PHYSICS REPORTS (Sectton C o f Physws Letters) 3, no 4 (1972) 193-260. NORTH-HOLLAND PUBLISHING COM"~,Ny
RECENT DEVELOPMENTS IN THE COMPARISON BETWEEN THEORY AND
EXPERIMENTS IN QUANTUM ELECTRODYNAMICS
B E. LAUTRUP and A. PETERMAN
C E R N - Geneva
and
E. de RAFAEL
lnst~tut des Hautcs Ftudes SclentqlqueL Bures.sur. YI ette
Received 1 August 1971
Abstract
This review is a survey of three main topics in quantulll elettrodynamlt,~ fundament,d I,ound ~ ,~lem,t anomalous mab'T~,I~
moments and hzgh energy expertments The emphasis hes parttcularly In recent developmenlz toncernmg the electron and tnu,m
anomalous magnetic moments
Single orders ]or thzs ~s'sue
PHYSICS REPORTS tSecuon C of PHYSICS LE'IffFRS~ 3, No 4 (1972) 193 -260
Copies of this issue may be obtamed at the price given below, All order~ ,~hould t.~e sent duecth to lhc Pubh,,htt Order~,
be accompamed by check
Single ~ssue pnce Hfl 20 - ($ 7 00). postage included
t._.
PHYSICS REPORTS
A review section of PHYSICS LETTERS (Section C)
.',. x "k
,-.
,,#i
V o l u m e 3C, n u m b e r 4, M a y / J u n e 1972
B.E. Lautntp[ A Peterman, E de Rafaei, Recent iei'eloptnents in the
com l~ntison betaveoi t,~eorv and expermients 11 qttantum electrodyrtamics
0 Introduction
1%
pp. lO3-260
'~ 2 "1he ~ixth order t ontilbutlon
226
O t h e r c o n l r l b u t l o n ~ , I,~ lhe m u o n anotll-
A. | undamental bound y~tems
1 The hydrogen atom
1 1 The Lamb ~dafft
i 2 The fine structure ,~E(2Ps,2 2Pc,l)
in hyctrogen
i 3 The hyperfine structure
2 Po~tromum
2 I The fine-~tructure interval of the
ground state of posttronmm
2 2 The anmhdatmn rates of orthoposflromum and parapo~ltromum
3 'quomum
3 I The h~ peril le-sphtun~ ol the
muontum ground state
198
198
198
204
205
209
209
aly
6 I 1 he hadtomc contrlbuil~,,~
6 2 The weak rattrap.Iron t.untrtbutnm
6 3 Measurements of the muon anomaly
"'['xotlc'" contrtbutmn~ to the electron
and mu(,n anoma|lc~
7 I GenerMltles
7 2 Modlt,catmn,, of quantunl electro-
212
7 ] ~tl~ge*,It'tl t o u r l I P
t,.' S,c~I I t.
parlltle,,
ttllll
8 I (. olli,.hn~ Dc.,ltllx
q 2 Ileih< |l,,.'llki l'..pt t \ l K l l l l l t t l l x
f
t
224
225
24 !
( Iilgh enetg,, exptrtmenls
8 S~.attcrlng experiment,, at l~lt~h m o m e n
the charged lepton~
213
4 t h e ele~.tion anoiual~
216
4.1 The ~econd and fourth order contxibutlons
216
4 2 The ~lxth order contribution
217
a 3 Measurements of electron anomaly 222
5 The qttantum electrodynamlc~ contrlbutmn to the difference between the apomelectron
5 1 The feurth order contribution
238
2"18
I, o l k.plon~ l o
B 1 he tntln,.ilou,,, llldgTlelll, llltltnent,, ot
alous m a g n e t i c r,,oment,~ e" m u o n a n d
236
24(I
d% n , i n l IC~,
21 I
2!I
233
233
235
[ 1,1 rl',,,l i." r
245
245
245
249
Conclusions
252
Acknowledgemenk
253
Review articles on quantum electrodynami~.s
References
253
--
Single orders for this issue
P H Y S I C S R F P O R T S ( S e c t i o n f' o f P I Y S I C S L E T T E R S ) 3, i~o 4 119721 1 L,
250
Copies o f this issue m a y be o b t a i n e d at t h e price given b e l o w All orders s h o u l d be
sent dtrectl) t o t h e P u b h s h e r O r d e r s m u s t be a c c o m p a m e d b y c h e c k
Single issue price Hfl 20 -- ($ 7.00), postage mcluded
196
B.E. Lautrup et aL, Comparison between theot9, and expenmems In quantum electrod),namics
~. Introduction
In this rev~e~ we shall be concerned wtth rece,t developments m the catculatton of qmmtum
electrodynanucs (QED) effects.and wtth recent cxpenments which test the vahdttv of lhc theor~
of QED. From the onset we wish to emphasize that by theory of QED we understand the precise
sdt"ofrules~whieh stem from the fundamental works* of Feynman, Schwinger, and Tomonaea;
and which allow us to predict observable effects o f the interaction o f light with charged leptons.
Possible breakdown o f the validity of QED has been searched for in two main d:rections: (i)
precision measurements in simple structures like the hydrogen atom, muonium and positromum,
and measurements of the electron and muon anomalous magnetic moments; (it) scattering expert.
ments at high momentum transfer. Experiments of the first type attain accuracies which in some
mstances** are of a few parts per million (ppm), and they are sensitive to higher order terms pre.
dlcted by the per!urbation expansion in the fine structure constant a(u -~" I/I 37). The second tyl~
of experiments involves large momentum transfers, typically of a few C~V/c. and they specifically
probe the vahdity of QED at small distances {of the order of a tenth to a hu,ldredth ol a fermi.
1 f m = I0 -t~ cm)
H~stoncally, ~t was the measurement of the 2St/~ and 2Pv~ level displacement in the h y d r o l ~ n
atom*** which led to the development of quantum electrodynamics. Since then, parallel progre~
has taken place in h~gher order ~.alculations and m the accuracy of expe:tments in their first e~.
pertment, Lamb and Rethertord [0011 showed that the 2S level of hydrogen, whtch tn the theory
of D~rac comcMes with the 2P~/~ level, was actually higher than the latter "by -bout °' 1000 Mltz
At present, the experimental accuracy of this Lamb shtft ts ~ 50 p m~ and ~,, ~cnslt~ve to correcttons* of relative order cJ Prectston expertment,, of energy level sl~h.,,n~ ha~c rtow been rn.Me m
hydrogen, in deuterium, tom,,ed heh,tm, tontzed hthlum, and also t n o t I,er h~ drogen-hke ~ ~tcn,s
m U O n l U m , positron,urn a r i d / d - n l e ~ l t atom~
The exl~.rime, tt of Kusch and Foley 10021 whtch I~rsl gave .hrc~.t In,ht.~tton ol an anom.th,u~
,nagnct~ n~ome~l tor the ele~tro~ wa~ ac~.urate enough (,~, = 0 0 I I I ~, -, 0 (1004~ to col~l'trm
Sch,lnger'~ t.alt.ulatton r'r it(e= a/2r, ) ~t pre~ent the .t~tur,tt~ ol tile Illt.d~.tlrt'llI¢ll| ol ,,c t', l t'~|)l!
and an adequate t.ompansoh with the tileory retlutres lhe tllcittstotl ol the st,~th order contrabu,.ton
to
ct~tt.
Does the muon have an anomalous inagnettc moment as predicted by QED" A positive answer
to this maportant question, whtch bears upon one of the greatests puzzles m physics, i.e. tile ongm of the muon-electron mass difference, has been given by beatitiful e,,perlment~ performed at
CERN. The accuracy o': the first g - 2 muon experiment 10041 was 4 x 10-~ The latest gu " 2 experiment [007,008] has attained an accuracy of 28 parts m lO s and has clearly confirmed the
vacuum polarization terms wtth ~trtual electron parrs predicted b7 QED + The accuracy ~ tll xcD'
likely m~prove b~ a factor o! 10 to 30 m the next generation of ~-~,- 2 experiments ++
*I here exist'., a ver~ useful compfl.~tl~ n o t the earle pa:,:ts on q u a n t u m el¢ctrod~ natnt~.,~ edited b~ S,.hwmger, ~ec .I S~ham~.'e~
mth,: h~t at ,e~e~, arttcle~ at the end, ref IR 1 ~,]
*"For ,he hyperttll~,-sphttlnS ot the h~d.,,gen ground ~tate the prtsent expertmevtal a t t u r a , x t, i tuall~ i 2 |,art~ t,~ 1o
***See Lamb and Retherford [001 ]. r e p r l n , - d m the compdatlc, n q u o t e d tn ref [R I $ ] . paper No I !
~See the d~scusslon m ~ectton I I
**See $ hwtnger [003] .r.~'prmted m the cornp.'atton o u o t e d , t re! IR 15], paper No 13
tt'~See the dascusston In section 4
+The calculatmn of tlus contribution was first dt ne b~ Peterman I OOS]. and by Suura and ~,tchman 1006]
see the dsscussaon of wctton 5 I
+'¢'Prwate commumeatlo,a from John Barley, ! rancts l-arle~ and ! mtho lhc.'ts~
I-or further ,h t.d~
B . E L a u t m t p ct ai . Compar:son bet ~vecn t h e o r y an t t pc, : m e n t ~ tn q u a n t u o t e h ' c t r o d v n a t n t c s
197
The understar, d m g of the g ,
2 experlme=~tal recult ha,, been a ch,=llenge to theoret=cJans whluh
h~J~ let] to new d e v e l o ment~ m computat~on.~l te.hn~quc,, An early d=~crepa,lcy between ~.heor.~
a n d c \ p e n m e n t m o t l ~ a t e d t h e c a l c u l a t t o n o t : l i t h e t e r m , , p r e d l c t e d b y Q E D w h l u h at ~lx,h o r d e r
make the ,inomaly el the IllttOll d , l l e r c n t from th,tt ~I the electron l h e rc,,t It el these ~.al~.t~lattota,,
has brought the theoretical p r e d l e t m n a g p m w~ttun t ae e r r o r s o f th~ e x p e r i m e n t . *
An m l p o r t a n t q u e s t m n a b o u t the prec,s~on tests o f Q E D is the accuracy t h e y can a t t a m b e f o r e
they are also sensitive to the e l e c t r o m a g n e t i c i n t e r a c t i o n s o f h a d / o n s T h e e l e c t r o n a n o m a l y a n d
systems like p o s i t r o n i u m and m u o m u m are stdl far f r o m being influenced by the e l e c t r o m a g n e t i c
interactions o f h a d r c ns. This m a k e s t h e m the more e x c e l l e n t candidates for f u t u r e precis~on test,
of QED. As we shall see the s i t u a t i o n is different for the L a m b shift, the h y p e l f i n e s t r u c t u r e m
hydrogen, and for the rouen a n o m a l y llere, an a d e q u a t e c o m p a r i s o n b e t w e e n t h e o r y and e,,per~ment requires the k n o w l e d g e o f c o n t r i b u t i o n s du¢ to the e l e c t r o m a g n e t i c mtt~,acl~on~ el hadron>
It I,, remarkable that m some cases, h k e the | n u o n a n o m a l } , tile hadronl~, c o n t n b t | t , o n s can be related to e m p | r | c a l informat=on already ,iva=lable from the lugh cnclgy e l e c t r o n c x p c r n n c n t s A very
mt,:rcstmg hnk b e t w e e n lugh energy e x p e r m a e n t s and pre~.~,,ion ',ow cnt.rgy experm~ents ts ti~t.rcby
de,,clopmg
I he lugh e n c q t y CXl~'rtment,, arc also t, selul l r o m a n o t h e r po=nt el v~ew ['hey can test the vahd=ty of the Q E D rules for the l e p t o n a n d p h o t o n p r o p a g a t o r s =n proce,,sc,, w h e r e the Born a p p r o x i mation Is a d e q u a t e yet the m o m e n t t l m transfer,, revolved arc large (ol a Icx~ (;eV,~ w~th the pre,,f i l l machines)
l he mare purpo,,e m writing this rev=ew was to dlscu.,,,, the re~.cnt dcvclopt;.ent,, t¢ nt.crnmg the
anon~alou,, magnetic, m e n , 'nt o f the r o u e n "lb=,, ~,, doric in ,~ct_tlotl,, 5 (~ and 7 (. lcarly th~s rcqu=rc, a parallel th,,cu,,,,lon el the e l e c t r o n a n o m a l y a,, well, ~ h l c h i,, m a d e =n ~cct~ 4 and 7 We
fctt. however, that m o r d e r to k e e p s o m e balance, there ,,hould ,11,,o b c a br=cl rcv=ev, o f , g t h e r tundamcntal Ol l) t o p t t s ~,,Iwre there ha,, been some re~.ent tleveh)pmci]t,, Wc hJ~c thcrctoic m,.lu ted
a Itr.,t part on low energy te,,l,, :'1 tund.t~zlcl~t,tl '~3,,tcm', ~,ct.tl()n., I 2 ,.tlltl 3 ,~lltt ,i third part ~n
el 'rt)rl ~nd phott)ll h,gh c11erg.',, cxpcrlmctlt~ ~,'~t~on ,'-;
' I , r c I'~ ,ll| 1111p(~rl,tllt t¢~p~ ~.t. ll]l IIll[~[It,lllt~ll,,. 1¢~1 (,)[ l ) ~.~,lll~_h I,, 11o1 t l l ' ~ t l s ' , , c t l Ill 1111,~ IC~,lC\\
I c
tl~: d t l c r t m n a t t o t ! el or from tt,c a t J o s c p h s o n c l t c c t . I he reason for tills ls that there alr¢ad3
e~t.,,ts an e x t r e m e l y detatled e~poslt~on el the s u b l e c t * * to wluch we have n o t h i n g to add Ot
course, m all the n u m e r i c a l estimates m the text we shall indicate the a p p r o p r m t e or~u~n for the
value o f a w h i c h ~s u ~ d
We should finally hke to p o i n t o u t t h a t there arc various cx~.ellcnt review article,, o r t h l f e r e n t
aspects of Q E D m t h e h t e r a t u r e *** In writing th,~ review we have a t t e m p t e d more to tt)/~t/~l~'t~c'Hf
th~ ,dready e ~ s t m g h t e r a t u r e r a t h e r t h a n to write an en~.,~clopaetl|c rcvlcw el Qt:l)
One c o m m e n t a b o u t i~otat~on,, We use tile ~ame mctrlt, and l)wac matr~ce', a'~ m .I D Blorken
11 d S D Drcli's t e x t b o o k , , (M~.(,raw i-hll, Ncw-York~
• I tu~ t~ JlSC'u~d in dctad =n ~ t i o n 5
*°See Table=. Langenberg and Parker IR IOI
" ' ~ ' e the list of re~mw -,tilde,, at tnc end
~
l.~ucrup et aL, Compa~on between theory and experlmentt in quantum ¢lecWo~nmn~
Part A
Fundamental bound sys:ems
.... In tim firs.t_part, v~e.shall constder three fundamental bound systems: the hydrogen atom,
muonhtm, and positmniurn.* We shall review the present status in the comparison between the
ffte6i'yand.experiments concerning: the Lamb-shift in atomic'hydrogen, the fine structure in hydrogen, the hyperfine structure of the hydrogen ground state, the f'me~tructure of the positronittm
8mand~atate, the annihilation rates of orthopositronium and parapositronium, and the hyperfine.
splitting of the muonium ground state.
1. The hydrogen atom
The basic features of the lower energy-level structure in the hydrogen atom are summarized in
fig. 1.1. In the Dirac theory, the degeneracy between the n - 2 PW and Pt/, levels is removed by
the spin-orbit interactaon. This leads to the fme structure ~E(2P,h - 2Pth ), proportional** to
(Za)*m, which corresponds to a level splitting of 10 969.1 MHz. The interaction of the electron
with the quantized electromagnetic field removes the degeneracy between the levels n " 2 S~ and
Pbn. The corresponding level structure: ZkE(2S~ - 2 P ~ ) - 1057.9 MHz is the Lamb-shift. It is
proportional to am(Za)41og(Za). Another cause of level splitting in the hydrogen-atom is the rater.
action of the magnetic moment of the orbital electron with the magnetic moment of the proton.
In the ground state n = 1, this leads to a hyperfine structure between the triplet F - I and singlet
F = 0 levels of 1420.4 MHz. The effect is proportional to (m/MXZa)*m.
1 !. The Lamb-shzft
The two dominant contributions to the Lamb-shift can be quahtatively understood m the following way.
On the one hand the electron, in the presence of the electromagnetic field of the proton, can
emit and reabsorb a photon (see fig. 1.2). This leads to a physical spreading of the electron charge
over a mean squared radius ( r 2) which for a free electron is
m 2
where ~, is an arbitrary small mas3 assigned to the photon. In the ~ ydrogen atom, however, there
is an effective lower limit (of the order of the hydrogen binding energy) to the energy of the photons which the bound electron can emit and reabsorb. Qualitatwely one expects that a correct
treatment of the binding effects will replace ), by the Rydberg which is the ionization energy ot
the ground state of the hydrogen atom
• For a review of other hydrogen-hke systems we recommend the reader the excellent review aTticles of Brodsky and Drell [ R 41
and of Wu and Wdets [R. 17].
• *Z is the atomic number, which for hydrogen is one. We shall, however, keep Z m our expressions as an mddtcattve of tt~¢binding
effects in contrast to the pun~l,, rachatwe effects
B.E. Latutrup et al.. Comparbon between theoo, and expenmenrs m quantum electrod),~, mw~
n,Z
19 9
p~
F~
t0tmOMl'~
~ruc~m
- iZ a ) ' m
h=2 q~ .S1,~ '---
he1 St~
~
[
~
I~Iz
F,0
2 Sv=
LambSh0ft
2 Ph
=(Z=P m
Hypecf,neStructure
~ (Z~l'm
r~
t'tg I l Lower energy-level ~tru~ture m the h y d r o g e n atom
X ~ Ry = ~ m ( Z a ) = = 13 6 e V .
an,i
(r2) ~ .¢rl
/ 2 ~loglZa)-',
1"he potential corresponding to this charge density d~stnbuhon diminishes the Coulom["~ binding
-Za/r and as a consequence the S levels are pushed htgher by an amount
£? ~ et(Za)*m log (Za) "! "- 1000 MHz
The detailed calculation l! 0 1 - 1 0 6 ] which also includes the c o n t n b u h o n from the ,momalous
magnetic moment o f the electron, leads to the result (see the fir.~t two entries m table 1 1 where
,.- ,~uut.~:u ma~,s correction has aiso been taken into account)
tlk~
-.AI
. . . .
J
._
_
(~lf-energy) = 1077 64 MH7
On the other hand, the effective potential seen by the electron is modified by the vacuum polafizahon due to virtual electron-positron pairs [ 107, 108] (see fig ! 3)
BE. ].~ulrup tt al., Comparison between theory anal expennx,nnr tn ¢uantum elecoorlymmtcs
"T"
•"il~ 1.2. Lowest order electron vertex contribution to the
-~
lamb-shift
Za
~,here (I/lr)lm
llm
Za+Z.~a 1
f
Fig. 1.3 Lo'swst order vacuum pohtd~llon contriOuth.n ro th¢
Laml~-thlfl.
tmnct,q _c
nit) is the vacuum polartzatton spectral function (which is postttve definite)
II(t) =-~- ~ ' ( i + - / - - 1
I -
Olt-4m~l.
I'he effec~tve potential seen by the electron is more attractive than the Coulomb potential and as
t consequence the S levels are lowered by an amount which turn~ out to be (w~thout tn~.ludmg the
reduced mass correctmn)
.~ (vac. pol )
=~(Za)*m(--l/30)
= - 2 7 . 1 3 MHz.
Experimentally, there are two accurate direct measurements of tile n = 2 Lamb-shift m hydrogen
l'nebwasser, Dayhoff and Lamb [ 1091 * .6'ex0= 1057.86 ± 0.06 MHz,
Roblscoe, Shyn [ 1101 (revtsed)! 1601 ~Oex~ 1057.90 ± 0 06 Mltz
Yhere are also three Independent measurements of the interval 2Ps,n - 2S,/2 Jn hydrogen:, wlu~h
combined wtth the theorehcal value for the fine structure mtcrval AE =
. . , P t ~ , g i v e tJtdttc~t
determinations of the Lamb-shift:
Kaufman, Lamb, Lea and Leventhal [ 11 ! ]
(AE - ~ ) e x p = 9911.38 ± 0.03, ,~= 1057.65 ± 0.05.
Shyn, William,~, Robiscoe and Rebane 1112 i
( A E - ~)exp - 9911.25 _+ 0 . 0 6 . . e = 1057.78 ± 0.07"
Vorburger and Cosens [ 1131
(AE - ~ ) e x p =
9911.17 _+.0.04: .12= 1057.86 ± 0.06.
The accuracies of these d~.tesmmatmns of the Lar~lb-shfft range from 66 ppm to 47 ppm The
terms ~tuch so tar ha~e been calculated are gj~,en m table i . i . A aetatled analysis of the different
contributions can be found m two arttcles** by Erickson and Yenme I 114, ! 15 I. O f particular
*Cc~rgected by Robiscoe and Shyn [ 160]
**For a chscusston of the nuclear recoil correctton~ ~ee also Grotch and Yennte [ I 16J
B.[.
Lailtr~p el aL. Conlpanlanbetween
theo~' and expert 'nents m quantut,, ole(
M)'namtts
201
Table I 1
Compilation of contrtbution~ to the Lamb-~htft 2S,/: - ?t'~/a ,n !,~ arogen* (a ' = 137 036 08(26))
rtkr
De,~criptlon and rc,eren, e~
(orret.t',m Itmlt~ l{/,.,), ~l)
Za)~m
2nd o ~ t
(_21o~(Za)+M+ , I .
Zal*n,
2nd otcki mlsnnellc moment 101331
~¢lf-enert~v [ 101 - 1061
Nunlcrtt.,tl~ Iluc
I%111/)
Ko(2,0)~
)i-'og ~o~.,~](l-3,mM)
~
1009 020 "~
1 - 2 75m/,tlq
I
67 720
} --0 0055
ZePm
-~( 1-3m/J4q
order ,m:. polalitallon 1107. 1081
- 2 7 084
Za)'m
2nd order blndin~ [ 125, ! 16 I 151
(Ta)q3ni(l* 11 1 .) 5
I~'2"8 -~h',g.+ ~-2 )
7 140
:la)' m
4th older btndlnll ÷ higher ordei
1127- 120. I i$. 1581
(ZaZl(a+b lug(Za)Z+c Iot,'z(/o):)÷(/oO31r.9 56**
0 372 ~ 0 (10491
/
HiEhe, o I'¢r, Zo and a unccrtllnty
: 0 '16568
qLaPm
4th orJer Jelf.eneq~ I 119. 123)
3(a/n}O 470
I) 444
II~a)im
4lh ordeI m l l ~ l k " moment 1406.407 i
lain)(
0 328)
0 102
ttTaPtr
4thorck-rvac polaruauon 1130. 5121
I, /ell
41/54)
II 2-~9
- 0 00341
Red n'~,~ ume,~lnty
(Zal' ,~
~m Hrtm
Retoti ~otrectlont [131 I t2 1161
I/m/Ml(a D*b~ I o g l / a P )
~d, ' ¢
Proton ~tze 1114, 116 15Xl
In2,mRtr~)"
~l 12 ~
I,~lO~,..'4
,it
~_ O 0006
PToton $liuclure uncertalnt)
lotal=
1057911 c O 0 1 2
Vl'hts tabl~ is i n Ul~lit~tl vernon of the ¢ompilatton made by A Peterman [R 13] ]'he value o f the Bethe logarithm
Iog~Knl2,1)l(£e(2,011 ii that (ntaluated by Schwartz and TJemann 11061 The constant~ a, b, c. a, and b, ,ire the follo'~.lntz
a= - ~ r . ' - 4 - 4 Iolt* 2 - 0 28~0.5. b ~ -~[ - log 2 . c = -314.b, = - I J 4 . a t = 2 IogfKo(2,1)/Ko(2.0))+97/12
'*'Thts fo mu t" Is vldkl lot the IS irate The state dependent.e gtvcs tt~e to tln~ corrections, of the order of+ 0 014 Mllz
m t e r e s I t o u s . b,2.catl~e ol- r e c e n t
changes
b u t t o n s w h i c h w e di,,cu.~.~ ne~,t
rile QED quantity
w h i c h is i n v o l v e d
lion to the Lamb-shift
in t h e i r e v a l u a t i o n ,
in t h e c a l c u l a t t o n
is t h e s l o p e o f t h e D i r a c f o r m
electron vertex definition
~s ! h e I o u t t h o m e r
o l tile I o u r l h
factor
order
o f tile e l e c t r o n
sell-ellerg3
~ o n It ~-
~ell-.'nergy contribu-
to order a 2 With the
B.E. l~utrup et ai., Comparlson between tbeo~ and exi~'Iment$in quantum electveclynnn~les
b)
c)
d}
e)
t~
g)
Fig. 1.4. Feynrnan dsagrams contributing to the 4th order electron vertex. These are the diagrams which contribute to the ~ ) ~ of
the Dirac form factor of the eie¢~zon (see section 1.1) and t,~ the anomalou¢ rasgn" t,c moment of the electron (tee sectmn 4 t).
~(P+q) ['"(p÷q,p) u(j~) = a(p+q) 3" F, (q') + ~ o ~ qo F:(q 2) u(p),
we are interested in the evaluation of
There are seven Feynman diagrams contributing to 0(4). They are shown m hr. 1.4 and thetr co,Itributions to the slope can be found in table 1.2. The first attempt to this extremely intricate calculation was made by Weneser, Bersohn and Kroll [ I 17] who gave analytic expre~ons for the
oontributions from the diagrams o f figs. 1.4b, ~e. 4f, 4g and bounds for the others. Later on, Sotc
|118] made a complete analytic calculation of o (4). Motivated by serious discrepancies between:
B.E. Lautntt tt al., Coml~ei~,,n between theory and expert nents m quantum electrodvnamws
203
Table 1.2
npdation of conlrlbuttons to the slope of the Dtrac form factor of ,he electron at )ourth order * The gnaphs reler to the
shogn ,r. fig 1 4
,rh~
Analytic rexult,~
~. l.'h)
-~1o1~
~lonlerl~.dl re,,ulls
+l-1"~"8+~*i'~w
~otlz- o
= 2.3925/
1122,
13.
a
-~'gnogg- +2 37~0 02 (ref 11191)
~Sgl
--I---~3tn2>-~~-2 3924-*0 0002 (ref [12.'] )
,30
~r
). 4b)
13. ..s./'319
l--grail^ * ~ - t ~ "
llolltl.).
1,
)1 s
= -!.710
.t.ll$11
)
;'09 : ~ ,
1~ 14c,
* 4d)
if-cwr'D'
.~logX-"
{ref~ 1118. 1591)
\
_ 1 _ ,~ - I 9328 t
i 691002(rcl
l•log•h
:-/Iogh
:
[1191)
1 91~O02ire!
[1191)
llo,:a~. ~-7-~logh-~-I 9 5 ± 0 0 5 ( r e ,
[1201)
( t t f 11211)
I.
,--t+ 1 .
.t ~' 1109+17 ~
)
I,~. - i ~ ~t ~ ~,o~x + ~ - ~
~2" " 1.6ss (~f, I 117,118,
121[)
l ~l')
.. rot4 ~ ! _ _[ v 1099.
O - ~ s ,77~ . ~ ~oo31 6) ,,el. i1,~.11s. 12o~)
-
Io~2x
a+l,
~.og~. a+l 68+-001 (ref [1191)
) 03,6±0 0002 (ref [1191}
ill v)e dtal~ams have been evaluated m the t e~ nman gauge 1 or the analytic resuhs of the vacuum polar~atlon, ',elt--eperFx and
add t dtagrams we have corrected the overall ~zgn error m rcfs [ 117. 1181 All the wesults ate given m umts (~/n)a
lco'~ and e x p e r i m e n t * , Appeiqut,,t and Brod,,ky [ ! I q l u n d e r t o o k a numerical *~ rec~alualton ol
l,). rhcy f o u n d an ovcrall-d,screpan~:y tn sign w~th tile previous calculations and d i f f e r e n t absottc values for the non-infrared divergent terms of the cross (fig. 1.4a) and corner (figs 1 4c, l a d )
raphs Their results concerning the everall-discrepancy in sign and the evaluahon of the corner
raph were confirmed by another numerical calculation made by Lautrup, Peterman and de Rafael
1201, and an analytic calculatior made by Barbleri, Mlgnaco and Remlddn [ 121 ] Recent ly,
eterman [ 1 221 has undertaken a systematic investigation of the work by Soto [ 118 ] and given
n analytic result fer the cross graph (fig. 1.4a) in good agreement with the numerical result of
ippelquist and Brodsky.***
The new value o f the fourth o r d e r self-energy c o n t r . b u t t o n to the Lav3b-shtft -~,
m |
"T j
48!9
"
Ac~
,
.
-,
log 2 - 3,
: 0 4c,9
;
"t ,,t a tev~w o1 the t o m p a r a o n ~ t w e c n tlmory and experiment before Appelqms! and Brodsky's reevaluanon of o(' ) see Broosl
Ia M.
* ' [ or a deK-t~tion of hh¢ lechmques uJed m thctr calculatmn see secuon 4 2
"*The ~ ¢ n t analytic resale of Batlnen, Mq;naeo and Rcmtddt 11591 confirms the results ot Peterman [ 122] and d~sagrees wtth
:el [i24l.
;The complete 4111 order trmqllnaty patti of Ft and F: have been calculated for all 6 : by Barblen, Mlgnaco and Rem,ddl
('quovo Cunento, to be pubhshed)
~,
Lautrup ct el,, C o ~
between theory and ¢xpcdments tn quantum ¢lectmdymmt~
Table 1.3
Determinations of a "~ obtained from bound systems*
Bound system
Discussion, section
~ ~
~FI
l ~ d m l e ~ (tl~tl " ~ H ) ( I )
ll~dmgen, (~i' H -.etl)(,i)
tb'dmlIen, hf$
Muonlum~ b.fs
Value of a '
} .2
! .2
1.2
1.3
3.1
13?.035 4S($9)
137.035 7~27)
137.035 4403)
137.035 91(35)
137.036 17130)
*Far reference, the value o f a -t obtained from the a.c. Joseph.son effect is a " • 137.036 08(26).
when added to the other contributions in table I .! leads to a theoretical value [ 1231
22,h = 1057.911 ± 0.012 MHz.
This includes the recent result o f Erickson [ 158]. The theorettcal and experimental value.,, for the
Lamb shift in hydrogen and hydrogemc atoms is given m table 1.4.
1.2. The fine structure AE(2P3/2 - 2P,12) in hydrogen
Let us denote AEH, the n = 2, P ~ - P ~ level interval in atomic hydrogen (see fig. I.I ) The
theoretical value of AE H Is well known
• the well known Dwac solution.* The appearance
The first term in the parenthesis I + 5i(Zor) 2 , ts
of the reduoed mass factor (1 + m/M)-' is explained m detail in G r o t c h and Yennie [ 116]. The
(m/M) 2 term, calculated by Barker and Glover [ 1331, is the effect of the DIrac m o m e n " , of lt, c
electron and the proton The a e term is the effect of the electron anomaly and ts the first tern, of
radtative on~n Its contribution is roughly 0.1%. whereas the last term, wluch ts also a rad~att~
correction, ,'ontributes only about 1 ppm. These radiative t.orrecnon:, have been calculated b) tth:
a~uthors of refs. 1127] and [ 128]. Bounds to the next uncalculated terms, O h a / ~ ' X Z a P 1, have
been estimated by Erickson.** Since these terms are comparable to terms of order O(a'im/M);
or(m/M)2; etc.) it seems more natural to expand the theoretical expression for AE H in powers of
,n/M and keep only the first powei in this parameter. Thus we get
°),,o.(,
1
16
-M
"
From this equation, and using the 'without q u a n t u m electrodynamtcs value" (WQI~D) or-' =
137.03o 08(26) obtained from the a.c Josephson effect,*** we get
AEH(th.) = 10 969,03 4- 0.04 MHz
Comoinmg the value of AEH(th ) w,lth the thcoretlcal value for the Lamb-shift (see section 1 I
we have that
*See e.g., B*~the and Salpeter [R. 2l
S'Quoted by Brodsky and Parsons [I 34]
***See Taylor. Parker and Lan~enber~ [ R lfil_
B.E. Lautrup etal,, Compa,t~on bet~ven theory and e~:per.mentt
m
quv~zturn elcctrod),namz¢ ~
205
Table 1 4
Larlb-shltt In hydrogent~. ~y~te-,t,, in Mltz units
~tel'n
I ~pcrtment
Theot3
1057911 t 0 O I 2
1 tn'3)
I ( n " 4)
) (,'2)
k ' l n " 2)
le'(n - 3)
le." ( n " 4)
714.896
133.084
1059.271
14044,765
418442
17(;9.088
j"(n-
6276341 ± 9 . 0 7
(783.678 ~ 0 251) x 10 ~
2)
• 0.003
± 0.001
z 0.025
:t 0.613
zOl8
t 0 0~6
1057.90
1057.86
314 810
133.18
1059.28
14045.4
4183.17
1776.0
1768 0
1769.4
630310
(744 0
Ret~
±006
± 0.06
", 0 0 5 2
:t 0 5 9
± 0.06
±12
±054
±75
±50
± I 2
±3270
± 7) x 10 m
t160,
[109, 1601
[170]
11611
[11~2] (revt~d)
[1631
[164]
[1651
11661
[1661 (rewsed)
11671
[1681
"he~ frequenctet ate the tame at thote quoted by G W. Ertckmn (ref [ 1581 ar, d prwate communl,.atton), bu usmg a "~ =
t '(WQED) - 137.036011 inltead of o ' = 137 03602. The Lamb-,thtft value~ for ~uperheavy elements ( / a > ! ) tan be found m
~f 1 I.',s!
(AEH"--°u)th
= 2Pv= - 2S,/a = 9911 13 ± 0 04 MHz
~s we p o i n t e d o u t in the p r e t e d e n t section, there are three i n d e p e n d e n t m e a s u r e m e n t s ot the m t t ,
,'al 2 P ~ - 2Sv~ in h y d r o g e n I! I 1 - 1 1 3 1 . We can c o m b i n e these e x p e r z m e n t s with the theoretical
~redlction o f AEI, - £'!t to obtain values For the fro,." strutlttrc Lonstanl cr
(1) the Vorburger-Co.,,ens m e a s u r e m e n t [i 131
'-fidr'H - £')l = 9911 17 ± 0 04 Mltz.
.t,mbined with the theoretical p r e d , c t l o n yields a 2 0 ppm a~urz~tc value ol c~-~
ai, ~ = 137 035 7 0 ( ? 7 )
(~,) the ~ ¢ t g h t e d average oI tl,c ~a!uc.s ol rc|s. [~ [31 a r i d [1121
AE'u - o , = 9 9 1 1 . 2 1 * _ G O 3 5 M H z .
combined with (A/z"a - . O n ) a , ' yields
a~
= 137 035 4 3 ( 2 3 )
(! 7 p p m )
I'he v a l u e s a ~ a n d a t-l, 1 are derived purely from r a d m t i o n theory and e x p e r , n c n t , and t h e y arL ~n
good a g r e e m e n t wtth the d e t e r m i n a t i o n o f a -~ from phrase c o h c r t n z e effects m s u p e r c o n d ~ c t o r s
At the pre~-nt trine, the a('crtrac], o n a -t o b t a i n e d w~th the new theoretical e'~prcs~lon for the m~er-.',tl 2P¥~ - 2S~p i,, about the same as that oo the value ot tz-~ d"rzved trom lht" t~,~(Iroeen hvperfine structure (tKcu~ed tn the n e x t section
I,~ table [ 3 we ha,'c h,,te.t ~artotts determmatlon,~ ot a ' o t 3 l , i t r e d l l o m d i f l c r L ' n t ~(,PtrLc', I hl~
t.~bic clearly shows the t.onststency o f QED in descr~blng b o u n d syst~'m,..
I 3 The hyper.fine strut ture
In h y d r o g e n i c a t o m s , the m t e r a c t t o n o! the m a g n e t i c m o m e n t of tl~c orbital ch'ctron with the
n~gneti,, m o m e n t o f the nucleus leads to a s p h t t m g o f a hnc strt,~.t',rL level with fJx~.d orb~t,~l a:a-
B,£. L a u m ~ et al., Compartmn bencc,en theory and ¢JCperlmentsin quantum electmdy~mk~
calm"momentum I and fixed total angular m o m e n t u m j into hyperfine structure (hfs) levels. To a
~'wst-approximation, the energy separation between the two outermost levels is given by the Fermi
Formula [ 1351. For the ground state of the hydrogen atom (see fig. !. ! ) this corresponds to a hvperfine frequency
~[~(Fermi) = ~ ( Z a P Ry ~tp ~- 1420 MHz
lie
gher¢ yp is the proton magnetic moment and/a~ the Bohr magneton
~ = etl/2mec.
The measurement of the hyperfine-splitting of the hydrogen ground state is probably the most
accurate number which is presently known in experimental physics [ 1361
A~exp(pe) = 1420.405 75 ! 7864(17) MHz.
it corresponds ~o an accuracy of 1.2 parts in 10 ~2 ! More than a test o f QED the hfs has become
yardstick to measure our progress m theoretical physics. As we shall sec below there is ,it pre~'nt a
gap of seven orders of magnitude between theory and experiment.
The framework for the formal treatment of the corrections to the Fermi formula is the BetheS~_lpeter equation. The dimensionless parameters which appear are:
re~M, electron to proton mass ratio;
R / a o, ratio of nuclear to atomic sizes;
a,
the fine structure constant;
Za, the strength of the Coulomb potential
A sequence of approximations in the correctln~as to Az,(Fermx) ~s estabhshed as follows * To fir,,t
approrlmation, with
Av(hfs) = Av(Ferml)(1+6),
the proton Is taken as a fixed point Coulomb potential
m / M ~ 0 and R / a o -~ 0;
radiative corrections are also neglected, and only relativistic corrections are taken into account
This yields the Breit correction [ 138 ]
3,
2
At the next level, one still takes t n / M -~ 0 and R / a o -* 0 but radmtive corrections are taken into
account as succe.,;sive powers of a, the dependence on (7,). which arises when b!ndLn.g is taken .-.~te
account, is not however smlply a power series in (ZoO Altogether. radiative plus binding corrcttions take the following form
8raa + ~b,nding=at ~+ctoot(Zot) +a2
+ a3
+ (Za)2[c22 iog2(Za)-a +ca, Iog(Za) -2 +c2oi +
+ Czoz o~CtZ~ + ...
71"
*An
excellent
detaflewl
~.~nnc*t,~.
,~¢ ~ h , - ~ , , , - , - , . - . , , . , ,
,,.a,,.,
^r
.........
,. . . . .
t..
¢. . . .
.~ . _
t~_~t_t.
....
.a ~ _ . _ t . . _ _
,,
-,-,,
B ~. l , ~ t r u p et ttl. Comparison between theo, v and expc~ tmcnt~ tn q u a n t , m ¢'leetrod~ t,amtes
207
he coeffi ~ n t s a t . a=, aa are those which give the correxpn,tdmg -trdcr contnbut=on to till anom
OU', =nagnet=c moment of the electron*
at = i / 2
~:
~- log 2 + ~ f ( 3 ) = -0.3285
107 ~ra
" 14~4 ÷ T ~ a~ = 1.49 ± 0.20
he coefficient c~o was first calculated by Kroll and Pollack [ 139 i and Karplus, Klein and
chwmger ! 1401
c=o " 5 + log 2.
41,
he coefficients c== and cat have been calculated by Layzer [ 1 4 1 l , Zwanzlgcr [1421, and Brod,,ky
;Id Erickson [ 1371. The latter author,~ have al~o calculated lilt.' doml:],lnt contribution to tlw .-officmnt c~,. The result~ are
c== = - 2 1 3 ,
c2,
281
360
8
3 log ~
ca0 = 18 4 ± 5
he higher order terms have not been calculated as yet. Wc rot.all that the ~.octt~clents c 2, ,rod c 20
re qate d e p e n d e n t . *'= In fact. the values for the n = 2 S-level h w e ,also been calcul,ated 1141. 142.
37, 1431.
The next level o f approxmmtlons take., into account the brute nu,,s (~:',1 ,,tru~.turc ol the proton
'he c~rrespondmg corrections can be classified as folios,,,,..
(t) reduced m a ~ ~.orrect~on~
lit) nonrelativlstlc ,,~ze contribution,, ~Snr.
{tltl additional recoil tcrm,, of order 0r(m/M)~Si~( ,
{iv) proton polanzattoh C()FIe*.|l¢)I1% ~p
?he reduced mass correction gwes simply a tactol IM/(m+M)] "~ m the Ferm~ formula Estimates ot
=onrelativistic size contributions were made by Brown and Arfken [] 4 4 ] , and, more elaborated.
~y Zemach [ 145]. The calculation of Zemach, in which the nonrelat~]stzc approxunat,on to the
~ave function was used, has been analyzed m detad by Grotch and Yenmc I 116] w=thm the f=amc
rerk of an effective potential model. It amounts to a correction
tin. = - 2 m aRp,
~hcre
Rp = f
lu-
rlpM'u)p~lrid3ud~r
.t:,,,.l #L(rhp~l(r)denote the t.laarge and m,ien-ttL dl,drtbutton ,)f the pr<>ton ,L ; n ' f i t h ,In( I Y u n n a ,
t~ ~,], u,,lng the e~pre,,s~on
p E ( T ) __ p r o ( , . ) = ( h 3 , 8 ; r ) ¢
~r
"See section 4
'%ee e.X Brodsky and Fnckson 1137! fer a d l ~ t , ~ ( ~ .
B
B.E. Lautrup et aL. Compari'on between theory and experamet,ts in quantum electrodvnamtcs
ruth A = 0.91M as suggested by the experimental determmatton of the proton form factors obain
R p = 1.02 F,
slueh amounts to a correction
~Sl~ = - 3 8 . 2 ppm
to the Fermi formula. Additional recoil corrections of relative order ,,, m/M have also been evaluate~
~, Arnowitt [146] using the method of Karplus, Klein and Schwinger [1471 and by lqewcomb
and Salpeter [ 148] using the Bethe-Salpeter equation, l'hese corrections, which in the case of
muonium can be calculated exactly*, have been done for a point proton with an anomalous magnetic moment i.e., the vertex corresponding to the absorption of a virtual photon of energy-momenta~m q by the proton is put equal to
r u = ?u + ~ i
oU~qP
The result ~s logarithmically divergent, due to the nonrenormahzability of the Pauli interaction.
However, as was shown by the calculations of lddings and Platzman [ 1491, the divergence dtsappears when the form factors of the proton are taken tnto account. Iddmgs and Ptatzman calculate
the correctioas to At, (Ferret) arising from two-photon-exchange (elastt¢ contribution). In fact. the'
calculate the contribution to the his from the difference between a coupnng w;th form factor,,
and the point-like coupling vertex quoted above. This, when added to the result of the ~.al~'t,laltorts
of Amow~tt; and Newcomb and Salpeter gives a finite correcUon of
tSp(elastlc) = 3.6 pl3m.
There are further corrections arising from the polarlzablhtv of the proton. ~ e contrtbuttt ns
to the hfs fron~ two-photon-e.,,change graphs with wrtual hadrom~, state,, other than tlz,e pro'on
~tself. The posszbd~ty of calculating these corrections from e,tpenmcntal data o~ mclast~,, ci,,.:fon
scattering from protons was first pointed out by Iddings [ i 501. The argument is analogous to
Cottingham's formulation of the neutron-proton mass difference [ ! 511 in terms of the prot )n
structure functions. However, in the case of the hfs of hydrogen, what is needed are the sp:t.
dependent structure functions of the proton. Theze are accessible from experiments on tnel,..,t,c
scattering of polarized electrons from polarized protons** The polarizabtlity contribution cut
to inelasUc contributions can be written.
8p(inelasric) - o t m
1
¢A~ + A2 ~,
rr 31 2(1 +~:p)
where gp Is the anomalous magnetic moment of the proton, and AI. A: the contributions lr ,m
the two spin depem ent structure functions G~(.t'-. v). G2(q a, v) defined in such a way that t ,r
real photons (qa = C k
*See ~,ctton 3.
**For a discussion of the spm dependent nucleon structure functions where earlier references can be round, see Donee[ ar,d
de Rafael [ 152]
B E . i . a l t t t u p e t aL, Corrtparl$on beh~t'ctt "rtc~-v a n d ~ xpcnt,tcnt~ it, d t u t tut,t ~'/c'( jr.~cl~ . a . m
v) = (/8~r:a)l,~,,~vJ
G,(O,
~
209
o~.(v)l
vhcrc o,~.piu) are the total pht,t<;ab~orl:tmn un',n ',c~.I~,)ns for photnrH~rott,n dnt]par.~llcl ( A }
~.+ralk'i (P) spin configuratm ~s t'.,,tm~.ttc,, .I A~ ,rod A h#,,c ah~,ty,, It ,td to ,,mtll ~v~tt~lbt.tt~ol~
8p(inelastic) -"
2 ppm.
I-
['he possible contributmns to the term A, have been discussed In detail by Dre!! and Shlhvan [ 154]
l'hese authors exploited the lnforn.ation prmlded by phystcai C o m p t o n scattering (q: = 0~ and
:oncluded t h a t i n all cases the contribution to 6p from the A~ term a m o u n t to nn more than
I-2 ppm. It has been recently pointed out b:, de Rafacl [ 160l that, from pog~tn,t~ mequal~ttes,
he contribution 6t, lA =) can be bounded mn terms ol the spin mdrpe,,tdent ~trt, cture ftmct:ons It
md W~ of the p r o t o n An estimate of the boumis from the regmn covered by prc~cnt melastk
~lectmn proton scattermg experiments g~ves
2 ppm ~, 8p(A~ ~ c 3 pp.n
From the ~ompan.,.3n h : l w e e n the cxpcrlmc ntal result II 361
A%~o(pe) = 1420.405 751 "/864(1"' MHz.
~nd [he Fermi formula with all correction,, d:,<.u,,,,cd ~lmv~'
n~h~ltt)' correctmn, and using the WQEI) vaI4~.' ol c~ ' , * *
o"
~n~l~
t~ralctl
~, ~ut'l~[
ll~c pr~)t,m p~)l ~-
= 137 036 ()8( 2~, ~.
A_Vt h_
: 2 C, + 4 () |)pill
6p Incl,Iql~.)
,or.,l,,l,..rlI,,v~th th,:~.,ttm.~t¢.,.t~, 6 I~m'l,.~,.,,~.+I I ~~
IS~
l,',ui
One can also use the theoret,cal' expreY.mn of Av (hfg) to obtam t h ' l sne structure constant
by comparison w~th the measured A~. (hfg) The value ~s
a -~ (hfs~ = 137.035 Olt35).
~.ons~stcnt with othcr detcrmm,,tmn.s o f o -~ (',ee table 1 3~
2. Positrontum
P o s l t r o r l.l l l m
.
.
.
.
t,.. !. h e.
.
I .! o !. ! ~ .
.
,..g!l~.!
.
.
t . . . . 6. . ".-',~ d l l
~ . ',., , , . t•~.v. l. t. .
d t i u j .1 I ) ~ r ~ l i l ( H l
[I
Vvd'-, t i l h k ~ ) x ' ~ l k . ' L I
- - ~
1]'~
lq51 e o ~ l t r o l l l t l l | l is h ' , t r l ~ ,, l~.t~d , l t ~ . ' l l t . l ] ,,\'~,lcl~'! l u I t ' d (.)J [) ill [,,tl [l~.llJ,J~ t , u l H i,k'r,,t?n,.ltng ~t llt.~: bmthng m~.ui': .r,m ,t,, d~'s~.rlb.'d h~, tlt.c Bcthc-bdll~ctux l~,rmdlinm ]2U ,. _'t.J4
D ' - u t s u h In
~Thc I - N 5 - ~ ' I V ¢ ! cO,ltq'~butlOl~ ha~ b c l n
c ~ ' l l i n a t c d b y ( . , u : r l n [ | 5 ~[ "t .<" ] n [ , m
m d tt ~. r ,'% n ,,m t,lt, ', :,~ ~,lv~ ,l t , m t t ' l b u l l ~ ' r
smaller than | p p m The Importance ol t',adronh, u~llql~tlum c o n t r l b u h n n s h,l~ oeen p,~rt:tul ,rl~ ernphau/ed b ~, Prull ,rod
~ul[Ivan [ |54] .and mole recently b3, ('hct,'uak.5trumln'~kland ZltlOleV [I 5 < ,, usnI: a quaslpotcn'ul rnt tl ,d ~'(~elnpccl h~
Lo[unov and Tavkhclflz¢ [ 1461 and I-atat~>~,[ 157 l
"°S¢¢ Taylol. Parlor ar,d Lan~nbcrg [R 16 ]
"%¢e
r t [ . [ 2 0 1 1 I o r a r e v i e w o f t h e c a , , m - f f X l ~ c r l I T l e r h , ,~c(" ~,(,,t'ln [ ) ¢ t l | s t h j 2f12 1 I o r ~ r m r u n . u . n t rc~ tu*~, v ' c j R I | I
~0
B.E. Lautrup et aL. Comparison between theory and e ~ q m e n t s in quantum electrodynamics
At the present time, the only measurements performed in the posttronmm system are on the
~ine-structure splitting of the ground (n = 1) state [205, 2081, and on the decay rates of ortho.
positronium (the ~St state) [209] and parapositror~ium (the t So state) The first mea,,ur,-mem of
Ihe decay rate of parapos]tromum F~ was obtained from a measurement value of I'p/i",, [ 2101.
~he.re I"o denotes the decay rate of o~ thopositromum, and using the value of F o quoted in [¢i
[209] (see also refs. [R. 11 and 2201 ). A direct measurement of l"p has only recen.'ly been reported
[2081. A compilation of results is given in table 2.1.
2.1. The fine-structure interval of the ground state of positrontum
The qualitative features of tile positronium energy levels, as predicted by the Schroedinger
~uatlon, are roughly 1/2 those of hydrogen because of the reduced mass for positromum wluch
~,me~2. Hence the ionizatmn energy of the ground n = I state o f positronium which is 6.8 eV.
The fine-structure in positronium ~s of order a 2 Ry and the features here are very different from
those of hydrogen Besides the dipole interaction, as in the hydrogen hfs, there is an exchange mteract'~oxt due to the virtual annihilation of the e'e" system in the triplet state into one 3' 1211 213.
221,222]. In fact, the largest contributmn comes from thts virtual annihilation interaction which
pushes the 3S~ level upwards with respect to the t So level. The contributtons to the triplet-stnglet
.~plittmg of the pos',Ironium ground state have been recently calculated up to terms o f relatnc order
a2 loga [2141. The result is
7 a
A v = a 2Ry[~-~---
([9.~ +log 2) -- 3 a ' log a +O(aa)].
I'he contributions of relative order a 2 log a represent recod corrections ansmg from low momen:urn components of the wave function assocmted ,vtth the Bethe-Salpcter equdt,on for posltro.
atom The techmques used m the calcula,lon by Fulton, Owen and Repko [2141 are an extcn~ton
3f those previously described by Karplus an,l Klein [ 2131 and by Fulton and Martin [ 215 i l-he
~erms of relative order a: have not been calculated as yet The theoretical prcdtctton from the ~at:ulated terms ~s thus
AVth 2.034 15 X l0 s MHz,
=
:o be compared with the most recent measurement [2081
Table 2 i
Compdatton of results m positromum
)bservable
Experiments
Theory
:me-structure o f
;round state
St - 'So
(2 033 80, 0 0 0 0 4 0 ) X 10 s M H z [206]
(2 033 3 0 , 0 000 40) y. IO s M H z [207[
(2 034 03 * 0 000 12) X lO s MHz [2081
203427x
)¢cav-rate
nthoposatromum (3S~)
3ecay-:ate
~araposRromum (~ So)
( 0 7 2 6 2 ". 0 0 0 1 5 ) x '0" ~e~ ' 1209. R I l l
(0 ?99 _-0011) x I0 '° sec" [2081
10 s M H z [ 2 I ~
:14r
07211 x 10" sec [ . ! :
,correcOon o f r e f [2171 not included)
0 798 X I0 i° sec "t 1218 21o]
B.E. Lautrup et al. Compa~son bert *een theory and erpertmet, t~ m qt,antttm eh'ctrod~'namu ~
2! 1
AV,, r = (2.034 03 -* 0.0'~0 I 2) X 10 ~ MHz (60 i'pm)
2 Tile a t l n l h d a t t ¢ m
rates of orlhopo.~tront,an
and
mr, q m ~ i t t : m t t , , : ,
The a n m h d a t m n rate of orthoposttronn,m (the ~S, c'c- ~tatc) has Occn recast.red to an accuracy
["0.2% [ 209 ! (S',:e also ref. [ R. I 11 )"
F o ( e x p ) " (0.7202 ± 0.0015) X 10 ~ sec -t
he theoretical, value for the 37 ray annihilation, which includes only the lowest order c c n t n b u on. was calculated by Ore and Powel11216].
fit 6
I',(thl=~ " m90ra
- 91=(0.721
12 ± 0 0 0 0 0 1 ) x
i0 ~ set. -t
:ecently, the correctmn term of relative order a due to the mterlerencc of the lowest order dla:ares with the htgher order dmgrams revolving photon-photon scattering has also been ~.ak.ulated
2171, using numerical integration techmques The effect t,, to lower the orthoposttromum decay
lie
U o ( t h ) - ~- m ~ - ( s a --9) LI - ~ - ( 0 . 7 4 1 +- 0 0 1 7 ) +
J
t must bc noted, however, that the photon-photon scattering correct,on is only a part of the con tete correction of relative order ~ The c~lculatlon of the other term,,, is clearly necessary for an
tlcquate comparison with the experimental value
[he theoretical value for the 2"), ~nnthilat~on rate of paraposltromum (tSo) Is known up to tern s
f relative order o The lowest order term was flr,,t cah.ulated b!' l.)lra~. [218] and the ~orlc~tJ(m~
I relat:ve order o by H a m s and Brown [219]
i'~lth) =rr- m
I
air( q
4
= 0 7c.~8 X 10 '° ~et.-t
~ecently, the first direct measuremt.nt of the paraposttronium anmhflatton rate has been made
...08]. The experiment, which is set to measure the fine-s*-~ :ture interval of the ground state of
~)sttromum, ;nvolves the measurement of an induced Zeeman transition between magnetic subLates of grouna-state posttromum. Detection o f coincident 27 ray anmhtlatlon rather than detecton of the "),-ray energy spectrum, as done m previous experiments, was used The natural hne
~adth of the Zeeman transition ymlds the value of Up
Ft,(exp}=(O,799 t 0 0 1 1 ) ×
1 0 t ° s e c - ' . ( i 4'/,)
~hcre a onc-standard-deviatton error ls given ~n e'~cel!ent agreeme:~t ~,,,!), the the.~.ret,ca! w.!ue
I
Muonium
Muonium ts the atom consisting of an elet.tron and a positive muon It was discovered by ttughe,
md collaborators [301 ] m 1960. The m u o m u m system provides an excellent ground to test our
mdetstandmg o f electromagnetic binding when two different masses are involved, and eventually
B.£, Lautrup et al., Comparison between theoo, and experiments in quantum elect~tvnamics
~12
:o detect a possible breakdown of electron-muon umversality. Recently. precision measurements
~f the m u o n i u m ground state hyperf!ne structure and the magnetic moment of the muon have
Fielded a new determination o f the fine-structure constant ,,. to an accuracy comparable wnth that
reached in measurements of a using the Josephson effect
3.1. The hYl~r~ne-splitting o f the muonium gound state
The lowest order splitting of the singlet and triplet levels of muonium ground state is given by
the Ferr.ni formula [ 135 ]
Av = ~ a 2 RY(tau/taeo),
where tau Is the muon magnetic moment and taeo the Bohr magneton
taeo = eh/2m~c
and
~ =tauo (I +a~,)
where tag = eh/2muc and au is the celebrated anomalous magnetic m o m e n t of the muon.
Qualitatively Av(tae) ~ 3Av(pe), as exp,zcted from the ratio o f muon to proton magnetic moments. Corrections to the Ferm~ formula have been calculated t,p to terms of relat,ve order ~Zal:
for the radiative and binding corrections 13021 and up to terms o f relative order (m,./m,, 10,~ log a
for the recoil corrections 1303]. Altogether. the theoret,cai expresston for the hYl~'rfine-sphttmg
of the m u o m u m ground state Av(tae) can be wrntten m the follownng ~.ny
Av(v.e) = T
+ a,'r(ZaF
Ry - - ! +--~,!
U,~,
3- 1°~2" (Za)-: +
! + a¢ + (Z0t)2 ~- + a(Za~
360-
'~ '- log 2
_
,
J log _• Io~.(Za_I-" + I,~ 4 -- .*, + 8.,
The factor [ 1 + (mdmu)]-3 ts a reduced mass correction The correctnons m curly brackets, e,~cept for the 6 . term, are the same as for the hyperfine splitting o f the I S level of the hydrogen
atom * The term a e ~s the anomalous magnetic m o m e n t of the electron. The term 6 . represents
the relativishc recoil corrections which for m u o m u m , unlnke the case of the hydrogen atom. ,.an
be calculated exactly. The expression for 6u reads
The leading term is well known it has been calculated by varnous authors 1 3 0 4 - 3 0 6 1 Hov, e~er
the correction term of relative order (me~ill u) Ot2 log a whnch represents recod effects artstng from
low-momentum components ot the muonlum Bethe-Saipeter equation ~s known oni~ xtnte recently. It has been calculated by Fulton. Owen and Repko 13031
The numerical estimate Av(tae) reqmres two quantities as input whtch have to be taken from c \
periments, the fine structure constant a. and the ratto ur--~ /Jue
The latter can be obtanned from
,---O •
*See sectton
1 3
B E Laurrup el al.. Con'lwrison between theory and cAi,cr:mcnts m quatttu,*t elecrrodvnamtcs
213
ncasurements o f the ratm ol the magnetic n~oment ol the tnuon to the magnetic m o m e n t ,)I the
~roton Bu/~t~, which ~eccntly have been performed ~o an ,:ccuracy of a few ppm by two group~
~07, 3081 it ~,, thanks to these new precise eteasur~ mont., ot tau/Bp that mUOnlUm ha~ bc~.omc
prc~.~ton test of Q! D Thc prcvmus n)ca,arcmcnts 130~) 311 I ot Bu//ap had errors ol 13 22
)pro, too large to profit fully lrom the more accurat~ determmat|ons of &u(~e) [31 2, 3 1 3 , 3 0 8 ]
(i) Precision measurements o f the magnetic mome~ ,t of the muon
The value o f the ratio pu/tap reported by a U m v e r m y of Washington-Lawrence R a d m t m n Laboatory collaboration ! 307 i is
/a~,/~p : 3 183 347(9) (2.8 ppm),
o be compared with pre~tou,dy reported values {,,ee table 3 I ) In term', ol the muon ma',s, tins
letermmation o f p J # p mlplies
mjm,
= 206, 7683('J) (2.9 ppm)
[he ratio ta./Pp measur..d by the authors ol re: [3071 was pertoltncd m three dacm~c,tI eu'~lrouncnt,, ,,howtng no ~,uhstanltal ddfcrcnccs Tht,, t,, ~n contrat ~t.t~on w~tti the suggestmn by Rudernan 13141 that a correction due to th ,r effect of dmmagneuc slueldmg on the muon m o m e n t
hould exist.
An i n d e p e n d e n t determination of ~ , , / ~ has been made by "l-clcgd~ and t.ollaborator.~ at the
Jntverstty o f Chicago [3081. In this experiment, both the hypcrllnc ,,phttmg Au(tac) and the
n,lon magnetic moment Bu are detcy ~med~from mcasurctncnls ot the Zecman (1"..,ll~) trans~t~on~
I.l)"ll.0)andl'l.
1 ) o (O.O) m t h c r c g m n o l mtcrnacd~ate~ouplmg l h e c x t c r n a l m a g N c t ~
"tcltl Is c h o s e n at a "lllaglL'" value, for which the fretlUCnt.~cs ol the two Zccman tr,m~t]Gn~ bc~.on~c
o t~r~t order, field independent The rc,~ult~ are
AL,(tzc) = 446 ~, .,0_
" 2~ ~')) .Xltl:
i,qd
,u.//at~ = 3
183 373t 13)
Fhc latter value has been o b t a i n e d assuming that the bound-state g factors arc not affected by
:olhstons with the host gas atoms If, on the contrary, as sugges'ed by cal~.ulatmr, s by Herman "~
1able 3 1
C u m p t l a t m n of results on the d e t e r m i n a t i o n of the m u o n magnetl~ moment
1-,~pertmental value
3 t , ~ n .tble
Rclcrentc
( o l u m b l a rcI [31)91
autt p
183 31'lOt4 O)
,1u it p
j 183 360('70)
B,:rkclex,ret
3 18~ 330o141
l'rmtLtr~a P, rm, r~l ]~111
3 t83 347cq)
"~, i,.hmt, l,m i RI
% up
13101
r~l I q,)71
'Ih/~
~ t h o u t pre,su~-xhlft correction)
11 p p m
p~'~ute-,~luft correctmn)
See ref 1308]. n o t e t 2) added m proo ~
3 183 373(13)
a 183 337113)
fluL~co, rcI [30~1
~14
R~. Lautrup el el., Comparison between theory and experiment: in quantum electrodynamlcs
Table 3.2
Compila~on of results on the hyperf'me sphtUng of muonmm glound state ~vOae) (All numbers ate nn MHz)
References
~.xpenments
Theory"
~de13.17t
tm 19tSi
,~463.i5(6)
4463.302(27) (from Kr data)
4463.220(33) (from At data)
446.L3 | 3421)
(ut'Nl/~/blp = 3.183 337( I 9})
D161
~'~ 13131
3tionSo[90Sl
¢m 13151
4463.317(2t)
,o~3.3~(,9)
4463.249(3D
4463.3022(89)
4463.311(12)
(utinll/~///p" 3.1113 347(9))
q'he ~rocentrecoil ¢onecuon of Fulton, Owen and Repko
13031 has been Included. The value of a " .ned is 137.036 02(21).
pressure shift of - 1 ! ppm corresponding to the experimental conditions of ref. [3081 is assumed.
then
,%/tap - 3.183 337(13),
~nexcellent agreement with the result obtained by the authors of ref. [ 307 !.
(ii) Comparison between theory and experiment"
The theoretical values for Av(#e) which are obtained usilt~ the recommended value of the fine
Rructur~ constant
a -t " 137.036 02(21),
and the values of #u/gp quoted abo~,e are
Avth(o~e)Ch,c,,o = 4463.313(21 ) MHz
Avth(oJe)wash./LgL = 4463.323(19) MHz.
The most precise expenmuntal determinations of Av(o~e) have been made by the Chicago group
[308] and by the Yale group [3151
Chicago, A~o~e) - 4463.3022(89) MHz; ref. 1308]
Yale,
At,(ge) -" 4463.311(12) MHz; ref. [3151.
Comparing these values to the theoretical predictions given above it can be seen that the Chtcago
and the later Yale experimental results are within one standard deviation of the theoretical values
The remarkable accuracy of the Av(tae) measurement obtained by the Chtcago group (2.0 ppml
combined with the new determination of OG/O~p allows, from the theoretical expression of Av(#el
an independent determmation of the fine structure constant a The value thus obtained 1308 t, ts
a -l = 137.036 17(30)
m excellent agreement with both the WQED ~alue a -t = 137 036 08(26) and the recommended
value (see table 1.3).
*We acknowledge a helpful d~scussmn on thts point wath Professor Thomas Fulton
B.I~" Laurrup et al., Compttrlson between theory and cxprr rvcnt~ tn :~uantum electrodv,~amtcs
715
Tab!¢ 4 1
Contributions to the theorcUcal values rot the lepton anomahes (a-~ ~ 137 036 08(26))
"ontt '~utton
(/~h ~x I0 v)
G~th{× 109)
~D;..ol,d ~
1 1614090±2.2
1 1.51409(I+ 22
~D 4. order
~D6 order
-1772.3
18.7 ± 2.5
0.0
I~honic
'oral
! 159 6 5 5 4 ± 3 . 3
4131 8
273 ± 14
65 ± 5
1 165 879
-- 15
Part B
the a:mmalous magnetic m o m e n t s of the charged ieptons
The anomalous magnetic m o m e n t s (the " a n o m a l y " a = (g
2)/2) of the t.harged leptons have
for many years offered one of the most interesting and ~mpo.tant challenges to bot.h theorehcal
md experimental quantum electrodynamics. The C E R N muon experiments are probably the most
~atstanding examples of high-precision experiments done with a hlgl', energy machine The ~.alcuLahons of the s,xth order QED c o n t r i b u h o n s now m progress around the world (and m one case
~mpleted) are the highest order experimentally significant radiative conect~or's t h a ' have been
~'aluated, and are formidable challenges to algebraic mampulahon and numerical ~,~tegratlon
~¢~hntques
Th: best experimental value for the electron anomaly has been obtained recently by Rich and
We.,,Iey [ 401 !
Cteexp =(1 1 5 9 6 5 7 7 + 3 5)X 10 -~
lhc best expcrm~ental ,,alue lor the muon an,~maly is the CbRN-stola~e ring value 10071
a~ ' ~ p - ' ( l I 6 6 1 6 * 3 1 ) x
10 -~
The electron anomaly Is a purely quantum electrodynamlcal q a a n h t y (see section 7). The theoretical value ~s
~ h - 0.5 (a/lr) - 0.328 48 (a/~r) 2 + (1.49 + 0,20) (a/Tr)3 ,
where the last term ts the sixth order c o n t n b t m o n with ~ts theoret~zal u r c e r t a m t y (see section ¢ 21
With a -t = 137.036 08(26) we obtain
a~h = ( l
1596554-+33}x
10 .9
The uncertainty has t~;o parts, one from the line structur" constant (± __22) drld one from th~_or~
I-_ 2 5~ F.xpenment and tbeo~' thus agree w~h~n one sta~dar, i dev~ah¢,r * It t~ mte-estJng to no~,'
that a slight improvement m experiment and t h e o r y might lead to a value for the fine ~tructure
constant which is better than the Josephson value **
• The Drdbi~gcL~Parlonl eittmate 1462, ~03! leads to 0.40(a/Tr) ~ and thus tt dr;agrees w~,l, the late',t expertment, which corresponds to a n x t h order term (I 67 t 0 331 (a/~) ~
• * s,t pre~cnt we obtain (dtm.gardmg the theoretical uncertainty) a~ ~. 2 = 137 03 ~ 8 2 ( 4 t ;
216
B.l~. lamtrup et al., Comparison between theory and experiments tn quantum electrodynamics
The muon anomaly is not a pure QED q u a n t i t y . The theoretical contribution from QED ~
( ~ c t i o n 5)
, aQED= 0.5 (~/lr) + 0.'16578 (a/~.)2 + (21.8 ± 1. I ) (a/Tr)3
which nffn~rically become~
a.QED" (1 165 814 :t 14) × 10-9.
TO this we add the strong interaction contribution (section 6. ! )
17"Hadr°ntc= (65 ± 5) X 10 -9
so that Uae full theoretical value is
at h = (1 165 879 ± 15) X 10 -9
in reasonable a~eement with experiment. In table 4. ! the different contribution~ are exhibited,
4 The elecfron anomaly
The electron anomaly has been calculated completely up to and including the r~xth order.
4.1. The second and fourth order contrtbuttons
lr~ second order there ~s only one dmgram (fig. 4. ! ) wluch gives the famous Schwmger ¢onln.
butlon [0031
cite:)_ I ~ _ 1 1 !(~1 4 0 9 0 ± ~ 2 ) x 10 -~
where we have ~sed the value
o~-~ = 13 "r 036 08(2(~)
l'here are seven dmgrams (fig. i.4) which contribute to a,, at fourth order m e. i he result is
rather n~ore comphcated than the second order expresston. Compared to the rational I/2, the
transcendentals lr2 , lra log 2 and ~'(3) now appear. These r e p r e ~ n t special values* o f the ditog
Lh(x) and the tnlog Li3(x) More precisely 1 4 0 5 - 4 0 9 1
i
Fig. 4.1 Second order c o n t n b u t l o n to the lepton vertex
"Seee.g ~.wm [4041.
B.l:: l.autrup et eL, Comi~arlson hetweetl theory a*~d e ~p,'ro~zc~lt~ m quantum elcc tn,d~ ~,a~,t(
217
hl, lund,H~acnt,~ic.|ltul,ltion I~v K.~rplt,.,il~,~i K~oll [-1051 (I,Ilct rc\l~ctt ~cc rot,, [40~'~1 dt~,l 140-~1!
~,=~the llr,~t to dCillOll~Lrdtc the ~.t'll~i~,Ic'l~.) t.~l the r llOlIIIdlle',ltit)tl l~l,~t .'dtlrc' ]~] Ilmellc= o r t l c r x o ~
w~I t,rb,it i<)l~ t h e o r y . Numcr~.a[l)
=
-- 0 . . . . 8
48
( e t ' / r ) ~ = - 1 7 7 "~ .~ X
l 0 "~
Ihe e l e c t r o n a n o m a l y shotlld m fact be e x p r e s s e d as an expansion ira the two paramet~'~s ~t a~d
/
t
n,,m=,. "l'hc leading ~orm m ~z~/m,. coming fron~ t h e second dmgram ol f~g 4 2, ~s. h o w e v e r . 1410l
o
t
- 3 x
lo,:
~'h~th. at the level ol approxm~atlon needed. ~.an he left out
4 2. The wxth ord('r ('ontrtht~tton
¢, o n s t d c n n g the increase m difficulty bctweei, tile ,,ct.ond al~d I o u r t h order it is not ,,t~rprl~Hag
Ihat the e v a l u a t i o n o f the sixth o r d e r a n e m a l y is m d c e d hard First o f all the n u m b e r o f grdphs Is
now 72 Ifig. 4.3) and the c o m p l e x , t y of thc graphs Is ~uch th,tt .in anal3,tlL evaluation if1 clo~ed
form I~ e x t r e m e l y difficult for virtually all diagrams Actu,llly d,agram~ 19 22 (tlg 4 3~ ha~c
been evaluated m c l o u d f o r m 141 ! I. In addition to the tran,,t.endentai,, e n c g u n t e r e d in a~4~ lhere
appear n o w special value~ of the p o l y l o g a n t h m [ 14(v) [4041 A~ the dla~ram~ lead tn tzcncrjl t~"
~.', on-fold integrals, one e x p e c t s that the analytic, re,,ult~ t.,ln mvolv~" ,,pceld', vdlues ol l.l,.(\ ) ~lth
~ rJn~jng from ! to 6 Luckdv. however, the dt,igrarn~ drc not more comph~.,~tcd th,tn the~ .dl~,.
lor n u n l e l ~ t a l
cvahiatlOla
Lcl us ,~tln|lnarl/tr the ~tu,atl<~n The 72 dtagrat~,, t . ( ) n t r l h t l l l n g t o ct~,4) l,tll Into ~;\ t l t l l c i ~ , ~ t
=la,~es, the tholt.e ot whl~.h ,~ t~a=nl~ b,txctl t)n g.it~gt lI)~,,ill,,llt t' ~.rltC=l '
Ulass II
Graphs
graph_,,
Cla,,.~ Iii G r a p h s
Cla~s IV. G r a p h s
f~la,~,, V Graphs
Class Vi. Graphs
7 - 1 8 . c o n t a i n i n g s e t o n d o r d e r ( b u t not fourth o r d e r ) vactium polar~zatlo,~ ~ub19- 22.
23 -28,
29--48.
49-72,
c o n t a i n i n g fourth order vacuum pol~trizut~on subgraph~
t h r e e - p h o t o n exchange
t w o - p h o t o n exchangc
o n e - p h o t o n ext.hange
\~
e
| lg 4 2 Mass d.-pendcnt fourth order c,~nt~tbut~n (, the. Icpt(~n vcr1~.cs
it8
g E . Lmitrtlli et aL, Comllrlion between I ~
t
~
exllii'tmenlll ill qllntum electrodynnmiii
,t
J
;,
£
A
A
t
l
A
E1
\
,;
62
\
,
\
,'
\
,'
\
/
Fig 4.?. Mass independent sixth order contributions 1o the lepton veitex
\
B.E. Lauttup et aL. Coml~rison between theot 9, and cxpertmetTt, n quantum eiectrodvnatmcs
219
Table 4 2
History of the ¢aKulatDons of tt, e sixth order contr~but~t,n~ o the electron anomal~
;r'H~h'~
Author~
~t~,ir
22
-6
t-22
t-22
'.5*29+30
i9-72
~3- 2S, 29, 30, 39, 40
.)3 72
Mlgnaco an ! Rcmtddt [41.11
Aldins et al [412,4131
Brodsky am Klnoshtta [4141
Calmet and Perrottet [416]
Levme and Wright [ 415 ]
De R6jula et al. [4171
Calmer 14181
Levine and Wright [4191
1969
1969
1970
1970
1970
197f,
197~
1971
_ T h e fi)st three classes contain fermion loops and comcMe with the class division for the differ,,nee au - a.. The last three clas,.es contatn no fermlon loops and have been classified according
to the number of photons crossing the mare vertex. This clasmficatton has the advantage of being
gatlge tnvariant. Inside some of the classes there are gauge mvartant subclasses We leave it to the
reader to pro~e that the following sets of graphs yteld a gauge mvariant anomaly {after renormahzatton}: I - 6, 7 - 1 0 , I l + i 2 . 1 3 + ! 4 + 1 7 + 1 8 , 15+!6, 19, 2 0 - 2 2 , 2 3 - 2 8 , 2 9 - 4 8 , 4 9 - 6 8 , 6 9 - 7 2
In table 4.2 the history of the sixth order calculations ts given. All s~x classes have been completely
evaluated, although detailed results are not yet available for the lost three The gq~phs of class III
have been evaluated analytically by Mignaco and Rctmddt 141 I] and checked numertcally [ 4 1 4 41o] All the o t h e r dtagrams have only been evaluated ntmlertcally In tables 4 3 - 4 6 the results
of ~artous calculations arc presented. We have chosen to present the results as much as posstble in
the term and detati vl which they were originally published The a n a l y h c results for cl0ss III [41 Ii
arc given m table 4 3 In table 4 4 a comparison )~ made betwe,:n the r~sults for cla%es ll and Ill o
Bred~k3, and KmoshJta [4141 and Calmet and Perrottet [416] The agreement t,, exceltel: " In
table 4 5 the results el De Rdjttla, Lautrup and Peterman 1417] for the gauge mvariant ~ubse!
It~t) 721 of the L.lax,, Vi graph,, is presented In table 4 6 the re.ent rcsult~ el Calmct 1418] ,~re
11.ted
We g~ve below the overall results
Class I
eJ
t•161
= 0 36(4t (a/~P
Aldmsetal
[412.4131
Cla.~s II
(6)
a ¢3I
=
f
- 0 . 1 5 3 f 5 | (c~/~r)~
Brodsky and Kmoshlta [4141
-0 151(31 (cdrr)3
Calm':t and Per)c, ttel [416]
!
"In rcf [411:I graphs 13 and 14 should be mterchan~d with graphs 17 and 18
~1]0
RE. Lautrup et ai., Comparison between theory and ex;x,rtments In quantum eleclroclymmlcs
Class I11
I 0.055 429 (oda')3
Mtgnaco and Remtdd: 1411 I
,,(6)
0.055 46(6) (t',Irr)
3 Brodsky and Kmoshlta 1414 ]
~'e.lll =
0 055(2) (a/~) 3
( ' a l m e t and Perrottct [ 4 1 6 ]
Oass IV, V and VI
at6),av-'-' %,v't~)+ ~,,vt"t°= 1.23(20) (a/lr) 3 Levine and Wright [4191
The uncertainty m this case is not obtained by statistical methods but it is rather an educated
guess.
The overall result for the sixth order electron anomaly is then
a t6)
= (1 49 + 0.20) (ix/n) 3,
P
Table 4.3
i
Resultsof Mtgq~aco and Remtdd11411 I. Ct4 ~ ~ _I._ ~ LI. { 112) " 0 517 48
" ' ! ~ln'
Graph
Analyttc form in untts
.N[tlWteftt ~ title ,or
of (a/,r)'
iiofl-tnftqlfcd t e t m ~
In units of (a/~) ~
19
943
4 na + 38.~.¢3)
324 -
20+2 I
1547
432
22
1145 +.i..~
n161 2 _ _ ~ l r , log2
432
o ()02 5585
3 ~
,
[ h X[II9
'
)
o 0 5 4 6,'~
+-~cZ~ + '
49
4 ),loga-,+~'tog
4
42
--~
[~" ~[ !19 +..r..n,'~~_
3_:
7
,
- 0 (,11 805
Table 4.4
Companson of results for classes II and Ill
Graphs
Brodskv and Kmoshtta [ 414 ]
Calmet and Pe.:ottcl [416'
7+8
9+10
11+12
!3-!4
15+16
17+18
19
20+21+22
-0
0
0
-0
-0
-0
-O OO31tlO|
0 0522{10)
0 0274(5)
0 04 74(20) * mtt,ued te,m
-0.1151(9)
-O 0653(5) + 0 031 60(521 loglk/m e)
0.O02 559(15)
0032{3)
0532(4)
0273{3)
051(3) - 0 0~14 Iog(h/m e )
1161(14)
064(3) + 0.0314 Iog(h/m e)
0 002 55(2)
0 052 91(6)
0.0522( 21 )
221
II.E Lautrup et aL, Comparison between theory a~,~~ expe, tment¢ m quantum electrod)'narmcs
Table 4 5
Result,; from De Rfijt, la et al 1417',
Iablc 4 6
Results trim (almct [4181
I
graphs
Value m urals of (~/.)~
,ib7
~ra
I = .~ I
}
1
" ~ It}e- + t(~) h)ga +
()t!ah
68 * 69
10
-33321111+
~-~--~-~- IogA,+
Graph.,
Value in umts
ol (a/n) a
23~-24
25
29+30
39+40
1 206(65)
- 0 136(211
- i 431(91)
- 0 476(64)
Iogah
I able 4 7
Experimental valuc,~ h)r tl~,, electron anomah,
I-olry and Ku~ah ( 19471 10021
Kt~nil:rt al (19~2) 14301
Berntl~r and Ileaid (1954) [4311
Franken and Licit,., ( 195 7) 14 ~2 I
hchuppet al (1961) 14331
Wilkinson and Crane (1963) 1434.4351
Farallo el al. (1963) 14441
Rich (correctlon to ref [4341 ) ( 1 g68) 14 ~61
(;tiff el al (1969) 14401
Wedcy and Rich (preliminary value) (I 970) 14261
Wedcy and Rich (1971) [4011
0 001
0 001
0 0(11
0 0(I l
0 On l
0 001
0 001
0 (I0 I
fl O01
0 O0 I
() (I()I
!5(4)*
146( 12)*
148(6)*
165111)*
160 9(24)
159 622(271
153(23)
159 549(~11)
159 66130)
159 644(7)
159 637 7(3_:)
"]hc,~expcr=mcnt,tattuall~, mea~urt I÷G e a n d not G t dtre~.tl~
++here we ha~,c u,,¢d the ,inalytl~. ~.1,,,,,, ill result, and the weighted ,t~,magc i t the ~.ia~,-, il re,,ult,
lk'taiNol thc~.al+ul,it=on b~, [ ¢ v J n c a n d Wrt,_,h) 141u I i t the thr~e~.la~',c~ IV V V1 are a ~ \ c t l~,)t
,,,.,liable I1 p, thcr..lorc lmt'~o,,,,|blc to ,,.on~p.Jrc with the three p,lrtl,ll ~.,=h_ulata~,,,, ,trot ha',c ap
D.'JFLd pr.'~l¢)u,d~ [ 4 1 5 . 4 1 "7.41~51 and that ,oxcrlap x~lth these ula~ac~
All the 72 graphs have n o w b e e n calculated However, as " m i r r o r graphs" gwe the ~ame anomal}. only 41 oi the 72 are i n d e p e n d e n t In view o f the c o m p l e x a y o f the calculatmns the ne~.ess~tx
for I n d e p e n d e n t t.hecks must be e m p h a s i z e d So far su,'h checks have only been carried out for
clas.~s !1 a n d ill. and in a few m~tances for graphs belonging to classes IV. V and VI
Techmcally the calculation ol the class I, !1 and III gr'lphs is not different from the calculatmn
ol the c o r r e s p o n d i n g graphs for a u
a¢ (see section 5 ) For the r e m m n m g graphs some simphfication can be o b t a i n e d lor those that contmn self-energy and vertex insertions by USlnL' properly
par.tmetrlzed "orm.,, ot these insertions ** For those that do not uontaln such insertions (the ~rrc~tuclble ones) there is no wa3, a r o u n d a full-fledged parametrlzatlon ol the three intcrlavcd loop~
llle ~.omple~lly o! the sixth order graphs ha_, necessitated cxtenswe u o m p u t c r ttsc toe ,Jlg,'bral~
In,illtpuldllOIl% died IlUlll~.'lI~.,II mtcgrat,,ons [hc y-algebla and vector st~b,.tltLttions 'n t}lc nu~mr.,to~
at a graph can lead to h u n d r e d s a n d s o m e t i m e s t h o u s , m d s ot terms So tar tou," d~ttercnt atgcbra~u
programs have b e e n i.l~ed in c o n n c , . t , o n with the calculatmns d ~ c u s s e d here anu m set.lions I l.
"11~ r~mlts of Le~=ne and, Writ(hi agree however wlth t h o r of De Rfijula et al (M J Levme, prwat~ ~(,rnmumcatlon)
"beccg reS 14171
22
BE. Lautrup et at., Comparison between theory and experiments tn quantum electroclynamics
L2 and 5.2. The interpretative program SCHOONSCHIP written by Veltman 14221 used m rcf.~
4 1 1 , 4 1 7 , 4 2 1 , 2 1 7 , 1 2 3 , 5 0 7 and 1591, the language-oriented LISP-based program REDUCE
vritten by A. Hearn [4231, used in refs. [ 4 1 2 , 4 1 3 , 4 1 4 and 1191, a LISP-program written bx
2almet [420] used in refs. [416 and 4181, and a program written by M J Levme 14241 used in
~efs. [415 and 4 t 9 ]
The parametnzations of Feynman graplu, gener,dly have a smgutdr tot a|mo~t ~,ngulat-) t~ehavour at s o m e pa/,ts of the border of the integration region. Gaussian integration methods a~e very
3recise for integrands with polynominal type behaviour, but lose rapidly in reliability for integrand~
~ith a steep rise towards the border The singularities can, however, be removed by means of polyaomial mappings, thereby smoothing out the function and allowing for Gaussian integration. Thm~
Iechnique has been used bv Levine and Wright [4191 but has the disadvantage that it does not
~eadily allow for an esUmate of error. Straightforward Monte-Carlo methods do not work well mr
integrals that have their main contribution from some odd corner of the integration region, because the integrand ~s not preferentially sampled there. The following technique* has proved ade:luate for many of the integrals met m QED. the integration refaon (the unit hypercube) i,~ ,,ttbdlvlded rote a set of subvolumes by dividing the umt interval on each axis. in each sub~,olumc the
contribution to the integral and to its variance is estimated by random sampling of the mteyrand
(usually only in two points). Using the variances found one calculates an improved ~ubd,vl~mn OI
the unit intervals and rmterates the above procedure. In this way the function Is explored and the
interval structure refined such that the interval density adjusts i t ~ i f to the rate of ~,an,itmn ol Ihe
mtegrand, thereby mtmmtsmg the total varmnce. A program implementing tins techmque was originally devised by C G. Sheppey [425] at CERN It was first used by Aldm~ et al 1412.41 ~1 in a
QED context, but has since proved invaluable for many calcul,tttons
4 3 Meas.zrements o / t h e electron anomaly
The best experimental value for the (negattvcly charged)elc~.tron .tnom.tl~ ha~ ,,,~ tar been ,,b
tamed by Wesley and Rich [401]
-,~,l,=tl
159(~57 7 + 3 q } X 10 -~
This value represents an mcrease of 14 X 10 -9 w~th respect to the prehminary measurement b~
the same authors [426] The error has decreased by a factor of two.
The first mdzcations that the electron possessed an anomalous magnetic moment were reported
:n 1947 [427--429] and Kusch and Foleb's experimental vah,e [0021 turned out to agree with the
calculahon by Schwmger [003]. Over the years the [ recision on the naeasurements has .qead,lv
improved [ 4 3 0 - 4 3 3 ] m particular with the fundamental experiment of W d k m ~ n and ( r a n t
1434-435 ] m 1963. Thezr valqe agreed with the theorehcal calculations t, ntd Rich 14361 and
others [437,438] reanalyzed the experiment and brought out a three standard deviation dl~rc
pancy With the new experiments [426.401 ] flus discrepancy ha.,,, however, again dmappca-cd **
Experiments along slmdar lines but with less precision have al~o been done by other group~ ** ~
*See ref [446]
* * T h e original value o f Wilkinson a n d C r a n e was a e = i 159 6 2 2 ( 2 7 ) X l 0 ' c o r r e c t e d b~ R , c h to ~'le = I 159 5 4 9 ( 3 0 ) x l 0 "
' * * S e e table 4 7
223
B E l,aula~p et a l , Comparison bet~,ern theor;' a n d e~permwr~t~ tn qlta~ttltm ch ( trodt natal, ~
11~e t e c h n i q u e used b y Wesley a n d R~ch [ 42¢,1 ~,, e,,,,entl,~lh the ,,,line a~ ih`ai tl',ctl b,. \~ Ill- ,!,,,,~,
tlld ('ralle 14341 altho.~ the a p p a r a t u s I,, c w m p l c l e l ) , e w `and tile nldgnctl~. I~cld ~,, an o~tler |~|
n,w-nltude , , t r o n g e r [-le~.:4~on,, w i t h cgt~r~ ol a r o u n d IO(/ keV ,,re p a r t u d l y pola~l, c d b.~ M o l t
t,tttcrlng at t~O° o n ,i gold "o11 `arid ,,ul~scqt~'nllx l r,l,~l~c|l m ,~ Ill,lent I~_ t~ottlc ( ~ I()00 (,1 It~l .I
t t t u r a t c l y n l e a , , u r e d Intc~'al ,q t l m c . \ l i c e t'~t'lllg CIC ted f r o m lilt' t'~Itlc the pt~l ~n/,~,~on of the
~lc~.tron~ in a l m t y , , ¢ d by mean.,, o l d s e c o n d 90" M o l t ~catterlng ~ h l l e t r a p p e d t h e a v e r a g e sp:n
norton o f t h e e l e c t r o n s can be d e . c n b e d as a p~cccss 9n o f t h e i r polarnzat~on relative to tllc~r ve0c~ty w k h a f r e q u e n c y
~,'hcre a is t h e a n o m a l o u s m a g n e t i c m o m e n t a n d ~ , , = eB/m,,c' ( I f til,? m a g n e t i c l leld j~, n o t h o m o tencous o r n o l p e r p e n d t c t t l a r Io t h e veiocfl} t t t h e e l e c t r o n s o r ~1 t h e r e ,arc t i c , In,. Iicids present
:his f o r m u l a m u s t be c o r r e c t e d a p p r o p r i a t e l y ) ,,,, a fun~.tlon ot tt,ipplllg t l m c lhc t,,, t " /`alton `and
h e r e b y t h e c o u n t i n g ride will bc m o d u l a t e d w~th tbn,, l r c q u c m . y I b i s pcrtn~ts dcte~m~llatl,m ~_,I
7 Fhc m a g n e t i c field is m e a , , u r c d b~, mean,, ,~t NIle I~',: M`agnct~t Rc,,on,ln~.c ( N M R ) probe,, ttctcnllllllllg t h e r e , , o n a n c c l'requcnc~ ol proton,, m ~ , t t c r In ; i~.t
_
t°vqll:O)
¢o,,
~
~/p.
~hcrc tl'p Is t h e m a g n e t l t lllOll'lCllt o t the p r o t o n in .i w`aict ,,,lnlpl,: I -~ I tt~l/Jl~ tht' i~,hl w~la~t~
ton r h e I m p r o v e m e n t ol t h e a~cura~.y o n t h e anomaly, t,, c'~',t'lllld'tl} , l i b IO t i l e 1 I l I ' t ' r lll,tgilt.'tlt_
twht u,,ed b y W e s l e y and R~ch
a. p r o m t x m g n e w e x p e r i m e n t a l t e t h l l l q t l e I1,1', hct'tl p r o p o s e d 14401 a n d `a I' cltr~,mary re,~;11 o b t , , m c d 14411 by a Bonvl gel,up I hey ,,tud} i~t~l.tli/t.'tl clc~.tron,, , t~.ttldtll~t: ,a a xU`a~nCtl~.
!~tttl at t h e ~.vclotron I r c q u c n ~ , c~, ll~t ,,lul~ ~.al~ i~t lhpl~t~i i,~ ,[,1,i.~ t ~u ,t t,ltl~ ivt'tl',cP.,.\ (I ~)
h~ld at tilt l a r m o r (,,pin lqipI trcqu..n~" ~,~ = ~.~11 + ~)anti thL' tra.a,,~,,m l,, ot~,,-rxcd .~,~ 1!~,
A~.tOlllpa11',llli/. depol,trl/.ttlt|ll ,,I tilt ~.i~,tlc)n,, It t II<>',x¢~.ci .ll,,t+ [+t+,~,,thlc 1~+ ,+h,,crxu ttl,' }a 't It,
tJ'Itqt~, t,.,31
',..,~
L . ' l l d a o I¢~'¢1~, i t ¢
G.~
,i~rr,.',,p¢:,lldlllk' Ill I ,,l',lllIl,,Ilt.i)l,x ',Illll flip ,III(.] I dll'.,llli~li 1'~.'I ', t I ,k ,
~]l,titg~" u l v l t~1i~ A I t l C , t ~ t | l C l l l ~ : t l t u t
buit~ W L dtl,J tx.~L -- C,,.)¢ ludtt'~ tt, ,t t i t
tLtllll-
nation el
l+a
cat_
~ ; c ad~.antage o f the. C X l ~ ' n m e n t o,.cr the prev;ou,, one<, ts t h a t t h e anom,;~. ~,, mva.,urc¢! rt ,q,e~tr,
s~oplLally, t h a t t h e clc,.tron.,, a r c q u i t e ntm-rctativtstlt, q- low c V ) a n d b e l l COl. and ~"t
t,.~ `ale
,I |
t , e t c r m m e d b y t h e . ~ m e m e t h o d In t h e sam,: l,c,t, I h~' p r c h m ~ n a r y value ~,,
= l 1 5 9 6 6 0 1 3 0 0 ~ g 10 ~'
ll.d is ¢ x p c ~ t e d to i r l l p r ~ ' , t : t.c~r',.I t c r J h ' 3 In 111." :'~l~;rt
ih,. aHoiHal), ot the p o s t t r o n v, as origmall} Hica,,ut~.d b~, R~,.tt .lr~d ( ,,t,,. I I t ' , '. xtI,
. i.',
~trtualiy i d e n t t t a l w i t h t h e o n e u ~ r d by W d k m , , o n a n d ( r a n t [4 ga! (,,co the dc,,~.lJptl~l, , t l ~ , ~ ~
7hc,, r o u n d th~ , c s u i t
a:=
0001
168(11)
I~E. £a,atup
et al,,
CompariJon between theory and experlmmt: in quantum ek, em~dy~mtes
More-r~cently Gilleland and Rich [443] have improved the accuracy by increasing the length of
time the positrons were trapped in the magneUc bottle. The result was
ae. = 0.00 ! 1602(11 ).
The equality of anomahes for the electron and positron is a test of
CPT in~armncc a hich lnlphes
a e- =.a.e,
The experimental values for the electron a n o m a l y have been tabulated in table 4.7.
$. The quantum electrodynamics contribution to the difference between the anomalous mallrletic
moments of muon and electron
The graphs o f the purely q u a n t u m electrodynamical contribution to ~the anomalous magnetic
mament o f a charged lepton, electron or muon, can be div'ded into two groups:
1. Graphs involving only one lepton
2. Graphs involving both leptons
The anomaly for a lepton with mass m can therefore be written
a = al(m)+ a2(m.m'),
where m' is the ma~, of the o t h e r lepton. Since a is dimensionless, it follows that as must be mass
independent, and that ct= can only depend on the ma~,; ratio We ma~ then rewrite this eqttalion
as
a = a~ + a~(mlm"t,
wh~ch speciahzed to electron and iI'luon becomes
a r = a n + a2(mJttt
),
Due to the smallness of the ratio mJ:n~, it is not necessary to evaluate a~(.r} for all ~:dues of r,
but only the asymptotic behav,our for small and large x. It follows from general arguments that
a:(x) vanishes as v --~ O. One may thus, in gcr, eral, disregard the contribution a2(mJm~,t to a,
The difference
a.
-
a,
= a;(m,/m~
-
a2(m./m,)
only invok,es a 2 and the evaluation o f au - a e is gerL '-ally easier than tile evaluation o f the co<r-
plete anomahes.
In the discussion below, v,e shall only mcl-lde "b~se graphs wb,ich gi:e a " " . . . . . . . . ~,,,, ...... ,,~,l,,,Lion after reP.ormalization Thus we leave out all corrections to external hnes. ~ t also le.~xe out all
graphs representing renorrnahzation cotmter-ter ns, assunilng them te bc ~mph~ttl) included The
difference a~, - a e has so far been computed up to (and mcludingt the sixth order. It vanishes at
second order.
B E l.al~lt,~p i t a l , Coll,i~li.~oll b t t ~ ¢ t l
225
~hoor~ ,,rid e ~ p e r t m e n t s ol q l t a n t u m e l c ( t r o d v n a m t c ~
/
f/~ ,k 4k~ ~
\1a
/
9.
3
i~
20
/
t
i
z
l
,,'
2'
"
/
'
//
\
23
22
Fig 5 '. Mass dependent sixth order contributions 1o the leptt.n vertex
; I The f o u r t h order c o n t r t b u t i o n
In f o u r t h order there is only one graph tl'tal cont~ ,butes to c~2, n a m e l y the oa,: ob~a~tacd t[om
lie Schwinger graph b3, insertn'lg a vacuum p o l a n z a l i o n loop In the p h o t o n hnc Lip to Ic~,a,, ot
he order o! ( m e / n : , , ? we tlave the contrlt',utlon to a u
/ ??I \
~rllel
"
i
P#u
DI e
3()
7[
~
4
t~,lla
u
\l;lla]
II~, e
'
" \ttlta,;
'#',~
[he first t w o terms were evaluated by Suura and Wlchtaann 1501 ] and by Pctcrrnan 150" I i tic
IL
B.E, 1,au:rup et aL, Comparison between theory and experiments in quantum electrodynamlcs
Table $.1
Hzstory of the calculatzons of the sixth otdez contribuUons to the.. difference.
. . of. muon
.
and electron anomahes
;faiths (,fig 5 1)
Authors aad relerences
Year
tr+ ~
tmoed~
t~?
.q-24
L9-20
Lautrup and de Rsfset [410l
Lautrup and de Rafael 1506]
Aldtns, Btod~ky, Dtdner tad Ktno~lta [412, 4131
Lautrup+ Peterman and de P..aftei [$0Tl
Lautrup [5081
Brod~] and Khzoshica [4141
L-.-6
[I-tG
?-10
~-24
ISOSl
1968
1969
1969
1970
19"/0
1970
~maining terms have been obtained by Elend [503] and by Erickson and Liu [5041. These
authors have calculated the function ct~4)(x) for all x. To the same accuracy, the contribution to
ae is [4101
2 ~m.]
45~,rn~/
+O
mr
,
which is vamshingly small, but has been im.,aded as a reminder of the fact that both a. and ct. depend on the mass ratio.
Hence the ft~orth order differencc (fig 4.2) is
(flu -- ae )(4) ""
( ~ ) 2 1 l " rn~ 25 ~.a m e
"31og-~ e -- 36 + 4 m.
4(m__~e)2 - - + ' ~ "134[m+~'
'n'~Jm" +o[m(-~ )']1
xmu/ I°gm¢
Numencally ~e have
(a~, - a.) (*) = 1.094 26 (c,/~r): = 5904 1 X 10 .9
5.2. The sixth order contribution
In sixth order there are 24 graphs* contributing to tta - ae, They are shown in fig. 5. I. These
graphs naturally are divided into three classes:
I Graphs 1 - 6 , containing photon-photon scattering subgraphs
tl Graphs 7 - 2 0 , containing second order (but not fourth order) electron vacuum polarizauon
subgraphs
Ill Graphs 21-24, containing fourth order electron vacuum polarization subgraphs.
Not all of these graphs are independent. Some are related to each other by charge conjugation,
thereby giving the same anomalous magnetic moment contribution. Two graphs give the same
anomalous magnetic moment if they arise from each other by reversing the directions of the muui
line Titus there are 14 independent sets of graphs, the only unpaired graphs belng 9, ]0, 21, and
24.
From table 5. l which gwes the history of the calculations of these graphs, it is seen that all
graphs except those of class ] have been evaluated twice (one-graph 21-even three times). The two
+Here we ~
the contnbu~on from
C~2(me/m~),
227
B.E. Lautrup et aL, Comparison bet ~een theory and erperlments ol quantum electrodvnamw~
Table 5 2
Sl~th order resuit~ from ref~ [410.5,q5,506,507 and 421 ]
Graphs .tnd
Semtanalyucal form m umt~ of (~/n) ~
'kl'll t t " t ~ . d l ~,dll.I~
Ill Urlll~i o'[ I ~
1 + II[4211
I~g2÷5~'(3 logL'~+ 2 0912)
rr~ 3
- 1 23(2)
?~Ie
9 14211
m,
io 14211
tl +
12 15071
-
m~ o
aol
IOS me
me
log ~ - - ~ . , o g . ~ e
I
~.
"j log ~ me
1'87
.'
2
~ 8
27]
[-ff.~ 2"7"t
))
4 962(4)
me -
logmat+
me
"%-
- 2 162(4)
298(6)
5 807(6)
13 * 14 150"/I
t5, 161~o7 I
1 387(6)
*
-I,o
Iog..-V- - 0 884t4)
-
-7 717(4)
~#ie
17 * 18
15071
0 67612)
1~÷ 2o !S061
2~ 1410.50~1
2 2 . 2 3 + 24 14101
0 100~
~-Io~ -'[~
me
"
1.
25 Iogm2U. 317 rr'
27
tn e ~ + -2"7
" l u , 1 .,)
2 7241
5
15173
independent re,,uit,, for graphs ! - 2 4 have been tabLdated Ân taN: 5 2 and table 5 3 Jllowlng 1or
d~talled comparison ol the values obtained fronl ca,:h graph "l hc unccrtamtlc~ ar)~ trom the
numerical integrations.* The agreement is generally very good, although there ~s a shght dltlerence between the two values tor graphs 7+8.
We can n o w q u o t e the overall results for the three ca~es**
('lass i [ 4 1 2 , 4131
(o_ u - a.e)l ~) = (18 4 .~ 1 1) (a/lr~ ~
Class II [ 5 0 7 , 4 2 1 , 5 0 6 , 4 1 4 ]
~(6) = t( - - 2 . 3 0 +~ 0 02) (et/~')3 t table 5 2
( a . - c%,.
[ ( - 2 32 +- 0 !0~ (a/~r) ~J table 5 3
Class III [410, 505, 4141
tau - a " t u
¢6) = t4"2414(°q '*rl~
1
14 21 _+ 0 03) (~/Tr)3
table 5 2
table 5 3
The overall agreement between the two |ndcpendent ~,~lculatlon~ 1~ c\cellcnr, tt-e ~m,~li d t l t c r ¢ ~ c
having largely cancelled out m the sum Fhe itnal result is
• S¢¢ the dhscusEion at the end of ~ t t o n 4.2
unecrtamtlesql~lclvarrcally because of the ~tatlst|cal nature of the errors obtained [~ the parucular integration,
t e c h t ~ u o ~hich has bccn u ~ d This ts m contra~t to the attttudt of the authors of ref [414l, ,vho add the error~ Imcad~ m
whmh cage one would haw obtained ( - 2 . 3 0 ,- o.05), ( - 2 32 ~ 0 20), (4 21 ~ 0 04) for the re~ult~ of classe~ II and I|1
** We add ~
B.F~lamtrupet a~,,Comparisonbetweentheoryand experimentsin quantumelectrodynamlcs
Table 5 3
2',
m,
25,3n'me'
Stxth order results from ref. 1414] f(o) = --~jtog ~ - ~ *
3~
m e 12
4
Semtanalytmal form
m umts of (a/~r) s
Num~rlcal
;raphs
r + tt
) + 10
t t + 12 + 15 + 16
2f(p)l-0.4671 + 0.76(1)
24~(#)[ 0.7781 - 0.53(6)
f ( p ) [ - 0 . 6 S 4 ] - 0.53(7)
-1.28(1)
2.88(6)
-1.96(7)
value
In unll~ ol (a]n) s
i3"~ 14
f ( # ) [ - 0 . 5 6 4 - log ~ ]
- 0.18(6)
-!.41(3)
17 + 18
f(,o)[-O.090 4- log..h-- ] - 0.45(3)
-0.650)
m~
1 9 , 20
21
22 + 23 + 24
0.101(2)
2.'/2(2)
! 4q(2)
( a . - ae) (6) = ( 2 0 3 ± 1.1 ) ( ~ / l r ) 3 -- ( 2 5 4 ± 14) X 10 -9.
In view of the fact that the class I contribution is by far the most ~mportant one, being an order
of magnitude greater than the rest, and at the same time the only part that has not been checked
by an independent group, we must emphasize the need for a recalculation It would also be desirable to lower the uncertainty because the planned experiments on the muon anomaly may
reach the precision of'-- 15 × lO -9.
We shall now dtscuss some of the techniques used in obtaining the sixth order result. ]'he amphtude for an arbitrary vertex graph can be written --teF),(p~, Pt ) where pt and/9, are the momenta
of the incoming and outgoing lepton The most general expression between two spmors ts
it21"#(l;:, lJl )ltl = if'2
t (F~
- Pt)~ F~
+ ~'2)3'. - -(P'" +P2)~
2m
F2 + (Pz ~,~
t
= 5"2 F t 3/. -
wluch defines the three form factors
alous magnetic moment ~s
f
u~
icr#a'(P:-P')~'
2m F2 + (Pa-Pt)u
2m F31u ~,
Ft[(p~ -
pz) a ] By defimtion, the contribution to the anom-
a = F2 (0)
A graph can be subjected to two kinds of gauge transformatmns, external and internal. A set of
graphs is mvanant under external gauge transformations when
(p: - p,k' ff:F~,(p:, Pl )ul = 0,
whzch mlphes that F~ = 0 Th~s condlhon (current conservatmn) ~s not m general satisfied for mdv
v~dual graphs. In fig. 5.1 the sets o f graphs mvanant under external gauge transformatmns are
I+3+5, 2+4+6, 7+11+14, 8+12+!3, 9+15+!6, 10+17+18, 19, 20, 2 1 , 2 2 , 23 and 24. The external
gauge transformations are, however, not relevant for the purely intrinsic quantity a, which will
mly bc mfluenccd
or ot the form
Yom which
one obtams
b> dlttcxntut~otl
ii,
l-1.
._ .I.
I Ilclrl.~y
uric
L&Iii
WrIti
h?).
'30
B.£. lautrup et aL, Comparison between theory and exper!mentz in quantum electrodynamles
Fig. 5.3, Vacuum polarization correction to $ - 2 .
[rt I'~,x, P2 can be put equal to pt because one power of q is already taken outside. Differentiation
3f a graph with respect to an external momentum acts like the insertion of zero momentum photons and decreases the degree of divergence by one, thereby removing the spurious logarithmic
J~vergence. The remainder of the calculation is in principle straightforward although very complicated. Aldins et al. used two different techniques for obtaining tlle parametric form of the integral, one being the stzndard Landau method* the other based on a method developed by
Nakanlshi [51 O] and Kmoshita [511] (double parametric representation). Part of the reduction
to parametric form was done by hand, and part was carried out by means of REDUCE, a programming language for algebraic manipulation developed by Hearn [423]. Finally the (7-dimen.
~ional) parametric integral was evaluated numerically by means of the special numerical integration program described at the end of section 4,2. It was found that the class I dmgrams contains
a logarithmic divergence for m e -* 0 contrary to expectations [505]. Writing
DI e
Mdm,, et al found that C~ could be expressed as a 5-dtmens~onat mtegral with the value
C,=64_+01
Flae remainder C2 was not determined directly but could be referred from the overall ~alue quoted
above
(72 -~: -16-+ l
I'he numbers Ct and C2 are surprisingly large. In all other cases purely numerical coefficients turn
out to be of the order of unity. This unexpected behaviour also stresses a need for a recalculat,on
of the class I diagrams
The remaimng dtagrams are all much sunpler to evaluate. All of the class Ill diagrams are examples of the graph shown m fig 5.3 where the insertion G is a fourth order vacuum polarizahon
graph. The vertex function fro.a this graph ts
I-,(G)(p~.. pt)
2 c d t,
u
= e 3(-T~)~ -y#(/92 - K - muYl 3,u(,~ 1 -- ~ - mu) -t 3,0 D~G)oo{A.),
where D (G)
po is the contribution to the photon propagator form the graph G This quantity can be
expressed m terms of a single spectral funchon
*See e.g Bjorken and Drell
{5091, sec. 18 4
B F. Lautr, Ap ¢t al, Compartson between tlteo O, and e.xper:ments tn qt~antum ele( trod1 namt( s
1
p-
231
kpkol fl(t~(X: )
~ "/
~ -,
Do
~attsl'ymg a once subtracted dispersion relahon
II(¢;~(/~:)
/,:
=~
o
The imaginary part
J
:
dt
/Italian')It)
-F
is given
-t
h ~---
by
--I
O(k)lmll(C)(k2) = ~'~ ~
(2~rp6(4)(k - k n) ( 01J~(0)In ) ( n l J ~ ( 0 ) 1 0 >
neG
where
the sum
goes over the possible mtermedmte states
mG
Acco~-dmgly we can write
o
where V(~)(p~. Pt. t) is obtained from the usual second order graph replacing the photo,~ propagator - ~gu,,/k a by a mass~ve transverse propagator
_i(g.v
/"./¢~
I
The anomalous magnetic m o m e n t satisfies tile samc t.'qtt,ttlon
:.W,, =
f ~m
li nc'"(r)c<t~('~tr',
0
here a(~)(t~ ~s the dnonldlot~ n]agnct,~, m o m e n t obtatncd l rom the sc~_ond order graph rcpla~lng
the l:,hoton propagator by a m,,~s~,,c propagator [ h e / , k tcrm~ t.an bc d~rcgardcd a,, the3 onl.
~ o n t r l b t ~ t c t o the renormahzatton c o l l M a l l I Z'.~ d l l t l d ~'~ )(t) I'~ glVt.'ll [3y the ~,cll-kno~n c x p r u , ~ ) n ~
~
dl:)¢t I = ~h(t)
/1"
1
•
5x"h/m:-
o
These expressions are vahd for any kind of m s e r h o n G, m parttcular for the hadromc vacuum pola
nzahort(see section 6 I I In the case of fourth order vacuum polar~zahon m,crhon~ the function
lm II(c)(t) was already known from the work ol K~illcn and SabD' [51 21, and the w h o l : cal~.ulatlon bodeo d o w n to a one-dimensional integral wtuch could be evaluated analytically w~th the result shown m table 5 2
For lhe analy'lc e~aluatlon ol the double bubble graphs 1<) 21, an e×,wc~tcm ~cry ~c~l ~t~ll:d
,~ 15061
I-x
*K(t) ts b o u n d e d , and monolonlcaUv decreasing
t
I m~, _
tI.E. lamlrup et aL, Compaei,von between theory and experiments tn quantum electrodynamics
vhich is ol;tained interchanging the order of integrations in x and t m the general expression for
~.(o:,. Here, the llt°Lfanction is rumply a product of two second-order ll-functions
For the class II diagrams it follows by the same arguments as abo,,e that the anomab for a
~aph is ~ e n by
f
Ct =
.dt
t
.1_ lm II(2)(:) a(t)
it
~hexe Ht2i(t) is the second order vacuum polarization by electrons, and a(t) is the anomaly from
the fourth order graph obtained from a sixth order graph by replacing the photon propagator con.
raining the vacuum polarization insertion by a massive propagator
-i
uv
k 1 I ki-t
~he kuk v terms can be disregarded because they are essentially gauge terms that cancel within the
following sets of graphs' 7+8+9+10, I 1+15, 12+16, 13, 14, 17, 18, 19.20. Thus at the expense of
ar~e extra integration relative to the fourth order calculation one can evaluate the anomaly from
:.he class II graphs [ 5 0 7 , 4 2 1 , 4 1 4 1 . It is even possible to perform the t-integration analytically m
the limtt mJm,, -+ O. For a graph (or set of graphs) for which a{O) exists* one has to order m,'mj,
a
n ' t t 3 l°gm--~
9 a(O)
3
tmul
where
<,: f
+
does not depend on the mass ratio. The quantity cc(ti is a multldmlensional integral (tip to 5-dr
merz,,~ions) of the form
,J
f "f
h n
c.+o,-
wtuch can be integrated analytlcall~ over t. For t = O. attl is the ordinary fourth order anomaly
from the graph (or set of graphs} obtained by removing the vacuum polarization.** These fourth
order anomalies have been given by Peterman, so that it is trivial to obtain the analytic form of
the coeffic:ent of log(mu/m ~) (see table 5 2)
The final stage of the class 11 calculations is a numerical Integratio,i of the parametric integral
again uswg the Sheppey program described at the, ~ld of section 4 2 The reason for the dlffcrcntx
m ;.he quoted uncertamUes of the two different calculations of the class II diagrams is probably
*If R(O) does not exist (as for graphs 9 - 1 8 ) it Is always possible to separate C((t) into a regat ,r part for which the analysis can be
carried out as shown, plus an irregular part which can be evaluated exphcltly
**If a(O) exists. Otherwise se~ the prevlou~ footnote
B lj'. Lautvup et al, Comparison between theory and cxpertment, m qventum ~lectrodvnamws
233
lue to the fact than the analytzc integratzon over t was t.arrzed oat ,n tots [50(~ and 5071 ,table
;2) b u t n o t t n r e f [414] ( t a b l e 5 3~
Other contributions to the rouen anomaly
~. 1. The h a d r o m c c o n t r i b u h e n s
The importance of hadronic vacuum polarizahon insertions m the seco~ad older rouen .,ertex
'see fig. 5.3) was first pointed out by Bouchtat and Michel [60! i They rentarked that resonances
,n the t'-,r system can lead to an e n h a n c e m e n t of the vacuum polarization corret.tlons which
:ould be observable in prectsion measurements of a~, The first estimates of flus effect ~ gave av
[hadrons) -.- 10 -*, t.e.. far below the error m the measurement of a v at that time wh,ch was [O0al
i 5 X 10 "*. Since then. a new precision measurement o f au has been made [007. 0081. and simultaneously, colhdin8 beam experiments have been performed at Novoslb~r~k 1605. ~061 a~a'~
Dr~y [607--609] which yteld prectous reformat,on on the i~adromc t.oHtrd~ut~ons to ~,
The hadrontc spectral functton which appears m the Kallen-Lehmann rcpresentatton el the
photon propagator (see the discussion of class Ill dtagranls 111se~.tlon 5 2) can be directly obtained
from measurements of the total e°e - anmhilatlo=a cross-section rote hadrons To 1o~ est o l d e r m
(~e fig. 5 3) and t ;~ 4ma,,.
4~raa
O~.e-_. hanron~ l , t ) - ~
im llOl)(t),
where t ts the total e'e - centre-of-mass energy ,~quared The hadron~c contribution., to c.,. due to
vacuum polartzatton insertions tn the second order rouen vertex, can thus be .tt~ectl3 related tt,
the anmhtlatton cros,,-sect,on measured m the ~.oll~dmg beams cxpertrnents A,~um,tag ll,.,Z the d~pcrston integral for the h,tdron~t , a c u u m polar~zatton only requires one subtractto,a ' ~h.t~,',.' l~'taotmahzatlon) we have
1
a,,(hadrons) = .la,3
f
dt o e.~-
hadrons
(t)Ku(t)"
,i
Onrr
where K v ( t ) is a purely QED function whtch results from the c o m b m a t l o n of the two ferm~on
propagators and the e, opagator of a " p h o t o n " w~th squared mass t m the rouen vertex of llg 4 I
!
Ku(t} --
f
x 2 ( l - x)
dx ~.2+( 1 - r ) t / m 2
o
o
The function K ~ t t ) ts posltt~e d e f i m t e m the integration region 4m2. -'-'- t <~ oo attd ttac~¢l(~re
au(hadrons) m u s t bc postttt.e Notice that, for large value,, of t, K~t ) ~lc~.rea~c, l~ t '
i
K u ( t ) = "3
t +ol m /tr"
*It dependent calcula~ton't were also made by Durand [602[ and later, usmd new mformatton on vet tot .'Tl~on~, b~ kmo,&lta and
Oakes 1603]. and by Bowcock [6041
I34
~
Lautrup et al., Co~avimn between t h ~ v and experiments in quantum electmdynamics
:t appears thus that the high ,~nergy contributions to au(hadrons) are depressed by the factor
g'a(t).* The au(hadrons) integral is dominated by the low energy region and, in particular, by
Ihose values o f t corresponding to the mass souared of resonances which have the quantum tRml~ers of the photon.
A calculation of a~,(hadrons) using the results of the Orsay coihdmg beams experiments has
~ n - d o n e by Gourdm and de Rafael 16101.1he total contribution to a~(hadrons)was separated
into an isosealar contribution t = 0 and an ~so,.eetor contribution I = 1. The isosealar part a~(l=0)
was estimated umng vector meson dominance, taking into account the contributions form the t,
md ~omesons:
V - . e ' e - )=
c~(/=O)"- v=,~.~,
~ 3Ku (M~:)r t......
Mv
(6.1 ± 1.2 + 5.0 ± 0.8) × 10 -9.
l'he isovector part au(l=l) was estimated assuming that the I = 1 hadronic system is dominated by
the :r- "r P-wave. The evaluation of the integral was made using the expression proposed hy
Gounaris and Sakurai for the pion form factor [6111. This expression, which is based on a generalized effective-range formula for n - n scattering, takes into account finite width effects and
fits well the Orsay data ranging from 644 MeV to 886 MeV with the following values for the
mass and the width of the p-meson
M o
= (770 _+4) I~-V,
F o - (! i ! + 6) MeV.
The result obtamcd is
ctu(t=l) = (54 + 3} × 10 -9.
Altogether au(hadrons) Is estimated to be
au(hgdrons) = (65 ± 5) × 1O-'.
Notice that the error quoted ai;ove only reflects the uncertainty m the Or~ay data used as input
The uncertamt:es due to the extrapolat,on of the Gounans and Sakural c ,prcss~on for the p~on
form factor below 644 MeV and above 886 MeV, as well as the uncertainty m neglecting other
contrtbut~ons to the isoscalar part than those coming from the ~0 and the x0 resonances could
be larger than 5 × 10 -9. Clearly, more experiments with colliding beams in the region just above
the ~ ' o p~on threshold and at h~gh energies will be extremely welcome to reduce these ur~certainties.
*In pa~tlculas, flus explams why hadromc vacuum polarization mserUons m the second order electron vertex can be neglec,cd
The~ ~ve a contnbutton
ae(hadr°ns) -~ ~
1
2
me
fdt
d
0 ÷
i"
ee"
~ hadrons (t)-
4m~r
In fact. using the mequahty (t¢/t)K~(to) g l~u(t) g-~m~t t o r t o g t. we get
l~me~ a
1
O'e0mdr°ns) 'g 1"2~'~] K#(4m~) a~a{hadr°ns) = 3 7 x 10 -s au(hadrons)
B.E. Lm~trup et ai , Comparison b e t ~ ¢ n theory and experiments tn qaantum ,.lectrodynamws
2"~5
There have been some atte"~pts to bound the hadromc va~.uum polarization contr~but~om, to
t~, An analysis by Bell and dr. Rafael [6121 shows that a proposed theoretical bound [6131.
~ased on the hypothesis of current-field Identity [ 6 1 4 ] , does not add usefully to the strict vc-'or
ncson dominance (V.M D.) estm~ate of a~,(hadrons) * In fact, the e~hmated botmd depe~ad~
tgam on a V.M.D. approx,mat~on, and th~s moreover t~ a quantity Q)r which that ,~l~proxlmat~on
s less rehabl¢ than for % ( h a d r o n s ) itself. It ~s possible, however, to bound ~.aOtadro,~) m ~t d~t'erent way by a q u a n t i t y which governs all sufficiently low energy vacuum polarization effects
,612]. Indeed, the hadronic contribution to the p h o t o n propagator IS
p(m(qa) =
f
~ (l/~r)Im H(t)
t-q a
md we have
%(haarons) < ~r-T
~ m~/~")(0) ~<-~c~
1 m~(l
Ior all negative q2. Large m o m e n t u m transfer electron-electron scattermg experiments can be u.,,ed
to set a limit on p n ( q a ) for particular values of q2 Present experimental results lead to an upper
bound
au(hadrons) < 9 X 10 -6
tht~ hmlt, however, could be lowered down with Hlaprov~ d cmpat~.al ~.nowledge on electron-eletIron scattering at large momentuna transfer and electron-positron annihilation at all avadablc c,aerg3es
b 2 The w e a k i n t e r a c t i o n o)prlrtbut o n
To gave a defimte prediction for the contrlbtttlon trotaa weak mterat.t~on,, ts not po,,,~lblc bc,.au~.
o: tile inherent theoretical dlfflcldtle,, with hlglael order weak co*rectiolt,, E~tlm,~tcs ~.an l.o~c~c!
be made.
If the weak interactions are of the local four-fermlon V-A type the dmgram m fig. 6 1 wdl gwe
n~e to an anomaly for the muon The corresponding dmgram for the electron ~s obtained by raterchanging e and/~. Power counting indicates that the anomaly wdl be quadrat,c m the cut-off and
by dlmensaonality arguments one would expect
~-
C
a
where G v is the Fermi couphng constant. A ts a cut-off and C a numerical con:t,mt Thi,, con.,ta~,t
surprisingly, turns out to be zero [ 6 1 5 ] . therefore, the dominant term.,, to bc expel.ted fxom tile
lour-fermlon interaction IS**
a #t~ ~ C G ~ m~, lo~(Aim,) "---i0 -~
'Using the e-~penmental input quoted w ref~ [ 6 0 7 - 6 0 9 ] , the strict V M D estimate ot a~(hadron~) ts (6~ ~- 5) ~
0 " ~
re1~
[610] and [612J
**C Jarlskog [634l has recently made a : i r e f u l evaluatJor cf the four-~ermlon contribution It appear~ that the quadratt~ tcHn~
~ti
RE. Lautrup et al , Comparzson between theory and experzments in ouantum electrodvnamics
:ig. 6.1. The four-fermton V - A contribution to the anomalous magILetic moment of the m u o n .
Fig. 6.2. Weak c o n t n b u t m n o f the muon anomaly via the intermediate vector-boqon W
rmch too small to be of ~mportance. If, however, the weak interactions are mediated by an mteravediate charged boson, W, the weak anomaly will only be of first order in G v = x/~, ga/mw . The
:orresponding diagram ~s shown m fig. 6.2. The histor.¢ of this dmgram has been particularly cona'oversml [ 6 1 5 - 622 I. Only the last two calculations ( Brodsky and Sullivan [ 621 !, Burnett and
Levine [6221 ) agree on the expressmn*
aw = Gvm:u ( 2 ( I - K w) log ~ + 10/3).
8~r=x~
Here K w ~s the anomalous magnetic moment of the W and ~ Is the regulanzer used m the ~-hmltlng
pro~.edure [6231. The merall factor is
GFm:~" ! 0X 10 -9
8~r:x,r2-2
The other factor ts m pr:nople u n k n o w n Brodsky and Sulhvan take ~ = a (the line strtu.ture ~on, W
~.
~tant) and K w = 0, obtaining "u
10 x 10 - 9
whale Burnett and Lewne get tzuw = - 2 0 X 10-9
usmg~ = (row/A) 2 = (2 GeV/300 GeV)-' and also K w = 0.**
6.3 Measurements oJ fhe muon anomal~
"Hie best experimental value for the muoq anomaly has been obtained at CERN [007, 0081 anti
the value lsi"
aexP = 0.001 lt',6 16(31)
*The ca~e A w = 0 considered by Schaffer [620] also agree; ~ath t' t'. cxprc~saon
**If the anomalous magnetu moment of W ts K w = I instead of 0, as suggested b~ a r c ~ n t l y rc~txed thcoD [636l, the v,cak
anomaly becomes fimte, posatave and small ~ 3 X 10 "9 (see iefs. [ 6 3 7 - 6 3 9 ]
tThts value contatns results for both u" and u*, thus assummg CPT Separating u ° and u- c o n t n b u t m n s one has ~'t~ - a u- =
(S0± 75) x 10-' (see t e l [R l)[ a~a te~t of CPT
B E L a u n ,~p et a l . Comper'~on b e t w e e n theory and e.~pe~ t m e , t s tn q uan ,um electrod~ ~a ,,~ ~
237
Table 6 1
Mea~urement~ ol the muon anomal~
atl,,~r~
Rcfcren~c~
\lltqllal\
irwin eta|. I,|960)*
aarlmk et al. (1961)
~arlmk et al. (1962)
~rk) ¢t al. (1966~
~de) ct at (1968)
ent'y et ai. (1969)
11~241
[625 ]
[004.6261
[6271
[007,008, R 11
16281
[)0LIx
0 001
0 001
0 001
0 00l
0 001
13114~
145(22)
162(5)
165(3)
166 16(31)
060(67)
I'h~ experiment determined g = 2 ( ! + a l rather than Ct,
l~e first d e t e r m i n a t i o n of the m u o n anomaly ~as the p~ec~sion measurement by Garwm et al
624] of the magnettc m o m e n t of the muon C o m b i n e d w~th the measurement ol the muon ma~s
~e anomaly could be calculated The prec~,o~on (see table 6 1) was however not sufficient to ,,ce
ae effect of the fourth order term (section 5 ! ) which d~ffcrs 1, om the co~re~pondmg term ~a the
lectron anomaly due to vacuum polar~zatton b~ electron~ l'he ~ubsequent ( LRN e \ p e r t m e n t
~25. 004, 626, 6271 although m u c h more precise, dtd stdl not allow an.~ deflmtc t t~ta,~lu.,,~on to
e ~eached about the fourth order term The second CERN experiment which gave the tmal ntma,or quoted above, however, establisi~ed that the m u o n anomal3 d~tlered flora the electron anomly by the a m o u n t predtcted by t h e o r y it was e~,en so prec~e that ~t wa~ ncces.,,ary to inch,ere the
~ t h order term m the theoretwal vahte m order to obtain full agreement ~ t h e , p e ~ m e n t
I'he CERN experiments measure the anomaly d~rectly (as m the electron e \ p e n m c n t , , ) lrom tl'e
~reces.,,~on o f the spin of the m u o n relatwe to ~ts m o m e n t u m In a uniform magnetw f~eld t3 the
~rctess~on rate (m the laborato;y) ~ 1(~35]
Cd a
=
a
t~lta~
ndependent o! the velottty ot the m u o n s
est m v a c u u m ~s
to,, = ( l + a )
I he spin (Lalluo~) pre~.es>lou llcquctl,.~ oI ,1,~,c~a~ at
~'--B
DILC
;uch that a/(l+a) = tOa:W~, The qua,atlty w , cannot be determined dtrectl,,, hut one ,.an mea~tl,'e
he precession frequenc) of protons ~n water COp Izy means of a nuclear magnclJ~ rc~onance tN\lt~ ~
nagnetometer m the same magnetic tteld ( ' o m b m e d with the measurement¢ [629. 030] of the
'atto X = w',,/co't, o f muo 1 to proton precession trequencles m water we find
t
a _C°a ! I
l÷a
' "-~- ~-t
(.Op
,~'here we have put to~,/ta~, 1 / ( l + e ) The Rudermann [6311 ~orr~'~t~on e ret~r,'~c,~l.~g 'I~,' d~.,'~, ~aetlc shielding o f the field of the m u o n m water has recently been e\perm'temall) ~ho~n to b~'
tleghglble [6291.
The frequency tOa~S Ineasured by observing the det.ay ot muons ~n a storage ring w~th a n almo,1
~g
RE. Lautrup et aL, Comparison between theory and experiments in quantum electrodynaml¢~
miform magnetic field ('-- 17 kG). The muons (--- 1.3 CeV) are obtained from the nearly forward
lecay of ~"= created when a proton beam (--- 10 GeV) hits a target reside the ring. The muons are
~n the average longitudinally polarized ("- 26%) when they become trapped, but whde they are
:.irculating before they decay the spin wdl turn relatively to the momentum wtth the frequency
a~a. The decay rate m the forward direction will be modulated with this frequent). The top end
3f the electron spectrum seen at the ms,de of the ring corresponds to near forward decays and wdl
thus ~ be'modulated with ~a.. For further details we refer the reader to for instance refs. [007,
008ahd-l~. i I.
A vewg~,-2 experiment [ 6 3 2 , 6 3 3 , R. ! ] is being planned at CERN as a continuation of the
~.~vious experiments. Ingeneous new features lead to an expected overall improvement of the uncertainty by a factor of 20.
Finally let us mention that Henry et al. [628] have also measured a~ with a technique resembling that of the measurements of the electron m ,maly (section 4.3). The result deviates from
theory by two standard deviations, but the precisioa is rather low (see table 6.1 ).
7. "Exotic" contributions to the electron and muon anomalies
In this section we present a summary of the various speculative contributions to the anomahes.
It is generally iraposstble to give definite predictions for such contributions due to the lack of
knowledge of the couphng constants and masses of the hypothetical particles or fields involved
Instead we shall turn the argument around and use the agreement between theory and experiment
to put limits on such parameters.
7 l Generalittes
I f a theory g,zves a certain contribution Aa to the anomaly of a charged lepton, th~s quantity ts
restricted* by the mequahty
I Aa+
a tla -
a exp I ~
C~/{oth) 2 + (oexP) 2 ,
where a t" __.o th and aexp -+ o'xP are the theo;etical and experimental lepton anomalies with assodated one standard devmtion uncertainties.** The constant C is related to the confidence limit of
the bound. We shall choose a 95,% confidence limit with
C = 1.96.
We have (sections 4.3 and 6 2)
aeCXp=tl 159657.7_+ 3.5)X 10-9,ateh=(l 1 5 9 6 5 5 4 : t 3.3)X 10 -9
Ct~xP = (1 166 160 + 310) X 10 -9 a~,h = (I 165 879 _+ 15) X 1(~,-9
U
--
•
stacb that the mequahty ,';Jove becomes
*We ~ssume that two or more exotic c o n t r i b u t i o n s do r o t consptre to cancel each other
**We ~ ave added these uncertamhes q u a ~ a U t . A l y although this ts n o t a u m q u e choice
239
B.E. Lautrup et al., C o , ~ , q s o ~ betu.ee,t t~.eoO , and experiments tn quantum electrodvnamw~
-7.1
X 10 -~ < A a e < i l 7 X 10 -9
-325X 10 -9 < z x a u ~ 8 8 7 x
10-*
Observe that the tnuon bound ts d c m l n a t c d by the e \ D nnaental uncerta~nt.~
Uany of the ex,)t~c contrlbutmn-, to be d~scussed bel ~w depend on a ma,,~ par,~meler ,\ ~tu~t~
~ n be ~amp|y it hagh mass cut-off or the mass of a h y p o &ettcal heavy parUcle, i he exotic, contrtbutions vanish in general when A + 0- and in most cases we have a quadratic dependence on 1 / A
i.e. for A > > m
A a = A ( m / A ) a,
where m is the lepton mass and A ts a quantity consisting of couphng constants, numerical constants and perhaps a slowly varying function of the masses (Let us stress, however, that there are
examples discussed below that do n a t have this form ", The bounds on A a lead to lower hm~ts on
A The superscripts + and - correspond to Ac > 0 and A a < 0, respectwely
electron
A; > 4 S GeV v/A
A;
S.9
,/-A,.
A;~
113 GeV ~fL
mtL(~Pl
A; > 183 GeV ~ - A ~
Thes,_" l i m i t s are s o m e w h a t bctttr For i l e g a t i x ' e ex(~ll~ t ( m t r d ) , ~ ! o n , , because the~ hl.lkC xAOl ~" tl,,
agreement between theors and e x p e r i m e n t
,,M/hough the ele~.tron expert ~ent Is about t~O t~mcs m(,rc prc~lsc than the muon cxpc~mc ~t
the l a t t e r in general p u t s m o r e
L r l n g e l l t hillllS Oll c \ o t l ~ _ . ( s l l l r l b t t t l o l l s
[ Ill,, ix ~ll tlt'tl\ d t , c tv, ',;
large xllas~ o f t h e m u o n
~ Ill~.tl ~ l,~w5 It t o
prot
c mt~,
h
sin,flit, dlxla,)v,,
lt~,,~) lht
,I.,
,
A very interesting case a m e s when we assume*
Ae=A ~ = A
A , ~ Au = A
m wtuch case we o b t a m
A* > 113 GeV
A-
l s3 C,eV , / 7 7
Since ,low
)2
ri'he~ ~luatlor~ could be called e-~-umversahty hut we shall refra,n from doing so here because ol the ambtguou'. ,n~.n -,._' ~!
ttus termmoloD
~YI~t
1~lL lamtrup et aL, C o ~ n
between theory and experiments in quantum elecirodynam~
we obtain an induced bound on the exotic electron contributions from the bound on the corresponding muon contribution
- 8 x 10 -12 < Acre < 21 X 10 -12
We_canconclude quite generally that for any exohc contribution .~tisfymg &a = A (m/A) a ,
A,~=,A.; A~ - A~ the agreement between theory and experiment for the muon anomaly guarantee.q
~ " t ~ e electron anomaly is not influenced by it. This is the meaning o f the usual statement: the
ele~:tron anomaly is a pure QED quantity.
7.2. Modifications o f quantum electrodynamics
In/order to measure the "goodness" of QED, modifications of a general ttature not specifying
special interactions or particles have been attempted. As shown by Kroll [ 701 ] such ad-hoc modifications are severely restricted by local current conservation giving rise to Ward-type identities. A
modification of the lepton propagator must be accompanied by a change of the vertex function.
and KroU showed that these almost completely cancel each other out, the only exoeption being
the pnapagators involved m closed Fermton loops. In the case of the anomaly, modifications of
the lepton propagator can first show up in fourth order and may be of the form
Aa ~ K(a/Ir) 2 (r,7/A)2
where K is a numerical coumtant and A is a cut-off characterizing the modification. For K = +- I
we, obtain
A~ >
11 MeV
A~ > 260 MeV
A~ >
14 i e V
A~ > 425 MeV
The a ~'>malous magnetic moment ts therefore not particularly sensmve to modifications of the
charged lepton propagator *
Vertex modifications, not due to plopagator modtficattons, wdl gwe rise (besides the obvlou,,
posz:bi.!tt), of an mtrinsac ~momaly) to second order effects of the form
Aa=K~m
2
7rA
For K = + 1 we obtain**
*Notice howe ~er, that the effect ot a modtficaUon of the electron propagator on the muon anomaly (vta the d t a ~ a m m ftg 6 2)
has not been mvesUgated. One m g h t expect an enhancement for A e < m~ such that the muon experiment could hmtt A e better
than the electron experunent.
**A recently [713] suggested modification o f the charge form factor o f the muon to explain shght devmt~.on from ~ - e umvegsaltty
Fta(qZ) = l - b + 1_q2~A'--~
with b 2 t,.04 leads to
and
t~eby
A
~ 1 d f?.~AT
,RE. Laueeup et al., Comparison bet~'een theory and evpertments tn quantu,~ etectrodynamws
A~ > 230 MeV
A; > 280 MeV
A~,~ 5 4 G e V
Au ~ 8 8 G e V
241
[-very modd~cat,on of the p h o t o n propagator c o n e s ~ontts to .t modification el tlk ~pcuttdl
'unution of the p h o t o n and the tntquence on the anom, ly ~ mo~t ea~13 e~.prc~ed' 1~\ l h : 1,~,~t~],~
see section 5 . 2 )
~- lm
A,X =
AII(t) K(t)
0
nhere
Ktl) =
i
xU(l-r)
~ 1 nF for t > > m:
dx x3+ll_ x l t / m :
3 t
0
~ince most modifications revolve large t values we can use the asymptotic form and write
-
£
•'here L/A 2 is defined by
o
For L = + ! we obtain
A" > 3.1 GeV.
A- > 5 I GeV
Observe that p h o t o n propagator modification,, obe3 ,_X,t, ~ 1,1,.,",1u 1: Ad~, and thcrctorc the\
cannot mfluent.e significantly the electron .tnonlal~.,
If the space of quantum mechanical states has a pos~twe dehmte metric lm All(t) rand thereby
L) must itself be positive defimte. The traditional p h o t o n propagator modfflcaaon
gtav
guv
-"
guv
-/.;
_A 2
does not satisfy this reqmrement 0t 'eads to L = - 1 ) Lee and Wick 1 8 0 5 - 8 0 9 1 . however, hax e
reuently proposed a theory to handle the problems assocmted with an mdefimte metric
7 3 Suggested couphngs of leptons to exotic .txtrttcles
ihe u'lldr~Ze(.[ I C D I O I I ' ,
[703-707] if the mas~ of tile external lepton i~ m, the n-,Jss of the internal lepton ,11 I\~hl,.tl max
or ma2~ hot be equal to ,zL the mass of the boson t~ iX, avd its uouphn : i,, t t,cc t~g 7 1 ~ \re haxe
for the case of scalar, pseudoscalar, vector and pseudovector coupling the follow mg gc;aer.d express~on
t , x qr'l. . . . .
* ,-,¢t . . . .
- , - - , , v c - t ~ a c, a e o , e t h ' ~ t
"See e g Femberg and Lederman [R 71
o , f '~ r ~ o t l t r ' l l
hc~,,nn
cotlplt'd
to
242
B.E. l, autrup et aL. Comparison between theory and experiments in quantum eleetrodynamtes
Fig. 7. I. Neutral boron exchange.
_ f2
Aa
ma
41# A a L,
L=fdx--
Q(x)
( l - x )( 1- ( m / h ) 2 x )
+ (M/A)ax '
where Q(x) is a p o l y n o m m m m x dependent on the t y p e of coupling. We list it for the four stan.
dard cases.
1 x2(l+e_x )
1 ) scalar Qs(x) = ft.
!
2) pseudoscalar Q ps(x ) = -ff x ~( 1 - e - x )
3) ~,ector Qv(x) = x(1-.~)(x- 2 ( 1 - e ) ) + ~1 x 2 ( ! - } e - x ) ~,2( l - e ) :
4) pseudovector Qp,,(x) = x(l - ~:)(x--2(I+e)) + ~1 x : ( 1 - e - v ) X : ( l + e ) :
where e = M/m and ~ = m/A. In the limit of a heavy boson, Le. m, M < < A we have in the four
cases
_M
A
a
1
L~--~
(log~-~-)
+~Lps=-m(l°gA-3)+lm
M
Lv- m
Lp,,-
2
3
M
n7
2
3"
These expressions are vahd for all m, M < < A.* We specmlize them to the following cases:
*Also assuming (M/A) 2 << m/M
B E. L a u t m t p e t al,, C o m p a r i s o n b e t w e e n t h e o r y a n d e r p e r t m e n t ~ m quantzttn c h ' c t r o d ; nantl¢ s
"~ "
a) if a n e u t r a l b o s o n exists c o u p l e d to only one d l a r g e d lepton [703 - 70(,I we h.~ c 3! = t~z ;
m , or m e and
L, = log A
7
12
m
A
11
= - l o g 7n +
1
3
Lv
5
Lpv = --~..
A,~ the l o g a r i t h m i c ts slowly varying, we can c o n c l u d e t h a t ff the , ' o u p h n g [ t s the same to m u o n
and electron, the e l e c t r o n a n o m a l y ts free of the i n f l u e n c e of such b o r o n s (sec ,,cotton 7 1 ) No
absolute h m l t , can, h o w e v e r , be put u p o n the mass because ot the lack o f k n o w l e d g e ol the ~ouphng constants.
For a n e u t r a l v e c t o r boson [ 7 0 8 1 , W °, one finds by c o m b i n i n g the ~ and pv case (the t.ro,~,,
terms d o n o t contr,.bute)
AaW.=
4 f 2 tn 2
V
II one writes/"m/Am =
Actw . =
3x
KGt/~/2 w h e r e
K ,s a n u m e r i c a l c o n s t a n t one c b l a , n s
10 -Q K
and one finds the rather t m t n t c r c ~ t m g Imut
b) if a n e u t r a l l e p t o m c b o s o n c o u p l e d to e and ta e x i s t s 1707]
e-# ÷ ..
Zo
then we have for e l e c t r o n and m u o n resr,ectl~ely the d o m i n a n t terms
LsU... !
~
- ~-
w
m
L,
6
m.
Lk,
•
~ - {Iot~-ttt{d
Ill(
"~
l~s u
3
L~, ~
(IogA
m
~
3
L~
P"
-
--me
3
4-)
*
)
!44
JRE, l,autrup et al., Compnrison between theory and experiments in quantum electrodynamics
3bserve that for the electron we have a strong enhancement. The induced limit on A a e does not
abe.y the inequality stated in section 7. ! but rather the following less restrictive inequalities
-9.6×
10 - g F t
26 × 10- g F
s
9.6 X 10 "9 F
ps
- 6 . 4 X 1 0 -9
2 . 4 x 10 -9
v
--2.4X
6.4X 10 -9
pv
- 2 6 X 10-gF
10 -9
A
where F = log my
3
-4"
c) Neutral ta-p resonance. If a neutral lepto-baryonic resonance o f non-zero spin exists with a
mass of about 1.9 GeV [7101, the contribution can be estimated (disregarding strong interactions~
from the general formulas given above by putting m = mu, M = rap. Assuming spin ! and a couplin[
[7! ii f( l +~.i~s )3,u, we obtain in the not quite justified approximation A > > I GeV
= f2 mu2(
Aa u
: ~
( I + X2) ~ 2- ]\ .
41r2A--q- ( 1 - ~ k ) mu -
In the experimentally favored case ~, = I we obtain
f 2 < 3X 10-*
which is far above the estimate of a lower limit (10 -'3 ) obtained from the neutrino experiment
~7111.
(ii) Neutral leptonic resonance. If a neutral 7r/a(~ 430 MeV) resonance %* exists 17 ! 21, and the
spin is 1/2. the dominating contribution in the not quite justified limit m u, m,, < < m,.. is
= 41r2 m~.
( 1 - X 2)
6 my. (1 +;k: )
for a coupling of the form f(~,s + ),). For h = 0 we obtain the limit on the coupling constant*
f 2 / 4 r t < 6 × 10 -s.
(iii) Magnetic monopoles. If magnetic monopoles ex|st, one may expect they contribute to the
anomaly of the leptons. However, because of their huge coupling, no trustworthy m e t h o d exist
for calculating their contribution. Perturbation theory is not valid, unless the effective coupling
turns out to be g m/M where g is the monopole coupling, tn the lepton mass, and M the .m.onopo!¢
mass. From general considerations, this seems to be the case if the contribution arises via the
vacuum polarization. Taking**
L - g2 1
47r2 15
*The Iov/er limit is 1.6x !0 -11. Private communication from A. De R•jula.
**This is ~ e value obtained for the vacuum polarization by heavy charged fermion pa~rs [4071 .
B . E L a u t m p et al . C o m p a r i s o n b e t w e e n t h e o r y ,rod e x p e r t m e n t ~ ~ , q u a n t u n , c l e c t r o d v n a m w ~
245
ve obtain ( w t t h M the m o n o p o l e mass)
A a ~ 4-5 ~r 4 rr~- ~,M ]
which at best is unreliable, T h e l o w e r b o u n d on M is 5 4 GeV w h e n eg/4~r .~ ! No serious s~,ntficanoe should h o w e v e r be a t t r i b u t e d to thts n u m b e r
~art C
~iigh energy experiments
~. Scattering e x p e r i m e n t s at high m o m e n t u m transfer
One of the a i m s o f these e x p e r t m c n t s ~s to test the v a h d t t y of Q E D at small i n t e r a c t i o n d ~ t a n ~ e .
!hgh energy s c a t t e r m g e x p e r i m e n t s w t t h veD' large m o m e n t u m transfc| q. ',a~ q ~> 2 G e \ ' ~ ,_an
probe i n t e r a c t | o n distances R "-. tffq < 10 -ta cm To o b t a i n ~ut.h large t:oomentum transfer~ tree
a.s resorted to the c o l h d m g b e a m s t y p e of e x p e r i m e n t and to the ~o-~vlled Bethe-Hettler t y & ' ot
2\pcrtment
S I ('oil,drag beam~
i-our types o f e x p e n n , ents have b e e n carr**d o r :
il) Moiler s c a t t e r i n g
e v- - c-e , ,ee t~. 8 I,
{Ill Bhabha scattcrtt g
c ' e - -. ¢'c
s~c Itg S 2
Ira) A n m h d a t t o n i n t o nluot'- l,.::_rs
c ' c ~/a'/.t , .,co t~g. 8 3,
(tv) A n n i t u l a t t o n i n t o p h o t o n pan ~, e ' e - - 7 ~
see fig 8 4
(l) MOiler s c a t t e r i n g was d o n e at the Prmcet,~n-~tanford storage ring at total enet~:e~ o f tl~e
colhdmg e l e c t r o n s E c M. = x/S-= 6 0 0 / v l e V [8011 and 1100 MeV 18n21 T h e e \ p e n m e n t mea~tl~c.!
the angular d e p e n d e n c e o f the cross s e c t | o n but not the absolute m a g m t u d c !tae o b ~ e n e d angular
dtstnbutlon was c o m p a r e d w i t h the MOiler formult~ [ 803 ] m o d i f i e d by r a d l a t w c corrcctlon~ [,~04i
A c o n v e m e n t w a y t o m a k e tht¢, c o m p a r t s o n ~s to a~sume a ~no,t~l'lcat|on factor for the p h o t o n p',,pagator of the t o r m *
q"
\q-
q" - " G /
lhe m a x l m u m - h k e h h o o d ~aluc ol ,\: from the I I00 McV Oata I~
I / A 2_ = - O 06 ± 0 06 (GeV -2} I~tatlstlt.al e r r o r only)
*The m o d f l i c a t l o n w l t h a r e m u s slgn has r e c e i v e d m u c h a t t e n t l o n lately In ~.¢~pnectlon w, th lht p r u b h ' m ol dizen,once dtttKult,~
m Phyracs In a series o f papers, Lee a n d %It.k h a v e r e e x a m i n e d the cla~slca[ dllt'iCUltl,.~ m h e r t nt to ~uth a m o d l t t c ,tz,m and h ~ e
p~c_ccnted a n e w theory, o f q u , t n t u m e i e c t r o d y n a m l c s [805 809]
246
B.E. Lautrup et al., Comparison between theory and experiments in quantum electrodynanucs
e" (p;)
e'(p,)
e'(p'i)
e'(p~)
,e'(p'~)
,e'(p2)
e'Lo,)
~e'(p,)
Fig 8 I. Lowest order Feynman ¢hagrams contributing to MeUer scattering.
e
__- ..... _ _~_e~'I
e"
e"
Fig 8 2 Lo~est o-'Ior Fevnman dlagrams contributing to Bhabna scattering
Fig 8 3. ~ae lowest order annihdatzon graph for e ' c --- ~ ' . -
'Y(Ki~,~,,
e~ ,)J
(Pl)
Fig g.4 The annthtlation graotls correspondmg to e*e" --. 2-~
8 E Lautrup et al, Cottipartson between theory and e ,peronents m qt~antutt~ ele, tmd)'namws
247
he corresponding hmtts of A~ are
A_>4GeVandA.>
24GeV.
c~c,~ confidence level
(ti) Large angle Bhabha scattering has been pertormt d at the Orsay storage nngs ~" at a total
aergy q2~r'= 1020 MeV. In this e x p e r i m e n t the absolute Bhabha cross secuon o ¢ _ was measured *~
he determination of Oe.e- implies a s]mttltaneous measurement of "xe n u m b e r of Bhabha scattertg events Ne. e- and the luminostty L of the storage rmg. smce
Ne. e. = L "qe"e'"
order to d e t e r m i n e the luminosity L, the double bremsstrahlun~ reaction ,: *c- ~ e "e- + 23, ot
no~n cross-section o...~ [ 8 1 3 - 8 1 6 ] was chosen. In the Orsay experiment, the two photc'a., c,t
ouble bremsstrahlung are e m i t t e d at very small angle (of a few mrad} Therefore, a breakdo,vn
f QED at large m o m e n t u m transfer cannot gtve an appreciable effect on o:v The su-,,tt.,~aeou.
~ea~atrement o f the two reactions led to the d e t e r m m a t t o n ol o .t. t2_.
ge"c"
Oe,t.e" - 02., r
',xpenmentally, ae. ,- is obtained from the corrected number of Bh,,bha exent~ and the ~,alue ol
hc iummostty integrated over data taking t~me.
o~.p = [ 1 971 -+ 0 09 (statist I *-- 0 10 tsystcm 1 ×
10 -at
t-l'fl2
he ~.orrespondmg theorctt~.al cro.,s s e c t t o n [ 8171 ob'.amed from the c,dcul,,tton ol lhe lo\~e~t
,rder F e y n m a n dtagrams shown m fig 8 2 anti modllted ~o as to take tnto a~ ~t,t,l the lacltat~c
orrecttons*** and the integration over at.t.epted ~ohd angle and avera~,,e o~er the energ: slat ~rum of mordent electrons ts found to bc
oth=2.13X
10-st cm *.
~ e n the compartson between t h e o r y and e x p e t t m e n t ts made by means of the cut-or f parametr~ztton indicated above, tt ts f o u n d that
A_>20GeVandA
>38GeV,
Lta 95% confidence level
Recently, electron-positron elasttc scattering has also been performed at the Vrascatx ,toragc
mg Adone t" The total energy ~/~-ranges ;tom 1 4 to 2 4 Ge'v The pubhshed result~ l,~421 ,ire
*I or a t e c h n t c a l d e s c r i p t i o n o f the Orsay storage nngs, see e g J L A u g u s t t n [~st01
*'I or a detailed d e s c r i p t i o n ot this e x p e r i m e n t , see ref [81 ! ] T h e result,, v ere "Jr, t report¢ A at the Ll~erpotd ( ~ n t e r t n~e ,t,
,.t [812l
' * ' T h e radtattve c o r r e c t i o n s to e ' e - --- c'~ - c a n b e o b t a m e d f r o m the calculattot,' ot l'sal 1804 J, t o r e e ~ ~ c t h~. app;~,p,t tt~
t r a n s c r t p t t o n , r e l e v a n t to the Orsay e x p e r i m e n t , has b e e n m a d e b y Tavernlet [81 81 T h e y lead to a dt.~.leasc of tbc Burn ,.to~sectton o f 7 3%. V a c u u m polattzalaon c o r r e c t t o n s are n e g h g l b l e ( < 0 1 ~ ) The radtatt~e c o r r e c t t o n ~ to e ' e ~ c~e + 2-I h t, e
[ 0 e n r e c e n t l y c a l c u l a t e d by Baler ( p r t v a t e c o m m u m c a t t o n f r o m Dr i Buon) T h e y are l o u n d to be small and d o n o t alter tt,c
t.oncluttons o f t h e Orsay e x p e r i m e n t We wish to t h a n k Dr B u o n for an m f o r m a t l ~ e dts~usston on t|'t|~ pore"
~'For a t e c h m c a l d e s c n p t t o n ot k d o n e , see e.g F A m m , n e t a ' {841 ]
~8
B.E. Lautrup et al.. Comparison between theory and experiments m quantum electrodvnamzcs
Table 8.1
.traits or, the brt~akdogaa of QED from colhdmg beams experiments Parametnzauon
1
mg'ator and pa m 2
Lxpenment
...4.
1
/~2 -m:
qa -" -
±
for the photon pro-
1
pa._m2_A 2 for the fermlon propagator
Tests
Reference
Total c.m. energy
(MeV)
Cut-off(95% c.L)
(GeV)
:-e" ~ e ' e "
_~ace-hke photon
[801 and 8021
600 and 1100
A. > 4. A. > 2,4
•: e - - * e*e"
space-hke photon
[811 and 8121
18421
1020
1400-2400
A > 2 0, A. : 3.8
A_ > 6
."e" -+ ~*~:+e- ~ 2~
ttme-hke photon a)
space-hke photon b)
[ 812 !
[ 819 ]
580, 644 and 704
! 028
A_ > 1,3
A(electron) > l,S
a)See also the results of ref [ 843 !
b)see also the results of ref [8441.
based on an analys~s of 3255 wide angle scattering events (WAS). At these energies, the WAS events
ire suitable to test the vahdlty o f QED.
In the Frascat~ experiment, the reaction chosen as a momtor to deternune the luminosity wa~
Bhabha scattering at small angle scattering (SAS). The SAS events mvolve small m o m e n t u m transfers and therefore they are unaffected by a possible breakdowa of Q E D at large m o m e n t u m transfer. The comparison between theory and experiment made by means of tl'e nunu.~ sign cut-off
l:arametnzatlon (see tal:le 8 1) leads to the result
A > 6 GeV
at a 95% confidence level
(111) The annllulatlon into a muon pair Ce- ~ ~a'~- hat also been perlormed at Orsay* at the
three total energies 580 MeV, 664 MeV and 704 MeV. The m u o n s were separated from the p~ons
on the basis of thetr different range m the thick plate chambers. T h e y obtained 62 events. Thts
experiment tests the time-like propagator of the p h o t o n since only the anmhilation graph shown
m fig. 8.3 is present. From the analyms ot the results In terms of a modlficaUon of the photon prcr
pagator it :s found that
A > 1.3 GeV.
at a 95% confidence level
Production of muon pairs has also been recently performed at Frascah [8431. The results are
presented in terms of the ratzo of cross-sect~ons
R = i e+e\ e'e-
+ la'la- )
--- c ÷ e -
.'
exp
QED
as a function of energy. Stratght-hne mterpolaUon of all the points gives"
R = 0.98 -+ 0.08;
*Se* Perez-~-Jorba [ 812].
2.5 ~< s < 4.0 GeV ~
B.E. Lautnup et al . Conlpa:tson between theory and ex'per:mentt tn quantum ele~ trodvnan'tcs
240
e•
a)
b)
8.5 Lowest order Feynman dr.,rams contributing to lepton pair photoproductlon, in fig. 8 5a there are two diagrams obtained by attaching the exchan3od photon to each lepton hne
(iv) The anmhilatton into "wo photons e'e- ~ 23, has been studied at the Nov,gs'blrsk storage
ngs [8191. The relative devtat~on o f the experimental cross section for 2")' aqmhdat~on from the
~lculated one ~s
Ao/o =0.1 ~ 0 2
lotice that tlus experiment tests the electron propagator (~ec fig 8 4) It ~s the ~lcctro,'a wh~O~ ~t~
ats process ~s olf-mass shell and c a m e s a large four-momentum When a cu:-ol f lot the lcrm~on
ropagator is introduced in the same way as the ( - ) photon propagator modff~catlon, and no co~ repondmg vertex correction*
!
1
1
p=-m ~ ~p~
_m2
p~-m~-A
~"
~t i~ found that
A(elect~oTl) > 1 5 GeV
at a 95% conftdence level
The reaction e'e- --, 3~' has also been recently studwd at Frast.at, [8441 I he experiment goc~
up to 1.6 GeV ~irtual-lepton 4 - m o m e n t u m A total ol 5~)7 event,, ,vcr~. analyzed I'hc QFI) te,t
~onslsts m comparing the c x p e m n c n t a l ratio ot w~dc-an,;Ic ,o ~,n,all-anglc ratc~ v, Llh ".he theoret~,.a
prediction. The two agree w~thm one standard devtat~on.
A summary o f the results obtained from colhdmg beams experiments has been compded in
table 8. I.
8.2. Bethe-Het tier type experlmen ts
Three types o f experiments have been performed
(i) Wide angle electron or muon parr p h o t o p r o d u c h o n (tee fig 8 5 ~,
(it) W~de angle bremsstrahlung of electrons or muons (see fig 8.6).
(m~ Trident production of leptons (see fig. 8.7)
These proces.,,es have m c o m m o n that, at lowest order ol pertttrbatlon thcq)~ the ~orr~'~I~o~d~t
Fcynman diagrams ln,,olve a fermlon hne wh~t~h ~s ol I-shell and possible naoct,ii~atl('n~ to the ~O~I
sponding propagator can be tested. Here, large off-shell lermlon masses are al tdinable because ~ c
proton target, or the low Z nucleus target, can be used to fix the t e n t e r of mass The nuclear s~ru
*See the dtscusslon in section 7 2 concerning the dfflqcultles with thts type .~f modlflc~t~,n of a fermlon propagator
B.E. Lautrup et ai., Comparison between theory and experiments m quantum electrodynamics
Table 8.2
.xrmts on the breakdown of QED from measurements of sym:netrtc wtde angle lepton pan photoproductton * Parametnzatton
j = oi:-~H[ 1 ±(Qa/,~a+_)2] where Q ts the mvanant mass of the final lept on parr. For each experiment, the ~tgnature retained ts the
one whtch ytelds the lowest cut-off
~periment'
Reference
lnvariant mass of lepton ~
(MeV/c = )
Cut-off (95% c.l )
(GeV/c I )
÷ C --* C + e*e"
r + C -* C ~ e~e ~ + p - , p + e÷e -
[8211
[8221
[8231
12 --<900
Q .< 444
12 < 4 9 0
A. >
A,. >
r + C " C + U*J', + C--~C + ~*~r+C.-.C+ldv-
[827]
{2 < 1225
A > 1.5
[828]
Q < 2100
A_ > 2,3
[829!
Q < 1770
A > 1,9
't
1.6
0.8
A, > 0.7
---
• For earher expeltments on electron pmr photoproductton, see refs [824. 825 I. For an earlier expertment on muon parr photoproducuon, see also ref [8261.
o
b)
0)
c)
F,g 8 o Lowes ~rder ! eynman diagrams contributing to bremsstrahlung ol leptons "I he olf-shell fermton m the dtagram ol It!
8 6a is tlme-hke, ~hfle m the diagram of fig 8 6b tt is space-hke
,j-
a)
b)
l'tg ~ 7 Lowest order Feynman diagrams contrtbutmg to trident productton
ture m the diagrams coJresponding to figs. 8 5a, 8.6a and 8.7a can be factored out, and is known
from mea~ arements of he form factors tn elastic electron-protcn scattering.*
(1) Mea.,,uremenls ol ~,ymmetrlc wide angle t.;ectron pair p h o t o p r o d u c t l o n have been done or
carbon at DES~-Miq t821 ] and CEA [822], and on hydrogen at Daresbury [823l In these e~penments a symm~'mc ,letectlon system with respect t'o the incident p h o t o n beam direchon ts
*For a recent revtew ot t h e , .tq leon form factors, see e g Rutherglen [820]
B.E Laurrup et el, Conlpar~scn between theor~ and eww.tments m quantum electrod~ naonc~
251
Table 8 3
muLs t,n the b r e a k d o w n t I QED trom experiment,, of w~de angle brems,,trahlung of lepton~ P,,rametr~z,,uo - = ol,,1.1[ I ~(Q~-/,,~ ~:
here Q t~ the mvartant ma~,~ of the final leplo ,-photon ~', stem I or c,, h experiment the ,,tgn.ture retained • tile one whtt.h
y,elds the lowest u , t-oil*
'.xpenment
Reference
l n v a r m n t mass of lepton
3' system (MeV/c ~ )
Cut-off ~95% c 1 )
(GeV/c a )
"+C-e'+C+'v
18331
18351
~ L l~30
O ~ 650
A > I5
A >073
:+C--."+C+'r
See a b e ref.
18341
~osen so as to eliminate the interference*of the Bethe-Hettler a m p h t u d e s (see fig 8 5a) w~th
te vtrtual C o m p t o n amphtude (me fig 8.5b). Wide angle detection is used to minimize the effect
f the virtual Compton-scattering cross-section ** ltd. these experiments, the mvariant mass of the
pton pair me,. goes up to 900 MeV/c 2
Measurements el' symmetric wide angle rouen pair photoproductton on carbon have also been
'ported [ 8 2 6 - 8 2 8 ] . The data corresponds to invartant masses of the muon pair mu~u_ up tG
770 MeV/c 2 .
A traditional way to represent a breakdown of the lepton propagator has been to '.nt~oduL.e
,it-off parameter A such as illustrated m table 8 1 Thet,, for values ot !~" small uompared with
3 and large compared with m : , the Bethe-Hu tier t.ross-section o~j5 [ 8 3 0 . 8 3 . ] has to be modlcd as follows [832]
0 =OttH(l+P:/&2)
lthough there ts certain arbitrariness tn paramutr,ztng the brt.akdox~,n of QI-I) the modtl'l,_atiol~
aggested above has the serious inconvenience o! violating the Ward-Takahashi identity An analynl
Kroli [701 1 shows that a rea,,onable mt~dlllt.at,on ol the Bethe-tteltler tyl)e anaphtttdu' le.ld,, Io
modfftt.atlon ot the ~.ro~-sc~.ttoix ot the t2, pc
o = oa. [l +(Q:/A~ )hi
'here n = 2 or a higher even number, and Qm ts the mvartant mass of the final lepton ~ystem
stag this parametnzar~on, w~th n = 2, the experiments mentioned above lead to the results shown
I table 8.2
(n) Experiments of wide angle bremsstr,ihlung of :lcctrons on carbon h,we been done at ('orncll
833], and on a hydrogen target ,it Fra~t.att [834] A n e x p c r i m e n t o f b r c m s s t r a l d u n g o f m t l o t l ~
rein 9 to 13 GeV/c on a carbon target has also been done by a llar~ard-Case-McGdI-SLA( ~o IJoration [835 ] Notice titat 111 I b i s l~ypc Oi" pruuc,,,c,, thu lu,vu~t ,,rd,.r lc~ ,~:;'.:;; d:~"r.:::~'. ::~.;' !, ..'
"qottee that the e ' e - parr produced by the virtual C o m p t o n a m p a t u d e and the e ' e parr ol the Bern, liettl< r ,,t~lpht,td "s, re
~tates which transfot-n oppositely under charge conjugation By a ~u,t,tblc ~.hOlt.t of dq a~ymmetrtc dtte~tton %v~tem, tht~ interference term can be measured and the real part of the virtual C o m p t o n amphtade can thus be obtained
'The Bethe-Hettler and C o m p t o n cross-sections differ m their angular dependence by a factor ~ t9 3 where ,~ t~ the .ingle b~
t~*en the l e p t o n and the mordent p h o t o n direction
$2
I~.E. Lautrup e~ al., Comparison betwee t the'or) and experiments m quantum electrodv~mmws
fff-shell leptons whrch are both spaceqlke (fig. 8.6b) and tlme-hke (fig. 8.6a). The ttme-like lepton
:an be far off the mass s.aell if the final lepton and p h o t o n energies and angles are large. The results
ff these experiments, winch have been summarized in table 8.3 agree well with the predictions of
~ED.*
0ii) Experiments on trident production (see fig. 8.7) are still on a preliminary stage to serve as
~recision tests of QED. There are, however, a number of very interesting features which have alr.ady been revealed by ~hea~ experiment,,. Production of a muon pair by incident electrons at
L9 GcV/c on a carbon target has been reported by a Northeastern-Austin collaboration [837]
duon pairs were observ.~d at invariant masses ranging from 0 4 GeV/c 2 to 0.9 GeV/c 2 ; the scattered
flectron was not observed. Th~ data are consastent with predictions o f a simple diffraction model
Mr the virtual Compton amplitude (see fig. 8.7b) which interferes with the time-like QED amplitude (see fig. 8.7a). Within this model, a heavy p h o t o n of mass less than 400 MeV in the time-hke
propagator is excluded by the data
Production of rouen pairs by incident muons at I 1 GeV/c on a lead target have been performed
it Brookhaven by a Harvard-U Mass.-McGdl collaboration [8381. The angles and m o m e n t a of the
incident muon and of all three final state muons were measured in optical spark cham,~ers. Runs
were made with positwe and negatwe incident muons. The total n u m b e r of events ebb'rued 75 2
~ 10.5 is ul good agreement w~th the theoreUt.ai predtctton 88 t~ _~ 2 6 which includes the interference term between the c~rect and exchange graphs.** The theoretical pred~ctton without mclustun of the interference term would be 120.9 ± 2 6 events In the e x p e n m e n t , the mvariant max~
spectrum of the two ~dentical final partmles clearly shows a depression at low mvartant ma~se,, ,ts
predicted b5 the Pauh exclusmn ormctple for fernuons
2onclusions
Quantum electrodynamlcs t,~ in very good shape No ,,ertous dtscrepancms ex~t between theory
:nd experiment The last year has wen mlprovements m st veral fk'lds In the h~gh energ.,, region
:ut-off masses are being pushed rote the region of several GcV. in the tow c n o g y region the prcasmn an both exl:eriment and theory is approaching one part per mdlion.
The anomalous magnetic m o m e n t of the electron has been remeasured w~th a p r e o s m n of 3
~pm. S~multaneously. the sixth order radmtive correction has been calculated theorettcally wtth
he same prec~saon. The two numbers agree beautifully
By now. the theoretical value of the a,mmalous magnetm moment of the rouen ts k n o w n with
preclsmn of ! 3 ppm. The crucml test of this number awaits the next CERN experiment
The thee,'et~cal value for Lamb shift m hydrogen has been ~mproved by analytic calculatmns
ff the fourth o.rder slope of the Dl~ac form factor of the electron, and by a calculatmns of the
~{Z,~)6 c o n t n b u t m n The theoretical uncertainty of 1 2 ppm is five tmles smaller than the expennental uncertainty, and the agreement between theory and experiment is excellent
The m;reased prects:on of QED calculatmns ,rid experiment leads to accurate purely radmtwe
,alues of the fine structure constant ~t. in one case (the 2Pah - 2S~/2 sphtting m hydrogen) the
~rectslon ts even comparable to that of the non-QED at- Josephson value.
*See ref. 18301, see also ref [8391
'* Nonce that m fig 8.7a, m the case v + A ~ ~ + v* + ~- + A, there are ai'ogethet etght F e y n m a n dtagrams the 4 dtrect ones ph'~
4 obtame~ by excr,.ange m the final ferraton lines o! equal charge. Tl,.eo~ettcal dtscuss~on of trtdent experiments can be found m
ref~ 1839] and [8401
t~F. L a l t t r l ~ l l e t a l
C~)mporL%o#l b(lli,l('lith~,.i atld~ W ~ t ~ , t n ~ , t ~ , n q l e a , l t t , , , A
/~<,~,l,,,,t~
~
~.cknowledgenlents
l'h¢ ,lulholk
B,illoy,
S
wlkh
l~rod,d~y,
It)th,illl,,
I
Bt!ol'l,
the' Iollo\~111, 2 p~.'l,~)tl~ lt~'
l
('dlnlol
A
l)o
Kulul
tlln~.tl,~,q~)t~,,, u o l l l H I C l l t n
. ( ; lily
I ri,.kklm.
I
I ~1
,lrlt.t u i l l i c l , , ~ q
i ,ilieV
I
t titlt>l~
tI.J. levine, E. Picasso, A, Rich and V . L . Telegdi.
Review articles on q u a n t u m electrodynamics
The foUowmg is a hst of review article.,, on ~,arlous a~p .'cts el q u a n t u m clcctrodynamlcs which
~,'c have found very useful to consult No attempt at giving a complete l]~t h<L,,been made We
lpologize for omission'L
lit.II J, Bailey alld ~ ~lt'a%lo. " ] h,a anoill.llou,~ lthlgrl¢llc mmllelll el the muoll ind rclalcd 1 ~p •,~ ", I'tt~t~tu,,,, ;1, Nu,.l,.,r l'h) sic.,
Vol. 12, part I 11970) p 43
lit -~1 tlan~ A, [lethe a n d l d',~,ln 1 S a l p c t e i , t ) u a n i u m n l c t h a n l t ,, ~>I ~me-and-t~ t,.-t,lt.,t, iron .it~)tu,, l Sprul#cr-Vcrl,tv, I ~)5 ~
lit31 Sianl¢~ J B r o d x k y , "Stalu~, ~1 qtlallltllll electro•l; nallllt~, , ' , P r o t 4tl, In• ¢~ ,ilpttMtlnl Oil I lc~ troll lad Ptl~i~m li~icrJ,.ilon,,
IR41
IriS1
al I h g h I nergle~ I p u b h , i h e d b~, Iht. Dar "sbur~ Nuclear Phs~lu~ l abet,•liar,, Ic)¢o9)
Stanley J Ilrod,k~, a n d Sidney D I)rell, "'TIw pre~cnl ,,laltt,, oi q u , m l u m clct.trt~d', i~,lml•,,'" &tail I{¢~ Ntiulu ~r 'qLl¢tlt?
2(I (Iq3'fl) 14 "~
Ll.l[~dl~,~ D. ~olllmlll% " A p p h e a h o n el archaic, beJill~, 1o eltlllcnlar% i,arlitlc ind i~uuel.ir p h ) i . s " %~n Rev NuLle,lr S~.lt.n•t'
17 (19671 33.
IR~I
IR
I J M. ! a t l e y , ' " i he ~tatus o f q u , i n t u n l d e c trod~ r i , l nl i t s" , Pri)t I ~,I ill, t ill, L, ill lht 1 u r t l p t . t q PI >xi< iI Ntlk It I% i I,,lt nut
(1969)
71 G<~rald l--clnbcrg and L e o n M L e d e r n l a n , " 1 h¢ p h ) ~lc~ ~1 lillitlll~ dlltl llitillll lit'lill+lll, ls %llil R.~ '4uclc,lr %t l e n t o 13
(1963) 431
IR.~l
Ri ch ar d P I e'~ n m a i l , " l he pre~ent M d i u l ot q u d l q l l l i ' u l t t tired', II,lillic', ' d ,ll~ I,I I h u o r l t ()u,lnt, lii~ d•', ( h,lilll~', Pro<
12~hSo',~a~, mcctlnt~llntt, rx~.lcnct lhabl Ncx~ ~tt~rk R %l<>op~ t~ditt'ur, Briixtll¢'~ 1961)
I r ~ l R ( , a t • o , " ' % n a k ~l,, el p t e ~ n t etldca~.c o n the ~,, ,,l,~ el quarlitinl tJ¢~.lrl},13, n a m l c s " , in t h l h I i~ert,x I>hv',lc ', I d ' It ',
| l u r h ~p, ~, o[ I 1 I \~.ademlt Pres,, 196NI T lit r ¢. I, L I f ' i t 'f'll 'Idol• n(lu,il" to till r, ,lt'x~ ,irtitl¢
R l u l V e r n o n % llugh~.,,, "Mu~,ntum'" 'kiln Ru~, Nuuk ir %tltllk. ](, I IO6(~1 44",
R i l l V e r n o n V v l l u g h e s "%lutllql l l i . l : l d f ) l l , , i i l l , l l i u r '" li~ l'~l~,,Its,~l I h t , ( l l l t illtl [x~.~ I I t t ' r , , l l \ l , m l , + \ , ~ r l h I1,,11 i , l ~','
pp 407 4 2 8
ri21 (, K,illen "'l,)uantt.ntlcki[tt,l~,n,ilulk" I l i n d b u • h d t [ Ph~ it %,,i \ I ' l t l I%tllii,,t \ • l i l t , I )f,b,~,~p 1~,- ,i,*
Petetnlafi
~lltlnlil. c n t r , ) l~.~tl~ in h ~ d r , - , l n ikt lit ill,,
I ,,ilk, Ill Pll~, (i ( I t l K b , l 411"~
rial
R 141 E lhca,,ao, " ( ' u ~ e n t d~.velopm-nts in t h c s t u d ) o f e l e c t r o m a g n e t i c propertle~ o f m u o n s " , In l h g h L m r.t.q' Phx x~t ,. jl'~d Nil
clear S t r u c t u r e , e d i t e d b.~ San',~cl De m,~ l P l a n u r l Prc~,,, i 971)) p p 615 635
R isI J u h a n S c h w m g e r , Selected papcr~ o n q t a n • u r n el 'ctrod~, naml~.~ (Dover P u b h c a t on~, "tnc , N e ~ - Y o r k 195 Si
R 161 B N T a y l o r , W Parker a n d D N. Linl'~.hF,'rz' "1) " t e r m i n a t i o n o f e,'h, u,,lllg inat.ro,,topit, q u , i n t u l h phase coher< r.. L ,n
,~upereonductors implltati~,n,~ l o t q u a n t u m e l e t t r o d ) n.um,.,, a n d thu luntLulle,li.d ph,. ,,i,.al t on,4.1n~'. ' Rt ~, \lt~tl Ph,. ,,
41 1 1 9 6 9 / 375
llTI C,S Wu a n d L a u , a r e n c e ~ t l t t , , , "Muonl~. ,ttorn,~ and nuclear ~tru~.turc" %nn Rc~ Nuclear St.lcnLc 19 119091 5 2 "
References
ltlOll ~,~tlll'~ i L, m b Jr a n d Robert
1oo21 P \ u s • h a n d tl l o l . : } , P h ~ s
( R e t h e r l u t d Phys Rc~ 72 i 1 9 4 7 ) 241
Rex 7 2 1 1 9 4 7 1 1 2 % 7~11 ) 4 ~ ) 4 1 2
l',ilt3 I J u h a n S c h ~ a n g e r , Phxs Rex "~'~ ~l q 4 ~ ) 4 I¢~
{li~141 t, ( h t r p a k , 1 J M I.trlc~ R 1 (,Jrx~,ul | Xlulltr J ( '-,tl,,, i111 \ ./,,In, l,i l'h,
10051 A P e l e r m a n n , P h ~ s Re~, 105 11957) 1931
10061 [1 S u u r a a n d E H W l c h m a n n , P h ) ~ Rc~, 1i"15119571 IQ'hl
10071 .I Bade:,',W Bartl. G w m B o t . h m a n n , R C A i}rox~r | I M l a r l c s 11 I o q l in I
I It,,. I Itm,~l I ~
28B (1968) 287
Itill81 J B a l l e y , W B a r t l , ( , ~,on B o c h m a n n , R (. A Bro,~n I J M | trle~,, M (,lcs~h II Itl,,ll~.lll 5 ~,dn d¢t M t t - I Pit l,,',,, llltl
R W W d h a m s , "Precise m e a s u r e m e n t el t h e dn~,malous i T I d g n k I' m~matnl ,~! tile 111,,it~n ' ( I RN Prcpimi I o ' 1
I l u l l t1 ~ B.,zthe, Ph~s Re~ 7 2 1 1 9 4 7 1 339
254"
B.E. Lautmp et ~ , Comparison between the~,, and experiments in quantum electrodynamics
[1021 N,M. Krcl! and W.E. Lamb, Phys. Rev. 75 (1949~ 388.
t1031 l.n, French and V.F. Wem~opf, Phys. Rev. 75 (19491 1240.
[1041 ILP, Fey~man, phys. Rev. 74 (19481 1430; 76 0949) 769
[10S] H. Fukvda, Y. Miyamoto and S. Tomonaga, Progr. Theor. Phys. 4 (19491 47° 121.
[106] C.L. Sc,|warz and J.J. Ttemann, Ann Phys. (N.Y.) 6 (19591 178.
[107] E.A. UehlL',.g,Phys. Rev 48 (19351 55
.[I08]_R. Se~ber~Phys, Rev. 48 (1935) 49,
11091 S. Tdebwasser, E.S. Dayhoff and W,E. lamb Jr., Phys. Rev. 89 0953) 98.
|1101 it.T, R6biscoe and T.W. Shyn, l~,ys. Rev. 168 (19681 4.
[1111 S.L. Kaufman, W+E. Lamb Jr., K.R. Lea and M, Ieventhal, Phys. Rev. Letter; 22 (1969) 507.
[1121 T.W, Shyn, W.L. wm,-ms, R.T. Robhcoe and T. Rebane, Phys. Rev, Letters 22 (19691 1273.
[113] T,V, V o r ~
and B.L. Cosens, Phys. Rev. Letters 23 (19691 1273.
[1141 G.W. Erlel~on and D.R. Yennte, A~n. Phys. (N.Y.) 35 (19651 271
[115] G.W. Ertdcson and D.R. Yennie, Ann. Phys. (N.Y.) 35 (19651 447.
[116] H. Crotch and D.R, Yennie, Rev. Mod. Phys. 41 (1969) 350,
1117] J.Weneset,R. Bersohn and N.M. Kroll,Phys. Rev, 91 (1953) 1257,
DtSl M,F, Soto Jr,,Phys. Rev, Letterq 17 (1966) 1153, Phys. Rev. A2 (1970) 734.
i1191 T. Appelquist and S J, Brodsky, h,ys. Rev. Letters24 (1970) 562; phys. Rev. A2 (1970) 2293.
112o1 B.E. Lautrup, A. Peterman and E a- Rafael, Phys. Letters 31B (1970) 577.
[1211 R. Barbwn, J.A. M ~ c o and E. Remtddl, Nuovo Cim. Letters 3 (1970) 588.
11221 A. Peterman, Phys. Letters 35B (1971) 325
11231 A. Peterman, Phys. Letters 38B (1972) 330.
[1241 LA. F~'~,Phys. Key, D3 (1971) 3228.
[1251 R, Karplus, A Klein and J. Schwmger, Phys. Rev, 86 (1952) 288.
[1261 M. B a ~ ,
H.A. Bethe and R.P. Fe3 ~nan, Phys. Rev. 92 (1953) 482.
[1271 AJ. I.~yzer,Phys. Rev. Letters4 (1960) 580.
11281 H.M. Fried and D.R. Yennie, Phys R 'v.Letter~ 4 (1960) 583,
[1291 AJ. Layzer, J. Math. Phys. 2 (19611 292, 308.
1i301 M. Baranger, F J . Dvson and E.E Salpeter, Phys. Rev. 88 (1952) 680
ll3t] E.E Salpeter, Phys. Rev. 87 (19571 328
[1321 T Fulton and P.C Martin, Phys. Rev. 95 (1~54) 811.
11331 W.A, Barker and F.N. Glover, Phys. Rev 99 (19551 317
[1341 S Brodsky and R. Parsons, Phys. Rev 163 (]967) 134.
[1351 E. Ferrm, L Physik 60 (19301 320
[136] R Vessot, H Peter~, J. Vamer, R. ~-.clJer, D. Halford, R Harrach, D Allan, D Glaze, C Smder, J Barnes, L Cu ,let and
L. Bodily, [EEE Trans instr M~.as IM-15 (196£) 165.
[137 Stanley I. Brodsk) and Glen W Erlckson, Phys. Rev 148 (19661 26
113B G. Breit, Phys. Rev. 35 (19301 1447.
[139 N. I(~olland F. Pollack, Phys. Rev. 84 (1951) .c97; 86 (19521 876.
[140 R. Karplu.%A. Klein and J. Schwmger, Phys. Rev 84 (19511 597.
[341 AJ. Layzer, Nuovo Cml. 33 (19641 1538.
!142 D.E. Zwanzlger, Nuovo Ctm. 34 (19641 77.
[143 D.E. Zwanz~ger, Phys. Rev 121 (19601 1128
1144 G.E. Brow1~and G.B Aft'ken, Phys. Rev. 76 (19491 1305.
[145 A.C. Zemach, Phys. Rev. 104 (19561 1771.
R. Amow~tt, Phys. Rev. 92 (1933) 1002.
11+7 R. Karplus, A. Klein and J Schwmger, Phys. Rev 86 (1952j 288
[+t'-;-,~ W.A Newcomb and E.E Salpeter, Phys. Rev. 97 (1955) 1146
11+9 C.K. Iddmgs and P.M Platzman, Phys. Rev. 113 (19591 192
115o C.K lddmgs , Phys. Rev 138 11965) B446.
it+~1 W N. Cottmgham, Ann, Phys. (N Y.) 25 (19631 424
[is~, M G Doncet and L de Rafael, Nuovo Ctm. 4A (19711 363
1t.~3 F. Guenn, Nuovo t~m. 50 (19671 211.
[I~4 S D. Drell and J.D. Sulhvan, Phys. Rev 154 ~.1967) 1477.
[155 V.L. Chermak, B.V. Strummskt and G.N. Zmovjev, Dubaa preprmt E2 - 4740 (19691.
[156 A A. Logunov and A.N. Tavkhehdze, Nuovo Ctm. 29 (19631 380
[157 R N. Faustov, Nucl. Phys 75 (19661 669.
O.g. Lautrup et al , Compart~on bet~eer, theo.y and erpenments m q,,antum electrodynam~cs
235
Glen W. Enckson, Phys. Rev. Letters 27 (197i) 780
R. Barbmrt, J Mlgnaco and E. Remtddt, Nuovo Cxm 6A (1971) 21
R. Robt~coe and T Shvn, Phys Rev. Letters 24 (1970) 569
C. Fabian, F. Ptpkm an, t M Sdverman, Phy~ Rev Letters 26 (1971 ) 347
B. Cosens, Ph} ~. Rev 173 (1968) 49
M Narar&tmham. Thest% Lhuverstty of Colorado (191)9), unpubh~h, d
O Mader,/vl Leventhal and W.E Lamb J r , Phy~ Rev A3 (1971t 1 ~32
L. itatt-ud~ and R, liushes, l'hys. Roy. 156 (1~67) 102
166 R. h¢obs, K. Lea andW.E. Lamb Jr., Bull. Am So¢ 14 (1969) 525
167 C. Fin, M. Garcia-Munoz and !. Sellm, Phys. Rev 161 (1967) 6
168 M. Leventhal and D.E. Mumtck, phys. Rev. Letters 25 (1970) 1237
169 E. de Rafael, phys. Letters 37B (1971) 201.
170 C.W. Fabjan and F.M. Pipkin, Phys. Letters 36A (1971) 69
201 M. Deutsch, Phys. Rev. 82 (1951) 455.
202 Matttr Deutsch, Pros. NucL Phys. 3 (1953) 131.
203 E.E. Salpeter and tl.A. Bethe, Phys. Rev. 84 (1951) 1232.
204 M. Gell-Mann and F. Low, Phys. Rev. 84 (1951) 350.
205 M. Deuts~h and S,C. Brown, Phys. Rev 85 (1952) 1047.
206 R. Weinttein, M. Deutsch and S. Brown, Phys. Rev. 94 (1954) 758, 98 (1955) 223
207 V.W. Hughes, S. Marder and C S. Wu, Phys Rev. 106 (1957) 934
21)8 E D. Thertot Jr., R H Beers, VW tlu~he~and K O !t Zto~k, Phys Rev A2 (1970) 707
209 ~ R.X. Beers and V.W. Hughes, Bull Am. Phy~ Soc 13 (1968) 633
2 1 0 S. Matdet, V.W. Hushet and C.S. Wu, Phys. Rev. 98 (1955) 1840.
1211 ] J. pitenne, Arch. Scl. Pltys. Mat. 28 (1946) 233, 29 (1947) 121,207,265
2121 R.A. Ferrell, Phys. Rev. 84 (1951) 858.
,2131 R Karplus and A Klein. Phys. Rev. 87 (1952) 848
[214l T Fulton, D.A. Owen and W W. Repko, Phys. Rev Letters 24 (I 970) 1035.25 (I 910) 782 Phys Re~ A, t~ be pubhshcd
L21~] T. Fulton and P C. Martin, Phys. Rev. 93 (1954) 903,95 (1954) 8 i 1
[216] A O t e a n d J L PoweiI, phy~ Rev 75 (1949) 1696
[217] P Pa.scual and E de Rafael, Nuovo Ctm Letter~ 4 (1970) i 144
[218] PA M Dtrac, Proc Cambrtdge Phd Sot. 26 (1930) 361
1219] I Harrlsand L M ~rown, Phys Rev 105 (1957) 1656
[2201 A L. Akhlerer and V B Berestet,sku (Nauka Publ , Mo~low 1969) SL~iI~,tl iX
[2,:1] V B Berestetskn and L D landau, JLTP 19 (1949) 673
[222[ VB Berestetskn. JFTP 19(1949) 1130
' , o l ] V W Hughes. D McColm. k Zlock and R Prepost, Phvs Roy Lt, tter~ 5 ~196f~) 63
[302] Starley J Brod~kv tnd Glen ~ l:rt~.k~on, Phvs Re~ 148 (1966) 26
[~'J1l Thoma~lult~a. Da,'ld A OttenandWa.vnt W Rcpko. Ph)s Rtv L, tter,'6(1971,p61
t3041 R. Atnowttt, Phys. Re',, 92 (1963) 1002
13051 T, Fulton and P C. Martin, Phys Rev 93 (1954) 90~t, 95 (1954) 811
1306l W.A Newcomb and E.E Salpeter, PI',y,L Rev. 97 t I955) 1146
[307] J F. Hague. J.E. Rothberg. A Schenck, ELL Wdham% R W Wtllmms. K K Young ar~,'l K M Crowc, Ph)'s Rev Letter~ 2~
(1970) 628.
[3081 R De Voe. P.M Mdntyre, A Magnon, D Y Stowcll, R A Swanson and V L Telegdt, Phy~ Rev Letters 25 (1970) 1779
[300l D P. Hutchmu3n. J. Menes, G Shapiro and A M P, tlach, Plays Rev 131 (1963) 1351
[310] G McD. Binsham. Nuovo Clm 27 (1963) 1"~5"~
[311 ] D.P Hutchiny,on, F L Lar*.en. N C Schoen, P, I St her and A S Kanofskt, Phys Rev Lettcr~ 24 (1970) 1254
[312] R D Ehrhch, H Hojer, A Magnon, D Stov, eli, R A Swan~onandV L Telegdt, Ph~s Rev Lette,s 24 (19693 5l'~
[31 ~] P Crane, J J Amato, ~,' W llughes, D M Lazaru% G zu Puthtz and P A Thompson, Bull ,'~,'rt Phy~ Soc 15 (19701 45
[314] MA Ruderman, Phys Rcv Letter~ 17 (1966) 794
[315l T Crane, D Caal3erson, P Crane, P Egan V ~V Hughes R Stambaugh, P A Thompson and (, zu Puthtz Phi,,, R~ t,z~t r
21 (1971) 474
['~'¢~1 R D Ehrhth, |t tlojer, .X 't,lgnon, D Y Stowell, R A ,wanson and 'v L lclcedt Phy', R,', I t. ttets 23 (1969) c,, 3
1317] WE Cleland, J M. Badcy, M ~.khause, ,' W Hughes, k M Mobley, R Prepost and J I R,~thberg, Phx's Roy Lctter~ 13
¢1964) 202
[3181 P.A. Thompson, J J Arrcxto P Crane, V W Hughes, R M Mobley, G 71, Puthtz and I I Rothberg, Phvs ge~ Ictters 22
(1969) 163
158
159
160
161
162
163
164
'~6
B.F.. Lamrup er aL, Comparison between theory and experiments m quantum electrodynam~cs
i319] D. Favatt, P.M. Mclntyre, D Y. StroweiL R.A Swanson, V L Telegdl and R Devoe. Phys Re~ Letters 27 (1971) 1336, 1340
[401] J.C. Wesley and A, R,ch, Phys. Rev A4 t1971) 1341
[402] S.D. Dreli and H R Pagels, Ph.xs Rev 140 (1965) B397
[4031 R.G Parsons, Phvs Rev 168 (1968) 1562
[a04] L Legm, Ddogarithms and a,,soctated functions (McDonald London, 1958~
I,¢05] R Karplus and N M Kroll, Ph~s Rev 77 (1950) 536
[~06] CM Sommerfield, Ph~s Re~ 107 (1957) 328
[ 407 i A. P,~trmann, Hel~. Phys Acta 30 ( t 957) 407, Nud. Phys. 3 (1957) 689.
[408l C.M. Somthertield, Ann Phys. (N Y.) 5 (1958)26
[409l M.V. Terent'ev, Zh. Expenm. 1Teor. Fiz. 43 (1962) 619; JETP 16 (1963) 444 (English translation) [see also ref. 1159[]
[410] B E Lautrup and E. de Rafael, Phys. Rev. 174 (1968) 1835.
[411] J. Mignaco and E Remiddl, Nuovo Ctm. 60A (1969) 519
|412| J. Aidms, S, Brodsky, A Dufner and T Kmoshtta, Phys. Rev. Letters 23 (1969) 441.
[413] J. Aldms, S. Brodsky, A. Dufner and T. Ktnoshlta, Phys. Roy. DI (1970) 2378
[414~ S. Brodsky and T gano~duta. Phys. Rev D3 (1971) 356
[415] M. Levine and J. Wright, Carnegie-Mellon prepnn' ~Aut~st lq70)
[416l J. Calmet and M. Petrottet, Phys. Roy. D3 (1971) 310I,
[417] A. De Rt~jula, B Lautrup and ~, Pet-.rman Phys. Letters 33B (19"~0) 605
[4181 J Calmet, Umverstty ot Utah prepnnt (April 1971).
[4191 M. Levtne and J. Wright. Phys Roy Letters 26 (1971) 1351
[420[ J Cahnet. These-Marseflle 70/P 336 (1970), Proc Colloquium on Computational Methods m Theoretical Ph~ ~IL~.Mar~dlc
(CNRS) 1970
[421] B.E. Lautrup, Phys Letters 3 .)B (1970) 627
[422] M V~.ltman, CERN preprmt (1967)
[423] A C Hearn, Stanford University Reprint No ITP-247 (unpublished), A C Hearn. m Interactive Systems for 1 xl~rtmcntal
Appl'ed Mathematics. Fds M Klerer and J Remft ld~ (A~ad:mlc Pre~. Ne~-3rork. 1968)
[424' M L~vtne. J 2omput Phys 1 (1967)454
1425~ W Czyz, G C S ,,:',,,ey and J D Walecka, Nuovo Clm 34 {1964) 420
[4261 J.C Wesley and ~, Rtch, Phys Rev Letters 24 (1970)1320
[4271 J E Nafe. E B Nelson and I I Rabt, Ph~s Re~ 71 (1947) 914
14281 D E Na#e, R S luhan and J R Zatllartas, Phvs Re~ 72 (1947) 971
14291 G Breit, Phxs Re~ 72 (1947) q84
{4301 S H Kocntg. a. G Plodell and P ku~ch, Phy¢ Rev 88 (1q52) 191
[431) R BcrmgerandVl Heald. Phys Rc~ 95 (1954~1474
[4321 P t ranken and S I.tebes, Ph.~s Re~ 194 (1957) 1197
14331 A \ S~hupp, R ¢, Plddand II R Crane, P h ~ Roy 121 (1961} 1
14341 D T Wtlkm,,on and li R Crane, Ph~, Rc~ 130 {! 9631 852
[4351 D.'I WflkLnstJ,. eh D 1-hegts, Lnl~er~at.~ of Mlt.htgan 1962 (Umvers~ty Mtt.rotttm~ ln~ , O !' b3 478, Ann Arb~..r.
Mtclugan).
1436] A Rich, Phys. Roy Letters 20 (1968) 967, 20 (1968) 1221 Erratum. Proc Third Int Conf on n,tomtc Masses, Ed R C
Barker IUm,,ersttv of Manitoba Pre~s, Wmmpeg)
[4371 G.R Henr~ and J E Silver, Phys Re~ 180 (1969) 1262
[438] F ].M Farley Cargese Lectures m Physics, Ed M Levy. \ el ~ (Gordon and Breach, N Y , 1968)
14391 E Klein, Z Ph~stk 208 (1968) 28.
[4401 G Grhff, F G Major, R W It Roeder and G Worth. Ph],s Rev Letters 21 (1968) 340
[4411 G Graft, E Klempt and G Worth, Z Phys~k 222 (1969) 201
[4421 A. Rich aiad H R Crane, Phys Rev Letters 17 t1966) 271
14431 I Gtllelandand A Rlch, Phy~ Rev Letters ..~ (1969)1130
14441 /'S Farago, R B Gardmer, J Mmr and A G ~ Rae. Proc Phys Soc (London) 82 (1963) 493
[445] 1 A Galbralth and R B Gardmer, J Ph,,s A! (1969) 104
[4461 B E Lautrup, Proc Colloquium on Computational Physics (Marseilles, 197 i)
[5011 H. Sut]ra and E Wtchmann, Pbvs lxev 105 (I 957) 19" t)
[5021 ~ Petermann, Ph)s Rcv 105.1957) 1931
15031 H a. Elend, Plays Lett,,r~ 20 ('966) 682, 21 (1966) 720
[504] Glen W Ertckson and ilenry H T L~u. UCD-CNL-81 report (1968)
[505] T. Fanoshita, Nuoso Ctm. 51B (1967) 140
15061 B.E Lautrup and E de RaL,el, Nt eve Clnl 64~ ~1969) 322
B . E L a u t m p er al , Comparison between theory a~ld experiments tn quantum ¢tcc trodl namtcs
257
507 B L Lautrup, A Peterman and !. de Rafael Nuo,,o('im IA (19711 238
ts (Mt.(;ra',~-I,iIl ~,¢~ ~ - r k 19(~51
509 J D Bjorken and S D I)rell, Rela~llvl'itlt. Qu,,ntul~
510 N Nak,ml'~hl. I ' r % "Sheer Pb'~,..~ ( k y o t o ) i 7 (195,4,
T Kin(,,~hnla, I Math Phv~ 3 ( I o 6 2 1 6 5 0
~t2 G k a l l e n . - n d A Sabr'~ k.,I D,insk:Viden~k Sekk Mat I~', Medd no 17119%1
Bouthiat and L '~|tuhcl I t ' h ~ R t d t u m 22 {19611 121
6fll
6,,|2 l u'.al Durand III Phv~ R ~ : 28 ( [ 9 6 2 ) 441
~03 I kanoshata and R.J O,tk'~. l % s Lettt.'rs 25B (19671 lZ3
6O4 J.E. Bowcock, Z. Phys. 21 ! (1968) 400
605 V L Auslat, der. G 1 Budker. Ju N Pe';tov. V A Sndoro~. A N Skrmsky and A G Khabakhpasht'v, Ph$ s Letters 25B 119671
433
6(161 V.L Auslander, £; 1 Budker. E V Pakhtusova. Ju N l:eslov, V A Sldoro~, A N Sknnsky
Nucl Phys. 9(19691 144
6('171 J.E Augustm. J,C Btzot, J Buon, J Ilal~,sm',,k~ D Lalam~e P M,mn, N Nguyen Ngo,., J
and S Tavernier. Phys Letter'; 28B (19691 508
6081 J L Augustln, D Benak,,aLJ Buon. V Grac~o,! ilai~sm,,kl, l) Lalann,' I lapl,md~e, I
I R u m p ( a n a l " SiIva. Ph~.~ L e t t e r ~ 2 8 B ( 1 9 6 9 } 5 1 3
,609] J.l Augu.,tln, J C BlTot, J Buon. B Delcoult, J tlalssinski, I Jeanl-'an l) [alanne P (
~,-Jorba, F Richard. i R,m'tpt arid D lreillc, Phy~ Letter,, 28B (1969) 517
610[ M Gourdm and I de Rat,tel. Nt, d Phy,,. B10 (196q) 667
f i l l G J (.ounatt~.,nd I ] Sakurai Phy,, Rev letter~ 21 (1968) 244
6121 J S Beililnd I ch Ratael. Null Ph'~ B11 (I96qJ 611
61 :l] II "ler,~rav~a, Pnil.rev, rheoret Phi', 39 119681 132¢.
614] N.M Kroli. T D L e e a n d B Zuniino. Phys Rev 157(19671 1376
615] R N Amado and L Itolioway. Nu@,o ( i m 30 (19631 1083.3(} (196"~) 1572
6161 N B y e r ~ a u d l Zarhartascn, N u u v o C t m 18(19601 1289
6171G Segre. Phy~ L e t t e r ~ 7 ( 1 9 6 3 1 3 5 7
1618l Ph Meyer and D 5thtfl. Phy,, L e t t e r ~ 8 ( 1 9 6 4 ~ 2 1 7
[619l tt Pnet~thmann. Z Physlk 17('1 t1964} 409
le,2(ll R A Suhafler Phi,', Rev 135 {19641 tl 1~7
1621[ ~J Brod,,ky ar'~, I I) bitlh~,an, PI..~ R~v 156 11967) 1644
[622] I B u t n e t t a n d % l l Levme, Ph~x Re~ 2 4 B i ' % ' r p 4 6 7
162~] [ D L e e a n d ( N 3ang, P h ~ Re~ 12V, (1962) 885
1624i R L (,attain, L~ P ilutdun.,on S Ptnman a n d ( , hliaplit~ Phxx P.t~ I IN, 19hill 2 '1
{62~] (., C h a r l ) a k , ( , J Xt l arh.~ R [ (,a,t~.ln I \ l u l k r ,
1~>2~,1(, ( h a r p a k I J %1 I,trle,, R I (,dr~,~.lii l \ l t l l k r
16271 I .1 Xl l arle~,,J l~hi,h. ~. R f, \ Ih'o,,~,n %1 (,it~,uh
( Inl
45 ~ 1966i
and A C Khabakhpashcv ho~ J
PereT-)-Jorba I Rumpt I bd,,a
letran~tu~ P Lehrl.mn P \1Jrln
Marln N N~ztt~.en Neot. J Perc,'-
I(
%tn~ X I I c l c i x h illtl A / , c h l ~ . h i l ' h ~ I~,tx I cit~ ,,f , l i t
i"
I ( ~ , t i ~ Illd \ / i . hluhl ~tio'~<,( II11 l ' ~ l l ' J f , gl 1741
I1 l l l g i c l i , ~ ",llltit. l XlCUl t I>k.ix'.,, Ilhi '~I [ i l l l l c l ] b i t l i l
k '"~ '
2'~1
16281 O.l Henry, O S c h r a n k a n d R A Sv, an,,on, N u o v o C I m 63.,%. ( 1 9 6 9 ) 995
[6291 J F Hague, J E l o t h b c r g , A Schenck, O L Williams, R W Wflh,imq, 1¢ k Young and k %4 (rox~t" Ph~,~ Rev l citer\ 2~
(1970) 628
16301 D.P Hutchinc,on, F L Larsen, N C S c h o . n , D I Sober and A S Kanotsk.v, Ph~ s Rex Letter~ 24 11970) 1254
[6~1] ~,1 A, Ruderman, Phy~ Rev Letters 17 119681 794
[6321 1 Baile.~, F J M 1 arley, I-! ]ostlcin, G Petrucu, l. Pluas,,o and I Wlck~n,,, Propo~<il tor I measurement o1 the alltllll lloll,
magnetic moment of tile muon at a level of 10--20 ppm, CI RN PIt I/COM - 69,/20 t%la~ 19691
[633] J Batley, G Bassomplerre, k Borer, I Comblev, P Ft,itterslev,G Lebee, G Pctruttl, I Plca,,so t l l P~el () ,(ullotl,,oia
and R, Tmgueh', The present status ol the g--2 project CLRN NP Internal Rcporl 70-73 (April 19701
1634l C Jarlskog, ( ERN Preprlnt TH-1363 ( i 97 i I
16351 V Bargman, L. Michel and V L 1. elegdl, Phys Rev Letters 2 ~1959i 435
16361 S Weinberg, Phys Rev Letters 19 (1967) 1264,27 ( 1 9 / 1 ) 1688
[6371 ! Bars and M Yoshimura, Berkeley preprint tmarcn tO721
16381 R Jackr~ and S Welnberg, to be published
16~91 ~ A Batdc'cn, R t;a~tiltan~ and t~ [ a u t r u p , (1 RN ptepitnt | 11-1485 I\la~ 1972i
17011 N M Kroll, Nuovo Ctm 45 ~1966) 65
[703] I Yu Kobzarev and L B Okun, Soviet Phy~ JLTP 14 (1962) 859
17041 R Sagano, Prog Theor Phys 28 (1962) 508
[7051 S J Brodsky and L de Rafael, Phys Re~ 168 (1968) 1620
58
B.E. Lat Irup er at..C o m l o ~ n between,t& zry and experiments m quantum electrodynamlcs
~,] K,M. Ct~, l~ys. Rev, 76 (1949) 1.
rO'~| S. Nakamuta, tL Matsumoto, N. Nakazawa and H. Uga;, Suppl. Progr Theor Phys. extra number (1968)
I08| M.L. Good, L. M~©neland E. de Rafoel, Phys. Ray. 151 (1966) 119#.
'10} I. Butlago~,D.C. Candy, C. Franzinetti, W.B. Fretter, H W.K. Hopkins, C. Manfredott], G Myatt, F.A Neznk, M Ntk~,h~,
T.B. Nove~, R.B. Palmer, J.B.M. Patttson, D H. Perkans, C.A. Ramm, B Roe, R Stump, W Venus, H W Wachsmuth a],.d
. * _ILXesh~l'k~.NuovoCun Letters 2 (1969) 689
? I t | A. DeR~ala and R,iLP. Zia, Nucl. Phys. B19 (1970) 224.
?t2|-C..A..Ra~nn, Nature 217 [1968) 913; 227 (1970) 1323.
?1~] W.T. Tbner, TJ. Brmmstein, W.L. Lakin, F. Martin, M.L. Perl, T.F. Z~pf,H.C. Bryant and I~ D. Dieterle, Phys Letters 368
(1971)2~t.
B011 W.C. Earbez, B, Gittelman, G.K. O'Nedl, B. Richter, Phys. Ray. Letters 16 (1966) 1127
S021 W.C. flaker, B. Gittelman, G.K. O'Neill, B. Richter, Proc. 14th Int. Conf. on High-Energy Physics, Vienna (CERN, Geneva,
1968).
803~ C.-M~olk'lr,,~m. Phydk 14 (1932) .531.
804] Y.S.Tsai,l'hys.Rev. 120 (1960) 269.
805] T.D. Lee and G.C. Wick, NucL Phys. B9 (1969) 209; BI0 (1969) I.
806] T.D, Lee, "A Relattvisttc Complex Pole Model with Indefinite Metric", m Quanta (Untvemty of Chtcago Press, 1970)
p. 260.
1~)7] T.D. Lee, Topical Conterence on Weak lnteracOons (CERN, Geneva, 1969) p. 427.
8091 T.D. Lee, "Feynman D~rams m a Finite Theory of Quantum Elect~odynam ca", lectures at the Ettore Majora~ International School of Physics, July 1970.
810] J.E. Augu$tm, These de Doctorat d'Etat, LAL 1212, Orsay (1969).
811] D. Lalande, These de Doctorat d'Etat, LAL 1235, Orsay (1970).
812] J. Perez-y-Jorba, Proc. of 4th Int. Syrup. on Electron and Photon lnteracuons at Htgh Energies, edtted by Daresbury Nuclear Physics Laboratory (1967) p. 213.
8131 V.N. Baler, V,M. Gahtskl, JETP Letter~ 2 ¢1965) 165.
8141 V.N. Baier, V.S. Fadin, V.A Khoze, Soviet physics JETP 23 (1966) 1073.
18151 P. Dt Vecchi.-- and M Greco, Nuovo Cim. 50A (1967) 319.
8161 M. l]ontebeyn~., These de 3eme Cycle, N 6E8, Fac. des Sciences, Bordeau~ (1969.
8171 H.J. Bhabha, Proc Roy. Soc. 15JA (1935) 195
18181 S. Tavermer, These de 3~me Cy¢,e, LAL, KI 68/7 (1968).
18191 V. S~dorov, Proc. ath Int. Syrup. on Electron and Photo~ Interactions at Htgh Energies, edtted by Daresbury Nuclear
Physics Laboratory (1969) p. 227
i820] ).G. Rutherglen, Proc 4th Int. Syrnp. on Electron and Photon lnteractvon~ at High Energies, edited by Daresbu~, Nucle,~
Physics Laboratory (1969) p. 163.
I821] H. Alvensleben, U. "B,r,ker, W.K. Bertram, M, Bmkley, K. Cohen, C.L. Jordan, T.M. Knasel, R. Marshall, DJ Qumn,
R. Rhode, G H. Sanders and S.C.G. Ttng, Phys. Rev. Letters 2;, (1968) 1501.
i822] J. Tennenbaum, A. Eisner, G. Feldman, W. Lockeretz, F.M. Lipkm and J.K. Randolph, Proc. 4th int. Syrup. on Elecuon
antl Photon Intcraclaons at High Energies, ethted by Daresbury Nuclear Physics Laboratory (1969) Abstract no. 145.
[823] P.J. Bisgs, D.W. Braben~R W. Ciifft, E. Gabathuler, P Kttching and R.E. Rand, Phys. Rev. Letters 23 (1969) 927.
[824] E. Etsenhandler, J Fetgenbaum, N. Mistry, P. Mostek, D. Rust, A Silverman, C. Sinclatr and R. Talman, Phys. Ray Letters 1
(196~ 425
[82.q] KJ. Collen, S. Homma, D. Luckey and L.S. Osborne, Phys. Re,'. 173 (1968) 1339.
[8261 J.K. de Pagter, J.! Friedman, G. Glass, R.C. Chase, M. Gettner, E. yon Goler, Roy Wemstem and A.M. Boyarskt, Phys Ray
Letters 17 (1966) 767.
[827| S. Hayes, R. |relay, P.M Joseph, A.S Ke,zer,J Knowles and P.C Stern, Phys. Ray Letters 22 (1969) 1134
[8281 S. Hayes, R. Imlay, P M. Joseph, A.S. Kelzer, J. Krtowles and P.C. Stean, Phys. Ray. Letters 24 (i970) 1369
[829i D R. Earles, W.L.Fmssler, M C,ettner, G Lutz, K M. Moy, Y.W. Tang, I|. ~on Bnese, J~, E. yon Goe~r ~,nd Roy We:nstem, Phys. Ray. Letters 25 (1970) 1312.
[830] H A. Bathe and W Hairier, Proc. Roy Soc (Londc :) A146 (1934) 83.
[831| J.D. Bjofkerl, S.D. D~J] and S.C. Frautschi, Phys. Rev. 112 (1958) 1409.
[832] J.A. McClure and S.D. Drell, Nuovo Cim. 57 (1965) 1638.
[833| RH. Siemann, W.W.Ash, K. Berkelman, D.L. Hartdl, C.A. Ltchtenstein and R.M. Ltttauer, Phys Ray. Letters 22 ' 1969)
42!
[834| C. Bernardini, F. Feli~tu, R. Querzoh, V. Silvestruu, C. Vtgnola, L. Meneghetu Vttale, S. Vttale and G Pens(>,Prec. 14'.h
int. Conf. on H~h Etaergy Physics, Vienna (1968) Abstract 707.
8.E Lauttup et al., Compae~son between theory aald experiments ~n qltantum electrodvnamtcs
259
8351 A.D Llberman, C M. Hoffman, E. Engels J r , D C irene, P G Innocent1, Rtchard X~tlson, ( Z,~dje W & l~lanpled I) ¢
Starts and D.J Dnckey, Phys Rev. Letters 22 (1969) 663
8361 S D. Drell and J D Walecka, Ann Phys IN Y ) 28 (1964) i 8
~371 D Earles, H yon Brtesen J r . R Chase, W. Falssler, M Gtttner, G Glass, E yon Goeh'r. G Lut7. R Par~h~ ' p Roth'~cU
and R Wemstem, Proc 4th lnt Syrup on Electron arid Pho'on lr~teracttons at l-hgh Lnert, lc~ cdttcd by 13aresburx Nu,lt tr
Physics Laboratory (1969) Abstract no 65
838[ a .1. Russell, R.C Sah, M J larmenbaum, W E Cl2land. D G Rya1~and I) G btalrg. Phvs ~,c~ Lcttcr~ 26 t 1971 ) 4¢,
8391 53. Brodsky and S.C C, Tmg, Phys. Rev. 145 (1966) 1018
8401 D,t.' F.,hn and G.R. Henry, Phys. Rev 162 (1967) 1722
8411 F. Amman, R. Andream, M. Bassettt, M. Bemarthm, A Cattom, V. Chlmentt, G F Corazza, D Fabmm. t~ Ferlenghl,
Massarottl, C. Pellegrml, M. Plaeldt, M Puglm, F Soso, S. Tazzart, F. Tazzloh and G Vlgnola, Nuovo Ctr~ Lette-s 1 (1969)
729.
8421 B Borgia, F. Ceradim, M. Conversl, L Paoluzl, W Scandale, G Barblelhm, M Gr,lll, P Spdlanttm, R Vl~cntsn arid A
Mulachle, Phys. Letters 35B (1971) 340
843J V Ages Borelh, H. Bernardmt, D. Boihni, T Massam, L Monan, F Palmonan, A Ztchichl, Nuovo Cim. ~ , (1972) 330, 345
8441 C. Baoet, R. Baldmt, G Lapon, C. Mencuccmt, G P Murtas, G. Penso, A. Reale, (; SalvmL M Epmettl, It Stecla Frascatl
Preprmt (unpubh~ed) 1971.