Turning the screws on the Standard Model: theory predictions for - PowerPoint PPT Presentation

Turning the screws on the Standard Model: theory predictions for the anomalous magnetic moment of the muon Christine Davies University of Glasgow Birmingham HPQCD collaboration February 2019

Outline 1) Introduction : what is the anomalous magnetic moment ( ) of the muon and how is it determined (so a µ accurately) in experiment? (recap) 2) Theory calculations in the Standard Model: QED/EW perturbation theory 3) Pinning down QCD effects, using experimental data and using Lattice QCD calculations. 4) Conclusions and prospects

have electric charge and spin e, µ, τ Interaction with an external em field p 0 has a magnetic component: q → 0 − ieu ( p 0 ) Γ µ ( p, p 0 ) u ( p ) A µ ( q ) Γ µ ( p, p 0 ) = γ µ F 1 ( q 2 ) + i σ µ ν q ν F 2 ( q 2 ) 2 m p Electric field interaction (charge consvn): F 1 (0) = 1 Magnetic field intn, equiv. to scattering from potential : µ i · ~ V ( x ) = �h ~ B ( x ) ⇣ e µ = e m [ F 1 (0) + F 2 (0)] ~ � ⌘ ~ ~ 2 ≡ g S 2 m g = 2 + 2 F 2 (0) Peskin + Schroeder

Anomalous magnetic moment a e,µ, τ = g − 2 = F 2 (0) 2 γ LO contribn is lepton mass independent µ µ Schwinger 1948 α µ γ 2 π = 0 . 00116 . . . many higher order pieces ….. γ New physics could appear in loops X X ∝ m 2 δ a new physics ` 1 TeV? ` m 2 µ Y X Motivates study of rather than flavour,CP-conserving µ e ≈ 10 − 8 ≈ 10 − 13 chirality flipping

CURRENT STATUS Keshavarzi a S M = 11659182 . 0(3 . 6) × 10 − 10 et al, µ 1802.02995 tantalising 3.7 σ discrepancy! details to follow … higher accuracy small-scale experiments possible (Penning trap) but discrepancies will be tiny … very hard since decays in 0.3picoseconds …. δ a τ = 5 × 10 − 2 (LEP) e + e − → e + e − τ + τ −

New determination of α (2018) : Mueller et al (h/M Cs ) Now ∆ a S M ≡ a e xpt − a S M = − 87(36) × 10 − 14 e e e 2.4 σ ‘tension’ and opposite sign to discrepancy for µ potentially adds excitement to the story! Davoudiasl+Marciano, 1806.10252 Aoyama, Kinoshita and Nio, 1712.06060 for QED calc.

Accurate experimental results + theory calculations needed spin 0 both helicity -1 in rest frame π p → π + → ν µ + µ + so get polarised beam pulse µ B field perpendicular to ring, spin precesses µ B measure frequency difference ω S − ω C

◆ 2 ! ~ " # � × ~ ✓ m ! C = − Qe E a µ ~ + .. ~ ! S − ~ B + a µ − m p c Q = ± 1 , µ ± from directly gives a µ need uniform possible stable B, measure electric field term vanishes EDM to sub-ppm with at ‘magic momentum’ ∝ ~ � × ~ NMR probes B p = 3 . 094 GeV / c calibrated using g p measure spin direction from e produced in weak decay µ + → e + + ν e + ν µ direction of highest energy e correlated with spin so N e µ oscillates at ω S − ω C

Status of experiment Muon g-2 E989 Fermilab goal E821 - 0.7ppm 4 α ⎛ ⎞ 2013: E821 ring moved 2004 Brookhaven + hadrionic + weak + ? ⎝ π ⎠ 3 α ⎛ ⎞ to Fermilab 1979 + hadrionic CERN III ⎝ ⎠ π 3 α ⎛ ⎞ 1968 CERN II ⎝ ⎠ π 2 α ⎛ ⎞ 1962 CERN I ⎝ ⎠ π α 1960 Nevis 2 π 0 0 0 0 0 0 7 1 0 0 0 0 0 E 1 0 0 0 0 1 , , , , 1 0 0 0 1 0 0 1 0 , 1 σ α μ ( × 10 − 11 ) Involvement from Germany, Italy, UK becomes E989 Aim: Much higher statistics with cleaner injection to ring, more uniform B field + temp. control : 0.15ppm i.e δ a µ = 2 × 10 − 10

Muon g-2 now running at Fermilab Run 2018 for 1-3 x E821, first results summer 2019 2017 commissioning run: 0.001% of final stats N e ( t ) = N 0 e − t/ γ t µ × [1 + A cos( ω a t + φ )] J-PARC future plan: γτ µ = 60 × 10 − 6 s slow µ in 1m ring - no need for ‘magic momentum’

Accurate experimental results + theory calculations needed QED corrections dominate - calculate in Perturbation theory higher orders depend on ratios 0 . 5 α of lepton masses: π subset of diagrams at α 5 integration challenging- use VEGAS Aoyama, Kinoshita et al PRD91:033006(2015), err:PRD96:019901(2017) For α use or Rb/Cs a e <0.5ppb

= α � α 2 � α 3 � � a QED 2 π + 0 . 765 857 425(17) + 24 . 050 509 96(32) µ π π � α 4 � α 5 � � + 130 . 879 6(6 3) + 753 . 3(1 . 0) + · · · π π Hoecker+ Marciano RPP 2017 a QED = 0.00116 + 0.00000413 … + 0.000000301 µ + 0.00000000381 + 0.0000000000509 + … using Rb α = 11,658,471.895(8) x 10 -10 uncertainty from error in α but missing α 6 (light-by- light) also this size

Gnendiger Electroweak contributions from Z, W, H et al, 1306.5546 ν H Z Z W is small - suppressed by powers of m 2 a EW µ H piece tiny µ m 2 at 1-loop; = G F m 2 W  5 � 3 + 1 µ a EW(1) 3(1 − 4 s 2 W ) 2 2-loops √ µ 2 8 π 2 γ (a) = 19 . 480(1) × 10 − 10 157 W 156 H 155 µ µ � 10 � 11 � a EW(2) = − 4 . 12(10) × 10 − 10 154 µ EW 153 a Μ 152 a EW = 15 . 36(10) × 10 − 10 151 µ 150 50 100 150 200 250 300 350 M H � GeV �

QCD contributions to start at α 2 , nonpert. in QCD a µ Blum et al, W 1301.2607 q γ ` Higher order Hadronic vacuum polarisation LO Hadronic vacuum (HOHVP) polarisation (HVP) Hadronic light- dominates uncertainty by-light, not well in SM result known but small Since QED, EW known accurately, subtract from expt and compare QCD calculations to remainder a E 821 = 11659209 . 1(6 . 3) × 10 − 10 µ a EW = 15 . 36(10) × 10 − 10 a QED = 11658471 . 895(8) × 10 − 10 µ µ

Hadronic (and other) contributions = EXPT - QED - EW a E 821 − a QED − a EW = 721 . 9(6 . 3) × 10 − 10 µ µ µ = a HV P + a HOHV P + a HLBL + a new physics µ µ µ µ Focus on lowest order hadronic vacuum polarisation (HVP), so take: “consensus” value a HLbL = 10 . 5(2 . 6) × 10 − 10 will return to this µ NLO+NNLO a HOHV P = − 8 . 85(9) × 10 − 10 Kurz et al, µ 1403.6400 a HVP ,no new physics = 720 . 2(6 . 8) × 10 − 10 µ a EW Note: much larger than µ

� � �� a HVP How to calculate - Two approaches: µ σ ( e + e − → hadrons) 1) + dispersion relations. 2) lattice QCD - “first principles” � Analyticity+optical theorem σ ( e + e − → hadrons) 1) 2 Z ∞ γ γ γ 1 had ⇔ a HV P ds σ 0 = had ( s ) K ( s ) � �� µ had 4 π 3 � ( ) m 2 π π 0 γ e + e − → γ ∗ → hadrons threshold q µ K(s) kernel emphasises low s - integral q ρ , π + π − dominated by . Use pert. QCD at high s. σ 0 is ‘bare’, with running α effects removed . Final state em radiation IS included - γ inside hadron bubble

Need to combine multiple sets of experimental data from many hadronic channels (+ inclusive) inc. correlations New data sets from KLOE, BESIII, SND(Novosibirsk) .. New results Keshavarzi, Nomura, Teubner √ s 1802.02995 : 70% redn in uncty since 2011. New data, more channels, correlations a H V P = 693 . 3(2 . 5) × 10 − 10 KNT18 agree well - µ 0.4% uncty Davier et al, a HVP = 693 . 1(3 . 4) × 10 − 10 1706.09436 3.5 σ from no µ a HVP = 688 . 8(3 . 4) × 10 − 10 new physics. Jegerlehner µ 1705.00263

2) Lattice QCD Blum, hep-lat/0212018 Z ∞ q = α i )ˆ a HV P,i dq 2 f ( q 2 )(4 πα e 2 Π i ( q 2 ) q µ π 0 µ q ‘connected’ contribution for flavour i Integrate over Euclidean q 2 – f(q 2 ) diverges at small q 2 with scale set by so dominates q 2 ≈ 0 m µ Renormalised vacuum polarisation function J J ˆ Π ( q 2 ) = Π ( q 2 ) − Π (0) vanishes at q 2 =0 This is (Fourier transform of) vector meson correlators. Can perform q 2 integral using time-moments of standard correlatorrs calculated in lattice QCD to determine meson masses. HPQCD,1403.1778

Lattice QCD: fields defined on 4-d discrete space-(Euclidean) time. Lagrangian parameters: α s , m q a 1) Generate sets of gluon fields for Monte Carlo integrn of Path Integral (inc effect of u, d, s, (c) sea quarks) 2) Calculate valence quark propagators and combine for “hadron correlators” . Average results over gluon fields. Fit for hadron masses and amplitudes • Determine to convert results in a m q lattice units to physical units. Fix from hadron mass *numerically extremely challenging* • cost increases as a → 0 , m u/d → phys and with statistics, volume. a

Using Darwin@Cambridge, www.dirac.ac.uk Allows us to calculate quark propagators rapidly and store them for flexible re-use. Inversion of 10 7 x 10 7 sparse matrix solves the Dirac equation for the quark propagator on a given gluon field configuration. Must repeat thousands of times for statistical precision.

‘2nd generation’ gluon field configs generated by MILC including HPQCD’s HISQ sea quarks. Physical u/d quark masses now possible. u/d (same mass), s and c sea quarks m u = m d MILC HISQ, 2+1+1 0.14 = m l HISQ = Highly 0.12 improved mass of u,d staggered quarks - quarks 0.1 very accurate 2 / GeV 2 discretisation 0.08 E.Follana, et al, HPQCD, hep-lat/ m � 0610092. 0.06 m u , d ≈ m s / 10 real 0.04 world physical m π 0 = m u , d ≈ m s / 27 0.02 135 MeV Volume: 0 0 0.005 0.01 0.015 0.02 0.025 0.03 m π L > 3 a 2 / fm 2