The strong coupling from hadronic tau decays

Transcription

1 The strong coupling from hadronic tau decays Maarten Golterman (IFAE, Barcelona & San Francisco State Univ.) Phys. Rev. D84 (20): D. Boito, O. Catà, M. Golterman, M. Jamin, K. Maltman, J. Osborne, S. Peris Phys. Rev. D85 (202): D. Boito, M. Golterman, M. Jamin, A. Mahdavi, K. Maltman, J. Osborne, S. Peris

2 Status of s (m 2 ) determinaqons a s Hm t 2 L Abbas et al. '2 Pich ' Caprini and Fischer ' Cvetic et al. '0 Menke '09 Narison '09 Maltman and Yavin '08 * Beneke-Jamin 08 Davier et al.'08 * Baikov et al.'08 World Average HBethkeL ' * Complete analyses from (mostly) ALEPH data

3 a s Hm t 2 L This work HCIPTL This work HFOPTL Abbas et al. '2 Pich ' Caprini and Fischer ' Cvetic et al. '0 Menke '09 Narison '09 Maltman and Yavin '08 Beneke-Jamin 08 Davier et al.'08 Baikov et al.'08 World Average HBethkeL ' Errors in non- perturbaqve part systemaqcally underesqmated Our analysis uses OPAL data (error in ALEPH correlaqons (Boito et al. 0))

4 Overview: Theory of tau decays Comparison of standard analysis with our analysis ALEPH vs. OPAL data Results of our analysis Conclusions

5 decays leptonic: hadronic: + R = (! hadrons) (! e e ) apple =3S EW V ud 2 + s +5.2 s use this to determine s (m ) ( q W m ) from non- strange decays see!! pions, not! jets = (770), (450), (700) (and others: incl. axial, kaons, ) relaqon with perturbaqve regime?

6 (! hadrons) R (s 0 ) = 2 S EW V ud 2 Z s0 0 ds s 0 s s 0 (opqcal theorem) 2 +2 ss0 Im (s) (s) = s =(q W ) 2 0 apple s apple s 0 = m 2 depending on how much momentum carries away

7 RelaQng s to decay data: complex s plane circle has radius s 0 posiqve real axis: spectral data ( Im (s) ) (s) analyqc everywhere except for real s>0 Cauchy, with a polynomial weight w(s) : Z s0 0 ds w(s) Im (s) = 2 i (Shankar 77,., Braaten, Narison, Pich 92) I z =s 0 dz w(z) (z)

8 master equa8on (FESR): Z s0 0 ds w(s) exp (s) = 2 i I z =s 0 dz w(z) OPE (z) Z s 0 ds w(s)im DV (s) w(s) polynomial weight (which should we choose?) exp (s) inclusive spectral funcqon from experiment: (non- strange) OPE (z) perturbaqon theory ( s (m 2 ): log z ) plus OPE condensates ( /z k ) DV (z) = QCD (z) OPE (z) Duality ViolaQons not small near Minkowski axis! (OPE spectral funcqon) Assume DV (z) decays exponenqally for z!: I dz w(z) DV (z) = Z ds w(s) Im DV (s) 2 i z =s 0 s OPAL π π 0 3π π 0, π 3π 0 MC corr. perturbative QCD (massless) naïve parton model s (GeV 2 ) Ansatz: Im DV(s) =e s sin ( + s) for s>s min

9 Compare standard vs. our analysis: Standard: (OPAL 99, ALEPH 05, Davier et al. 08) - Assume no duality violaqons, weights with pinching factor ( s/s 0 ) n, n =2, 3-5 weights of degree 3 to 7 at s 0 = m 2 assume dim. 0-6 condensates vanish - Fit with 4 parameters ( s (m 2 ), dim. 4,6,8 condensates) to 5 data points - Inconsistent dependence on s 0 for other pinched weights (Maltman & Yavin 08) - 4 out of 5 moments have bad behavior in pert. theory (Beneke, Boito, Jamin 2) I This work: ( dz z n ) z k / n,k - Main fit: w =, hence no OPE condensates, but DV contribuqon instead - Take s 0 2 [s min,m 2 ], determine s min from quality and stability of fit - Fit with 5 parameters ( s (m 2 ), DV parameters) to more (correlated) data - Vector channel only in main fit - Test with other weights (treat OPE consistently) and including axial channel 4 s z =s 0 Both: perturbaqon theory to order, CIPT and/or FOPT (Baikov, Chetyrkin, Kühn 08)

10 RPM RPM2 FOPT CIPT Borel sum Moments in perturbaqon theory CIPT and FOPT (Boito, Beneke, Jamin 2) 0.28 FOPT CIPT Borel sum w(z) = FOPT CIPT Borel sum FOPT CIPT Borel sum ( z) 2 ( + 2z) FOPT CIPT Borel sum FOPT CIPT Borel sum ( z) 3 z( + 2z)

11 OPE: X k=0 C 2k ( s) k (up to logarithms) ALEPH 08 analysis (Davier et al. 08) neglect duality violaqons choose weights with - double zero at s 0 - use only data at s 0 = m 2 - OPE terms up to dim. 6 but assume C 0 = C 2 = C 4 = C 6 =0 - fit s, C 4, C 6, C 8 Weights shown (top to borom) Non- strange vector channel (Maltman & Yavin 08) z( z) 2 ( z) 2 ( + 2z) ( z) 2 ( + z/2) ( z) 2 (all with degree apple 3 )

12 Ansatz: Im DV(s) =e s sin ( + s) Oscillatory: duality violaqons due to resonances ExponenQal decay: finite width / /N c, small Argument of sine linear in s : Regge- like (daughter) trajectories Model for (s) with such behavior does exist! For example: (z) / (z) = d log dz z = s (z) = X n=0 z + n + constant AnalyQc for 0 < <, except for cut on (negaqve) real s axis

13 imaginary part and absolute value of DV (z) = (z) OPE (z) asymptoqc behavior along real axis has the form of the ansatz, with =2 2 ( ) / /N c small. (Blok, Shifman & Zhang 98, Bigi et al. 99, Catà, MG & Peris 05 & 08, Gonzalez- Alonso, Pich & Prades 0, Jamin )

14 ALEPH vs. OPAL correlaqon matrices (a) ALEPH (b) OPAL Figure E.: Correlation matrices for the vector spectral functions from aleph (a) and opal (b). The correlations are given in %. (Boito, Ph.D. thesis )

15 ALEPH vs. OPAL spectral funcqon data (vector channel) top: borom: experimental data Monte Carlo data generated with covariance matrix ALEPH OPAL s [GeV] 2 (a) s [GeV] 2 (a) 3 toy data set: ALEPH 3 toy data set: ALEPH 3 toy data set: OPAL 3 toy data set: OPAL s [GeV] 2 s [GeV] 2 (b) ALEPH (c) s [GeV] 2 s [GeV] 2 (b) OPAL (c)

16 RESULTS

17 Fit to w FOPT s min =CIPT OPE only s 0 GeV OPAL 98 data Not a fit! FOPT / CIPT s 0 GeV 2 Vector channel fit with weight w(s) =and s min =.5 GeV 2 Result: s (m 2 )=0.307 ± 0.08 ± ± (FOPT) =0.322 ± ± ± (CIPT) Errors: () fit error, (2) stability wrt s min 2 /dof = 0.36, e =0.02 ± 0.0, (3) truncaqon of pert. theory

18 Checks: R V +A;ud FOPT CIPT s 0 GeV 2 Fits with weights w =, (s/s 0 ) 2, ( s/s 0 ) 2 ( + 2s/s 0 ) and axial channel data give completely consistent results Weinberg sum rules and DGMLY sum rule saqsfied within errors Fits describe R V +A,ud extremely well Take vector channel with w = as main result (only 5 parameters)

19 Compare with OPAL: OPAL s original results: s (m 2 )=0.324 ± 0.04 s (m 2 )=0.348 ± 0.02 (FOPT) (CIPT) This work: s (m 2 )=0.307 ± 0.09 s (m 2 )=0.322 ± (FOPT) (CIPT) Same data, central values shived downward by about 0.02 Errors previously underesqmated; larger than difference CIPT and FOPT

20 Update OPAL data 98 OPAL spectral funcqons constructed by summing over exclusive modes, normalized with 98 PDG values for branching fracqons Rescale by using current branching fracqons from HFAG Vector channel update: [new data] [old data] bin

21 Markov- chain Monte Carlo analysis of 2 distribuqon vector channel, w = χ 2 projecqon from 6d to 2d plot α s complicated landscape! 2 has two minima, with 4 or 2 δ V - sigma contour 2- sigma contour model favors (Catà, MG & Peris 08) log F 2 M 2 (also absolute minimum) log (0.2 2 )=4.2 α s at the edge!

22 Results and comparison: OPAL 99: This work, updated data: s (m 2 )=0.324 ± 0.04 s (m 2 )=0.348 ± 0.02 s (m 2 )=0.325 ± 0.08 s (m 2 )=0.347 ± (FOPT) (CIPT) (FOPT) (CIPT) Agreement of central values purely coincidental! Note larger errors; for instance compare: This work: NP = ± 0.02 (FOPT) NP = ± 0.02 (CIPT) vs. previous esqmate: NP = ± (Pich, from ALEPH analysis) in which R V +A,ud (m 2 )=N c V ud 2 S EW + pert.th. + NP

23 Conclusions New value of s (m 2 ) from hadronic tau decays Larger error (±0.02) than previously assumed because of non- perturbaqve uncertainqes (OPE and DVs); supersedes earlier values Fits to OPAL data at the edge of being possible 2 Best fit with 5 parameters and good values Expect that significant progress (more stringent tests!) is possible if errors are reduced by a factor 2 or 3 BaBar and BELLE: please produce inclusive spectral funcqons! Theory: berer understanding of CIPT vs. FOPT? Duality violaqons?