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LIBRARY

Michigan State
University

,H»_'nl".

 

This is to certify that the

thesis entitled

RECOIL CORRECTIONS TO THE HYDROGEN HYPERFINE
SPLITTING OF ORDER (me/mp)a61na-1

presented by

Robert S. Cole

has been accepted towards fulfillment
Of the requirements for

Ph . D . degree in Phys iCS.

 

Z/az/né W Wfiff'"

/ Major professo/

Date July 28, 1972

0-7639

 

ABSTRACT

RECOIL CORRECTIONS TO THE HYDROGEN HYPERFINE
SPLITTING OF ORDER (me/mp)a61

By
Robert S. Cole

A calculation of the (me/mp)OL61n0L'1 recoil corrections
to the hydrogen hyperfine splitting is presented. The result

is

2 -5
8 me me In me
Av — — 1+— ONCR lnoc'l 3-K(13+7_e.)+1<2(7-14——)
H 3<hp) ( mp) yw [ mp mp

= 2.899 X 10‘4 MHz (0.2 ppm)

The calculation is performed using a perturbation theory
based on the Bethe—Salpeter equation and on a wave function
Obtained by a single iteration from the nonrelativistic
Pauli wave function. Form factors were included to account
for proton structure, but it is shown that the
(me/mp)a61na'1 is totally insensitive to the details of the
proton form factors.

The recoil corrections calculated here, when combined
with similar results for muonium, can be used to set an

upper limit on the proton polarizibility.

 

RECOIL CORRECTIONS TO THE HYDROGEN HYPERFINE

SPLITTING OF ORDER (me/mpmelna‘1

’Y

W"

Robert SiVCOle

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1972

ACKNOWLEDGMENTS

It is a real pleasure to thank Dr. Wayne Repko who
suggested this problem. He has Offered advice and encourage-
ment in many conversations through the course of this in—
vestigation, and it has been a genuine pleasure to work with
him.

While a graduate student, I have received financial
support from Michigan State University in the form of a
National Science Foundation traineeship (three semesters),

a National Defense Education Act Traineeship (six semesters),
and part-time Teaching Assistanceships (three semesters).
I gratefully acknowledge this support.

Finally I wish to acknowledge my wife, Cynthia, for

the emotional support and encouragement she offered during

our stay at Michigan State.

ii

 

TABLE OF CONTENTS

 

Page

LIST OF TABLES . . . . . . . iv
LIST OF APPENDICES. . . . v
I. INTRODUCTION . . . . . . . 1
II. BACKGROUND . . . . . 5
III. FIRST ITERATED WAVE FUNCTION . . 14
IV. CALCULATIONS . . . . 20
A. Coulomb Recoil - . - - . . . . . . 20

B. One Photon Exchange. . . . . . . . . 25

1. Charge Component . . . . . . . 26

2. Linear Dipole Component . . . . . . 35

C. Two Photon Ladder Exchange . . . . . . 47

1. Charge Component . . . . . . . 49

2. Linear Dipole Component . . . . . . 62

3. Quadratic Dipole Component . . . . . 74

D. Two Photon Crossed Exchange . . . . . . 80

1. Charge Component . . . . . . . 81

2. Linear Dipole Component . . . . . . 86

3. Quadratic Dipole Component . . . . . 97

E. Non-Contributing Diagrams. . . . . . . 101

V. SUMMARY AND CONCLUSIONS- - - . - . . . . 103
REFERENCES . . . . . . . . . . . . . . 108
APPENDICES . . . . . . . . . . . . . . 110

 

LIST OF TABLES

 

 

Table Page
I. Time integrations for one photon exchange. . 30
II. Spin averages for one photon exchange . . . 32
III. Momentum integrals for one photon exchange . 34
IV. Spin averages for two photon ladder diagram . 52
V. k0 integrations for ladder diagram . . . . 57
VI. Momentum integrals for two photon exchange . 61
VII. Spin averages for two photon crossed exchange 87
VIII. k0 integrations for crossed exchange - - - 89

 

IX. Summary of all ocslnoc"1 contributions for

charge component - - - - . . . 104
X. Summary of all delnoc~1 contributions

for linear dipole component . . . . . . 104
XI. Summary of all Otelnoc—1 contributions for

quadratic dipole component. . - . . - . 104

 

 

LIST OF APPENDICES

Appendix Page
A. Derivation of Interaction Kernels . . . . . 110
B. Spherical Averages . . _ . . . . . . . 116

C. Derivation Of Integrals . . . . . . . . 119

 

 

I. INTRODUCTION

A calculation of the (me/mp)d21n&1 recoil contributions
to the triplet-singlet splitting of the hydrogen ground state
(the hydrogen hyperfine splitting) is presented. The result

to this order is

J(K%4-I=zL/m;lw W%“('+% H){9~K(13+7 :W 44¢);

 

 

(1.1)

where K is the proton anomalous magnetic moment (K 2 1.79).
The calculation was done using the perturbation theory treat—
ment (developed by Karplus and K1einC1) and Fulton and
Martin(2)) Of the Bethe-Salpeter (BS) equation(3). The wave
function used was a perturbation approximation to the Coulomb
wave function Obtained by a single iteration of the BS equation
containing only the Coulomb interaction. In addition, the
calculation included proton form factors to account for proton
structure.

The theoretical expression for the hydrogen hyperfine

interval, vH, can be written as

..3 /
7/:‘—°’IZ A‘cfltw’fle’) 1+ 0“ f 12
H ... + + (.)
where up and “B are the proton magnetic moment and Bohr mag-

neton respectively, and where O represents the radiative cor-

rections to the electron magnetic moment, 6p the nuclear size

L...“ ,.

and recoil terms, and 6E the proton polarizibility (proton
structure) terms.

Equation (1.2) takes the form of the Fermi frequency
(namely the first term in the expression) due to a point
magnetic dipole in the proton interacting at zero range with
the electron moment in a spherical S state, modified by radia-
tive corrections, nuclear size and recoil corrections, and
polarizibility corrections. The leading term of the radiative
corrections to the electron magnetic moment is of order
a (21/137.) relative to the Fermi term. These radiative
corrections have been completely calculated to order
a(Za)leg2(Zd)_2 relative to the Fermi frequency. and the
dominant as terms have been computed. The leading recoil
correction from the static magnetic moment and charge distribu-
tion is of order (me/mp)d relative to the Fermi term, and hence

I correction is of comparable magnitude to

the (me/mp)d21nd'
the d3 terms in p.

The nuclear polarization (having tO do with the ”non—
rigidity” of an extended proton) is an extremely important
yet difficult term to deal with rigorously. It cannot be
calculated since it involves, in principle, most of the
uncertainties of proton dynamics. Attempts have been made
to calculate 6% using inelastic electron - proton scattering
data(4), but at present such calculations are highly model

dependent. Most of these model dependent calculations find

the proton polarizibility to be Of the order of Ill to [2| ppm

(a recent calculation by Jensen, Kovesi-Domokos, and
Schonberg(5) places Oé at 1 ppm).

While the direct calculation Of the proton polarizi—
bility appears to be far in the future, new experimental
measurements Of the hyperfine interval Of muonium,

(u+ e" bound system), when coupled with the calculation Of
the recoil terms, provides an exceedingly accurate measure
Of 6'. Specifically the theoretical expression for the

P
muonium hyperfine interval is

721.??? Afl/i— 1+ ) ;/+f+/§ (1.3)
fl

where, again, 0 represents the radiative corrections to the
electron moment, 6H the recoil terms, and both are, in
principle, calculable. Taking the ratio Of (1.3) to (1.2),

we Obtain:
W , 473.3 r
.../#:A (+471 3 f,+¢.fldgflfi//’)/} (1.4)

The important point to note about (1.4) is that the radia-
tive corrections tO the electron moment have cancelled out,
as has the leading a dependent coefficient. Thus, the ratio
is free Of uncertainties in a and affords an Opportunity

to directly measure the (calculable) recoil terms and the
(non-calculable) polarizibility terms. Hence, although the

1

absolute size Of the (me/mp)d21nu' recoil correction pre—

sented here is small, it is nonetheless significant because

its effect can be compared directly with experimental data.
Moreover a precise calculation Of the recoil terms pro-
vides an accurate measurement Of the proton polarizibility.
Another noteworthy feature of the (me/mp)oc2lnoc'1
recoil correction is that, as will be shown, it is totally
insensitive to the details Of the proton form factors.
This is due to the fact that the logarithmic dependence
arises from the low momentum limit Of the exchanged quanta.
Hence, no further ambiguity or uncertainty from the proton
form factors is introduced into the theoretical result for
vH.
Finally, although the calculation is presented here in
the Lorentz gauge, it was also done in the Coulomb gauge
as a check on gauge invariance Of the perturbation formalism.(6)
In the following calculations we denote the space-time

coordinates by x (x,ix0) and take yp tO be Hermitian

u

matrices satisfying {yu,yv} = dev. The symbol ml W111
refer to the electron mass, and m2 will refer to the proton
mass. M will denote the total mass Of the system, (ml + m2),

and u the reduced mass Of the system, u = mlmZ/M.

II. BACKGROUND

The starting point for the calculation Of the hyper-
fine splitting tO our order is the Bethe-Salpeter(3) equa-
tion. We use the perturbation theory treatment Of the Bethe—
Salpeter equation as developed by Karplus and Klein(1).

The essential idea is to separate the instantaneous Cou-
lomb interaction, which accounts for the major part Of the
binding, from the total interaction. Using an approximate
solution to the Bethe-Salpeter equation containing only
the Coulomb interaction, the remaining time dependent
interaction terms are treated by perturbation theory.

We start with a two body Green function defined by:

GONE/y, 41,1) : <0/TfWJknflr’fa.)IZ}¢,)%‘Z;QJP[O> (2.1)

Note that the spinor indices are suppressed, that the sub-
script H denotes the Heisenberg Operator, and the particles
are labeled (1) and (2) respectively. Using well known
techniques, (2.1) can be transformed tO the interaction
picture as:

(O/TZHPUR.) Rabblfiigz.) 3/2211.) 3/00, —oo,“/ 0)

[(4779 753/ 1 (22)
7 , 044/ (O/S'(m/-oo)/O>

 

where the w are the interaction picture field operators,
I0) is the unperturbed vacuum state, and

S

w t t
S(tt’/=E<-2')A,;L; at, Jag, T{H1(t.) Hid“); (2.2a)
AUG ‘ t’ t’

where HI is the interaction Hamiltonian given byzc7)

(2)

~2'e 1!)"fo éflflfl)%%¢ ‘26 7.9% X1, 4/» 2,0 (x) (2. 2b)

Using the Wick theorem(8) on (2.2) one derives the perturba-

tion expansion:

(1)

’ , ll (2; 3 (I)
é<¢u 12/ x3, Y4): SFYX,‘ X3J§F(y1“'x4)“ 62/d363d4x6 SF (71.4%) SF (yz'yc X
0) (2/ I (3’-
’ a A” D (XS-7(6) Sig/(“5'79“) gefléqg) 1‘

V ,asz

+ (2.3)

which can be exhibited diagramatically as:

 

 

 

 

 

 

Formally we write the infinite series in (2.3) as:

(70191., 13,1.) .—_ Sg)(7<,-x,)‘§g(x.-y.) + filfx‘d‘x/‘x? $71945. )

(2) (2)

x g; (X) ‘ 7(6) 1(7‘5, 11/ X7, X3”) gFm/X7 ' 151SF {15’ - I)

4

(2.4)

where I is the sum Of all graphs contributing tO the two
particle interaction. One casts this in the form Of an
integral equation be denoting I as that part Of I which

consists Of the sum Of all irreducible graphs(3). Then

(2.4) can be written:

(I 1 '4 4 4 W .
G081.) x5, 74/ = §p/(1,-9za) Silk 7:.) + [dry/”Mair: ‘3; (ms)

01

X SF <11 -Xé) T(X5, ¢él 7Z7, ijééXZ Y; 79/ ’14) (2 ' 5)

which is schematically shown as

 

 

 

 

 

 

3 4
3 4 3, 4
\
‘ v
if...
a T— + 7/“f y
I

5 [1' //¢

/ Z I 1

Equation (2.5) is the Bethe-Salpeter equation(3). From

equation (2.5) one can rigorously derive a form for the two

particle wave function Of a bound state(9):
a. \. 4 4 4 4 0) O.)
Wot/(1U XZ/z'. fd'fifdxéd’x7élyg g: (11'15’) Sp (Xi-X6)
XEOXSJ «(U 17/ 0(3) 3%)(17/ 7(8) (2'6)

For convenience we will henceforth use the convention
that all repeated indices are to be summed (integrated)
over. Equation (2.6) is the basic equation used in calcu-

lating the hyperfine splitting. Introducing center Of

mass and relative coordinates X and x, where:

X EZ/xlf'72—7Zl ’L/5X4’7L’X

 

 

7' : «I -1; ¢L :l -— ' [IX
7X, : ”’4' . 7? : 444,:—
WIf/M; ) 2 ”41,747an (2'7)

we can write (2.6) as:

fla): Cuber/1‘“ (4’ XI) (Ink/y") (2.8)

h —JI§'X: _2
w ere WW) 2 e {'77) 774(74/7‘2) 1 (2.83)
/ —?Z(X‘X) O
Ska/X7: f2? .6 SPI[X’XI+ 71(X’XQJI
(2.8b)
2/
X 37: [X~X’ + 7,0070]
. , {XXX—X) ——
and I“ (1] 17 :: ffX e I<’X,‘XL’X-Xi) (2.8c)

We have used momentum K as conjugate to center of mass posi-
tion X. For convenience we will work in the system where
the center Of mass is at rest so that Ku = (0,iKO). We
will also take p to be the relative momentum, conjugate to

x. Thus we can rewrite (2.8b) as

d4 {74(1’1) .(U (2)
6,092 : ($5 SFW’Z/KKM'W’N) (Md)

We now begin the perturbation theory by separating

the instantaneous Coulomb interaction from the interaction

kernel: IK = :[Qa- IKI (2.9)
C ' ’ f U) yo}

where I =—— ~2a/{1-r) We) 5:, 4 a (2.9a)

and 1k the remaining part Of the interaction kernel.

Hence (2.8) can be written:

fled :: 42(61, (7/, 17/5/4924 0) It'll/,0) (WWW

fl

 

(2.10)

+ 6.44.01”; «7 find

10

For the unperturbed (Coulomb interaction) equation.

having total energy KC, we take:

gem): —20<[7 “(a X)/(x’-— Maw 3:“ 04% (’x”) (2.11)

 

We then want tO evaluate the energy difference
AE 5 KO - K8. In order tO get at this energy difference,

it is useful to factor GK(x,x') into the following form:

(71,091? = g. (79;) Adyfl’) 3107401
0) 2/
: Aka/Z) gKQ‘j/X’) 43 314‘
(2.12)
where
1' {V'i/ ’/
fix“ )z/fi g f (I; (2) .- O
y (2 )4 [Ha/W7, 7/ k] (2.12a)
and
I, ( '1’ 1 'TI (2) TI.
(2.12b)

where H(i)(p) = u(i 1) p + 8(i)ml . is the Dirac Hamiltonian.

A similar factorization holds for GKC(x,x'), Then we note
that:
d4 z. ,«(m-y) (2) -I -/
g“ (IX/y).jk6(IX/;) : A E/(Z—ébe [/((1”)+//('I)’/G][H%) + #{fif/ - 467
(2.13)

Multiplying (2.13) by AK(y,x') we obtain:

11

3.21%.), «'1 — g“. (cap/123,29 = Mag/29a magma, 2')

= w, «MW fixes/Ms“) = AE GWWQWW

(2.14)
In order to utilize this equation we define a hybrid wave

function, (which differs from ¢KC(X) by terms of order AE):

fl’flx/ 2 g (2,?) AM} 29%.? 0pm,) (PLAN) (2.15)
: 'iAK(“/fl)0fl(go)j((fl) (2.15:1)
Where 077.) 2%); «fig/w) d’m') (mac ’)/,2 (2.151))

Then multiplying (2.14) by -ia5(xov)¢Kc(X.)/r, we have:
éx(«/’XI)IKQ(IX:17(PKI(’XH) _ (PKKX) : AE'. GK 69;) X20} x40) gxcg/ 7")

p I
x<}ifl'if?)9%4012)

 

(2.16)

where we've used fifldfllfl/A =£10)fl6(¥)//1 which follows

because 6(XO)AK(x.y)6(yO) is independent of K. Equation
(2.16) then becomes:

(2)742 — gm, «7 1%; x 7%?“ ’2 -——= 2' A26“ W) )1,"’11”’4:9.)p7)

(2.17)

We then write (2.17) as:

(I)

[9477]“ ‘ 1W “71%.?“ == Z‘AEBS 91’”an In)

(2.17a)

12

whose adjoint is

(PK/(XI) [Gilt/'2‘)“ 1.791)]: ME 422.)XE) (2.18)

 

We then multiply by ¢K(x):

I , -I. , C / . ‘ .
%(X)[GKUW)*IK(1,1)Y(PK{1) - 2A,;
x
Where we've taken the normalization 2((1/‘0Q¢0)4PK{1)= I

Adding and subtracting Ik(x.x'):

{MMML'M x) — MW) + 1K'(rx,’rx)]gpk(o<) : 2 AE

/ r I I . .. . __
= (P«@)Ik(fxx7‘)$pu(y " ZAt (2.19)

where we've used (2.8) and (2.9). We have already seen

that ¢k differs from ¢KC by terms of order AB, which is negli—
gible to our order of accuracy. Hence we wish to express
(2.19) in terms of ¢k which we will then replace with ¢KC.

To find the relationship between ¢K and ok we write (2.17a)

as:

[CM-Yd, ’X’) - I: (’X, ’X’) ”1“,“, ’X’) +IK’(7(, Visa/(Ix? 2 ME XL“, 31(2) (7(0) jQIX)
(2)

I.» agmm’m — @2217de + 1.121991%) = .12.— 31"); 091.17%

so that

92:00 — flex) + G“@,7’)I“(7<l’.x”)$’j(’(x~j : {AEG(’X»;)X,”>‘4WO£;.)7Q7/ (2. 20)

13

Expanding the exact Green function in terms of the Coulomb

Green function:
9092’) = G 42.1%? Gil/2(7) Rig/jyéucgfiy) +

and neglecting the term in AB in (2.20) we write (2.19) as:
A; z .1 W2) [1‘1xe + Ixxyékcggfiggj «QJPKA’X’J (2. 21)

This expression is the one that will be used in the follow-
ing calculations. But first we must derive an approximation

to the Coulomb wave function.

 

III. FIRST ITERATED WAVE FUNCTION

In order to make use of the expression for energy shift,
eq. (2.21), we need an approximation to the Coulomb wave
function ¢KC(x). This may be obtained by using (2.11) as the
basis of an iteration procedure(10) beginning with the non-
relativistic Pauli wave function ¢p(x). Specifically the first

iterated wave function is written as

U) I/
v,.<«1=fa*.'w6. (4, «71'... (1’, «'9 W“ (3.1)
I [[4 2F(1 1/ (I/
Where G“(’X/’Yj: (El-rd“ e gF 02+? V)S,;(- P1“ 7Z2. “)
m (I 2: .z I,
and Icmq") : -2 4nd 344 A” /(x 27/th :13) ii

'Lgl‘

,, all ” y x <0)
and WU 2 49“)” IXCM’)

where y E ud, and¢p(0) = ys/Zn‘l/Z(spin ftn) E ¢(O)(spin ftn)
The x” integration is trivial using 6(x'-x”). The x' inte-
gration yields (2w)3o( —p1-g). Hence the now trivial q

integration yields

 

(I) (I) M 4 3 I ' .- __,‘ 0
Wm)— -.mw 2MP elm, rO£S~F(’/,+.(,K)§;’g,a+w)

" W)" WWI-27‘
(3. 2)

14

15

but 42:; fl ,
(22+r7/2-74’/12wyw)

SO

 

(II M 12/ {/1225 -2‘ of
(42X 344 A! g (b ('J (L)
C JZ’ _____.——---"""' «Z _____ _ 4,
CF“ (2 {m4 fl¢iw SN? (”7’10 $477 7‘“)?OP(O)
I (3.3)

In order to do the pO integration, we expand the propagators:

U) (2)

u) <2) __[/1 My) A+ A—
g + 4K 36 +72K) :: +
PO” 7 ) F 70 {I +7,K— —,E)(-( Z+7.K E) (FoWZzKo‘EJU’ o+q.k.+e)

 

 

 

 

 

 

 

 

('l (2)
+_A—(1/J/\+(’¢’J + A- (12M- (,2)
(22222.1222+.) (22.22) W. ...)

 

(3 4)
The pO integration is then accomplished by closing the inte-
gration contour in the lower half of the complex plane if
t>0 and in the upper half of the complex plane if t<0. For

example: (++ term):

-5 .t " "
(9/2?“ f M -(~m+€w-éeJ1 [20"(74‘o‘5W/WJ]

 

 

 

{(53- [VJ/é/ -2. E2- 0 '
21/(965/‘67 (901:) e ( i’k/k/f
z: + y

El +53 ' K.

E +51 'ko

I

pffl / t‘>o

t<o

where (96 f

and using 714—7, =
(3.5a)

Similarly the (+-) term yields

16

 

._ _ 9(2)
Ez'E/ +Ko ElvE, + “0

 

-(agJ/JME’ ”W” e ”(Ewwofltw

and the (-+) term

-[(E,+7z,xo)ft/ 4021-43, u,))t/
e e. #__¥fl__ 6%49
—<2m‘J{ -

EZ‘E'+K° El-E’ +K0

 

and the (--) term

—z'(E.+4[,Ko)}t/ ~2' (51+{L “oJ/tl ‘
(iwdzf§ffl e +-9&J e {

 

 

Cl+El+M° E.+E; +‘(O

(3.5b)

(3.5c)

(3.5d)

Substituting (3.5a - 3.5d) into (3.4) and rearranging we have:

 

EELEQZTZE*J*%U[&d—Mtflg -i@fl-%VOH/
4 Ely/Exp) 4—- 42

“ £3

__ [E’+HI%VJ] E,-H(2£J {equEn’ZlkoJ/t/ ,3'{5,+I[..1(0)/t/‘

EL‘EI + [(0

[(2, * HVPJM «6‘2 (5, +7, malt/J?

EI+E1 + K0

 

+ 2219(3) 313+ Win] [a + MIX-in] -1 (5., (MoJJt/
1»— e

4 5.1/45.sz 51" 5-. - K.

—.

[E'H"5vl/1[E. + H‘W) (£405. +2219
E), ’ E) +Ko

“H” 1' (21’ -.- E: 7('z(o)]t/
1W) *< +
EmE; tVo

‘ L— (Ez '7; KoJ/t7

(3.6)

 

17

This equation can be rearranged by using 9(tJ=-§(/+e(ty

/ t>0
and Q(~t') = é.- (/ - 6(ij where 6(tJ = 2/0 tzo
./ t4

Then we use H(i)(p) = aCiJ.p + BCiJmi, (1=1,2) and the fact
that B(i)¢p(0) = ¢p(0) so that we can effectively set
8(i)=1(i=1,2). We combine the exponentials according to the
formula:

1‘. #122? ‘2 7/ “W "‘ “(J (“W/2N

(3.7)

Terms of order 92 in the numerator are neglected
(i.e. K0 2 M - qu/Z = M) while the denominators are expanded

in exact powers of momentum:

 

____i,——— 2:._E:;EJ::J§;____2_._e-
(E,+E;): We1 [(6 +6, )l-“Ol }[E 5);- 4(6)]
(3.8)
(Ez-EV'M‘ + WK’J
‘4M’ (#11347 +15%”)
and similarly
I _ (E.+5,)‘-M* + fi/ANJ (3 9)

 

 

(aw—w ‘ ~ 4M“(F‘+(J+ZW<7

After a bit of manipulation we obtain

(0

W 2

\H

¥:M—+cfif —r0€o/ (3.10)

 

18

where

 

- pc’“-4//M{/+fl){/ 5 _—izf¥’:)[pe/+ e(£J((£/] M W”

ZIJB/‘M (1+!)2

 

and

e 2 JPN 1%)de I“ W”
fl/JO 7b [ t +é é ](’sz/%M(/’l+rj)
and

 

{0/
J29, : ‘10(aw/Zida/jp/FKEHéKtJC/‘géi/]@rr/::;JZ:*X/

with

Fer: [($5va + 0 62)]
aw [(2419 eff") — 092)]
m; [Jig/W; ’ ———-—e”—‘1’+-£"’ If” + pay]
me = [Jag—m2. If” _ If) —— owl]
/F(3:/ = [gm 04.72)]

mm: [4% - (we—ea]

(3.10a)

(3.10b)

(3.10c)

(3.10d)

 

 

 

 

19

It is also useful to note at this point that the static wave

function (i.e. at t = O) is given by

 

M l 2 (0)
0 : F- “(5/ (2/ flufl (/ 1’de
(19%} [MM (70,. 42 F644 1’— éfl/ 7M1 [fig/WV”? (3.11)

where

WJ'ZfiMfi/fiA4—M/1hg MJ—+~)+3(/flr HW)/g—J—)]
F491-2M2(?+ ":J [”65] *1)

HQL»£W, J;+-Z% +/9(E é/) ($12)

 

6?):(M/Lfig 444

Both do and 5¢' are proportional to p2

and hence repre-
sent those terms of the first approximation to the true
Coulomb wave function which may be in error. Only a second
iteration process would resolve the uncertainty in these
terms, but a second iteration calculation appears to involve
a prohibitive amount of unnecessary work at this stage.

The only other uncertainties or approximations made
in the calculation of ¢KC(1) are that terms of order o2 were
neglected in the numerator and terms of order a“ were neg—
lected in the denominator. In each case such terms would

contribute only to an energy splitting of order d7 or higher,

hence we are justified in dropping them.

IV. CALCULATIONS

Chapter IV consists of the details of the calculations

'1 hyperfine split-

involved in the computation of the delno
ting. For convenience this chapter will be broken up into
five sections: A. The Coulomb recoil, B. The one photon

exchange, C. The two photon ladder exchange, D. The two pho-

ton crossed exchange, and E. Non—contributing diagrams.

A. Coulomb Recoil

’1 contribution

The first source of potential aslnd
comes from the Coulomb recoil. In order to determine AE
(eq. (2.21)) to order delnoc'1 we must first determine the
”unperturbed” energy, KOC, to the same order. The aslna—l

component of KOC is called the Coulomb recoil.

We begin by writing (2.11) as(2):

{Ki- HMW’ #73710] (PW (12/0)

 

5/ 3 k-/ Q »/
+ffizjgéé~ ~{AL/H— )(-/\’J( ,} fit»)
(4A.l)
Adding and subtracting (a/r)?KC(r,O) and using the property
of projection operators that
HAT + A’wa Ai"/\(f+ AWAY”
we can write (4A.l) as:

20

 

21

[K:—- ngz) — MIX—7:7) + %]‘f“c(£/0J

(4A.Z)
efdjz'fl(£,~);fi 39“ g ’0)==o
where
:JJZEZC *3 41(2) A} .15) + AWN/M1. L)
, (4A.3)
I (u ‘ g,(27~)
+ £A(_)L£)/L(-k)fez 0'
Equation (4A.2) is very similar to the Breit equation:
[K06 _ HUJfi/‘HAU + loijzfgaz/O) :_ O (4A.4)

The energy levels of the Breit equation can be expressed as

a power series in q2(11) with no onslnof1 term in the expan—

sion.

We then take the 9(q/r) term of (4A.Z) as a small per-

turbation of (4A.4) and compute the difference of KOC and

KOC' to order aslnq’l. Since KOC' contains no oaslnoc—1 term,

the difference mentioned above will be the Coulomb recoil to
‘1

order delnq

Hence we write:

AECU 11“" ‘ “0
on. :(::)D(3

 

(70H i’J 11(7) 303(7) (4A.5)
142:4!)

       

22

where 9(p') is the Fourier transform of (4A.3) and where

(I (2] c/ ~I d3, 4,134] I.
(p30,): [H J40) + 146/) - [(0 (5733/, e [7? 533)ij

(4A.6)

which is the Fourier transform of (4A.4). Taking the first

iteration of (4A.6) by inserting
, -Y‘Ifl’}
SDpQ/LJ: e6 30PM)

on the right hand side. we obtain:

 

“I
. (I, ‘ C’ (OJ
1/ ) 7:. H ) + “(i j -' K ft dip—'—
CPB fl J W 1 0 ](.Zvr)é/79‘+X'l)

as the first iteration. The operator [H(1)(p)+H(Z)(-p)

-KOC']—1 can be explicitly written as:

[HO/12) + Hfifl) ' K0] 4 :

 

 

 

o) (2) w (l/ “J ‘1’ up (2)
__ A, A. + AA. + A- A. A_ /\_
E,+E; -—Kc Ere; ,KU E,*E:"Ko E,+E,_+t(o

Tdius from (4A.3) and (4A.7a) we have:

 

T (1!
* / 4 . (W (J (u “’ (H w
femorweflipfl -Aia;_+ M. - A-/\..

(tha/lép'gxl) E"- EoI‘TK" ELLE!” K° E, I‘fEil‘f’ [(0

Seatting 8(i) = l (i=l,2) the projection operators are

efEfectively:

(4A.7)

(4A.7a)

(4A.8)

23

(I) (2.)

I " \ ‘ I I
A+ A+ ~4E'EIL{CEI+MI)(EZ 17le) +<El+ml)%lllf _(E’+m,)ga’/£ __'_ [051"; gfl/‘z?
h, (2) I (I) (2) (I) 2)
A4} A- = E {\(E, +MJ(E1‘M;) '1- (92‘ Way/“V ~£ +(E.+W)/L8 "f + 2x, wglzfl ?

u; (2 I (I (2/ {l/ 2/
/\_ A )— ——-— gemyefim} (Eg—mu)“ Ifqgtmjg .13 .24.”);

* "45,5.

AWA‘Z) _. ___L._ {ZE’~4M’)(EZ-W1)‘(El-Wl)g0{£ +(El‘WJsz/f _, 34/323,592?

‘ ‘I’ -4EsE2
(4A.9)
Since we will need only the fgmkg’,@nzfl/ term in

computing the spin-spin term (see Section 4C and Appendix B)

we can write (4A.8) as:

T / / / ‘
J3 r (mi/war) 4E.’E.’(E.’-E,’-%J ”raft/Et'el-“J
4EE3’@;QE/+KJ

Similarly the only part of ¢B(p) that can possibly con-
tribute to the spin-spin interaction is:

47rd (flJOJ -93"! a”! (I‘/ “0-)
9W : __..,J£_J . _ )5 A .2

Cszf'I’ (141+ a“) 45.5, (5+5, —— K.) 451202,: N.)

 

 

(4A.ll)

I/ 2
+ J”’M"‘2 - .- W J

%;EE;:;T2) 4afifa+a+ufl

24

(1/

Using the fact that <€§U-:f’g -3 0(13 4 49> ~+ —- ‘guflxl JIM fm>

(4A.5) becomes:

 

 

 

_ __ 4nd 7(pé2 <O-()_.—O:—-—-(2/> 0);!)3’7/ Jfl/J‘fl/z
AECOUL A W) W W} 3 (I’M)? (I'M) 1.2-2'!’

 

 

 

x {5 “ ‘_ +___ ' - I
.- / r I
Elltl (EL .5! I+k0) E" El (E) I- E""K0 I I

E. El’ (E2 +E,'+ 1(0)

 

 

 

 

WM

- I I
x [ r “t ‘ " '
EIE2(EI+EL'L(°) EIE1(EL-EI+L(0) EIE1(E1-Ele°> EIEL(EI+E1+K"J Y

 

 

 

 

 

 

(4A.12)
Now
’ I
:3 I
:2’_ E! /+ KO (EL/+Mz> -(E,/-—I1M./)"’2':/M“l 62'4”; + W’L)
) .. I
ELI—El, ~Mb 2M1’ 4- 3(74/2)
I J _,_Z/:/
E.+E;-Ko '- (E.~’m.)+(E.'4"J + é/aal 43’ + X1

The most singular term in (4A.12) (the term involving
(El + E2 - KO)‘1) has dimension [momentum]O at small mo-
Inentum and hence(12) will not contribute to order qslnq'l.

Thus, to this order, the Coulomb recoil is zero.

 

B. One Photon Exchange

1 contribu-

The simplest interaction to yield an dslnd-
tion is the exchange of a single photon. We separate the in-
stantaneous Coulomb interaction (which accounts for the major
part of the binding) from the covariant single photon ex-
change, and then utilize (2.21) from perturbation theory to

approximate the remaining time dependent contribution.

The single photon exchange kernel is (see Appendix A):

 

.. , fl?” m a, .
IMxyz-zllvx/(ozw7‘flW/FW; )1 + {lg/mea/jyf (413.1)

flmu.fl.fly

where ku is the four momentum of the exchanged photon. Sub—
tracting out the instantaneous Coulomb interaction,

i(4no)yk(l) y4(2)/{k|2, we obtain:

I?“ 1) =-2'47ro<0? x”) M. ygnyyD/{ji 11/sz
/ IX“ (17:)4 g; F; 4 4 )2!) "25.“,

(413.2)

‘+ Ed? y“v_fl2y

2””; % fl),

 

Thus we substitute this kernel into

AE, : 4/;(41141' fit/(X) INM/X’) 30;: (at) (413.3)

 

U}
Where garb“ is the first iterated Coulomb wave function.
The kernel (4B.2) very naturally separates itself into a

”Charge" part IC(x,x') and a ”dipole” part Id(x,x'). We

26

compute each part separately for computational ease.

1. Charge Component
For the charge component it is sufficient to set

Fl(k2) = F1(O) = 1(13) since the onslnoa‘1 contribution arises

 

 

from low momentum recoil terms. Indeed for any term to give
a logarithmic contribution, we can take k<<m1(14). Equa-
tion (4B.3) in momentum space is
d4 (”C / U)
”W ——-.-*/§0 77’11 (ff)? (209 (413.4)
M2 J
m it {fit 1;) I?) _
where “f (p): (figé 9093f) ; faint) given by (3.10)
and
IIvM: i(47( “) flux M[ “I. a(1/) /& E —P/

Thus we can write

6 4 / t i €~?' at +1. alt, I (H
.455 :r-4n'a .//{f;:fgt{ é( ’0 e. f $nglfl 2? X?

 

TY) do the p0 and p6 integrations we invoke the following

fO‘rmulas (based on the outgoing wave prescription):

__,_.——— ___,."————~_-_

 

27

00 -ifot -. L t:
/#0 £‘_______.. : — ’72’2. ____,___—————‘e l LVN / (4B. 58)
-—w %% ' lk/l ’2’

0° 10026-1'f0t ' ..z/l‘lt/

1% wj—IfQTO/‘W—Tl/L/g E (4B.5b)
-00 fa ~ /é/

and quickly obtain

v4” 344””? Hm/ It/

AE, -4.C M W“
(Fl/NJ 27/

(27V)

1» 2 ”Alt!

«XE/’15“; %_ @9464} f) (4B.6)
It is worth noting that want) ==- jofth) 25‘1””) (413.7)

which can be seen by examination of the steps leading up to
(3.5). We also have ¢1(p,t) = oé + do + do' where both do
and do' are of order p2 compared to ¢é. Since they are of
higher order in momentum they won't make a logarithmic con—
tribution, and hence we need examine only oéKmé

We first examine the part of (4B.6) that contains

 

28

w ; iii/KEN?” __ 7t 0‘ MW
(At: / (fir-[74 fifz+x72(flz+rzjz M {27(3) {/M M)

4w ,. H— .77- Li?»
flit/fl”; f.” {"74 (92277347147 (92)]

._€ Ké/[/:’,Z:M+ 740/7; ”HZJ/fi? “(I/+7110) _ {#4sz

-2‘ MM

x -—————-———~ (4B.8)

11/

We have eliminated all terms linear in e(t) as they would
lead to an integrand that is an odd function of time. The
time integration can then be put in the following form
(using the fact that €2(t) = 1 except for a point discon-

tinuity at t = O):

ffl/Méf31WLff2/‘W— 277/5?” 1“?)

+ Mmz+EE {Lilia/+1 [07+ +<___+:: _ (ft/Ina I In

 

)k/ (4B.9)

 

 

29

We note that we encounter integrals of the form

fmdt e'ialtle'iklt‘e'ibltl which, since the integrand is an

even function of time, can be written as
medt e'ialtle‘ikltle'ibltl = -Zi(a+k+b)"l.

Hence we can draw up Table 1.

Thus (4B.9) will yield

 

-Z{ 1

 

(mi—EH) + . 1 (42.224414224242411
[H EE’ k+a+5fl ZEE' k+a+a941
(43.10)
‘ E ‘ El” 2. ’ ‘
+_.fl——I(_LJ_ZQZ£.~::3 + <[é9227
fie/EL k+E)+‘E;/+ l“

We then expand the denominators according to
I

__.i___ 2 4._4/4 - _i€11?4__\ (43.11)
h+El4€7 k \ £+E¢+&’ ’

so that equation (48.11) can be written as

 

 

 

'1: 22.22124 1 7244247 _ 72444444442212.1221
(E152, E15,, E,E,’(‘L+E)+E,7

 

QEJ%7(L+E,+EJ*AU

73“Mu ’7 -m4 B12
.. W .1 (,m) (4 >
2E1E1’(¢+EI+ELI+M)

Referring to Table 2 we see that the spin averages (see

Section 4C and Appendix B for the details of their computa-
tion) are

(4pM)‘1(® + Q + Z?) (4B.13)

 

30

Table I. Time Integrations for One Photon Exchange.

 

 

-2‘ £1th

 

 

71 4:24;"- 11192

: _— ZT:(I¢+6.'15/)

'00 (J (, ~II<IU
[M1 1 I ”173...... o

 

 

 

 

 

 

fl/ 4 f: f2" 8-21M . / I
I: + = 1 ’
Te 11 <L+EI+E2,"M + E+EI+EII+M)
’0 < (I) (2/ '2 NH '
fit/[MI {ad-ff w: ;( / __ /
.. (7 ’4 14%,.»ch £2 5121444)

 

 

 

 

 

 

31

Thus we can write (4B.8) as

0/‘70(O____)_]1;”/ 4M(§S+D+27?)
AE :- ..L’j
( J: M “(z—7:7 (741:7 447/:21 lag/[IL

(43.14)

X fM—BJZ) m BKACESE

The only terms that will yield a logarithmic dependence
are those terms with an integrand of dimension [momentum]—2

at small momentum and with a coupling between the p and p'
integrations(12). Inspection will show that the
I).
E , z L’+ 1 . . .
f {’hm/+ f k, m] term Will yield a potential loga-
25a’fl+&+g_q)
rithmic contribution. Since the integrand of this part has

dimension [momentum]_3 at small momentum, we expand in powers

of momentum to extract the [momentum]'2 behavior

Z..— : 172___ 1 — Ez/+E,’~M
k+E.+E;’—M bra—EX L+E,+e/_M
4: i (1_ L
k

gut (4B.15)
SO that

1::451:32&:EC£E13:232§;__Jgf/_L___E:_1_Ll (4B 16)
251 EL/(k+EI+52/_M) Ell k 2 kill?” fill)

11 2

 

Wlth the potential logarithmic contribution underlined. We

are thus left with

 

Table II.

Spin averages

32

for one photon exchange.

 

 

spin average

Coefficient of <§flicuj>

 

 

1H

.—
—_

/ (If 7);" .7)/ /// /
\fl a a .g 5 'Z>

:ra<’w[ (I/

/2)
<01 74 a

/l/
<42

“1w KE> T Hf“ 7112/7
flan/gig: {VI/2 {(23) ——+— éfllfl‘qP/f‘ fill/17487]

(27

d" f7a’ “max

 

 

 

 

 

 

     

 

(E5 <“n70‘m1i‘z’3fl-jfl’> a - 33— 77'7-
:_<%M fl “(1a(2/fl/2)fl> > gf'fl/
<%Qfl a/ Ixavxqgj>
<2ngae7?amkaw >; —:g 1

1” fl) 2 / ,2
QM IN £4Ufl> t ‘élwz/

m; 2] W / {2 ‘ X I L
<7 £42341? If 77> fl" * é‘lflxz/
<11? 453$ 31,4 3 74> muff é)! ‘ ,1
/ II} ,a/ M/l/é (27 / ‘ [ X I};
\\4 f“? 17» ~,g 2, %> _; 1?,fl
/ 0/ W / / ,2
14,424 é fiffl” t 25441221

 

 

9‘ '§‘(f‘f"f’7

 

W ——>- 2477:7171

 

 

 

 

 

 

 

33

—:z____‘a/WOJ/ 5. WW y x; [51 .___£_/1
44/47:) [3/ M< >24” WW(7”+U} 2-44 “H

(413.17)

)1’ {L2 «1» “M + (I Hz)
E,’ 249-447 euEIM'z’P

Using Table III (see Appendix C for a derivation of the

integrals) we obtain:

'7; 7c M 3 / WOJ/L <0‘”’- 0"”) A a " (4B.18)
3 //u-A4
We now return to that part of (4B.6) that has no u

matrices in the kernel. The only possible spin-spin contribu-

tion comes from‘fl so that we have

(1157. : W 4212/41.):
“777 WWJ/I: +44] M [217) ?

 

 

(43.19)

m
X/fl [Ht/74 é/tJG/t/ffi’étH aim @f/y/g/ 2 i/é/lé/
7g /£/ ”j?

LU
(413.19)

The integral over the 6 function is trivial, while the inte-
gral over the other term is identical to that done above.

Inspection of (4B.12) shows that only the

”mm: 441 3 . . .
———— + ——4— portion Will be Singular enough to
Euil aew

yield a logarithmic contribution, so that we have:

 

34

 

 

Table III. 0161mm“1 momentum integrals for one photon exchange.
fwdgpéli: [0110,15 4(4 £4“;
‘+3‘/‘ 12 +11) 142 121 M”
015W 121212191L : li‘flutX-z
(P ‘21") 12’123‘12 2’1””
0190(1) 10 44' V16) 17:4 ‘1
2 _ Ad
11221‘11‘11‘1112H‘E21 222‘
0‘19} AF}
I 1.12 ('4’) 175* -1
,1 -——. £401
S1231?“ 112 +1“? 112 2’1”“
{Pd-51” 17"],122179’1‘ M" -1
4 *—1«0<
“SH—K‘j ( ll M1

 

 

 

 

 

 

 

 

               

 

 

 

35

 

 

(AE4) : 474244 41‘ 4(‘2 4=«‘/2/‘221/‘§_
1272/4 f‘W {4“4 12y M‘Kznfi

\
” ZZZ; 222‘ .2 2 _ 2
X6 5:52, +515, +0Hy 2'/!5/‘ 71.4%]

which vanishes identically.

(4B.20)

 

Hence the total contribution of the charge component of

the one photon exchange is

NJ: 3; 4 4131220412<ww22> 2:5" (413.21)
I 3 //a.fi4

 

2. Linear Dipole Component
We now turn to the linear dipole component of (4B.2).

We have

fir _4,,41 «2
4/42/2 fi. 221 l (m1 1% 12/ (413.22)

where I(MEX,X’/~14rmc/fi 744/1:ng Z/ LZZ/XH WA) (4B.23)

24w /" 5”

Equation (4B.22) in momentum space is
4 14 ti;-
24/442] VK/1I17;/2’1K'//4 (43.24)

If a t {/l ”I

Where jo(gp1=fl§%,aeft4(/1QU ,' fiflél/fl given by (3.10)

 

36

and kU E (pH-pi)
Thus

<1 //1 -5045 x 121 (21 fl 4 2‘ a,
12 .4. 2 212221244441 2 15,22,112: 4124,14, 42

(27:1 127:)
(48.25)
We can do the p0 and p5 integrations by first expanding
/ (l/ I 2 l/ {2] 2
117% = 1: 221 2,2211 2 242 121-2121] (22222

#222
and then invoking the following formulas (based on the outgoing

wave prescription):

L/:%H ,y K/lwe If
1112-1121 2111

z ‘2/°t ~2L/f/ 64(é1-r-AJA/t/
[17/740002 “Ag )=+K7€2/£ - / (4B.28)
-.. 11212211121 12 £12121

6&1 (4B.27)

. ..z' é/f/ -4 (1.2+ Alf/7w)
—— K17 (£3 2—.e

 

 

For F2(k2) we have written KAz/(A2+k2) where K is the proton
anomalous magnetic moment(131. This particular form factor
is quite suitable for low momentum processes, which is what

‘1 contribution is. It will be seen that the exact

the delnd
functional form of F2(k2) is unimportant, and indeed, the
results to our order would be unchanged if we had used a
point proton (i.e. let A2 + 00). This is not surprising since

1 contributions come from low momentum recoil

again all delnd-
effects.

Thus we can write (4B.36) as

 

37

d 01 /z‘ w
(AEljzzlztafii; mflfizfl ( t)??? Hijf 1(an [~”1é_ %({%_§11311203$]K

, {NH —-2' (121+ A1174
x 23—— ~ 2
( It (1212219 V‘)

12/ a JUt’ -5 [21+ {At/I 1 I
_KTL—ix‘fl/%2g)_€ _£ ( A)/ ew‘ffiDzfl/{J

 

 

(43.29)

We now do the trivial t' integration and insert the wave

function (3.10) recalling that Lfl'gg/t): 1%"1T@t)x"‘y"’
and 3*(t1 = '€(t). Again we keep only even powers of e(t),

as e(t) is itself an odd function of time, and the rest
of the time integrand is an even function of time. As was
the case with the charge component we separate out the

part from the remainder terms (involving do or 5¢'). First

the 9:” K (fem part:

d 31% ___K____-n:[ 7:01 [80(01013/2’10113 4 M12
(AE') : ”7’1 (Zrcf W afl+r>ji<rn+xvl (/

 

-2111 15211 11 2221112211121 — 112221]?
22121 -z'(1*+/v)"*121
X k _. £2 (1‘2Aflz 1 (43.30)
+«12%)(1—U111W11’Y‘W2‘”11+——a€1(-é;’f1>
x flotflgflfmver (12.2)][13134 151—112-2127.?

H ’m ”2 W 222212111
— 11221”+2112<1H11112122 21 +1 l

. 1+ sz
X [6-5121121“ 3-1(12 A) ] E

(4B.30)

39

The spin average for the first term can be written (see

Table II)

(SM/M1 1111 1222/ 221121W1 2221122221 2122 /22211;2+i1

(4B. 31)

while the spin averages for the second term can be written

Mum'lm - <2) (413.32)

The momentum integrations in (4B.30) thus become

fé’ffid} 1+1) (fl/21%,) [22m /2/ 11222.,371 1‘?) Xm/M/12w/j (413,33)

 

dyriEy

A4 Q ~
and /1/"’+1“1’1r”+1‘1‘[4/ 1 $1] (413.34)
Next we turn to the time integrations. The first time inte-

gration in (4B.30) has been done previously (see (4B.12))

and can be written as

 

 

2.{./m'/m1 + 471,1 + +111, .— 491(E1'1‘m») ‘f' f’2(E‘+/VH‘,Z
.— I ——’-- __——————-— f I» / .
5,551} 5.5/1; E111: +551 25,5512 (2+5. +52 “ M1

 

26.5; Ezaed—EfiEj—rM

jo‘(E2"sz+/II(E"WJ) _‘_ (1 H2); - OE <—> (L14— A117,);

(4B.35)

The second time integration in (48.30) is

“21 {{fl— (__¢___ _1_// _ £|+M,)<El'+mu1 + (E‘-/m.1<E,’_2mj

1222.283 ’5 E2 25.5;(2+E,+E;—M) 2E,E,’(le+E,2E,’+M)

I

 

 

42221} — 122—2122217}

(4B.36)

 

40

Combining (4B.33) and (4B.35) and taking the spin average

terms one at a time we have for the (S-T) term,

Y/mti 01201? 0‘“ 0'12) {...—— m m. /m z
3 L117”??? WW“) 1‘11 ’11 1< >1 é‘E E. Pae/

+123 +79 5 21011EL1+M1)+1¢11(E,+/m,)
+LEE,1}2,+E+E’)— 2E.E;’Lz([¢+ E,+-E"-M)

 

 

 

 

 

yer-mnpwa—m ... _ .... rm" §
zeagwuefiggw) +0 2)!) (ta k H

(48.37)
Only the underlined terms are of the proper dimension to
give a logarithmic contribution. Hence the relevant inte-

gral in (4B.37) is:

 

 

 

 

61%? 12x17? mm M.
111101+510111011+le E,E,'H£_£ill + E’E'r)fl_fllll T 11 2)
(4B.38)
MI 4472, 471.1 a )2
1 I 1‘ I — I ‘* 6+2
E" 1111311 +A1 E;E.1]p—£’/1+ A‘)

Since A is of the order of m2, the A dependent terms will not

1 since they are not singular

coritribute to order delno‘
enough to give a logarithmic contribution. The remaining
terms can be integrated with the help of Table III and we
obtain for the J-j term:

w Ne/ ‘“’<o~ WM“

3 M’ (413.39)

 

41

The next term of (4B.33) and (4B.35) is the y+§termz

gm‘. 013F1131 2’1 . I, ,1 n I I 2
——-..:—5— piazza.” w m n w a: xmkww

 

x {—1111 + —-————/”“'l + #547“? ,_ 1»‘(Ea’+m)+r”<e+w
5,1351? E,E,' k’ 5,5/ V( we, HEX) 21:5; {81L + E, + E.’ -—M)

 

— fL(E11‘-WL) + 7/‘(5‘ v/m,) +1}e—92)§— (E 6—)- 1L1+A91Aj§

25.13160. +E':,+E,’+M)

(4B.40)
Again the terms singular enough to contribute to order
ocelnoc'1 are underlined, and as before the A dependent terms
will not contribute as they are not singular enough. The

relevant integral is

 

 

[dioxin/[Plffi- (rial/’— @"2+7711€1fl)]/MM. + M.‘ + 082))

 

(P;+Yi)z(pqz+ 5(le 11-«311; EIE11 E,E.1
(4B.4l)
Using Table III we obtain for they”; term:
K1%#WWZWM¢X%K4(M,
3’W7’ ””51 (4B.42)

Finally the’X+ufterm in (4B.33) and (4B.35):

42

32”";flMZ. 0.0). $111 91;” A3P1 ' v / 2. ”in/”11 M1,;
3 < >é (f’+3“7‘1r"+r*)1111 111 + —_

 

 

E, E; 18 E. E,’ 18

+ 4231+ fag, _ fl(5&f,.m,7)+fll(g‘+xm,) (43.43)
E‘EI'E11‘Q+E‘+E,’) ZE.E1’[g"1{a+E,+EL’—M)

MW’WVIWW

 

- 111531-0111 111115-4111 + 0911}- 11 H (Lu/‘91)}

25511111 (11+ E, + E" + M)

(4B.43)
The only term singular enough to contribute to our order is

the underlined term in (4B.43). Again the A dependent terms

do not contribute to order aelnd'l. The remaining integral

is

01; (15V [\F‘1E211‘W3J4- [0"1E,+rm,) _
[1,111512111P11'1’5“); Z 2 + OH 2); (4B.44)

E|E11<b+EI +EL1- M)

 

which is identical with the one leading up to (4B.16) so
that we obtain for the x+uftermz

2. «m ‘J swarm. 0"”),2‘ zx "

_ 13.5
3 ”M (4 4)

 

In order to evaluate the remaining linear dipole con-
tribution, namely the second term in (4B.30) we combine

(4B.34) with (4B.36) to obtain:

43

—/lo:'/,LM <f1’1-0‘“> 61310013P1W.P){{LM1 <__1_____1_j

3 W2+191VW+EH+y

 

 

 

1E+MJ1Ef+rmJ + (E‘~W.)(E‘z1—rm3)

.. 2E, E;1(k+EI+El1-M) ZE'E111krl-E, +EtQM) '- 0"“sz <L H1LI+A) 7);

 

 

 

 

 

(4B.46)
Only the underlined terms in (4B.46) are singular enough to
contribute to our order. The first term can be written:
,m 9.1-5, 1 2 E’eE. l ~.__ 5,45.
I I I ”47’ ' l
5.5' 1Q+Er+E’ C,E,' k h+EI+E1
E’-—E ’2 ’
= xm, _I__I__ .. m f -/ (413.47)
EEI1E’ E1k*EI+EII)
3.1- 1 _ m‘(k1l_F)
Z LE‘E'1 121E: +671)

Both terms are of dimension [momentum]+1 so they will not
give a singular enough integrand in (4B.46) to yield a log-
arithmic contribution. The A dependent term in the form of

(4B.47) is even less singular. The second term in (4B.46)

can be written:

 

44

 

 

 

 

 

(E; 29-4911) (E. 4")“,2 : - (E, +MJ(E +0M’+) P‘2(E|+Ml) + P2 (Egg/’71,)
ZEIE1'<k+E‘+Ez'“M)2-,ZLEE’ ZkE.E,' (L+E,+El'-M/
(4B.48)
Q4, _ (E.;”‘:I}:zE/+7MJ + P0207“ + _E1 20";
. . 2 185. 5,} 2435 5/

 

In (4B.48) the underlined terms represent the potential log-
arithmic contribution from the A independent terms, while the

only A dependent term that could possibly contribute is

feta???” LP'L- F1) {(EWM')<E’/+m‘) __ 0H2) (413.49)

(F1+xx)7<rn+x))l ZE|E;I(kL+Ag)'/L

 

 

Again using the fact that A 2 m2, and keeping only the lead-

ing terms in the numerator, we could write (4B.49) as:

 

 

3 I
dpd )0 (,flm “J{ E: F’ _ ff (413.50)
(FM‘HF M?) A 5'5, EVE;

Letting pa» p' in the last two terms we obtain:

fl ' / )2—
4m ,m. ”(3" ”PF f F __ F‘ (413.51)
(F2+b")2(P"+K’)l/\ Z E,’El 5,5;

 

 

 

the leading term of which is

4/m M 04 35W F"(W*'W4<P"V7 (413.52)
(101124”) PM” 25.E,’E..E.’

 

 

 

45

which has an integrand of dimension [momentum]0 at low mo-

mentum and hence won't yield a logarithmic contribution.
Returning to the A independent terms of (4B.48) that are

underlined we find the potential logarithmic contribution

of (4B.46) to be:

 

Maw ~ )0) (_ m -05.);
1'») (Pn4‘XJ2 Ergzib} EIEL‘L}

which can be written:

d3 413’ ’l 1
0. PP M.<_L_P)_wp_‘_( L)
1+ 11 )2 17’ E|'Ez 5.5"
(F X] (p 1+V) E'E E 5; (4B. 54)

The leading behavior of the expression in braces is

 

 

 

Mz<mm[P”~/v’+ £24.... +'])

E ElE E;

(43.55)

t4__ 4 \

(Amwu[P"Fl+ f%5??'+"']) >

“~nM' .
E‘s—l 5351’ J

The only terms in (4B.55) yielding an integrand of proper

 

dimension (at small momentum) are the lowest powers of mo-

mentum. Thus (4B.54) is to our order:

 

46

 

Qf OPP ”PF, 4’". W2. (Qt-m,)(p”— pl)
07

+X‘X)1<P|1+xujl E‘El £315.},
(4B.56)

: c. ”71/771; (M1 -m/ Jammy, (’0’+ F):
(ftka‘jYp'la'a‘fiEfzefla’

Using the symmetry of p' and p the integral in (4B.56) can

be written:

3 ’ .1
5W A? P (413.57)

2
(PW A”); (P"+W)‘ E, Ez EfElr

1

and (4B.57) contains no lna' dependence whatsoever. This

can be seen by writing:

/p”'+EX1-'J“)‘\
fuming. {/{F’jatfla j/

 

 

The p' integration yields a number independent of a, while

-/

the p integration can be approximated as nnf'JuPP<P‘+XU‘zE.

 

wig? WWW—W] 2,. + 22L. m-|[§m.‘—XL)V’]

”14,1 - x1)” Y 4443'- r; (“‘1' x33!) 3‘

which has no logarithmic dependence whatsoever.

 

47

Thus we can finally write (4B.30) as equal to (see
(4B.39), (4B.42) and (4B.45)):
d M’WONKWW W>fl 0( ‘2' -
K M l

+

l ' .L.Zfll 4B.58)
—.- ...—3..)

I

M.
/A

Thus the total oc61n0f1 contribution from the exchange of

a single photon is (see (4B.21 and (4B.58))

(43.59)

 

AE! I7<§% + K743”); *5 “5i ”:4

WNW W44” (Tm>1¢ oz'

h 2
w ere‘? __ M

C. The Ladder Diagram

The ”ladder diagram" contribution to the energy shift
comes from the second order perturbation formula of the one

photon exchange. Specifically we have(12):

u}
~2i fd‘kd?’ (199(0) I. (7nd) “PW (1') won
where IK(x,x') is derived in Appendix A and is:

(471“): d1 ATL’fl ”to: ifil’l’ 0, a) A: w 12/ Rafixm 3%“
(27:)? j 7"2 “g nnG—[fd‘fl ”A”!

,L 0/ (z) ,'
xéj’x-x’) { Who/("21‘ - 2a).. Rd) BA“ 031’ Lt" (4C. 2)

He!

‘5

 

109%): ~

K
20‘";

Z/m2

4 /
Where Qk(x4[’/ = {—j’é‘é { F (”X y )S(l)(f+ 7%): ("P + 7' K)

Inserting (4C.2) into (4C.l), the integration proceeds as

 

48

follows: do the x integration obtaining (2w)“6(-k-p'); do
the p' integration using the 6 function; do the x' integra-
tion obtaining (2W)”6(-k'-k-p)6(ké+k0). Then change var-
iables: k' + -k' and we obtain:

The charge component of AEL

AEC_ ELM) d} 014mg JQ-L 72/0/22 j)

 

 

 

 

 

 

 

" .. <sz: ,2 fi” (”0)
m
m m £;__.Wq(ug E+ 1(Ejz E K'X“ W A?-d 0’”) ¥%¢@g}
”{V‘MW .. )3 (— 12,) b?) AGE; g9) '67—
(4C.3)
the linear dipole component of AEL:
AEdr— 22(4nfi’ fiféd‘é’ a/Zélréu?) J&M_£Q LIP o)
L (flKj‘, £2— £1+ P(
x{r"-’X;”(li—15rl{ozj”o()§;”ewg «)3: (My BWM’» 0% @5319
24m (271)?“
FJL7YMa'LZJEU (I . (l) w (2) I U (z)
_. 2:141” ggr.7.«)sp(w,x)m/£W W1" 3/
H5] 2 rc) *
( 4C.4)

the quadratic dipole component of AEL:

49

1312:”: 2i (47:1 if”

(inf j if (fie?

               

 

 

 

x‘fEQL/ij ”(3) ”31% MM W)‘§F)Uc+”z K) Fa/Mmr’ (21k Pkg} (a i

2/71'112/"1; (271' )3/t

(4c.5)

1. Charge Component
We turn out attention first to the charge component
given by (4C.3). The k6 integration is trivial using
6(kO-kb). Inserting the first iterated static wave function

(3.11) we have:

Wm fL/t/
L. 7t} M (27y4 Ail/all (IQ-hf X2):

202 M (u )U} (2) m (1) 2‘ (ll (2)
X<(\r;z*“c'“é/§F““’Z'“)SF “MW (117‘ “9)“

 

 

 

v l W) + F <12) .26 7Q) - WM"? - WM"? MU)

(4C.6)
The product of propagators can be expanded as a series of

projection operators times yiyi

SF(1)SF(Z) = [ ]y:yi (see (3.4)). Thus the expression in

the angular brackets can be written as the sum of four terms:

SO

 

 

L4. Hm} —- ,g d“’a<"’{w ...-i} 721d“) “’[]{ L}

(4C.7)

t ”(Skim [ 14;)d;){w. {E

where [ ] is the product of projection operators associated

with sF(1)sF(2) (see (3.4)) and {w.f.} is
[W + Fez/Me ~ fcmw — GWWJQ 25%;]

When we insert the product of projection Operators,
(3.4) into (4C.7) and anti-commute the 8(1) (i=1,2) toward

the left we will get different results depending on the d

matrices multiplying [ ] on the left. Specifically we can

write:
For d(l)o(2) on the left:
I ()
[(Ep/VM'NEa-mfifl -<E;"rmi Q‘Uk + (5‘ —M,)Q( lye

[]';4EWEE )i (,/&.+fimu—afléyr7zh'5)

—[%5%&
(El ‘W')(VEL+/Mlj "‘ (E2 *W2)g”‘j,é _ (E; 'm’h)4021§ ‘f‘ [@972 £032
(-4Qc + 7' K0 ’E/)<220 + {x “0 +51)

(E.w.)(a—W+<ez Maw Mawwerflé + as“! 219.1.

(“A + 7,10 “afljo r q, we .. a)

 

 

(E,+MJ(E,+ m1)+(El+/m,7)£(“3k — (E.+/m,)£0).g — gwk @5ng
(~30 + 4z.l(o+E.)(-£., + 7; 14— 5,)

+

(4C.8)

51

and for no a matrices on the left:
H: *' {<e.+m.)(e.+m.) -— (E.+m)2<"-Us + (Emmazg — 20% 2M
45.045114 (.430 + 7.1%“ 5,) ( i, + 41; Mo — Ea)

 

(El+/m,)(61 - xml) ~ (E; -mzjd(’2£ .... (E. +4305“! + 26% fi%

(”L + 7.1a wax}. + 1m + E.)

+

 

E.‘M.)<Ez+/mz +(Ez+fm1),0.(”‘lk +<E)'m"ljf\(r(Z/;é 1" Ext, XII/é

éfi+¢gl¢+ejfi£ +7glt/o—EJ

 

 

(Ei’m)(EZ-/’fib) + (Ez'ijgqlfé "(EF'MJéaié “,go‘ié gojé}
(‘zac +41: KO+E,) k/Qo + (IMO‘i‘Ez)

(4C.9)
The spin-spin interaction portion of AB: is extracted
in the following manner: Repeated use of the relation

a = §.. + igi'kgk is made to reduce a product of a

.Q.
l J 13 J
matrices to a 0 matrix. Then use is made of the relation

1
éCl). A 0(2). > + _ A.B /O(l).g(2)>
3 \

(see Appendix B) to separate the momentum vectors from the
spin variables. It is important to note that in order to
yield a spin-spin contribution, and given product of a
matrices must have an evee number of a matrices for particle
(l) and an exee number of a matrices for particle (2). An
odd number of a matrices will not result in a <é(1)'0(zt>

term. An example of this method follows:

52

Table IV.

Spin averages for two photon ladder diagram.

 

 

Spin average

Coefficient of <W".(7-(l)>

 

A

\H

<ghlga)%nz gm>
(WWW “(22¢ flWMa/a’néf >
<a//') “(ya/fl/éda’é 17(1/ (Xa/(X/UZA [241/ ¥>

<a/M'0fll/flmfé flag “(z/angé/>

m

l l

H

E <K/1.)X(2/0<Mamfl/mé[fit/mf>

E <d/léd/9'Kfljé “(l/é am “11/

L: @rllfiMA/Ué A/Mdp’lflmé /A/Md fen} qfl>
M <a~<o<~w1m2 a/’?a(Va/‘?Lé’a «23>
N : (Oi/H- owl/“(2% “(C/éfipfk WWW (I/ ,fl>
O

P

B
C
)5
p
g)

3 <9( n_/ 0(11/ gag If 050.1 g ”(x/{da/Mzzfééémfl>

= m: WW m w a’fiW

”Wei 9245“? 7
E <0(M‘ am a W; K/z/ffl>

AWL 0W”. «w 45> : (1H2)
@%£%WW$
@WM 96”.» : Us”)
'3 (a/”.p(a/6”’!£/¢V$> : (14—72)
Qgig 0("401‘1’4‘144 />
<a‘"<d“/e<"£é M >

N

”1

m

H!

 

&
.26
¢
fl
[9
2"
L7

 

"'3 Hkiflré “0421’
Z V (be)
mm

mm

-2 I.
g?
2:

'4‘

3

l
w
s“-

ah

WW WHO

 

 

 

 

53

Table IV (continued)

 

 

Spin average

Coefficient of <70). UJZ)>

 

 

2,7 2: a”. 40/4014 gflé’> : am) -§- #1:
,g/ 5 Maw/4% 0% ww > -% W‘
“(a WWW/w x‘” a“? 72>: (162) —§— Mal)
: '5 <o<"?a“*a“ié Me w W) “:— V Us ‘49
01/? <fl"-’d”’%”’é&’“é 4‘72 2%.”? > =0”? a: L‘Lk’ad
t/E (New ‘éa’fla’k > 062/ +3; 1(er
0/ E <2C”£95“,é WWWW”) 062/ ~§ Llp‘
m 2 (05% as“: 4% 95'3”} a Low:
”f" z <01”; 3’12”? 915’" > :GHZ) -g‘ lim)‘
“5” 2 @511 95% 95”: 59> =0”? 0
Z“; <z”’ié w &('/.é/€<.’2¥> ””21 "% 1W"
to (M M M 26% O
y (M 01% 5% 25%? O

 

 

 

 

 

54

(1 HIV %(fl “fl/(v (U
<2X ’gg “B:>i '<:AIB” m1 ”h’afiu
: ‘ .. ' (2)
Z. 3 < . ‘ ,. (I) (2) .— ' I
A184“ 62:); €2ms 0:, 03 >“ <A2-Bm. (dwaS—[sagm)q H 0;.(2)>
: A c B a—Il)‘ (2)> __ _l_ A o B 0-0). 2) _ 2 (I), 2)
,~ A,<: 0‘ 3 {g ~<< 0“;> ”‘7? £.j§<fir 04:)
Tabulation of all spin averages for the ladder diagram
occurs in Table IV.

We now extract the k0 integration from (4C.6) and apply

it to (4C.7):

fdé< m» l-

 

Mm I;

       

 

(’J JJJU/[w‘f w}

«Loyal? )dj(’@(2/fW-£} (4C. 7a)

Since both [ ] and {w.f.} contain terms independent of a,
dependent on d(1), or on a(2), or on d(11d(2), one can ex-
tract the spin-spin contribution (remembering to keep an

even number of a matrices for each particle) by inspection.

 

,QHHJHE [-2]: 1%)” term:
Clearly the only spin average to contribute to the

spin—spin interaction is <a~(”./£ 4032 95’qu {32> E T

 

55

Thus the above term contributes

 

 

~I LN) III)

I;+ +-I+' %CEI}T (4c.10)
4eMeWJI WXIEI ,
where IIEI: jojko £0 E I

:3 L2. 114M.- 1H) [L4 (7! K. :E, :2e)]IL (”Mat EIEZIg 11)

All such k0 integrations for the ladder diagram are tabulated
in Table V.
In like manner the -Q01Ue] [ JMIII awful A} term will

contribute

I I “ZII <5 w x5. Wqém + (E +"" )§2)B *5 WW3)? 355%? I

 

 

4E5
W
+ {-(e.+m«,Xs.~m)4M + ewe/z - (E..+rm.)rm73 + @2211.
+ {-(e,-m,ye.+m)é,¢p)a — (ammmy +Le,-~».)x:m3 + we}? I.
+ {"(E,‘M"¢>(Ez‘/7"2)C7{¥)a - (Er/WJELEJZ’ - (EnM/Fmfi owefif. A?
(4c.12)
Similarly the -fio’/k’[ 24 II Wqufw. ()2 term will contribute:

..M”
4E E2 {{.. (E «WXEL' mz)G(}?)d +<E M)f{£),8+ (E~ "M'JFQJQ‘ DIIEIZZII;

 

+{.(e,_m.) swam + (Ezwgfww -<E,-w,);:ma + mg} L”-

+ {— (E,+m,XEl-m1)Q(£)a — (E; —/m9E(g)/8 +(E, +442) @798 + Q33} [A

+ Irma...) em —- (El-WW - Gamma — D40 2 g z -

(4c.13)

 

56

, (2/ “I (2}
And the 06)“; [3a]; 9; {ML}? term contributes:

 

 

‘ _ ,
43 E1 {{(E‘-MXE1-W’)D/B/A '— (E2 ”4%)pr " (E,-W.)Hg)3 + CV2“? I++

 

 

+ {LE,—m,)(E;+M,)3)Q/A - {BMW ROB +(Ec’WDE42)B -‘ SCENE I2+-

 

+flEW<Ermbw + (EL—m) Hens —- (amamg —- mach-..

 

+ {(E’+M1‘XEZ+W1)R£)A + (EL.+/W!1)F(1)B + LE.+4M)E@)B + G(£}Cg In}

 

 

(4C.l4)

Previous results(12) have shown that the only terms
which can give an delnd’1 contribution are those with an in-
]-2

tegrand of dimension [momentum in the limit of low mo-

mentum. Such terms are underlined in equations (4C.10),
(4C.12), (4C.13), and (4C.14). The term underlined twice
in (4C.l4) is of dimension [momentum]'3 in the low momentum
limit. The remaining. non—underlined, terms are not singu-
lar enough to give a logarithmic contribution and hence
will contribute to order as or higher. Thus the entire

1

oelno' contribution from the charge component of the ladder

diagram is:

57

Table V. k0 integrations for ladder diagram.

 

 

Ifn)=- :? imum) I —| l
:1 '00 £3" (15,!) 261 ’ “1."; [20" (411K. 71- 511.1%)? [/40 '(”l&“of5a $13)]

 

 

.... m y. _ mutuu'mu‘g
In 2 {wmwuy‘w

alt/wh’awu)
_ ..t
1* " 1'“ §4m.lkll\k‘(’§

. “I
1-+ ”firm [4014, “41 “all?

I-_ ~* 1’“ g I E

3...»... MM UL'MH)

 

 

 

I .7} Y E/a/m" ‘
1++ XRLC 1H1 (I£(1+347 + zlk‘ltlLll;

 

I ’_ “3' ru’ 4
* 2 5 4.. Wow

I

I -—)- It —/
_., 9“ {4M [UHLMW‘HU‘UE

I I #1! -l ' ' _|
-- " hemzmaww ’* MMMMHW

 

 

 

1+:'-+-2ngg:—- I + CM/E”ME)(LII
alk’/It(((L'I+/L/) (W (MW a“);
1+“. 9 Q‘ru‘ \T_l\

3%«3<(k%+lé0§

l_+-+2‘r([j E, g

30"} (WWW)

I: ——+ an.’ {fidf};

 

 

 

58

Table V (continued)

 

 

 

m x . Q4/4491?) (ӎ/k g
l++ "7‘47“; (k‘I‘UH‘JrX’)

1+: ’4? gang? &(Q°)§
1:: ~7- am‘{&'(fc‘9§

I" -—+ flni {mk‘)§

101)_+ flfi‘[ I g
** Mir/ML!)

(w)

11‘ —+ aur‘ffléjg

’—

(w)

I“ ~—> am gflg);
I?)—+ firm {fiKé9i

 

 

 

 

 

59

(AE5) = m’llef flax. 43/; “54-42)
M (470‘ trim"? 5,005.00

x [--(E.+m,)(El +M))G,(2)a + (52+ml)f¢)3 +(_E‘«+mJFm’B - HQ)Q]%
+ [ (E.-w.XE.—»«.)pg;A - (a.-m.)&g)B—(E.-mfig)3 + two] I++

- Li
D422 :1? + [05,- was. +xm.)1w A -(E.+Im) m3] I. ,

+ [(EWXEz—mhxem -(E,+n«.)E(.L)B:( I- .

+ [(EfifmrXEzt’W) 'D(£)A]I-_ }

(4C.15)
Since it is the low momentum region that will make the log-
arithmic contribution, we can make certain approximations

in the above expression(14).

Namely we can expand the Ei
in the numerators, and keep only the leading terms as they

occur. Thus (Bi + m-) = Zmi (i=l,2) and (E1 - mi) = kZ/Zmi.

1
We must be a bit more careful in the denominators. Here we
have the product El(k)E2(k)E1(p)E2(p), where E1(k)E2(k)
comes from the projection operator expansion of the particle
propagators, and E1(p)E2(p) comes from the wave function co-
efficients of the matrices. It can be shown (Appendix E)
that it is sufficient to set E1(p)E2(p) = mlmz in the de-

nominator so that the only explicit energy dependence left

is El(k)E2(k) in the denominator. Thus for purposes of

 

60

1

calculation to order aslno’ we can effectively set:

13(9) 2 8
12(9) 2 4mZ

F03) g ml (4C.16)
C(p) 2 2

Hence we can reduce (4C.15) to the point where we can then
use the table of integrals, Table VI.
For example, the first 11+ term inside the braces of

(4C.15) can be written:

LEE/UM <U'II).0*2~)> 7:72 (%QZF2+ 51ij + [£2] (4C.17)

where we have used only the term with dimension [momentum]-2
from 11+, since the other terms would not give a logarithmic
contribution. If we now integrate over (4C.17) using Table

VI we obtain

d3Pd3‘1[P)+ 221+sz
<P2+WZE.(UEz/L) LEV H'tfll’

 

to order delnd_1. Similar methods are applied to the other
terms in [4C.15). Only the I__ term deserves special men-
tion. I__ contains terms of dimension [momentum]'3 and

[momentum]”1 but none of dimension [momentum]_2. The term

 

 

61

Table VI. oc‘slnof1 momentum integrals for two photon exchange.

 

 

 

 

fwd» F‘ S 421%”;

 

 

 

 

 

 

 

 

 

 

 

 

(F‘J‘rW‘EKU em I2 w: W.
XL 0W UM?) : .. 1‘4 Ax”
(Fwy: Ema—m L‘lgm‘ ”‘WL
faikd? ’21 ~ 0
+5» E (ME (L)LL “9th
deT Hfifiv :: ln4flaap/
(p‘+-X‘)"E,(L)E.(L) L4\l+,gll 4",”"1
dgL ((3? “0(4)." : ,Z‘7f4 /& A/d

(WW awed) t1 gym/4 M141,

 

 

 

62

of dimension [momentum].3 contributes to order as and the term

of dimension [momentum]'l contributes to order as. but there

is no on‘slnoc'1 dependence to this term. Hence summing up all

of the terms in (4C.15) we obtain

 

s z 1% v w
(AEC)= 7t“ LSD/0)] <0‘ W>AM [3+13‘fl] (4C.l8)
L M1 [A
2. Linear Dipole Component

The linear dipole component of AEL, (4C.4), is computed

 

in the same way as the charge component, although the situa-
tion is slightly more messy. For computational convenience
we break (4C.4) up into two terms:

a. _ 2.1422): MALL/T Witt-e
(AEL/G— (2K)4 jag”

            

 

 

m. (2 rr} 3"

’IMW ‘ “w m 7
{fi% [XL/\S'fl ~M MS (ta—4x) FILLLWW [if W49)

(4C.4a)

 

2 5 Lx‘L’Ja/s’f a” .- ’
(EA L121)? fli‘fl” #J (L g) W0)

 

. (HF/‘2) U (2 I) ('I
X {- FILL}; U’W I” SFL-Lqm) Spay 7Z2“) 5“ 4(2)(T—L; -— (X 0(W M

(27d 3/2.
(4C.4b)
We deal first with (AEi)1. The matrix product

is reduced using OUV = -iyuyv + iopv, and Yukv = y4ik0 — iy4ai.ki

63

We readily obtain:

X2203? L2: baa/[fig [0901+ 1X7” 205% gag/£27 (4019)

We again expand the propagators in terms of projection oper-
ators (see (3.4)): SélLSF(Z) = [ ]yidi. Incorporating the

matrix portions of the wave functions, we can write the

matrix portion of (4C.4a) as:
<y4m44 ”(W/3” M2) (IT!) 1.2401“ £22)[ :(XM) (2)( (”[5152]—gz‘lfié/tgfl'lfln/(Vé/‘flo‘! 9])

X [222 + 222) 2“+’2 5532229 ~ €722 2‘2 3’2?

 

392122512 - 25% — ,52222'+ 2222]

    

2W +F<2M -2 +2312 26’-2+2222<’”M 2‘32}

(4C.20)
where we have anti—commuted a yi to the right through the
wave function.

We then integrate over k6 using (kO—ké) and explicitly

indicate the k0 integration so that we can write (4C.20) as:

64

V.)

 

 

H

00
fit, {:1 [ l<¢”%’+g‘@7{w.£§ -— £71 [ 1051’ an;
‘00 .~ ~

 

“‘ {€021M’ugmgazé’fw.fol ~ Wm W4 sea-1mm;

I) 1 ,l ’1 2) I
had/05. 25% ]gflodm[w.£§ + Pia/5% [lécka/K‘iégwdfig

(4C.20a)
When we insert the product of projection operators, (3.4)
into (4C.20a) and anticommute the 8(1), (i=l,2) toward the
left we will get different results depending on the a ma-
trices multiplying [ ] on the left. For the calculation at
hand we can use (4C.8) and (4C.9). As before we extract the
spin-spin contribution be retaining only those terms with
an even number of a matrices for each particle. We thus

obtain

/éOZ[]<%M’,é/ + %(2)',é7{w,¥,§ term;
£{<E+ )( TI+E+AMC"Y\I n / Z/fI/l
45,51 ‘ ‘ ”m ”7(3) k. W742) —-—~ Fgejf- F42) fl

‘1”[-(Ez-‘MJGCWT/”(E.+W.)§Q)Tl+ quf’+ ng’fgfl

1!

+ {(E1+M;)éwg)7f/+(Er’m!> €742) 7N+ Reg/Jr F9920 371-),

+

{(EL-m/Cwmw— (EI‘WJG¢)T/” E49)?” PM? {I}

Similarly the —,Z03)H_2[ ]%('.Igm[w.f.; term is

65

 

”1'2 __ m
‘4 a E; wxaw aw 4am») was WW - WEI“

+ {(E,+M4,)(El~/ml)é7(2>a ~ (E2- mfiwg ~*(E,+wjfig)$ +M)QZ 1::

+ {(5 ”“XBWGM + mem +92 Mme ma? 1;"

+ {(EI’IMJ(E1‘MI)G{¥K+CEZ‘ ”JP-(2)3 -(Ec'm’)a£)£ "DW’Q'qu—E

and the - 2,1 lb|-1[ Y9!!! eff) all}? ((60. if term

Ffl‘fl—TME +rm,() Ez+’mL)F(£)B (E +m;)D¢Q)€ TGE +W)C—,Q2)7Z- Rpffifl:

4E E1
,V‘WMfl/WM/

 

+ {(Efi/mJflEl-mt) F6213, _ (Eivfl’tszQJEg " (El+m) QLEYIL + FQfifg I+
+ ZRE‘v/mMErf/mi) mag, + (El—Hm.) 1142)}? + (5,441,) GAB/71 * Swift;
+ {(5. 'MXEz-WJFLEUQINEUM‘JDGUE’ " (57“) 67¢”: ‘ E@)152I':2>

and the - gwog<fy[]guiflél [an fig term

    

(EI‘WI)<El-MIJE¢Q)I8/ — (E2 ‘MJGIW-g I+ (EK’MJDWQI_ FQ/W/;I++
W
+' {(Et ' W‘)(Ez+”"¥) t@)9/ — (81+ flJGQ/fl I — (E"W')DC£)Q 1+ Fm)W/J71+'

+ [ware-Mme +<Emaqma+Lamawzu rmwf I,+

 

+ [(E'W‘XEfiWJF—(w/g’fifi we.) Cage) y/ —- (Emmy)? — Hoof/371E

66

and the —- 930% Qfa)[ ] 9522710. Lg term

_ l {{f' WMa-mfimb’ ~— (gummy? +02%) €7de ’— @J/ 371“

4E.Ez

 

 

+ {(3. rm.)[EL+mL)FLB) ,GL (E;+rm;)])(g)j’ —-(E, —rm,)égz)é/l+ EQJW’gI+_

 

+ {Er/macro?» Fm 8‘ +<Ez , MW); ’+ (a W W3 2.. ELM/J21»,

+ {(E«+M')(Ez+m)mz)8’+(EL-+ WW2)? — (awmwt TQM/fr"?

and the koglc’gq ]g1-~2 {w.f.} term

i

43E; {{(Er/m.)(Ez-mz)])®A -— (Ea—MJFGy B +(E‘_W)E(¥)B ‘QaZ)C§I++/

 

+£(ENWMELWJDée/A - (E;+W2)Fm)E-(E.-MJE(¢)B +§Q2Cf I)“
+ {<E.+m.)(g,myw + tel—m) Ems + (mm)? + QQJC§1,'+

+ [(Ecw/Efimabam 452%) RM: (aw-m 42263 :13

and finally the @1'd2[ ]d1-d2d2ok'{w.f.} term:
l

42.5 ”(Er ”New Few r: - (El-mam waww —- a» M E In

 

 

+ {(EfW‘MEfi/miijg)‘: — (52+WL)C1££)L - (El—«49mm + film} 1+-

“ 2- 1 C L. E+m,D )D+FL2)M I,
+{<E\+Wu)(Ez‘W1)FW)P+QE 4") ‘75?) ‘4‘ )‘P E +

+ {(E,+W4)(EL+MZ)EWI [I +<E1+M4JC7¢1>L " (51+M4)'M)D ‘ R?)M§I-,g

 

67

As before, all terms underlined once are of dimension
[momentum]'2 while those underlined twice are of dimension
[momentum]'3. All other terms are less singular in mo-
mentum and hence won't contribute to order aean-l.
Collecting all terms with a potential logarithmic contri-
bution we have

4) 2‘ WWW)” 1?de 439’ OWN/'1‘. '44)
(A CD ZWLMWWJ" (Pl-+3“)z EMU E111}

 

 

L

I” [f <a+mxez+wrma — (Elmawzl 11.1 22!“

‘ (EFWJM) 3 1H - ('5 .MMJDmJ EL, + (EI'WIJWYQIIH

+<Ez+m)'1>(2/Z '1; — HP) M I++ + (Ermmwb 1+,

+LEI+4M,)‘]X2)EI_+E (4C.Zl)

Only the 11+ term merits special attention as it is of di-
mension [momentum]”3. A careful calculation of 11+ shows

that it contains a term of dimension [momentum]'3 plus a

term of dimension [momentum]_2. Explicit calculation of the

integral involving the integrand of dimension [momentum]-3
shows no logarithmic dependence, while explicit calculation

of the integral with the integrand of dimension [momentum]-2

dBkdBP UN 215'"? + k2]
(pwwzewa—m 2142 {kHz/1

 

reveals the integral which,

68

from Table VI, is zero to order delnd'l.

The remaining terms in (4C.21) are quickly calculated
using the low momentum approximations mentioned before equa-

tion (4C.l7) and Table VI. We obtain

3 Z /
(A54) 2-x 296 /W0)/<0410*2’>%4 / M, {M
L @ M2 3 Z',’ gal
(4C.22)
We turn now to (AE%)Z as given in (4C.4b). As before
we write y iofivkv = yi[dl-k—d2°k+d1-a2(k0-d2-k)] and expand

the propagators as in (3.4) so that the matrix portion of

(4C.4b) can be written as

<MW 3’2 25W 2. r‘l’flfl [ (If). « «1’62“.

 

~

(4C.23)

beifl) + C(gjaf/fi - Ryan .45..- Qw/a”? {nifj

When we insert the product of projection operators (3.4)

into (4C.23) and anti-commute the 8(i), (i=l,2) toward the
left we will get different results depending on the d
matrices multiplying [ ] on the left. For the present calcu-
lation we need (4C.8), (4C.9) as well as, for

a(l)(or d(1)d(2)d(2)) on the left:

 

69

[] __: ‘ {(Er/144,)(EL4—MJ—(E;+4mjgfljé +(E,-mjg’fl.ié —— QN/ggQLL:
Wigwam Q. A + 43,110 43%!“ 42; K6 ~51)

«El-Maw — «will: ~<E.-m)2<“-Ll,c+ «Mg
(t’eo +71 K0 'El)(‘Qd + ’Z; ((0 +51.)

 

(Ext/"4X52 +411; + Err-rm. "i I? +(E.+/m. Mid 0("1L 96%
~ A. A ‘f’ A, ~ ~ ..

(‘ £0 +7Z.t(,, Ha) <L+ 7]; K0 'Ez)

 

 

+ (EWXE, 4m) + (5—2 'WIJQN/‘xé —— (“wail - 954% 25%
6&0 + 02‘ MO +EI)(/Qo + [’12 “a +E2)

(4c.24)

and for d(2) on the left
H" I { (awe—m) —<EL-m),o.<“~’t + (aw/4% - New
4E,(E/E1(U GA, + ’Z' ,(o ‘E‘X L t 45‘ K0 - El)

 

kE'+/VM|)(E1+4M1)~ (E1+M1)gwoé -- QE.+/m.)gu.)15 + 9C).)“; and:
('VZO + 74 W¢~E,) (“20 + ”Z; u° +E’)

 

(EI’WI)(El‘/m1 +(E2—M41/gufk +KE'_/ml)Q/r(2/Aé + ,Q‘Wé {(2.2

(,20 + 1"(0 +Et>( Qg+4zl “0"E2)

 

 

(Ermxew Ham) 05% — «El—mad‘ié — 9:“; 5:14
<~Vfl<3+¢Z‘I(O+E/)(/Qo+7zzwo +52)

(4c.25)

70

Again we extract the spin-spin content by retaining only
those terms with an even number of a matrices for each

particle. Doing the trivial k6 integration. we obtain:

iozlk’IJXQALYSw'L} term

Hs'l‘l

45E;

{{"(Ei-w) €702)?" + 1742) >30; 1+: +{<E"m')é‘(fl)7— EC?) 5321;"
+ (HE. +M‘Jééfi/TtEGQgE 33—: “L §(E’+M’) Q4137?“ EWHILE

Similarly the ‘Lllk‘l'gfljflwwfig term is

iii—.... {{ (ermmmfi mg in” + {enmemfiw Z3213".

4E,EL

+ {-31-33337333333WMy};
and the if i ”15,1 dig/(@9591 Y {w‘f‘g term

IVY-‘2

4E‘El

2Q“ (EI'MMEz-xmaéitm + (Ea-WQEW/kg Hepwpmfi $3,233? 1:;

+ {"(E,’M,)(E;+W;) QWQ + LEZ'+M42)-E@)187 “(E , ‘4’“) FéR/fi +D¢£fllji+ul

+ [‘(Ei+m)(a-M)QW {EL-WEN +0534") PW8 ”mg 1+

+51“ (Efl-M’h) <Ez+m41) Q(¥)a —<Ez+/WI2)EE¥)% ‘ (EFF/W” Febg _M)gf]:-_§

and the ~23 kiligf'jgaKag/él ngff term

 

71

i F333 19 flew/mama); — t,-w6¢24/+ Fee/WI;

                    

 

 

+[(E,'W')<Ez’/MIJFLP)§ “(El-M;)D(¥)¥ +(E"M4‘)€7Q)'y~ {fauw E15.

+ {(El-+W')(Ei +4141) FWS +(Ez-+rm2)M/‘Z “(Efimjégug/ _. fiWZ/fl_';

+ (ME‘ W‘XEWMQF‘M “5me ~21 + (EMJWM + WWIILE

and the — “Wok [ ] %(’{g<2/gquf‘i term

                               

(ENLMQQZQL Wampum/<1 — Fge/ 11+
+{—(E,—m)(Ez-wf33):5 «3 (323-sz 9mg 453'me + FW/JZL-
+{—<E.+»«J(E,w13fi2,33 ~(E.+m/6m} +(53+MJM)Q «WW/$21.3
+ (f-(e.+m,ia-m)fi3u;8 «53-334633»; *(EM'WM “W41";

and the “(2)4 [ ] “(Igda/[w‘fig term

(,E+m,()e M2)F(,2)B (Ea W>)D£J)9 (E+xrn)€ QJé+ngjb/EIH

    

 

E+ZF(E3+/m.)(ET+/mi) rpm - (E3 +m4m}fl' + (E, +434) (742)01 — Pm )J? 1+“
W W
+[(EI””"3)(EPM;)F/p)$ +(E; vmjbmfl — (El-4M,)€7(p/O{ — EQ)JJ)I- +

+ §(E3'W3)<El+m-1)C¢/(B +(Ez+m4))(£)fi +(EI'WI)C7(¥)0£ +—&(«?)0/§I_-§

72

and the “/20 flf-IIOKQ)[ ]d(:)0((2);w.7f.; term

 

4E E, {[(E -,)/m )(E -mu)'D’42)A— (Ez- 4%)ng- E ('”)%B +C7WC§I++

+§(E.-w.)(E.i-m)Dcw/1 -— QEi»+mi)FLP)B+QE,-m.)3ci)3 —(;,Li)c§j:
* {U5 MAME MODEM +(E- -m)F¢2)B (E +M)E(#)B 4422ch

+ §(e.+,m,)(ezm,>i><2m «Ea/mamas +(E, +3333szng +C7m>C§IL§

and finally the duldmgajkl: ]d"./o(‘2’ refif term

                  

+mjf¢1flB +(E2 +Wz}§4’)c +<E, ’MQ‘WG ‘FéQ/NEI-H.

'WMW n44¢u(<«1144‘44”

 

+ {‘(EF’M'XEI‘M‘JEQJB +05 —/mz) ch ~(E,-—w)% + F92) “fit,“

+ [-(EfiMMEL-rmgfipfi - (E. WJC—‘zmc +(E3+Ml12£)6 Hi2)” €1-33

 

 

4. {’(E‘+M.XEI+M1)E¢)B - (52"WI)GC?)C " (53+M3)Dg2)€7 -F@)MEI--3

As before, all terms underlined once are of dimension

‘2 while those underlined twice are of dimension

[momentum]
[momentum]_3. All other terms won't contribute to our

order. Thus we can write:

 

73

d : -idM/X/ “2/ A319 0‘ Uflz (JP/,1, 9—K) {(5,+m./)J>{p)2:
<AEL)@ Z/WMMQ‘ [12(9ow ems—3d) ”fit I

 

[(5 +w)é34e)04- Pam/713.3% wXEWEwfi WMWL
.3 [(E,+rm)(Ez "7"»)ij 3 $3 mama ‘(EWW‘JGWH +54?) 6013”
+ [-(E‘ Em,)(E,+m1)'f-:m) B +(Ei+maz)é1¢)c +(E, "rigging, ‘F(1P}M]I++

,3. [(E+w)(E-Z+4¢11)F(2)$‘(Ez +M41)D1£) Q] Ii-

+ ["(E\+M')(Ez+mz)fip)3 +(E3+m,)b.42)£]1'+ 3 (4C- 26)

As in the (ABE)1 case the term with integrand of dimension

[momentum]'3 (the I:+ term above) will contribute zero for

exactly the same reasons as in the (AE‘E)l case. The remain-

ing terms in (4C.26) are quickly calculated using the low

momentum approximations mentioned before (4C.l7) and using

Table VI. We obtain:

 

 

 

A\__ fl“vWWZWMw%>AaN £M4_f4
@EL/G)- -K M2 E [3+ 57,11; 472.] (4C.27)
Adding (4C.22) and (4C.27):
3 2 m 1) -/
A3234: _K 73a /Wo)/<a.ai>jma [wifl __,fl (4028)
M1 5 4'1; 3/)",

74

3. Quadratic Dipole Component
The quadratic dipole component of AEL is computed using

identical methods as before.

AEM‘ 230133001 d3’pd’1id4l apt/5 A
L — (2%)“ i 1"

 

         

“(9(0)

 

{7 KEa/"fy L3» ngQE+023K)SS)(k+%k/)K X) A? ”L3: 2 (PM/4?)

X _ 2””). J (27f)3/7

(4C.5)
Using (4C.l9) and (3.4) we can write the matrix part of

(405) as:
WNW Ma’éfli'é”’£'+9<.”’é[]£((flé’ 9501116214
._ 3w 3 333 .337 may 3 ii 2 «WW. 3 33223)
33/233 WWW 3 WW/A-.3<“BUZM”3!
-1353: mww - 12.51’20’[lZ’1’a”’,a/"£é’
33.7

¢(‘)a(ydfi(1)é[]0(“~(Z/+ “(ljoffz)a(fi[]du')£(2)aagil>;w‘Jug

(1} r/ (2)
Where {W3 {12% {1)(94’ Fw)p((' 33-9 + FCQJK ,2 1" CQ)“ £K° z;
Agéiin using the same techniques as before we can extract the

Spiin—spin content.

The ZSM'AZE [] gl’jé’gw‘f.§ term

75

I
45.6;

 

{{(EWMJEQJzC‘ éflWG? 1+4. 1' g— (E,- m,)f¢p)Z/+ Qm/Qg I+-

3. 33..33e333’+ e333§3,+ + §~<E3WE33$51 33,333,;

while the -— gal} [ j 963?," $333.30? term yields

 

-— I {{'CE.—wz)rcp3§ /— 763229} 1+4 + ;'@2*”"‘)F(P)§/+ C74" 9; 1+'

415.5.
+ { (5.393373%; QWEI-.. + hem»; F3333”- 93339318

and the - ESQ-{é [ jg”’5£“[w.f.} term; since the spin averages

are of dimension [momentum]4 we need examine only I++,

/ -— / ‘
-’ (E3+W3)(Ez"mz) GWXY" (Ez'fl/nz) PW): + (EFHMJ RWW _ DQ)V§I++

...—f.

432,5.

 

and the W7£[]pv<W-£fwvl-% term: since the spin averages

are of dimension [momentum]4 we need examine only I++ ,

 

 

4E'E {(E'vmxgfim/QWT;<52+MJEWW +£5"”’"'JF£¢Z/gl‘ prle

and the —- £0g('3£[1g(/{’1fl0/fw,_f. { term

 

-3 __ . 3
45,5; {(53-W,XE,+/m,) HWS ’(E2+/m.}éz({2)<%+(g, -..3,)D33)Q ~ r3323 “£133.

1

All terms in IL+ are of dimension [momentum]_ so all

' or I:_ will be of the same dimension or

, ' '
terms in I+_, I_+,

eVen less singular. Hence no ocelnof1 contribution.

The 10 3752).}, [ J 93’”, 611;“). [g term,

76

 

\ {(E3+M3)(EI*WIJFQZ)3 —<Ez,~/m=)m7/Q +(E‘+rm')éw7&”f(¥)6/§I+l+

4E,E3

There is no contribution to our order here for the same rea-

son as in the previous paragraph.

The - 20”.}; i :1 “WW/9593”! 21)er term]

 

-— I {{ (53-...335ewee333e - (33-W33e37z+<e~m3fi331 «33735:...

4E! E2 MAN‘MAAMAWW

 

‘5' §( El ' ”ENE-:2 ’meJE " (E2 ‘ ”71)ij K ’(EW’M’JECP)1 + GQJwL 3 I43 -

+{{EI+W,)(EZ+W1)D€Mg +(E2‘1‘W;)F()Q}.K +(EI+”WI)E(P/j +Q®M§I_+

 

+ 3‘33; +/m,)(El+ 4,49%); .3 (52 - W2)H?)7{* (E3 Hmyfwi " €363)?! fl”?

and the {WA/é [ ] at”). Kayla/{fey} term ,

f f , r _,

 

+£(E,+W.)(Ez+m3)(76pflt 452—3333.)me —(E,+MJF¢)'P + Dame? 13.

+;(E.~/m.)(E.v W3)€7Q/7(+(E2—ML)E@)0'~/-(E,- WW3)?” M3Q§I-+
+ §(E3'W3)(E£+M;)C—l(flj77 +(Eg‘f/mijféfl7fi ' (E/I’Mr)fé3)p “ MJQ§I,,?

and the ‘4/z{lé[ jfilqzalfiwé/ {at [E term/

\ Q / / /,
45,5; {(Erw)(€rw1jF(fl)/8’_(Ez—.Mi/G7(,p)y + (5,-— mJM)?‘ Fifi/W {IH

 

77

the most singular term here is of dimension [momentum]-1

so the terms in I;_, 11+, or lj_ will be of the same dimen—

sion or even less singular.
Identical arguments apply to the Ago/7dw[]aflljlm£f term
whose most singular term is:

l

4?; {(E, “M3)(EL‘W2)EQ) I8 "‘ CE?" mam)Q/+(El ' 4M.)€7@2jy/‘ Ema/£14.:

The — fig/(”50.21:]g£"5£’[w,7(3§ term contributes

._ 1

46-. E1; {EV/MWEIT’W'JDW :1 / - (53+W1)F¢)WI+(E,—m,)fm)? - 6769/6/§ I++

WWW

 

+ {(EI~W.)(E2—Im1/‘Dwg3_ (Elemz) FwW’_ (El—MJE—QOJ’Pi‘QQ/Jj I+_

3 { (63333.3(533333333332’ 3 35.3333.) FWW/ + (EWJEw? + 6333/ f 1-3

4 {(EWJ‘thWJMJQ“ + (Eu/M3) WWW filly-3w.) E7481? —€73?/Jf1--57
and the -wdflg?ghzé[:]qeé’§uxfig term,

._ I

 

4535.

{{(53'm37<&2+%)6@/¢//“ (31+m11)E62/J/’+(El—4a,)FQ/W- szK/fI-H
+ *5 'W‘lgrwééw’— (weaned - (8333313 7? + 33331371..-

+ flEW’MXEdwfléwa” 3 (5.333.)?3331/ + 0?me Ftp/f +D<w7<§13

. $3.333233333ee—333.33 33333333 3333--

and the _£0203r3{03wzj “(lawn/{Q0707 term

78

4351 {{(EpMME-z—wa) Day/4 -(E;—M;)FQ)B +(E|—rm,)l-t(,g)B “GéOJC Y I“;

 

J‘fig'w'x‘ifimdbwd "(Eat/’"QFGQJB ”(53‘M’I)E£2)B 4343236? 1.".
+ [(Eamxez—w.n>33m «er/33.331333 3.333333333353333 3mm? :1;

33; (E,3M)(E3W2)MM +(53W3W338 “(5+M3)f4£)3 ”4W3 11".?

and the _/go fill/EDT ]f’:q}gmflf)é/;w”£g term
”I Pv/W _ - I
4e.e. {[(t' “Erma?” '<E3—WG32)L Hemmer — P332331 fig
3, {(E-W3>(Ei+’m-L)EQ}F ~ (E.-+/mz)é(p/L -(3:-. 'WJDWD + Hob/1371.3].
+ {(EWMXErWJEJM +(93'M‘LMQJL + (93* ““me J" @ij14”
+5<E3#m)(Et+/MI)E(Q)F +(EL+/m2)éll(g)L —(E,+m,)b(¢)b " CCFJM§ I_: g
and the Z gill3gwaejfé’27A[yap/gral}:g term
,1... V ~ 3
4E3E2 Z{(EI‘W3)<E¢4/mlj F963 *r (Fa—ram) QQQ}C +<E,-4M3)W4Q)6 - F(£)M§I+*

+22E,*’M,)<E4-WQ ECQB " (E2-lm*) Q3406: ‘(E3’”’U)D¥)6 + FCP)N? 1+]!

3 {(3,333,)<3,+33,fa335 +<E33w33é3<3>c «err/3339mm + PM? I;

+ {(Emflez-m) imagermaewc -(E3+/w3):a3)e, — Pam/321-- 5

79

and finally the QUJXZQ/‘f’fiéi 160333flgflfé'fw. if); term

(
42.393

 

(fpr/mflfil—wmljmw — KEZ+MJF992C +(E,-4M)E{Q)O ~44»; I.”

 

+[(E,—M3)(E3 mam» — (Evmflmc {Emu-We) 0 +écpJP§ I,”

+{(E3+/m.)(53+w3)DCQfl> + (523—3333.),t3333c +LE333/333Jfiy0 +6352)P§I_+

 

+ §(E3+3m,)(Ez-4n.) Dam) + (E2 -M3) F@)C - (E,+ 444,) EC?) 0 " 41¢) PEI"?

Collecting those terms that will contribute to our
order, we have:
3331 3' 3(3/ WOW #3 3333 d33’ 0% ’- it 3)
AEL :- "' 7 1 i z .— [(E ..M‘)(EI+M1)D4)E]I
(04-73 M; (VHF) E, (13) 53(13) “H

 

- [(E’df-meFéajw +(E;‘+WL)E@) wJI++ ‘- (E,'+M3XEz+M3)DQZ 21+,
3. [3e,...,)e33m .. 3333-3231” +[—<e,-33.)<s.+m3)D/33254533334933301“

3 (E3+m)'faz)¢/'- Dank +<ErN<E3+mflz39b - (E3+m/P@)C]I++

Jr [(ENWXELTM’”) D@)D]I_ + E

Using the same procedure as before, we quickly find that

 

A556!d - 331 no/j/W/‘<0"W“’>fi W .3: (4029)
L 7 ML 3

79

and finally the Q‘ng’gfld ”0332203,? Era/.793? term

 

( b
435. ffle.—m>(e.+m)m>p — W +(E,-m.)r:33g)o -—é,qz)?)z I”

 

+ [(E3-MI)(EL 'MZ)ZXQ)D ' LE1 ‘0“ t) ch “(E)’W.)‘fi1) 0 + QC?) PE I4...

+[(E3+xm.)(63+/M3)D£Pfl> + (En/3344mm +LE,+M,)‘1:’(,Q)0 +éee/P§I_,

 

3, {(E,+m,)(EZ-/mi) D33») + (El-4m) R32); - (E,+ 433,) Fe?) 0 - COMPEL}

Collecting those terms that will contribute to our

order, we have:

33 333/303“ 33333333333233-33)
3.3333333; <31+We<33e<33 [ [<5 'XE ”W13.

-[(E+WMJH%MU+fé+flmdfb2481;+ —-@fi+quErhm5HlyE:I+,

_3_ [(E3W3)F¢W .. 333342] 1+, +[—~<e,—m.)(é.+m3)1>(3)313(53+m)&pl¥/’]I+,

.3- (EL+VA1)E@)Z//_ M)? '+<El’%|)<EL+/WY))M)% —- (E1+M1L) Féfl) C]I++

+ [(63+m,)(EH/mz) 3901273; + 3

Using the same procedure as before, we quickly find that

AEde : K3 ”0/3/90(0)//\:/<La-”4y4“>/& A’" 134 (4C.29)

80

In summary the total ladder diagram contribution is:

:7{3+%fij—K<4t%%fi§ .)+“’é"5> (“'30)

i3:

3 / 2 n). ) -/
where 7 ___ eta “Pd/M4? fivxéa

D. Crossed Diagram
The crossed two photon contribution to the energy
shift comes from the irreducible crossed two photon kernel,
and is given by(12):

N

33) ,
AEC :1 Jiffy/"73’ appw/ 15(1’17WK‘C753 0/ (4D.l)

 

where T~(x,x') is derived in Appendix A and is

A¢ 1 d4 41/ - £3, 3 3 2." _ 143 3
1.1 (¢«/—— Egg/0:3 f/Zéélnte 82M 7514916 ((£7 7%
K n

M

 

w 0) fl)(1 H
3([2fufy Qfi/xr17xy&)+5¢e 7W ”ff ZJHHAy/OZ

It
1m 3

 

— ¢1(29)f0:"o’fl£/;G~KX oz) BMW”)

Wflwzymf 33W f3, ”ff/£— J‘ML (413.2)

/m 7" fi
, _ d4 ’ 2f77eyJ J” , “' ~”‘ 3 IV
where GE (x—X) —f&7t)fi; g p&+?/ZJSF6ID+7ZIK) (4D.Za)

Doing the x,x' and p' integrations when we substitute (4D.2)

into (4D.l) we obtain:

81

 

AE : Mfr new We re-eM/E w?!)
EC ITmy4 ,2 ,é/‘

,( ngjAX’va’ng'Q—sz, er) Sigma ’+"Z. If) ”XXX”

 

+ W“ W’X s"’gt..z,K/‘s‘”(. We») ""1

£014).

.. Eigijxu ”X 0:0)L/5/ SFU“ 8é+4ZIk/SP<LI +721 kjyy/Igju)

52/7")

- EXQ‘XF1XXLI7&UIO;(1)L/ 30%!“th K) 5932/6 7; UXXIVUJOrO/Lxg
44MB (BXé X57

(N 3/1
X L? W42“) (413.3)

The expression for ABC rather logically breaks itself up
into a charge component, a linear dipole component, and a
quadratic dipole component. We consider first the charge

component.

1. Charge Component
The matrix portion of the charge component of (4D.3)

can be written as

m (2) (”X U) (z/ (2/
(X, X 31X! 2: 5F X X [442+ WWW E424"? 6e415’346“§e§>
(413.4)

We expand the product of propagators according to (3.4) so
that (4D.4) can be rather quickly reduced to the following

expression in brackets:

82

<3:ng I} - JO‘I 3021“quan — “/71“: {Cut {f + “2'0?in 1K;/)X‘-a/[w. (3>

(4D.5)

where [ ] contains the projection operators and {w.f.}
{Dew + Boyd — @9502; «(miggmflzflg . The k6 integration is
trivial using oCkO—kb), so that we can readily extract the
spin—spin contribution using identical methods to those de-
scribed earlier. All that is needed is the expression for
the projection operators when various combinations of d

matrices multiply it from the left. These are given by:

for a(1)d(2) on the left:

 

L :1 r; __‘.___. { (Eq'Wi)(Egl'/mi)“(Ei’wmjfili’i—é _(E’_,/m')g’(z}£/+ 9(3’15 49),?"
4EMS<W [lo/(mtg- gt)j[jo_(% ((0 - 55)]

<e. mace; my - (E2 way/ls + arm/law ’ — 5% a?!

 

 

 

 

+
[£0 — (my-EUMO - (7.: “e +50]
$dm¢¢;VmJ+¢gflwjgflé-Q;+migy£5'Qfléxfié,
[[0, (41.1(0+E.)j[£c - (¢g,_ Mo -El)]
+ (5W, flan/”1945240"; 5% “away! 1; +g’fi’ég"-’£’

 

We "(7'10 ”HI/Z, (a We +59]

(4D.6)

for d(1) on the left:

83

 

[-12 {(E,‘ m/Mlk IW) (E +'/m1)am£ -(E %)Q$(z)£ +(X“’L9§‘”_13
4EE [L (’Z'h‘gflbfl. (7 k 51)]

E “XE ”"1 (521,)” +<e -4")de g’rééw'

he (’ZW'E )j[,2 (f/«gwiyj

 

 

(E|+”"")(Ez' +09%) + (51' +442) {Ufa/E " (54 *M'JQSW é“ Q} IE 9!. of!

[non ~ fisz/flM , (”an '51)]

+

 

 

+ (EmmXE/«xmd + (EJ—mQQKJQ/fé +(E,+m,)g(f’g’+ gig/54;?
[20 "(71’s tEJ‘Yl/Q‘ ’— ({LK+E,’)]

(4D.7)

for G(2) on the left:

 

[ 1 ._ _‘__ (E,+m,j/E/mu,/ fialméjgfié mam/5% + 21% Kwi/
c 51' [£_ /”(.A/o ,ngz a .- (”Za (a "E;)]

+LE +M’)<EL +Mz)” LE; +047 )d 2 +(EJ—4’MIJKOJL (“WAX/2);

we“ (40/0 WJZ/Z- é’Z K +51)7

 

-m.)(g,c 4441/ Hex—m/Q’Cé figs/"1M3!" 4’72 49’!
[ll - (We *5)?! A ~ Wréfl J

+3; KM'MEI‘W; +(E Jami/a1 + (p5 WJRIMLJr Xf'zéq’E/é/
[£4 /7,t«o+g,)][£o( («L/(0+5 )]

(4D.8)

84

for o(2)o(2), or no d matrices on the left:

H = ’ ‘E'W'J‘EWJ deamw’f’ - <5,+m.)4’i’l'+i":ég(il'
45' E" w. , ran-521% Ma. '57]

 

 

(EWWEJ-m)—LEz’—Wa“-’£ +‘(E.+m.)Q$aZ,LI- 01'!“ 25:)!

[,Q. ' two-FUN. 41.1. +5.9]

 

(Ex/MME; +01”) +(Ez’+/mi)Q/qié ”(EPWJXI/‘kl—gqlga‘i/

+ -*—‘ [fie _ (7'1“ +E,)J[,£o ’(7,(0 ’51)]

(EI‘M‘MEJ’WJ +<e’—m)a'3’é t (E. —/m~)9§ly‘ Q +£"féa’f’é’

[12 MM. HEML m. w. + 5)]

(4D.9)

From (4D.S) we now extract the spin-spin terms.

[ ]{w.f.} term

r\

18f”: ——1 +1--§ -
4EEJ 4* +' the most Singular

1

'1'
term here is of dimension [momentum]' so there is no con—

tribution to our order.

The - a;u[-] ayflyzuif'} term yields:

85

 

4E: (Em, {{"(EI fwd (Ez"'m1) €769)§ .+ (ELI'M1)EEP)g'(EI+MJFfl/fi 1' Ina/IE]? If

W

+ {— (tfimXEJr/MJQéPfl +(Ez't/ml) 3mg HEM/"VFW? —1¥N§}f+-

 

+ {'[Et'mixezl'wJéflji ‘(ng'MJE/fp/g -—(E"M,)%1/E - mjfifff'+

+ {-(E’ 'm.)(E;+Mi)(—(¢)§ — (Ez’i—zmijEéa/g Ha HmQFga/Z +DCQJ/5J'; [Hg

and the —- N-(I/[]0<{y{rW./‘.f term:

 

AEE ,fi (E 'W)(Ez +/mi)éép/fl +(E +001)Fé;1)$ ~<e WjFéQ’Z' +b@/,8EI++

 

+ [-(Erm)<a’%)6<efl H‘s/'Wfiwf gamma? “192%? 55+-

+f—(E'+mJ<ELI+/mz/C7gp)z ~(El’+mQE(p)§ ‘(E‘+WI)F60)ZU ’ E976 gfr-r
WW . W

 

 

+§’(E,+/m,)(E.’-4mj€7wff «(E.”mi)fég/§+(E.+m‘jF6Q/7 +M)§§E—-f

(ll 2 (I (a)
and the did/()[jad 1% ( [w [5’ term:

 

mp)? -(Ez’w»)Fw§ +(Ermffgo/é" 4mg} I...

               

+[(e,-w.)<e.w.mpfl »—(E.’+.om)Hp/§ —<E,—m)5g7/B’ + (saw/553+-
WWW W

+/[ (E.+»~.)(E.’—m)m/Z meat/m) Ftp)? +(Ewmfwfi’ + Cvefcvffq

W W,
+ {LE +441, )(lE “FMJMJA +(Q +m17)F(2)3 ~<C.+/m,>PQ)§/- 7Q}C $1-,3
«WW

 

86

Thus we can write the entire charge component of the

crossed diagram as:
(A5 c) .. 2' a3/WQ/l 45px]: ASL, afld‘. ’2. '43
C _ M (21% (piflr‘j‘ 5,04 54(4)

 

) [[(Et 'hmJ 9W 2; " M)’5]f++

+ [(5. +M.XE;I+ML)6(¥}Z[ ~(E. if'.M1;/E(I?) f _ (E; +MIJFKE/E +1152) g? 133+ ,
+ [(E‘+WJ(EL'+/MQC7Q)& +Etl+wqfiw $9 +Efim) Fag/z +32) §]f_+

-[(Ea’+mf5u)§+ Bowl?” + 312)ng + )(p)§I+_
T [(ErmNEJmJDQ/z ~(E/+m)§?)§jffl +(E.+m)E<;2)g it?

+ [(E‘ +M,XE;I+W;)M/f] E-_ E

(4D.10)
The first two I++ terms, with integrands of dimension

[momentum]-3

, will give zero contribution to our order.
The remaining terms in (4D.10) are quickly calculated using

Table VI, and we obtain:

 

A E" ._ _ m’Ww/KO‘W 0.09% a/ " M (413-11)
C M 1 fl
2. Linear Dipole Component

The linear dipole component is given by the second and
third terms of (4D.3). For computational convenience we

break the expression up into two terms, (AE3) and (AEg)2.

l

87

 

 

 

 

Table VII 8 '
. pin averages f
f or two photon crossed diagram.

Spin average
N Coefficient of (OW-64$
A s «46%,? -‘”>
Kg ‘5 <0((‘(//J(2)0{m£J WW (2,; %(I)£> + 2
i9; ‘<K(WJGQfUé/‘“Mfll(y — %>!£.¥

m
d $> ' ‘2' V
2.: : <a/(I2J (1} (I) (2} j P— ”g

_ o( M MW ”oz"? oz‘ 7g)

:75 r @791”) fl my)

-— 0/0/41 “Xxx (Z/X(2/é>
E 3 <0“??? “a (4 _% E.“

J“ 25/21? p((z/ £> A, .~
g E <a/(‘m% Uflmé/ MW Oz/l/ fl/(‘IU> 06“)
(I =<?"/@/:;F'“/l~ Elli}

, X % LN “/aaféy’ng ’jp> :(L/
N, (H 2 H / 4
M: <4 é”: a); “mica/(5(a) gwéflm 49>
£7; <N((I aga/‘é/dm' “(z/L, “0/ (1] 97(é4)

A ~ azv,v'~ ‘X- y.
5: <a’f’3m/drz/ , my (u wot (%[> my?

I (J~~/O\£~dja/I ~(1:%(2;g>
P E <d,."¢jyp(a)L/flun W2! I W m. (94"

A N r N A ~ 0:; we 4% Wm”
22/ E a” WM W > i ..m 05> we)
3 : <0/(2/0(”:é 02/1/“(2:£> '32— Pl
4:; , m - N + 3:— ta
3 = <% M W?» 3 A “9
W /~—
Q <o;"%i’§fl’a”%> A i— k Ii
4% I '- 3’ (
Q : <O/(f/ad/ IXKz/a/U > 3 k "B
N N N ( A
/V, - a -+E;k’
a : <04 MW > 3” «a
2.
~ ”3' J5 fl

 

 

 

 

 

 

 

88

Table VII (continued)

 

 

 

’5 e <06“ [05% N"*£’I9"’>

“j; I 2 <94” 9521 ’95 "L W5

; (W JI’I M20

E :- <dfflé “ff/D

"i E <I<~"ie<”fé Iowa/"9

i 2 s <0§2)~(, ’9 K505519550 >

Z : <w(”’;’Ie’I<,-“’z“*k I’” I?” >

if: <II’I'I III—I MI I?»

W: <XIMQUL,’5W "9534 9X” I>

w <I ww>

77 @110). I , 052/! If M I?»

IV <I<”I~<”L’I,"’~’ W 6‘ ”Ia W

Z: <Ig(”2<“’I’I’:I Ira? WSW
; <x“g%’wrié 4"“2’ IVE/e519

\

m

<0Ww wig/M a“?! 01%: ’>
<4(_"/ KICZLMvL/g fl/é/a/ Ia/fl ,Q5$>
<0Q/a/Iré “(I14 99% (gel/N Mm ¥>
<d01da/fl/ULfl/Mé%(UL/%/gfl::€>

A, N N

§> is?) R? H?

’—
’
’—
.—

N
__,.
I"
I

 

 

 

+
ow
“T
1":

+
VIN MN

 

 

 

 

 

 

 

 

 

 

 

89

Table VIII. k0 integrations for crossed diagram

 

 

NM if“) I l '
If? @121 (SC-1.1.4149 WM‘ [12 (Mme-ted] W (’lz’ME‘ ”U

 

 

 

2 Tu. ML + )L‘HH th‘ I
“7+ 4 NR'PM’UUIHW “ UAM‘M‘E

%{4/M: lk'l‘kl‘g
f_+ amirt —+i[4% [kl M1}

N

1,, “Pam ’ .
{3M.Mz[k‘llkl((k'l+‘kog

z» / . 1
I“ ”75““ {mama—g

. ‘ vi
+- #1 L §4mzlk‘((lgl(llc’l+ltf) 3

 

 

 

 

 

 

I ‘ ~I
li~+ fixtz {9LM‘[Q%\kJ(\kJ\+‘QL)g

 

 

f’: ——7 17M? ‘1 ”I
:ewjmguwmm + - '

[6491’le (HUI-i- It!)

N H ‘
I: —$-2fié I
++

—.___ + (
m‘hll ”’12 E2)

 

(
.. (
{2|le MULAIHH) ' ?<

N
l I

I I ’
*’ '+ 1“ {YA/m (\‘k\+(kl)§

(in ’5‘1th‘ (ye-"1"at g
+ g4“ {\k(+\L1)

 

 

 

 

 

 

 

 

 

90

l 3 3 3 é _ +__.
éfiEjlf: :/mf;md4JféiPd Ad L 0/(:2%2¢)6Q2 1L) ¥%@» Flfiéy

xfgxny12)s} WGL+4Z,K)<§FM _ 4&th Kit/q; :Wédffl: “(21)

(4D.12)
Using (3.4) and (4C.l9), we can write the matrix portion

of (4D.12) as

<[M'Ww-f-f 09"? lot/”’5 (2%wa — “Hy/W +wé)fw..%f
W ”W +45" %[]«"””// 42.1mm— weM’zflg’m>

(4D.13)
where {w.f.} = {D42} +ng gm-fl +E¢)’oyl)¥ +Gwa”? way}
We use (4D.6) through (4D.9) where appropriate and do the
trivial k6 integration as well as the k0 integration (see
Table VIII). We then extract the spin—spin contribution.
[ ]g(2)'k{w.f.} term: the spin average are of dimension
]4

[momentum while the most singular k0 integration, I

++’
is of dimension [momentum]_5, so the integrand will not be

singular enough for a contribution to our order.

The - éymijaimgflié fluff term:

91

i

i _, , N , ~ _ .r-v N
EIELI {fiE,+wJ(t,*W;jRfl/f ~(E$'W1)D4¥)¥ -<EI+M;)6(P/Q+ fifi/J/fl—w

 

+ {(EWJE’WJHWE —(E.’+m)mfj + (EM) (7425-- Wwifl 5..

WWW.-

 

+ {(E, 'wv)(Ezl'Wz/\FL,Q)E +(E¢I—/M;)_ng)§ -(E(‘MJGGP/g- Ejjf f_+

+ EOE , mflEJ-I—mflfpjg +£5£+m§l¥fl§ + (e, -44.] gpjj+qu¢7f EH}

and the —A’(-M[]qfl/Z flag/2:4) [cuff term:

—

I _ / Au / {:5 ”A: {V
4E'E‘/ [[02 '/m,)<Ez+4m/€7(F/a —-(E2 +m1jfwjg ‘(EI’WJ fie/fl “Flip/3 ff;

 

+ {<e.-wze.um)é<m&' — (Eu-mfrfwg +<e.-»«.)v=;az§ axe/3} ii-

*{{E‘*”“')<Ez'*m»Jé/«Pfi +(E»’+WFW§ -<E. MUM): $405; f}

+ SIEMJ<awemi wax-Mame art-Ammo)? +1225} ’5
m. I... p ~ - E4... Wfi ve—wawnawfifi

w“ )(EL‘f / a2): (W :7)

+ {(Eum/(eu-W.)Fw§’ ~(EJ-mjflaz/é? +—(E,-w.)6<,o)é/~ Fez/#EI _

N ”(:3 f4; . Q7 ~
+ {(e.+m,)(e,1+m) Hp): flag/m4”); -—(E,.+M.,J €79“? - Halli/qu

+ {(E+zm,)(El/Jmt) ng/g +{E,’..M1/W§ +kElf/m‘EéP/g-f a?)¢/7§2f.-3

and the g(l)-§[ ]{w.f.} term: Again the spin averages are

of dimension [momentum]4 and the most singular of the k0

92

integrations is of dimension [momentum] 5, so there will be

no contribution to our order.

The 34“} A?” [ ] CY/fl/[aL-fij? term yields:

 

- I / ~ N , g N N ,V N

45’ E; {{(El -mJ(Ei—xm.) Pep) “B - (E. '0'"leij —(El-—m/Dfp)/9 +qul'71f I”
J“ {(E.-w(e;+m) Hp) f” (Ez’wcmjégfll +<E~mfl>02f§ ~9—P/;CVJ?IN+_
+ [(EI—rmfla’wmj Em 3:; +0?! —/m) Gil/4;: *{En‘IMJDWE ~FCP/51VEE-+

+§<E+mflezwmmwi +(eumaéw’i Mammy/s + ova/41,?

and finally the mm May] m A’l-M/Z+~ “(y/4) ).{w )5)? term

431E {{(Ep IMKE WJDCWN 4 ‘(ELI'_,m3)F-(?)g - (E,— 4",] [342ng +nggf 3+;

+ {(Epwanz'MWW/f “<E&'+”"‘)FZP)’§ ”Fur-1'4“) a)? _ (76?)??? E l“

 

1

.+§(E.w<vs.:m/1>cpzi Hex-mm? -<E.+wfirfi’ 6W Nil

‘2

t§<EIIWvXEZIIMWPJK +(Ef+.m,)t¢p)r§ +(E.+mjfip)§’ + QWC «JUL
+E{< w/(er WJFWE (e. mJGW/Z (E mmpfb" + FM in

1» {(5, —/M,) (El’+mz)ELp)/E‘<EL’ +me€7ééfz +<El"m)m)‘$ -— F92);)J§ E+_

+ {{Efl/Mfle‘fl mg) a) F 4‘ E; -4441} Cit/fl: “(E‘+M.)D,D)3 “' ROI/$14ij

+ {(E, +xm.)(61’+ «42) fig)? +(E.’+mr) (.74): +(E,-+4«.]D(P)3 + fill/”iii”;

93

Collecting the terms with a potential contribution to

order delna'l, we have

5" : ff/wov‘ d?d3é [13F g/Zéévfl
(A C/(D ZmlM/Xn)¢ ([01+r91 EMQEZU‘?

 

fig; [+4119 D62] i (ft;
— [(5. mXa’WW/E ~<exwmfijf
._ [/EI+W)(Ei/fM1)Fga/f +(El/rmJD¢/§ ‘(El—f-m/Dgojg —(E,+m/w/5]E: f

.. [PCP/g - @L'Irmfipé 2i; 3
(40.14)

3 (the last I+ term

The integrand of dimension [momentum]- +

in (4C.l4)) yields zero. The remaining terms in (4C.l4)

are readily evaluated using Table VI:

d 3 ()2 0—0470) V ~/
(AEc)®:‘KTK/SDO/f7; >AIX (_§%t+i3%) (411.15)

We now turn to (AEg) which is given by

z'a‘ ’ M44“ é’- - / -- ’
OAE:%C>::—v:mtfi%P¢jnd/9 Jéjiziz é fly éz7’él/ ¢yw

M (1} (I) (. (I/ ' 3
1M 0% é/§F€é+7,l(/S}46él+7zk)yv ”Mfg/(27:) /’

/M.

 

 

KW

(4D.16)
Using the same method outlined before (4D.l3) we can write

the matrix portion of (4D.16) as

94

<26"4£’I]{w. I} + «:74 "-451wa Mm + Ie"’,I.I,°-’i’£7 IFVI-I?

+ H 5% {at} - «fay/g ’- {IL/5’) [ ] 09/949?“ ff — 4]? MW}; mm}?

(4D.17)
where {w.f.} = [Dm)+Fw¢m-£ — Erma"). ~G2¢PJQIUQE£2C€§
Using (4D.6) - (4D.9) where appropriate. and doing the k6
and k0 integrations, we then extract the spin-spin contribu-

tion.
The 9(21k'[ ]{w.f.} and [ ]g(l)°§'{w.f.} terms have

spin averages of dimension [momentum]4 and the most

~

singular of the k0 integrations, I++, has dimension

[momentum]—5, so there won't be any terms singular

enough to yield an aslnd-1 contribution.

The ,angy[:]qgugan¥.§ term

43.5; [{”{E’+”"J(EV'“’)G‘W WEI'WJFQJ i5- (51+WJF69/Z +bw§§f
W W 1'4}

an).

 

+ §‘/E.+IIIJ(E.’+M/C7¢)§ +(Ez’+-I"2>Ecp)§ dawn-JUPIS - hwy fig

+ §-<E/ “’7"I)(Ez/'/””) Gwi '- (ELI'MJ Fifi/g — (EV ”/F‘p) a, "mjggf;+
+ f—(Epmj (Egg/ydéwg _ (Ez’vafF—‘m 557+ (Ex’WI)/:WE + DUI/g)? EL?

and the - “fa/€5,049 [ ] 0101“) fa), {f term

95

({<Ea+/’"J(Ezl+”m) Fay/E -(E;’+W1)W 9 +(E' +091.)qu W ‘Ejiffwu

MAMA/WW
WA

 

 

*{trmrauwawa «am/1W —<a+«)e<m? VIII/iii-

+ {(6,‘W,XELI+ Map/Z +(EJ+MI)DW§ H514“) QW’K + ENE §I—+

 

~ ~ N -—‘ 52’ ii
’i“ [(E.-m.)(E;/?MJFQQ/ £1 + LE;’-m._)_IXP)/B - (E‘ -— 4",)QQZ/K P49) 3?

and the Q-Wfimk/ [ j A’('/Z)fw. 5.} term

4EJ E ’ ”(El-MOE. £4”)ij Z ’<EII'W’)D¢)§ + (E’ ” ”jaw 76 _ EQ/gE f“

 

+ §(E‘-m.)(EzI+/mz)fip/CQ —(El/+/m9)_D(pJ/§J ~(E.—MJQ62}£ “Fffipjgf'f .-

4.

+ § (awex—mwfi than)??? +<E. WWW +54pr If...

. {Kai-Myeu—mm? +<eu+mm§ — (mm-aw ~'F@)%EI--§
and the — “(An/i ]€<LHI/é’ Q/(‘mgfw.[.§ term
- l

55 65/ 9'6, fr? A“
E, f{~{E,-m)(e’+m/fiyfi’ + (EJWMWW “(em/1W9 Hie/i 121..

+5: (E;‘WJ{ELCm2jFéo/£/ f(Ez’-/mle76p) 97?: +(E, ~mJM/x/gl — l-Yé/f’ j); -

+;“(_E/ +WJ(E¢'+M1/Eéfl)lgl ‘ (EL I‘M“) GGPfiZ/l ’ (Ex ”MI/W/Ig/ — ”Vi/K714

 

+ [’(EIWJ/EX-IMJF-Zugj 452" WGév/W/ +(EIWJ960/9' +F6b/f’521"3

96

and finally the —- “mam/j -a’a:é/[]/(:”{ 5” w. ff term:

[ {(5 4H,)(E1- -/m):D(j?)/I-

a {(E— IMMEL +mi}_D(,0)/l "(El +449 F903 "(EF’mJ F605] +C/(7D/C $1 .,
5 9 Wm )(ELLMmQQMN «(Eb-m) Fee)? + (5, 7%,} 7599/3 ”WW/552 121+

IM)ng/B +(E —/m,)F¢-,v)8’ 641C511;

 

.4EEZ

—- g (E‘+M,)(EL’+/mzj up)? 46—214"wa ~(E,+m,)73(p)§7 — 960/0 f1

4' j“ (EV/ml)(El/+’mz)-€,Qj/BV/ +(E;I+Mg/C7g£/E "(E/’me/é 7" @Nf
g- (E .. (Barfly? lg, Maw: (Emmi - emf,

? (E +Mm)(52,l+ 4:1,) EQI’BV/ ’ (Ez’+aor2)é;(pj Z .. (E, +414.) DWC; - FQ/UEI
WWW

 

§’(E +4“ (Ell'W1)E—Q)E/ "(El/“MIEQ/E’ +(E/+M’)D¢WE +F6VVEI--§

Collecting all terms of potential contribution to our order,

we have:
4 _;o<’J\o(o))’ dgpd3éd3L/0p(,é/'ié ,2) (V. N, m N f,
(“3% 2”" MM ow) Ewen) (PM (w WM]
- (E.’+»«.)D,2)§f_ , — 02wang B ’i’, ,

+ [(6.W,)(E. wow ?I HE, ”4mm 3., +(E.'+m) 11.2)]; i,

-,[(E+’M)(Ez+/m2)F(Q)B’+(E(#491,).ng)(j I.4 -(.E+m,)(EzLIm,)F@Z I g

(4D.l8)

-3 ~
(the I++ term above)

The integrand of dimension [momentum]

97

‘1

makes no contribution to order delnq , and the remaining

terms are quickly calculated to be:

M). 3 471—: .011,

(13!;le : K Tag/(mow <¢m. 0.(2)>%a~/ M +3Lfl) (4&19)

Thus the total crossed diagram contribution to the

linear dipole component is (see (4D.lS) and (4D.19)):

 

A154: K Ids/Way <0-m 7429,41" (6 fl 1— fifl) (413.20)
C M: 5 m; 5 on,
3. Quadratic Dipole Component

The quadratic dipole component of the crossed two photon

diagram is given by the last term in (4D.3):

M_ m: duawuflcé’.é—, a”. g __
”ma/J a W 2 WW

Mia/1% §FI€4+%K/S}w{—é’+¢m)yy"’ "’4‘“;qu
(4D.21)

Using now familiar techniques we write the matrix portion

of (4D.Zl) as:

98

+ ”ff-”.4? MW - d-ié “it?! my” - «5.715714 6152‘?
, M a“’£’[ J — [ 1457,!5‘21é J. aging/at; J. 04"”[J9s’1’é’fl
+ fia(-"’5"23£’[ )Wfié 45% {XXI/23.” fl war (fl )5;

,/Z A/(‘ft/0(J(21[JXJW%(2/Ag(zjé + £0 “(aw/(17’ g(Z/é/[7aé “(\(z/

a/'/ /(l) ECO/é [WU a/fl(z(é];wv If)

(4D.22)

where {w.f.} = [My + Ftp/gwfl + EWQSW-J’ + (”kW/25%? Zia/‘4?

The spin—spin contribution is extracted using identical
techniques to those discussed earlier. In all of the fol-

lowing we need consider only spin averages of dimension

[momentum]2 or less as the k inte rations have dimensio-
0 g

5, [momentum]-4

-2

[momentum]— , ..., and we want an integrand

of dimension [momentum]
The — [2’59 [ 703% {wig and - Q’QLYK'QTmfif terms:
No contribution from these terms as spin averages are
at least of dimension [momentum]4
The-agg’z“f,g[]og“z‘iié [w.f.§ and [xi/W“ N m {We} fzterms
All spin averages are of too high a dimension to con-

tribute to our order.

99
(I) (2/ (2) / (II
The ' 2‘ 15021 K .4 i. Y“; 220-va term: we consider only spin
averages of dimension [momentum]2:

\

'—

4Eflh'

 

{(E’ —m.}(E;I+ml)]xp)$i/++ +(E,—/m,}(E-;’-W>)D(¥)3 f4,

+ (E. + m,)(E.'+m)1w3f. + + (5, WM Ez-"W2M)3f--3

and the -— Q’XIT jaw-[é fi/{wflflfwxf’f term:

I , xv, c N
45.5; (5 (5.-....)(5. WW) 3 I... +.<E~»«J<eu--)w:a”f.-

 

WWW

+<E'+/m,>(gz[+W‘>D(p)3/E—+ +(EI+M14)(E_L,'M;JW$I jig-1’3

and the ng-k g‘flg’fliw.“ and fi"%f€‘%l]fw.f.§ terms:
There is no oce’lnoc'1 contribution because spin averages
are of too high a dimension.

The ’50 “JEEYOQVZWE [MIX term: Spin averages are of
at least dimension [momentum]2, so need to examine
l;+ only:

I , ~ , . "’ ”V ~ ~,
45,51'flE‘Jr/M‘MEI MM) F(P)€ ~(Ez ~Wx/D‘WQ- *(E,-+mj(7(7p/Jy + FGP/W El.”

 

The ~~ fl. dc‘mi] flfl'jé/dz'mfwotg term
——/ ‘ ' 96, N (a; ”I,
45%, §<a MME, +4...) H.022 w; 3.4...) any — (E, WWW/93 R42) 1 51+;

 

The L fill/Kai? [ ] 0Y1” fab/f term

4EE [{(EF’M.)(E1’, out) PCP) z ~<Ell~4441)b$a}b' ~(EI’WJC760/z‘f'E-(JU if rf+'+

 

The Q EEO/é MJJZ ] OgIIFW/f.§ term

 

 

100

' are, mxex-mjfcw‘é’ «a; «#6601557 'e-MWEV W373i!

4&5

 

The - 20 am (”I 159”? (2 «2.15.; term: Since 11+ is of dimen-

sion [momentum] 3 the only spin average which could

possibly contribute is A:

 

~ r
“A __, / N I, / z ” "1;
45‘51/ 5(8, 441,)(E2. “M1)I++ + (EI‘WI)<E; +4"; J. +_ +(EI+MJ(EZI— 4".)1 -+

+(EIT/M,)(ELITM¢;/ £1. E

The .. fl 0((«(/:¢é(2/[]9(/"%‘2210/é fag/1)? term:

 

..l C , ~ N I ~ N N N!
ABE; Z<Eo’w')<61”m‘)F{-P)F ‘(EI’WJ (7981' ”(EF’WIJD‘P):D +FLPIM E 1+4.

The in M i an”: M] ”Ia/[w ..E/ term:

Finally the (ZQ’(ré(/(Uo(‘”é I 709% W {W fw.1(.§ term:
\

45,5;

 

F . , a
2 (E, — we. +mmw IH + E, —....,)(E;_..,:) mfii,
+(E, +m/(Ez '+M.):DQD/jf.+ + (E, +w,)(éz”m) Rafi/{,3

Collecting all the underlined terms which will poten-

tially lead to an aelnd_1 contribution, we have:

101

 

AEM: _m<’)gow;ll L 0% d3? fill-é 32)

 

 

)n’
Mall/VHM)‘ (y $1ij ((4 E— (L)
{[{EI+M.XEz/+MJ DC?) [3): :8)— 3,131+ %
(4D.23)
Using Table VI we find:
AEjdr. ~91: 7(a3/W0)/’<¢”)-0‘U)Ad"(i £1) (”'24)
Ma. 3 0";

Thus the total crossed diagram contribution is (see

(4D.ll), (4D.Zl), and (4D.24)):

 

M "E‘ 2 M. 2 /_Z-.
AEC Z —7Z/-"‘— *K7Z(3 %L+§R)+ K 7\3 4m)
(4D.25)
where «z; ”Ks/Woj/zfl’fiaw>/fl\fl-z
Ml
E. Non—Contributing Diagrams

Radiative corrections to the one photon exchange repre-

sent a potential source of aslna_1 contributions, but de-

tailed calculations(14) have shown that in fact they do not
contribute to our order.

Since terms involving three photons are at least of
order as, we conclude that the ocf’lnoz—1 terms are those cal-
culated in this Chapter and give by (4B.59), (4C.30), and

(4D.25):

102

I '— 4 ——_'
1315:73 - K7(§+§’%)+K‘t(é—%1) (4E.1)

T43 / ”(0// 1<an 7.91:4 N -,
M2

 

V. SUMMARY AND CONCLUSIONS

The (me/mp)d21noc—1 recoil correction for each of the
components is listed below, diagram by diagram. n repre-
sents the quantity no3l¢(0)I2<ol°oz>lnd-1/M2 which can be
readily factored (see (5.4)) into the Fermi frequency times
a term with the ”correct” mass dependence (i.e. doesn't
diverge for m2 + m). It should be noted in Tables IX, X, XI,
terms with the ”wrong" mass dependence (e.g., n(M/U) cancel
out so that we are left with only terms with the ”correct”
mass dependence.

The total a6lna-l contribution is thus

AE:%[?-K{l3+7%l)+t(2(7- I44%)§ (5.1)
where, again, 7 : 10(5/WW/‘47W- 0"”)44" M” (5.2)
Using M = (m1 + m2), l®(O)I2 = yaw-1, and Ry = a2m1/2 we

can write (5.2) as

: §(”7?.)29<4Ld"figw (5.3)
<1 + a)

This can then be rewritten with the Fermi coefficient

 

7

as a factor
4F a“: “‘Qaw‘flflwafiflf’ %{r#)“2%“"/’*%N
(5.4)

103

104

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table IX. Charge component
1%
One Photon Two Photon Two Photon
Diagram Ladder Diagram Crossed Diagram
2M 1M _ M
“‘3'? “(3+3‘1T) n 11
Table X. Linear dipole component
One Photon Two Photon Two Photon
Diagram Ladder Diagram Crossed Diagram
KUZM-tl-iml —KnO+—2-m1 -Kng.M+£+iln_l
3/4 3 3mg 3 3m2 3/ 3 3m2
Table XI.

Quadratic dipole component

 

 

Two Photon

Two Photon

 

 

Ladder Diagram Crossed Diagram

 

 

 

 

 

105

where the later factor in parentheses has the value
1.070 X 10‘4 MHz (where we have used d'1 = 137.03608).

Thus the frequency shift corresponding to (5.1) is
0H = 2.899 x 10‘4 MHz (0.2 ppm). (5.5)
It is interesting to note that, numerically, the linear
dipole term and the quadratic dipole term almost exactly
cancel each other (the linear dipole term being -8.324 X 10'4
MHZ, and the quadratic dipole term being +7.913 X 10'4 MHz).

The present theoretical value(15), including this work, of

the hydrogen ground state hyperfine splitting is
(0H)th = 1420.4026 (l + 65) i 0.0057 MHz (4.0 ppm) (5.6)

where the quoted uncertainty is the root sum square of the
following errors: 3.8 ppm in aWQEDZ’ 0.6 ppm due to the un—
certainty for the uncalculated terms in o, and 0.9 ppm in
6p(15). Hence the uncertainty in the determination of the
fundamental constant, a, represents the bulk of the uncer-
tainty in (VHJth‘

The experimental value for the hydrogen hyperfine split-

ting of the ground state is(l6)

(0H)exp = 1420.4057517864 (17) MHZ (1.2/1012)

which is over six orders of magnitude more precise than the
theoretical calculations.
The uncertainty in.d does not affect the ratio of muonium

to hydrogen hyperfine splittings (see (1.4)) as this ratio

106

is insensitive to the value of q:

 

3
7/ _/< (H $4?) { ,
ffi}: (1+ &)3 l+0//;“0/P'O(/;? (1'4)

The greatest difficulty in using (1.4) is the uncertainty

involved in the determination of the ratio uU/u At

p.
present there are two determinations of the ratio, a Uni-

versity of Washington/Lawrence Radiation Laboratory (UW/LRL)

(17)

collaboration which measured the muon and proton pre-

cession rates in a liquid, and a University of Chicago re-
su1t(18) which measured Zeeman transitions in muonium.

The two results are

(MU/UP)UW/LRL = 3.183 346 7 (82) (2.6 ppm)
(Up/Up)Chi = 3.183 338 (13) (4.1 ppm)

From these values we can put an upper limit on 6p.

using (1.4). The present value (including this work) of

the proton recoil correction, 6 is (-34.4 i 0.9) ppm(15).

p,
the uncertainty coming primarily from uncertainties in

the form factors in the (me/mp)d correction. The muonium

recoil correction. 6U, is (~174.0 ppm)(19), and the muonium
is 4463.3023 (35) (0.7 ppm)(17).

hyperfine frequency, Vm’

These numbers then lead to an upper limit of dp':

(8 S (6.5 i:22.8) ppm

l
p )UW/LRL

l+

(310')Chi é (3.8 4.3) ppm

107

The uncertainty in each case was computed as the root sum
square of the component uncertainties listed above. Clear-
ly, the measurement of the “u/“p ratio affords a most accurate
determination of the proton polarizibility.

The calculation presented here is the logical culmina-

1

tion of a series of delnq recoil correction calculations

that have included the positronium calculation(12), the

(19)

muonium calculation and finally this calculation which

has been the most involved of all.

REFERENCES

 

[\J

10.
11.

12.

13.

14.

15.

16.

17.

REFERENCES

R. Karplus and A. Klein, Phys. Rev. 81, 848, (1952)

T. Fulton and P. Martin, Phys. Rev. 88, 903, (1954);
Phys. Rev. 88, 811, (1954)

E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232,
(1951)

S. D. Dress and J. D. sullivan, Phys. Rev. 154, 1477
(1966)

D. Jensen, 8. Kovesi—Domokos, and E. Schomberg,
Preprint #NYO 4076—17, Johns Hopkins University

D. Romer, PhD. thesis (unpublished), New York University
(1971). Mr. Romer calculated the charge term in the
Lorentz gauge for positronium. This served as a check
on the positronium calculation previously performed in
the Coulomb gauge by Fulton, Owen and Repko, Ref. 12.

Equation (2.2b) treats the proton as if it were a point
Dirac particle, although the results that follow this
equation would be the same if we were to include a
phenomenological magnetic moment term for the proton.

For example, see Schweber, An Introduction to Relativistic
Quantum Field Theory, Row, Peterson, and Co. (1961)

M. Cell-Mann and F. Low, Phys. Rev. 88, 350, (1951)
E. E. Salpeter, Phys. Rev. 81, 328, (1952)
T. Ishidzu, Prog. Theo. Phys. 8, 154, (1951)

T. Fulton, D. Owen, and W. W. Repko, Phys. Rev. A8, 1802,
(1971)

C. lddings, Phys. Rev. 138, B446, (1965); or F. Guerin,
Nuovo Cim. 50A, 1, (1967) and 50A, 211, (1967); or

R. Faustov, Nucl. Phys. 18, 669, (1966)

0. Owen, Ph.D. Thesis (Johns HOpkins University, 1970)

B. N. Taylor, W. H. Parker, and D. N. Langenberg, Rev.
Mod. Phys. 41, 375, (1969)

R. Vessot, et. al., IEEE Trans. Instr. Meas. IM-15, 165,
(1966)

K. M. Crowe, R. W. Williams, et. al., Phys. Rev. 88,
2145, (1972)

108

109

18. D. Favart, V. L. Telegdi, et. al., Phys. Rev. Lett. 11,
1336, (1971)

19. T. Fulton, D. Owen, W. W. Repko, Phys. Rev. Lett. 18,
61, (1971)

APPENDI CES

APPENDIX A

DERIVATION OF KERNELS

1. Single Photon Exchange
¢,‘/K’

 

79 94’
Using well known Feynman diagram techniques, the above

diagram is written:

1019/79; X: 71,) : - el/fl4yfl1,*X/7X.l/l('l a.) p {7" Jag/Z: (bk-51), 045))

; (".62 fiffid?d4f/fll4é A?” [/flny/y Ik/X/‘;) (ya/(111;)
" (272’: 4 4 4

y 1' VII/y- IX}? H/ / / //
'X g; gikF.M0};(/Offi/
(A.1)

We introduce the center of mass and relative coordinate

system (see (2.7)):
’ ' ’ _ (‘6‘ 4 4 4/ 4 ,/ (r) . I H
1697/ XX ) (27" fdgd/WZX/ /” 3:“ IBM/"55‘4”
' ’{1/ “ ' k '; - ' ’ . ,,
xelf ”XV?! /7?(;jvezz/ /’[,X%i/Oel/ 9717,17

x gly’lé'fflzfl'i/‘WFVX”

(A.2)

110

111

-zrfl'X/

But I(x,x') Ef/X’e 109/1212C) (2.8c)

so performing the indicated integrations on (A.2) we obtain:

( +—é+ +}W .292:
(70 (Zr/“WWW— 1)

Ibex) '-' [AF-)5
Now do the trivial p integration. Then do the y integration

obtaining (2n)“6(k+p'—p”). Then do the p” integration:

1(« 1): (aw/f1,” ’flp

MW

fl) FIZ/FéZ/glL“ £"(F/’7Z.K/(’xexy

 

 

 

(A.3)
Now change variable p' ’11_,.—»- F + 7,1; and do the p'
integration:
0 féx'
. ’ (2)1364 42 a »

I“ X) : (1:)4/dé a“ b/uA/F 2 ) JYK ’X) (A.4)
Insert DF = SUA/k2 and let Fk(k) include a phenomenological
magnetic moment for particle 2:

__ . 0’ R/fl‘) <12 (A.5)
50/ : (712/21) — 2% 03,1. I
Thus:
4:4) 4;? [w (I/ 2
{my} ’ 2/ 4‘6 1) Mat/1 ’2’ 8(1) w (2/

 

112

2. Ladder Diagram

‘7‘ ’Y;
Y 1
l l
' 1
I I

m
”AW W74"

As was seen in Chapter II. the two photon ladder kernel is
a second order perturbation expansion of the one photon
interaction kernel, Kk (see (2.21)). 1k is that part of the

one photon interaction kernel (A.6) that remains when the

Coulomb interaction is separatekd out, namely:

1(‘/éx/77:24TC0(C/‘6('Xj(;{7:jwefFZZ/XUI/w

(I)

   

   

       

7

 

The two photon ladder kernel is then given by:

SEA/>919 : I 651) ng;)1‘”,x ' (A.7)

4 , l
where é«(1,«7:fld—é «Q ,0(7< ”5} 0+ mew PMZJZ')
Thus

. I (4475“) dm Iéqz’e {Ll , 2 2
IK{1,1):-———//é7ufi(zflg 12(Mé()24()%:_51’357+

 

Wm 2M :31 ) 1W

(A.8)

 

113

which is the desired result.

3. Crossed Diagram Kernel

 

 

 

 

The indicated idagram is written:

Ibo/Aw 7/) =e fdgdzd”; ’A‘e’ Tiff) (X27 (7;; 93% 1, 7-,)

W (2) / n , ,
$372 -29 1x ’2. (2733 'x‘2/2‘%F(7(17)Y>)H(51-79)

(A.9)

4; / 2o - /
We set F1676 0” j Ejzf/ffl‘ffi3(11‘g)£l§(y-Q)I}"/éji)

(it/3

and take the appropriate Fourier transforms:

I(’X.j¥z)xl;xl/) _ (ii/:2 dydy/qu”(42J43’d4ft714/[471’Zé/4 y’éflfl/

x 5/222; 2’) Sitwsfép/ WF {’7 7/12 W, m p... . (W
e 2 {(x. W21“; 4217473 Irlj’e {/{2 -272 2'1/y2y7

1' ' l' J {1((14' '1 " - '
€¢Hj°fl£ WfizHg 7,)
(A.10)
The y.y',z,z' integrations yield:
{Zr/m/y'é’+§/+[)//-7+f-Z/a/(‘f/t/y/A/fll" ’7)

Doing the now trivial 5,5', n, n' integrations, we have:

114

(2/

I(m,o<.,x,”)x. = ‘2‘“ If”; I’d WW}; ”77 7:04} /)§p(f)S (NAM

2(k+,n}’xl

 

r' 0.; M29.» 41.) b. (m

{(P+kl)?(ll Z. (P/+k/)’X) l'(f/+L/¢z./ (A-ll)
2

<2 J2

We transform to the center of mass using (2.7) and then in-

tegrate over X' according to (2.8c):

 

 

1m)- 43f; J“ UM I;V,4;;a)sébsfi M,“

(I l I 2k 4» '_
Ctr/yzéjjgahébpgficw/fi (+P+r L 71)}:

fifl’lt/"(I/F 7“ -7z,1€)’x ,2 [Mn/5%,!) +1,/< -7,471’

, , (A.12)
(27;)4 0/0 +f+é +é ’14:)

Do the p' integration and change variables p1.» [9 +”Z/<K'£"£9

4 , , 2/ (I/ rv z N
:m : fgjfmm W was. misst-ww)

 

Karly—521‘!) guy, It {/2} DM’, r M} £17014: “(UL 1,52%
£1‘(’7tk '71L7/X/
(A.13)

where i: a K—je-fi’

We now insert the appropriate vertex functions:

 

m« . 9 m 2(fl) R
rig/fingefia/y) _ LIV ’L)

2m

and obtain:

(41m)z [1141’ [hi-75L) [Mi'tth’
Z ,2

 

 

167977: ‘/7Jt)? W

X {E&17F'(£7 y’lrv(z)gf(¢_,x) 341103140/
, HZarM 311/35“) 63: (My 26% L4
+ wmu Mr (Ty "CL' (.1 m “W

flan; ’3 PC
_ (fl/W X100. ML; 4.1L?! ~76) raw/«01L 3
4%,
(A. 14)
where

- ' = l l r J’) / N (2 N

 

APPENDIX B

SPHERICAL AVERAGING

We outline the proof(l4) of the statement
<§m£ °“"'.§> —-> 34,625 (“to") which was used extensively in
the calculations. Consider an integral of the type we en-

countered in Chapter IV:
5 a ( 1 3‘ AB 041/1 0*” (3,1)
M: MAM A, 1M, ~~M~2

where A and B are momentum vectors, and f an arbitrary
function bilinear in A and E. If we subject all momentum
vectors to a simultaneous rotation, R(¢.w,e), where ¢, ¢, 6

are the usual Euler angles, remains invariant. Thus:

J:/d%dgfi {043; Ag, -~)<Q‘"’.2r4)<,aj”. 2g) (13.2)
We integrate both sides of (B.2) over the invariant volume

for Euler angles, dQR = sine d6 d¢ dw
J/ (mg = fd1d3Bf(A331iii/"IJ/dIZASTfl-jflfiflq‘?£3)
R e

(B.3)

1 ”l {2/
z 521%: fA’AfB-RAiB ,fi'§,~)(7;§ 4942411234.
e
The integral over R can be written as a general second rank
tensor:

(H (1)

0:0; {C,J}J-+C,A;BgԤ (B-4)

116

 

 

117

where Cl and C2 are constants. However C2 is zero which can

be shown by the following argument:
fang (rag-mg}. = My. MA; 83- (3.5)

Now perform a further rotation R' such that

 

A + R'A
(B.6)
B + R'B
Then (B.5) becomes:
faz'glwee'yjo/ng = faa'gmae'yflnm,
(B.7)

= C, of! + C, A; 33-

The second term in (8.7) is not invariant under (B.6), so
C2 must be zero.

Thus (B.4) becomes:

/” m f I (B.8)
Off] C,/ 2 C, 0480.12)

(/ M N

To evaluate C1 we take the trace of (B.S):

71 [(6/1], (Eggdfig -——- Ta C /~ = 34,

I [J
,flflfl' (€95) “IQ/L

xii/Mr
A3: (YT?)

H

r I

H

E?I‘

—:>C,: 3

A2
(13.9)

 

118

Thus using (B.l) and (B.3) we obtain:

(Taxi? gin-32> .4, 3LA '3 <0“". am) (3.10)

The remainder of this section lists some general
spherical averages used in computing the values listed in
Tables II, IV, and VII. Anticommutation relations can be

applied to find related spin averages.
< om Man/{egg —-+ i A .3 (NH-0))
<0” Kalgwm W») —;> -— (A xc) .Q Q) (N. W)
(N mg «{(U # ax - i .
@5112: f.“ 9‘ 6C £5 3) + {-3— (A'QXS‘D) 3i (4%) (92)
+ are ya -.2) $0.1 My); <°"”- 0“”)

<d"?o<"’o(".’/I%"’.J§> —> -§- .423 <<rW>

 

<0(”5M‘)0(”O4 a”?o<(2)p((’/.B > —+ g A «,3, <0“. 00>
<0” a“) dry/4 “(2230({1/“(0 > ..., EA ~ 0.0,. 041D
éqrfibwrw> ~+ graewm»

<09” (”,5 afiflazf> 9 - 3g 5 .3 <V”/.V"/>
<0< 5' ’09“’r“wg“’m ”a (”12> .2 a a w (70>

<mfl/j/QSWA km}; Kan/A”, ”(2} -——> é A .5 <U‘rtl.o-(z>

)Sl

wwwb> fracmwaoww
+§ (52%;? .c) —§—(A'2)C§'2J§<°“"0">

APPENDIX C

INTEGRALS

In this Appendix the derivation of the k0 integrations
and the momentum integrations used in this calculation will
be given. We start with the k0 integrations.

The basic k0 integration for the ladder diagram is of
the following form:

( ~(¢(’A’ A
All—IAMW/r, mA/Wm

(c.1)

Iced.

+4

-a°

 

where we have included one form factor of the type
KAz/(k2 + A2). We will show that, in fact. any form factors
are superfluous to the oaslnof1 calculation because terms
with the form factor parameters (in this case A2) are not
singular enough to yield a logarithmic contribution.

The integrand in (C.l) has poles in the upper half
plane at k0 = ~Ikl + 18, k0 = -Ik'| + is,
k0 = —(k2 + A2)1/2 + is, and k0 = anO - El + is. We then
close the integration contour in the upper half plane to

obtain

119

120

A3—

I “r: K 2 L‘ I" 1

Al
+
. ~2IL') (La [UH-N)? L" (”ZIVO’EW'VVZJVE’)

l

A
+ war)“ (it. v ww- (kw:<ww)(<v+w"‘-«<Ma

A‘ .
+ N
[(71% ”E’)‘ “(41][(’ZIKO ‘5‘)" Lu J!”Z.VO_E')2“ k1_Az] ['7' [(0 —El+{,§ 4:72];

 

(C.Z)
Since all logarithmic contributions come from the low mo-
mentum limit of momentum integrands of dimension [momentum]'2,
we can set an0 — E1 2 ml - E1 - uaz/Zml 2 ~k2/2ml - Yz/Zmz.
Moreover, A is at least as large as m2 so that we can expand
in powers of kz/A2 to obtain the leading low momentum behavior

of I++.

 

 

_ _ . 2 I'm 114/2w M
I _. K 21 fl - l
H ( 7.) {IN/mun a 2M 12290ka

(03)

 

.. a + i
A’ [HWY-W9 ZA’ lkl’
I

The important point in (C.3) is that the form factor

121

dependent terms (i.e. the A dependent terms) are two powers

of momentum less singular than the A independent terms.
Explicit calculation has shown this to be true for all k0
integrations encountered in this thesis. If the momentum
integrands are never more singular than dimension [momentum]'3,
then the most singular A dependent terms will always be of
dimension [momentum]‘l, and hence will never give a logarith-
mic contribution. This is in fact the case. The [momentum]‘3

5

terms in the two photon diagram give part of the a contri-

bution, but there are no terms more singular than [momentum]'3

in either of the two photon diagrams. Thus the aslnoc‘1

 

contribution from the two photon diagrams will be totally
independent of the proton form factors. This is a valuable
result.

A few examples of other k0 integrations where special
care must be taken to isolate the two most singular momentum

terms will be given below.

 

l2k! " l
fd’é) ’ M. (75,16 51%)] [I +QZJ. E. +29]

Poles in the upper half plane are at k0 = -|k[ + is,

k0 = -|k'| + is, and anO - E1 + is.

 

122

 

I“'=;Zf m{1 -/k/
H 2W< Mpg L'Wkw, 72%)(14- (E.- 1.14))

_ {L’l’
214.4 (at flaws-72m.) (Lg- (E. «M.»

 

 

 

_ MI. E) E
((IZ/za—E/Z-I‘XQZM 'E/Z— [flan—E, MAI. ~15)

 

’ E'— K K I /
but F———— : _‘-< _ ————’ IO )3 "‘ ~»--—
IVE, - ’ZlKa k ‘ k-f E,’q,‘(o k 02’7";
using 71%., —: ”11,: gr: / E,~zm, c ki/Qrm,

Keeping only the two most singular terms of I++, we obtain

In"? Z( l #1” file)
+.‘ ’t‘ 2mm erw + m + I (C 4)

 

The calculation for the crossed diagram is similar.

 

#2]on I I 1
-/£/ I [U1 [I-(r,m—a+xe)] [I*(%l(.-E.’+ze)]

Poles in the lower half plane are at k0 = Ikl - is, and

= )k'l — is

123

“H?

 

 

I :3
+2“? )Zrzw mm ...) Wa— w m a” an...) t... -...)

. ___”L__( I (El-WI _ 5"“
z—(fim)[1(k'tlfi) L‘ I Wit-“Jo m)

L' <- E , ’
4‘» _—z— 1‘— I’Ml — ... #2.;
2 (W‘- W k' )1'+E.~M.>(I HES-m)

 

 

Keeping only the two most singular terms at low momentum,

we have,

fl/22fi__r~___, . .L -. .
” {2&1meer Ar”... +,.M;)+0(L)§ (cs)

And in like manner,

 

0&4 l f
151% w [12. mama -..); 1.4. (12./(.... a);
Poles are in the upper half plane at k0 = -|kl + is, and

k0 = ‘lk', + :18

j: l

 

,_:l2ni
{PQL ([214;ch — (E,+m,})(~(¢ {Ea'may}

+ 1 I
'2‘; (r- V) (—L’—€,+/m,/){.L” (5; my

 

124

but

 

(1+5, + m, Emu. lad-Er-r/m,

‘ l:
Eff/M. £E|+M,)x

 

U}
H10

 

‘)

+ , r0 .1 o- .
-ZQI (kit h9<EHMJ E,’+,».,) E, 4m, "57:

 

 

 

which for small momentum becomes

 

ii”: lnr‘éA A , _ A /( +_;_ ‘
gmrm’u 1&1HLI (lk'l‘k (Ll) gMIM1(kn-—l¢z) \EI‘M": Ellflflx)

__‘_____ (,L. w -.
+ X’V’Lv’mt (kl: La) Elva": + E;'+nm) + flé)i

 

:- Tzi I ‘1
2 {Yumltk’flk/(IEHILI) ““91“; (m

The rest of the k0 integrations are listed in Table V and
in Table VIII.
We turn now to the momentum integrals (Tables III and

VI). The general form of the momentum integrals encountered

125

in this calculation is

f 3M3", F049
(FEW ‘(r“+3‘7’E.<we;,(rm~.er

 

(1':sz (c.7)

where F is a function involving the spin averages.

The first thing to note is that for both p and p'
>>y(=ua), (C.7) is simply of order us because the inte-
gral itself is independent of a. Thus the logarithmic be-
havior arises from the low momentum region. We also note
that by putting on of the wave functions at the origin in
the two photon calculation, we have forced the other wave
function momentum, p, to be small to insure a logarithmic

contribution. Hence we are justified in setting

E(k+p) = E(k) in the denominators of the two photon momentum
integrals. Similarly we are justified in setting
mi/Ei(p) = l as was done repeatedly in the use of the static

wave function.
The momentum integrations depend upon the use of the
formu1a(12)

/

l __ I 4!: -3A
W1 ”Efl,-w.[“+a”“j”] (C.8)
0

 

which is used in conjunction with the usual Feynman para—

meter integrals to obtain the following expressions:

 

126

 

 

 

 

 

I
ah _ ‘ij‘w I ahL (r-u)
alC (I-W‘ [afipdw+@%fl'@fluv1%
(C.9)
V‘ l 2 L93. d“ M {l 7")
0* I} C‘ [(9 o-(l LIV)" 01:“ J'Qr-OJ'U' +(C— aju {juggmy-j q/z ((3.10)

Equations (C.9) and (C.lO) are used to expand the denomina-
tors of the momentum integrals. For example, (i=l,2)
‘3 3 ‘ 1 (1
:0: drdf Fr
(PZ+X))1(PIL+X11)1JJI_41{L E

’ijjr) I!” P Fig/”‘6 “) (“MI/i0? W”) M1171

where Ai = (l—u)[p2u + miz + (y2 - mi2)v]
= [l—u)[p2u + A1] 2 0.
Let p'2¢(p' + up) and integrate over p'.

v<17/M/sz/zf(%a 4::ry:)4 TC ELM + M?

:sAf

 

127

The integral over p is then done, yielding

4M l- usaJ‘W 3J‘ at elvufi Lt] _££_ ¢Ln
:%:f ( ) I’d/)4 2( 4190. A"1 +(I—W‘EZ‘7/l‘1’r

The parameter integration over u is trivial. The V inte-

 

gration is

I'm!» ’ 44.;

(«’01 _. —- _‘_I_
o w) ... —»~

aql'xl 2 U
+ fime-VMww- d

 

expanding in powers of (y/mi)2, since y << ml, the leading

logarithmic behavior is
i 2.41. -/
”he [Ag 7 + Affl 7

Since the coefficient of the momentum integrals contains an

as, we need keep only the Inca"1

term. All other terms of
the V integration contribute to order a6 or higher. Hence

we can write

I
Mlv _ 2 g, p
f, WWI/LA .— ELI/g4” (1:47)) (C.ll)

128

The other momentum integrals for the one photon ex-
change (Table III) are done in a similar manner. The general
method is to integrate over p' (using the Feynman para-
meters), integrate then over p, and finally do the parameter
integrals using (C.10). Rather than go through all of the
details of the remaining integrals in Table III, we list a
few helpful p' integrations that are necessary to the re-

maining derivations.

13’
09'1 M lfl W

djp' MM"! I_u __(f
LIP +34] )4 M45 ”(I‘d/VJ /f4,9'/;1:\L£1

2d;j:£____,' %' i. .
f(V3X7‘}¥-¥I‘fi’ ’ IZMfi-u) d0 iii.-

-4/jl/z AL).

OFF, \
[whwwrrra' fwfl) If% 7?

0

 

-mfia(2% fi,[fls —Jij

 

 

\:\

$ij12: [MIC] zi/dlufi, fi/Uwér/LA‘ J.

 

129

where A; :(I-MJH‘M +w;’ +(a‘l-mfpr] = U'WEP‘“ +Ai] 20
Similar methods are applied to the momentum integrals
For example,

for the two photon diagrams (see Table VI)

dig/001% 'pl
(p‘+3‘j ‘ 5.0:) ad.) it [ Lug/t

djé 1L1 71‘ rt2 -/ k
: fewefi) la 1 7C _ — + i W (F);
03% few" (W

x { (4’43“) X
: EML - 4“} l+<25
$0 €130,544) 8‘ o EAL) EAL) L of Elk) 51¢) k
0

 

 

The first two terms contain no logarithmic dependence.

while the later integral is evaluated by integration in the

complex plane.
-, (‘1— W).

5f; wit—1L :Lfo __:(+eoe/r
56912 f A: a, Mnj/z

. fiw , if" .4. (mm)
2'0 ///l°‘”“‘/‘ 20 fo‘wnW‘

Change variables in the later integral, q-z, q

A (I— z‘z/r)

I : fi/XZ fight/my”

,¢>
In the complex plane we have poles at q = 0, and q ~

 

130

The square root branch cut is indicated by 0000000000
while the logarithmic branch cut is indicated by A-MAW~V

and there is a logarithmic singularity at q = —iY

   

' 1““ CAM/LE?
PLANE

 

 

 

(h l
’1 K4 I
2< w
>
<1
<
<7
‘ \
‘ég-lrm
28
7O
O
0

We close the integration contour in the upper half plane,

and use the residue theorem.

 

2%5- ( ff“ (- f/rl/ Kyla- 2W) 3; M0 zf/N
5*;{;+«/* it] Mammy“ . 1 ECWQWyg

QURD
co @m

:‘éZEJEZJ E22, —
1 .

We note that the function just to the right of the branch
cut is the negative of the function just to the left of the

branch cut. so we have

131

. ’° ”albeit/J”) L_____(/-zr/N _
#ij f”; ;[;+——TJ’ '0

 

Now let y = eiq in the later integral
L !¢(/— lat/g4) A0 + 47W)
210:? f/f1+%]/a:/;-?(Tl_mt)yt

AWL m [A4/r)+A(I+X—)
__—___T“‘—7_—‘ ——~——————______1L____
Pffl? '7 /g12',”44j :l/é; ?(?"£l’ m1; 7;

MEI/M (,0 y
20°]th J/ Zfi'Wm

Comm 30725 To 01%?

fl w
”'*’\r wZ1xfllfléfwh 4— la X
4* 7/?” M imam/"1

 

kfl—fi‘l
CbflfluswaiTa fig?
Tr -
“‘5’ ;;h 12a?’

Thus to order ocslnof1

we have

tam/w T -,
I fd;——— W h 12%“ (IE/,1)

(C.lZ)
An identical procedure is followed in performing the

remaining integrations listed in Table VI.