Charm Mass Determination from QCD Sum Rules at O(α )

Transcription

1 Charm Mass Determination from QCD Sum Rules at O(α ) 3 s Vicent Mateu MIT - CTP Cambridge - USA PANIC 11 - MIT Taskforce: A. H. Hoang MPI & U. Vienna V. Mateu MIT & IFIC S.M. Zebarjad & B. Dehdadi Shiraz University arxiv:

2 INTRODUCTION

3 Heavy quark masses Renormalization and scheme dependent object Only interested in short-distance schemes, which do not suffer from the O( Λ ) QCD renormalon problem inherent to the pole mass scheme. MS scheme Short distance scheme. Standard mass for comparison: m ( m ). And free of renormalon ambiguities. q q Why high precision? Strong dependence in flavor processes Constrains new physics B X γ S Strong charm mass (scheme) dependence in NLO matrix elements Misiak & Gambino K + + π νν NNLO QCD computations for charm contributions

4 Determinations of mc

5 Determination of m c from sum rules Fixed order analysis (correlated variation) μ α = μ m Kühn et al ( 08)[3] Boughezal et al ( 08) [4] Maier et al (08) [5] mc( mc) = ± 0.009exp ± 0.009α ± 0.002μ 1.295± ± ± exp ± ± ± exp α α μ μ Only for n = 1 [3,4], 2 [5] 3-loops in pert. theory. Updated experimental data.

6 Determination of m c from sum rules Fixed order analysis (correlated variation) μ α = μ m Kühn et al ( 08)[3] Boughezal et al ( 08) [4] Maier et al (08) [5] mc( mc) = ± 0.009exp ± 0.009α ± 0.002μ 1.295± ± ± exp ± ± ± exp α α μ μ Only for n = 1 [3,4], 2 [5] 3-loops in pert. theory. Updated experimental data. Tiny errors! ( underestimated? ) Need for more general analysis Similar for bottom mass determinations

7 Relativistic sum rules Total hadronic cross section σ Rs () = σ + ( ee hadrons) + + ( ee μ μ ) Vacuum polarization function Moments of the cross section z = s 2 4m ds 1 dz M n = () () 4 2 s R s 1 cc = R z m n+ n 1 n+ 4 1 cc z 2 ( m ) Vector current (electromagnetic) 2 ix q ( gμν qμqν ) q i xe { Jμ x Jν x } Π ( ) = d 0 T ( ) ( ) 0 J ( x) = c( x) γ c( x) electric charge 2 R s = Q Π s+ i Π q Π = cc () 12π Im ( 0) μ q ( ) (0) d π μ R () s 2 cc s Q m s( q s) 1 2 Π( q 0, m) = M q 2 2n 2 2 n 12π Q n= 0 Π() s Mn = 6π iq ds s n+ 1

8 Relativistic sum rules m c Effective energy range: E eff = (asymptotically correct for large n) n m c n Λ QCD Since we want to apply perturbation theory for Wilson coefficients. Otherwise the OPE converges badly. n = 1 is the cleanest moment, and we will focus on it for the analyses presented in this seminar. (n = 2 is also fine)

9 Experimental data

10 Experimental data: charm Narrow resonances Narrow-width approximation

11 Experimental data: charm Sub-threshold and threshold BES 1999 * Above threshold experiments measure the total hadronic cross section: u+d+s+c We are interested in the charm cross section. Subtract u,d,s background Below threshold only background * Means that there is no information on the splitting of systematic errors in correlated and uncorrelated

12 Experimental data: charm Sub-threshold and threshold BES 2001

13 Experimental data: charm Sub-threshold and threshold BES 2004

14 Experimental data: charm Sub-threshold and threshold BES 2006 (I)

15 Experimental data: charm Sub-threshold and threshold BES 2006 (II)

16 Experimental data: charm Sub-threshold and threshold BES 2009 *

17 Experimental data: charm Sub-threshold and threshold Crystal Ball 1986

18 Experimental data: charm Gap region Crystal Ball 1990 (I)

19 Experimental data: charm Gap region Crystal Ball 1990 (II)

20 Experimental data: charm High energy region CLEO 1979

21 Experimental data: charm High energy region CLEO 1998

22 Experimental data: charm High energy region CLEO 2007

23 Experimental data: charm Sub-threshold and threshold CLEO 2009 *

24 Experimental data: charm High energy region MD

25 Experimental data: charm Threshold and high energy PLUTO 1982 *

26 Experimental data: charm Threshold region MARKI 1976 *

27 Experimental data: charm Gap region MARKI 1977 *

28 Experimental data: charm Gap region MARKII 1979

29 Experimental data: charm Threshold and gap regions Mark-I I 1981

30 Experimental data: charm Perturbation theory Only where there is no data Assign a conservative 10% error to reduce model dependence M 1 6% M n>1 < 1%

31 Experimental data: charm Data used in Kühn K et al, Boughezal et al, Bodenstein et al and Narison Use perturbation theory right from here! Even though there is data available... 30% of the first moment! Finite energy sum-rule? Underestimates errors! (they assing only naive theory error)

32 Fit procedure

33 Fit procedure 1. Recluster data. Clusters not necessarily equally sized. Number of clusters and size of cluster according to the structure of the data

34 Fit procedure Cluster energy 2. Calculate the energy of the cluster. One weights the energy of the data points inside the clusters with their errors.

35 Fit procedure R n 3. Fit the value of R for each cluster. Data is allowed to move within its systematic error. The method renders errors and correlations among various clusters. One can then calculate errors and correlations for the moments.

36 Fit procedure Method inspired by a similar one in Hagiwara, Martin & Teubner. 2 Avoids the problems of a regular χ in which the systematic errors are 100% correlated One can fit for signal and background simultaneously Prediction for moments M n = m n 10 n+1 GeV n+1 M 1 = ± 0.20 stat ±0.46 sys M 2 = ± 0.18 stat ±0.29 sys M 3 = ± 0.19 stat ±0.25 sys M 4 = ± 0.19 stat ±0.23 sys We also predict correlations among the various moments, useful for simultaneous fits.

37 Fit results Below threshold 1 cluster

38 Fit results Below resonances 2 clusters

39 Fit results First resonance 20 clusters

40 Fit results Second resonance 20 clusters

41 Fit results Continuum data 10 clusters

42 Theoretical developments

43 Methods in perturbation theory Fixed order i αs( μα ) ab, a mc( μ ) b m mc( μm) exp n = i log log 2 n 2 2 = n 4 m ( ) i 0 π μ c μ = m m μα M C M Expanded 1 i 1 α ( μ ) ab a ( ) b ( ) out 2n α m μ m μ M = C log log M = 2 mc( μm) i= 0 π μm μα s, c m c m exp 2 n ( n) i ( n ) Iterative m (0) c 1 exp 2n n ( ) 1 exp M 2 n M = = 2C 2C n,0 n,0 m m C n i (0) 2 (0) 2 (0) αs( μα ) ˆ ab, a mc b mc c( μm) = c 1+ n, i log log 2 2 i= 1 π μm μα

44 Methods in perturbation theory Fixed order i αs( μα ) ab, a mc( μ ) b m mc( μm) exp n = i log log 2 n 2 2 = n 4 m ( ) i 0 π μ c μ = m m μα M C M Numerical solution for mass: sometimes there is no solution Expanded 1 i 1 α ( μ ) ab a ( ) b ( ) out 2n α m μ m μ M = C log log M = 2 mc( μm) i= 0 π μm μα s, c m c m exp 2 n ( n) i ( n ) Iterative m (0) c 1 exp 2n n ( ) 1 exp M 2 n M = = 2C 2C n,0 n,0 m m C n Analytic solution for mass always has a solution! i (0) 2 (0) 2 (0) αs( μα ) ˆ ab, a mc b mc c( μm) = c 1+ n, i log log 2 2 i= 1 π μm μα

45 Methods in perturbation theory Fixed order i αs( μα ) ab, a mc( μ ) b m mc( μm) exp n = i log log 2 n 2 2 = n 4 m ( ) i 0 π μ c μ = m m μα M C M μα and μ independent m Expanded 1 i 1 α ( μ ) ab a ( ) b ( ) out 2n α m μ m μ M = C log log M = 2 mc( μm) i= 0 π μm μα s, c m c m exp 2 n ( n) i ( n ) Iterative m (0) c 1 exp 2n n ( ) 1 exp M 2 n M = = 2C 2C n,0 n,0 m m C n i (0) 2 (0) 2 (0) αs( μα ) ˆ ab, a mc b mc c( μm) = c 1+ n, i log log 2 2 i= 1 π μm μα

46 Methods in perturbation theory Fixed order i αs( μα ) ab, a mc( μ ) b m mc( μm) exp n = i log log 2 n 2 2 = n 4 m ( ) i 0 π μ c μ = m m μα M C M μα and μ independent m residual μ and μ dependence α due to truncation of α series m Expanded 1 i 1 α ( μ ) ab a ( ) b ( ) out 2n α m μ m μ M = C log log M = 2 mc( μm) i= 0 π μm μα s, c m c m exp 2 n ( n) i ( n ) Iterative m (0) c 1 exp 2n n ( ) 1 exp M 2 n M = = 2C 2C n,0 n,0 m m C n i i residual dependence μ α renders correct μ dependence to the order of truncation i (0) 2 (0) 2 (0) αs( μα ) ˆ ab, a mc b mc c( μm) = c 1+ n, i log log 2 2 i= 1 π μm μα m

47 O α s first moment 3 ( ) analyses Contours in the μ μ plane α m Exclude regions with μ, μ < m ( m ) m α c c m ( m ) μ, μ 4GeV c c m α

48 O α s first moment 3 ( ) analyses Contours in the μ μ plane α m Kühn et al path! μα = μ m m ( m ) μ, μ 4GeV c c m α

49 Various error estimates Kühn 2GeV ( μ = μ ) 4GeV α m μ m = 3GeV 2GeV μ α 4GeV 3 O( α s ) analyses, first moment Double variation mm ( ) μ, μ 4GeV m α μ = m ( m ) m c c 2GeV μ α 4GeV

50 Results

51 Results Result for α ( m ) = ± s Z m ( m ) = ± ± ± ± ± c c stat sys th α GG = ± 0.025

52 Comparison to similar analyses 4-loops lattice data psedoscalar Sum rules 3-loops weighted finite energy QCD sum rules Sum rules 4-loops

53 Situation for bottom? Perturbation theory Perturbative QCD Aren t we comparing theory to theory? 10% error gives a huge error to the total moment 65% of the first moment for bottom sum rules!! + same issues with perturbative analysis

54 Conclusions and outlook It is essential to have a reliable error estimate for the charm mass. Concerning relativistic sum rules, a revision of perturbative errors was mandatory. Experimental input must be treated with care (combining various sets of data, correlations, systematic errors ) Perturbative QCD should be used only where there is no data, and asigning a conservative error. - For charm PQCD is only a small fraction of the moment small impact. The analysis can be easily extended to other correlators connection to lattice It can also be used to determine the bottom mass Stay tunned for updated numbers on charm, and for results on bottom mass and pseudoscalar correlators. Result for α ( m ) = ± s Z m ( m ) = ± ± ± ± ± c c stat sys th α GG = ± 0.025