Self-Energy Correction to the Bound-Electron g Factor of P States

0.18 MB·English

by  U. D. Jentschura

Self–Energy Correction to the Bound–Electron g Factor of P States
Ulrich D. Jentschura
Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409-0640, USA
The radiative self-energy correction to the bound-electron g factor of 2P1/2 and 2P3/2 states in
one-electron ions is evaluated to order α(Zα)2. The contribution of high-energy virtual photons
is treated by means of an effective Dirac equation, and the result is verified by an approach based
on long-wavelength quantum electrodynamics. The contribution of low-energy virtual photons is
calculated both in the velocity and in the length gauge and gauge invariance is verified explicitly.
0 The results compare favorably to recently available numerical data for hydrogenlike systems with
1 low nuclear charge numbers.
0
2 PACSnumbers: 12.20.Ds,31.30.js,31.15.-p,06.20.Jr

n
a
J I. INTRODUCTION (high-energy part) and the low-energy photon contribu-
tion of the same order.
2
2 Whenaboundelectroninteractswithanexternal,uni- Previous studies of the bound-electron g factor for P
form and time-independent magnetic field (Zeeman ef- states in hydrogenlikesystems include Refs. [1–5]. Quite

] fect),theenergeticdegeneracyoftheatomicenergylevels recently,theproblemhasreceivedrenewedinterest[6,7].
h
p with respect to the magnetic projection quantum num- For few-electron ions, the bound-electron g factor has
- ber is broken, and the different magnetic sublevels split been investigated in Refs. [6–10]. For the 23P states of
m according to the formula helium, there is still an unresolved discrepancy of theo-
retical and experimental results (see Refs. [10–12]).
o
at ∆E =gjµBBµ (1) The expansion of the quantum electrodynamic radia-
tive correction to the electron g factor, which is an ex-
.
s where gj is the bound-electron (Land´e) g factor, µB = pansion in powers of α for a free electron, is intertwined
c
−e/(2m) is the Bohr magneton, and B is the magnetic with an expansion in powers of Zα for a bound electron
i
s fieldwhichisassumedtobeorientedparalleltothequan- (this fact has been stressed in Ref. [13]). For an nP
y j
tizationaxis. Finally, µ isthe magneticprojectionquan- state in a hydrogenlike system, we can write down the
h
p tumnumberoftheelectron;i.e.theprojectionofitstotal following intertwined expansion in powers of α and Zα,
[ angularmomentum(dividedby ~)ontothe quantization
g
1 axis. δg(nPj)=g00+(Zα)2 n220 +O(Zα)4
In leading order, the bound-electron g factor is deter-
v
mined by nonrelativistic quantum theory andis equal to
3 α b
1 a rational number for all bound states in a hydrogen- + π b00+(Zα)2 n220 +O(Zα)4 . (2)
9 like ion. Both relativistic atomic theory as well as quan- (cid:26) (cid:27)
3 tum electrodynamics (QED) predict deviations from the
The coefficients g and g characterize the relativistic
. nonrelativistic result. The relativistic effects follow from 00 20
1 effects, whereas b and b are obtained from the one-
00 20
0 Dirac theory and can be expressed in terms of a power loop radiative correction. The nonrelativistic result for
0 seriesintheparameterZα,whereZ isthenuclearcharge the Land´e g factor reads
1 number and α is the fine-structure constant. The QED
: effectsarecausedmainlybytheanomalousmagneticmo- 2 4
v g (nP )= , g (nP )= . (3)
i ment of the electron, which is turn in caused by the ex- 00 1/2 3 00 3/2 3
X
change of high-energy virtual photons before and after
The relativistic correction follows from Breit theory and
r the interaction with the external magnetic field. Here,
a the Dirac equation in an external magnetic field [3, 14],
by “high-energy” we refer to a virtual photon with an
energy of the order of the electron rest mass. A second 2 8
g (nP )=− , g (nP )=− . (4)
source for QED effects are exchanges of virtual photons 20 1/2 3 20 3/2 15
with an energy commensurate with the atomic binding
energyscale,whichissmallerthantheelectronrestmass The leading correction due to the anomalous magnetic
energy by a factor (Zα)2. Here, the electron emits and reads as (see Refs. [1, 3]),
absorbsavirtualphotonbefore andafter the interaction
1 1
with the external magnetic field, undergoing a virtual b (nP )=− , b (nP )= . (5)
00 1/2 3 00 3/2 3
transition to a excited atomic state in the middle. For
P states, the latter effects lead to a correction to the We are concerned here with the evaluation of the b
20
bound-electron g factor of order α(Zα)2. The complete coefficient of nP states, which is determined exclusively
j
result for the correction of order α(Zα)2 is obtained af- byself-energytypecorrections(vacuumpolarizationdoes
ter adding the anomalous magnetic moment correction not contribute).
2
We adopt the following outline for this paper. In turbative expansion of E in a magnetic field read
D
Sec.II,wereexaminethecontributionofhigh-energyvir-
tual photons (see also Ref. [3]). Two alternative deriva-
tionsarepresented,whicharebasedonaneffectiveDirac ∆E = ψ −e α~ · B~ ×~r ψ (9)
equation (Sec. IIA) and on an effective low-energylong- 2
wavelength quantum electrodynamic theory (Sec. IIB) D (cid:12)(cid:12)e (cid:16) (cid:17)(cid:12)(cid:12) E
which is obtained from the fully relativistic theory by a − (cid:12) κ ψ βΣ~ ·B~(cid:12)ψ
2m
combined Foldy–Wouthuysen and Power–Zienau trans-
D (cid:12) (cid:12) E
formation [10]. The low-energy part is also treated in (cid:12) (cid:12)
eκ (cid:12) (cid:12) Q
two alternative ways. The velocity-gauge calculation in + ψ i~γ·∇~V α~ ·(B~ ×~r) ψ .
2m 2m
Sec. IIIA is contrasted with the length-gauge derivation (cid:28) (cid:12)(cid:12)(cid:16) (cid:17) h i(cid:12)(cid:12) (cid:29)
in Sec. IIIB. Conclusions are reserved for Sec. IV. Nat- (cid:12) (cid:12)
ural units (~ = c = ǫ = 1) are used throughout the (cid:12) (cid:12)
0
paper. whereQ= 21(1−γ0)isaprojectorontovirtualnegative-
energystates,andψ istherelativisticwavefunction. An
evaluation of the first term on the right-hand side of (9)
with Dirac wave functions confirms the results for g
00
II. HIGH–ENERGY PART and g given in Eqs. (3) and (4). The second term on
20
theright-handsideof (9)yieldstheresultforb asgiven
00
A. Effective Dirac Equation in Eq. (5). When the Dirac wave functions are properly
expandedinpowersofZα,thesecondandthirdtermson
theright-handside of (9)yieldthe followinghigh-energy
In Ref. [3], the contribution to b due to high-energy
20 contributionb(H) totheb coefficientdefinedinEq.(2),
virtual photons was obtained on the basis of the two- 20 20
body Breit Hamiltonian. Here, we perform the calcula-
tionusingasimpleapproach,basedonaneffectiveDirac
1 1
Hamiltonian (see Ch. 7 of Ref. [15]). For an electron in- b(H)(nP )=− , b(H)(nP )= . (10)
teracting with external electric and magnetic fields, this 20 1/2 2 20 3/2 10
equation reads
TheseresultsareinagreementwiththosegiveninEq.(5)
H =α~ · p~−eF (∇~2)A~ +βm+F (∇~2)V
rad 1 1 of Ref. [3].
h e i
+F (∇~2) i~γ·E~ −βΣ~ ·B~ . (6)
2
2m
(cid:16) (cid:17)
We here take into account the Dirac form factor F and
1
the Pauli form factor F . The matrices α~ = γ0~γ and
2
β =γ0 arethe standardDiracmatricesinthe Diracrep- B. Long–Wavelength Quantum Electrodynamics
resentation [15], m is the electron mass, and e = −|e| is
theelectroncharge. Uptothe orderrelevantforthecur-
rent calculation, we may approximate both form factors It is instructive to compare the fully relativistic ap-
in the limit of vanishing momentum transfer as proachoutlined above to an effective nonrelativistic the-
ory. In Ref. [16], a systematic procedure has been
F (∇~2)≈F (0)=1, F (∇~2)≈F (0)≈κ≡ α . (7) described in order to perform a nonrelativistic expan-
1 1 2 2
2π sion of the interaction Hamiltonian for a light atomic
system with slowly varying external electric and mag-
The vector potential A~ corresponds to a uniform exter- netic fields. This procedure involves two steps, (i) a
nal magnetic field, i.e. A~ = 1 B~ ×~r , and the electric Foldy–Wouthuysen transformation of an interaction of
2 the type (6), suitably generalized for many-electron sys-
field E~ is that of the Coulomb(cid:16)potent(cid:17)ial (eE~ = −∇~V). tems, and (ii) a Power–Zienautransformationto express
Finally, V =−Zα/r is the binding potential. So, the vector potentials in terms of physically observable
field strengths. The result is an interaction, given in
e
H ≈α~ ·p~+βm+V − α~ · B~ ×~r Eq. (30) of Ref. [16], which describes a nonrelativistic
rad
2 expansion of the atom-field interaction in powers of Zα
(cid:16) (cid:17)
and can be used in order to identify terms which con-
iκ e
− ~γ·∇~V − κβΣ~ ·B~ . (8) tribute at a specified order.
2m 2m
If we are interested in evaluating the corrections to
Diraceigenstatesfulfill(α~·p~+βm+V)ψ =E ψ,where theg factoruptoorderα(Zα)2,i.e.allcorrectionslisted
D
E is the Dirac energy. The first few terms in the per- in Eq. (2), the relevant effective interactions for a one-
D
3
electron system are where p~ is the electron momentum. In an external mag-
neticfield,itisthephysicalmomentump~−e (B~×~r),not
3 the canonical momentum p~, which couples2to the quan-
H =H + H , H =µ (L~ +~σ)·B~ , (11a)
mag M i M B tized electromagnetic field. This amount to a correction
Xi=1 δJ~ to the electron’s transition current given by
µ
H = − B p~2(L~ +~σ)·B~ , (11b) e i e
1 2m2 δJi =− B~ ×~r =− ǫijkBjrk, (14)
2m 2m
(cid:16) (cid:17)
µ (1+2κ)Zα
H = B (~r×~σ)·(~r×B~), (11c) Because of the symmetry of the problem (Wigner–
2 4m r3 Eckhart theorem), we may fix the axis of the B field to
bealongthequantizationaxis(z axis)andtheprojection
µ κ
H3 = − 2mB2(p~·~σ)(p~·B~), (11d) of the reference state to be µ = 12. This procedure al-
lowsonetosimplifytheangularalgebra. Itisinspiredby
where µ =−e/(2m) is the Bohr magneton. We denote the separation of nuclear and electronic tensors that are
B
the Schr¨odinger–Pauli two-component wave function by responsible for the hyperfine interaction. Such a separa-
φinordertodistinguishitfromthefullyrelativisticwave tionhasbeenusedinEq.(1)ofRef.[19]andinEqs.(10)
function ψ. Specifically, φ reads as φ(~r) = R(r)χµ(rˆ) and (11) of Ref. [20]. In the case of the g factor, the
κ
in the coordinate representation, where R(r) is the non- magnetic field of the nucleus is replaced by the external,
relativistic radial component of the wave function and homogeneous magnetic field of the Zeeman effect.
χµ(rˆ) is the standard two-component spin-angular func- Wedivideoutaprefactor−e/(2m)fromboththemag-
κ
tion [17]. An evaluation of the perturbation netic HamiltonianHM andfromthe currentδJi andob-
tain the perturbative Hamiltonian h and the scaled
M,0
∆E =hφ|H |φi (12) current δji. In the spherical basis, this procedure leads
mag 0
to the operators
confirms the results of Eqs. (3), (4), (5) and (10) for
the high-energy part. No second-order effects need to hM,0 =L0+σ0, δj0i =ǫi3krk, (15)
be considered in this formalism up to the order in the
which are aligned along the quantization axis of the ex-
Zα-expansion relevant for the current study.
ternal magnetic field. The index zero of the operators,
in the spherical basis, denotes the z component in the
Cartesian basis (see Ref. [17]). The following shorthand
III. LOW–ENERGY PART
notation for the atomic states with magnetic projection
µ= 1 proves useful,
A. Velocity Gauge 2
|j1i≡|nP (µ= 1)i, hh i≡hj1|h |j1i. (16)
2 j 2 M,0 2 M,0 2
Themosteconomicalapproachtothecalculationofthe
low-energycontributionoforderα(Zα)2tothegfactorof Finally, we can proceed to the calculation of the per-
P states consists in a calculationof the orbital g factor, turbed self-energy. The nonrelativistic (Schr¨odinger)
ℓ
witha conversionofthe orbitalg factortothe Land´eg Hamiltonian of the atom is
ℓ j
factor in a second step of the calculation. In the order
p~2
α(Zα)2, one may indeed convert the spin-independent H = +V , (17)
NR
2m
correction to the orbital g factor to a spin-dependent
ℓ
correctiontothegj factorsofthe2P1/2 and2P3/2,asde- and the nonrelativistic self-energy reads
scribed in Appendix A. However,a more systematic ap-
proachtotheproblem,whichisalsoapplicabletohigher- mǫ
2α 1
order (in Zα) corrections, is based on a perturbation of δE = − dωω j1 p~ p~ j1 .
3πm2 2 H −E +ω 2
the nonrelativistic self-energy of the bound electron by Z (cid:28) (cid:12) NR NR (cid:12) (cid:29)
0 (cid:12) (cid:12)
the magnetic interaction Hamiltonian (11). This is the (cid:12) (cid:12) (18)
(cid:12) (cid:12)
approach outlined below.
We thus investigate the perturbation of the nonrela- The wave-function correctionto the self-energy reads
tivistic bound-electron self-energy [18] due to the mag-
4α mǫ
netic interaction Hamiltonian δE =− dωω (19)
ψ 3πm2
e Z0
H =− L~ +~σ ·B~ (13) ′
M 1 1
2m × j1 ~p ~p h j1 .
givenin Eq.(11a). In the vel(cid:16)ocity ga(cid:17)uge,the interaction * 2(cid:12)(cid:12) HNR−ENR+ω (cid:18)ENR−HNR(cid:19) M,0(cid:12)(cid:12) 2+
(cid:12) (cid:12)
of the electron with the vector potential A~ of the quan- Here, [1(cid:12)/(E −H )]′ is the reduced Green fun(cid:12)ction,
(cid:12) NR NR (cid:12)
tizedelectromagneticfieldisgivenbytheterm−p~·A~/m, withthereferencestatebeingexcludedfromthesumover
4
virtual states. The contribution of virtual nP states A numerical evaluation leads to the following results,
j
(with j being equal to that of the reference state and
n≥2) vanishes because of the orthogonality of the non- δg (2P )= 2α(Zα)2 1 ln ǫ +0.25134 ,
relativistic radial wave functions. The interaction h H 1/2 3π 6 (Zα)2
M,0 (cid:18) (cid:18) (cid:19) (cid:19)
couples 2P and 2P states, but the contribution of (25a)
1/2 3/2
virtual states with different j as compared to the refer-
encestatevanishesafterangularalgebra[17]becausethe 4α 1 ǫ
δg (2P )= (Zα)2 ln +0.15907 .
self-energyinteractionoperator{p~[1/(HNR−ENR+ω)]p~} H 3/2 3π 6 (Zα)2
is diagonal in the total angular momentum. Virtual (cid:18) (cid:18) (cid:19) (cid:19)
(25b)
stateswithdifferentorbitalangularmomentumthanthe
referencestatearenotcoupledatalltothereferencestate The correction to the current is given by
by the action ofthe perturbative Hamiltonian h . Be-
M,0
cause all contributions vanish individually, we can thus 4α mǫ
δE = − dωω
conclude that δEψ =0. C 3πm2
Hence, we have to evaluate first-order corrections to Z0
the Hamiltonian and to the energy corresponding to the 1
replacements HNR → HNR +hM,0 and ENR → ENR + × j21 piH −E +ωδj0i j21 , (26)
hh i in Eq. (18). Furthermore, we have a correction (cid:28) (cid:12) NR NR (cid:12) (cid:29)
M,0 (cid:12) (cid:12)
to the current corresponding to p~/m→ ~p/m+δ~j0. The wherewetakeinto(cid:12)(cid:12)accountthemultiplicit(cid:12)(cid:12)y factordueto
energy correction reads as the current acting on both sides of the propagator. The
corresponding correction to the g factor is
2α
δE =− hh i (20)
E 3πm2 M,0 δEj
δg (nP )=g (nP ) . (27)
C j 00 j
hh i
M,0
mǫ 1 2
× dωω j1 ~p ~p j1 ,
Z0 * 2(cid:12)(cid:12) (cid:18)HNR−ENR+ω(cid:19) (cid:12)(cid:12) 2+ We obtain the following numerical results,
(cid:12) (cid:12) 2α
where mǫ is an uppe(cid:12)(cid:12)r cutoff for the photon en(cid:12)(cid:12)ergy [18] δgC(2P1/2)= 3π(Zα)2 0.24607, (28a)
(the scale-separation parameter ǫ is dimensionless). It
corresponds to the following g factor correction, 4α
δg (2P )= (Zα)2 0.06151. (28b)
C 3/2 3π
δE
E
δg (nP )=g (nP ) . (21)
E j 00 j
hhM,0i Summing all low-energy corrections, the spurious loga-
rithmic terms cancel, and we obtain
A numerical evaluation of this correction according to
established techniques [21] yields b(L)(2P )=0.98428, (29a)
20 1/2
2α 1 ǫ
δgE(2P1/2)= 3π(Zα)2 −6 ln (Zα)2 −0.12831 , b(2L0)(2P3/2)=0.49216, (29b)
(cid:18) (cid:18) (cid:19) (cid:19)
(22a)
as the spin-dependent low-energy contribution to the
bound-electron g factor. The result for 2P is equal to
4α 1 ǫ 3/2
δgE(2P3/2)= 3π(Zα)2 −6 ln (Zα)2 −0.12831 . half the correction for 2P1/2 (this fact is independently
(cid:18) (cid:18) (cid:19) (cid:19) provenalsoAppendixA). Wedenotethelow-energycon-
(22b) tribution to the b coefficientdefined in Eq.(2) by b(L).
20 20
For the correction to the Hamiltonian, we get
2α mǫ B. Length Gauge
δE = dωω
H 3πm2
Z0
In the length gauge, the interaction with the quan-
1 2 tized electromagneticfield is givenby the dipole interac-
×*j21(cid:12)(cid:12)~p(cid:18)HNR−ENR+ω(cid:19) hM,0p~(cid:12)(cid:12)j12+ , (23) Ttiohneg−aeu~xge·-Ei~n,vawrhiaenrte[E2~2]isnothnereelaletcivtriisct-ificesledlf-oepneerragtyorin[2t2h]e.
(cid:12) (cid:12)
(cid:12) (cid:12) length-gauge reads
where we ha(cid:12)ve used the relation [HNR,hM,(cid:12)0] = 0. This
translates into the following correction for the g factor, 2α mǫ 1
δE = − dωω3 j1 ~x ~x j1 .
δgH(nPj)=g00(nPj) δEH . (24) 3π Z0 (cid:28) 2(cid:12)(cid:12) HNR−ENR+ω (cid:12)(cid:12) (23(cid:29)0)
hhM,0i (cid:12) (cid:12)
(cid:12) (cid:12)
5
Inthelengthgauge,thecontributionofthewave-function and with the help of a somewhat lengthy calculation, it
correctionvanishesbecauseofthesamereasonsasforthe is possible to show analytically that the velocity-gauge
velocitygauge. Also,there isno correctionto the transi- and the length-gauge forms of the low-energy contribu-
tion current, because the canonical momentum does not tions are equal. The calculationfollows ideas outlined in
enter the interaction Hamiltonian in the length gauge. detail in Ref. [23] where the more complicated case of a
We only have corrections to the Hamiltonian and to the relativistic correctionto a transitionmatrix element was
energy. We start with the energy perturbation, considered. Here,weareinterestedmainlyinthenumeri-
calvalueofthecorrection,forwhichthegaugeinvariance
2α
provides a highly nontrivialcheck. Note that the matrix
δEE =− hhM,0i (31)
3π elementsgoverningthetransitionsofthereferencetothe
virtual states are completely different in the length and
mǫ 1 2 in the velocity gauges, and the final results are obtained
× dωω3 j1 ~x ~x j1 .
Z0 * 2(cid:12)(cid:12) (cid:18)HNR−ENR+ω(cid:19) (cid:12)(cid:12) 2+ athfteerspseucmtrmumingofovveirrttuhael sdtiastcerse.te and continuous parts of
(cid:12) (cid:12)
(cid:12) (cid:12)
The subscript E inste(cid:12)ad of E serves to differen(cid:12)tiate the
length-gauge as opposed to the velocity-gauge form of
IV. CONCLUSIONS
the correction. Indeed, the numericalresults for the cor-
responding correctionto the bound-electron g factor are
different from those given in Eq. (22) and read In our approach to the calculation of the bound-
electron g factor of P states, the contribution due to
2α 1 ǫ high-energy virtual photons can be obtained using two
δgE(2P1/2)= 3π(Zα)2 −2 ln (Zα)2 −0.88488 , alternativeapproaches,basedeitheronaneffectiveDirac
(cid:18) (cid:18) (cid:19) (3(cid:19)2a) equation or on a low-energy effective Hamiltonian. The
contribution due to low-energy photons is treated as a
perturbation of the bound-electron self-energy [18] due
4α 1 ǫ
δgE(2P3/2)= 3π(Zα)2 −2 ln (Zα)2 −0.88488 . to the interaction with the external uniform magnetic
(cid:18) (cid:18) (cid:19) (cid:19) field. CorrectionstotheHamiltonian,tothebound-state
(32b)
energy and (in the velocity gauge) to the transition cur-
renthave to be considered. The final results for the low-
For the correction to the Hamiltonian, we get
energy parts in the velocity- and length-gauges agree al-
2α mǫ though the individual contributions differ (including the
δEH = dωω3 coefficients of spurious logarithmic terms).
3π
Z0 Adding the high-energy contribution to the g factor
correction given in Eq. (10) and the low-energy effect
2
1
× j1 ~x h ~x j1 . (33) given in Eq. (29), we obtain the following results for
* 2(cid:12)(cid:12) (cid:18)HNR−ENR+ω(cid:19) M,0 (cid:12)(cid:12) 2+ the self-energy correctionof order α(Zα)2 to the bound-
(cid:12) (cid:12) electron g factor of 2P states (b =b(H)+b(L)),
(cid:12) (cid:12) 20 20 20
A numerica(cid:12)l evaluations leads to (cid:12)
b (2P )=0.48429, (37a)
20 1/2
2α 1 ǫ
δgH(2P1/2)= 3π(Zα)2 2 ln (Zα)2 +1.25399 , b (2P )=0.59214, (37b)
(cid:18) (cid:18) (cid:19) (cid:19) 20 3/2
(34a)
wheretheb coefficienthasbeendefinedinEq.(2). Both
20
above results compare favorably with recently obtained
4α 1 ǫ
δgH(2P3/2)= 3π(Zα)2 2 ln (Zα)2 +0.97716 . nAupmpeenridciaxlBda).taAfnoroblovwio-uZshgyenderoragleinzalitkieonioonfsth[2e4f]o(rsmeealaislsmo
(cid:18) (cid:18) (cid:19) (cid:19)
(34b) outlined here to the 3P and 4P states yields the results
Adding the length-gauge corrections, the logarithmic b20(3P1/2)=0.40500, (37c)
termscancel,anditisstraightforwardtonumericallyver-
ify the gauge-invariancerelation b20(3P3/2)=0.55250, (37d)
δgE +δgH +δgC =δgE +δgH (35) b20(4P1/2)=0.31331, (37e)
andthus,thenumericalresultsalreadygiveninEq.(29).
b (4P )=0.50665. (37f)
20 3/2
Let us finally discuss the analytic proof of the gauge
The two main results of the current investigation can
invariance. Using the commutator relation
be summarized as follows. First, in Sec. III we formu-
pi late a generalizable procedure for the calculation of low-
m =i[HNR−ENR+ω, xi], (36) energycorrectionstotheLand´egj factorsinone-electron
6
ions, applicable to P states andstates with higher angu- Institutes of Standards and Technology (NIST).
larmomenta. Thisprocedureisbasedonchoosingaspe-
cific referenceaxisforthe externalmagnetic field. Inthe
future, it might be applied to include higher-orderterms
from the Hamiltonian (11) which couple the orbital and TABLE I: Higher-order remainder functions i1/2(Zα) and
spin degrees of freedom. Second, we resolve the discrep- i3/2(Zα) for the self-energy correction to the g factor of
ancy reported in Ref. [24] regarding the low–Z limit of 2P1/2 and 2P3/2 states, respectively, as obtained recently
in Ref. [24]. The value of α employed in the calculation is
the α(Zα)2 correction to the gj factor with previous re- α−1 =137.036,andthenumericaluncertaintyoftheall-order
sults reported in Ref. [5] for this correction [see Eq. (37) (in Zα) calculation due to the finite number of integration
and Appendix B]. In Appendix A, it is shown that the points in thenumerical calculation is indicated in brackets.
discrepancy to the results of Ref. [5] can be traced to
the final evaluation of the logarithmic sums over virtual Z g1/2(Zα) g3/2(Zα)
1 0.121258(21) 0.148104(21)
states, while the angular momentum algebra is in agree- 2 0.121715(22) 0.148294(24)
ment. Thatisafurtherreasonwhythecross-checkofour 3 0.122414(19) 0.148567(24)
calculationinthelengthandvelocitygaugesappearedto 4 0.123280(14) 0.148851(22)
be useful. 5 0.124305(10) 0.149338(18)
Regardingthe experimentalusefulness ofthe obtained 6 0.125473(7) 0.149816(14)
results, we can say that recent proposals [25] concerning 7 0.126803(5) 0.150350(11)
8 0.128186(4) 0.150933(7)
measurements of the bound-electron g factor of low–Z
9 0.129711(2) 0.151561(4)
hydrogenlikeionsarebasedondouble-resonanceschemes
10 0.131336(2) 0.152231(4)
that also involve transitions to 2P states in the presence
of the strong magnetic fields of Penning traps. In order
to fine-tune the double-resonance setup, the results ob-
tained here might be useful. Also, the results reported
hereserveasageneralverificationfortheanalyticformal-
ismusedinthetheoreticaltreatmentofα3 correctionsto
the g factor of 23P states in helium, for which an inter-
estingdiscrepancyofexperimentalandtheoreticalresults
persists (see Ref. [12] and Sec. V of Ref. [16]).
We conclude with the following remark. Our calcu-
lation concerns the g factor of P states, and we con-
firm that in the order α(Zα)2, low-energy virtual pho-
tons yield an important contribution. From the treat-
ment in Sec. III, we can understand physically why
there is no such low-energy effect of order α(Zα)2 for S
FIG. 1: (Color online) The higher-order remainder function
states. Namely, a Schr¨odinger–Pauli S state happens
to be an eigenstate of the Hamiltonian1/2h . We have i1/2(Zα) is shown as a function of Z. Numerical values for
hM,0|nS1/2(µ = ±21)i = ±|nS1/2(µ = ±21)Mi.,0This prop- it1h/e2(cZoeαffi)caireentgiv41ebn20i(n2PT1a/b2)leaIn.dTihseapppoirnotacahteZd s=m0ooisthgliyv.enby
erty holds because an S state carries no orbital angular
momentum, and therefore is an eigenstate of the third
component of the spin operator σ , and b therefore
0 20
vanishes for S states [26]. For P states and states with
higherorbitalangularmomenta,thesituationisdifferent:
thesestatesarenoteigenstateofh irrespectiveoftheir
M,0
angular momentum projection, even though h com-
M,0
muteswiththenonrelativisticHamiltonianH . There-
NR
fore, there is a residual effect of order α(Zα)2 due to
low-energy virtual photons for states with nonvanishing
angular momenta.
Acknowledgments
FIG.2: (Coloronline)SameasFig.1,butforthe2P3/2 state.
Valuable and insightful discussions with Thehigher-orderremainder function i3/2(Zα) is plotted as a
V. A. Yerokhin and K. Pachucki are gratefully ac- fbulencIt.ioTnhoefpZo,inwtiathtZnu=me0riicsallimvaZlαu→es0fio3r/2i(3Z/2α(Z)=α)41gbiv20e(n2iPn3/T2a)-.
knowledged. This project was supported by the
The limit is approached smoothly.
NationalScience Foundation (GrantPHY–8555454)and
by a precision measurement grant from the National
7
Appendix A: Remarks on the Low–Energy Part
j(j+1)+ℓ(ℓ+1)−s(s+1)
g =g
First of all, let us remark that our results reported j ℓ
2j(j+1)
in Sec. III can be expressed as logarithmic sums over
the spectrum of atomic hydrogen. After performing the j(j+1)+s(s+1)−ℓ(ℓ+1)
angular algebra [17], we can write the total low-energy +gs
2j(j+1)
correctionδg(L)(nPj)=δgE(nPj)+δgH(nPj) as follows,
3j(j+1)−ℓ(ℓ+1)+s(s+1)
≈ . (A4)
δg(L)(nP )= 8α (E −E )2 hmS||~r||nPi2 2j(j+1)
1/2 9π mS nP
m
X In our notation, the analytic result given in Eq. (14) of
Ref. [5] for the correction to the orbital g factor reads
2|E −E | 8α
×ln mS nP − (E −E )2
(Zα)2m 9π mD nP α
(cid:18) (cid:19) m δg (nP)= (E −E )2 hmS||~r||nPi2
X ℓ 3π mS nP
m
2|E −E | X
×hnP||~r||nDi2 ln mD nP (A1)
(Zα)2m 2|E −E | α
(cid:18) (cid:19) ×ln mS nP − (E −E )2
(Zα)2m 3π mD nP
(cid:18) (cid:19) m
X
and
2|E −E |
×hnP||~r||nDi2 ln mD nP . (A5)
δg(L)(nP3/2)= 49απ (EmS −EnP)2 hmS||~r||nPi2 (cid:18) (Zα)2m (cid:19)
Xm The prefactors multiplying g in Eq. (A4) read 4 for
ℓ 3
×ln 2|E(mZSα−)2mEnP| − 49απ (EmD−EnP)2 Pfo1r/m2ualandob32tafionredP3in/2Esqt.at(e1s4,)aonfdRtehf.er[5ef]oirseinthaegareneamlyetnict
(cid:18) (cid:19) m with our approach for both P and P states, after
X 1/2 3/2
the correction to g is converted into the correspond-
ℓ
2|E −E |
×hnP||~r||nDi2 ln mD nP . (A2) ing modification of gj. However, their numerical result
(Zα)2m δg =−0.24α3 disagreeswithourresultbothinsignand
(cid:18) (cid:19) ℓ
in magnitude. Indeed, expressed in terms of our b(L) co-
20
Here, hmS||~r||nPi and hmD||~r||nPi are reduced matrix efficient,theresultsindicatedinRef.[5]wouldimplythat
eElenmP)en∝ts(iZnαth)2e annodtathimonS|o|~rf||RnePf.i[∝27](.ZBαe)c−a1u,steh(eEambSov−e b(2L0F)i(n2aPl1ly/,2)le=t u−s4n.0o2teanthdabt(2L0th)(e2Pca3/lc2u)l=ati−on8.0o4f.the orbital
correctionstothegfactoraremanifestlyoforderα(Zα)2. correction to the g factor is only applicable to order
j
The sums over m extend over both the discrete as well α(Zα)2, not α(Zα)4, because the higher-order terms in
asthecontinuouspartofthehydrogenspectrumandcan themagneticinteraction(11)couplethe orbitalandspin
convenientlybeevaluatedusingbasis-settechniques[28]. degrees of freedom. The formalism outlined in Sec. III
Because it may not be completely evident from the generalizes easily to the calculation of higher-order cor-
presentation in Ref. [5], we reemphasize here that the rectionsthatcouplespinandorbitalangularmomentum,
authors of the cited article evaluate a correction δg to which might be needed in the future, whereas the sepa-
ℓ
the orbital g factor according to the definition ration into spin and orbital g factors only holds up to
order α(Zα)2.
H =g µ S~ ·B~ +g µ L~ ·B~ (A3)
eff s B ℓ B
Appendix B: Comparison to Numerical Data
for the effective interaction of a bound electron with an
external magnetic field (S~ = 1~σ measures the electron We parameterize the one-loop self-energy correction
2
spin, and we have g ≈ 2, g ≈ 1). Here, an inter- δ(1)g to the g factor of P states as
S ℓ
esting analogy to the description hyperfine splitting can
α
be drawn, because the interactionwith the nuclearmag- δ(1)g(2P )= b +(Zα)2 i (Zα) , (B1)
j 00 j
π
neticfieldcanalsobeseparatedintodistinctcomponents,
(cid:8) (cid:9)
namelythe orbital,spin-dipole,andFermicontactterms
where i (Zα) is the nonperturbative (in Zα) remain-
j
given, e.g., in Eqs. (24)—(33) of Ref. [29].
der function. The remainder functions i (Zα) and
1/2
Returningtothediscussionoftheg factor,weseethat i (Zα) for 2P and 2P , respectively,have recently
3/2 1/2 3/2
as a combination of g and g , the Land´e g factor is ob- been evaluated in Ref. [24]. Numerical values of i (Zα)
s ℓ j
tained as for Z = 1,...,10 are given in Table I. Note that these
8
dataimplyaspin-dependence ofthe higher-ordercorrec- AsshowninFigs.1and2,thislimitisbeingconsistently
tion term b beyond the spin-dependence of the high- approachedby the numerical data.
20
energy part given by Eq. (10). The limit as Zα → 0 of
the remainder function i (Zα) is
j
lim i (Zα)= 1b (2P ). (B2)
Zα→0 j 4 20 j
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