1. 1    hadrons Revisiting the Tau/ee Discrepancy: Consequences for the Muon Anomaly Michel Davier Laboratoire de l’Accélérateur Linéaire, Orsay with A. Höcker (CERN), X.H. Mo, P. Wang, C.Z. Yuan (IHEP), Z. Zhang (LAL) Muon Magnetic Moment Workshop October 25- 26, 2007, University of Glasgow davier@lal.in2p3.fr

2. 2 Improved Determinations of the Hadronic Contribution to (g –2)  and  (M Z )2 Energy [GeV]Input 1995Input after 1998 2m  - 1.8Data Data (e + e – &  ) (+ QCD) 1.8 – J/  DataQCD J/  -  DataData + QCD  - 40 DataQCD 40 -  QCD Eidelman-Jegerlehner’95, Z.Phys. C67 (1995) 585 Improvement in 4 Steps: Inclusion of precise  data using SU(2) (CVC) Extended use of (dominantly) perturbative QCD Theoretical constraints from QCD sum rules and use of Adler function Alemany-Davier-Höcker’97, + later works Martin-Zeppenfeld’95, Davier-Höcker’97, Kühn-Steinhauser’98, Erler’98, + others Groote-Körner-Schilcher-Nasrallah’98, Davier-Höcker’98, Martin-Outhwaite- Ryskin’00, Cvetič-Lee-Schmidt’01, Jegerlehner et al’00, Dorokhov’04 + others Since then: Improved determi- nation of the dispersion integral: better data extended use of QCD Better data for the e + e –   +  – cross section and multihadron channels CMD-2’02 (revised 03), KLOE’04, SND’05 (revised 06), CMD-2’06, BaBar’04-06

3. 3 The Role of  Data through CVC – SU(2) hadrons   W  e+e+ e –e – CVC: I =1 & V W: I =1 & V,A  : I =0,1 & V Hadronic physics factorizes in Spectral Functions : Isospin symmetry connects I=1 e + e – cross section to vector  spectral functions: branching fractions mass spectrum kinematic factor (PS) fundamental ingredient relating long distance (resonances) to short distance description (QCD)

4. 4 SU(2) Breaking Corrections for SU(2) breaking applied to  data for dominant  –  + contrib.: Electroweak radiative corrections: dominant contribution from short distance correction S EW to effective 4-fermion coupling  (1 + 3  (m  )/4  )(1+2  Q  )log(M Z /m  ) subleading corrections calculated and small long distance radiative correction G EM (s) calculated [ add FSR to the bare cross section in order to obtain  –  + (  ) ] Charged/neutral mass splitting: m  –  m  0 leads to phase space (cross sec.) and width (FF) corrections  -  mixing (EM    –  + decay) corrected using FF model m  –  m  0 and   –    0 [not corrected !] Electromagnetic decays, like:     ,    ,    ,   l + l – Quark mass difference m u  m d generating “second class currents” (negligible) Electromagnetism does not respect isospin and hence we have to consider isospin breaking when dealing with an experimental precision of 0.5% Cirigliano-Ecker-Neufeld’ 02 Marciano-Sirlin’ 88 Braaten-Li’ 90 Alemany-Davier-Höcker’ 97, Czyż-Kühn’ 01

5. 5 e + e  -  Data Comparison: 2006  problems: overall normalization shape (especially above  )

6. 6 Requestioning the Procedure  spectral functions unchanged final ALEPH results (Phys. Rep. 2005) CLEO, OPAL still waiting for final Belle data; also BaBar coming how to relate  and ee spectral functions revisit corrections for SU(2) violation

7. 7 At What Level to Apply CVC?  e  e   V 0 involves lowest-order  -V0 coupling (bare  ) + vacuum polarization (VP) in photon propagator (dressed  ) question: should VP be included or not in the definition of the V0 hadronic state? if V0 is a resonance, does the Breit-Wigner lineshape apply to the bare or the dressed cross section? in our previous analyses we assumed that VP should be left out: the    V  spectral function was related to the bare e  e   V 0 cross section we now argue that it was incorrect: CVC should relate physical (dressed) quantities, therefore one should use the dressed ee 

8. 8 Magnitude of the VP effect  (0)  (  )     1  2 (1+  FSR ) bare +FSR dressed VP FSR at s = m  2 leptonic VP 2.5% hadronic VP  1  4% mass shift from resonant VP: m R  m R (0)  3  R  ee / 2   1.4 MeV for 

9. 9 Direct Test with J/  and  ‘ Masses difference between dressed and bare masses: J/  1.14 MeV  ’ 0.50 MeV accurate measurements of dressed masses by KEDR: 0.01-0.025 MeV also measurements from pbar-p (FNAL/E760) (gluons exchange) compare pbar-p and e-e masses under 2 hypotheses for the ee masses dressed ee masses  mJ/  = -0.01  0.03 MeV  m  ’ = -0.13  0.10 bare ee masses  mJ/  = +0.67  0.04  m  ’ = -0.99  0.10 clearly favours dressed masses in ee annihilation

10. 10 Testing the Non-resonant VP Effect non-resonant VP slowly varying across resonance  no mass shift only way: compare partial widths (bare or dressed) to total width not possible with narrow ccbar/bbar: total width only accessible through sum of partial widths,except FNAL, but not enough precision possible with  but precision on leptonic width just at the limit best test so far: Z 0 at LEP (dressed) partial widths measured by peak cross sections total (physical) width measured directly  invisible width consistent with 3 with 0.3% precision if bare widths used: 3% discrepancy would show up

11. 11 Test with  0   Mass Difference  resonance wide: mass ill-determined, but mass difference OK  0 and  ± accessible in ee annihilation and  decays: perform combined fit of spectral functions with free  ,  ± parameters but same for  ’,  ’’   m  = m  0  m  ± =  2.4 ± 0.7 MeV bare ee  1.0 ± 0.7 MeV dressed ee also measured by KLOE in  decays 0.4 ± 0.9 MeV theoretical estimate (mostly EM) Bijnens-Gosdzinsky  0.4  0.7 MeV both KLOE and theory favour ee dressed mass in ee/  fit

12. 12 SU(2)-breaking Corrections Revisited (1) more precise value of V ud  very small change better calculation of the long-distance radiative corrections G EM (s) Lopez Castro et al.  vertex,  not accounted for in previous calculation (  PT, Cirigliano et al.)  interference: better ee data, interference better determined ee fit with 4 parameters: amplitude, phase, m ,   (last two in agreement with PDG  3  ) m  ±  m  0 effect in cross section and   (opposite effects) m  ±  m  still taken to be 0 ± 1 MeV, consistent with all results use  PT dependence for    m  3   3 / f  2 (stronger effect)

13. 13 SU(2)-breaking Corrections Revisited (2) main change: effect of EM decays on   ±,   decay modes -- previously only calculation (Singer): hard  bremsstrahlung + guess for divergent piece -- new calculation just out (Lopez Castro et al.) hard  + soft/ virtual   finite result, much larger than estimated before      ±  = 1.83 MeV (0.4 MeV) as in all calculations of this type: photon coupling to mesons point-like

14. 14 SU(2)-breaking Corrections Revisited (3) ±±

15. 15 e + e  -  Data Comparison: 2007 (1)  agreement in overall normalization shape much better still not perfect (region around 950 MeV, but small impact)

16. 16 e + e  -  Data Comparison: 2007 (2)  disagreement with KLOE reduced, but still strong

17. 17 Integral #1 : B CVC Test integrating over the ee  spectral function with the  factor + correcting for the SU(2)-breaking effects  compute B CVC compare to measured B(     ) = (25.50 ± 0.10) % essentially insensitive to the shape of the  spectral function B CVC computed using bare (before) or dressed (now) ee SF bare ee SF (24.95 ± 0.19 exp ± 0.12 SU(2) ) %  2.6  (was  4.5  with previous corrections) dressed ee SF (25.57 ± 0.19 exp ± 0.12 SU(2) ) % in agreement with  BR within 0.9% (± 0.24%)

18. 18 Integral #2 : a  had,LO [ ,  ] (10  10 ) update the  based calculation of a  had,LO with new VP prescription and new isospin-breaking corrections  contribution threshold  1.8 GeV 501.0 ± 3.5 exp ± 3.1 SU(2) (was 520.1 in DEHZ03) VP correction also applied to 4  spectral functions also update ee  contribution (published CMD-2 since Tau06) 502.5 ± 3.6 exp ± 1.0 rad good agreement  / ee at last, justified to combine the 2 approaches careful! only 77% of hadronic contribution is  /ee independent, remaining 23% comes only from ee (mainly I=0 component)

19. 19 Comparison with BNL-E821 3.1  3.5  3.6  (hadVP)(LBL)(EW)

20. 20 Conclusions (1) Comparison of  and ee spectral functions completely revisited Previous basis relating bare ee SF to  SF found invalid CVC should apply between dressed (physical) quantities Several tests performed, which confirm validity of new approach physical masses of J/  and  ’ are dressed, bare are excluded sum of dressed partial widths is the physical total width (Z)  ±/  mass difference favours the dressed mass in ee annihilation VP correction is the largest change (  10.0 units in a  ) Isospin breaking corrections reconsidered better knowledge of  interference long-distance radiative corrections more complete (  2.9 units)  contribution to  ±/  width difference includes now soft/virtual part: the next largest change (  5.2 units)

21. 21 Conclusions (2)  Results from the new procedure BCV now in agreement with the direct  measurement within 0.9%  contributions to a  from  and ee (CMD-2+SND) agree within 1.2% comparison with KLOE still problematic for the SF shape  Combined  /ee prediction disagrees with BNL measurement by 3.6   Combined uncertainty for hadVP now at the level of error estimate for LBL  Total theory uncertainty (5.2) significantly smaller than experimental one (6.3)  A new more precise g-2 measurement is desperately needed, as present precision will overshadow any progress on the theory side