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

Z

hadrons

Units: 10-11

QED 116 584 719 (1) hep-ph/0402206 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 Total SM 116 591 858 (82) Experiment − SM Theory = 222 (102) (2.2σ deviation)

Muon g-2: new data

Brookhaven, January 2004: µ- measurement.

exp SM 11

222 102 10 2.2 a a

µ µ

σ

−

− = ± ⋅ →

exp SM 11

123 89 10 1.4 a a

µ µ

σ

(based on e+e-)

−

− = ± ⋅ →

(tau) from A. Vainshtein

QED contributions: muon vs. electron

m

e

g g

e

g g

m

Enhancement factors:

2 ln n e

m m

µ

π lnn

e

m m

µ

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

( )

( ) ( ) ( ) ( )

1 2 1 2

1 2 2 2 2 2

1 , 2 1 2

D a a D a a

J a a d k k k kp p d k k k k kp

µ µ

= + ∂ = ∂ +

∫ ∫

a1 a2 k p

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 2 11

5 24 2 195 10 G m

µ µ

π

−

⋅ ⋅

• ( -1 +2)

Higher-order electroweak effects

Most important: photonic corrections → large logs

2 2 2

ln 23% of one-loop

W

M G m m

µ µ µ

α π − ∼

Kukhto et al. AC, Krause, Marciano Heinemaier, Stockinger, Weiglein

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(mµ/MZ) 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

( )( ) ( )( )

2 2 2 T L

w q q F q q F q q F w q q q F

σ σ σ µν µ σν ν σµ ν σµ

− + − +

• ∼

Perturbative result:

2

1 2

L T

w w q = ∼

Anomaly:

2 L

q T q w q T

µν ν µν µ

≠ = ∼

Vainshtein’s non-renormalization theorem for wT V A

µ

ν

q

( )( ) ( )( )

2 2 2 T L

T w q q F q q F q q F w q q q F

σ σ σ µν µν µ σν ν σµ ν σµ

− + − +

• ∼

In the chiral limit, wT has no perturbative corrections Idea of the proof: ImT q q F q q F

σ σ µν µ σν ν σµ

+

• ∼

(symmetric)

( ) ( )

2 2

2

T L

w q w q =

Guidance from one-loop calculations!

wT,L in QCD (chiral limit)

2 2 2

2 2

L T

w w Q q Q = = ≡ −

Perturbative: Non-perturbative: Small Q2 Large Q2 ( )

1 1 1

2 2 2 4 2 2 2 2 6 8 2 2 2 2 2 2

2 2 0.7G 1 eV 1 1

L a T a a

w Q Q m m m m w O Q Q m m Q Q m Q m

π ρ π ρ ρ

⎛ ⎞ − − ⎛ ⎞ − + − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − + + ⎝ ⎠ ⎝ ⎠

(pion pole) (model for wT)

Contributions to g-2

2

2 2 2 2 2 2 2 Z L T Z Z m

m M a dQ w w M M Q

µ

µ µ

α π

∞

⎛ ⎞ ⎛ ⎞ ∆ + ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠

∫

∼

Asymptotics:

2

2 , 3 2

1

Q T L f f f f

w I N Q Q

→∞

⎯⎯⎯ →

∑

2 2 2 2 2 2

: diverges ln

L Z T Z Z

dQ w M dQ w M M Q

∞ ∞

+

∫ ∫

∼

theory inconsistent unless anomalies cancel

“Pure” hadronic contributions Recent progress Updated studies of g-2 using e+e- data Novosibirsk results tested by Daphne Shift of the light-by-light prediction

Davier, Eidelman, Höcker, Zhang Hagiwara, Martin, Nomura, Teubner See talks by Kühn, Leone, Shwartz Melnikov and Vainshtein

Vacuum polarization: τ decays vs. e+e–

From M. Davier, A. Hoecker

Vacuum polarization: e+e–

|Fπ|2

• CMD2

— KLOE

0.5 0.7 0.9 20 40 10 30 M GeV

ππ 2 2

( )

|Fπ|2

• CMD2

— KLOE

0.5 0.7 0.9 20 40 10 30 M GeV

ππ 2 2

( )

e+e– data have greatly improved New results from KLOE confirm Novosibirsk CMD2

From A. Denig

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 Nc

Quark box: pQCD asymptotics

Axial

+ → The same structure as in the EW-hadronic loops. Dominant contribution: π0 pole Crucial observation: 1/Q2 asymptotics → no formfactor in π*γ*γ if one of the photons soft.

( )

PS 11 PV 11

76.5 2 18 10 22 10 a a

µ µ − −

∆ + ⋅ ⋅ ∆ ⋅

What about terms subleading in Nc?

Example: pion box. It is chirally enhanced,

2 2

/ 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 perturbative QCD, at asymptotic momentum transfer. Large contribution of high virtualities.

From Melnikov and Vainshtein

Dominant in Nc→∞: pion pole Still room for improvement: subleading terms

α π

Pomeranchuk and Sakharov on g-2= 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 Measure 2 2 from NMR: from 4 / 2 Master formula: 2 / /

a p p a p p a p

B B g B e g m m e m g e

µ µ µ µ µ

ω µ ω µ ω ω µ µ ω ω − = = ≡ − = −

• Measured by E821

From muonium

Muonium spectrum determines µµ/µp

U +

e

_

µ

ν12 ν34 B

34 12

/

p

B

µ µ

ν ν µ µ µ − ⇒ ∼

Measured to relative 1.2•10-7 (like 15•10-11 in aµ) Will need improvement for the “next g-2”

Mu: also mµ/me and tests of QED