amplitudes. Here we show
that simple analysis with the use of the analytical ratio
Ref. [9], the result for Yb+ is in excellent agreement
of the matrix elements of the weak interactions supports
withthemany-bodycalculationsofRef.[36]fortheYbF
the findings of Refs.[15, 16] leaving the problem of sign
molecule. We stress that the Table compares the ratios
disagreement unsolved.
of the single-electron matrix elements obtained with a
Ifwekeeponlys p matrixelementsoftheweak
1/2− 1/2 simple analytical formula to the ratios of the EDMs ob-
interactionforbothSIandSDinteractions,thenthegen-
tainedwithsophisticatedmany-bodycalculations. These
eral expression (16) can be reduced to
results provide an unambiguous link between the sign
′ and value of two different contributions to the EDM of
c(F,F )
′
R(F,F )=r , (21) atoms and molecules which have a heavy atom from the
C(F,F′)
Table III.
where r is one of the ratios r (15) which corresponds
i
′
to the s p weak matrix elements, c(F,F ) and
1/2 1/2
′ − A. EDM of polar molecules
C(F,F ) are corresponding angular coefficients. Com-
paring (13) to the parameterization used in [16] (
R ≡
Im(E1PNC/M1)) Polar molecules have strong inter-atomic electric field
which enhance the effect of electron EDM and lead to
(F,F′)=C(Z)[Q 6κξ(F,F′)], (22) molecular EDM which are several orders of magnitude
W
R −
larger than those in atomic systems. The experimen-
and substituting (A10) into (21) we get for the parame- tal search is in progress for YbF [18], PbO [21, 22], and
ters ξ ThO [23] while other molecules are also discussed in the
literature (see, e.g. [34]). Interpretation of the measure-
′
c(F,F )
ξ(F,F′)= 0.817 1 0.073Z2α2 . (23) ments requires molecular calculations. Table IV shows
− − C(F,F′) the results of most recent or most accurate calculations
(cid:0) (cid:1)
for some polar molecules. More detailed data are pre-
Correspondingvalues of ξ arecomparedin Table II with
sented for YbF molecule for which the EDM measure-
the resultsofthe many-bodycalculationsof[16]. We see
ments were recently reported [18]. The effects of elec-
that both calculations give very close results leaving no
tron EDM and scalar-pseudoscalar CP-odd interaction
room for a sign error.
areconsidered. The results arepresentedinterms of the
CP-odd parameters W and W
d c
IV. EDM OF ATOMS AND MOLECULES
Wd = Ψ0 He Ψ0 /de, (24)
h | | i
Wc = Ψ0 HSP Ψ0 /CSP. (25)
Table III shows the ratio of the s p matrix ele- h | | i
1/2 1/2
−
mentsofthe electronEDMoperator(A15)to thescalar-
To compare with other works one should keep in mind
pseudoscalar CP-odd operator (A13) for eight different
that most of them present W instead of W , where
S c
atoms calculated using analytical formulas presented in
the appendix. The results for cesium and thallium are 2
in good agreement with the many-body calculations of WS = k hΨ0|HSP|Ψ0i. (26)
s
6
violating interaction. The values of the electron EDM
TABLE IV: CP-odd interaction constants W , and W for
d c extracted from the experimental data for Tl [19] and
some polar molecules and their ratios. Effective electric
YbF[18]under assumptionthatthere is noothercontri-
field Eeff is presented together with Wd (hHei = Wdde =
bution to atomic/molecular EDM read
−Eeffde). The results of present paperare shown in bold.
Eeff Wd Wc Wd/Wc Tl: de = (6.9±7.4)×10−28 e cm, (27)
Molecule (GV/cm) (1024 Hz/e cm) (kHz) (1018/e cm) YbF: d = ( 2.4 5.7 1.5) 10−28 e cm. (28)
e
BaF 6.1a -1.5a -10 152 − ± ± ×
6.1 -1.5 -10b 152 In fact, there are other contributions from the CP-odd
YbF 31c -7.5c -59c 127 electron-nucleus interactions. Here we consider only the
19d -4.6d -41d 112 scalar-pseudoscalar interaction (A11). Other contribu-
26e -6.3e -55 114 tionsshouldbesmallduetotheconstrainsobtainedfrom
26f -6.2f -54f 115 the EDM measurements for mercury [43]. Using data
24g -5.8g -54g 108 from Table III one can rewrite (27,28) as
25h -6.1h -53 114
32a -7.7a -68 114 Tl: d +1.2 10−20CSP =
e
21 -5.2 -46b 114 (6.9 7.4×) 10−28 e cm, (29)
HgF 95a -23a -226 90 ± ×
PbF -31a 7.5a 85 88 YbF: de+8.8 10−21CSP =
×
PbO a(1) 25i -6.1i -69 88 ( 2.4 5.7 1.5) 10−28 e cm. (30)
PbO B(1) 33i -8.0i -91 88 − ± ± ×
PbO a(1) 23a -5.6a -64 88 Solving these equations for de and CSP leads to
ThO 104a -25a -336 75
ThF+ 90a -22a -290 75 de =( 2.8 3.0) 10−27 e cm, (31)
− ± ×
CSP =(3.0 3.0) 10−7. (32)
aMeyerandBohn,Ref.[34]. ± ×
bNayakandChaudhuri,Ref.[35]. The limit (31) for d is significantly weaker than those
cKozlovandEzhov,Ref.[37]. e
presented in (27) and (28). This is because the values
dTitovet al,Ref.[38].
eKozlov,Ref.[39]. of the ratios of the matrix elements of the two operators
fQuineyet al,Ref.[36]. are very close for Tl and Yb (see Table III). The most
gParpia,Ref.[40]. fortunatecaseleadingtostrongestlimits ford andCSP
e
hMosyaginet al,Ref.[41]. would correspond to very different values of the ratio,
iPetrovet al,Ref.[42].
preferablywiththedifferentsign. However,theformulas
(A15)and(A13)andthedatainTableIIIshowthatthis
is not possible. The ratio is always positive and slowly
Theconstantsk andCSP ofthestrengthoftheCP-odd
s decrease with Z. A slightly different value of the ratio
interaction are related by Zk =ACSP. Factor 2 in the
s shouldbeexpectedfortheEDMofradiumintheexcited
definition of W (26) is absent in some of the papers.
S metastable 7s6d 3D2 state. The EDM of radium in this
Most of the calculations of the molecular EDM in-
state is strongly enhanced [44] due to proximity of the
clude only one of the CP-odd effects: that of the elec- 7s7p 3Po state (the energy interval is 5 cm−1) and
1
tron EDM or the scalar-pseudoscalar interaction. We ∼
is proportionalto the 7p H 6d matrix element
use the relationsbetweenthe matrixelements of the two h 3/2| CP| 3/2i
of the CP-odd interaction H . However, the value of
CP
CP-oddHamiltonianstofillthegapsinthetable. Corre-
this matrix element is strongly dominated by the core
spondingresultsareshowninbold. Forexample,accord-
polarization effect which in turn is mostly due to the
ing to Ref. [34] the effect of electron EDM in the YbF
s p matrix elements between core and excited
molecule is W = 15 1024 Hz/e cm. Using the ratio 1/2 − 1/2
W /W = 114d 1−018/e×cm for Yb+ from Table III we states. Therefore, the ratio of the two contributions is
d c × notverydifferentfromwhatwasconsideredaboveforTl
found that the effect of the scalar-pseudoscalar interac-
and YbF.
tion is W = 132 kHz.
c
−
C. EDM of Tl and Fr
B. Extraction of the electron EDM and the
parameter of the scalar-pseudoscalar CP-odd EDM of thallium due to electron EDM and scalar-
interaction from the EDM measurements for YbF pseudoscalarCP-odd interactionwas recently calculated
and Tl
in our paper [9]. The ratio of the two contributions
was found to be 83d /(CSP10−18e cm) which is in
e
Recent measurement of T and P violation in YbF good agreement with the analytical result of this pa-
molecule [18] combined with the Tl EDM measure- per: W /W = 89/(10−18e cm). The EDM enhance-
d c
ment [19] and the data from Table III allows to obtain ment factor is 582 [9] which is in very good agree-
−
independent limits on the electron EDM and the CP- ment with the value 585 found in earlier calculations
−
7
by Liu and Kelly [8]. The EDM enhancement factor
for thallium was also calculated in more recent work of
Ref. [6] and found to be -466 (dTl = 466de) which is
−
about 25% smaller than the results mentioned above.
The EDM of thallium due to the scalar-pseudoscalar
CP-odd interaction calculated by the same group earlier
(dTl = 4.06 10−18CSPe cm [7]) is also smaller than
− ×
in other calculations of Ref. [9] and [20]. On the other
hand, the ratio of the two contributions is 115 (in units
ofd /(CSP10−18ecm))whichissignificantlylargerthan
e
the analytical and numerical values of 89 and 83 (same
units, see Table III).
EDM enhancement factor for francium was calculated
in Ref. [49] and found to be 910(46): d(Fr)=910(46)d .
e
This value is in goodagreementwith the value of 894.93
found in more recent calculations [50]. Using the value
d(Fr)=910(46)d andtheratiooftheelectron-EDMma-
e
trix element to the scalar-pseudoscalar interaction pre-
sented in Table III we can now reconstruct another con-
tribution to the EDM of francium:
d(Fr)=11 10−18CSPe cm. (33) FIG. 1: The 5d core function of Tl calculated in VN−3
× 5/2
(solid line) and VN−1 (dashed line) approximations.
It is interesting to note that the calculation of the nu-
clear spin-dependent PNC amplitude between hyperfine
components of the ground state of francium presented
and VN−1 approximations. The difference between the
in Ref. [29] involves the same sum as the expression
functions is very small. This is due to the fact that the
(2) for the EDM of the atom. Therefore, we can re-
valence6s and6pelectronsarelocatedoutside ofatomic
construct the EDM enhancement factor for Fr using the
core. Their charge distribution creates almost constant
results of Ref. [29] and proportionality between matrix
potentialand no electric field inside the sphere where all
elements of the nuclear spin-dependent interaction (A9)
inner electrons are located. Therefore, the valence elec-
and the electron EDM operator (A15). This leads to
trons have practically no effect on the core wave func-
d(Fr)=854d whichis ingoodagreementwiththe value
e
tions (see [45] for a detailed discussion). The change is
d(Fr)=910(46)d ofRef.[49]. Thisisagoodconsistency
e
evensmallerfor othercorefunctions. The corefunctions
test of both calculations.
enter the configuration interaction (CI) Hamiltonian via
Reading the papers citing our works (see, e.g. [6, 50])
corepotentialV towhichallcoreelectronscontribute
reveals the need to clear some points about our method core
(from 1s to 5d). The difference for V in the VN−3
of calculations. For example, the atomic electric field in- core
and VN−1 approximations is very small [45]. Moreover,
teractingwithelectronEDMsiscalculatedin[9]and[49]
the corresponding corrections to the configuration inter-
as a derivativeofthe totalpotentialwhich includes both
nuclear and electron parts. The formula E = Zer/r3 action (CI) Hamiltonian have been included in [9] using
the many-body perturbation theory approach.
for the leading contribution to the atomic electric field
presented on first page of [49] may make an impression Asitiswellknown,theeigenstatesofaHamiltoniando
that only nuclear field is included. However,the formula notdepend onthe basis one uses. The valencestates are
(2) few lines below clearly includes screening functions indeeddifferent inthe VN−3 andVN−1 approximations.
Q(r) and P(r) for both the nuclear Coulomb and the However, this should have no effect on the final results
external electric field. By the way, the inclusion of the aslongasthe saturationofthe basisforvalence statesis
electron electric field change the matrix elements of the achieved. There are only two conditions the basis states
electronEDMforthalliumby0.4%only. Thisisbecause must satisfy: (a) they must be orthogonal to the core,
main contribution comes from short distances where the and (b) they must constitute a complete set of states.
electronelectric fieldis smallsince the electronpotential Therefore, it does not matter whether valence states are
rapidly tends to a constant inside the 1s orbital. There- calculatedintheVN−3orVN−1potentialorbyanyother
fore, we include only nuclear electric field in the ana- method (e.g., a popular basis ψn(r)=rnψ0(r) [46]), the
lytical analysis (formula (A15) in the appendix) while final results should be the same. If there is any differ-
keepingbothcontributionsinthenumericalcalculations. ence in the results, the most likely reason for this is the
The authors of [6] claim that atomic core is strongly incompleteness of the basis set.
contracted in the VN−3 starting approximation used in In spite of no difference in final results there is a good
ourcalculations[9]. Infact,itisnot. Fig.1showstheout- reasonforthe use ofthe VN−3 approximation–the sim-
ermost 5d core function of Tl calculated in the VN−3 plicity and good convergence of the many-body pertur-
5/2
8
bation theory (MBPT) for the core-valence correlations. Acknowledgments.
This is the only approximationwhich has no subtraction
diagrams [47]. To be more precise, we should state that The authors are grateful to E. A. Hinds for stimulat-
theconditionfortheabsenceofthesubtractiondiagrams ing discussions. The work was supported in part by the
isthatthepotentialusedtocalculatesingle-electroncore Australian Research Council.
statesandthepotentialofthecoreintheCIHamiltonian
areexactlythesame. Inthecaseofthalliumatomtreated
as a three-valence-electrons system this corresponds to Appendix A: Matrix elements of the weak
the VN−3 approximation. When any other approxima- interaction
tionis usedone has to include the subtractiondiagrams.
Large energy denominators suppress the value of the 1. Wave function
correlation terms in the VN−3 approximation ensuring
goodconvergenceoftheMBPT[45]. Theremustbelarge We use single-electron wave functions in a form
cancelation between subtraction and other diagrams to
ensure the same final results if any other initial approx- 1 fv(r)Ω(n)jlm
ψ(r) = , (A1)
imation is used. This is very similar to the well known njlm r (cid:18)iαg (r)Ω(n) (cid:19)
v jlm
factthattheHartree-Fockbasisisthebestchoiceforany
e
MBPT calculations. Zero-order results might be better where n is the principal quantum number and an index
in some other approximation, however, strong cancela- v replacesthethree-numbersetn,j,l;αisthefinestruc-
tion between subtractionand other diagramswould lead ture constant.
to poor convergence of the MBPT. We need s1/2 and p1/2 wave functions inside the nu-
cleus to calculate matrix elements. Following Ref. [51]
The authors of [6] claim that the major drawback of
we assume uniform distribution of electric charge inside
our work [9] is the inclusion of the core-valence correla-
a sphere of radius R . Taking formulas for the s and
tions in the secondorderonly. However,the correlations N 1/2
p wave functions from Ref. [51] and keeping terms up
between the valence electrons and core electrons below 1/2
to Z2α2 we come to the following expressions
the 6sstatearesmallwhichisevidentfromthe factthat
their inclusion change the EDM of Tl by 3% only [9].
3
Therefore, only the correlations between three valence fs(x)=Asx 1 Z2α2x2 ,
(cid:18) − 8 (cid:19)
electrons should be treated to all orders. This is done
in [9] to a very high precision using the CI technique. x2 x2
g (x)= A Z 1 , (A2)
s s
− 2 (cid:18) − 5 (cid:19)
x2 x2
f (x)=A Zα2 1 ,
p1/2 p 2 (cid:18) − 5 (cid:19)
3
g (x)=A x 1 Z2α2x2 .
V. CONCLUSION p1/2 p (cid:18) − 8 (cid:19)
Here x = r/R , R is nuclear radius. A and A are
N N s p
We have demonstrated the proportionality rela- normalization factors. They can be found by matching
tions between the nuclear-spin dependent and spin- (A2) to atomic wavefunctions at shortdistances outside
independent PNC effects in atoms and the scalar- ofthenucleus(see,e.g.[52]). Theuseofthesemiclassical
pseudoscalarand the electronEDM contributions to the wave functions presented in [52] leads to approximate
EDM of atoms and molecules. The relations are given expressions
by the simple analytical formulas and can be used to
express one symmetry breaking effect through another. (1+γ)(2ZR )γ
N
A =
Using these relations and accurate calculations of the s (1 0.375Z2α2)√ZΓ(2γ+1)ν1.5
spin-independent PNC we have confirmed earlier inter- − s
√Z(2ZR )γ
pretationsofthenuclearanapolemeasurementsincesium N
A = , (A3)
and thallium. We have also confirmed the ratio of the p (1 0.375Z2α2)Γ(2γ+1)ν1.5
− p
scalar-pseudoscalarCP-oddandelectronEDMcontribu-
tions to the EDM of Cs, Tl and other atoms and some where γ = √1 Z2α2, Γ is gamma-function, ν is effec-
−
polarmolecules. Usingtherelationswefoundthescalar- tiveprincipalquantumnumber: single-electronenergyof
pseudoscalarcontributionto the EDM of francium atom state n is ǫ = 1/(2ν2). Note that normalization fac-
n − n
andthe scalar-pseudoscalarandelectronEDMcontribu- tors(A3)areneededonlytocomparethematrixelements
tionstotheEDMofmanypolarmolecules. Usingexper- ofthe electronEDMto othermatrixelements. Forcases
imental limits on EDMs of thallium and YbF we found notinvolvingelectronEDMthereisexactcancellationof
model-independent limits on the electron EDM and the the normalization factors. Therefore, their uncertainty
constant of the scalar-pseudoscalarCP-odd interaction. does not contribute to the uncertainty of final results.
9
2. PNC matrix elements 3. Scalar-pseudoscalar CP-odd interaction
Matrix element of the SI PNC HSI interaction (first The Hamiltonian of the scalar-pseudoscalar electron-
term in (4)) is nucleon (T,P)-odd interaction can be written as [1]
G Q
hκ1|HSI|κ2i=i 2F√2W × (A4) HSP =iGFACSPγ0γ5ρN(r), (A11)
√2
where G is the Fermi constant, A = Z +N is the nu-
αδ−κ1,κ2Z (f1g2−g1f2)ρ(r)dr. clear maFss number, Z is the number of protons and N
is the number of neutrons. CSP = (ZCSP +NCSP)/A,
p n
Ssuumbsintigtuutninifgor(mA2n)ufcoleratrhcehsa1r/g2e−dipst1r/i2butrtaionnsitwioengeatnd as- where CpSP and CnSP are the parameters of the scalar-
pseudoscalar (T,P)-odd interaction for protons and neu-
trons, γ are the Dirac matrices.
G Q n
F W
hs1/2|HSI|p1/2i=i 2√2 × (A5)
G
αρ0AsApRN 1 0.34Z2α2 . ha|HSP|bi = −√F2αACSP
3 −
(cid:0) (cid:1)
Here As and Ap are given by (A3) and ×δ−κa,κbZ (fagb+gafb)ρNdr.(A12)
3
ρ0 = 4πR3 . (A6) Substitution of the functions (A2) leads to
N
Matrix element of the SD PNC b operator (11) is s1/2 HSP p1/2 = (A13)
h | | i
G ACSP
hκ1||b||κ2i=−iG√F2καh−κ1||C1||κ2i× (A7) − F3√2 αAsApρ0RN(cid:0)1−0.56Z2α2(cid:1).
((κ2 κ1+1)f1g2 (κ1 κ2+1)g1f2)ρ(r)dr. Here As and Ap are given by (A3) and ρ0 is given by
Z − − − (A6).
The reduced matrix element of the spherical harmonic
C is
k
4. Electron EDM
κa Ck κb =( 1)jb+1/2 (2ja+1)(2jb+1)
h || || i −
jp j k
ξ(l +l +k) b a . (A8) The Hamiltonian for the electron EDM interacting
× a b (cid:18)−1/2 1/2 0 (cid:19) with internal atomic electric field Eint can be written
1, if x is even as [52]
ξ(x)=
(cid:26)0, if x is odd
He = de(γ0 1)Σ Eint, (A14)
Forthes p transitionanduniformnuclearcharge − − ·
1/2 1/2
−
distribution we have
where
G κ
s b p = i F (A9)
h 1/2|| || 1/2i − √2 × Σ= σ 0 ,
(cid:18) 0 σ (cid:19)
2
αr3ρ0AsApRN 1−0.41Z2α2 .
(cid:0) (cid:1) and Eint is internalatomic electric field. The s1/2 p1/2
−
The ratio of (A9) to (A5) is matrix element can be found in Ref. [53]:
hs1/2||b||p1/2i/hs1/2|HSI|p1/2i≈ (A10) s H p = 4Z3α2de . (A15)
6 2 κ 1 0.073Z2α2 . h 1/2| e| 1/2i −γ(4γ2−1)(νsνp)3/2
r3( Q ) −
− W (cid:0) (cid:1)
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