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 <M ft; -9 % -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 <<n Hit structure of the eichth- crder contribution to f\ f ^£ „/_ ") , which wo denote A f ^V ^ , 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.