Hadronic Vacuum Polarization Confronting Spectral Functions from e + e Annihilation and τ Decays: Consequences for the Muon Magnetic Moment γ Michel Davier ( ) LAL - Orsay γ γ Lepton Moments Cape Cod, June 9-12, 2003 hadrons ( ) work done in collaboration with S. Eidelman, A. Höcker and Z. Zhang [hep-ph/0208177, Eur Phys J C] µ
Outline The Muon Magnetic Anomaly... and Hadronic Vacuum Polarization a µ [had] α QED [had] e + e versus τ Data Isospin Breaking and Radiative Corrections Standard Model + Data Predictions for (g 2) µ
Q E D γ γ µ Magnetic Anomaly QED Prediction: Computed up to 4 th order [Kinoshita et al.] (5 th order estimated [Mohr, Taylor]) a QED µ ie Γ = eγ + a σ q 2m α a = = 0.001161 2π µ µ µν ν n Schwinger 1948 α 11614098.1+ 41321.8 = 10 = 1 π n + 3014.2 + 36.7 + 0.6 10 QED: Hadronic: Weak: SUSY...... or other new physics?
Why Do We Need to Know it so Precisely? Experimental Progress on Precision of (g 2) µ Aiming at: σ exp ( a ) 4 10 µ σ a 10 had ( aµ ) weak [SM] µ The fine structure constant at M Z is an important ingredient to EW precision fits
The Muonic (g 2) µ Contributions to the Standard Model (SM) Prediction: g 2 a = a + a + a QED had weak µ µ µ µ 2 µ The Situation 1995 Source QED Quarks and Hadrons Z, W exchange σ(a µ ) ~ 0.3 10 10 ~ (15 4) 10 10 ~ 0.4 10 10 Reference [Schwinger 48 & others] [Eidelman-Jegerlehner 95 & others] [Czarnecki et al. 95 & others] Dominant uncertainty from lowest order hadronic piece Cannot be calculated from QCD ( first principles ) bu we can use experiment (!) 2 had α Ks ( ) aµ = ds R s 2 3π s 4m 2 π ( ) h a d γ Dispersion relation had µ... γ γ
The R Data Situation (around 1995) Data density and quality unsatisfactory in some crucial energy regions
Improved Determination of the Hadronic 2 Contribution to (g( 2) µ and α(m Z ) Situation 1995 [Eidelman-Jegerlehner] and since... Eidelman-Jegerlehner hep-ph/9502298 (1995) Energy range [GeV] 2m π -1.8 Input 1995 Data Input after 1998 Data (e + e & τ) + QCD Davier-Hocker hep-ph/9805470 (1998) 1.8 J/ψ J/ψ - ϒ Data Data QCD Data + QCD ϒ -40 Data QCD Improvement in 3 Steps: 40 - QCD QCD Inclusion of precise τ data using SU(2) (CVC) Alemany-Davier-Hocker 97, Narison 01, Trocóniz-Ynduráin 01 Extended use of (dominantly) perturbative QCD Martin-Zeppenfeld 95, Davier-Hocker 97, Kühn-Steinhauser 98, Erler 98, + others More theoretical constraints from QCD sum rules Groote, Körner, Schilcher, Nasrallah 98, Davier-Hocker 98, Martin-Outhwaite- Ryskin 00, Cvetič-Lee-Schmidt 01, + others
A New Analysis of a µ had Motivation for new work: New high precision e + e results (0.6% sys. error) around ρ from CMD-2 (Novosibirsk) New τ results from ALEPH using full LEP1 statistics New R results from BES between 2 and 5 GeV New theoretical analysis of SU(2) breaking CMD-2 PL B527, 161 (2002) ALEPH CONF 2002-19 BES PRL 84 594 (2000); PRL 88, 101802 (2002) Cirigliano-Ecker-Neufeld hep-ph/0207310 Outline of the new analysis: Include all new Novisibirsk (CMD-2, SND) and ALEPH data Apply (revisited) SU(2)-breaking corrections to τ data Identify application/non-application of radiative corrections Recompute all exclusive, inclusive and QCD contributions to dispersion integral; revisit threshold contribution and resonances Results, comparisons, discussions... Davier-Eidelman-Hocker- Zhang hep-ph/0208177
The Conserved Vector Current SU(2) W: I =1 & V,A CVC: I =1 & V γ: I =0,1 & V τ W ν τ e + γ hadrons hadrons e Hadronic physics factorizes in Spectral Functions: fundamental ingredient relating long distance (resonances) to short distance description (QCD) Isospin symmetry (CVC) connects I=1 e + e cross section to vectorτ spectral functions: 2 ( I= 1) + + 4πα 0 σ ee ππ = υτ ππν τ s 0 BR τ π π ν 2 1 dn 0 τ m υ τ π π ν τ BR τ e ν N eν 1 s/ m 1 + s/ m 0 ππ τ 0 ds 2 2 2 ππ ( ) ( ) τ τ τ branching Fractions mass spectrum kinematic factor (PS)
τ Spectral Functions Hadronic τ decays are a clean probe of hadron dynamics experimentally in many ways complementary to e + e hadrons: τ e + e Excellent normalization (branching fractions) due to high statistics, large acceptance, small non-τ background Shape subject to bin-to-bin resolution corrections (unfolding) Good relative cross sections (correlated systematics, but large rad. corr.) Overall normalization subject to radiative corrections, systematic uncertainties from acceptance and luminosity V A ALEPH ZP C76, 15 (1997) ALEPH EPJ C4, 4 (1998)
QCD Results from τ Decays Evolution of α s (m τ ), measured using τ decays, to M Z using RGE (4-loop QCD β -function & 3-loop quark flavor matching) shows the excellent compatibility of τ result with EW fit: α s (M Z ) = 0.1202 ± 0.0027 (ALEPH 98, theory dominated) α s (M Z ) = 0.1183 ± 0.0027 (LEP 00, statistics dominated) The τ spectral function allows to directly measure the running of α s (s 0 ) within s 0 [~1...1.8 GeV]
τ π π 0 ν τ : Comparing ALEPH, CLEO, OPAL Shape comparison only. Both normalized to WA branching fraction (dominated by ALEPH). Good agreemen observed between ALEPH and CLEO ALEPH more precise at low s CLEO better at high s
SU(2) Breaking Electromagnetism does not respect isospin and hence we have to consider isospin breaking when dealing with an experimental precision of 0.5% Corrections for SU(2) breaking applied to τ data for dominant π π + contrib.: Electroweak radiative corrections: dominant contribution from short distance correction S EW to effective 4- fermion coupling (1 + 3α(m τ )/4π)(1+2 Q )log(m Z /m τ ) subleading corrections calculated and small long distance radiative correction G EM (s) calculated [ add FSR to the bare cross section in order to obtain π π + (γ) ] Charged/neutral mass splitting: m π m π0 leads to phase space (cross sec.) and width (FF) corrections m ρ m ρ0 and ρ -ω mixing (EM ω π π + decay) corrected using FF model Electromagnetic decays, like: ρ ππγ, ρ π γ, ρ η γ, ρ l + l Marciano-Sirlin 88 Braaten-Li 90 Cirigliano-Ecker- Neufeld, hep-ph/0207310 Alemany-Davier-Hocker 97, Czyz-Kuhn 01 Quark mass difference m u m d generating second class currents (negligible)
Mass Dependence of SU(2) Breaking Multiplicative SU(2) corrections applied to τ π π 0 ν τ spectral function: Only β 3 and EW short-distance corrections applied to 4π spectral functions
SU(2) Breaking Corrections for isospin violation applied to τ data Source π π + (simple) π π + (improved) a µ had (10 10 ) π π + 2π 0 2π 2π + Short distance radiative corrections to τ decays (S EW = 1.0233± 0.0006) [Marciano- Sirlin 88, Braaten-Li 90, new evaluation DEHZ 02] 10.3 ± 2.1 12.1 ± 0.3 0.36 ± 0.07 0.18 ± 0.04 Long distance corrections - 1.0 - - m π m π0 (β in cross section) 7.0 7.0 + 0.6 ± 0.6 0.4 ± 0.4 m π m π0 (β in ρ width) +4.2 +4.2 - - m ρ m ρ0 (± 1 MeV/c 2 ) + 0 ± 0.2 + 0 ± 2.0 - - ρ -ω mixing (exp. uncertainty) + 3.5 ± 0.6 + 3.5 ± 0.6 - - EM decay modes 1.4 ± 1.2 1.4 ± 1.2 - - Total correction 11.0 ± 2.5 13.8 ± 2.4 + 0.2 ± 0.6 0.6 ± 0.4
e + e Radiative Corrections Multiple radiative corrections are applied on measured e + e cross sections Situation often unclear: whether or not - and if - which corrections were applied Vacuum polarization (VP) in the photon propagator: leptonic VP in general corrected for hadronic VP correction not applied, but for CMD-2 (in principle: iterative proc.) Initial state radiation (ISR) corrected by experiments Final state radiation (FSR) [we need e + e hadrons (γ) in dispersion integral] mostly, experiments obtain bare cross section so that FSR has to be added by hand ; done for CMD-2, (supposedly) not done for others
Correct τ data for missing ρ -ω mixing (taken from BW fit) and all other SU(2) breaking sources Comparing e + e π + π and τ π π 0 ν τ Remarkable agreement But: is it good enough?...
Comparing e + e π + π and τ π π 0 ν τ Relative difference between τ and e + e data zoom
Comparing the 4π Spectral Functions τ data not competitive e + e π + π 2π 0 Large discrepancies between experiments e + e 2π + 2π CVC relations (from isospin rotation): σ ( I= 1) 2 4πα = 2 υ 0 π 3π ντ s 4πα = s 2π + 2π 2 ( I= 1) σ υ υ ( ) 2π π π ν π 3π ν π + π 2π 0 + 0 0 τ τ τ data not competitive
Testing CVC Inferτ branching fractions from e + e data: 2 m 0 6 π Vud SEW SU(2)-corrected CVC ( τ π π ντ ) = υ 2 m τ 0 BR ds kin( s) ( s) 2 τ τ π π 0 ν τ Difference: BR[τ ] BR[e + e (CVC)]: Mode τ π π 0 ν τ τ π 3π 0 ν τ τ 2π π + π 0 ν τ (τ e + e ) +1.48 ±0.32 0.08 ± 0.11 +0.91 ±0.25 Sigma 4.6 0.7 3.6
Specific Contributions: Low s Threshold Use Taylor expansion above π + π threshold: 2 3 πα β 2 σ ππ = Fπ and: 3s F 1 2 2 3 4 1 1 2 ( ) 6 r s cs c s O s π = + + + + π exploiting precise space-like data, r 2 π = (0.439 ± 0.008) fm 2, and fitting c 1 and c 2 Fit range: up to 0.35 GeV 2 Integration range: up to 0.25 GeV 2 Excellent χ 2 for both τ and e + e data strong anti-correlation between c 1 and c 2
Specific Contributions: the Charm Region New precise BES data improve cc resonance region: xxx Agreement among experiments
Specific Contributions: Narrow Resonances Use direct data integration for ω (782) and φ (1020) to account for nonresonant contributions. However, careful integration necessary: trapezoidal rule creates systematics for functions with strong curvature use phenomenological fit SND, PRD, 072002 (2001) CMD-2, PL B466, 385 (1999); PL B476, 33 (2000) Trapezoidal rule creates bias
use data Results: the Data & the Theory Agreement between Data (BES) and pqcd use QCD Better agreement between exclusive and inclusive (γγ2) data than in previous analysis
Results: the Compilation Contributions to a µ had from the different energy domains: Modes Energy range [GeV] e + e a µ had (10 10 ) τ Low s expansion 2m π 0.5 58.0 ± 1.7 ± 1.1 rad 56.0 ± 1.6 ± 0.3 SU(2) π + π 2m π 1.8 440.8 ± 4.7 ± 1.5 rad 464.0 ± 3.2 ± 2.3 SU(2) π + π 2π 0 2m π 1.8 16.7 ± 1.3 ± 0.2 rad 21.4 ± 1.4 ± 0.6 SU(2) 2π + 2π 2m π 1.8 14.0 ± 0.9 ± 0.2 rad 12.3 ± 1.0 ± 0.4 SU(2) ω (782) 0.3 0.81 36.9 ± 0.8 ± 0.8 rad - φ (1020) 1.0 1.055 34.8 ± 0.9 ± 0.6 rad - Other exclusive 2m π 2.0 32.2 ± 1.6 ± 0.3 rad - J/ψ, ψ (2S) 3.08 3.11 7.4 ± 0.4 ± 0 rad - R [data] 2.0 5.0 33.9 ± 1.7 exp ±0 rad - R [QCD] 5.0 9.9 ± 0.2 theo - Sum 2m π 684.7 ± 6.0 ± 3.6 rad 709.0 ± 5.1 ± 1.2 rad ±2.8 SU(2)
Discussion The problem of the π + π contribution [shifts given in units of 10 10 ]: Experimental conspiracy: new CMD-2 data produce downward shift [ 1.9], with much better precision new ALEPH BRs produce upward shift [+3.5] CLEO & OPAL spectral functions produce upward shift [+5.9] previous difference was: [τ e + e ] = (11 ± 15) 10 10 we could use average Who is wrong? e + e is consistent among experiments, but error dominated by CMD-2; large radiative corrections applied τ is consistent among experiments, but error dominated by ALEPH SU(2) corrections: basic contributions identified and stable since long; overall correction applied to τ is ( 2.2 ± 0.5) %, dominated by uncontroversial short distance piece; additional long-distance corrections found to be small At present, we believe that it is unappropriate to combine τ and e + e : [τ e + e ] = ( 24.3 ± 6.9(exp) ± 2.7(rad) ± 2.8(SU(2))) 10 10
A few checks detailed π + π contribution [in units of 10 10 ]: e + e consistency (contribution 610-820 MeV) CMD-2 313.5 ± 3.1 CMD 320.8 ± 12.6 OLYA 321.8 ± 13.9 τ consistency (contribution > 500 MeV) ALEPH 460.1 ± 4.4 CLEO 464.7 ± 9.3 OPAL 464.2 ± 8.1 test of ALEPH τ ππ 0 ν τ branching fraction τ 323.9 ± 2.1 global analysis of τ decays: π π π 0 π π 0 π 0 rad. corr. and SU(2) breaking not included in errors e + e 440.8 ± 4.7 predicted by τ-µ universality from πππby isospin B π B uni π = ( 0.08 ± 0.11 exp ± 0.04 th ) % B π2π 0 B π2π 0 3π, iso = ( +0.06 ± 0.17 exp ± 0.07 th ) % excludes a 1.1 % bias in the ππ 0 fraction
Final Results [DEHZ 02] a had µ [ee] a had µ [τ ] a µ [ee] = = = (709.0 ± 5.9) 10 10 (exp and theo errors added in quadrature) (684.7 ± 7.0) 10 10 692.4 ± 6.2 [τ e + e ] = (3.5 ± 1.1)% (11 659 169.3 ± 7.0 had ± 3.5 LBL ± 0.4 QED+EW )10 10 [DH 98] a µ [τ ] = (11 659 193.6 ± 5.9 had ± 3.5 LBL ± 0.4 QED+EW )10 10 including: Hadronic contribution from higher order : a µ had [(α s /π) 3 ] = (10.0 ± 0.6) 10 10 Hadronic contribution from LBL scattering : a µ had [LBL] = + ( 8.6 ± 3.5) 10 10 Observed Discrepancy: a µ [exp] a µ [SM] (10 10 ) = 34 ± 11 9 ± 11 [e + e ] [τ ] Effect on α had (M Z2 ): [τ e + e ] = (2.8 ± 0.8) 10 4 M Higgs 16 GeV/c 2 for τ
Conclusions/Perspectives Hadronic vacuum polarization creates dominant systematics for SM predictions of many precision measurements, such as the muon g-2 New analysis of leading hadronic contribution motivated by new, precise e + e (0.6% systematic error for e + e ) and τ (0.5% error on normalization) data New theoretical analysis confirmed the rules to correct for SU(2) breaking Radiative (VP and FSR) corrections in e + e are major source of systematics We have re-evaluated all exclusive and inclusive as well as resonance contributions We conclude with two incompatible numbers from e + e and (mainly) τ, leading to SM predictions that differ by 3.0 σ [e + e ] and 0.9 σ [τ ] from the experiment The key problem is the quality of the experimental data... Future experimental projects are: CLEO & BES as τ /charm factories B factories: will improve the line shape from τ, but not the normalization ISR production e + e hadrons + γ @ KLOE, BABAR (systematics?)
Preliminary Look into BaBar Analysis
ISR Method at BaBar : e + e - γf dσ ( s, x) dx [ s(1 x) ] = W ( s, x) σ f ; x = E * 2 γ s W ( s, x) β 1) = β (1 + δ ) x 1+ (δ 0.067 at ϒ(4S) ) 2 ( x 2α s β = (2ln 1) ( 0.088 at ϒ(4S) ) π m e Cross Section for final state f (normalized to radiative dimuons) µµ dn fγ ε µµ (1 + δ ) FSR rad σ f ( s') = σ + + ( s') f e e µ µ dn ε (1 + δ ) µµγ Detection efficiencies f FSR rad γ detected at large angle in BaBar Corrections for final state radiation effective c.m. energy = s(1-x) dl(s ) ISR luminosity
BaBar ISR : e + e - γµ + µ normalization and important test sample Dimuon mass vs. Detected ISR Photon Energy 1 µ ID 2 µ ID γ µ tag particle ID - measurement of µ ID efficiency - kinematic fit - muon sample purity > 99.9 % - kin. fit improves resolution: 16 8 MeV at J/ψ - muon detection weak point of BaBar
BaBar ISR Luminosity µµ ( dn µµγ ( M ) µµ inv dl M inv ) = µµ ε (1 + δ ) σ ( M µµ ) µµ FSR rad e + e µ + µ inv 1+FSR fraction equivalent e + e - luminosity : BaBar s 90fb -1 corresponds to an e + e - scan with ~700 nb -1 /0.1GeV at 1GeV and ~4 pb -1 /0.1GeV at 4 GeV M Statistically very competitive continuum sample
BaBar ISR : e + e - γµ + µ at J/ψ (1) 7800 signal events
BaBar ISR : e + e - γµ + µ at J/ψ (2) 5.26 ± 0.37 kev PDG2002: 87 ± 5 kev
BaBar ISR : e + e - γπ + π N N ππ µµ Fπ ( s 2 ) Boost: acceptance down to threshold easier particle ID Ratio cancels: luminosity ISR and VP radiative corrections many efficiencies (photon, tracking) Small corrections: trigger efficiency (track and EMC triggers) FSR corrections, can be studied exp. Major work: particle ID efficiency matrix (P,θ,φ)
BaBar ISR : e + e - γπ + π BABAR AR preliminary Events e + e γπ + π ππ ID ρ (770) Finite mass resolution visi ble in sharp ρ -ω interference. Requires unfolding. The data correspon to an integrated lumino sity of 88 fb 1. µµ ID Background from e + e γµ + µ events i at the level of 1%. ρ (1450) ρ (1700) The present statistics competes well with the latest results from CMD-2. Large mass range coverage Work in progress on the control of the syste matics, in particular fo particle identification. Invariant π + π mass [GeV/c 2 ]
BaBar ISR : e + e - γf higher masses require the study of many channels particle ID essential (DIRC) for correct mass determination work in progress K + K, pp, π + π π 0, 4π, 5π, 6π, ππη, KKπ, KKππ, 2K2K, KKη R will be reconstructed from the sum of exclusive channels up to 2.5 GeV inclusive approach also tried, but limited by photon energy resolution at lower recoil masses
BaBar ISR : e + e - γ2π + 2π (*see note below) BaBar 89.4fb -1 * Very preliminary: Not finally normalized. very clean sample (background~2% ) whole mass range is covered large statistics (~75k events)
BaBar ISR : e + e - γf examples BaBar preliminary K + K m(k S Kπ) BaBar yield = 2963 events K s Kπ m(φη) φη DM2 yield = 367 events Ge π + π π 0 ω J/ψ φ