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Christine Davies University of Glasgow HPQCD Collaboration Dirac Day, Edinburgh Sept The HVP contribution to the anomalous magnetic moment of the muon Christine Davies University of Glasgow HPQCD collaboration DiRAC Day, Edinburgh Sept. 2016 Methodology Interac$on)between)) The muon is a heavier cousin of electron with same electricmagne$c)moment)(spin))with)B?field.) charge, e, and half integer spin. e Its magnetic moment is µ~ = g S~ Spin)precesses)around)B)at) 2m a)frequency)determined)by)“g”) Difference of g from 2 by coupling to virtual B))field)(photon)) particles gives anomalous magnetic moment g 2 aµ = − 2 Fermilab Muon g-2 Experiment Mark Lancaster : PPAP 2015 : p26 Experimental measurement at sub-ppm level allows test of the completeness of the Standard Model. The largest corrections to g come from QED effects but weak and strong interaction (QCD) effects non-negligible. eB For polarised muons circulating perpend- ! = icular to a B field, cyclotron frequency c γm Difference of spin precession and eB ! ! = a cyclotron frequencies is prop- s − c µ m ortional to anomaly at a ‘magic !a momentum’ ⌘ ⌫µ ⌫ Muons decay to electrons via e parity-violating weak interaction so electron emission direction µ e⌫µ⌫e tracks muon spin. ! µ Number of electrons measured as a function of time then modulated with e cos(!at) G. W. BENNETT et al. PHYSICAL REVIEW D 73, 072003 (2006) 10 representative electron decay time histogram is shown in 2001 BNL result (av. muon and antimuon): Fig. 2. To determine a, we divide !a by !~p, where !~p is the expt 10 1 measure of the average magnetic field seen by the muons. a = 11659208.9(6.3) 10− The magnetic field, measured using NMR, is proportional µ to the free-proton precession frequency, ! . The muon -1 p ⇥ 10 anomaly is given by: Million events per 149.2 ns !a !a=!~p R 3σ -2 Discrepancy with Standard Model ~ a ; (11) 10 ! ! ! =!~ ! =!~ R L ÿ a L p ÿ a p ÿ where !L is the Larmor precession frequency of the muon. -3 hint of new physics? 10 The ratio R !a=!~p is measured in our experiment and 0 20 40 60 80 100 the muon-to-proton magnetic moment ratio Time modulo 100µs !L=!p 3:18334539 10 (12) FIG. 2. Distribution of electron counts versus time for the NOW: experimental ring moved to Fermilab9 is determined from muonium hyperfine level structure 3:6 10 muon decays in the R01 ÿ data-taking period. The data is wrapped around modulo 100 s. measurements [12,13]. The BNL experiment was commissioned in 1997 using the same pion injection technique employed by the CERN ratory frame (n N, A) (here, E 3:1 GeV ! ! max III experiment. Starting in 1998, muons were injected and A is the laboratory asymmetry). As discussed later, directly into the ring, resulting in many more stored muons the statistical uncertainty on the measurement of !a is with much less background. Data were obtained in typi- inversely proportional to the ensemble-averaged figure- cally 3–4 month annual runs through 2001. In this paper, 2 2 of-merit (FOM) NA . The differential quantity NA , we indicate the running periods by the labels R97–R01. shown in the Fig. 1(b), illustrates the relative weight by Some facts about each of the runs are included in Table II. electron energy to the ensemble average FOM. Because the stored muons are highly relativistic, the B. Beamline decay angles observed in the laboratory frame are greatly compressed into the direction of the muon momenta. The Production of the muon beam begins with the extraction lab energy of the relativistic electrons is given by of a bunch of 24 GeV=c protons from the AGS. The protons are focused to a 1 mm spot on a 1-interaction E E pc cos E 1 cos : (9) length target, which is designed to withstand the very lab high stresses associated with the impact of up to 7 Because the laboratory energy depends strongly on the 1012 protons per bunch. The target is composed of decay angle , setting a laboratory threshold Eth selects twenty-four 150-mm diameter nickel plates, 6.4-mm thick a range of angles in the muon rest frame. Consequently, the and separated by 1.6 mm. To facilitate cooling, the disks integrated number of electrons above Eth is modulated at rotate at approximately 0.83 Hz through a water bath. The frequency !a with a threshold-dependent asymmetry. The axis of rotation is parallel to the beam. integrated decay electron distribution in the lab frame has Nickel is used because, as demonstrated in studies for the form the Fermilab antiproton source [14], it can withstand the shock of the instantaneous heating from the interaction of Nideal t N0 exp t= 1 A cos !at ; (10) ÿ ÿ the fast beam. The longitudinal divisions of the target where N0, A and are all implicitly dependent on Eth. For reduce the differential heating. The beam strikes the outer E989 (FNAL) will reduce exptl uncty to 1.6,athresholdenergyof1.8GeV( improvingy 0:58 in Fig. 1(b)], the radius of the large-diameter disks. The only constraint on asymmetry is 0:4 and the average FOM is maximized. A the target transverse size is that a mis-steered proton beam systematics, starting 2017. TABLE II. Running periods, total number of electrons recorded 30 s or more after injection having E>1:8 GeV. Separate systematic uncertainties are given for the field (!p) and precession (!a) final uncertainties. Theory calculation also needs improvement.Run Period Polarity Electrons [millions] Systematic !p [ppm] Systematic !a [ppm] Final Relative Precision [ppm] R97 0.8 1.4 2.5 13 R98 84 0.5 0.8 5 R99 950 0.4 0.3 1.3 R00 4000 0.24 0.31 0.73 R01 ÿ 3600 0.17 0.21 0.72 072003-4 Hadronic corrections to the muon g 2 from lattice QCD T. Blum − Table 1: Standard Model contributions to the muon anomaly. The QED contribution is through a5, EW a2, and QCD a3. The two QED values correspond to different values of a, and QCD to lowest order (LO) contributions from the hadronic vacuum polarization (HVP) using e+e hadrons and t hadrons, higher − ! ! order (HO) from HVP and an additional photon, and hadronic light-by-light (HLbL) scattering. QED 11658471.8845(9)(19)(7)(30) 10 10 [2] ⇥ − 11658471.8951(9)(19)(7)(77) 10 10 [2] ⇥ − EW 15.4(2) 10 10 [5] ⇥ − QCD LO (e+e ) 692.3(4.2) 10 10, 694.91(3.72)(2.10) 10 10 [3, 4] − ⇥ − ⇥ − LO (t) 701.5(4.7) 10 10 [3] ⇥ − HO HVP 9.79(9) 10 10 [6] − ⇥ − HLbL 10.5(2.6) 10 10 [9] ⇥ − The HVP contribution to the muon anomaly has been computed using the experimentally measured cross-section for the reaction e+e hadrons and a dispersion relation to relate the real − ! and imaginary parts of P(Q2). The current quoted precision on such calculations is a bit more than one-half of one percent [3, 4]. The HVP contributions can also be calculated from first principles in lattice QCD [8]. While the current precision is significantly higher for the dispersive method, lattice calculations are poised to reduce errors significantly in next one or two years. These will provide important checks of the dispersive method before the new Fermilab experiment. Unlike the case for aµ (HVP), aµ (HLbL) can not be computed from experimental data and a dispersion relation (there are many off-shell form factors that enter which can not be measured). While model calculations exist (see [9] for a summary), they are not systematically improvable. A determination using lattice QCD where all errors are controlled is therefore desirable. In Sec. 2 we review the status of lattice calculations of aµ (HVP). Section 3 is a presentation Standard Modelof our results theory for aµ (HLbL expectations) computed in the framework of lattice QCD+QED.flavour Section 4 gives our conclusions↵ and outlook for future calculations. Contributions QED =0.00116 and CP 2⇡ from QED, conserving EW and QCD ... Blum et al, interactions. Z Z 1301.2607 QED W dominates. QCD contribs LO Hadronic vacuum3 polarisation (HVP) start at ↵2Figure 1: Representative diagrams, up to order a , in the Standard Model that contribute to the muon QEDanomaly. The rows,dominates from to top to bottom,uncertainty correspond toin QED, SM EW, result and QCD. Horizontal solid lines represent the muon, wiggly lines denote photons unless otherwise labeled, other solid lines are leptons, QEDfilled loops denote quarks (hadrons), and the dashed line represents10 the higgs boson. a = 11658471.885(4) 10− µ ⇥ EW 10 a = 15.4(2) 10− 3 µ ⇥ E821 10 a = 11659208.9(6.3) 10− µ ⇥ Hadronic (and other) contributions = EXPT - QED - EW expt QED EW 10 a a a = 721.7(6.3) 10− µ − µ − µ ⇥ HV P HOHV P HLBL new physics = aµ + aµ + aµ + aµ Focus on lowest order hadronic vacuum polarisation, so assume: HLbL 10 a = 10.5(2.6) 10− µ ⇥ HOHV P 10 NLO+NNLO aµ = 8.85(9) 10− Kurz et al, − ⇥ 1403.6400 HV P,nonew physics 10 a = 719.8(6.8) 10− µ ⇥ Leading order of hadronic contribution (HVP) Dispersion relations and VP insertions in g 2 ! Hadronic vacuum polarization (HVP) − Starting point: 2 2 Ì Optical TheoremVμ (unitarity)Vν =( for theq g photonµν propagatorqµqν )ΠV (q ) − s + Vµ = Qf f¯γµf quark’sImΠ⇥⇤ ( sEM) = current⌅tot(e e −: anything) ⇤ f 4 ⇥ X ! Optical Theorem Ì Analyticity (causality), may be expressed in form of a so–called (subtracted) dispersion relation s + ImΠ (s)= ⇤ (e e− X) V 4⇥ tot ! ! Analycity 2 ⌅ 2ImΠ (s) 2 k k ⇥⇤1 ImΠV (s) Π⇥⇤ (k ) Π⇥⇤ (0) = ds .
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