536

R. KARPLUS AND N. M. KROLL

section of the plane of w and with the plane normal v

to

Dot multiplication by t annuls the right-hand

curl

w.

side, so that we obtain the following not inelegant
theorem : Let the curves e be everywhere tangent to the

plane of and and everywhere normal to

then

w

v,

curl w;

along these curves, which are determined by the instan-

taneous velocity field only, in any flow of a viscous

incompressible fluid of uniform viscosity if the vorticity be

steady and the extraneous force be conservative Bernoulli's

theorem in the classical form

U+V+½v2+(p/p)=F(t)

(7)

is valid. Special cases when (7) holds for wider classes of curves or for special types of curves or for surfaces may be left to the reader; among these are the results of Sbrana and Castaldi. 6

e 8 In the case of plane or rotationally-symmetric flow the curves are the vortex-lines, as indeed is obvious from symmetry, and

The foregoing theorem exhibits the non-uniformity of
the limit µ-0 in a strikingly simple dynamical form:
since the curves along which the pressure obeys the C,
relation (7), are determined by the instantaneous ve-
locity field, they remain fixed as µ-0, while at the limit
µ= 0 of an inviscid fluid they spread out discontinuously
into Lamb's Bernoullian surfaces.
The results given here constitute an application of a
general theorem of pure kinematics, to be published
elsewhere. 7
only if perchance the stream-lines and the curves of constant vorticity magnitude form an orthogonal net (Sbrana's case v•curl w=O) does (7) yield a non-trivial result, namely a theorem of type (B).
7 C. Truesdell, "The kinematics of vorticity," Memorial des Sciences Mathematiques (to be published). I am obliged to Dr. P. Nemenyi for having suggested the problem of finding a kinematical generalization of Bernoulli's theorem valid in motions where Kelvin's circulation theorem does not hold, and for discussion of the present paper.

PHYSICAL REVIEW

VOLUME 77, NUMBER 4

FEBRUARY 15, 1950

Fourth-Order Corrections in Quantum Electrodynamics and the Magnetic Moment of the Electron
f ROBERT KAJI.PLUS* AND NORMAN M. KROLLt
Institute for Advanced Study, Princeton, New Jersey
(Received October 17, 1949)

The covariant S matrix formalism of Dyson has been applied to the calculation of the fourth-order radiative correction to the magnetic moment of the electron. Intermediate results for the covariant Ll.-functions which describe the interaction of virtual electrons and photons with the vacuum are given to order a. The addition to the magnetic moment to order a1 is found to be finite after the charge of the electron is renormalized consistently. This correction amounts to -2.97a'/r Bohr magneton so that the magnetic moment of the electron is µ.= 1.001147 Bohr magnetons.

RECENT developments in the techniques of quantum electrodynamics, and in particular the general considerations of Dyson,1 have shown that the radiative corrections to the motion of the electron can be made finite in all orders by the consistent use of the ideas of charge and mass renormalization. The renormalizations are, of course, infinite, so that one is forced to regard the present form of the theory as provisional. Still, the fact that one can give an unambiguous, consistent, and sensible prescription for dealing with this situation, and the excellent experimental verification accorded the second-Oider effects already computed, suggest that an investigation of a fourth-order effect might be of value: first, in order to make possible a sensitive test of the agreement of the theory in its present form with experiment and second, to demonstrate in a complete calculation of a particular example
t* Frank B. Jewett Fellow. National Research Council Fellow.
75, f Now at Columbia University New York, New York.
1 F. J. Dyson, Phys. Rev. 486, 1736 (1949), henceforth called I and II, respectively.

the feasibility of Dyson's program. The magnetic moment of the electron was chosen for investigation because it promised to present the least difficulties of computation while it does contain those points of theoretical interest which are relevant to the difficulties of quantum electrodynamics. Furthermore, in view of the success already achieved in the measurement of the anomalous moment of the electron,2 it appears that the fourth-order effect may be accessible to experiment.
METHOD OF CALCULATION
We shall begin with a discussion of the fourth-order corrections to the elastic scattering of an electron by an an external electromagnetic field. The question of isolating that part of the scattering which may be attributed to an anomalous magnetic moment will be discussed in a later section.
In evaluating the matrix element describing the scattering, the methods of Dyson have been followed quite closely. We, therefore, require the fourth-order
1 P. Kush and H. M. Foley, Phys. Rev. 74, 250 (1948).

QUANTUM ELECTRODYNAMICS

537

u part, 1<•>, of the transformation matrix U1 given by
Dyson3 as
whence

and Dyson.5 The matrix element in question is given by a sum of terms each of which may be described by a graphically represented transition scheme. The diagrams for our process appear in Fig. 1 and will be discussed in the next paragraphs. To each diagram there corresponds an integral over the variables xo, x1 •• •, x,; the integrand can be written down by inspection and gives the contribution of the associated transition scheme to the matrix element. In these integrals the effect of the ordering operator P[ ] has been absorbed into the SF(x) and DF(x) functions, so that this operator no longer appears explicitly. These functions do of course contain an implicit dependence upon P[ ] in view of the relations

The variables x, refer to particular space-time points and are thus to be understood to have four components; P[ ] is the chronological ordering operator of Dyson.4
In the above expression, H•(x0) describes the interaction with the external electromagnetic field, whose vector potential is denoted by A,.-(xo), and is given by
H•(xo) = - (1/c)j,..(xo)A,.e(xo)
= -ieft(xoh,.1"(xo)A,.e(xo), (3)
while H1(x,) describes the interaction with the photon field and is given by

½SF,.i3(X1-X2)= (P[if/p(x1), 1Ya(X2)])0E(X1, X2), (5)
(lic/2)DF,,,(x1-x2)= (P[A,,(xi), A,(x2)])0
= (lic/2)8,,.DF(Xi -x2), (6)
where { )o denotes the vacuum expectation value. To complete this summary of the method of calcu-
lation, it will be convenient to specialize the discussion to the problem at hand. For this reason we turn now to a discussion of the diagrams in Fig. 1 and the corresponding integrals.

It is to be observed that the operator -iec{l(x,)-y,,if;(x,) is the usual unsymmetrized current density operator for the Dirac particle field. The term - 8mc2if/if; appearing in H1(x,) implies that the interaction representation in which the theory was originally cast has been modified so that the mass appearing in the equation of motion of the electron states is the mass of the electron as corrected by its interaction with the radiation field (i.e., presumably the experimental mass) rather than the mass of a hypothetical uncharged electron.
The matrix U has the property that when it is applied
+ to the state vector of the system at - oo, it produces
the state vector at ao. U1 is the part of U corresponding to a first Born approximation and is the limiting form of U for a weak external field, while U/4> is that part of U1 which describes processes involving four interactions between particles and photons and one interaction between a particle and the external potential. As such it describes a great many processes in addition to those in which we are interested. In particular, we shall be concerned with the "one electron"
u part of U1<•>, i.e., that part of 1<4> connecting states
consisting of a single electron and no photons. A simple and elegant method of extracting from U any portion in which one is interested has been given by Feynman
1 See reference 1, II, Eqs. (6) and (7). 'See reference 1, I, Section V.

A ' x, d x,

0
--oA

~~ ~

A.~ --<Y--o-

YL .

A ,

~

F1G. 1. Feynman diagrams for the fourth-order radiative corrections to the scattering of an electron by an electromagnetic field.

1 See reference 1, II, Section II.

538

R. KARPLUS AND N. M. KROLL

ץI I
~

I
_A
x. 2 x,

I
I I
- r A ,__,,
3

Fm. 2. Feynman diagrams for the second-order radiative cor-
rections to the scattering of an electron by an electromagnetic field.

included in
Xd4x2d4x2'd4x/'A/' (xo)DF' (x2-x1) Xif'(xi')r,(xi-xi', xi''-x1)S/(xo'-x1") X r µ.{xo-xo', xo" -xo)SF1(x2'-xo")

The diagram I gives rise to the integral

This diagram is irreducible since it cannot be repre-

sented as a lower order process corrected by modified

interaction functions. It contains a logarithmically

divergent charge renormalization plus finite physical

effects, of which the magnetic moment, to be extracted

later, is one.

Integrals analogous to M1 can be written down for

the diagrams grouped under II. For example, we might

observe

f ea2,r2
Mll•=--

d4xod4x1d4x2d4xsd4x4

4hc

XTr['YASF(xs-X4),,,Sp(X4-xa)]

XA/J,•(xo)ift(x1h.SF(xo-x1hµ.

XSF(X2-xohAi/;(x2)DF(X2-X4)DF(xs-x1), (8)

where Tr[ J indicates the trace of the bracketed ex-
pression. These diagrams are all reducible, however to the second-order diagram 2 in Fig. 2. Since the second-order diagram 2 is given by

Xr,(x2-X2', X2"-x2)y;'(x2"), (10)
where the primed functions and the current operator
r ,(x, x') are as defined by Dyson.8 The presence of nine
rather than five variables of integration is associated with the fact that (10) contains terms of all orders in a
except the zeroth. The primed functions and r µ. are to
be obtained as expansions in a, and inserted in Eq. (10). The terms of order a 2 will then include all of the diagrams in class II. When this is done, integration over four of the variables will be trivial, as these variables will appear only in the arguments of Ii-functions.
We should like to emphasize that for the evaluation of the diagrams of class II, the use of the reduction Eq. (10), rather than expressions like Eq. (8), is of great assistance in the unambiguous elimination of the effects of charge renormalization. In any order, radiative corrections to scattering processes must be expected to include terms which merely renormalize the electronic charge occurring in lower order corrections. Thus, for example, a strightforward evaluation of MII from expressions like Eq. (8) would yield infinite corrections to the magnetic moment for just this reason.7 Were the charge renormalizations finite, this would cause no difficulty as these terms could then be readily subtracted out. They are, however, infinite and one would therefore have to exercise the greatest care to guarantee that no finite remains of charge renormalizations had been included in the true higher order correction. On the other hand, in using the reduced diagram method one explicitly separates out all renormalization effects, infinite and finite, at each order, so that the isolation and removal of the entire contribution of renormalization to the moment is simple and unambiguous. For the purpose of illustration the renormalization terms will be retained throughout so as to exhibit at the conclusion the renormalized second-order moment. This, of course, is not really necessary since the under• standing that all effects be given in terms of the experimental charge allows one to drop renormalization terms as they appear.

all higher order corrections as well as (9) itself are
Fro. 3. Feynman diagram for the second-order terms of D/(x).

6 See reference 1, II, Sections III and IV. 7 One can avoid the use of the reduction Eq. (10) if one is
willing to introduce Pauli regulators in such a way as to make all charge renormalizations finit i. The application of regulators to
higher order processes is discussed by J. Steinberger, Phys. Rev. 76, 1180 (1949). The true fourth-order correction obtained after
the now finite contributions from charge renormalization are recognized and removed is the same as obtained by our procedure.

QUANTUM ELECTRODYNAMICS

539

integral

~

X+X

,

/

X

I

---

I

Fm. 4. Feynman diagram for the second-order terms of Sp'(x).

DF11) 2>(x)= (-l)(-i/lic) 2J d4x2d4x3 X (P[A,.(x1), A1,.(x2)])0 X (P[(l/c)j1,.(x2), (1/c)j.(xa)])o

Diagrams of class III are all reducible to the secondorder diagram 1 of Fig. 2. Therefore, methods similar

Xo.,.Dp(Xa-x1+x)

to those described in the last paragraph are to be used in their evaluation. These diagrams include corrections

= -(a1r/2) J d4x2d4xaDF(X1-x2)

to the polarization of the vacuum by an external field and charge renormalizations applied to lower order

XTr['Y.SF(X2-xah,.SF(Xa-x2)]

vacuum polarizations. As such, the observable effects

XDF(xa-x1+x)

which they represent are modifications of the external

(11)

potential and not of the properties of the electron. This

implies that they cannot contribute to the magnetic

moment, so they will not be considered in any further

detail.

The diagrams of class IV can all be regarded as

reducible to the second-order diagram 3 of Fig. 2,

without including any modifications of the external

potential at the vertex xo. They can therefore contribute

nothing but a charge renormalization of the zeroth-

order scattering. They are of interest only if one wishes

to investigate the actual form of the fourth-order renor-

malization.

Our discussion of the diagrams may be concluded

with the remark that by the Furry8 theorem diagram

Va and Vb exactly cancel.9

The remainder of the paper will be concerned with

the evaluation to order a of S/, DF', and r,.. and the

extraction of the magnetic moment correction from the relevant integrals M1 and MII. No error has been in- Here

curred by the neglect of the supplementary condition.90

SECOND-ORDER FUNCTIONS
The function DF'(x), which describes the properties of a virtual photon as modified by its interaction with the electron-positron field, must be obtained to second order in e. The leading term, of course, is

The integration over k must be carried out carefully,
because the integral is divergent. This has been effectively carried out by Schwinger and yields10

The corrections to this function arise from the ability of the virtual photon to create pairs. The first term is simply due to the creation and annihilation of one pair, as described by the Feynman diagram Fig. 3, or by the

8 Wendell H. Furry, Phys. Rev. 51, 125 (1937). 9 No reference has been made to diagrams and associated matrix elements arising from the term -6mc2,f,(x),f,(x) in H 1. As described by Dyson, diagrams containing these interactions are to
be placed in one-to-one correspondence with the diagrams containing self-energy parts. Their effect is taken into account in the evaluation of the SF', ,f,'(x), and ,t,'(x) functions.
11a F. J. Dyson, "Longitudinal Photons in Quantum Electro-
dynamics," Phys. Rev. (to be published).

after the requirements of gauge invariance have been imposed. Since the electromagnetic potentials obey the
10 Julian Schwinger, Phys. Rev. 76, 790 (1949), Afpendix; Schwinger's result has been multiplied by 2n because o slightly different definitions of the singular functions.

540

R. KARPLUS AND N. M. KROLL

Lorentz condition, the term (p"p.) may be dropped. Then
and

= _-3!_ fe-;P,,d4p{~(1+~A)

(2,r )4

p2

211"

f +a-

1 2v2(1-½v2) } d•----

2,r o 4K2+p2(1-v2)

= ( 1+ 2: A )DF(x)+.DF<2l(x),

(13)

where

4( A=-- limlPno+--P-1 ] .

3 P➔ r,,

K

This infinite constant, however, has no observable consequences because the term in which it occurs is indistinguishable from the original DF(x) function. It merely means that the matrix element in which DF(x) occurs is multiplied by a factor [l+(a/2,r)A] and that the quantity which measures the intensity of the dynamical interaction described by the matrix element must be renormalized.
By a very similar calculation, the function A •'(x), the external electromagnetic field modified by second-order interaction with the pair field, may be calculated. Thus,

f A,,e'(x)= eipxd4pA,,e(p){ ( 1+ 2: A)

ft +ap-2

2v2(1-½v2) }
d.71------

211' o 4,c2+p2(1-v2)

= ( 1+~A )A,,e(x)+~A,,e<2>(x).

(14)

21T'

21T'

The function SF'(x) describes the behavior of a virtual electron as modified by its interaction with the electromagnetic field. The relevant diagrams, in this case, are in Fig. 4, while the appropriate integral is11
J SF<2l(x) = (-i/hc) 2(-ie) 2 (P[f(x1), if(x2)])o'Y"
X (P[f(x2), 1f(xa)])o'Yv(P[Aµ{x2), Aha)])o
XSF(x1+ x- xa)d4x~4xa
J -(-i/hc) d4x2(P[f(x1), if(x2)])0
X omc2S F(x1+ x- X2)
f =e2/8hc SF(X2-x1h~F(Xa-x2)
X 'Y.SF(x1+x-xa)o""DF(xa-x2)d4x~4x3
f +i/2hc d4x2SF(x2-x1)omc2SF(x1+x-x2)
f i'YP-K
= (a/21T'3)(21T')-4 d4pe-iP"-pz+ic2
{f i'Y(p-k)-,c 1
X d4k'Y,.----~,.-(p-k)2+"2 k2+~2
-(4i1T'3/a)(omc2/hc) }-i'Y-P-K. (15) p2+"2
Here again, the integral over k diverges and therefore must be evaluated carefully; furthermore, charge renormalization must be exhibited explicitly. This identification can only be done simply, however, after the integrand has been rearranged considerably so as to write it as a function of i'YP+ "· Thus,12

f f -2

1
du

d4k pi)" u - - - - - -i'-Yk- - - - - - - .

(16)

o

ak" [k2+K2u2+(p2+K2)u{1-u)+~2{1-u)]2

11 The DF(x) function here is replaced by a .dF(x) function with mass A>./c to avoid an anticipated infra-red catastrophe in the physically significant part of the S:r'(x) function. sFm(x) does not diverge in the infra-red.
a The Lorentz invariance of I assures that it is a function of (i-,p), hence of (hP+K).

QUANTUM ELECTRODYNAMICS

541

Here the second integral represents a surface term that must be added to take into account the effect of the
displacement k"-k"+p,.u at large values of k2, where the integrand does not tend to zero rapidly enough. It should be emphasized that all integrations are undertood to be symmetrical with respect to the origin of the variable of
integration; i.e., the angular integrations are to be carried out first, and are followed by the integration over Ik j.13
With these points in mind, the operator becomes

where and
Using the facts that14
and that

(18a) (18b)
(19)

f J one obtains

1 { (hP+K)(1-u)+K(l+u)

i1r2 i1r2

I(i-yp+K)=-2 du d42----------+-u(i-yp+K)--K----u

o

(k2+Ao2) 2

2 2

+i1r2[(hP+K)(1-u)+K(1+u)](hP+K)(i,yp-K)u(1-u)f1 dz

1

}· (21)

o Ao2+(p2+K2)u(1-u)z

Only the still remaining integral over momentum space is divergent here, and it will become apparent that it consists of renormalization terms only. After a slight rearrangement of the finite parts, the operator assumes the form

I(i-yp+K)=-2f 1 du{[Jd4k (l+u)K (i1r2/4)K]+(i-yp+K)(1-u)[fd4k 1

2i1r "2u(l+u) i1r2]

o

(k2+ Ao2) 2

(k2+ Ao2) 2

Ao2

2

1 K(l+u)+ (i-yp-K)(l-u){ 1-[2K2u(l+u)z]/[u2K2+X2(1-u)]} ]} +i1r2(i-yp+K) 2u(l-u) [ f d z - - - - - - - - - - - - - - - - - - - - . (22)

o

K2u2+X2(1-u)+(p2+,c2)u(1-u)z

The first term in this expression is equal to (4i1r3/a) (limc2/lie) and is therefore canceled by the mass renormalization term, Eq. (15). The rest of the integral can now be inserted into the expression for S/(x),

S/(x) =SF(x)+sF<2>(x)

= -2i/(21r)'fe-•'P"'d'p{i,yp-"(1-~B)+(a/21r)f1 duu(l-u)

p2+,.2 21r

0

f 1 K(l+u)+(i-yp-K)(l-u) {1-[2K2u(1+u)z]/[K2u2+X2(1-u)]J
X ~:---------------------

0

K2u2+X2(1-u)+(p2+K2)u(l-u)z

= [1-(a/21r)B]SF(x)+ (a/21r)SF<2>(x),

(23)

13 This implies J k"J(k2)d4k=O, J k"kd(k2)d4k= J ¼8",k2J(k2)d4k, etc. 14 R. P. Feynman, Phys. Rev. 76, 769 (1949).

542

R. KARPLUS AND N. M. KROLL

(24)

The fact that Bis infinite, however, is not a source of difficulty since it can be interpreted as a charge renormalization just as the constant A in the treatment of the D/(x) function.
It must now be observed that the physically significant term of sFc2>(x) diverges logarithmically as :>.-0. Since this divergence is associated with the vanishing mass of a photon, it is an infra-red catastrophe. It is introduced by the separation of real and renormalization effects in SF<2>(x). One must hope, of course, that the logarithmic dependence on A will cancel when all contributions to a certain scattering process are added together. The work of Bloch and Nordsieck16 indicates such a cancellation will actually occur.
We shall merely note now that the modification of the electron wave function brought about by virtual interaction with the electromagnetic field is obtained from the same diagram as the SF'(x) function, if one of the electron lines is taken to be an external line. Then, since the wave function obeys the Dirac equation,17'

,t,'(x)=[1-(a/4'll')B]iJ,(x)

(25a)

and

f,'(x) = [1- (a/4'll')B]j(x).

(25b)

As pointed out by Dyson, the effect is merely one of renormalization so as to preserve the unitarity of the matrix U.

Some explanation is required for the necessity of replacing the renormalization factor Z2in Eq. (23) by Z2l=Z2/Z2I in Eq. (25), a substitution which is equivalent to dividing by Z2" the matrix element of U between states containing n electrons; for as long as the scattering matrix U is defined between two specific surfaces .,-1 and .,-2 in the remote past and distant future, its unitarity is guaranteed. Thus, it should not be necessary to apply an explicit renormalization. Furthermore, the use of the eigenstates of noninteracting fields to specify conditions at a-1 and a-2 must be justified, since experimental conditions would lead one to assume

Fm. 5. Feynman diagram for the second-order terms
of r ,.(x, x').

16 By the use of

J. f 1

(1-2u)k2

0 du d4k(k2+ Aa2)3

J. f l d d•~ -2uAa2+2~2(2-u)

ou '

(k•+ Ao2) 3

j ' -J.1

4 -2(1-u)Ao2+2~2(2-u)

- o du d k

(k'+Ao•)a

which is the result of an integration by parts. 15 F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937).
17 See reference 1, II, Eq. (99). Z2= 1-(a/21r)B+ •• •.

that the one-particle eigenstates (i.e., essentially the BlochNordsieck states) of the interacting electron and photon fields (or combinations thereof for several particle problems) ought to be used. The replacement of a-1and a-2 by - 00 and + 00, however, together with Dyson's computation rules imply a certain averaging of the matrix elements over Jong sequences of surfaces in the past and future. One can readily show that the averaged matrix element is just equal to the matrix element between BlochNordsieck states, multiplied by a constant which depends only upon the number of real particles in the initial and final states.
For simplicity, we confine our attention to the one-electron part of the scattering potential and restrict a-1and a-2to be surfaces of constant time, t1 and t,. The requirements that the state vector cf>1(t) corresponds at time 11 to an uncoupled or "bare" electron of momentum k1 and that cf>2(t) corresponds at 12 to a bare electron of momentum k2 are contained in the relations,
y,,,+(x,.)<1>1('1) =ak1,,+ exp[ik1,.x,.]<1>1(t1), y,,,+(xµ)<l>2(t2) =ak,.,.+ exp[ik2µx,.]<l>2(t2),
where the a+•s are annihilation operators, together with the annihilation of cf>1(t1) and cf>2(t2) on application of if,-(x,.) and A,.+(x,.). The possibility that <1>1(t1) and cf>2(t2) have 11 or t2 dependent phase factors is eliminated by the requirement
(<1>2(/2), ak/ak/<1>1(t1)) = 1
for all 11 and t,. To interpret the procedure of calculation used it is convenient to expand <1>1, and cf>2 in the exact eigenstates, '11,.(t), of the coupled electron and radiation fields thus,
<l>1(t1) =2: A ,.(t1)'11,.(t1), <l>2(t2) =2: B,.(t2)'11,.(t,).
We might observe that these relations serve to determine the behavior in time of the states <1>, for zero external field. That is,
cf>1(t) = Uo(t, l1)<l>1(t1) =2:,. A,.(t1)'11,.(t),
<l>2(t) = Ua(t, t2)<l>1(t2) =2:,. B,.(t2)'11,.(t).
The important point is the fact that the t1 and t2 dependence of the A's and B's as determined by the boundary condition is given by
A,.(ti)=a,. exp[(i/h)(E,.-E1)t1], B,.(t2)=b,. exp[(i/h)(E,.-E2)t2],
Thus the matrix elements of U(t2, ti) between <1>1('1) and <1>2(/2) are related to matrix elements between eigenstates of the coupled system by
(<1>2, U(t2, t1)<l>1=l: b,.a.. exp[-(i/h)(E,.-E,)t,] Xexp[(i/h)(Em-E1)t1]('11,.(t2), U(t,, t1)'11,,.(ti)).
Since the matrix elements between the exact eigenstates wiJI not depend upon t1 and t, if these occur respectively before and after the application of the external field, we can average over these times explicity, thus obtaining
{(<1>2, U(t2, t1)cf>1))Av=b,.2*a..1('1',.,(t2), U(t2, t1)'1'..1(l1)),
where Em1=E1, E.2=E2, From
((cf>:, U(t,, l1)cf>1))Av=a..1*a..1('1'm1, Uo(t,, t1)'1'.,1) =a,,.1*a..,
where k1" is taken equal to k2p, one finds a,,.1*a,,.1=Z2 and similarly b.,1*b..1=Z2.
If there are n widely separated electrons in the initial and final states, the appropriate factor is clearly Z,", because the state vectors <1>1 and <1>2 can be factored into states corresponding to the presence of a single electron, and for each of these the above analysis can be carried through.

QUANTUM ELECTRODYNAMICS

543

The vertex operator rix', x") describes the scattering of a virtual electron by a potential. The second-order
contribution to it is given by the diagram, Fig. 5, or by

A,.<2l(x', x") = (-i/Ac)2(-ie)2[·n.(P[1Y(x0-x'), ~(xo)])o-y,.(P[1Y(Xo), ~(xo+x")])o X-y,(P[A:>.(Xo-x'), A,(xo+x")])J

(26)

where (27)

Thus one obtains the operator L,.(p', p").18 This must now be rearranged so as to display explicitly renormalization

f l fl I terms. First the denominators of the three ~-functions are combined, 'Y,[i-y(p'-k)-K]-y,.[i-y(p"-h)-K]'Y, L (p' p")=(ia/41r3) 2udu dv d4k , - - - - - - - - - - - -

,. '

o

o

{[k-u(p'v+p"(1-v))J2+A'2} 3

f f = (ia/21r3) f1 udu 1 dv d4k{'Y.[i-y(p'(1-uv)-p"v(1-u)-k)-K]-y14

0

0

where

X [i-y(p"(l-u+uv)-p'uv-k)-K]'Y,} /(k2+ A2) 3, (28)

A'2=u2[K2+ (p' -p")2v(1-v)]+X2(1-u)+u(1-u)[(p'2+K2)v+(p"2+K2)(1-v)];

(29)

a change of variables, k,.-k,.+u[p,.'v+p/'(1-v)] has been made. On extensive rearrangement, the numerator of the integrand can be brought into the form19

(30)

where

K,.(p', p"; u, v) = (1-u)(i-yp'+ K)-y,.(i-yp"+K)- (i-yp'+K)[K(1-u2h,.+i(1-u)(1-uv)(p'+ p"),.

-i(l-u+ 2uv)(1-uv)(p' -p"),.]-[K(1-u2h,.+ i(l-u+uv)(l-u)(p'+ p"),.

+i(l-u+uv) (1+u- 2uv) (p' - p"),.](i-yp"+ K)+ (1-uh,.[(p'2+ K2) (1-uv)

+ (p"2+K2)(1-u+uv)]-iK(P' -p"),.u(l+u)(l- 2v)

Thus, with

J f f l

I

{

L,.(p', p")= (ia/21r3) udu dv d4k

0

0

+-y,.(p' -p")2[1-u+u2v(1-v)]+ Ku,.,(p' -p"),u(l-u). (31)

+ Ao2 = K2u2 X2(1-u),

-y,.[k2 -4K2(1-u-½u2)] 2K,.

[k2+ Ao2]a

[k2+ 4'2Ja

(18b')

(32)
where B is defined by Eq. (24).
18 See reference 1, II, Eq. (26). 19 This expression may be compared with Julian Schwinger, Phys. Rev. 76, 790 (1949), Eq. (1.94). The expressions differ only in the definition of v and in the fact that certain terms, zero for a real electron, are included here.

544

R. KARPLUS AND N. M. KROLL

Finally, therefore, rix', x")=-y,.6(x')6(x")+A,.<2>(x', x")
f =-y,.6(x')6(x")[1+(a/21r)B]-(a/21r)(1/(21r)8) d4p'd4p" exp[-ix'p'-ix"p"]

Here again the separation of the renormalization term B has made the physically significant correction divergent in the limit X-0.

CALCULATION
Now that the singular functions have been calculated to second order, it is possible to proceed with the evaluation of the matrix elements written down earlier.
Thus,
f MII = - (ea1r/2hc) d4xad4xo'd4xo''d4x1d4x1'd4xi''d4x2d4x2'd4x2"A/'(xa)DF' (xo-x1)if;(xi')

xr.(x1-xi', X1"-x1)SF(xo'-xi'')r,.(xo-xo', Xo"-xo)SF(x2'-xo'')r.(x2-X21, X2"-x2)i/l'(x2"). (34)

On substitution of the last part of Eqs. (13), (14), (23), (25a), and (33) one obtains, to order a2,

MII = M IIo+ Ji?IIa+ Mm+ Ji?IId+Ji?m+Ji?llf, 20

(35)

where

f MII0= -(1+~A)ea1r d4xod4x1d4x2A,.•(xo)DF(X2-x1)ifi(x1h,SF(xo-x1h,.SF(x2-xoh,i/l(x2) 1r 2/ic f = - :,( 1+;A) d4xod4x1d4x~,."(xo)ifi(x1)A,.<2>(xo-x1, x2-xo)i/l(x2),

(36a)

Finally,

(36b)
(36c) (36d) (36e) (36£)

XSF(Xo-x2h,.SF(xa-xoh.SF(X4-X3)-y.,,,if,,(x4). (37)
IO The bar on R 11•, etc. indicates that the renormalization terms have been removed. These are incorporated in MII0• Since M 11b contains only renormalization, £rtn is zero.

QUANTUM ELECTRODYNAMICS

545

It is convenient to continue the calculation in momentum space. The momentum Pi will be used to denote the momentum. of the final state and P2 the momentum of the initial state:

(38a)

and

(38b)

Then e a2

f

f

i-y(Pi-k)-K i-y(pi-k-k')-K

MI=- -(21r)4 d4Pid4p2A/(P1-P2) d4kd4k' (1/k2)(1/k12)y;(p1h,-----'V1.-----

lic l61r6

(Pi-k) 2+K2 (Pi-k-k1) 2+K2

i-y(p2-k-k1)-K i-y(p2-k')-K X-y,,------v,.---~1.t/t(P2), (39)
(P2- k- kI)2+ K2 (P2-k')2+ K2

+ f £ £ MllO= ( l+;A)(

2: : J ( 2 1 r ) 4

d4pid4p2A/(Pi-P2)

1
udu

1
dv

_ {u,.,(p1-P2),u(l-u)+ (pi-P2)2-y,.(l-u+u2v(l-v))
+ Xi/;(Pi) - - - - - - - - - - - - - - - - B-y,.+-y,.(pi-P2) 2u2v(l-v) K2u2 X2(1-u)+ (p1-p2) 2u2v(1-v)

x[£ 1 K2u2+x2(l-u)+ ;;i-h) 2u2v(1-v)z

[J. X

_ ,/

;

(

i-y(p1-k)-K{K,.(p1-k, P2-k; Pih,,---- --------+-

u, v) -y,

,

(

A

'

2-

Ao2)

1

dz

(P1-k)2+K2 A12(Pi-k, P2-k; u, v)

o Ao2+(A'2 -Ao2)z

2K2(1-u-½u2)] }i-y(p2-k)- 11.

- - - - - -----v,if;(P2), (41)

f I f f. _

iea 2

AfII0 =--(21r) 4

d4p1d4p2A,,•(p1-P2)

d4k

1 -

-

A' 2Ao 2

(P2-k)2+K2

1

1 _

i-y(p1-k)-K i-y(p2-k)-K

udu dvt/t(Pih,.----'V,.----

41r4kc

k2+x2 o

o

(P1-k)2+K2 (P2-k)2+K2

[J. X

{

K.(p2-k, p2; u, v) -------+--y,(A'2-

Ao2)

1

dz

2K2 ( 1 - u - ½ u2) ] }

- - - - - - - - - - if;(P2), (42)

A'2(p2-k, P2; u, v)

o Ao2+(A'2-Ao2)z

A12Ao2

AfIId= - iea2 (21r)4f d4pid4p2A,,2(p1-P2)f d4k_l_ f.1 u(l-u)duf i dzy;(Pih,i-y(P1-k)-K "

41r4lic

k2+x2 0

0

(P1-k)2+K2

+ (i-y(p1-k)- K)(l-u)(l-[2K2u(l+ u)z]/[K2u2 X2(1-u)])+ K(l+u)

+ x - - - - - - - - - - - - - - - - - - - - ' V , i / l ( P 2 ) , (43) K2u2 X2(1-u)+[(P2-k)2+ K2]u(l-u)z

f f f AfII•= ---ie(a221r) 4

d4P1d4p2A/(P1-P2)

d4k

1 d

2v2(1-13v2)

1

81r4kc

O

1-v2 k2+(4K2/l-v2)

_ i-y(pi-k)-K i-y(p2-k)-K X i/l(Pih,----'V,.------v,if;(P2), (44)
<Pi-k)2+"2 (P2-k)2-K2

546

R. KARPLUS AND N. M. KROLL

(45)

Now, the interaction energy density of an anomalous magnetic moment µek/2mc with the electromagnetic field is

H(xo) = - µ(ek/2mc)!F,,.(xo)f;(xo)rr,,,y;(xo).

(46)

Since the calculation is being carried out in momentum space, it is convenient to have the Fourier transform of this expression. Its contribution to the scattering of an electron is

(47)

In calculating the correction to the magnetic moment of the electron, therefore, one must seek to bring the matrix elements into this form by rearranging the Dirac matrices occuring in them and by using the Dirac equation to simplify the momentum dependence of the integrand. This, of course, can only be done after the integration over the directions of the virtual momenta has been made trivial, so that these variables no longer conceal a dependence on the initial and final momenta. In this process, any terms that contain a factor (p1-P2)2may be dropped from further consideration, because they represent derivatives of the quasi-constant electromagnetic fields. Hence, MIIJ does not contribute to the magnetic moment. Further, MII 0 gives (}-2=0)

The second-order part of this expression clearly is due to the well-known anomalous magnetic moment

(a/21r)(eli/2mc). 21

(49)

This quantity, however, depends on the "bare" charge e of the electron. The measured charge of the electron is e1=[1+(a/41r)A]e to second order. Furthermore, the external potential Ai,•, whose source is a current, must also be renormalized, (Ai,•)1=[1+(a/41r)A]Ai,•• Equation (48) may therefore be rewritten, to order a2,

(SO)

and is due to a magnetic moment (51)

which depends on the renormalized charge.

Aflle can be evaluated quickly by observing that the integration over d4k and subsequent rearrangement of the

matrix element is idential with that in Li,(pi, p provided one sets X2 =4K2 /(1-v2 Hence,

2

),

).

_

e a 2 f 1 f 1 2v2(1-½v2)

u(l-u)

M11• - - udu d • - - - - - - - - - - - - ( 2 1 r ) 4

2licK 21r2 0

0

1-v2 u2+4/(1-u)/(1-v2)

JX d4P1d4P2A/(P1-P2)f(P1)rr,.,(p1-P2),,/;(p2). (52)

The magnetic moment responsible for this scattering is

at f i 11 u2(1-u)v2(1-½v2) a2(119 7r2)

a2

µII•=- du d•>-------=- - - - ""0.016-.

(53)

7r2 o o u2(1-v2)+4(1-u) 7r2 36 3

7r2

11 Julian Schwinger, Phys. Rev. 73, 415 (1948).

QUANTUM ELECTRODYNAMICS

547

The expression for Mm will be examined next. The first task now is to simplify the integration over k. The
situation here again is very similar to that encountered in the evaluation of L,,(p', p"); simplifications can be made, however, because of the equations satisfied by P1 and P2 and because (P1-P2)2 may be neglected. The scattering becomes

1 X -y, {i-y[p1(l-vw)- P2v(l-w)-k]- K )-y,,---
u(l-u)z {i-y[P2(l-v+vw)-p1vw-k]- KI (1-u)(1-[2u(1+u)z]/[u2+ (X2/ K2)(1-u)])+ K(l+u) X - - - - - - - - - - - - - - - - - - - - - - - - - ' V , v , , ( p 2 ) . (54)
{k2+v2K2+ X2(1-v)+v(1-w)[K2u2+ X2(1-u)J/[u(1-u)z]} 3

Here, obviously, the kx vectors in the numerator contribute no magnetic moment, because the linear term vanishes and the quadratic one is independent of P1 and P2,
It is useful at this point to discuss the extraction of magnetic-moment terms from these more complicated momentum-dependent spinor matrices. Thus, with the neglect of charge renormalization terms (independent of P1 and p2) and terms representing higher derivatives than the first of A,,•(x0),
i(P1) ,,{I(P1)f(P2) = i(P2),,f;(p1)v,,(p2) = -½f;(P1)u,,,(P1- P2),,f,,(P2) = -½m,,,

/t(Pih,,i-yp1f(P2) = f;(Pi)i-YP2'Y,.f(P2) = - m,,,

if;(P1h.i'YP1'Y,.-y.,f,,(P2) = {,(Pih.i'YP2'Y,.-y.,f,,(P2) = - 2m,,,

if;(P1h,'Y,hP1'Y,Y1(P2) = if;(P1h,'Y,.i-yP2'YvV1(P2) = - 2m,,,

(55)

if;(P1h.i'YP1'Y,,i-ypn,v,,(P2) = f(P1h,i-YP2'Y ,.i-YP2'Y,V1(P2) = - 2Km,,,

if;(P1h.i'YP1'Y,,i'YP2'YvV1(P2) = -4Km,.,
f;(p1h,i-YP2-Y,,hP1-Y,f(P2) = o.

The magnetic moment contribution to AfIId is due to a moment

,,,_·a_\)f µ.IId= (-

i u(l-u)duf i dzf i vdvf i dw·i-·1r_2 __1__

1r4 o

O

o

o

2 u(l-u)z

2K[2(1-v)(1-u)(1-[2u(1+u)z]/[u2+ (1-u)A2/ K2])-(1+u)(l-v)J -2K(1-v)(2-v)(1-u)(1-[2u(1+u)z]/[u2+ (1-u)X2/ K2])
x------------------------------------

(56)

As was already mentioned earlier, this expression may diverge logarithmically as X-0. It is easy to verify that this catastrophe occurs only in the term which has two denominators, and that is associated with the integration over u. After integration over z, only one simple term is left which is afflicted with this difficulty. The photon mass may then be set equal to zero in all others, and the integration can be easily completed to yield

µ

.I

It

l

a
=

2
-

(

1-1 -

-1r+ 2

-1

l

nx-2)

'.::::'.

(

-0.090+1- lnx-2)a-2.

(57)

11'2 24 18 2 K2

2 K2 1r2

The expressions for µII", µ.110, µ.1become successively more complicated and very much more tedious to evaluate
and cannot be given in detail here. The contributions from group II are all treated in a manner similar to L,.<2>. The presence of two virtual momenta in M1, however, and the symmetry of the integrand suggest that this quantity

548 be evaluated by noting that

R. KARPLUS AND N. M. KROLL

x

f

-

-

-

-

-

-

d 4kd
--

4k'/k 2k'
--

2
-

-

-

-

-

-

-

-

}

(58)

[(p-k)2+µJ[(p' - k-k')2+µ'J[(p" -k-k')2+µ''][(p"' -k')2+µ"'] pf, :i:,;a!j,,

and evaluating the integrals over k and k' before carrying out the other indicated operations.22 The result will clearly involve five variables of the type of u, v, w, Eq. (54), to be integrated from zero to one. The other two remaining terms also involve five variables, but in these the variables tend to separate into two groups, because they were introduced in connection with two independent momentum integrations. The magnetic moments may now be deduced as before. They are integrals of rational functions of the auxiliary variables.23
After one trivial integration, µIla involves the same type of functions as µ.IId. It is found, however, that the infra-red catastrophe introduced into the X"<2> operator is compensated by one that arises in the integration over k, Eq. (41). In other words, the terms depending on the photon mass all go to zero as this quantity is made to vanish. Thus

- + - a2(11
µ.11a = -

11"2)"'0.778a2-

(59)

71"2 48 18

r2

and no longer involves X.

After a term -(a2/2r2) ln(}.2/,c2) is separated from µ.II•, this quantity is finite in the limit x-o, so that the

integrand may be accordingly simplified. A typical term, which happens to involve only four variables, is

f i duf i dvf i dtf i dwf I dz

2wt(1-t)(1-uv)

.

(60)

o o o o o {vw[1-uv+ut(1-v)]2+ut(1-uv) }2

If the first three integrations are carried out in the order indicated, each can be done analytically by virtue of simplifications that occur when the limits are inserted in the preceding integration. The order of the last two integrations must be determined by inspection for each term; with two exceptions they can be carried out with the help of well-known formulas. The value of µII• is therefore given in terms of two integrals, L1 and L2:

Here and

ii

oo 1

Li= [ln(1-x)]2dx/x=3 L -=2.4041138· • •

0

-1~

0.3005655 .... 24

(61) (62a) (62b)

A typical term of µ1 is

f f J. f 1 du 1 dv 1 dt I dwwt2u4v2

0

0

0

0

1
x - - - - - - - - - - - - - - - - - - - - - , (63)
[1-wt(l-uv+u2v2) J{u2t-w[(1-t+tuv)2-u2t2(1-uv)- 2tu2v(l-t+tuv)J) 2

u The integration over the virtual momenta is accomplished by combining the six denominators in the manner of Eq. (16). There
are many equivalent ways of introducing the auxiliary variables; some of these, however, are much more convenient than others for carrying out the subsequent integrations.
13 The details of two independent calculations which were performed so as to provide some check of the final result are available from the authors. The work is made lengthy by the large number of integrals over auxiliary variables.
24 Note ad4ed in proof: Using the results of H. F. Sandham, J. Lon. Math. Soc. 24, 83 (1949), one can show that L2= ¼L1.

QUANTUM ELECTRODYNAMICS

549

where a trivial integration over one variable has been carried out. After two integrations, which again can be carried out analytically by virtue of some remarkable simplifications, the functions of u and v obtained are very similar to those encountered in the calculation of µ.m. The final result, in terms of the integrals Eq. (62) is

(64)

SUMMARY OF RESULTS

The five contributions to the fourth-order radiative

correction to the electron's magnetic moment are, Eqs.

l (53), (57), (59), (61), and (64), µ.1=a-2[1-3+-113r2--51r2 ln2+5-L2+-5L1 = -0.499a-2,

11'2 96 36 6

3 12

1r2

here L1and L2 are the integrals of Eq. (62a) and (62b), respectively. Hence, the total radiative correction to
the magnetic moment of the electron, to fourth order in e, is

a1

a12

~--2.973 ~0.001147 in units (ed't/2mc).

21r

11'2

ACKNOWLEDGMENTS
It is a pleasure to acknowledge much helpful dis-
cussion with F. J. Dyson and to thank Professor
Oppenheimer for his continual encouragement.