Muon g-2 : a theoretical review Tau04 Nara, September 2004 Andrzej - PowerPoint PPT Presentation

Muon g-2 : a theoretical review Tau04 Nara, September 2004 Andrzej Czarnecki University of Alberta

Outline QED: present and future ( T. Kinoshita ) Electroweak loops Hadronic effects * vacuum polarization ( J. Kühn, S. Eidelman, D. Leone, M. Davier, B. Schwarz, K. Hagiwara ) * light-by-light scattering Summary and outlook

Muon g-2 : Standard Model update Units: 10 -11 QED 116 584 719 (1) hep-ph/0402206 hadrons Hadronic LO 6 963 (72) hep-ph/0308213 NLO − 98 (1) hep-ph/0312250 LBL 120 (40) tentative, see hep-ph/0312226 Electroweak 154 (3) hep-ph/0212229 Z Total SM 116 591 858 (82) Experiment − SM Theory = 222 (102) (2.2 σ deviation)

Muon g-2 : new data Brookhaven, January 2004: µ - measurement. − − = ± ⋅ exp SM 11 222 102 10 a a µ µ → σ 2.2 (based on e+e-) − − = ± ⋅ exp SM 11 123 89 10 a a µ µ → σ 1.4 (tau) from A. Vainshtein

QED contributions: muon vs. electron m e g g m e g g Enhancement factors: m m 2 ln µ µ π ln n n m m e e Leading five-loop effects must be included!

QED contributions: problems at 4-loop order Traditional approach (T. Kinoshita and M. Nio (Nara)): numerical problems, digit deficiency New approach (various groups, in progress): Combine numerical and algebraic methods Reduce all integrals to a smaller basis Evaluate the primitive integrals numerically, with high accuracy.

Example: integration by parts a 1 k p a 2 1 ( ) ∫ = , D J a a d k ( ) ( ) 1 2 a a + 2 1 2 2 2 k k kp ∂ 1 ∫ = 0 D p d k k ( ) ( ) µ ∂ a a + 2 2 1 2 2 k k kp µ

New approach to the QED part Obstacles: Very large number of integrals Reduction to primitive integrals Evaluation of master integrals Recent progress: algorithmic reduction (Laporta, 2001)

Electroweak effects: pure and hadronic Small part of the total g-2: 154(3)×10 -11 2 5 G m µ µ ( -1 +2) ⋅ π 2 24 2 − ⋅ 11 � 195 10

Higher-order electroweak effects Most important: photonic corrections → large logs α 2 Kukhto et al. M − 2 ln ∼ 23% of one-loop W AC, Krause, Marciano G m µ µ π 2 Heinemaier, Stockinger, Weiglein m µ

Muon g-2 : hadronic loops Hadronic effects dominate theoretical uncertainty: Vacuum polarization Light-by-light scattering Electroweak triangle diagrams (numerically small)

Electroweak-hadronic effects Large logs ln(m µ /M Z ) appear in individual fermion contributions; But cancel in the sum for each generation – like anomalies. This cancellation between leptons and hardons was contoversial. AC, Marciano, Vainshtein vs. Knecht, Peris, Perrottet, de Rafael Useful illustration: similar techniques used in light-by-light

Structure of the triangle V A µ ν q external V ) ( ) ) ( ) ( ( � � � � σ σ σ − + − + 2 2 2 ∼ w q q F q q F q q F w q q q F µν µ σν ν σµ ν σµ T L Perturbative result: 1 = 2 ∼ w w 2 L T q Anomaly: µν ≠ 2 ∼ 0 q T q w ν L µν = 0 q T µ

Vainshtein’s non-renormalization theorem for w T V A µ ν q ) ( ) ) ( ) ( ( � � � � σ σ σ − + − + 2 2 2 ∼ T w q q F q q F q q F w q q q F µν µν µ σν ν σµ ν σµ T L In the chiral limit, w T has no perturbative corrections Idea of the proof: � � (symmetric) σ + σ Im T ∼ q q F q q F µν µ σν ν σµ ( ) ( ) Guidance from one-loop = 2 2 2 w q w q calculations! T L

w T,L in QCD (chiral limit) 2 Perturbative: = = ≡ − 2 2 2 w w Q q 2 L T Q Non-perturbative: Large Q 2 Small Q 2 2 2 (pion pole) w 2 2 L Q Q ( ) ⎛ ⎞ 4 2 − 2 − 2 2 ⎛ ⎞ 0.7G eV 1 1 1 m m m m π ρ π − + − ⎜ ⎟ a ⎜ ⎟ 1 w O ⎜ ⎟ − + + 2 6 8 2 2 2 2 2 2 T ⎝ ⎠ Q Q Q m m Q m Q m ⎝ ⎠ ρ ρ a a 1 1 (model for w T )

Contributions to g-2 2 ∞ 2 ⎛ ⎞ α 2 ⎛ ⎞ m M ∫ µ ∆ + 2 ∼ ⎜ Z ⎟ ⎜ ⎟ a dQ w w µ π + 2 2 2 L T ⎝ ⎠ ⎝ ⎠ M M Q 2 Z Z m µ Asymptotics: 1 ∑ 2 ⎯⎯⎯ →∞ → 2 Q w I N Q , 3 2 T L f f f Q f ∞ theory inconsistent ∫ 2 : diverges dQ w unless anomalies cancel L ∞ 2 M ∫ 2 2 ∼ ln Z dQ w M + 2 2 T Z M Q Z

“Pure” hadronic contributions Recent progress Updated studies of g-2 using e + e - data Davier, Eidelman, Höcker, Zhang Hagiwara, Martin, Nomura, Teubner Novosibirsk results tested by Daphne See talks by Kühn, Leone, Shwartz Shift of the light-by-light prediction Melnikov and Vainshtein

Vacuum polarization: τ decays vs. e + e – From M. Davier, A. Hoecker

Vacuum polarization: e + e – e + e – data have greatly |F π | 2 |F π | 2 improved — KLOE — KLOE 40 40 New results from KLOE • CMD2 • CMD2 confirm Novosibirsk CMD2 30 30 20 20 From A. Denig 10 10 0 0 0 0 0.5 0.5 0.7 0.7 0.9 0.9 2 2 2 2 M M ( ( GeV GeV ) ) ππ ππ

Hadronic contributions: outstanding problems How to reconcile e + e - and τ data? Can we improve the light-by-light prediction?

Light-by-light scattering Recent evaluations: Knecht, Nyffeler 80(40) Hayakawa, Kinoshita 90(15) Bijnens, Pallante, Prades 83(32) Melnikov, Vainshtein 136(25)

Effects enhanced by N c Quark box: pQCD asymptotics Axial + → The same structure as in the EW-hadronic loops. Dominant contribution: π 0 pole Crucial observation: 1/Q 2 asymptotics → no formfactor in π * γ * γ if one of the photons soft. ( ) ∆ + ⋅ ⋅ − PS 11 � 76.5 2 18 10 a µ ∆ ⋅ − PV 11 � 22 10 a µ

What about terms subleading in N c ? Example: pion box. 2 2 It is chirally enhanced, / m m µ π Numerical effect: small. Previous estimates: -4.5(8.5)×10 -11 HLS Hayakawa, Kinoshita, Sanda -19(5)×10 -11 VMD Bijnens, Pallante, Prades Melnikov & Vainshtein: 0±10×10 -11 (cancellations with higher orders in the chiral expansion)

Summary on the light-by-light scattering Matching of hadronic model with From Melnikov and Vainshtein perturbative QCD, at asymptotic momentum transfer. Large contribution of high virtualities. Dominant in N c →∞ : pion pole Still room for improvement: subleading terms

α Pomeranchuk and Sakharov on g-2= π If this is true, it’s exceptionally important; if it isn’t true, that, too, is exceptionally important. (Pomeranchuk after Sakharov’s talk, 1949) I felt like the messenger of the gods . (Sakharov)

How do we determine g-2? − 2 g e ω = Measure B a 2 m µ µ 2 B ω = from NMR: p B p � � e e µ ≡ from g µ 4 m m µ µ Measured by E821 ω ω − / 2 g = Master formula: a p µ µ − ω ω 2 / / µ p a p From muonium

Muonium spectrum determines µ µ /µ p U + µ ν 12 ν 34 _ e B ν − ν µ ⇒ µ µ ∼ / B µ µ 34 12 p Measured to relative 1.2•10 -7 (like 15•10 -11 in a µ ) Will need improvement for the “next g-2” Mu: also m µ /m e and tests of QED