Radiative corrections for many-electron atoms in perturbation theory

ANNALS

OF PHYSICS

184,

231-253 (1988)

Radiative Corrections for Many- Electron Atoms in Perturbation G. FELDMAN Department The Johns Hopkins

Theory*

AND T. FULTON

of Physics and Astronomy, University, Baltimore, Maryland

21218

Received October 12, 1987; revised February 1, 1988

The techniques of Erickson and Yennie for treating the Lamb shift are generalized to obtain finite expressions for additional radiative corrections. These terms were previously formally generated by us in their unrenormalized form in perturbation theory, and are of significance for the case of many-electron atoms. For the sake of brevity, only log corrections (terms proportional to In(Za)2 and the analogues of the Bethe-log term) are considered. Explicit finite expressions, suitable for immediate numerical evaluation, are obtained. The effect of these corrections is first to replace the unshielded nucleus by one that is shielded by the core electrons. A secondary effect accounts for the change of the energy of a given orbital state due to the radiative effects on the core electrons during their interaction with that orbital state. Because of the simple physical interpretation of the perturbation theory results, the corresponding radiative correction contributions to the Dirac-Fock approximation can be obtained in the simplest approximation (neglect of Bethe-log terms) and can be conjectured in the general case. 6 1988 Academic Press, Inc.

1. INTRoD~CTI~N

The present work is a direct continuation of the program begun in our most recent paper, “Relativistic Many-Electron Atoms in Perturbation and Dirac-Fock Theory” [l] (hereafter referred to as I). Our current focus is on a further consideration of the perturbation theory treatment presented in I, and on its generalization to include the Dirac-Fock (DF) approximation. Our starting point in that reference was the one-lepton propagator (two-point function), GN, which we had extensively considered previously [2, 31. GN is an expectation value between lepton states of total lepton charge’ Ne of a time-ordered product of lepton field operators ij and tit in the presence of a nuclear Coulomb potential, I’,,,, of charge -Ze, considered as an external potential, and of the electronelectron interaction, which involves the exchange of virtual photons. We expanded this propagator in terms of gN, the two-point function in the limit of no electron-electron interaction, * Supported in part by the National Science Foundation. ’ N must refer to a non-degenerate state, in order for standard QED methods, such as Wick ordering and Feynman diagram representations, to apply.

231 OOO3-4916/88 57.50 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

232

FELDMAN

AND

FULTON

and the photon propagator, Dpv, through second order in D,,. This expansion can also be represented by Feynman diagrams.’ The next stage of the analysis was to isolate terms containing radiative corrections from those which were finite. We accomplished this by successively using an identity3 connecting gN with g. g is the two-point function which is a vacuum (or zero lepton charge, rather than lepton charge Ne) expectation value. We continued to use the identity, Eq. (1.2.10), until one of two results was achieved: either the expressions we obtained at a given stage of this process were finite, or the only lepton propagators which remained were g, rather than gN. The latter terms included all the radiative corrections. The remainder of Section 2 of I, which dealt with perturbation theory, was devoted to the consideration of the intrinsically finite terms in greater detail. Further treatment of the radiative correction terms was left to the present paper. We begin Section 2 of this paper by listing once again the Feynman diagrams of I which will be of current interest. We shall illustrate the Feynman rules, as applied to these diagrams, by giving the algebraic expressions which correspond to three of them. This will also serve as a review of the relevant notation of I. (These algebraic expressions, corresponding to radiative corrections, were not explicitly written down in I.) Since we are at present concerned with corrections to the standard Lamb shift results which are due to the many-electron features of the problem, we shall not consider diagrams involving closed lepton loops. These contributions are a small part of the usual Lamb shift, and are expected to be a small part of our present corrections as well. In the same spirit, we shall consider only the log terms of the self-energy and vertex terms which remain. At this stage, we make our arguments more transparent by taking the log term to be a constant, ln[1/(Za)2]. Since we are following standard QED procedures, we can renormalize the charge, mass, and wavefunction, and insert these renormalized values for all finite terms. We then connect the evaluation of the self-energy and vertex terms to the Lamb shift calculation of Erickson and Yennie [4] (hereafter referred to as II), by a trick: we identify all non-radiative photon exchanges as part of an “external” potential, which now can be time-dependent. The analysis of II then applies in the present case, with slight modifications. Just as in II, only a mass renormalization counterterm must be introduced. Furthermore, just as in II, the terms we consider are gauge-invariant combinations in every case. Also, as expected, the effects of Fermi statistics are automatically contained in our results. We remind the reader that, in II, the Lamb shift is expanded in a series, the terms of which are successively smaller for small Za. Accordingly, the Lamb shift to lowest order, Z4a5(a ln(Za)-2 +b) can be read off from II (or any standard textbook). The radiative corrections which we calculate in this paper include the electronelectron interaction to lowest order and are of order l/Z times the standard Lamb shift. In Section 3, we reformulate and further simplify the already simple result of * See Figs. 2.1-2.4 of I. We shall subsequently use the notation Figs. 1.2.1-1.2.4. 3 See Eq. (2.10) of I. We shall subsequently use the notation Eq. (1.2.10).

RADIATIVE CORRECTIONS FOR MANY-ELECTRON

ATOMS

233

Section 2. This final form has a natural physical interpretation and leads, in the simple approximation of Section 2, to an obvious conjecture for the radiative correction terms of similar type in the Dirac-Fock approximation. The Bethe-log terms cannot be neglected in the log approximation. They are treated in detail in the Appendix, which completes our consideration of all log terms. While terms other than log terms can be treated as well, there is no need to carry out the calculations required to obtain them for our present accuracy. We summarize and discuss our results in Section 4 of the main part of the paper. Our discussion includes a consideration of some obvious non-log contributions. We emphasize that all of our final results are finite and are easily accessible to numerical methods of evaluation, which are in principle no more difficult than the evaluation of standard Bethe-log terms for Lamb shifts.

2. CALCULATIONS We begin a brief outline of our calculations by listing the four groups of relevant diagrams with which we shall be concerned in Figs. 1-4. They are taken from the lists of diagrams given in Figs. 1.2.6-1.2.10. We have made one change in notation. The corrections we are calculating are to the single-particle energies of the state labeled n. We used the label s in I. The diagrams in Figs. l-4 are the only ones which remain after we have excluded those which give finite contributions to the energy, and those yielding radiative corrections of vacuum polarization type (i.e., those which contain closed fermion loops). We have treated the former effects in I. We shall not be concerned with the latter effects, since they are small relative to the log terms on which we shall concentrate. Those interested in the details of the arguments which led to the diagrams we have listed should refer to I. We limit ourselves here to a brief summary of our notation. For the purposes of the present paper, these diagrams should be considered as given, and our only concern here is in the evaluation of their contributions to the energy of atomic levels.

FIG. 1. Diagrams which contribute to electron shielding corrections to the Lamb shift for state 595/184/2-2

n.

234

FELDMAN AND FULTON e-w 1 Cc)

,‘klf&-\\

’

w-b ,

4;A

5;P

3; Y I e , I

”

(4

(4

7-‘, (‘4

,’

3;v

’

(‘4

‘5;pw;w4;x

S~(X)(o;PC)

,/

‘7

a-&J’ A-., wI

a4;X

\

(8)

5iP

’

’ 10 I l,Z;!J

IO I 1,2;P

I” 1,2; Ir

3;”

S~(x)(O;Pa)

%(X)(@‘b)

FIG. 2. Diagrams which contribute to radiative corrections for the state n, due to core-state shielding by the electron in state n.

The symbol S appearing in the figures stands for self-energy. Indeed, all the diagrams of Figs. l-4 at the initial stage of the analysis, where only g” appear, are self-energy diagrams involving two virtual photons. The self-energy diagram associated with one virtual photon yields the Lamb shift (excluding the Uehling effect). We have nothing to add to its evaluation, and will not list it until we collect all our terms below. The self-energy diagrams where the g appear, which are associated with two virtual photons, are higher-order Lamb shift corrections. That is, they will be of order CI times the usual Lamb shift, and thus will be neglected. The subscript labels X(T), X(X), XT, etc., in the figures indicate the Feynman diagram (containing only gN) from which the diagram in question originated. Thus, X(T) indicates an origin in the first diagram of Fig. 1.2.2 (an irreducible diagram involving a single lepton line and the insertion of a vertex correction, (r), into an exchange, (X), diagram). X(T), T(X), and X(X) are associated with the second, third, and fourth diagrams of Fig. 1.2.3 and represent, respectively: the insertion of a “tadpole,” (T), into an exchange, (X); an exchange into a “tadpole”; and an exchange within an exchange. All of these diagrams are irreducible. The diagrams labeled TX, XX, and XT are associated with the second, third, and fourth diagrams of Fig. 1.2.4. These are reducible Feynman diagrams, containing a succession of a “tadpole” and an exchange, two exchanges, and an exchange followed by a “tadpole,” respectively. A curious feature of our result is that triplets of diagrams which arise from different origins are grouped together in the samefigure. It is not difficult to show that each of the three diagrams in a given figure forms a gauge-invariant set. To continue with the definition of our notation: The arguments of the lb&’

a-w’

(4

,f$‘,

a .I

I

%(X)(O;pa’ FIG.

(a)

,7---. ,'a-II+ w

Rd’

w'\

(4

3;v n

L-Y, 0'

‘X X ( ‘a)

sx (r ,P;Pa)

3. Exchange contributions

,'

to Figs. 1 and 2.

\

RADIATIVE

sxx(pc) FIG.

CORRECTIONS

FOR

MANY-ELECTRON

Sqr )(o;P,)

ATOMS

235

%(x)(*pc)

4. Complex conjugate of the exchange contributions in Fig. 3.

S-functions in the figure labels are relics of I and are included only for purposes of identification. Now, it is only necessary to know that a lepton line indicates a g (rather than g”‘), and a lepton line with an x through it indicates g’ (rather than g). This is a propagator with the singular term removed. All indices a, 6, c in the figures are dummy summation indices and thus can be replaced by the single label a. These indices take on the values 1,2, .... N; i.e., they are the labels for the singleparticle core states. The symbol P, stands for the appearance of the projection operator P,, where p*= l@>(4.

(1)

la) stands for the ket of energy E,, and is represented in the figures by a line with a vertical “stop” on the right. The corresponding bra is a line with a vertical “stop” on the left. The labels over lines (dashed or solid) in the diagrams indicate the corresponding energies. Thus, n - c + o’ in Fig. 4 stands for E, - E, + o’, etc. The labels 1; p, etc., are space-time labels and Lorentz index labels for y,,. One final point before we proceed to write down the algebraic forms which correspond to a sample of diagrams: Not only do the three diagrams of each of the figures form a gauge-invariant subset, but, also, different gauges can be used for the exchanged photon on the one hand and the radiative photon (i.e., the photon in a self-energy or a vertex part) on the other hand [lo]. It is convenient to choose a covariant gauge (to match II, we shall choose the Feynman gauge) for the latter and the Coulomb gauge for the former. We can then neglect the transverse parts of the exchanged photons, since they yield higher-order contributions [l]. Alternatively, we could choose the Feynman gauge for both self-energy/vertex and exchange photons, keep only the O-0 component of DPy for the latter, and also neglect retardation effects in Doe. We illustrate the transcription of the diagrams into algebraic expressions in perturbation theory, and in the approximation and gauges discussed above4,5 with 4 We have now allowed for the time-dependence of states and energies. ’ See Table (1.1) for the applicable Feynman rules. For the Lorentz metric and y-matrix conventions we use, see footnote (1.4).

236

FELDMAN

AND

FULTON

the self-energy contribution from the three diagrams of Fig. 3. We note that the diagrams are remarkably similar to those corresponding to lowest-order self-energy corrections in the presence of a new “effective potential,” which depends on the Coulomb propagator and the states (nl and la). Thus, from Fig. 3, we have in the space-time representation, 27d(&,

- En,) &,X(X) = (4 &,x,@;Pu)In’) E - ie2f J (a I4) yoypg(43)yoywwww

yoyv

“==l

I n’), (2) - E,,)E,X(Q = (4 x&o; Pa)In’> = - ie2f J (a I 4) Y0Y2 d43) Y0Yv4..(3) d32)YOYp x A$

27d(&,

.(2)<2

OS1

I n’), E,XX= (nl S&P,) In’) x

27&y&,-E,,)

DAp(42)(2

(3)

= - ie2 agl j (a I 3 > YOYyAl, .(3)g’(34)Y0Y2 g(Q)YOYp x DAp(42)(2 1 n’).

(4)

The labels 2, 3, 4 are space-time variables, where, e.g., 2

=

(f2,2)

=

(f2,

(5)

r2),

and integration over all the space-time variables is understood. The lepton Green’s functions, as well as the wavefunctions, are space-time functions. Typically, we have (1 ) n)~e-i&,tl~n(rl)-e-‘E”‘l(l

( n),

where the E, and the (P” are the unperturbed single-particle wavefunctions, respectively. The photon propagator is given by Dp”( 34) = - m The “effective potential”

1

gp” I

(time-dependent)

e-ik.(3-4)

6 d4k.

(6)

energies and

(7)

is given by

4, a(2) = e2 1 d41 (n I 1) yoyp( 1 I a) D&(12),

(8)

RADIATIVE

CORRECTIONS

FOR

MANY-ELECTRON

ATOMS

where, in our approximation, we keep only the CM component propagator in the Coulomb gauge, i.e., s e

-ik.(i-2)

4tt -a

$ d4k=

4n 11-21’

237 of the photon

(9)

This, in turn, gives A;,.(2)=

(ei(en-6.)t= u,,,(2), O),

(10)

,J2)=cc

j- d31 ‘n ‘,:“;,’

(11)

where6 V

?

We have defined the operator’ A;,, or A”“, ,(x) so that it enters into Eqs. (2~(4) in a way similar to that of the local external nuclear potential operator, A+&,(x), defined below. Thus, in the space-time representation, we have (11 4.12)

3 (11 4,.(x)

12) =4.w~4(1

-2).

(12)

In the same way, v,,, or v,,, (x) is a static local operator where (11 tJn,AXf 12)=%.u)~3(~-2).

(13)

It is important to note that the way terms have been collected together in Eqs. (2)-(4) makes it appear that we are calculating off-diagonal (a c-) n’) corrections to self-energies in the presence of a perturbing local time-dependent potential A; ,(x). However, the time-dependence in this potential, taken together with the timedependences in the propagators and states, combines, to produce a diagonal element in the correction to the single-particle energies, as indicated by the b-function on the left side of Eqs. (2)-(4). We use notation similar to that in II to define, formally, the electron propagator in the presence of the nuclear Coulomb field. Thus,* 9(43)y0=(41$-*(

3),

(14)

where JJ.17=:pZIp,

l7p =pp - Ak”,(X).

(15)

’ Since the four-momentum operator, p,,, will appear in later equations as a derivative, we shall reserve the labels ~4’ (sometimes displayed explicitly) for those quantities which do not commute with p,,. All space-time variables over which integrals are performed will be labeled by numerals, 1, 2, etc. (see Eq. (5)). * Cf. Eqs. (11.2.1) where the electron propagator as it enters into the energy correction is displayed, and also (11.2.5a) and (11.2.5b).

238

FELDMAN AND FULTON

The symbol pP is the single-particle four-momentum operator and Ai;,, is the static nuclear Coulomb field, not necessarily that of a point charge: 4uc(x)

= (~Nuc(X)9 0).

(16)

In II, the operator p” is replaced by the energy eigenvalue En because, in the energy representation, only diagonal elements of the operator appear. This is not the case here in Eqs. (2k(4). We are now ready to take the key step which will alow us to evaluate the sum of Eqs. (2), (3), and (4), using the formalism of II. (The sums corresponding to the other figures can be similarly evaluated.) We use the device of formally adding the “effective potential,” given by Eq. (10) to the nuclear potential. In other words, we define

(nf n.a)J‘,IZ@-f/f" lI,a

(17)

This allows us to sum the expressions, Eqs. (2)-(4), for E(3) n

s

~W) ”

+

EXu-) n

+

EXX n

(18)

3

into’

(19) The states in this modified w,

t,;

potential a)

=

(1,

are (to lowest order in

f) given by (204

11 I aa,>,

(2Ob) (214 @lb) They satisfy (to lowest order in (Y * R, (I -m) We are following

f) the equations

l@!J~)>=o

II in planning

and


(Yq.--N=Q

(22)

to evaluate our expressions in space-time. The

g Cf. Eq. (II.2.1), but note the overall sign difference.

RADIATIVE

CORRECTIONS

FOR

MANY-ELECTRON

explicit time-dependence of the “effective potential” that the components of pP must be taken as

.a

Pj’

a-1 2~

ATOMS

239

is a new feature in this paper, so

.a

po=1,,.

(23)

To evaluate Eq. (19) it is necessary (just as in II) to introduce a mass counterterm. When we do so, we obtain an expression which corresponds to (11.2.1), and which can also be manipulated formally in exactly the same way as Eq. (11.2.1) is in II. The expressions associated with Figs. 1, 2, and 4 can be treated in a similar way, with “external” potentials and states corresponding to those defined in Eqs. (lo), (20), and (21). We summarize the different A, ,,, bras and kets which appear in the respective energy contributions:” E(l). n .

A a.07 (4,

E(2).

A n,n’9 (4, Ia>; A n,a, (4, In’>;

Wb)

A a,n’7(43 Ia>.

Wd)

n.

E(3).

” .

E(4).

n .

In’>;

Pa)

WC)

Because of overall energy conservation (the appearance of c’?(E,- E,,)), the “external” potentials in Eqs. (24a) and (24b) are actually time-independent (see Eq. (lo)), and therefore so are the corresponding wavefunctions (I$-!Jn)) and its conjugate in (24a), and 1${(a)) and its conjugate in (24b)). Let us now return to the evaluation of Eq. (19), with a mass counter-term added. To lowest order in Zcc, the formal manipulations of II lead directly to the analogue ’ i of (11.3.27a) (the superscript NR means that the non-relativistic limit is taken), dz P(z, u)

with

v<,.(x) = vNuc(x) +fu&x)

eicEneEa)‘.

(26)

” The expressions for &!,I)and E!,*)will differ in overall sign from those for si3) and eL4).This can be traced back to I and is a consequence of the Fermi statistics of the electrons. ‘I Note that we use a sign convention different from that of II for NR bound-state energies. We take the eigenvalues of the corresponding stationary state equations to be +syR, with &FR-K0. In contrast, Erickson and Yennie, in Eq. (II.3.6), take these eigenvalues to be -&FR, with sFR>O.

240 The polynomial

FELDMAN

AND

FULTON

P(z, U) is given by (11.3.27b). Equation (Hi, o)NR = 2m

[

-PO+&+

(11.3.28b) is replaced by

V{,.(x)

1 .

(27)

The NR wavefunctions satisfy time-dependent Schrodinger equations (the analogues of the stationary-state Schriidinger equation (11.3.32.a)) which are the NR limits of Eqs. (22),

<4G)Iwi, JNR=0.

(29)

d< p is given by (11.3.28a), with HNR replaced by (H{,JNR. kince we are dealing with a correction of relative size l/Z to the Lamb shift term, it is sufficient for us to consider only the contributions which become logarithmically singular when (Hi, JNR is neglected. This we call the log approximation. Accordingly, we make the approximations qz, u) z P(0, u) = -2U(l

- U) + 1,

(30)

and A< Jz, u) N zm’ + (HL, JNR.

(31)

The integrals over u and z are now trivial, and yield the result

x K..dil

Iv%w>“”

I f=O

.

(32)

We next note the identity

CvL.a7Pi1= CtHL,aJNR, Pi17

(33)

and use (28) to rewrite the operator, the matrix element of which appears in (32), in the form pi. >

(34)

We have also taken advantage of the fact that (H{,.)NR commutes with itself to symmetrize this operator. We can now differentiate with respect to the parameter f: This differentiation in

RADIATIVE

CORRECTIONS

FOR

MANY-ELECTRON

241

ATOMS

the log term will lead to a contribution which is not of log type, and thus can be neglected. We carry out the time-integrations for the surviving terms, and now explicitly obtain the overall energy conservation factor, 2718(~,-Q), which we cancel from both sides of our equation. After some tedious but straightforward algebra, we can cast the energy shift in the form (35)

(36) where h = p2/2m + VNuc,

(37)

and k, Y, m, and q stand for the labels a or n. The primes appearing in customary significance: singular terms in the denominator are absent. h is the NR Hamiltonian in the absence of the e-e interaction, which in the NR limit of the equations for g or gN. The logarithmic operator is given by 6p(r)=ln

(36) have the The operator also appears Y(r) in (36)

m 2 /E, - h( ’

(38)

For purposes of identification with analogous terms in II, it is useful to cast Eq. (36) into a different form. We note the identity 2piY(r)(-E,+h)pj =

Substitution

Ir>

{(-Er+h)Pi~(r)Pi+

C

P,,

hl Y(r)pi-pi~(r)Cpi.

of (39) in (36) leads, after some manipulation, “it&=

(4 {~m.q[-&]

p; -piT(k)Cpiv

(39)

to

([Pi, hl ~P(r)pi-p,~(r)Cpi,

+ ([pip hl ~(k)pi-pi~(k)Cpi, + [pi, II,,,+rlJ Y(r)

Al> Ir>.

hl)

[k 1

hl) &

um.q I}

’ urn,,

IO

(40)

242

FELDMAN

AND

FULTON

We gain some insight into this result by replacing the logarithmic Y(k) by their orders of magnitude. First, we write

operators

2’(r),

1

= In (zcr)2 + yB(r),

Y(r)

where (42)

Next, we substitute the first term on the right of Eq. (41) for Y(r) and Y(k) into Eq. (40) and call this approximation the log (LL) approximation and so label all of the resulting matrix elements. We obtain 1

c (4 %,(X) In’> (4 n,Zr E, - E,, i

u%,;L=ln(ZCo2 + c

(kl VVNuc(X) In’)

V*~dx)

Ir>

(n” umsq(x) Ir) Ek -

n’ + k

E,,

+ (4 V2~,,,(x) I r 3 I

(43)

where we have used the result that (44)

From the definition

of u,, ,(x), given by Eq. (1 1 ),

(kll><2lq>(1ln’> (4 um,y(x) In’> =a J d31 d32 11-W 3

V&;

(45)

n’y.

Thus, in the contribution to E,(31, Eq . (35), the matrix element (al v,Jn’) is just the exchange potential energy between the two electrons with energy labels n and a. Further, using Eq. (1 I), we obtain

(4 V2cn,q(xl Ir> = -ana

J d31(k I l>(mI I)(1

I q)(l

I r>,

(46)

i.e., a “contact” term. Evaluation of the energy contributions corresponding to Figs. 1, 2, and 4 is completely analogous to the process for Fig. 3. Indeed, we introduced arbitrary labels in

RADIATIVE

CORRECTIONS

FOR

MANY-ELECTRON

ATOMS

243

the definition of the matrix element O:,t in order to be able to write down these results immediately. In analogy with (35), we have

(35), (47c) (47d) We define the total contribution

arising from these diagrams as (d&“),,=

i i=

(48)

(4%. 1

Accordingly, we see from Eq. (48) that, in this approximation (replacing the logarithmic operators by their orders of magnitude), the contributions from the contact terms cancel, and we find

+(“un:.,n-“,n;nd E, - E,!

(n,, l(a)+cc

. 7 I

(49)

where I is the local “Lamb shift operator,” (50) so that the various matrix elements in Eq. (49) are

x 5 &l(d In this same (LL) approximation,

( 1) V21/N,,(1)(1

1 n), etc.

(51)

the usual Lamb shift is given by

Cfd& LambMLL= (4 1In>.

(52)

244

FELDMAN

AND

FULTON

Further insight is gained by replacing the nuclear potential nucleus. Thus, if we take

by that of a point

vNUC(x)+ Vpoin*(X) = -ZCr/lXl,

(53)

so that V*Vpo,“,(X)

= 4nZc163(x),

(54)

&

(55)

then

(4 zIk) -51,

(qlO)(Olk).

It is worth pointing out a number of features of this limit. The wavefunction at the origin, (0 1 n), for the state of interest, n, is not a common factor in (dsJLL and therefore there are contributions to states n which are not s-states. This is in contrast to the usual Lamb shift, which affects only the s-states in the approximations we have made. (Only the wavefunctions at the origin enter for the intermediate states labeled n’, and therefore this sum is entirely over s-states.) Although substitution of Eq. (43) into the energy formula, Eq. (48), will give a large part of the contribution, we must, in general, also compute the “Bethe-log” terms. l2 That is, we must include J&(T) (Eq. (42)) in Y(r) in Eq. (40). Accordingly, the total single-particle energy shift is made up of three parts: (AsLamb),,, the usual Lamb shift correction, the new leading-log (LL) correction (d~,,),~ given by Eq. (49), and the new Bethe-log (BL) contribution (d~,)~~ given by

- (4 p In”> . 1 In 2 lE,_ E,,,, + f (um;n’n- van;.d) C(n’l iGuc In”>. E,- E,! n*=,

- (n’l p In”> e(0 i&k Ia>1 In 2’f~E~,l a + it&“,,;nca-8aionin.rr).(n’l + it&,, nfn I2 See, for example,

(11.3.33)

p In) In

(Za)2m

2 I% -

&“,I + h.c.,

(56)

RADIATIVECORRECTIONS

245

FOR MANY-ELECTRON ATOMS

where c?& is given by Eq. (44), and matrix elements uanionSr etc., are defined by Eq. (45). Further, the electronelectron force field matrix elements are given by the expressions

=a

I cl31d32(k I l>(m

3. REFORMULATION

l2)(2

I q)
I r>(-zT,)

&.

OF THE RESULTS AND GENERALIZATION THE HARTREE-FOCK CASE

(56a)

TO

In this section we wish to show how rearrangement of the terms in Eq. (40) will lead to a better understanding of the meaning of our results. This will immediately suggest the generalization necessary to proceed beyond perturbation theory (in the electron-electron interaction) to the DF approximation. We make the rearrangement so that all terms are written as diagonal matrix elements-i.e., expectation values of operators in the single-particle state of interest, n. The quantities that actually appear in the expression for de, are 0:: :, 0;;;, CJ;;;, 0;:; (see Eq. (47)). 0;:; and O;$, as expressed by Eq. (40), appear, in the position representation, as off-diagonal elements (a H n) of local operators (including v,, (I or u,, .). However, using the property of the matrix element of u,,,~, given by Eq. (45), we can also write 0;; and 0;: as a combination of diagonal elements (n t, n) of nonlocal operators. The effects’ of these rearrangements are best exemplified by working with the approximate expression (~9~3~ m,y)LL given by Eq. (43). We leave for the Appendix the more accurate results when Eq. (40) (including both terms in (41)) is used. For example, consider the contribution to eL4) defined by Eq. (47d). We have, using Eqs. (43), (45), and (46)

N4%L= - &

f (oZ:%L=-j-$ ln1

(za)

a=1

x f [ (nl l>{W 2)<2ln) V,,(1-2)(11gb13)V2VN”,(3) u=l [ +V2bdKll g:, 13>(a I2)<2 I n> U-2)1(3 I a> -4na(n

I l>
1

I n>(l I a> .

An integration over all the space variables 1 to 3 is understood. I’,,(1 - 2) is defined as V,,(l -t)=a/II

-21.

(57) The new symbol (58)

246

FELDMAN

AND

FULTON

To avoid overloading symbols with many subscripts and superscripts, the states In >, Ial ), etc., and the propagators, g and g’ are from now on taken to be the NR analogues of those used earlier in Figs. 14 and Eqs. (2~(4), etc. Thus, for example, the NR Green’s function in frequency-space, g,, is g,s-

1 o-h’

where h is the NR Hamiltonian (Eq. (37)). Likewise, the primed quantities gb (gi) appearing in Eq. (57) have the usual meaning, i.e., the singular contributions to g, for w = a (n) have been removed. Equation (57) can be rearranged so as to appear as a diagonal matrix element in the state labeled n. Thus,

(&(4))LL n

=

-

7

37lzn

In

&

g J(nll: (1 I

x w sb13) V’~Nuc(3)<3 I a>CU I 2) ~~~(3-2) -4m(l (u)(u (2) S3(1-2)}(2 1n).

(60)

This result is best displayed as a diagram which is pictured in Fig. 5, along with the rules associated with each line and symbol. A sum over a is always understood. Again we note the change in meaning from Figs. 14 of the symbols drawn. All the wavefunctions (i.e., (q I 1)) and Green’s functions (g, g’) are NR, associated with the Hamiltonian h. Also, I’,,(1 - 2) is restricted to be the Coulomb interaction between two electrons. In a similar fashion, the expression for (sL’))rr, (.sc2))rL, (E(39LL can be rearranged and their associated diagrams drawn. We are inteiested in’(d.s,),, defined by Eqs. (47) and (48). First, as already mentioned, all the contact terms disappear because of the Fermi statistics. The other terms can be simplified

FIG.

5.

Rearrangement

of the eee perturbation

theory

terms

of Fig. 4 in the simplest

approximation.

RADIATIVE

CORRECTIONS

FOR MANY-ELECTRON

by reintroducing the propagator g,“, i.e., the N-particle two-point lepton operator, in the NR limit. 2,

(1 I a>(a I 2) =I,

g

247

ATOMS

expectation

value of the

(11 g,” l2),

where the contour, C, is in the upper half plane [2]. Similarly, f,

((11 s: 13) V’b.J”,(3)<3 = cg I

I a>(a I 2) + (1 I a><0 13) V2h”,(3)<31

(11 g,” 13) V2G”C(3)(31

s,” 12).

We collect all the remaining terms contributing in this approximation Lamb shift, ((4ambL)LL, represented by the diagrams of Fig. 6, is

=$p&j

(n I 1 >P’h”C(1)

+v2~iv”c(1K11 s:, l3)(3lC + I ,-g

(62)

to (ds,,)rL and also add the usual (Y(r) z ln[ l/(Za)2]). The result,

d3(1 -2)

12) +
~3l~,Nl~~~2~,,,(~)~~lg~l~~c1/,,(3-1)63(5-3)63(2-1)

- Ve,(3-5)63(1-3)63(2-5)]}(2

==y 1

FIG. theory)

s: 12))

1 n),

(63)

i
6. Representation of the leading-log contributions (through first order to the radiative corrections. The Lamb shift is included, but all Bethe-log

in e-e perturbation terms are neglected.

248

FELDMAN AND FULTON

where

x [Ve,(3-1)63(4-3)63(2-1)-Ve,(3-4)63(1-3)63(2-4)].

(64)

The diagrams of Fig. 6 are immediately suggestive of the generalization required for the corresponding NR DF radiative corrections, which will involve HartreeFock (HF) wavefunctions and energies. We see that (in this approximation) the gauge-invariant (GI) radiative corrections can be represented by an effective local Lagrangian, namely the local Lamb shift operator defined in Eq. (50). The generalization to the HF approximation will proceed in the same manner as previously carried out [2] for radiative transition amplitudes. The Lagrangian in Ref. [2] involved the local current operator, jJx). We saw in [2] that the correct GI transition amplitude could be written as a matrix element of a non-local effective current between HF states. The same procedure leads us here to introduce a non-local Lamb shift operator, L, which, in the position representation, satisfies the integral equation (11 L 12)=2(1)63(1-2) -

d33d34(11 s,“” 13)<31 L 14)<4 g,“’ 12) ve,u -2) d33 d34 d35(31 g,“” 14)(41 L IS)

+a’(l-2)jc$j

x (51 ET,“” I3 > Ve,(l - 3).

(65)

This equation is represented diagrammatically in Fig. 7. The propagators g,“’ are now the HF propagators, i.e., an approximation to the exact G”’ in the NR limit. The radiative correction in this approximation is given by

s

d31 d32(nHF

1l)(ll

L (2)(2 I rF>,

(66)

where (2 / rrHF) are HF wavefunctions.

FIG. 7. Representation of the integral equation for the generalized Lamb shift operator in the HF approximation.

RADIATIVE

CORRECTIONS

FOR

MANY-ELECTRON

ATOMS

249

Thus, the radiative corrections involve not only a matrix element of the local Lamb shift operator, I, but also terms which arise due to the non-local effects of L. The latter can be thought of as screening corrections in the HF approximation. 4. CONCLUSION

AND

DISCUSSION

We have calculated the log contributions of the radiative corrections to the energy levels of certain atoms, where we considered the electron-electron interaction, V,,, in lowest-order perturbation theory. This implies a restriction to those atoms for which N, the number of electrons, is much smaller than Z, the nuclear charge. In calculating the radiative corrections, we have followed the procedure of Ref. [4] and have kept only leading terms in Za. We may thus expect the results to be accurate for values of Z up to about 10. The effect of including the corrections represented by Figs. 14 is best displayed by examining Eq. (49), which gives the quantity (d~,)~~ when Bethe-log terms are neglected. One can see from this equation, or the more accurate Eq. (A5), that the do, corrections are of order l/Z times the Lamb shift. The first set of terms on the right of Eq. (49) represents a shielding effect, due to the core electrons, on the single electron labeled n. For a point nucleus, these contribute only to s-states, just as for the usual Lamb shift. There is an additional effect given by the second set of terms in Eq. (49), which represents a change in the states of the core electrons, due to their interaction with the electron in state n. For a point nucleus, these will contribute to states other than s-states. In fact, they will most likely provide the dominant log contributions to the radiative corrections for higher angular momentum states of many-electron atoms. For a similar effect in helium, see, e.g., Ermolaev [ 111. If we include the Bethe-log terms (see Eq. (A5)), then, in addition to the effects already mentioned (and now contained in the terms proportional to BNuC in this equation), there is an effect which is not proportional to the local electric field of the nucleus, &,,,, but to the non-local electric field associated with two electrons, CC?‘“,,. The sizes of all of these Bethe-log terms will, most likely, be comparable. The Bethe-log terms cannot, of course, always be neglected. A good guide for estimating their relative importance is again provided by the usual Lamb shift. We note from (11.3.35) that In a-* zz 10 and the Bethe-log is approximately - 36,0. (An important property of the Bethe-log is that it is independent ofI Za.) An estimate of the relative error we make in any given term by leaving out Bethe-log terms is thus

Even in the presently uninteresting case of hydrogen, the error is 30 %. For the atoms of interest within the range of applicability of our perturbation theory (Z = 3 to ll), the error ranges from 40 to 60 %. ‘I See (113.42)

595/184/2.3

and the sentence

above

this equation.

250

FELDMAN

AND

FULTON

What about our ignoring all but log terms in our calculations? We can again use the Lamb shift as a basis for estimates. The relevant ratio for s-states is 0.633/2(7-In

Z’),

0.633 =&+

;- j,

7-lncl-*-3,

(68)

where we have included not only the fact that these errors occur in order l/Z in perturbation theory, but also the Uehling term ( - l/5) in the numerator. For Z = 3, the estimated error is 4 %. For Z = 11, it is 3 %. This error is probably acceptable at the presently needed level of accuracy. We must emphasize that we can evaluate all terms of this order in perturbation in V,,, and not just log terms. For example, in evaluating the expression given by Eq. (25), the log assumption was not made until Eq. (30) was used. This was done to simplify the algebra. However, just as in II, the integrations over the Feynman variables can be performed (see (11.3.31)). The result in our case will be to replace the factor ln[ 1/(Zcrj2] by In [( l/(Z~r)~] + 1l/24. There are other sources of non-log terms, the “M-terms” (see Eqs. (11.3.14a) and (111.3.22)). These depend on the nuclear electric field EINuC.There will also be M-terms which depend on the field &. Finally, there will, of course, also be the vacuum polarization corrections. All of these non-log terms can be computed directly, but our numerical arguments above persuade us that such a complete calculation is not yet required. The present calculation of log terms is the end product of a rigorous derivation. It corresponds, for the usual Lamb shift, to the results of Welton [S], which were based on semiquantitative arguments. One of our goals in calculating the radiative corrections to lowest-order perturbation theory in V,, was to direct us towards the generalization of these results to include the HF (or DF) approximation in a gauge-invariant (GI) way. As already pointed out in the text, in the limit of neglecting Bethe-log terms, the radiative corrections in e-e perturbation theory are given in terms of an effective Lagrangian proportional to the local Lamb shift operator, 1, defined by Eq. (50). The generalization is immediate. Replace this local operator by a non-local operator, L, which satisfies Eq. (65). The GI radiative correction in the HF approximation is then given by the diagonal matrix element of L between HF wavefunctions. On inclusion of the Bethe-log terms, the local operator I is replaced by a more complicated one. However, the generalization to the HF approximation may still be conjectured. We see from Eq. (A5) and Fig. 8 that the required replacement is

+ terms proportional

to c?&.

(69)

The conjectured generalization then is to replace the inhomogeneous term 1(1)?i3(1-2) in the equation for (11 L)2) (Eq. (65)) by the (11 to 12) matrix element of Eq. (69). We hope to provide a proof of this conjecture in a subsequent publication.

RADIATIVE

FIG.

CORRECTIONS

FOR

MANY-ELECTRON

ATOMS

251

8. Additional contributions to Fig. 6 due to Bethe-log terms.

The need for electron shielding effects in radiative correction contributions to DF energies has been documented by several researchers [6,7]. Heuristic arguments have been made [S J to include the effects of such shielding, or to estimate their order of magnitude [9]. The generalization of the present paper for the DF case is much more detailed and specific and arises directly and naturally from a rigorous perturbation theory result.

APPENDIX

In Sections 2 and 3 we evaluated, in lowest-order perturbation theory in I’,,, an approximate result for the leading-log contributions to radiative corrections for single-particle states. The result was represented by the diagrams of Fig. 6. We obtained this approximate form by replacing all logarithmic operators, L?(r), by ln[l/(Z~)2]. In this appendix we give the results, which will include YB(r) (see Eq. (41)), the “Bethe-log” terms. Two modifications will occur in Eq. (57) for Bethe-log contributions. First, the matrix element of the local Lamb shift operator, l(x), defined by Eq. (50), is replaced as follows,

(II~l2)=&ln& -$

(11v2ylvuc12) (11ehc’ -%(a)

P- P . S(w)

6.l”, 12L

(Al)

252

FELDMAN

AND

FULTON

where gNUc is the nuclear electric force-field defined in Eq. (44). Second, the contact terms no longer cancel exactly. Thus the contact term in Eq. (57) is replaced by -&

f,

j (n 1 l){i~,(l-2)~

- 01 p%(n)

13) .ie,(3-2)0

which, on rearrangement,

-&f -

(a l2)(2 I 2x2

13) (AZ)

becomes

l3)(3

p-%(n)

%(U)P

I d I(3 I a>,

j (nIl>{i~e(l-2).(ll~‘,(a

(11

I n>(ll

) P l3)(3

I a><~ 12) +%,(3-2)}(2

la>(al2>

I n>,

(A3)

where & is the electric force-field between two electrons, i.e., &“,( 1 - 2) = -v,

V,,( 1 - 2).

(A4)

Continuing now as in Section 3, the additional contribution to (AsLamb + LIE),, due to Bethe-log terms is represented by the diagrams of Fig. 8, plus diagrams in which the symbols are written in a reverse order to those in Fig. 8. Symbols not already defined in Figs. 5 and 6 are defined in Fig. 8. Combining the results from Figs. 6 and 8 gives, for the total log contributions of the radiative corrections, (A&Lamb

•t

A&In

=$--&

j

(n 11)

+ (11 &I,,

i

.2(n)

(11 sv”c.an)P

12)

P 13)(3l s:, 14)(4l~

+ (11 Jc 13)(31 g:, 14)(41 ~~N”C.~(~) +Jc$

(3lg:l4)(4l

12) P 12)

G”c.a~)Pl5)

x (51 g,” ~6)[Ve,(3-1)63(6-3)63(2-1)+Jc$

V,,(3-6)d3(1

-3)d3(2-6)]

(31 ~(~)~14)(4lg~l5).C~~,(3-1)~~(5-3)~~(2-1)

-e.(3-5)63(l-3)02-5)1+~c~ x [i&.(1 -3) 63(1-5) x (2 1n) +c.c.,

~31~7~14x51~m-w 63(4

- 3) -i&(4

-3)

63(4

-5) 63(1-3)] WI

RADIATIVE

CORRECTIONSFOR

MANY-ELECTRON

ATOMS

253

where

x [V,,(3-1)63(4-3)63(2-1)-

1/,,(3-4

)S3(1 -3)a3(2-4)].

(A6)

The expression corresponding to Fig. 6 is obtained by replacing all 9 by ln[ l/(Zcl)*], in which case the last two terms ( +c.c.) of (AS) become contact terms and cancel.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

G. FELDMAN AND T. FULTON, Ann. Phys. (N.Y.) 179 (1987) 20. G. FELDMAN AND T. FULTQN, Ann. Phys. (N. Y.) 152 (1984), 376. G. FELDMAN AND T. FULTON, Ann. Phys. (N. Y.) 172 (1986), 40. G. W. ERICKSON AND D. R. YENNIE, Ann. Phys. (N.Y.) 35 (1965), 271. T. WELTON, Phys. Rev. 74 (1948), 1157. P. J. MOHR, in “Proceedings, NATO Advanced Study Institute on Atoms in Unusual Conditions, Cargese, 1985” (J. P. Briand, Ed.). Gordon & Breach, New York, 1986. W. R. JOHNSON, S. A. BLUNDELL, AND J. SAPIRSTEIN, University of Notre Dame preprint, August 1987. M. H. CHEN, B. CRASEMANN, M. AOYAGI, K. -N. HUANG, AND H. MARK, A,. Data Nucl. Dara Tables 26 (1981), 561. S. S. LIAW, G. FELDMAN, AND T. FULTON, preprint JHU-HET 8707, 1987. J. SUCHER, Phys. Rev. 107 (1957) 1448; G. FELDMAN, T. FULTON, AND D. L. HECKATHORN, Nucl. Phys. B 174 (1980), 89. A. M. ERMOLAEV, Phys. Rev. Lert. 34 (1975), 380.