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Physics with e + e  and  Spectral Functions

Physics with e + e  and  Spectral Functions. Michel Davier (LAL-Orsay). Flavianet Workshop, November 2 4, 2006, Barcelona. Rev. Mod. Phys. Dec.2006; hep-ph/0507078 MD, A. Höcker, Z. Zhang. Outline. ALEPH  Hadronic Spectral Functions Theoretical Framework

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Physics with e + e  and  Spectral Functions

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  1. Physics with e+e and  Spectral Functions Michel Davier (LAL-Orsay) Flavianet Workshop, November 24, 2006, Barcelona Rev. Mod. Phys. Dec.2006; hep-ph/0507078 MD, A. Höcker, Z. Zhang M. Davier : Physics with ee and  Spectral Functions

  2. Outline • ALEPH  Hadronic Spectral Functions • Theoretical Framework • Measurement of the Strong Coupling Constant • Precise Test of Asymptotic Freedom • ee Spectral Functions: BaBar ISR program • Hadronic Vacuum Polarization • Problems with ee and  Spectral Functions M. Davier : Physics with ee and  Spectral Functions

  3. Tau Hadronic Spectral Functions • Hadronic physics factorizes in(vector and axial-vector)Spectral Functions : branching fractionsmass spectrum kinematic factor (PS) Fundamental ingredient relating long distance hadrons to short distance quarks (QCD) • Isospin symmetry connects I =1 e+e– cross section to vectorspectral functions: M. Davier : Physics with ee and  Spectral Functions

  4. Spectral Functions: V, A A V M. Davier : Physics with ee and  Spectral Functions

  5. Inclusive V+A and VA Spectral Functions • Results from ALEPH and OPAL  and their comparison Of purely nonperturbative origin ALEPH, Phys. Rep. 421 (2005) OPAL, EPJ, C7, 571 (1999) M. Davier : Physics with ee and  Spectral Functions

  6. Hadronic Tau Decays and QCD • ALEPH, CLEO and OPAL carried out tests of QCD using the tau hadronic width • The opportunity to precisely measure s(m) has triggered theoretical research in particular on the perturbative expansion of the tau hadronic width, affected by a truncation at NNLO = O(s3) : • contour improvement of fixed-order perturbation theory (FOPT) • effective-charge and minimal-sensitivity approaches • large 0 expansion (renormalons) • The measurement of inclusive spectral functions lead to direct studies of nonperturbative hadronic properties as a function of the mass scale s0 < m2 • Nonperturbative contributions were found to be very small at m2 • Direct measurements of the running of s(s0 < m2) could be performed M. Davier : Physics with ee and  Spectral Functions

  7. Contribution from QCD radiative corrections  21% nonperturbative contributions < 3% Tau and QCD: Naïve Considerations neglecting QCD and EW corrections • The tau hadronic width: • Separate into appropriate currents: R = R,V + R,A + R,S • Using ALEPH branching fractions (and the WA tau lifetime): Beuniversal = (17.810  0.039) % BS = (2.85  0.11) % [  2.2 per mille on absolute measurement !!! ] M. Davier : Physics with ee and  Spectral Functions

  8. Tau and QCD: Towards a Theoretical Prediction • The optical theorem relates J /aJ(s)  ImV/A(J)(s), where V/A(J)(s) are 2-point polarization functions describing the creation of hadrons out of the QCD vacuum • The vector current has spin J=1 (CVC), while axial-vector has J=0,1 • The tau hadronic width for current U is then given by: • Problem: ImV/A(J)(s) contains hadronic physics that cannot be predicted in QCD • However, owing to the analyticity of (s), one can use Cauchy’s theorem: kinematic factor • The contour integral can be solved using perturbative QCD for sufficiently large s0 M. Davier : Physics with ee and  Spectral Functions

  9. Perturbative quark-mass terms: EW correction: Perturbative contribution Adler function to avoid unphysical subtractions: Nonperturbative contribution Tau and QCD: The Operator Product Expansion • Full theoretical ansatz, including nonperturbative operators via the OPE: (in the following:as = s/) M. Davier : Physics with ee and  Spectral Functions

  10. Complex s dependence of as driven by running: The Perturbative Prediction • Perturbative prediction of Adler function given to NNLO, but how should one best solve the contour integral A(n)(as) occurring in the prediction of R? (1) • Four ways to solve Eq. (1): Fixed-order perturbation theory (FOPT): insert Taylor expansion in Eq. (1), evaluate integral, collect all terms of equal powers in as, and reject all powers n > 4, leads to: Contour-improved fixed-order perturbation theory (CIPT): perform numerical solution of integral in Eq. (1), which corresponds to keeping all known terms in Eq. (2) Effective-charge perturbation theory (ECPT): express  (0) as physical observable (“effective charge”) so that it is renormalization scheme (RS) independent: Large 0 expansion : renormalon chains M. Davier : Physics with ee and  Spectral Functions

  11. However: obtaining K4 this way assumes negligible K5 dependence  not true; even K6 contribution is not negligible ! • The error on the ECPT prediction: K4 ~ 27 is of similar order • Similar conclusions for R-scale modification method (methods are correlated) • Direct calculations of K4 are underway: partial results agree with the magnitude found by ECPT or R-scale variation • K4 = 25  25 used in the analysis weak K4, K5 dependence Baikov-Chetyrkin-Kühn, Phys.Rev. D67, 074026 (2003) Kataev-Starshenko, Mod. Phys. Lett. A10, 235 (1995) Le Diberder, Nucl. Phys. (Proc. Suppl.) B39, 318 (1995) Detailed discussion in: Davier-Höcker-Zhang, Rev. Mod. Phys., 78 (2006) strong K4, K5 dependence Estimating Unknown Higher-Order Coefficient: K4 • Trick: modify R-scheme (ECPT instead of FOPT) or R-scale ( in FOPT or CIPT) to obtain less convergent perturbative series so that sensitivity to K4 is enhanced, and K4 can be obtained from comparison with experimental observable M. Davier : Physics with ee and  Spectral Functions

  12. Comparison of the Perturbative Methods • Predictions of  (0) for the various perturbative methods and orders: (*) (*) (*) (*) (*) • CIPT converges faster than FOPT • the difference between FOPT and CIPT is not due to the Taylor approx. in FOPT, but because FOPT erases the known higher orders of the Taylor series • ECPT and Large-0 exp. have slow convergence and hence large truncation errors • CIPT has best convergence and should be used to predict  (0) • the difference in  (0) between CIPT and FOPT is not a useful measure of the perturbative uncertainty: it is artificially increased due to the neglect of large known terms in FOPT ! (*)assuming K4=25 and geometric growth for the unknown coefficients Kn>4 and n>3 M. Davier : Physics with ee and  Spectral Functions

  13. Four unknowns: s(s0), sGG, sqq-bar2, O8 Quark-mass and Nonperturbative Contributions • other terms in the OPE: • D=2 (mass dimension): quark-mass terms are mq2/s0, which is negligible for q=u,d • D=4: dominant contribution from gluon condensate sGG • D=6: dominated by large number of four-quark dynamical operators, reduced to an effective scale-independent operator sqq-bar2 • D=8: absorbed in single phenomenological operator O8 Shifman-Vainshtein-Zakharov, NP B147, 385, 448, 519 (1979) Braaten-Narison-Pich, NP B373, 581 (1992) • Exploit shape of spectral functions to obtain additional experimental information: Le Diberder-Pich, PL B289, 165 (1992) M. Davier : Physics with ee and  Spectral Functions

  14. ALEPH Fit Results • The combined fit of R and spectral moments (k=1, =0,1,2,3) gives (at s0=m2): (*) (*)second errors given are theoretical • correlations between s and nonperturbative parameters between 0.3 and 0.6 • theoretical error on s dominated by unknown K4(used: 25 ± 25) and R-scale (m2± 2 GeV2) • still keep additional syst error from difference between FOPT and CIPT • agreement between independent vector and axial-vector fits • nonperturbative contributions negligible for V+A, significant, but small (and opposite) for V, A M. Davier : Physics with ee and  Spectral Functions

  15. Running within the Tau Spectrum • The spectral functions allow to measure R(s0 < m2); the previous fit allows us to compare the measurement to the theoretical expectation assuming RGE running running strong coupling: evidence for quark confinement • assuming quark-hadron duality to hold, this is a direct evidence for running strong coupling • also: test of stability of OPE prediction, and hence of trustworthiness of s(m2) fit result M. Davier : Physics with ee and  Spectral Functions

  16. Final Assessment of s(m2) and Evolution to MZ2 • Experimental input (1): Beuni = (17.818  0.032) % WA / ALEPH (2): R,S = 0.1686  0.0047 WA / ALEPH  R,V+A= 3.471  0.011 still use ALEPH spectral functions • Theory framework: tests  CIPT method preferred, no CIPT-vs.-FOPT syst. • The fit to the V+A data gives: • Using 4-loop QCD -function and 3-loop quark-flavour matching yields: M. Davier : Physics with ee and  Spectral Functions

  17. Tau provides most precise s (MZ2) determination Z result tau result Overall Comparison QCD  Result + QCD running: good agreement with measurements ! M. Davier : Physics with ee and  Spectral Functions

  18. Test of Asymptotic Freedom • Need precise determinations of s at widely different scales: here m and MZ • Tau hadronic width: s(m2) = 0.345  0.010 error dominated by theory Z hadronic width: s(MZ2) = 0.1186  0.0027 global EW fit; experiment  s(Q2) runs • RGE evolved tau result s,(MZ2) = 0.1215  0.0012  S(Q2) runs exactly as predicted by QCD • Defining evolution estimator r (s1,s2) = 2 (s1(s1)  s1(s2)) / (ln s1 – ln s2) rexp (m2, MZ2) = 1.405  0.053 rQCD(m2, MZ2) = 1.353  0.006  Most precise test of asymptotic freedom in QCD used by DELPHI EPJC 37 (2004) 1 M. Davier : Physics with ee and  Spectral Functions

  19. ee Spectral Functions • Long tradition to measure (ee hadrons) and R • Precise measurements only in recent years (<1.4 GeV) : CMD-2, SND, KLOE • Crucial transition region 1.42.5 GeV still very poor M. Davier : Physics with ee and  Spectral Functions

  20. ISR ISR Program with BaBar • systematic program underway using ISR from (4S) energies, taking advantage of high luminosity (B-factory) • statistics comparable to CMD-2/SND for Ecm<1.4 GeV, much better than DM1/DM2 above • full energy range covered at the same time: from threshold to 3-4 GeV X = 2E /Ecm H is radiation function M. Davier : Physics with ee and  Spectral Functions

  21. SND BaBar DM2 BaBar ISR: some results 22  2220 33 BaBar BaBar M. Davier : Physics with ee and  Spectral Functions

  22. QCD analysis of BaBar ISR data • Inclusive hadronic cross section (R) will be available in 1-2 years • Precision will be comparable to ALEPH  data at masses <1.4 GeV already the case for  data from Novosibirsk will be checked by the independent ISR method with BaBar • No limitation at m : upper integral limit can be varied up to 3 GeV • Precision at larger masses much better than for  (phase space) • QCD analysis will be performed • Comparison between  and ee results M. Davier : Physics with ee and  Spectral Functions

  23. Vacuum Polarization Define: photon vacuum polarization function (q2) Ward identities: only vacuum polarization modifies electron charge with: Leptonic lep(s) calculable in QED. However, quark loops are modified by long-distance hadronic physics, cannot (yet) be calculated within QCD (!) Way out: Optical Theorem (unitarity) ... ... and the subtracted dispersion relation of (q2) (analyticity) Im[ ]  | hadrons |2 M. Davier : Physics with ee and  Spectral Functions

  24. Muon anomalous magnetic moment Contributions to the Standard Model (SM) Prediction: Hadronic contribution dominated by LO vacuum polarization (one should also add HO and LBL contributions) had   had   M. Davier : Physics with ee and  Spectral Functions

  25. Recent steps in the determination of ahad Eidelman-Jegerlehner’95, Z.Phys. C67 (1995) 585 Since then: improved determi-nation of the dispersion integral: • better data • extended use of QCD • Inclusion of precise  data using SU(2) (CVC) Alemany-Davier-Höcker’97, Narison’01, Trocóniz-Ynduráin’01, + later works • Extended use of (dominantly) perturbative QCD Martin-Zeppenfeld’95, Davier-Höcker’97, Kühn-Steinhauser’98, Erler’98, + others Improvement in 4 Steps: • Theoretical constraints from QCD sum rules and use of Adler function Groote-Körner-Schilcher-Nasrallah’98, Davier-Höcker’98, Martin-Outhwaite-Ryskin’00, Cvetič-Lee-Schmidt’01, Jegerlehner et al’00, Dorokhov’04 + others • 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 M. Davier : Physics with ee and  Spectral Functions

  26. Isospin-breaking Corrections Applied to  Data • Electroweak radiative corrections: • dominant contribution from short distance correction SEW • subleading corrections calculated and small • long distance radiative correction GEM(s) calculated [ add FSR to the bare cross section in order to obtain  – + () ] • Charged/neutral mass splitting: • m–  m0leads to phase space (cross sec.) and width (FF) corrections • m–  m0 and –  0 • - mixing • Quark mass difference mu  md (negligible) Marciano-Sirlin’ 88, Braaten-Li’ 90 Cirigliano-Ecker-Neufeld’ 02 Lopez Castro et al’ 06 Alemany-Davier-Höcker’ 97, Czyż-Kühn’ 01 M. Davier : Physics with ee and  Spectral Functions

  27. CMD-2 revision 2002-2003 SND revision 2005-2006 The Radiative Correction Struggle in ee Data 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 corrected by expts • Final state radiation (FSR) [we need e+e–  hadrons () in dispersion integral] • usually, experiments obtain bare cross section so that FSR has to be added “by hand”; done for CMD-2, (supposedly) not done for others M. Davier : Physics with ee and  Spectral Functions

  28. Problems in ee Spectral Functions ee results compared to fit of SND data band indicates relative systematic error CMD-2 KLOE M. Davier : Physics with ee and  Spectral Functions

  29. Problems between  and ee Spectral Functions M. Davier : Physics with ee and  Spectral Functions

  30.  and ee Discrepancy in one Number Infer branching fractions (more robust than spectral functions) from e+e– data: Difference: BR[ ] – BR[e+e– (CVC)]: ee data on  –  +  0 0 not satisfactory M. Davier : Physics with ee and  Spectral Functions

  31. Contributions to ahad[in 10 –10]from the different energy domains M. Davier : Physics with ee and  Spectral Functions

  32. Update for ICHEP-Tau06 (Preliminary) Hadronic HO – ( 9.8 ± 0.1) 10–10 Hadronic LBL + (12.0 ± 3.5) 10–10 Electroweak (15.4 ± 0.2) 10–10 QED (11 658 471.9 ± 0.1) 10–10 inclu-ding: .0 Knecht-Nyffeler,Phys.Rev.Lett. 88 (2002) 071802 Melnikov-Vainshtein, hep-ph/0312226 Davier-Marciano, Ann. Rev. Nucl. Part. Sc. (2004) Kinoshita-Nio (2006) BNL E821 (2004): aexp = (11 659 208.0  6.3) 1010 Observed Difference with Experiment: M. Davier : Physics with ee and  Spectral Functions

  33. Conclusions (1) QCD • Hadronic tau decays: one of the most powerful testing ground for QCD • Despite low scale, dominated by perturbative QCD with good fortune: analyticity, weight factors, inclusiveness, V, A, N2LO + estim of N3LO, systematic treatment of NP contributions (OPE), lowest dim NP terms suppressed • Detailed studies on the perturbative series and its truncation: CIPT method preferred • Good quality data for the spectral functions and their normalising branching fractions (ALEPH). Data from other experiments used to improve precision • Joint QCD fits of total rate and spectral moments permit the extraction of the strong coupling constant and the main NP contributions • s(m2), extrapolated at MZ scale, yields the most precise value of s(MZ2) • s(m2) and s(MZ2) from Z decays provide the most precise test of asymptotic freedom in QCD • These studies will be carried to ee data (R program with BaBar) M. Davier : Physics with ee and  Spectral Functions

  34. Conclusions (2) Vacuum Polarization • Hadronic vacuum polarization is still the dominant systematics for SM prediction of the muon g –2 • Significant step in precision from new experimental input CMD-2 + SND for 2 BaBar for multipion channels • Precision of SM prediction (5.6)now exceeds experimental precision (6.3) • precise KLOE not included, awaiting full understanding • Discrepancy with  data (ALEPH & CLEO & OPAL) confirmed • Until  / e+e – puzzle is solved, only e+e – data used in dispersion integral • SM prediction for a differs by 3.3  [e+e – ] from experiment (BNL 2004) • It is very unfortunate that the upgrade of the BNL experiment will not be funded • What is behind the 4.5  discrepancy between BR() and the isospin-breaking corrected spectral function from ee    ? M. Davier : Physics with ee and  Spectral Functions

  35. Backup M. Davier : Physics with ee and  Spectral Functions

  36. Tau Branching Fractions measured by ALEPH  (7525)% used (+CLEO) ALEPH, Phys. Rep. 421 (2005) M. Davier : Physics with ee and  Spectral Functions

  37. Lepton Universality in the Charged Current Beand B g / ge = 0.9991  0.0033 Be , B and  WA (290.6  1.1) fs g / g = 1.0009  0.0023  0.0019  0.0004 g / ge = 1.0001  0.0022  0.0019  0.0004 (Be , B)()(m) B and  WA g / g = 0.9962  0.0048  0.0019  0.0004  0.0007 (radiative corrections) Beuni = (17.810  0.039) % M. Davier : Physics with ee and  Spectral Functions

  38. All terms known to n=3, unknown 4 coefficient only enters at n=6 Four Ways to Solve Eq.(1) : FOPT and CIPT … • Fixed-order perturbation theory (FOPT): insert Taylor expansion in Eq. (1), evaluate integral, collect all terms of equal powers in as, and reject all powers n > 4, leads to: (2) • Contour-improved fixed-order perturbation theory (CIPT): perform numerical solution of integral in Eq. (1), which corresponds to keeping all known terms in Eq. (2) Also:Taylor expansion inaccurate for large positive and negative angles Pivovarov (1991,1992) LeDiberder-Pich (1992) M. Davier : Physics with ee and  Spectral Functions

  39. The evolution of the effective charge obeys the RGE: where the -function is obtained via variable exchange: (4) Using Eq. (2) for a at l.h.s. and r.h.s. of Eq. (4) and equating all coefficients of same orders in as gives: Four Ways to Solve Eq.(1) : … ECPT … • Effective-charge perturbation theory (ECPT): express  (0) as physical observable (“effective charge”) so that it is renormalization scheme (RS) independent: (3) Grunberg (1980), Stevenson (1981), Dhar (1983), Maxwell-Tonge (1996), Raczka (1998) In ECPT the energy evolution of the observable is iden-tical to -function evolution (running) of the coupling M. Davier : Physics with ee and  Spectral Functions

  40. Following this line, (0) can be expanded as: neglected in large-0 expansion large-0 expansion: comparison w/ FOPT: Four Ways to Solve Eq.(1) : … Large-0 Expansion • Consider fermion-loop insertion into qq-bar vacuum polarization diagram: Beneke Phys.Rep. 317 (1999) The “large-0 expansion” consists of resumming this chain, and only keeping lowest order (0as)n terms At large orders, this series diverges; this divergence can be encoded in singularities of convergent “Borel”-transformed series, giving Results agree not too badly; however, could be by chance: worse agreement for Adler function  powerful idea, but not suited to a precision analysis M. Davier : Physics with ee and  Spectral Functions

  41. Inclusive S=1 Spectral Functions • Results from ALEPH and OPAL  and their comparison ALEPH, EPJ C11, 599 (1999) OPAL, EPJ C35, 437 (2004) M. Davier : Physics with ee and  Spectral Functions

  42. Vus and ms determinations from strange SF M. Davier : Physics with ee and  Spectral Functions

  43. Contributions to the dispersion integral 2 3 (+,) 4 > 4 (+KK) 1.8 - 3.7 3.7 - 5 (+J/, ) 5 - 12 (+) 12 -  < 1.8 GeV ahad,LO 2 2[ahad,LO] 2 M. Davier : Physics with ee and  Spectral Functions

  44. Final remarks on main  contribution The problem of the  + – contribution : • Experimental situation: • revised SND results in agreement with CMD-2 • data without m() and () corr. in strong disagreement with both data sets • ALEPH, CLEO and OPAL data in good agreement, preliminary BELLE less so • ee spectral functions have now reached the precision of  data • Concerning the remaining line shape discrepancy (0.7- 0.9 GeV2): • SU(2) corrections: basic contributions identified and stable since long; overall correction applied to  is (– 2.2 ± 0.5)%, dominated by uncontroversial short distance piece; additional long-distance corrections found to be small •  lineshape corrections can improve the situation, but cannot account for the difference above 0.7 GeV2 • The agreement between SND and CMD-2 invalidates the use of data until a better understanding of the discrepancies is achieved (an interesting question as such) • Discrepancy between KLOE and CMD-2/SND results: not safe to include KLOE in the calculation and take advantage of decreased error M. Davier : Physics with ee and  Spectral Functions

  45. BaBar ISR:  huge discrepancy with DM2 SND BaBar DM2 contribution to ahad (1.05-1.8 GeV) : all before BaBar 2.45  0.26  0.03 all + BaBar 2.79  0.19  0.01 all – DM2 + BaBar 3.25  0.09  0.01 x1010 M. Davier : Physics with ee and  Spectral Functions

  46. BaBar ISR: 22 contribution to ahad (<1.8 GeV) : all before BaBar 14.20  0.87  0.24 all + BaBar 13.09  0.44  0.00 x1010 M. Davier : Physics with ee and  Spectral Functions

  47. BaBar ISR: 33 BaBar contribution to ahad (<1.8 GeV) : all before BaBar 0.10  0.10 all + BaBar 0.108  0.016 x1010 M. Davier : Physics with ee and  Spectral Functions

  48. BaBar ISR: 222 BaBar contribution to ahad (<1.8 GeV) : all before BaBar 1.42  0.30  0.03 all + BaBar 0.890  0.093 x1010 M. Davier : Physics with ee and  Spectral Functions

  49. Evaluating the Dispersion Integral use data Agreement bet-ween Data (BES) and pQCD (within correlated systematic errors) use QCD Better agreement between exclusive and inclusive (2) data than in 1997-1998 analyses use QCD M. Davier : Physics with ee and  Spectral Functions

  50. Partial solution to the lineshape problem Isospin breaking in the   masses zoom observed ± –0 mass difference: 2.3 ± 0.8 MeV; width 0.2 ± 1.0 MeV Davier, hep-ex/0312064 Jegerlehner, hep-ph/0312372 M. Davier : Physics with ee and  Spectral Functions

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