Theoretical status of the g 2 −

Andreas Nyffeler

Institute of Nuclear Physics Johannes Gutenberg University, Mainz, Germany [email protected]

Service de Physique Th´eorique Universit´eLibre de Bruxelles, Belgium January 29, 2016 Outline

• Basics of the anomalous magnetic moment

• Electron g 2: Theory, Experiment at Harvard, Determination of α − • Muon g 2: − • Brookhaven Experiment • Theory: QED, weak interactions, hadronic contributions (HVP, HLbL) • Current status of the muon g − 2

• New Physics contributions to the muon g 2 − • Complementarity between precision physics at low energies and searches for New Physics at high-energy colliders • New Physics contributions to g − 2: general observations • • Dark Photon

• Conclusions and Outlook Basics of the anomalous magnetic moment Electrostatic properties of charged particles: Charge Q, Magnetic moment ~µ, Electric dipole moment ~d For a spin 1/2 particle: e 1 ~µ = g ~s, g = 2(1 + a), a = (g 2) : anomalous magnetic moment 2m | {z } 2 − Dirac Long interplay between experiment and theory: structure of fundamental forces In Quantum Field Theory (with C,P invariance):

γ(k)   iσµν k  µ 2 ν 2  =( ie)¯u(p 0)  F (k ) + F (k ) u(p)  γ 1 2     − 2m  | {z } | {z } p p’ Dirac Pauli

F1(0) = 1 and F2(0) = a 2 ae : Test of QED. Most precise determination of α = e /4π. aµ: Less precisely measured than ae , but all sectors of Standard Model (SM), i.e. QED, Weak and QCD (hadronic), contribute significantly. Sensitive to possible contributions from New Physics. Often (but not always !):  2  2 m` mµ a` 43000 more sensitive than ae [exp. precision → factor 19] ∼ mNP ⇒ me ∼ Some theoretical comments • Anomalous magnetic moment is finite and calculable Corresponds to effective interaction Lagrangian of mass dimension 5:

AMM e`a` ¯ µν eff = ψ(x)σ ψ(x)Fµν (x) L − 4m`

a` = F2(0) can be calculated unambiguously in renormalizable QFT, since there is no counterterm to absorb potential ultraviolet divergence. • Anomalous magnetic moments are dimensionless To lowest order in perturbation theory in (QED): α = ae = a = [Schwinger ’47/’48] µ 2π

• Loops with different masses ae = aµ ⇒ 6 - Internal large masses decouple (not always !): " !# 1  m 2 m4 m  α 2 = e + O e ln µ 4 45 mµ mµ me π - Internal small masses give rise to large log’s of mass ratios:

 1 m 25  m   α 2 = ln µ − + O e 3 me 36 mµ π Electron g 2 − Electron g 2: Theory −

Main contribution in Standard Model (SM) from mass-independent Feynman diagrams in QED with electrons in internal lines (perturbative series in α):

5 n SM X  α  a = cn e π n=1 12 +2.7478(2) 10− [Loops in QED with µ, τ] × 12 +0.0297(5) 10− [weak interactions] × 12 +1.682(20) 10− [strong interactions / hadrons] ×

The numbers are based on the paper by Aoyama et al. 2012. QED: mass-independent contributions to ae • α: 1-loop, 1 Feynman diagram; Schwinger ’47/’48: 1 c1 = 2 • α2: 2-loops, 7 Feynman diagrams; Petermann ’57, Sommerfield ’57: 197 π2 π2 3 c2 = 144 + 12 − 2 ln 2 + 4 ζ(3) = −0.32847896557919378 ...

• α3: 3-loops, 72 Feynman diagrams; . . . , Laporta, Remiddi ’96:

28259 17101 2 298 2 139 239 4 c3 = + π − π ln 2 + ζ(3) − π 5184 810 9 18 2160     83 2 215 100 1 1 4 1 2 2 + π ζ(3) − ζ(5) + Li4 + ln 2 − π ln 2 72 24 3 2 24 24 =1 .181241456587 ... • α4: 4-loops, 891 Feynman diagrams; Kinoshita et al. ’99, . . . , Aoyama et al. ’08; ’12:

c4 = −1.9106(20) (numerical evaluation)

• α5: 5-loops, 12672 Feynman diagrams; Aoyama et al. ’05, . . . , ’12:

c5 = 9.16(58) (numerical evaluation)

Replaces earlier rough estimate c5 = 0.0 4.6. ± Result removes biggest theoretical uncertainty in ae ! Mass-independent 2-loop Feynman diagrams in ae

1) 2) 3)

4) 5) 6)

γ

µ γµ γ e τ 7) 8)7) 9) Mass-independent 3-loop Feynman diagrams in ae

7) 8) 1) 2) 3) 4) 5) 6)

9) 10) 11) 12) 13) 14) 15) 16)

17) 18) 19) 20) 21) 22) 23) 24)

25) 26) 27) 28) 29) 30) 31) 32)

33) 34) 35) 36) 37) 38) 39) 40)

41) 42) 43) 44) 45) 46) 47) 48)

49) 50) 51) 52) 53) 54) 55) 56)

57) 58) 59) 60) 61) 62) 63) 64)

65) 66) 67) 68) 69) 70) 71) 72) Electron g 2: Experiment (a) Latesttrap experiment: −cavity Hanneke, electron Fogwell,top Gabrielse,endcap 2008 n = 2 electrode ν δ quartz spacer c - 5 /2 compensation νa n = 1 electrode n = 2 fc = νc - 3δ/2 nickel rings ring electrode νc - 3δ/2 n = 1 n = 0 0.5 cm compensation νa = gνc / 2 - νc ν - δ/2 bottom endcap electrode c electrode field emission n = 0 point ms = -1/2 ms = 1/2 microwaveCylindrical inlet Penning trap for single electron (1-electron quantum cyclotron) Cyclotron and spin precession levels of electron in Penning trap Source: Hanneke et al. Source: Hanneke et al. g ν ν¯ − ν¯2/(2f¯ ) ∆g e = s ' 1 + a z c + cav ¯ 2 ¯ 2 νc fc + 3δ/2 +ν ¯z /(2fc ) 2

νs = spin precession frequency; νc , ν¯c = cyclotron frequency: free electron, electron in 2 9 Penning trap; δ/νc = hνc /(me c ) ≈ 10− = relativistic correction 4 quantities are measured precisely in experiment: ¯ 3 g fc =ν ¯c − 2 δ ≈ 149 GHz;¯ νa = 2 νc − ν¯c ≈ 173 MHz; ν¯z ≈ 200 MHz = oscillation frequency in axial direction; ∆gcav = corrections due to oscillation modes in cavity

aexp = 0.00115965218073(28) [0.24 ppb 1 part in 4 billions] ⇒ e ≈ [Kusch & Foley, 1947/48: 4% precision]

Precision in ge /2 even 0.28 ppt 1 part in 4 trillions ! ≈ Determination of fine-structure constant α from g 2 of electron − • Recent measurement of α via recoil-velocity of Rubidium atoms in atom interferometer (Bouchendira et al. 2011): 1 α− (Rb) = 137.035 999 037(91) [0.66ppb] This leads to (Aoyama et al. 2012):

SM 12 ae (Rb) = 1 159 652 181.82 (6) (4) (2) (78) [78] 10− [0.67ppb] |{z} |{z} |{z} |{z} × c4 c5 had α(Rb)

exp SM 12 a a (Rb) = 1.09(0.83) 10− [Error from α(Rb) dominates !] ⇒ e − e − × Test of QED ! → exp • Use ae to determine α from series expansion in QED (contributions from weak and strong interactions under control !). Assume: Standard Model “correct”, no New Physics (Aoyama et al. 2012):

1 α− (ae ) = 137.035 999 1657 (68) (46) (24) (331) [342] [0.25ppb] |{z} |{z} |{z} | {z } had+EW exp c4 c5 ae The uncertainty from theory has been improved by a factor 4.5 by Aoyama exp et al. 2012, the experimental uncertainty in ae is now the limiting factor. • Today the most precise determination of the fine-structure constant α, a fundamental parameter of the Standard Model. Muon g 2 − The Brookhaven Muon g 2 Experiment − The first measurements of the anomalous magnetic moment of the muon were exp performed in 1960 at CERN, aµ = 0.00113(14) (Garwin et al.) [12% precision] and exp improved until 1979: aµ = 0.0011659240(85) [7 ppm] (Bailey et al.) In 1997, a new experiment started at the Brookhaven National Laboratory (BNL):

µ ⇒ ⇒ ⇒ Storage

Ring ⇒ ⇒

eB

ωa = aµ

mµ ⇒

spin ⇒ ⇒ ⇒ momentum

actual precession 2 Source: BNL Muon g − 2 homepage ×

Angular frequencies for cyclotron precession ωc and spin precession ωs : eB eB eB eB ωc = ,ω s = + aµ ,ω a = aµ mµ γ mµ γ mµ mµ p γ = 1/ 1 − (v/c)2. With an electric field to focus the muon beam one gets: e   1   = a B~ − a − v × E~ ~ωa µ µ 2 ~ mµ γ − 1 p Term with E~ drops out, if γ = 1 + 1/aµ = 29.3: “magic γ” → pµ = 3.094 GeV/c The Brookhaven Muon g 2 Experiment: storage ring −

Source: BNL Muon g 2 homepage − The Brookhaven Muon g 2 Experiment: determination of a − µ

 −t  Histogram with 3.6 billion N(t) = N0(E) exp γτµ decays of µ−: × [1 + A(E) sin(ωat + φ(E))]

10 Exponential decay with mean lifetime: τµ,lab = γτµ = 64.378µs 1 (in lab system). Oscillations due to angular frequency -1 10 ωa = aµeB/mµ. Million Events per 149.2ns

-2 10 R ωa µµ aµ = where R = and λ = λ − R ωp µp -3 10 0 20 40 60 80 100 Brookhaven experiment measures ωa and Time modulo 100µs [µs] ωp (spin precession frequency for proton). Bennett et al. 2006 λ from hyperfine splitting of muonium + (µ e−) (external input). Milestones in measurements of aµ Authors Lab Muon Anomaly Garwin et al. ’60 CERN 0.001 13(14) Charpak et al. ’61 CERN 0.001 145(22) Charpak et al. ’62 CERN 0.001 162(5) Farley et al. ’66 CERN 0.001 165(3) Bailey et al. ’68 CERN 0.001 166 16(31) Bailey et al. ’79 CERN 0.001 165 923 0(84) Brown et al. ’00 BNL 0.001 165 919 1(59) (µ+) Brown et al. ’01 BNL 0.001 165 920 2(14)(6) (µ+) Bennett et al. ’02 BNL 0.001 165 920 4(7)(5) (µ+) Bennett et al. ’04 BNL 0.001 165 921 4(8)(3) (µ−) World average experimental value (dominated by g 2 Collaboration at BNL, − Bennett et al. ’06 + CODATA 2008 value for λ = µµ/µp): exp 11 aµ = (116 592 089 63) 10− [0.5ppm] ± × 11 Goal of new planned g 2 experiments: δaµ = 16 10− E989: partly recycled− from BNL: moved ring× magnet ! (http://muon-g-2.fnal.gov/bigmove/ ) First beam in 2017, should reach this precision by 2020 (?). J-PARC E24: completely new concept with low-energy 11 , not magic γ. Aims in Phase 1 for about δaµ = 45 10− by 2019. × Theory needs to match this precision ! For comparison: Electron (stable !) (Hanneke et al. ’08): exp 12 a = (1 159 652 180.73 0.28) 10− [0.24ppb] e ± × Muon g 2: Theory − In Standard Model (SM):

SM QED weak had aµ = aµ + aµ + aµ

In contrast to ae , here now the contributions from weak and strong interactions 2 (hadrons) are relevant, since aµ (mµ/M) . ∼ QED contributions • Diagrams with internal electron loops are enhanced. • At 2-loops: vacuum polarization from electron loops enhanced by QED short-distance logarithm • At 3-loops: light-by-light scattering from electron loops enhanced by QED infrared logarithm [Aldins et al. ’69, ’70; Laporta, Remiddi ’93]

e   3 3 (3) 2 2 mµ  α   α  aµ = π ln + ... = 20.947 ... + ... lbyl 3 me π π

µ

• Loops with tau’s suppressed (decoupling) QED result up to 5 loops Include contributions from all leptons (Aoyama et al. ’12): 2 QED  α   α  aµ =0 .5 + 0.765 857 425 (17) × π |{z} × π mµ/me,τ

 α 3  α 4 + 24.050 509 96 (32) + 130.8796 (63) |{z} × π |{z} × π mµ/me,τ num. int.  α 5 + 753.29 (1.04) | {z } × π num. int.

11 = 116 584 718.853 (9) (19) (7) (29) [36] 10− |{z} |{z} |{z} |{z} × mµ/me,τ c4 c5 α(ae )

• Earlier evaluation of 5-loop contribution yielded c5 = 662(20) (Kinoshita, Nio ’06, numerical evaluation of 2958 diagrams, known or likely to be enhanced). New value is 4.5σ from this leading log estimate and 20 times more precise. • Aoyama et al. ’12: What about the 6-loop term ? Leading contribution from light-by-light scattering with electron loop and insertions of vacuum-polarization QED 11 loops of electrons into each photon line ⇒ aµ (6-loops) ∼ 0.1 × 10− Contributions from weak interaction Numbers from recent reanalysis by Gnendiger et al. ’13. 1-loop contributions [Jackiw + Weinberg, 1972; . . . ]: a) b) c) γ

W W µ νµ √ Z H 2G m2 weak, (1) µ µ 10 2 2 11 a (W ) = + O(m /M ) = 388.70 × 10− µ 16π2 3 µ W √ 2G m2 2 2 weak, (1) µ µ (−1 + 4sW) − 5 2 2 11 a (Z) = + O(m /M ) = −193.89 × 10− µ 16π2 3 µ Z weak, (1) 14 Contribution from Higgs negligible: aµ (H) ≤ 5 × 10− for mH ≥ 114 GeV. weak, (1) 11 a = (194.80 0.01) 10− µ ± × 2-loop contributions (1678 diagrams) [Czarnecki et al. ’95, ’96; . . . ]:

weak, (2) 11 2 α MZ aµ = ( 41.2 1.0) 10− , large since GF mµ ln − ± × ∼ π mµ Total weak contribution: weak 11 a = (153.6 1.0) 10− µ ± × Under control ! With knowledge of MH = 125.6 ± 1.5 GeV, uncertainty now mostly 11 hadronic ±1.0 × 10− (Peris et al. ’95; Knecht et al. ’02; Czarnecki et al. ’03, ’06). 11 3-loop effects via RG: ±0.20 × 10− (Degrassi, Giudice ’98; Czarnecki et al. ’03). Hadronic contributions to g 2 − The strong interactions (Quantum Chromodynamics) • Strong interactions: quantum chromodynamics (QCD) with quarks and gluons • Observed particles in Nature: Hadrons 1 Mesons (quark + antiquark: qq¯): π, K, η, ρ, . . . 2 Baryons (3 quarks: qqq): p, n, Λ, Σ, ∆,... • Cannot describe hadrons in series expansion in strong coupling constant of QCD with αs (E = mproton) ≈ 0.5. Particularly true for light hadrons which consist of three lightest quarks u, d, s. Non-perturbative effects like “confinement” of quarks and gluons inside hadrons.

Source: NIKHEF Source: NIC J¨ulich • Possible approaches to QCD at low energies: 1 Lattice QCD: limited applications, often still limited precision 2 Effective quantum field theories with hadrons (chiral perturbation theory): limited validity 3 Simplifying hadronic models: model uncertainties not controllable 4 Dispersion relations: extend validity of EFT’s, reduce model dependence, often not all the needed input data available Hadronic contributions to the muon g 2 − Largest source of uncertainty in theoretical prediction of aµ ! Different types of contributions:

                               Z  γ     

(a) (b) (c) Light quark loop not well defined Hadronic “blob” → (a) Hadronic vacuum polarization (α2), (α3), (α4) O O O (b) Hadronic light-by-light scattering (α3), (α4) O O 2 (c) 2-loop electroweak contributions (αGF m ) O µ

2-Loop EW e u,d Small hadronic uncertainty from triangle diagrams. Anomaly cancellation within each generation ! γ Z γ Z µ µ Cannot separate leptons and quarks ! Hadronic vacuum polarization (HVP) Hadronic vacuum polarization

HVP aµ =

µ µ Optical theorem (from unitarity; conservation of probability) for hadronic contribution → dispersion relation: 2 Im σ(e+e γ hadrons) ∼ ∼ − → ∗ →

Z + 1  α 2 ∞ ds σ(e e− γ∗ hadrons) aHVP = K(s) R(s), R(s)= → → µ 3 π s σ(e+e γ µ+µ ) 0 − → ∗ → − [Bouchiat, Michel ’61; Durand ’62; Brodsky, de Rafael ’68; Gourdin, de Rafael ’69]

K(s) slowly varying, positive function ⇒ aHVP positive. Data for hadronic cross section √ µ σ at low center-of-mass energies s important due to factor1 /s: ∼ 70% from ππ [ρ(770)] channel, ∼ 90% from energy region below 1.8 GeV. e− Other method instead of energy scan: “Radiative return” γ at colliders with fixed center-of-mass energy (DAΦNE, B- Hadrons Factories, BEPC) [Binner et al. ’99; Czy˙zet al. ’00-’03] γ e+ Measured hadronic cross-section

2 Pion form factor Fπ(E) (ππ-channel) | |

3  2  2 1 4mπ 2 R(s) = 1 − |Fπ(s)| 4 s 2 2 (4mπ < s < 9mπ)

R-ratio:

Jegerlehner, AN ’09 Hadronic vacuum polarization: some recent evaluations

HVP 11 Authors Contribution to aµ × 10 + Jegerlehner ’08; Jegerlehner, AN ’09 (e e−) 6903.0 ± 52.6 + Davier et al. ’09 (e e−) [+ τ] 6955 ± 41 [7053 ± 45] + Teubner et al. ’09 (e e−) 6894 ± 40 + Davier et al. ’11, ’14 (e e−) [+ τ] 6923 ± 42[7030 ± 44] + Jegerlehner, Szafron ’11 (e e−) [+ τ] 6907.5 ± 47.2[6909 .6 ± 46.5] + Hagiwara et al. ’11 (e e−) 6949.1 ± 42.7 + Benayoun at al. ’15 (e e− + τ: BHLS improved) 6818.6 ± 32.0 + Jegerlehner ’15 (e e−) [+ τ] 6885.7 ± 42.8[6889 .1 ± 35.2] • Precision: < 1%. Non-trivial because of radiative corrections (radiated photons). HVP • Even if values for aµ after integration agree quite well, the systematic differences of a few % in the shape of the spectral functions from different experiments (BABAR, BES III, CMD-2, KLOE, SND) indicate that we do not yet have a complete understanding. • Use of τ data: additional sources of isospin violation ? Ghozzi, Jegerlehner ’04; Benayoun et al. ’08, ’09; Wolfe, Maltman ’09; Jegerlehner, Szafron ’11 (ρ − γ-mixing), also included Jegerlehner ’15 and in BHLS-approach by Benayoun et al. ’15 (additional BHLS model uncertainty can lead to maximal +15 11 shift in central value of 27 × 10− ). − • Lattice QCD: Various groups are working on it, precision at level of 5-10%, not yet competitive with phenomenological evaluations. Hadronic light-by-light scattering (HLbL) Hadronic light-by-light scattering in the muon g 2 − QED: light-by-light scattering at higher orders in perturbation series via lepton-loop:

γ γ γ In muon g − 2: e ⇒ γ e γ µ Hadronic light-by-light scattering in muon g − 2 from strong interactions (QCD):

γ π+ 0 π ,η,η′ had.LxL aµ = = + ... + ...

µ γ Coupling of photons to hadrons, e.g. π0, via form factor: π0 γ

View before 2014: in contrast to HVP, no direct relation to experimental data → size and even sign of contribution to aµ unknown ! Approach: use hadronic model at low energies with exchanges and loops of resonances and some (dressed) “quark-loop” at high energies. Problems: Four-point function depends on several invariant momenta ⇒ distinction between low and high energies not as easy as for two-point function in HVP. 2 2 Mixed regions: one loop momentum Q1 large, the other Q2 small and vice versa. HLbL in muon g 2 − • Only model calculations so far: large uncertainties, difficult to control. • Frequently used estimates:

HLbL 11 a = (105 26) 10− (Prades, de Rafael, Vainshtein ’09) µ ± × HLbL 11 a = (116 40) 10− (AN ’09; Jegerlehner, AN ’09) µ ± × Based almost on same input: calculations by various groups using different models for individual contributions. Error estimates are mostly guesses ! • Need much better understanding of complicated hadronic dynamics to get 11 11 reliable error estimate of 20 10− (δaµ(future exp) = 16 10− ). ± × × • Recent new proposal: Colangelo et al. ’14, ’15; Pauk, Vanderhaeghen ’14: use dispersion relations (DR) to connect contribution to HLbL from presumably numerically dominant light pseudoscalars to in principle measurable form factors and cross-sections: 0 γ∗γ∗ → π , η, η0 + 0 0 γ∗γ∗ → π π−, π π Could connect HLbL uncertainty to exp. measurement errors, like HVP. • Maybe in future: HLbL from Lattice QCD. First steps: Blum et al. ’05, . . . , ’14, ’15. Work ongoing by Mainz group: Green et al. ’15. Approach to HLbL in muon g 2 up to now −

Classification of de Rafael ’94 2 Chiral counting p (from Chiral Perturbation Theory (ChPT)) and large-NC counting 2 as guideline to classify contributions (all higher orders in p and NC contribute):

  k +     π , Exchange of   0 Q  k = p’ − p  π , η, η          ρ  other reso- ρ         = + + nances +

µ−(p’) µ−(p) (f0, a1, f2 ...) Chiral counting: p4 p6 p8 p8

NC -counting:1 NC NC NC pion-loop pseudoscalar exchanges quark-loop (dressed) (dressed)

Relevant scales in HLbL (hVVVV i with off-shell photons !): 0 − 2 GeV  mµ ! Constrain models using experimental data (processes of hadrons with photons: decays, form factors, scattering) and theory (ChPT at low energies; short-distance constraints from pQCD / OPE at high momenta).

General analysis of four-point function Πµνρσ(q1, q2, q3) relevant for g − 2: Bijnens et al. ’96; Bijnens (Talk at g − 2 Workshop, Mainz, ’14); Eichmann et al. ’14, ’15; Colangelo et al. ’15. HLbL scattering: summary of selected results

  k +     π , Exchange of   0 Q  k = p’ − p  π , η, η          ρ  other reso- ρ         = + + nances +

µ−(p’) µ−(p) (f0, a1, f2 ...) Chiral counting: p4 p6 p8 p8

NC -counting:1 NC NC NC

11 Contribution to aµ × 10 :

BPP: +83 (32) -19 (13) +85 (13) -4 (3) [f0, a1] +21 (3) HKS: +90 (15) -5 (8) +83 (6) +1.7 (1.7) [a1] +10 (11) KN: +80 (40) +83 (12) MV: +136 (25) 0 (10) +114 (10) +22 (5) [a1] 0 2007: +110 (40) PdRV:+105 (26) -19 (19) +114 (13) +8 (12) [f0, a1] +2.3 [c-quark] N,JN: +116 (40) -19 (13) +99 (16) +15 (7) [f0, a1] +21 (3) ud.: -45 ud.: +∞ ud.: +60 ud. = undressed, i.e. point vertices without form factors BPP = Bijnens, Pallante, Prades ’96, ’02; HKS = Hayakawa, Kinoshita, Sanda ’96, ’98, ’02; KN = Knecht, AN ’02; MV = Melnikov, Vainshtein ’04; 2007 = Bijnens, Prades; Miller, de Rafael, Roberts; PdRV = Prades, de Rafael, Vainshtein ’09 (compilation; “Glasgow consensus”); N,JN = AN ’09; Jegerlehner, AN ’09 (compilation)

Pseudoscalars: numerically dominant contribution (according to most models !).

−11 Recall (in units of 10 ): δaµ(HVP) ≈ 45; δaµ(exp [BNL]) = 63; δaµ(future exp) = 16 Data-driven approach to HLbL using dispersion relations (DR) Strategy: Split contributions to HLbL into two parts: I: Data-driven evaluation using DR (hopefully numerically dominant): 0 (1) π , η, η0 poles (2) ππ intermediate state II: Model dependent evaluation (hopefully numerically subdominant): (1) Axial vectors (3π-intermediate state), . . . (2) Quark-loop, matching with pQCD Error goals: Part I: 10% precision (data-driven), Part II: 30% precision. HLbL 11 To achieve overall error of about 20% (δaµ = 20 × 10− ). Colangelo et al. ’14; ’15: Classify intermediate states in four-point function, then project onto g − 2.

π0 FsQED ππ Πµνλσ = Πµνλσ + Πµνλσ + Πµνλσ + ··· π0 Πµνλσ = pion pole (similarly for η, η0) FsQED V Πµνλσ = scalar QED with vertices dressed by pion vector form factor Fπ ππ Πµνλσ = remaining ππ contribution 2 Pauk, Vanderhaeghen ’14: Write DR directly for Pauli form factor F2(k ). HLbL;P 0 0 aµ , P = π , η, η : impact of precision of form factor measurements AN: work in preparation In Jegerlehner, AN ’09, a 3-dimensional integral representation for the pseudoscalar-pole contribution was derived. Schematically: Z Z Z 1 HLbL;P ∞ ∞ X aµ = dQ1 dQ2 dτ wi (Q1, Q2, τ) fi (Q1, Q2, τ) 0 0 1 − i with universal weight functions wi (for Euclidean (space-like) momenta: Q1 · Q2 = |Q1||Q2|τ, τ = cos θ). Dependence on form factors resides in the fi .

Weight functions wi : • Relevant momentum regions below 0 1 GeV for π , below 1.5 GeV for η, η0. • Analysis of current and future 4 0.35 3.5 0.3 3 measurement precision of = 0) = 0) 0.25 2.5 , τ , τ 0.2 2 2 2 2 ∗ ∗ ,Q ,Q 0.15 single-virtual F (−Q 0) and 1 1.5 1 Pγ γ , Q Q ( 1 ( 0.1 1 2 w 2 w 2 0.5 0.05 1.5 1.5 double-virtual transition form factor 0 0 0 1 0 1 0.5 Q2 [GeV] 0.5 Q2 [GeV] 2 2 1 0.5 1 0.5 ∗ ∗ 1.5 1.5 F (−Q , −Q ), based on Monte Q1 [GeV] 0 Q1 [GeV] 0 Pγ γ 1 2 2 2 Carlo study for BES III by Denig, Redmer, Wasser. • Data-driven precision for HLbL 0.6 0.25

0.5 0.2 = 0) 0.4 = 0) pseudoscalar-pole contribution that , τ , τ 0.15 2 0.3 2 ,Q ,Q

1 1 0.1 0.2 could be achieved in a few years: Q Q ( ( 1 1 0.05 w 0.1 2 w 2 1.5 1.5 0 0 0 1 0 1 0 0 0.5 Q2 [GeV] 0.5 Q2 [GeV] HLbL;π HLbL;π 1 0.5 1 0.5 1.5 1.5 δa /a = 14% Q1 [GeV] 2 0 Q1 [GeV] 2 0 µ µ HLbL;η HLbL;η 0 ◦ δaµ /aµ = 23% Top: weight functions w1,2(Q1, Q2, τ) for π with θ = 90 (τ = 0). 0 0 Bottom: weight functions w (Q , Q , τ) for η (left) and η0 (right). HLbL;η HLbL;η 1 1 2 δaµ /aµ = 15% Muon g 2: current status − 11 Contribution aµ 10 Reference × QED (leptons) 116 584 718.853 0.036 Aoyama et al. ’12 ± Electroweak 153.6 1.0 Gnendiger et al. ’13 ± HVP: LO 6907.5 47.2 Jegerlehner, Szafron ’11 ± NLO -100.3 2.2 Jegerlehner, Szafron ’11 ± NNLO 12.4 0.1 Kurz et al. ’14 ± HLbL 116 40 Jegerlehner, AN ’09 ± NLO 3 2 Colangelo et al. ’14 ± Theory (SM) 116 591 811 62 ± Experiment 116 592 089 63 Bennett et al. ’06 ± Experiment - Theory 278 88 3.1 σ ± HVP: Hadronic vacuum polarization HLbL: Hadronic light-by-light scattering HLbL 11 Other estimate: a = (105 26) 10− (Prades, de Rafael, Vainshtein ’09). µ ± × Discrepancy a sign of New Physics ? Hadronic uncertainties need to be better controlled in order to fully profit from 11 future g 2 experiments at Fermilab and J-PARC with δaµ = 16 10− . Way forward− for HVP seems clear: more precise measurements for× + σ(e e− hadrons). Not so obvious how to improve HLbL. → Muon g 2: other recent evaluations −

HMNT (06) JN (09) 3.3 σ Davier et al, τ (10) 2.4 σ Davier et al, e+e– (10) 3.6 σ JS (11) 3.3 σ HLMNT (10) HLMNT (11) 3.3 σ experiment BNL BNL (new from shift in λ)

170 180 190 200 210 a 10 µ × 10 – 11659000 10 Source: Hagiwara et al. ’11. Note units of 10− !

exp SM 11 Aoyama et al. ’12: aµ − aµ = (249 ± 87) × 10− [2.9 σ] exp SM 11 Benayoun et al. ’15: aµ − aµ = (376.8 ± 75.3) × 10− [5.0 σ] exp SM 11 Jegerlehner ’15: aµ − aµ = (310 ± 82) × 10− [3.8 σ] New Physics contributions to the muon g 2 − Tests of the Standard Model and search for New Physics

• Standard Model (SM) of very successful in precise description of a huge amount of experimental data, with a few exceptions (3 4 standard deviations). − • Some experimental facts (neutrino masses, baryon asymmetry in the universe, dark matter) and some theoretical arguments, which point to New Physics beyond the Standard Model.

• There are several indications that new particles (forces) should show up in the mass range 100 GeV 1 TeV. − Tests of the Standard Model and search for New Physics (continued) Search for New Physics with two complementary approaches:

1 High Energy Physics: e.g. Large Hadron Collider (LHC) at CERN + p q¯ µ Direct production of new particles e.g. heavy Z 0 resonance peak in invariant ⇒ + mass distribution of µ µ− at M 0 . q Z′ Z p µ−

2 Precision physics: e.g. anomalous magnetic moments ae , aµ Indirect effects of virtual particles in quantum corrections γ Deviations from precise predictions in SM ⇒  2 m` For MZ 0  m` : a` ∼ MZ 0 Note: there are also non-decoupling contributions of heavy New Physics ! Another example: new light vector meson (“dark Z′ µ µ photon”) with Mγ0 ∼ (10 − 100) MeV.

ae , aµ allow to exclude some models of New Physics or to constrain their parameter space. New Physics contributions to the muon g 2 − Define: exp SM 11 ∆aµ = a a = (290 90) 10− (Jegerlehner, AN ’09) µ − µ ± × Absolute size of discrepancy is actually unexpectedly large, compared to weak contribution (although there is some cancellation there):

weak weak, (1) weak, (1) weak, (2) aµ = aµ (W ) + aµ (Z) + aµ 11 = (389 194 41) 10− − 11− × = 154 10− × Assume that New Physics contribution with MNP mµ decouples: 2  NP mµ aµ = 2 C MNP α where naturally = π , like from a one-loop QED diagram, but with new C NP particles. Typical New Physics scales required to satisfy aµ = ∆aµ: 1 α ( α )2 C π π +0.4 +21 +1 MNP 2.0 0.3 TeV 100 13 GeV 5 1 GeV − − − Therefore, for New Physics model with particles in 250 300 GeV mass range − and electroweak-size couplings (α), we need some additional enhancement O factor, like large tan β in the MSSM, to explain the discrepancy ∆aµ. Generic 1-loop New Physics contributions to g 2 Possible New Physics contributions: − a)γ b) c) d)

+ + M M H− H X− X f f m m 0 0 µ M0[S,P] µ M0[V,A] X X Neutral boson exchange: a) scalar or pseudoscalar and b) vector or axialvector, flavor changing or not. New charged bosons: c) scalars or pseudoscalars, d) vector or axialvector.

Neutral boson of mass M0 coupling to muons with strength f from diagrams (a) and (b) [Lautrup, Peterman, de Rafael ’72; Leveille ’78]: f 2 m2 1 Z 1 Q(x) ∆aNP = µ L L = dx µ 2 2 , 2 2 4π M0 2 0 (1 − x) (1 − λ x) + (λ) x where Q(x) is a polynomial in x which depends on the type of coupling: 2 Scalar : QS = x (1 +  − x) 2 Pseudoscalar : QP = x (1 −  − x) 2 2 Vector : QV = 2x (1 − x)(x − 2 (1 − )) + λ (1 − ) QS 2 2 Axialvector : QA = 2x (1 − x)(x − 2 (1 + )) + λ (1 + ) QP with  = M/mµ and λ = mµ/M0. For mµ, M  M0 one obtains, for instance,   M M0 3 1 M=mµ M0 7 LS = ln − + = ln − mµ M 4 6 mµ 12 In accordance with Minimal Flavor Violation requirement it is more realistic to assume a flavor conserving neutral current M = mµ (second form). Expressions for all L for mµ, M  M0 and mµ  M ∼ M0 (important for SUSY), see Jegerlehner, AN ’09. Generic 1-loop New Physics contributions to g 2 (continued) −

(a) mµ = M  M0 (b) mµ  M0 = M Single particle one–loop induced NP effects from neutral bosons for f 2/(4π2) = 0.01. 2 2 2 (Note, a typical EW SM coupling would be e /(4π cos ΘW ) = 0.003). In panel (b), the large chiral combinations LS − LP and LV − LA are rescaled by the muon Yukawa coupling mµ/v in order to compensate for the huge pre-factor M/mµ. New physics contributions due to charged S,P,V and A modes from the diagrams (c) and (d): f 2 m2 1 Z 1 Q(x) ∆aNP = µ L L = dx µ 2 2 , 2 2 4π M0 2 0 (λ) (1 − x) (1 − − x) + x where again Q(x) is a polynomial in x which depends on the type of coupling: Scalar : QS = − x (1 − x)(x + ) Pseudoscalar : QP = − x (1 − x)(x − ) 2 2 2 Vector : QV = −2 x (1 + x − 2) + λ (1 − ) QS 2 2 2 Axialvector : QA = −2 x (1 + x + 2) + λ (1 + ) QP Expressions for L in the limits given earlier can be found in Jegerlehner, AN ’09. aµ: Supersymmetry a) b) Supersymmetry for large tan β, µ > 0:  2 χ˜ χ˜ µ˜ µ˜ SUSY 11 100 GeV a 123 10− tan β µ ν˜ χ˜0 ≈ × MSUSY (Czarnecki, Marciano, 2001) 11 Explains ∆aµ = 290 10− if MSUSY (93 414) GeV (2 < tan β < 40). × ≈ − In some regions of parameter space, large 2-loop contributions (2HDM): γ a) γ b) γ c) γ d)

τ,b W h,A γ H νµ Z µ h, A µ µ µ µ H Barr–Zee diagram (b) yields enhanced contribution, which can exceed the 1–loop 2 2 result. Enhancement factor mb/mµ compensates suppression by α/π 2 2 ((α/π) × (mb/mµ) ∼ 4 > 1).

all data 70 mŽ , mŽ > 1 TeV Μ1,2 ΝΜ 60 full result improved one-loop

D 50 10 - 10 @D @ 40 SUSY Μ

a 30

20

10

100 200 300 400 500 600 700 MLOSP@D@GeVD Constraint on large tan β SUSY contributions as Allowed values of MSSM contributions to aµ as function of mass of Lightest Observable SUSY Particle M (MSSM parameter scan function of MSUSY . LOSP with tan β = 50. Courtesy D. St¨ockinger,’09.) aµ: Supersymmetry (continued) Pre-LHC parameter space in constrained MSSM (CMSSM) (Courtesy K. Olive, ’09):

tan β = 10 , µ > 0 tan β = 40 , µ > 0 800 1000

700 mh = 114 GeV mh = 114 GeV

600 mχ± = 104 GeV

500 (GeV) (GeV) 0 0 mχ± = 104 GeV m m 400

300

200

100

0 0 100 200 300 400 500 600 700 800 900 1000 100 1000 1500

m1/2 (GeV) m1/2 (GeV)

(m0, m1/2) plane for µ > 0 for tan β = 10 (left panel) and tan β = 40 (right 11 panel) in CMSSM. aµ favored region at 2σ level [(290 180) 10− ] in pink. ± × Region excluded by b sγ in green. Allowed region by cosmological neutral dark matter constraint→ is black parabolic shaped region. Disallowed region

where mτ˜1 < mχ in brown. aµ and Supersymmetry after first LHC run

• Global SUSY fits (EW precision tests, flavor physics (aµ, b-decays), mh, Dark Matter relic density, Dark Matter searches, LHC searches) by several groups (with year of last publication): The BayesFITS group (2012); Fittino (2012); The MasterCode Project (2015); SuperBayeS (2011).

• Note: muon g 2 and SUSY searches at LHC only lead to tension in CMSSM or NUHM1− / NUHM2 (non-universal contributions to Higgs masses).

• LHC so far only sensitive to strongly interacting supersymmetric particles, like squarks and gluinos (ruled out below about 1 TeV).

• In general supersymmetric models (e.g. pMSSM10 = phenomenological MSSM with 10 soft SUSY-breaking parameters) with light neutralinos, charginos and sleptons, one can still explain muon g 2 discrepancy and evade bounds from LHC. − SUSY after first LHC run K.J. de Vries et al. (The MasterCode Project), arXiv:1504.03260 [hep-ph] Parameters in pMSSM10 (specified at low renormalization scale close to EWSB scale: mean scalar top mass M ≡ qm m ): SUSY ˜t1 ˜t2 3 gaugino masses : M1,2,3 2 squark masses : m = m 6= m q˜1 q˜2 q˜3 1 slepton mass : m `˜ 1 trilinear coupling : A Higgs mixing parameter : µ Pseudoscalar Higgs mass : MA Ratio of vevs : tan β

Constraint d.o.f. pMSSM10 CMSSM NUHM1 NUHM2 best fit low m low m low m low all ˜t1 q˜ g˜ LHC8 1 0.1 1.0 0.8 1.0 0.4 - - - Jets+E/T 1 - - - - - 2.0 0.0 0.5 Mh 1 0.0 0.0 0.2 0.2 0.0 0.1 0.1 0.4 MW 1 0.0 0.1 0.1 0.5 0.1 0.0 0.0 0.4 + − Bs,d → µ µ 1 0.2 0.0 0.0 0.2 0.0 0.5 0.3 0.4 BR(b → sγ) 1 0.1 0.0 0.2 0.1 0.1 0.5 0.0 0.0 BR(Bu → τντ ) 1 0.2 0.2 0.2 0.2 0.3 0.2 0.2 0.2 Other B physics 5 3.3 3.0 3.2 3.3 3.0 3.2 3.3 3.3 Ω h2 1 0.1 0.0 0.3 0.0 0.1 0.0 0.0 0.0 χ˜0 1 SI σp 1 0.0 0.1 0.0 0.1 0.5 0.0 0.0 0.0 A/H → τ+τ− 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Nuisance 3 0.0 0.1 0.0 0.1 0.8 0.1 0.0 0.1 (g − 2)µ 1 0.0 0.7 0.0 0.0 0.6 9.3 10.6 8.4 Z pole 13 16.3 17.0 17.1 16.8 16.4 16.8 16.5 16.7 Parameters 10 + 3 10 + 3 10 + 3 10 + 3 10 + 3 4 + 3 5 + 3 6 + 3 χ2/d.o.f. 20.5/18 22.2/18 22.0/18 22.3/18 22.2/18 32.8/24 31.1/23 30.3/22 χ2 probability 0.31 0.22 0.23 0.22 0.22 0.11 0.12 0.11

χ2 breakdowns at pMSSM10 best-fit and low-m , low-m and low-m points and in CMSSM, NUHM1, NUHM2. For each set of ˜t1 q˜ g˜ 2 constraints, the χ contribution and the number of non-zero contributions is provided. Nuisance parameters: mt , αs (MZ ), MZ . SUSY after first LHC run (continued)

K.J. de Vries et al. (The MasterCode Project), arXiv:1504.03260 [hep-ph]

Profile likelihoods for the SUSY contribution to (g − 2)µ

9 92 pMSSM10 8 NUHM2 NUHM1 90 7 CMSSM 6 88

2 5 2 χ χ ∆ 4 86 3

2 84 1 0 82 1 0 1 2 3 4 5 1 0 1 2 3 4 5 g 2 1e 9 g 2 1e 9 − − ∆ 2 ∆ 2 ³ ´ ³ ´

2 Left panel shows ∆χ contributions from (g − 2)µ to global likelihood functions of fits to CMSSM (blue dotted line), NUHM1 (blue dashed line), NUHM2 (blue solid line) and pMSSM10 (black solid line), as well as the assumed experimental likelihood function (solid red line). Right panel displays global χ2 function calculated without (dashed line) and with (solid line) the contribution of the electroweakly-interacting

sparticle searches implemented via LHC8EWK. ae , aµ: Dark photon

In some dark matter scenarios, there is a relatively light, but massive “dark

photon” Aµ0 that couples to the SM through mixing with the photon: ε µν mix = F F 0 L 2 µν A0 couples to ordinary charged particles with strength ε e. ⇒ µ · additional contribution of dark photon with mass mγ0 to the g 2 of a lepton⇒ (electron, muon) (Pospelov ’09): − α Z 1 2x(1 x)2 adark photon = 2 dx ` ε  − 2  2π m 0 0 (1 x)2 + γ x m2 − ` ( 1 for m` mγ0 α 2 = ε 2m2  ` for m m 0 2π × 3m2 ` γ γ0  3 For values ε (1 2) 10− and mγ0 (10 100) MeV, the dark photon ∼ − × ∼ −11 could explain the discrepancy ∆aµ = 290 10− . × Various searches for the dark photon have been performed, are under way or are planned at BABAR, Jefferson Lab, KLOE, MAMI and other experiments. For a recent overview, see: Dark Sectors and New, Light, Weakly-Coupled Particles (Snowmass 2013), Essig et al., arXiv:1311.0029 [hep-ph]. Status of dark photon searches

Essentially all of the parameter space in the (mγ0 , ε)-plane to explain the muon g 2 discrepancy has now been ruled out. −

From: F. Curciarello, FCCP15, Capri, September 2015 Conclusions

• Over many decades, the (anomalous) magnetic moments of the electron and the muon have played a crucial role in atomic and elementary particle physics.

• Experiment and Theory were thereby often going hand-in-hand, pushing each other to the limits.

• From ae , aµ we gained important insights into the structure of the fundamental interactions (quantum field theory).

• ae : Test of QED, precise determination of fine-structure constant α. aµ: Test of Standard Model, potential window to New Physics. Outlook • Current situation: exp SM 11 aµ − aµ = (278 ± 88) × 10− [3.1 σ] Sign of New Physics ? Hadronic effects ? Does g − 2 experiment measure something different from what is calculated in theory ? • Two new planned g − 2 experiments at Fermilab and J-PARC with goal of exp 11 δaµ = 16 × 10− (factor 4 improvement) • Theory needs to match this precision ! • Hadronic vacuum polarization + Ongoing and planned experiments on σ(e e− → hadrons) with a goal of HVP 11 δaµ = (20 − 25) × 10− (factor 2 improvement) • Hadronic light-by-light scattering HLbL 11 aµ = (105 ± 26) × 10− (Prades, de Rafael, Vainshtein ’09) HLbL 11 aµ = (116 ± 40) × 10− (AN ’09; Jegerlehner, AN ’09) Error estimates are mostly guesses ! Need a much better understanding of the 11 complicated hadronic dynamics to get reliable error estimate of ±20 × 10− . - Better theoretical models needed; more constraints from theory (ChPT, pQCD, OPE); close collaboration of theory and experiment to measure the relevant decays, form factors and cross-sections of hadrons with photons. 0 - Promising new data-driven approach using dispersion relations for π , η, η0 and ππ. Still needed: data for scattering of off-shell photons. - Future: Lattice QCD. And finally:

g-2 measuring the muon

In the 1950s the muon was still a complete enigma. Physicists could In 1959, six physicists joined forces to try to measure the muon’s The second g-2 experiment started in 1966 under the leadership magnetic moment using CERN’s first accelerator, the Synchrocyclotron. of Francis Farley and it achieved a precision 25 times higher than not yet say with certainty whether it In 1961, the team published the first direct measurement of the muon’s the previous one. This allowed phenomena predicted by the theory anomalous magnetic moment to a precision of 2% with respect to of quantum electrodynamics to be observed with a much greater the theoretical value. By 1962, this precision had been whittled down sensitivity — vacuum polarisation for instance, which is the momentary was simply a much heavier electron to just 0.4%. This was a great success since it validated the theory of appearance of ‘virtual’ electron and antielectron pairs with very short quantum electrodynamics. As predicted, the muon turned out to be lifetimes. The experiment also revealed a quantitative discrepancy a heavy electron. with the theory and thus prompted theorists to re-calculate their or whether it belonged to another predictions. species of particle. g-2 was set up to “g-2 is not an experiment: it is a way of life.” John Adams test quantum electrodynamics, which predicts, among other things, an

A third experiment, with a new technical approach, was launched in anomalously high value for the muon’s 1969, under the leadership of . The final results were published in 1979 and confirmed the theory to a precision of 0.0007%. The g-2 muon storage ring in 1974. They also allowed observation of a phenomenon contributing to the magnetic moment, namely the presence of ‘virtual hadrons’. After magnetic moment ‘g’, hence the name of 1984, the United States took up the mantle of investigating the muon’s anomalous magnetic moment, applying the finishing touches to an edifice built at CERN. «The science I have experienced has been all about the experiment. imagining and creating pioneering devices and observing entirely new phenomena, some of which have possibly never even been predicted by theory. That’s what invention is all about and it’s something The first g-2 experiment at the SC, sitting on the quite extraordinary. CERN was marvellous for two experiment’s 6 m long magnet. From right to left : reasons: it gave young people like me the opportunity A. Zichichi, Th. Muller, G. Charpak, to forge ahead in a new field and the chance to develop J.C. Sens. Standing : F. Farley. The sixth person in an international environment.» was R. Garwin. Francis Farley, Infinitely CERN, 2004

Source: CERN “g 2 is not an experiment: it is a way of life.” − John Adams (Head of the Proton Synchrotron at CERN (1954-61) and Director General of CERN (1960-1961))

This statement also applies to many theorists working on the g 2 ! − Backup slides Anomalous magnetic moment in quantum field theory Quantized spin 1/2 particle interacting with external, classical electromagnetic field 4 form factors in vertex function (momentum transfer k = p0 − p, not assuming parity or charge conjugation invariance)

γ(k)

   µ   ≡ ihp0, s0|j (0)|p, si  p p’ " µν µ 2 iσ kν 2 = (−ie)¯u(p0, s0) γ F1(k ) + F2(k ) | {z } 2m | {z } Dirac Pauli µν # 5 σ kν 2 5 2 µ µ 2 +γ F3(k ) + γ k γ − k/ k F4(k ) u(p, s) 2m

µ 2 k/ = γ kµ. Real form factors for spacelike k ≤ 0. Non-relativistic, static limit:

F1(0) = 1 (renormalization of charge e) e µ = (F1(0) + F2(0)) (magnetic moment) 2m a = F2(0) (anomalous magnetic moment) e d = − F3(0) (electric dipole moment, violates P and CP) 2m F4(0) = anapole moment (violates P) HLbL in muon g 2: summary of selected results − Some results for the various contributions to aHLbL 1011: µ ×

Contribution BPP HKS, HK KN MV BP, MdRR PdRV N, JN π0, η, η0 85±13 82.7±6.4 83±12 114±10 − 114±13 99 ± 16 axial vectors 2.5±1.0 1.7±1.7 − 22±5 − 15±10 22±5 scalars −6.8±2.0 − − − − −7±7 −7±2 π, K loops −19±13 −4.5±8.1 − − − −19±19 −19±13 π,K loops − − − 0±10 − − − +subl. NC quark loops 21±3 9.7±11.1 − − − 2.3 (c-quark) 21±3 Total 83±32 89.6±15.4 80±40 136±25 110±40 105 ± 26 116 ± 39

BPP = Bijnens, Pallante, Prades ’95, ’96, ’02; HKS = Hayakawa, Kinoshita, Sanda ’95, ’96; HK = Hayakawa, Kinoshita ’98, ’02; KN =

Knecht, AN ’02; MV = Melnikov, Vainshtein ’04; BP = Bijnens, Prades ’07; MdRR = Miller, de Rafael, Roberts ’07; PdRV = Prades, de

Rafael, Vainshtein ’09; N = AN ’09, JN = Jegerlehner, AN ’09

• Pseudoscalar-exchanges dominate numerically. Other contributions not negligible. Cancellation between π, K-loops and quark loops ! • PdRV: Analyzed results obtained by different groups with various models and suggested new estimates for some contributions (shifted central values, enlarged errors). Do not consider dressed light quark loops as separate contribution ! Assume it is already taken into account by using short-distance constraint of MV ’04 on pseudoscalar-pole contribution. Added all errors in quadrature ! • N, JN: New evaluation of pseudoscalar exchange contribution imposing new short-distance constraint on off-shell form factors. Took over most values from BPP, except axial vectors from MV. Added all errors linearly. HLbL: recent developments • Most calculations for neutral pion and all light pseudoscalars agree at level of 15%, but some are quite different:

HLbL;π0 11 a = (50 80) 10− µ − × HLbL;PS 11 a = (59 114) 10− µ − × • New estimates for axial vectors (Pauk, Vanderhaeghen ’14; Jegerlehner ’14): HLbL;axial 11 a = (6 8) 10− µ − × Substantially smaller than in MV ’04 ! • First estimate for tensor mesons (Pauk, Vanderhaeghen ’14): HLbL;tensor 11 a = 1 10− µ × • Open problem: Dressed pion-loop Potentially important effect from pion polarizability and a1 resonance (Engel, Patel, Ramsey-Musolf ’12; Engel ’13; Engel, Ramsey-Musolf ’13): HLbL;π loop 11 a − = (11 71) 10− µ − − × Maybe large negative contribution, in contrast to BPP ’96, HKS ’96. • Open problem: Dressed quark-loop Dyson-Schwinger equation approach (Fischer, Goecke, Williams ’11, ’13): HLbL;quark loop 11 a − = 107 10− (still incomplete !) µ × Large contribution, no damping seen, in contrast to BPP ’96, HKS ’96. Supersymmetry (pMSSM10) after first LHC run

K.J. de Vries et al. (The MasterCode Project), arXiv:1504.03260 [hep-ph]

Particle spectrum and dominant decay branching ratios at best-fit pMSSM10 point

4800 b˜2 t˜2 b˜1 ˜ 4000 t1 Mass / GeV

3200 q˜L q˜R g˜ 2400 0 H H A0 ± 1600

800 0 `˜R τ˜2 χ˜4 0 χ˜2± `˜L ν˜τ χ˜3 χ˜0 ν˜L τ˜1 2 χ˜ h0 0 1± 0 χ˜1 ± ± Note near-degeneracies betweenχ ˜0, χ˜0,χ ˜ , between sleptons, betweenχ ˜0, χ˜0,χ ˜ , betweenq ˜ ,q ˜ , between the heavy Higgs 1 2 1 3 4 2 L R bosons, and between stops and bottoms. The overall sparticle mass scales, in particular of colored sparticles, are poorly determined.

Summary of mass ranges predicted in the pMSSM10

4000

3500

3000

2500

2000

1500

1000

Particle Masses [GeV] 500

0 0 0 0 m 0 m 0 m 0 m 0 m m m˜ m˜ m m m m m˜ m˜ m˜ m˜ m Mh MH MA MH ± χ1 χ2 χ3 χ4 χ1± χ2± lL lR ˜τ1 ˜τ2 ˜qL ˜qR t1 t2 b1 b2 ˜g Light (darker) peach shaded bars indicate 95% (68%) CL intervals, blue horizontal lines mark values of masses at best-fit point. Supersymmetry (pMSSM10) after first LHC run (continued)

K.J. de Vries et al. (The MasterCode Project), arXiv:1504.03260 [hep-ph]

Particle spectra and dominant decay branching ratios at the four benchmark points

4000 4000 q˜L 3600 A0 q˜R g˜ H 3600 H0 ± 3200 Mass / GeV 3200 Mass / GeV A0 2800 2800 H H0 ± 2400 2400

2000 t˜2 2000 ˜ g˜ b2 b˜1 1600 1600 q˜L ˜ χ˜0 t1 4 χ˜ q˜R χ˜0 2± 1200 3 1200 t˜2 800 b˜ 2 800 χ˜0 ˜ τ˜2 b˜ 4 `L 1 ˜ τ˜ 0 χ˜2± 0 t˜1 `L 2 χ˜ 400 ν˜L ν˜τ χ˜ 3 2 χ˜± 400 `˜R ν˜τ ˜ τ˜1 χ˜0 1 0 0 `R 1 ν˜ τ˜ χ˜2 h L 1 0 χ˜± 0 h0 χ˜ 1 0 1

2000 4000 t˜2 b˜2 q˜L ˜ q˜R 3600 b1 1800 t˜1 g˜ Mass / GeV 3200 Mass / GeV 1600 A0 H t˜ H0 ± 1400 0 2 2800 H H ˜ A0 ± b˜2 q˜L b1 1200 t˜1 2400 q˜R

2000 1000 `˜ τ˜ 1600 800 R 2 `˜L ν˜τ g˜ ν˜L τ˜1 1200 600

800 400 χ˜0 0 `˜ 4 χ˜4 R τ˜2 0 χ˜2± 0 χ˜± ˜ χ˜ χ˜3 2 400 `L ν˜τ 3 200 χ˜0 0 0 ν˜L τ˜ 2 χ˜ h χ˜ χ˜1± 0 1 χ˜0 1± χ˜20 0 h 1 0 1

Upper left panel: the low-m pMSSM10 point, where stops and bottoms are relatively light. Upper right panel: similarly for low-m ˜t1 q˜ benchmark point, where all the squarks are relatively light. Lower left panel: similarly for low-mg˜ benchmark point. Lower right panel: 0 0 similarly for the point where all squarks and the gluino masses are < 2 TeV. Note in each case the near-degeneracies betweenχ ˜1, χ˜2, ± ± χ˜ , between the sleptons, betweenχ ˜0, χ˜0,χ ˜ , betweenq ˜ ,q ˜ , and between the heavy Higgs bosons. 1 3 4 2 L R