Muon ( g – 2) : State of the Theoretical Art Andreas S. Kronfeld f Fermilab 24 September 2013 Fermilab Academic Lecture Series Tuesday, September 24, 2013
a = µ 2 m e ~ − 1 = 1 Feel Like a Number? 2 ( g − 2 ) 10 14 a e = 115965218073 ( 28 ) expt ⇒ better α 115965218178 ( 76 ) SM with α from 87 Rb 10 11 a µ = 116592089 ( 63 ) expt 116591802 ( 49 ) SM with HVP from e + e − 10 8 a τ = between − 5200000 and 1300000 expt 117721 ( 5 ) SM 2 Tuesday, September 24, 2013
Outline • Recap from last week: ( g – 2)/2 in quantum field theory. • QED-EW & QED-BSM contributions to ( g – 2)/2 : • on the one hand, the discrepancy is evidence for susy; yet, on the other, ... • … the agreement provides a strong constraint on susy [Bechtle et al. , arXiv:0907.2589]. • QED-QCD contributions to ( g – 2)/2 : γ • hadronic vacuum polarization; γ μ • hadronic “light-by-light” scattering. μ 3 Tuesday, September 24, 2013
Magnetic Moments in QED (+ EW + BSM) 4 Tuesday, September 24, 2013
Electromagnetic Vertex γ µ F 1 ( q 2 )+ i σ µ ν q ν � F 2 ( q 2 ) u ( p 0 ) = e R ¯ u ( p ) 2 m • Static quantities—electric charge and magnetic moment—obtained as q → 0 . µ = e ~ • Magnetic moment . 2 m 2 [ F 1 ( 0 )+ F 2 ( 0 )] • By definition of e R , F 1 (0) = 1 . • So a = F 2 (0) : as Prateek discussed, algebraically intensive methods can be automated. 5 Tuesday, September 24, 2013
Four-loop QED PHYSICAL REVIEW D VOLUME 41, NUMBER 2 15 JANUARY 1990 moment of the muon QED contribution to the anomalous Eighth-order magnetic T. Kinoshita, B. Nizic, ' and Y. Okamoto Laboratory of Nuclear Studies, Cornell Ithaca, New York 14853 Newman University, (Received 27 September 1989) of the eighth-order to the muon anomalous We report a calculation QED contribution magnetic moment a„"' coming from 469 Feynman all of which contain electron loops of vacuum- diagrams, Our result is 126. 92(41)(a/m) . The error polarization type and/or scattering light-by-light type. accuracy (90% confidence limit) of the required the estimated integration. We numerical represents to a„. Combining also report an estimate of the tenth-order contribution these with the lower-order value for the electron anomaly a„we find that the QED contribu- results and the latest theoretical is given by a„D =1 165 846947(46)(28) X 10 ', where the first error is tion to the muon anomaly in a. In- an estimate of theoretical and the second reflects the measurement uncertainty uncertainty for a„available the best theoretical the hadronic and electroweak contributions, prediction cluding X10 ", where the error comes predominantly is a„'""'" =116591920(191) from the ha- at present dronic contribution. 6 and other theories. I. INTRODUCTION AND SUMMARY account of our cal- In this paper we present a detailed Tuesday, September 24, 2013 to a„. In culation of the eighth-order QED contribution moment of the muon a„pro- The anomalous magnetic of the tenth-order QED we report an estimate addition tests of the renormaliza- vides one of the most stringent of the The long in the publication contribution. delay of the standard elec- tion the model, unified program until result was caused by the unavailability, eighth-order This is in strong contrast to troweak sector in particular. the last couple of years, of computing power which could electron a„ of the the anomalous moment magnetic some of the huge involved. integrals handle adequately to strong and weak interac- effect on a„was reported is rather which insensitive of the hadronic Our evaluation for the offers the best testing tions, and hence ground elsewhere. "pure" quantum electrodynamics. to the The QED anomalous contributing diagrams Much of the theoretical for elec- is identical analysis moment of a charged lepton (electron, muon, or magnetic trons and muons except that the effect of the electron on con- tauon) can be divided into three groups: (i} diagrams a„and that of the muon on a„via vacuum polarization, only one kind of lepton; containing (ii) diagrams taining are The electron, asymmetric. much less quite being of leptons; all containing and (iii} diagrams two kinds for a lepton of mass m„be- create a virtual than the muon, cannot massive readily The anomaly three leptons. (and all heavier parti- muon-antimuon pair. Thus muons in the gen- can be expressed ing a dimensionless quantity, effects on a, . The muon, cles} have little observable on eral form the other hand, can create a virtual electron-positron pair a = At+ A2( 1/m2)+Ax( t/m3) or- ease. relative Indeed, in the fourth and higher with Simi- electron loops dominate. ders, diagrams containing + A3(m, /m2, m & /m3), effects of strong the interactions are and weak larly, in a„ than in a, . much more important For m2 and m3 are the masses of other leptons. where for a„experimen- In testing the theoretical prediction the electron and the muon we have tally, it is crucial to know all these contributions precise- a, = A &+ A2(m, /m„)+ A&(m, /m, ) We have therefore carried out an extensive calcula- ly. to a„, and managed tion of terms contributing to reduce + A3(m, /rn„, m, /m, ), (1. 2} value of 10X 10 error from the previous the theoretical to 2X10, which a„= A &+ A2(m„/m, )+ Az(m„/m, is of the same order of magnitude as ) effect on a„. A preliminary result of the weak-interaction + A3(m„/m„rn„/m, ) . (1. 3) in Ref. 1. It has provided a this calculation was reported g — for the motivation A „A2, strong muon 2 experiment new of QED guarantees that The renormalizability E821 which at the Brookhaven is in progress National series in a/~ with A3 can be expanded in power and Laboratory. When this experiment and associated exper- finite calculable eoeScients: to improve to iments needed the hadronic contribution '2 3 a„are completed, a ~6~ a our theoretical results will enable us to + + Q + WI" 0 ~ ~ test the prediction of the standard model at the one-loop l l l 7T In addition, it provides level. useful constraints on possi- i =1, 2, 3 . ble muon internal structure as well as supersymmetric 1990 The American 41 593 Physical Society
T. KINOSHITA, B. NIZIC, AND Y. OKAMOTO 596 41 e e (o) (a) (b) Tr γ odd = 0 (c) ~~er P FIG. 3. Six of the diagrams to subgroup I(b}. contributing (c) I(a). Diagrams obtained three by inserting Subgroup second- second-order vacuum-polarization a loops in I I I to this subgroup. vertex. order Seven l / diagrams belong I & in Fig. 2. The other four are obtained Three are shown of Figs. 2(b) and 2(c) by permuting elec- from diagrams tron and muon loops along the photon line. FIG. 1. Typical eighth-order vertex diagrams from the four I(b). obtained one Diagrams by inserting to a„. Subgroup groups contributing and one fourth-order second-order vacuum-polarization be- in a second-order vertex. Eighteen loops diagrams corrections scattering with further radiative Six are shown in Fig. 3. subdiagram long to this subgroup. Tuesday, September 24, 2013 of various consists of 180 diagrams. This group kinds. fer- I(c). Diagrams two closed containing Subgroup in Fig. 1(d). are shown Typical diagrams dia- the other. There are nine loops one within mion Six of them to this are that belong subgroup. grams Group I shown in Fig. 4. of insertion These diagrams can be classified further into the fol- I(d). Diagrams obtained Subgroup by sixth-order electron vacuum-polarization lowing gauge-invariant loop) subgroups. (single (a) (c) I I (c) FIG. 2. Three of the diagrams FIG. 4. Six of the diagrams I(a). to subgroup to subgroup I(c). contributing contributing
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