w^. m.-m?^iw f'tf- 1 -I ÛG 0 Zi
Tt;: ANOMALOUS KAGNETiC i-'.OMEKT OF THE MUOM :
A REVIEW DF THE THEORETICAL CONTRIBUTIONS
j. CUV.ET*, S. NARISON**,
H. PERROTTET* and E. de RAFAEL14
ABSTRACT : we review the possible contributions to the muon g-factor anomaly, o^ = — t^ -Z) which arc potentially significant for a comparison between theory .^nd experiment at present. This includes purely ÇF.D perturbation theory effects up to eighth-order ; hadronic effects at fourth end sixth-order ; and weak interaction effects. From these sources we'get a total theoretical contribution
O-.JTheory) = (1 155 920.6 ± 12.9) x 10"9 , to be cc.'pared with the latest CERK expc-rinental result £l] ,
a- (Experiment) <= (1 !G5 895 ± 27) x 10"9 .
J'JNE 1«6 76/P.333
K Zïr.*.rz de Pnysique Théorique, CNRS Marseille. *" Boursier de la CEE (Coir.r.unautè Economique Européenne) Postal Address : Centre de r'nysique Théorique - uiRS 31, C'.OT.ir, -'cseph Aiguicr P-Wfc-
i - INTRODUCTION
The latest published value of the anomalous magnetic cornent of the rcuon ir&ccured by the CCRN \-\uor. Storage Ring Collaboration [ll is
The accuracy of this rce a s urcr.G nt represents an improvement by an order of magnitude with respect to tne previous result '. Since then, the Ci'\\ tfuon Storage Ring has been in operation for a certain time and o still further improvement in the accuracy is soon expected. At this level of precision there are contributions to <*• ^ , other than those gevernce uy the quantum electrodynamics of nuons ar.d electrons (Gr-D), like hadronic effects and possibly weak interaction effects, 21 which play an important role '.
Progress in obtaining a theoretical value for a- worthy of the improving experimental accuracy has been achieved in four directions :
i) A more accurate determination of scn-.e cf th» sixth-order QLD contributions has been ir„ide, either by analytical calculctiore in so'iic cases [-1 , 5] , or by improved numerical techniques in ccners [c-9] .
ii) it has been shown [1.C, lij thai certain closes of rcyn^an diagrams contributing to C^, 2re governed by a sin^e renorr.ia lization group equation. As a result, it has been possible to evaluate a l^-ye set of eighth-order contributions to ct.^ with pewers cf fr. ™ *. r i ,V1- factors |_11 j . Many oi the revoking eighth-orcer contributions with potentially important - C>\ Zl£ factors have also been estimated r "i rrtt numerically [12J . iii) Hadronic contributions to o-^, which trkc into account the new structures in the hadronic e e~ annihilation cross-section OT-tt) have also been calculated fl, 13, H! . More recently, the hadronic contributions o» order \ —J have been estimated |_1^ I » partly from the expcrinental information on CT^Lt) , portly in a r.iodel dependent way. An estinate of the induced contrioutior. to o, ^
76/P.833 ûvr. the JSyrrptotic behaviour of G"H (, J"t > 7. ^ GeVy based on asymptotic freedom ' has also been incorporated [l4j .
iv) The contribution to CX^ from weak interactions uf \%'toiô (ire new unambiguously calculable v/i thin the framework of joi-ge theory models with spontaneous symmetry breakdown [l7 - is] .
All these contributions will be reviewed in the following sections. Sect. 2 is devoted to QLD contributions up to sixth-order ; sect. 3 to the eighth-order CED contributions ; sect. 4 to the hadronic contributions ; and sect- 5 to the weak interaction contributions. The final results with comments upon their significance in the comparison bctv.cen theory and experiment are given in sect. 6.
76/P.833 2_~ qc_D_CQNTRlUUTIQMS UP TO SIXTH-ORDER
The latest WQE3 value of c< (oC = 137.035S37 (29) ; see ref.[20j) leads to the following value for the second orde>* Schwinçcr torr,
s~lZ) - a.Z) = — = U 161 405.535 1 0.246)*10*9 . (2.IJ °V ~ c- ill
The error quoted here represents an improvement by sn order of nûcnitude with rer.pect to the previous one K Note that the value of Cl ' has -9 *^ changed by about onu jnit of 10 The fourth-order contributions ( p< contributions) are ai^o well-known [_3] . They include the Peterman, Sorr-iierfield terra plus mass ratio corrections due to vacuum polarization :
= («YÏ i?Z ^ i1,. a. ciij.sT^-l *J-(^.V Vit/ L »'''• ll "l ' * «.5 \"l(t /
3nd
where Ç|?)= 1.202056S03 is the Riemann S function of argument 3 . From these analytic expressions we get
at =. .Ï] (-0.323478445) = -1W2.303(1)X10 ; (2.2)
«0 i 9 ( C .t_a )= (i] (1.094260675) = 5904.074(2)«!0" . (2.3)
The errors in (2.2) and (2.3) come from the uncertainty on c<~
76/P.033 -5-
The contribution from the mass correction tern /^s[y^\) in eq.(2. is larger than the uncertainty due to °< ' . Numerically '
fi^-L (H^t S 0.003.10"9 .
z In general, the o< contribution to the muon anomaly from a heavy lepton of mass H ( f\ »wi„) (see Fig. 1) is [zz] ^(HeavyLepton^^ff-L^)*-. n^r-^jî • !2-'l)
If the anomalous & U. events obse ved in the e e" annihilation at SlAC [23, 24] are due to the production and decay of a M M" pair with .mass M £ 1.8 GeV/c then we have
CLp^M % 1.8 GeV) = 0.4 x 10"9 . (2.5)
Ï In order to describe the sixth-order contributions ( •< contri butions) we shall use the sair.e notation as in ref. [3]. a) C13SS I (Fins. 2 and 3) : 12 graphs involving scattering of light by light subgraphs. Their contributions are respectively '
O^. * (0.368 i 0.010) (21/ ; (2.6) and [9] 7>
C °>- ae)1 • (21-32 Î. 0.05) [&) . (2-7)
In both cases, the errors come from the integration procedure.
b) Class II t Class in (Figs, la, b and 5a, b) : they
76/P.833 -6-
include second order electron vacuum polarization subgraphs and fourth order electron vacuum polarization subgraphs» These contributions are
known analytically [lO , 22 4 41 . Their numerical contribution is
O. „ ^ =-0.09474 I 2-) • (2.8)
(*•,* - <*e) = 1.94404 (_^)3 . (2.9)
c) Class IV + Class V + Class VI (Figs. 6-8) : three acrcss- type photons ; two across-type photons ; one acrcss-type photon. Thes.e sraphs do not contribute to (aij-ae). and the more precise evaluation we have gives [_5l '
<" I et \* CL __ . = (0.915 i 0.015) I _ . (2.10)
Adding (2.6), (2.8) and (2.10) we obtain the sixth order 9) contriDution to the electron anomaly '
a!? ~~ (1-183 i 0.025) {£) , (2.11)
This, together with the second and fourth-order contributions leads to the result Qtî> U) 111 (fc> „-g
o-t a a *ae+&e « (i 1.59 652.4 i o.ejx 10 , (2.12)
to be compared with the experimental result [27]
CC*P' = (1 159 656.7 i 3.5)x!0"9
The corresponding value of the muon anomaly obtained from the previous results is
ill % -t û-e , a * (1 165 848.1 ± 1.2)x 10 . (2.13) 76/P.833 -7- 3 - EIGHTH-ORDER QED CONTRIBUTIONS The eighth-order contribution to the muon anomaly may be potentially significant for a comparison between theory and experiment because of the presence of powers of -tv\ (W^/MJ terms which in spite of the overall [ fa ) factor may lead to large numerical values. Since J£w("ir/mc J terms can only appear in the difference (a.^- <* e | it will be sufficient to limit * discussion to the class of diagrams which contribute to this difference. Altogether, ti-ere are 469 Feynman diagrams which, up to terms which vanish as JUi- —> ° , may contribute to ( au - O-e) . They all have characteristic subgraphs of the vacuum polarization type and/or of the light by light scattering type with electron type loops. Of these 459 diagrams, there are 304 which are governed by a simple renonr.alization group equation of the Callan-Symanzik type [_28, 29] . They are schematically represented in Fig. 9 . Let us denote by A(^Zil ,x) the contribution to the muon anomaly from the class of diagrams obtained by replacing all internal photon lines in a renormalized muon vertc-x by dressed phcton propagators with fermion lines of the electron type only. It has been shown by Lautrup and do Rafael [ll] that at the limit !H^-»oo the corresponding asymptotic expansion of A (^!S fe<) which we cal 1 J\ ^__£ s oc J obeys, order by order in perturbation theory, the differential equation Here p(o(J is the Callan-Symanzik function, known in perturbation theory up to sixth-order [30] 10' , equation (3.1) summarizes the constraints on the muon anomaly due to the renonnaliration group property to all orders in oi ' . In particular, it predicts the complete < 76/P.833 in teres of the lower order constant terms 3. = A~ ( 2£. = ') * = «.*>» j !3'31 •- UO V rrt / and the coefficients ft , /I , A . More precisely, A~ f-=t ) = {if { \ + C, JL Sifi , JJ.W *£,£ ^.«j with J The explicit expressions for the constant terms B- 1=1,2,3 are ( a , (X and Q.c are the second, fourth- and sixth-order contributions to the electron anomaly given in (2.1) , (2.2) and (2.11)) B - ^ a1" - ' • (3-6a> B, - (7M* a."0 _ il - _i.oii3i ; <3-6b> and, from the analytic calculations of refs. \\Q, 22, 4J , B3 = [ï) a"' + I.54ÊOZ : 2.7S4 i 0.045 . (3.5c) Altogether, we find that the total numerical contribution from the in ""if terms of the 301 diagrams represented in Fig. 9 is 3 7 (c, ?« =£ +T> fe + Eu 4a !=* l&W {\7.za t .n){±i <-> Let us next consider the remaining 165 diagrams which may 76/P.B33 -9- r , ,, tcrm5 a contribute to powers o The eighteen diagrams in Fig. 11a) have been evaluated numerically by Calmet and Peterman [12] . As expected ' , they give a large contribu tion ^(fïj.uo,) = Im-.tJ-Oli)" . (3.B) , w and These diagrams give rise to ?'^l "f/t><-) 1 '*» l f/«e) constant terms. In fact, the <« (""('/«(I contribution can be calculated using renorma- lization group techniques W a ltf«. lia)) -t^hl (fl--6.Z3or .osi)H*!?} (3.9) ? (1 ' " vir./ { 5 TOt + "unpredicted J-v\ ^l*- tcirms where A is the coefficient of (37) *•* ^£ in the contribution to a.^ from the diagrams in Fig. 3 . A comparison between eqs.{3.3) and (3.9) shows that the size of "unpredicted -Av> ïl£ terms" + Cte is rather large. The diagrams fron Figs. llb),c), d) can give at most <*\ I^i. terms. They haven't yet been calculated. A beautiful application of Kinoshita's theorem on mass-singula rities [331 shows that the 6 diagrams of Fig. lie) give no •*« ^£ terms. r -, mo. The same c/.^ren can be applied |_36J , with some care because of vertex renormalization, to the diagrams in Fig. Uf). Again no -«w ïlt terms arise from these 12 diagrams. All the in ^V and <« WK. terms of the eighth-order mt **t- ft w contribution to the muon anomaly are therefore known. The in 1?£ terxs are also kn.Twn except for those which may arise from the light by light srntering corrections shown in Figs, lib), c) and d). 76/?.833 -lO- As a limit of error due to the lock of knowledge of these contributions we suggest ± 3 « * »2l !iL^ =-±65(JLV : (3-10! % Vul ^ -re / i.e., we have taken the value of the sixth-order contribution to the rauon anomaly from the diagrams in Fig. 3 (see eq.(2.7)) times ?L it for the radiative correction, times a factor of 3 since there are three independent classes of diagrams. The overall eighth-order contribu tion thus obtained is {*.) (123.3Î 8.4 Î 63) = (3.72 Z.l)x 10"9 . (3.11) For a comparison we give the results of previous educated guesses by Lautrup [35] (100-200) (1) ; (3.12) and by Samuel [37l 230 (±f - (3.13) 76/P.833 -11- -1 - HAOPOUIC CONTRIBUTIONS The dominant hadronic crrrections to the muon anomaly are due to hadronic vacuum polarisation as shown in Fig. 12 . They have been discussed in detail elsewhere (see e.g. ref.[3] ). Here we shall limit ourselves to a review of their most recent numerical evaluation. Their contribution to the muon anomaly is most conveniently expressed in the form CO I + wnere G"Hlt) is the total one photon e e* annihilation cross- section into hadions. Clearly their contribution to the anomaly is positive definite. Three recent evaluations of the integral in eq. (4.1) give 9 , ref. [l] ; (4.2a) (73 + 10) x l(f , ref. [l?1 ; (4.2b) (66 t 10) x 10"9 ? , reft. [l4,3s] • (1.2c) (70.2 ± 8.0) x 10" Tr.e di f ro*-onces between these results are due to different evaluations 'jf the \c>i energy region as well as to different appreciations of the I'acr^. o-.Tid contributions ; and also to different estimates of the contri butio :rc ~ 0<=; -.iT(<«i(3ZQ;)[u? _L_ ) ,(4.3, o A- t -12- where Ct; denotes the electric charts of the relevant constituent quark in units of c and S" is a numerical factor depending 3 on the choice of gauge group.. For ( 'côi0* ^ ^'flavor S -- __± . (4.4) U- |N" The parameter yC in eq.(-..3) is an arbitrary mass seals. It corres ponds to the subtraction point chosen to renormalize the theory. This expression for the asymptotic cross-section induces the following contribution to the muon anomaly (4.C-) where t> is the largest energy for which G?.l^0 *s experimentally known ( i|t>=7.« GeV at present), and tl (x ) = Jo j,,/^^ If the mass scale W in eq. (4.3) is fixed in such a way that G"„lt>) = Sl'~M'('t>) • wt"ch ct""resP°r'd5 t0 /<• = 5-4 10.6 GeV, su one obtains for (4)fiavor Higher order virtual hadronic effects have also been recc-rt/y J• '- mated [ll, 3SJ . lhese hadronic effects give contributions of ordtr (.%) and can be car.t ir. five classes which are shown diagramaiatically in Fi^.13. The contributor-! from the clas: a) (twc diagrams) is exactly given by the expression r r' <«* L in H ,K l ^"ÏÏ^TL '-» ^/(4.?) wheree ;5/P.333 -13- From these expressions, it is clear that the contt ibution to cX^ from class a) must be positive. The contribution from the class b) (fourteen diagrams) can be written in a form analogous to the well-known fourth-order expression (M » (.ukj)» -L.- fdb eH|t) K (i-x) , (4.9) ATT e* / V- l where K* ( % p j is a QED function which has been calculated analytically, by Barbieri and Remiddi, ref. [4*1 , as an intermediate step in the evalua tion of some of the purely QED contributions to «3-t* . Its asymptotic behaviour for t >> vn* is as follows In fact, K (%,M1S negative definite throughout the integration region in eq. (4.9) and the contribution from class b} must therefore be negative. The contribution from diagram c) can also be written in a closed form in terms of ^"H^J , which clearly exhibits the positivity property. Numerical estimates of these contributions (eqs. (1.7), (4.9) and (4.11)) have been made using the present experimental information on the e e —> Hadrons cross-section. The results can be found in Table I . There is an important cancellation between the contributions from classes a) and b) ; the contribution from class c) is negligible. Diagrams of classes d) and e) are topologicals different to those from the other classes. They involve the hadronic light by light 76/P.833 -14- fcattering amplitude for which no empirical information is available at present. One is obliged to make a model dependent estimate of these contributions. For this purpose, it has been assumed that the bulk of the hadronic effects in diagrams of the type d) and e) is effectively characterized by the sum of quark-1 loop contributions of different colors and flavors [39j . It is also assumed that the effective quark masses P1^ are larger than the muon mass m- so that an asymptotic expansion in ""H/n- is justified. The contribution to o. from class d) is then governed by the low \< behaviour of the vacuum polarization quark-loop with an internal photon correction ; K is the energy momentum of the virtual photon attached to the external muon line. For three colored quarks with flavors : i = u , d, s , c, ... one g.ts ,<*»*)*(#** *.vt*f%)& (4-12) Similarly, the contribution to O-^. from class, e) is governed by the low energy behaviour of the virtual light by light scattering amplitude with a quark-loop. A numerical evaluation of this contribution gives the result where the error is due to the numerical integration procedure. Using the SU(4) flavor scheme and typical ' quark-mass values H^ï W^ •= 0.3 GeV ; Mj =.0.5 GeV ; and V\ = 1.5 GeV , one obtains the results shown in Table I . These results do not change significantly if new quark-flavors with higher and higher masses are added, however they are rather sensitive to the u and d quark masses. From all these calculations we consider tl at the best estimate at present of the hadronic contributions is a„(Hadrons) = (66.7 + 9.4) x 10"9 . (4.14) It corresponds to the sum of eq. (4.2c) plus the total of Table I. .76/P.833 -15- TAGtî I Contribution to the muon g-factor anomaly <*-*. = i- ( <Î -A) from the five classes of higher order virtual hadronic effects represented in Fig. 13. Contribution Result Eq.(4.7) ; Fig. 13a) ; 2-diagrams (1.10 2 0.14)_ x 10"9 Eq,(4.9) ; Fig. 13b) ; 14-diagrams (-2.07 i 0.29) x 10"9 Eq.(4.11) ; Fig. 13c) ; 1-diagram 0.Q2 x 10"9 Eq.(4.12) ; Fig. 13d) ; 3-diagrams 0.09 x 10~9 Eq.(4.13) ; Fig. 13e) ; 6-diagraras (-2.6 Î 1.0) x 10"9 Estimated total (-3.5 Î. 1.4) x 10"9 76/P.833 -16- 5 - WEAK INTERACTION CONTRIBUTIONS '.J-;th the advent of renormalizable spontaneously broken gauge theories, it TS now possible to calculate unambiguously the contribution to the muon anomaly from the weak interactions. In fact, many calculations have already been done in various models £l7 - 19] . It turns out that one predictive model, in the sense that it gives finite upper and 1ov:er bounds to the muon anomaly, for any values of the parameters, is the popular Weinberg-Salam model J41, 42] . In the other models so far studied (Georgi-Glashow[43J , Lee-Prentki-Zumino [.44, 45] ) tnc contri butions can be extremely large for particular values of the parameters. In these models, it is better to use the muon anomaly to put bounds to some of their parameters. We shall review in this section the results for the models mentioned above. i) Weinberg-Salam's model. He have three Feynman diagrams (see Figs. 14a), b) and c)) which give the following contributions [l7] (5.2) (5.3) a.(f;s.iH)) 5 a-1 = iL2& * U* x*U-x). where 76/P.S33 -17- The contribution £*-„ + °~u. can be written in terms of the **• r z Weinberg angle only- More precisely, we have a quadratic form in Co$. 6W (JO It is represented in Fig. 15 . Fig. 16 shows the contribution Q-u w as a function of r . Altogether, 1.9 x 10~9 ^ <\ £ 8.9 x 10~9 . (5.41 If we now take into account the experiirental limits on A'\n 0^ [l6J and *f one accepts Weinberg's argument on the mass of the Higgs boson [47] , then CL/J is quite negligible and the bounds 0.1 GL become 1.9 x 10"9 ^ o^ ^ 2.3 X 10"9 . (5.5) il) Georgi-Glashow's model. We have also three Feynman diagrams (Figs. 17a), b) and c)) wnich give the following contributions [_19l v 0 ' v- sir MI 3 VJ fr <1 ' t* yn: n^ (5.7) 1 (5.8) ' w where 76/P.833 -13- We have in this model •) so that the contribution °^h*- ^s the StiIne fls in thc" ^inberg- Salam's model. The neutral hea\y lepton mass tTlya is related to the charged heavy lepton mass my+ l m yo cos (b = yn y+ + rvy CO V yo J and this allows one to express ex „ ^ a... + o-,, » a.1 as a function of cos. (1 i for various values of *• , at fixed charged heavy lepton mass inly» (Fig. 18). It can be seen that for r small the contribution **-l,>0 dominates whereas for y large, O-^A becomes negligible and a.J are imposed on the parameters Col f> , vfly+ and w c© iii) Lee-PrentkWumino's model. The contribution to the muon anomaly in this model can be obtained from the contribution calculated in the Salam-Weinberg's model by*simply redefining the coupling constants [l8J ; we get a - & w lit [ iS. _ iL ^^"(iv^flcas'afl (5.9) Jtr With (see ref.[44] ) G_ _ i i-?Z ,_, 151 Q.j_ depends only on Cos a and Ji. ' . Thus, putting 76/P.833 f -19- and with the constraint X ^ 4 , we have LP2 i=k [if -|l, ^fl)] (5.10) a /» tli .LPÎ In Fig. 19, we have plotted O. as 1 function of cos1 9 . for various values of * , that is of A . We car see that the upper curve (X-4) provides an upper bound for tiie contribution to the [iiuon anomaly Lfi - 3 (5.11) i.-i \0 There is no lower bound unless sone of the parameters are fixed. 76/P.S33 f "20* 6 - FINAL RESULTS Let us summarize the various theoretical contributions to the muon anomaly, discussed in the previous sections. QED contributions including all the second, fourth and sixth- order diagrams give (see eq. (2.13)) a. (QED "2" + "4" + "6") = (1 165 B4S.1 £ 1.2) x 10"9 . The error is mostly due to the numerical method of evaluation of some of the sixth-order diagrams. The contribution tD the error from the -9 present knowledge on the fine structure constant is Ï 0.2 x 10 From the eighth-order contributions we get (see eq.(3.11)) a. (8th order) - (3.7 i 2.1) x 10"9 r v> All eighth-order diagrams which contribute to vn {' f«/r>ie'\an'l/or ti'lVmt ) 'erros have been evaluated. The error JK 2.1 x 10 is due partly to the numerical method of evaluation of the calculated diaqrams (i 8.4 f— )"* in eq.(3.111) ; partly to an estimated limit of erroc of the uncalculated diagrams (i 63 [ —) in eq. (3.11)) . Our estimate of the hadronic contributions is (see eq,(4.14)) CL (Hadrons) •= (66.7 •+. 9.4) x 10~9 . This includes fourth and sixth-order contributions. The result is partly mudel dependent because on the one hand we have used asymptotic freedom to evaluate the contribution from the high energy behaviour cf the e e —> Hadrons cross-section ; on the other hand we have used a simple quark-model to evaluate some of the sixth-order contributions. The weak interaction contributions calculated within the Xeinberg-Salam's moael give lower and upper bounds which we shall adopt as characteristic of weak contributions (see eq. (5.5)) -Q a (Weinberg-Salam) = (2.11 0.2) x 1C 76/P.833 -21- The total of the theoretical contributions is therefore a. = (1 105 «20.6 ± 12.9) x 10"9 'r The comparison between theory and experiment can be best seen in Fig. 20. It is clear that the hadronic corrections are required for an agreement Detween theory and experiment. 75/P.333 -22- - FOOTNOTES 1) For a review of previous measurements of g-2 see the review articles by Comb ley and Picasso [2a] and Farley [2b] . 2) For a discussion of the different theoretical contributions, where earlier references can be found, as well as for a review of the comparison between theory and experiment prior to August 1971, see ref. [3] . 3) For a clear exposition of asymptotic freedom, where earlier references can be found, see re fs. [l5. 16] . 4) See for example ref. [3] . 5) The mass ratio J11/^ is also known with an amazing precision [211 ; J mt —•£• = 206.7G927 (17) (O.Sppm) 6) This value has been obtained as a weighted average of results given in refs. [6, 25, 26] . 7} Previously published values for this contribution are (18.4+ ] .1) ( ~ ) [25] and (19.79 ± .16)^) [8] . The three calculations agree on the algebraic part of the calculation. The differences are due to the numerical integration procedure : differ r.t choice of integration variables ; differences in the integration subroutines and machine time. 8) The result quoted in (2.10) comes from ref. [5] . It has been obtained by combining known analytic values with numerical value; for graphs not yet known analytically. See ref. [5] and references therein. 9) The error quoted in (i.ll) is the sun of the errors in (2.6) and (2.10). The error quoted in ref. \5] is calculated quadratically. 10) In fact, the eighth-order contribution to ^WJ from fourth-order vacuum polarization insertions is also known [31] 76/P.633 -23- H (- ±i_ + JLTT^I $-(-*;)[*f-I-*ss (*) . 11) The fourth- and sixth- order constraints on the muon anomaly due to the renormalization group property were first pointed out by Kinoshita TlOj. For a discussion of various choices of effective coupling constants relevant to the muon anomaly problem see réf. [32j , 12) The importance of this class of diagrams was first pointed out by Lautrup [35J . 13) lhese a.-e typical mass values used in quark model calculations of weak decays. For à comprehensive review see ref. [40] . H) We have taken fro^i ref. [46] ; Q.l 4 sin2 © < 0.5 . w 15) M and AT" are the vacuum expectation values of Higgs scalars in this model. 76/P.833 #• -M- - REFERENCES - [I] J. BAILEY et el., Phys. Letters 5_5B , 420 (1975). [2a] F. C0MI3LEY and E. PICASSO, Rhys. Reports 14C , 1 (1971). [2b] F.J.M. FARLEY, Contemp. Phys. , 16 , 413 (1975). [ ] B.E. LAUTRUP, A. PETERMAN and E. de RAFAEL, Phys. Reports 3C , 193 (1972). [4] R. BARBIERI and E. REHIDDI, Duel. Phys. MO , 233 (1975). [5] H.J. LEVINE, R.C. PERISHO and R. ROSKI'S, Phys. Rev. _D13 . 997 (1976). |_6j J. CAINET and A. PETERMAN, Phys. Letters 47B , 369 (1973). [7] P. CVITANOVIC and T. KINOSHITA, Phys. Rev. Dl£ , 4007 (1974). [o] J. CALMET and A. PETERMAN, Phys. Letters 58B , 449 (1975). [9] H.A. SAMUEL and C. CHLOUBER, Phys. Rev. Letters 36 , 442 (1976). [lo] T. KINOSHITA, Nuovo Cimento 518 , 140 (1907). [II] B.E. LAUTRUP and E. de RAFAEL. Nucl. Phys. b70 , 317 (1974). 76/P.833 -25- J. CALMET and A. PETERMAN, Phys. Letters 568 , 383 (1975). V. BARGER, W.F. LONG and H.G. OLSSON, Phys. Letters 60_B , 39 (1975). J. CALMET, S. NARISON, H. PERROTTET and E. de RAFAEL. Phys. Letters B1E , 2c.3 (1976). D.J. GROSS and F. HILCZEK, Fhys. Rev. D8 : 3C33 (19731 H.D. POLUZER, Phys. Reports KÇ , 129 (1974). W.A. BARDEEN, R. 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CALMET and E. de RAFAEL, Phys. Letters 56B , 181 (1975). [32] R. BARMERI and E. REMIDDI, Phys. Letters 578 , 273 (1971;). [33] T. KIN0SHITA, J. Hath.Phys. 3 , 650 (1972). [34] E. de RAFAEL, Unpublished. [35] B.E. LAUTRUP, Phys. Letters 388 , 408 (1972). [36] B.E. LAUTRUP, Private communication. [37] M.A. SAMUEL, Phys. Rev. 09 , 2913 (1974). 76/P.833 -27- [38] S. NARISOîl, Thèse de 3e cycle, Marseille (1976). [39] H. FR1TZSCH and H. GELL-MANN, Proceedings, XVI International Conference on High-Energy Physics (1972), vol. 2, p. 13:». [10] Mary K. GAILLARD, Phenomenology of Weak Interactions, Lectures at the Cargêse Summer Institute, 1975. Ûlj S. WEINBERG, Phys. Rev. Letters _19 , 1264 (1967). [42] A. SALAM, Proceedings of the Eighth Nobel Symposium, 196B, ed. N. Svartholm (Wiley, New York, 1963). [4.3] H. GEORGI and S.L. GLASHOW, Phys. Rev. Letters ZB , 1494 (1972). [44] B.H. LEE, Phys. Rev. D6 , 11BB (1972). [45] J. PRENTKI and B. ZUMINO, Nucl. Phys. B47 • " (1972). [46] J.G. M0RF1N, Proceedings of the 1975 International Symposium on Lepton and Photon Interactions at High Energies, Stanford University, ed. W.T. Kirk. [47] S. WEINBERG, "Mass of the Higgs Boson", Harvard Preprint (1975). 76/P.833 t -II;- - FIGURE CAPTIONS - Figure 1 : P< contribution to the muon anomaly from a heavy lepton of mass H (H » m^. ). {»» Figure Z : Two of the six graphs of class I contributing to CL _ . Figure 3 : Two of the six graphs of class I contributing to [a.-. <»e) • Figure 4 : a) One of the twelve graphs of class II contributing to «f?Lr- b) One of the four graphs of class III contributing to oiw-f • figure 5 : a) One of the fourteen graphs of class II contributing to b) One of the four graphs of class III contributing to Figure 6 : One of the six graphs of class IV contributing to Q.* -^ . Figure 7 : One of the twenty graphs of class V contributing to CX . Figure 8 : One of the twenty-four graphs of class VI contributing to a«" Figure 9 : The 301 diagrams governed by a simple renormalization group equation of the Callan-Symanzik type. There are six classes, and the number of diagrams in each class is indicated under a typical diagram of the class. Figure 10 : One of the three diagrams with a nuon loop inside an electron loop. Figure 11 : a) One of the eighteen diagrams obtained by inserting a single electron loop in the photon-photon scattering (with electron loop) sixth-order graphs. 76/P.833 -29- b) One of the eighteen diagrams obtained by inserting a single muon loop in the photon-phoLon scattering {with electron loop) sixth-order graphs. c) and d) Two of the 103 diagrams obtained by attaching a single virtual photon line to the photon-photon scattering (with electron loop) sixth-order graphs. e} and f) Two of the 18 diagrams of the following type : photon-photon scattering graphs in which all 4 vertices on the single electron loop involve a virtual photon. Figure 12 : Hadronic vacuum polarization correction to lowest order contribution to a. . ru Figure 13 : Hadronic contributions to ec^ of order (i) a) Hadronic vacuum polarization corrections to diagrams with two fermion lines H and c. • There are two diagrams of this type. b) Cxample of a correction to a fourth-order diagram with only muon lines. Altogether, there are fourteen diagrams in this class. c) Improper fourt'n-order hadronic vacuum polarization correc tions to the second order QEO diagram. d} Internal radiative corrections to the usual hadronic vacuum polarization correction. e) Hadronic light by light scattering contributions. Figure 14 : a) Neutrino exchange diagram contributing to ct in the J* Salam-Weinberg's model. b) Z-boson exchange diagram contributing to CL.„ in the same model. c) Higgs scalar exchange diagram contributing to CL^ in the same model. 76/P.833 -30- Figure 15 : The contribution °~/± -* a*- as a function of co\ c^ (Salam-Weinberg's mcJel). ' H [ iti»\ Figure 16 : The contribution Ck, as a function of the ratio r - I —) (Salam-Weinberg's model] Figure 17 : a) Neutrino exchange diag'ram contributing to cx„ in the Georgi-Glashow's model. b) Neutral heavy lepton exchange diagram contributing to o.„ in the same model. c) Higgs scalar exchange diagram contributing to cx„ in r the same model. Figure 18 : The contribution O- _ as a function of coi fh for various values of r''- - f[mm»fA.\?L\ . at the rmas s vr> = l.e GeV/c V In fact, we havve quoted rr\ Figure 19 : The contribution a.- as a function of coi é9 for various values of the vacuum expectation value ju. . In fact, we have quoted x s. V^. rather than JLA, . Figur.e 20 : Final comparison between the theoretical contributions and experiment. 76/P.833 f -31- FIG. 1 FIG. 2 FIG. 3 1 -32- FIG.A FIG. 5 + ... FIG. 6 i -33- FIG.7 FIG.8 p )i«r e ^J» (15+3 =18) FIG.10 4 -35- fi fi fi fi o) (3x6) b) (3x6) JJ Ji d) (10x6) FIG. 12 -36- bJ àJ -37- C) FIG. K 9 9 AQJfhO A(aV+J)-1G 800 600 4 00 cos2e. 200- *>cos2P AoJ.109 4 -200 -400 i i - > i i br>r 20 40 60 80 100 .500 FIG.16 FIG. 18 4°j;pz.109 X = U .1 .2 .3 M .5 .6 .7 .8 .9 ->cos26 X = 10 X=20 FIG.19 ~ ["2"+"4"+"6") QED i— ('•2•V'4•' + ,•6"•^-•'8',) QEO. QED + HADRONS i > QED + HADRONS+WEAK r EXPERIMENT 75 100 116580CN Fig. 20