Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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440 Part B Atoms

Table 28.4 Values of the function G

(α)

n S

1/2

3/2

5/2

1 − 30.290 24 (2)

2 − 31.185 15 (9) − 0.9735 (2) − 0.4865 (2)

3 − 31.0477 (9)

4 − 30.912 (4) − 1.165 (2) − 0.611 (2) 0.031 (1)

6 − 30.82 (8) 0.034 (2)

8 − 30.80 (9) 0.008 (5) 0.034 (2)

12 0.009 (5) 0.035 (2)

low-Z limit of this function, G

(0) = A

,inthe

cases where it is known, together with extrapola-

tions of the results of complete numerical calculations

of F(Zα) (28.90) at higher Z [28.76, 77]. There is

a long history of calculations of A

[28.2], lead-

ing up to the accurate values of A

for the 1S

and 2S states obtained by Pachucki [28.73, 78–80].

Values for P and D states have been reported subse-

quently by Jentschura and Pachucki [28.62], Jentschura

et al. [28.81, 82]. Extensive numerical evaluations

of F(Zα) at higher Z, which in turn yield val-

ues for G

(Zα), have been done by Mohr [28.83],

Mohr and Kim [28.84], Indelicato and Mohr [28.85],

Le Bigot [28.86].

The dominant effect of the ﬁnite mass of the nucleus

on the self energy correction is taken into account by

multiplying each term of F(Zα) by the reduced-mass

factor (m

)

, except that the magnetic moment term

−1/[2κ(2l + 1)] in A

is instead multiplied by the fac-

tor (m

)

. In addition, the argument (Zα)

−2

of the

logarithms is replaced by (m

)(Zα)

−2

[28.56].

The uncertainty of the self energy contribution to

a given level arises entirely from the uncertainty of

(α) listed in Table 28.4 and is taken to be entirely

of type u

28.3.5 Vacuum Polarization

The second-order vacuum-polarization level shift, due to

the creation of a virtual electron–positron pair in the ex-

change of photons between the electron and the nucleus,

(2)

(Zα)

H(Zα)m

, (28.92)

where the function H(Zα) is divided into the part cor-

responding to the Uehling potential, denoted here by

(1)

(Zα), and the higher-order remainder H

(R)

(Zα) =

(3)

(Zα) + H

(5)

(Zα) +···, where the superscript de-

notes the order in powers of the external ﬁeld. The

individual terms are expanded in a power series in Zα

(1)

(Zα) = V

+ V

(Zα) + V

(Zα)

ln(Zα)

−2

+ G

(1)

(Zα)(Zα)

, (28.93)

(R)

(Zα) = G

(R)

(Zα)(Zα)

, (28.94)

with

=−

πδ

=−

. (28.95)

The part G

(1)

(Zα) arises from the Uehling potential,

and is readily calculated numerically [28.76, 87]; val-

ues are given in Table 28.5. The higher-order remainder

(R)

(Zα) has been considered by Wichmann and Kroll,

and the leading terms in powers of Zα are [28.33,88,89]

(R)

(Zα) =



−





−

31π

2880



π(Zα)δ

+··· .

(28.96)

Higher-order terms omitted from (28.96) are negligible.

In a manner similar to that for the self energy,

the leading effect of the ﬁnite mass of the nucleus is

taken into account by multiplying (28.92) by the fac-

tor (m

)

and including a multiplicative factor of

) in the argument of the logarithm in (28.93).

There is also a second-order vacuum polarization

level shift due to the creation of virtual particle pairs

other than the e

−

pair. The predominant contribution

for nS states arises from µ

−

, with the leading term

being [28.90, 91]

(2)

µVP

(Zα)



−









(28.97)

Part B 28.3

Tests of Fundamental Physics 28.3 Hydrogen and Deuterium Atoms 441

Table 28.5 Values of the function G

(1)

(α)

n S

1/2

3/2

5/2

1 − 0.618 724

2 − 0.808 872 − 0.064 006 − 0.014 132

3 − 0.814 530

4 − 0.806 579 − 0.080 007 − 0.017 666 − 0.000 000

6 − 0.791 450 − 0.000 000

8 − 0.781 197 − 0.000 000 − 0.000 000

12 − 0.000 000 − 0.000 000

The next order term in the contribution of muon vacuum

polarization to nS states is of relative order Zαm

and is therefore negligible. The analogous contribution

(2)

τVP

from τ

−

(−18 Hz for the 1S state) is also

negligible at the level of uncertainty of current interest.

For the hadronic vacuum polarization contribution,

we take the result given by Friar et al. [28.92]that

utilizes all available e

−

scattering data:

(2)

had VP

= 0.671 (15)E

(2)

µVP

, (28.98)

where the uncertainty is of type u

The muonic and hadronic vacuum polarization con-

tributions are negligible for P and D states.

28.3.6 Two-Photon Corrections

Corrections from two virtual photons, of order α

,have

been calculated as a power series in Zα:

(4)





(Zα)

(4)

(Zα) , (28.99)

where

(4)

(Zα) = B

+ B

(Zα) + B

(Zα)

−2

+ B

(Zα)

−2

+ B

(Zα)

ln(Zα)

−2

+ B

(Zα)

+··· . (28.100)

The leading term B

is well known:



3π

ln 2 −

10π

−

2179

648

−

ζ(3)





ln 2

−

197

144

−

3ζ(3)



1 − δ

κ(2l + 1)

(28.101)

The second term has been calculated by Eides and

Shelyuto [28.93], Pachucki [28.94], Eides et al. [28.90],

Pachucki [28.95] with the result

=−21.5561 (31)δ

. (28.102)

The next coefﬁcient, as obtained by Karshen-

boim [28.96], Yerokhin [28.97], Manohar and Stew-

art [28.98], Pachucki [28.99], is

=−

. (28.103)

For S states the coefﬁcient B

has been found to be



− ln 2 + γ + ψ(n)

− ln n −



(28.104)

where γ = 0.577 ... is Euler’s constant and ψ is the psi

function [28.100]. The difference B

(1) − B

(n) was

calculated by Karshenboim [28.101] and conﬁrmed by

Pachucki [28.99] who also calculated the n-independent

additive constant. For P states the calculated value

is [28.101]

− 1

. (28.105)

This result has been conﬁrmed by Jentschura and

andori [28.102] who also show that for D and higher

angular momentum states B

= 0.

The single-logarithm coefﬁcient B

for S states has

been given as [28.99]

39751

10800

4N(n)

55π

−

616 ln 2

135

3π

ln 2

40 ln

−

9ζ(3)



304

135

−

32 ln 2





+ γ + ψ(n) − ln n −



(28.106)

Part B 28.3

442 Part B Atoms

where N(n) is a term that was numerically evaluated for

the 1S state by Pachucki [28.99]. Jentschura [28.103]

has evaluated N(n) for excited S states with n = 2to

n = 8, has made an improved evaluation for n = 1, and

has given an approximate ﬁt to the calculated results in

order to extend them to higher n. Values of the func-

tion N(n) for the states of interest here are given in

Table 28.6. There are no results yet for P or D states for

. Based on the relative magnitude of A

for the S,

P, and D states, we take as uncertainties u

) = 5.0

for P states and u

) = 0.5 for D states.

Recent work indicates that there may be an add-

itional contribution to B

and/or B

[28.104, 105].

The effect of such a contribution would be to change

the S-state energy levels by an amount that is likely

to be less than half the uncertainty of the nuclear size

correction due to uncertainty in the rms radius of the

nucleus.

The two-loop Bethe logarithm b

, which is expected

to be the dominant part of the no-log term B

, has been

calculated for the 1S and 2S states by Pachucki and

Jentschura [28.106] who obtained

=−81.4 (3) 1S state (28.107a)

=−66.6 (3) 2S state . (28.107b)

An additional contribution for S states,

N ,

(28.108)

was derived by Pachucki [28.99], where N is given in

Table 28.7 as a function of the state n. These contri-

butions can be combined to obtain an estimate for the

coefﬁcient B

for S states:

= b

N +··· ,

(28.109)

where the dots represent uncalculated contributions to

which are at the relative level of 15% [28.106]. In

order to obtain an approximate value for B

for S states

Table 28.6 Va lu es of N

n N

1 17.855 672 (1)

2 12.032 209 (1)

3 10.449 810 (1)

4 9.722 413 (1)

6 9.031 832 (1)

8 8.697 639 (1)

with n ≥ 3, we employ a simple extrapolation formula,

= a +

(28.110)

with a and b ﬁttedtothe1Sand2Svaluesofb

,and

we include a component of uncertainty u

) = 5.0.

The results for b

, along with the total estimated val-

ues of B

for S states, are given in Table 28.7.ForP

states, there is a calculation of ﬁne-structure differences

[28.107], but because of the uncertainty in B

for P

states, we do not include this result. We assume that for

both the P and D states, the uncertainty attributed to B

is sufﬁciently large to account for the uncertainty in B

and higher-order terms as well.

As in the case of the order α self-energy and vacuum-

polarization contributions, the dominant effect of the

ﬁnite mass of the nucleus is taken into account by

multiplying each term of the two-photon contribution

by the reduced-mass factor (m

)

, except that the

magnetic moment term, the second line of (28.101), is

instead multiplied by the factor (m

)

. In addition,

the argument (Zα)

−2

of the logarithms is replaced by

)(Zα)

−2

28.3.7 Three-Photon Corrections

The leading contribution from three virtual photons is

assumed to have the form

(6)





(Zα)

[

+ C

(Zα) +···

]

(28.111)

in analogy with (28.99) for two photons. The level shifts

of order (α/π)

(Zα)

that contribute to C

can

be characterized as the sum of a self-energy correction,

a magnetic-moment correction, and a vacuum polariza-

tion correction. The self-energy correction arises from

the slope of the Dirac form factor, and it has recently

been calculated by Melnikov and Ritbergen [28.108]

Table 28.7 Va lu es of b

and B

n b

1 − 81.4 (3) − 61.6 (9.2)

2 − 66.6 (3) − 53.2 (8.0)

3 − 61.7 (5.0) − 50.1 (9.0)

4 − 59.2 (5.0) − 48.4 (8.8)

6 − 56.7 (5.0) − 46.7 (8.6)

8 − 55.5 (5.0) − 45.8 (8.5)

Part B 28.3

Tests of Fundamental Physics 28.3 Hydrogen and Deuterium Atoms 443

who obtained

(6)





(Zα)



−

868a

25ζ(5)

−

17π

ζ(3)

−

2929ζ(3)

−

217 ln

−

103π

270

41671π

ln 2

540

3899π

6480

−

454979π

9720

−

77513

46656



, (28.112)

where ζ is the Riemann zeta function and a



∞

n=1





= 0.517 479 061 .... The magnetic-

moment correction comes from the known three-loop

electron anomalous magnetic moment [28.109], and is

given by

(6)





(Zα)



−

100a

215ζ(5)

−

83π

ζ(3)

−

139ζ(3)

−

25 ln

25π

298π

ln 2

239π

2160

−

17101π

810

−

28259

5184



κ(2l + 1)

(28.113)

and the vacuum-polarization correction is [28.110,111]

(6)





(Zα)



−

8135ζ(3)

2304

4π

ln 2

−

23π

325805

93312



. (28.114)

The total for C



−

568a

85ζ(5)

−

121π

ζ(3)

−

84071ζ(3)

2304

−

71 ln

−

239π

135

4787π

ln 2

108

1591π

3240

−

252251π

9720

679441

93312





−

100a

215ζ(5)

−

83π

ζ(3)

−

139ζ(3)

−

25 ln

25π

298π

ln 2

239π

2160

−

17101π

810

−

28259

5184



1 − δ

κ(2l + 1)

(28.115)

An uncertainty in the three-photon correction is

assigned by taking u

) = 30δ

and u

) = 1,

where C

is deﬁned by the usual convention.

The dominant effect of the ﬁnite mass of the nucleus

is taken into account by multiplying C

in (28.115)by

the reduced-mass factor (m

)

for l = 0orbythe

factor (m

)

for l = 0.

The contribution from four photons is expected to

be of order





(Zα)

, (28.116)

which is about 10 Hz for the 1S state and is negligible

at the level of uncertainty of current interest.

28.3.8 Finite Nuclear Size

At low Z, the leading contribution due to the ﬁnite size

of the nucleus is

(0)

= E

, (28.117)

with





(Zα)



ZαR



(28.118)

where R

is the bound-state root-mean-square (rms)

charge radius of the nucleus and

is the Compton

wavelength of the electron divided by 2π. The lead-

ing higher-order contributions have been examined by

Friar [28.112], Friar and Payne [28.113], Karshen-

boim [28.114](seealsoMohr [28.115], Borisoglebski

and Troﬁmenko [28.89]). The expressions that we em-

ploy to evaluate the nuclear size correction are the same

as those discussed in more detail in [28.1].

For S states the leading and next-order corrections

are given by

= E



1 − C

Zα −





Zα



+ψ(n)+γ −

(5n + 9)(n − 1)

−C



(Zα)



(28.119)

where C

and C

are constants that depend on the de-

tails of the assumed charge distribution in the nucleus.

The values used here are C

= 1.7 (1) and C

= 0.47 (4)

for hydrogen or C

= 2.0 (1) and C

= 0.38 (4) for

deuterium.

For the P

1/2

states in hydrogen the leading term is

= E

(Zα)

− 1)

. (28.120)

Part B 28.3

444 Part B Atoms

For P

3/2

states and D states the nuclear-size contribution

is negligible.

28.3.9 Nuclear-Size Correction to Self

Energy and Vacuum Polarization

In addition to the direct effect of ﬁnite nuclear size

on energy levels, its effect on the self energy and vac-

uum polarization contributions must also be considered.

This same correction is sometimes called the radiative

correction to the nuclear-size effect.

For the self energy, the additional contribution due

to the ﬁnite size of the nucleus is [28.116–119]

NSE



4ln2−



α(Zα)E

, (28.121)

and for the vacuum polarization it is [28.117, 120, 121]

NVP

α(Zα)E

. (28.122)

For the self-energy term, higher-order size correc-

tions for S states [28.118] and size corrections for

P states have been calculated [28.103, 122], but these

corrections are negligible for the current work, and are

not included. The D-state corrections are assumed to be

negligible.

28.3.10 Radiative-Recoil Corrections

The dominant effect of nuclear motion on the self energy

and vacuum polarization has been taken into account by

including appropriate reduced-mass factors. The add-

itional contributions beyond this prescription are termed

radiative-recoil effects with leading terms given by

α(Zα)



6ζ(3) − 2π

ln 2 +

35π

−

448

π(Zα) ln

(Zα)

−2

+···



. (28.123)

The leading constant term in (28.123)isthesumof

the analytic result for the electron-line contribution

[28.123, 124] and the vacuum-polarization contribution

[28.125,126]. The log-squared term has been calculated

by Pachucki and Karshenboim [28.63]andMelnikov and

Yelkhovski [28.64].

For the uncertainty, we take a term of order

(Zα) ln(Zα)

−2

relative to the square brackets in (28.123)

with numerical coefﬁcients 10 for u

and 1 for u

.These

coefﬁcients are roughly what one would expect for the

higher-order uncalculated terms.

28.3.11 Nucleus Self Energy

An additional contribution due to the self energy of the

nucleus has been given by Pachucki [28.126]:

SEN

α(Zα)

3πn





(Zα)



− ln k

(n, l)



(28.124)

This correction has also been examined by Eides

et al. [28.2], who consider how it is modiﬁed by the

effect of structure of the proton. The structure effect

leads to an additional model-dependent constant in the

square brackets in (28.124).

To evaluate the nucleus self-energy correction, we

use (28.124) and assign an uncertainty u

that corre-

sponds to an additive constant of 0.5 in the square

brackets for S states. For P and D states, the correc-

tion is small and its uncertainty, compared to other

uncertainties, is negligible.

28.3.12 Total Energy and Uncertainty

The total energy E

nL j

of a particular level (where

L = S, P,... and X = H, D,...)isthesumofthe

various contributions listed above plus an additive cor-

rection δ

nL j

that accounts for the uncertainty in the

theoretical expression for E

nL j

. Our theoretical esti-

mate of the value of δ

nL j

for a particular level is

zero with a standard uncertainty of u



nL j



equal to

the square root of the sum of the squares (rss) of the

individual uncertainties of the contributions, since, as

they are deﬁned above, the contributions to the en-

ergy of a given level are independent. (Components of

uncertainty associated with the fundamental constants

are not explicitly shown here.) Thus we have for the

square of the uncertainty, or variance, of a particular

level

(δ

nL j

) =



(XL j) + u

(XL j)

, (28.125)

where the individual values u

(XL j)/n

and

(XL j)/n

are the components of uncertainty from

each of the contributions, labeled by i, discussed above.



The factors of 1/n

are isolated so that u

(XL j) is

explicitly independent of n.



Part B 28.3

Tests of Fundamental Physics References 445

Table 28.8 Measured transition frequencies ν in hydrogen

Authors Laboratory Frequency interval Reported value ν/kHz

Hagley and Pipkin 28.127 Harvard Univ. ν

(2S

1/2

−2P

3/2

) 9 911 200 (12)

Lundeen and Pipkin 28.128 Harvard Univ. ν

(2P

1/2

−2S

1/2

) 1 057 845.0 (9.0)

Newton et al. 28.129 Univ. Sussex ν

(2P

1/2

−2S

1/2

) 1 057 862 (20)

The covariance of any two δs follows from (F7) of

Appendix F of [28.1]. For a given isotope X,wehave



,δ





(XL j)

)

, (28.126)

which follows from the fact that u(u

, u

) = 0and



, u



= 0forn

= n

.Wealsoset



,δ



= 0 ,

(28.127)

if L

= L

or j

= j

. For covariances between δsfor

hydrogen and deuterium, we have for states of the same n



nL j

,δ

nL j





i=i

(HLj)u

(DLj) + u

(HLj)u

(DLj)

(28.128)

and for n

= n



,δ





i=i

(HLj)u

(DLj)

)

(28.129)

where the summation is over the uncertainties common

to hydrogen and deuterium. In most cases, the uncer-

tainties can in fact be viewed as common except for

a known multiplicative factor that contains all of the

mass dependence. We assume



,δ



= 0 ,

(28.130)

if L

= L

or j

= j

. These covariances correspond to

correlation coefﬁcients as large as 0.991.

Since the transitions between levels are measured

in frequency units (Hz), in order to apply the above

equations for the energy level contributions we divide

the theoretical expression for the energy difference ∆E

of the transition by the Planck constant h to convert it

to a frequency. Further, we replace the group of con-

stants α

/2h in ∆E/h by cR

∞

to calculate the

uncertainties.

28.3.13 Transition Frequencies

Between Levels with n = 2

As an indication of the consistency of the theory sum-

marized above and the experimental data, we list values

of the transition frequencies between levels with n = 2in

hydrogen. These results are based on a variation of the

2002 least-squares adjustment in which the measure-

ments of these particular transitions are not included

[28.1]. The calculated values are

(2P

1/2

− 2S

1/2

) = 1 057 844.5 (2.6) kHz ,

(2S

1/2

− 2P

3/2

) = 9 911 197.1 (2.6) kHz ,

(2P

1/2

− 2P

3/2

) = 10 969 041.57 (89) kHz .

(28.131)

These results compare favorably with the most recent

experimental values given in Table 28.8.

In addition, in He

, a recent experimental value of

the Lamb shift is S = 14 041.13 (17) MHz [28.130], and

the current theoretical value is S = 14 041.474 (42) MHz

[28.131].

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(2000)

28.2 M. I. Eides, H. Grotch, V. A. Shelyuto: Phys. Rep.

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28.3 P. J. Mohr, B. N. Taylor: Rev. Mod. Phys. 77, 1 (2005)

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28.5 T. Kinoshita, B. Ni

c, Y. Okamoto: Phys. Rev. D 41,

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28.6 T. Kinoshita: Present status of g − 2 of electron and

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Part B 28

449

Parity Noncon

29. Parity Nonconserving Effects in Atoms

Until 1957 the invariance of the laws of physics

under the process of parity inversion was assumed

to hold. The concept of this invariance has its origin

in atomic physics, where Laporte introduced it to

explain certain aspects of the iron spectrum. The

underlying theory of atomic structure, quantum

electrodynamics (QED), is an example of a theory

that has this invariance. However, it was shown

in 1957 that weak interaction processes are not

invariant under parity inversion. At that time

the only known weak interactions were charge

changing and thus did not affect the spectrum of

a stable atom. However, if charge conserving, or

neutral weak interactions exist, these can lead to

29.1 The Standard Model............................. 450

29.2 PNC in Cesium ..................................... 451

29.3 Many-Body Perturbation Theory ........... 451

29.4 PNC Calculations .................................. 452

29.5 Recent Developments........................... 453

29.6 Comparison with Experiment................ 453

References .................................................. 454

parity nonconserving (PNC) processes in atoms.

These include electric dipole transitions between

states of the same parity and optical rotation.

It is now well established that such interactions do exist,

and are mediated by the Z boson. Unfortunately, be-

cause of the extremely small ratio of the energy scale

of atoms to the Z mass, PNC processes in atoms must

be correspondingly small. In 1974, however, the Bouchi-

ats[29.1], in a paper that laid the foundation for the ﬁeld,

showed that parity nonconserving (PNC) transitions in

heavy atoms with atomic number Z were enhanced by

a factor of Z

. While still very small, this effect has been

observed in a variety of heavy atoms, speciﬁcally ce-

sium (Z = 55) [29.2–4], thallium (Z = 81) [29.5,6], lead

(Z = 82) [29.7], and bismuth (Z =83) [29.8]. The accu-

rate calculation of the electronic structure of such atoms

is of course a very challenging atomic physics prob-

lem, and such calculations must be carried out before

the experiments can be interpreted in terms of particle

physics. This problem is not present for hydrogen, but

experimental problems have impeded progress in this

direction.

At the time of the Bouchiats’s paper, the question

of whether or not neutral currents existed was an im-

portant problem in weak interaction physics, and even

a qualitative observation of atomic PNC would have

been of great interest. However, now that neutral cur-

rents are well established, what now has become the

most important aspect of atomic PNC is the precise

measurement of the effect. This is because of the present

intense interest in possible modiﬁcations of the standard

model. While this model, described in more detail in the

next section, is extremely successful, it is incomplete in

some ways, in particular, in the way particle masses are

treated. In the standard model, for example, the mystery

of the fermion mass spectrum is simply transferred to

another mystery of the sizes of Yukawa coupling con-

stants that couple Higgs ﬁelds to fermions. It is hoped

that some deviation from the standard model will be un-

covered, generally referred to as ‘new physics’, that will

provide a hint as to a more satisfactory theory. At the

present time, since all standard model predictions are in

agreement with experiment at the few percent level, the

hoped for deviation will be small; and for this reason

high precision experiments combined with calculations

of similar accuracy are needed. For this reason atomic

PNC, while apparently a somewhat esoteric branch of

atomic physics, actually is involved with one of the most

central problems of the ﬁeld, the accurate solution of

the Schrödinger equation for many-electron atoms. We

will describe an approach to this fundamental problem

below.

Another weak interaction effect important in atomic

physics has to do with CP noninvariance. While the

weak interactions are not invariant under parity inver-

sion (P), until 1964 it appeared that they were invariant

under the simultaneous operations of P and charge con-

Part B 29

450 Part B Atoms

jugation (C), or CP. In 1964, however, measurements of

the neutral K meson system showed that CP invariance

is slightly broken. While this does not directly affect

atomic physics, it has been suggested that this still in-

completely explained breaking in the quark sector may

have a counterpart in the lepton sector, which could

have, as a consequence, a nonvanishing electron elec-

tric dipole moment (edm). This opens the possibility of

using precise atomic experiments to search for such an

effect. In this case, as with ‘ordinary’ PNC, a calcula-

tion of atomic structure is necessary. The requirements

of precision are not as important in this case, as the

simple detection of a non-zero atomic edm would be

a discovery of importance comparable to the detection

of CP noninvariance in the kaon system. Because of lim-

itations of space, we do not discuss this interesting ﬁeld

further, referring the interested reader to the review of

the subject given in [29.9].

29.1 The Standard Model

The Standard Model is the present theory of the strong,

weak, and electromagnetic interactions. We are con-

cerned with the part of the standard model that uniﬁes

the weak and electromagnetic interactions, which is re-

ferred to as the electroweak theory. This theory involves

the interactions of leptons and quarks with four vec-

tor bosons, the photon ( A), a heavy neutral partner of

the photon (Z), and two charged bosons (W

−

). In-

teractions involving the photon conserve parity, while

interactions between the other bosons do not. The neu-

tral currents affecting atomic PNC are described by the

following Lagrangian:

L =−e



−

sin2θ



− A

)ψ

, (29.1)

where i ranges over three types of fermions, the elec-

tron



=−1, V

=−

+2sin

, A

=−



, the up

quark



, V

−

sin

, A



,andthe

down quark



=−

, V

=−

sin

, A

=−



The weak angle θ

is a fundamental parameter

of the standard model and is discussed further

below.

The physics in the above Lagrangian that leads to

atomic PNC is the exchange of a virtual Z, either be-

tween a quark in the nucleus and an electron, or between

two electrons, though calculations of the latter effect

show it to be negligible [29.10]. PNC arises when the

Z matrix element is a vector on the nucleus and an ax-

ial vector on the electron (V

), or vice versa (A

The dominant PNC contribution comes from the for-

mer case, because all the quarks contribute coherently.

The latter case, however, cannot be neglected and is dis-

cussed further below. Because the nuclear current is of

vector nature, one can introduce a conserved charge, the

weak charge Q

= 2Z(2V

) +2N(V

+2V

). (29.2)

Here Z is the number of protons and N the number of

neutrons. Putting in the above values of V

and V

then

gives

= Z



1 −4sin



−N .

(29.3)

It would appear that the dependence of Q

on the

weak angle would allow its determination from atomic

PNC. While this can be done, it is important to note that

the above discussion has been at the tree level. One of

the most important features of the electroweak theory is

that radiative corrections can be calculated. These cor-

rections enter at the percent level, and must be included

for an accurate interpretation of experiments sensitive

to electroweak effects. Until recently, of the parameters

affecting the electroweak theory only the ﬁne structure

constant and the muon decay rate were known with high

precision, leaving the weak angle a free parameter. How-

ever, now that the Z mass has been measured with very

high precision, it is now possible to use that measure-

ment to determine θ

, and to then predict a large set of

radiatively corrected electroweak processes, including

atomic PNC [29.11]. While there is still some sensi-

tivity to the top quark and Higgs masses, as discussed

further below, using the Z mass allows the prediction for

cesium of, assuming a Higgs mass of 100 GeV,

=−73.20(0.13). (29.4)

Therefore, the importance of atomic PNC for funda-

mental physics is not so much its ability to determine

the weak angle, but rather the fact that the standard

model makes a deﬁnite prediction for the size of the ef-

fect, and any disagreement indicates new physics; and

conversely, agreement to within a given precision puts

limits on new physics.

Part B 29.1