Determination of Vxb

Transcription

1 Determination of Vxb Thomas Mannel, Siegen University Workshop on Flavour Dynamics Chamonix, France. October 8-15, 2005

2 Contents Motivation: Why is Vxb of interest? Exclusive Vcb Heavy Quark Symmetries Lattice determinations Perspectives Inclusive Vcb OPE Scheme Dependence Parton Hadron Duality Perspectives

3 Exclusive Vub Light Cone Sum rules Lattice determinations Perspectives Inclusive Vub OPE Shape functions Perspectives Summary

4 Motivation Determination of the Unitarity Triangle: V ubv ud + V cbv cd + V tbv td = 0! 3 "m s! "m d CKM f i t t e r EPS 2005 Vcb: Relative Normalization of the input from kaons! 2 sol. w/ cos 2! 1 # 0 (excl. at CL $ 0.95) Vub:! 2 V ub ρ 2 + η 2! 3!

5 Vcb

6 Exclusive Vcb Main ingredient: Form Factors dγ dw ( B D l ν l ) = G2 F 48π 3 V cb 2 m 3 D (w 2 1) 1/2 P (w)(f(w)) 2 dγ dw ( B Dl ν l ) = ( G 2 F 48π 3 V cb 2 (m B + m D ) 2 m 3 D (w2 1) 3/2 (G(w)) 2 ω = v v P (ω) : Phase Space Factor (5)

7 Heavy Quark Symmetries (Isgur, Wise, Shifman, Voloshin, Bigi, Grinstein, Uraltsev, Georgi, Falk, Luke, Neubert,...) Absolute Normalization of the form factors is known at ω=1 Corrections can F be calculated / estimated [ [ ] F(w) = η QED η A 1 + δ1/µ 2 + ]+(w 1)ρ 2 +O((w 1) 2 ) (6) G(1) = 1 + O ( mb m D m B + m D 1 µ η QED = 1.007, η A = ± µ = 1 m c 1 m b : Parameter of HQS breaking )

8 b c Form Factors from the Lattice Unquenched Calculations become available! Heavy mass limit is not used Lattice calculations of the deviation from unity F(1) = (Hashimoto, Kronfeld...) ± G(1) = ± ± (from the CKM05 talk by A. Kronfeld)

9 σ π B = 0 BPS Limit (Uraltsev) µ 2 π = µ 2 G * HQS Relations for the B D form factor hold to all orders * Expansion in µ2 π µ 2 G µ 2 π * Estimate G(1) = 1.04 ± 0.01 power ± 0.01 pert.

10 ] -3 F(1)! V cb [ # " = 1 OPAL (part. reco.) ALEPH DELPHI CLEO OPAL (excl.) AVERAGE DELPHI (part. reco.) BELLE BABAR ] -3 G(1)! V cb [ # " = 1 ALEPH BELLE CLEO AVERAGE HFAG Winter05 prel. 2 " /dof = 30.4/ ! ± 20 HFAG Winter05 prel. 2 " /dof = 0.3/ V cb = (41.2 ± 1.9) 10 3 (exclusive) 2!

11 Perspectives Further Progress in Lattice Calculations Full QCD Values at the level of 2-3% uncertainty Targeting 1% needs more analytical work Exclusive channels finally more precise than inclusive?

12 Inclusive Vcb (Bigi, Uraltsev, Shifman, Chay, Georgi, Grinstein, Manohar, Wise, Bauer, Ligeti... Basis: Operator Product Expansion (OPE) Yields a an expansion in 1/m Γ = Γ m Q Γ m 2 Q Γ m 3 Q Γ also for the spectra

13 Total rate Γ = V cb 2ˆΓ0 m 5 b (µ)(1 + A ew)a pert (r, µ) ( ) ( µ 2 π m 2, µ2 G ρ 3 b m 2 + z 3 (r) D b m 3 b [ z 0 (r) + z 2 (r) Non-Perturbative Parameters:, ρ3 LS m 3 b ) ] +... Λ = M B m b µ 2 π = B b(id ) 2 b B µ 2 G = B b(id µ )(idν )σ µν b B ρ 3 D ρ 3 LS = B b(id µ )(ivd)(id ν )b B = B b(id µ )(ivd)(idν )σ µν b B

14 Heavy Quark Parameters Use the information from the spectra Expansion in singular functions Consider Moments of the Spectra Charged Lepton Energy Hadronic Invariant Mass Mixed Moments... Simultaneous fit for HQ parameters and Vcb and the charm mass

15 Scheme Dependence Use of the pole mass = large radiative corrections Presence of Renormalons ==> short distance masses Two schemes on the market kinetic scheme (Bigi, Uraltsev, Shifman...) m kin (µ) : defined from a sum rule for the kinetic energy 1S scheme (Manohar, Hoang, Bauer, Ligeti...) m 1S (µ) : Defined from the (perturbative) calculation of the Upsilon(1S) mass Both schemes yield comparable and small uncertainties

16 Extraction of Vcb: Kinetic (Plots von CKM05, P. Urquijo)

17 Extraction of Vcb: 1S

18 ations are consistent with each other: 3 Vcb = (41.5 ± 0.7) 10 (inclusive) 3 Vcb = (41.2 ± 1.9) 10 (exclusive). this consistency may be viewed as a valida case further reduction of the uncertainty is, we nevertheless quote an average value, Vcb = (41.5 ± 0.6) 10 3.

19 Parton Hadron Duality (Shifman, Bigi, Uraltsev...) As an example: correlator of two (scalar) currents: T (q 2 ) = d 4 x e iqx 0 T ( J(x)J (0) ) ds 0 = 2π Naive notion of duality: ρ(s) q 2 s + iɛ T hadronic = T partonic, ρ hadronic = ρ partonic... : avaraging with a smooth weight function w... = dq 2... w(q 2 )

20 One possible definition of Duality Violations Truncation of perturbative series and OPE yields an estimate of the natural size of the corrections to be expected: O(α n+1 s (Q 2 ), (1/Q 2 ) m+1 ) Any Contribution larger than this natural size: Violation of Duality!! Analytic Continuation Q 2 Q 2 = m 2 b!! Negligibly small contributions in the Euclidean can become large in the Minkowskian

21 Check this with the help of data: Moments of various spectra... in various processes No indication for any Duality violation in any process No additional uncertainty can be / is added for a possible Duality violation

22 3 and exclusiv he values obtained from inclusive Vcb = (41.5 ± 0.7) 10 (inclusive) Summary oneach Vcb inations are consistent with 3 other: Vcb = (41.2 ± 1.9) 10 (exclusive 3 Vcb = (41.5 ± 0.7) 10 (inclusive) 3 consistency may be viewed as a vali Vcb = (41.2 ± 1.9) 10 (exclusive). his se further reduction of the uncertainty le this consistency may be viewed as a validati we nevertheless quote Average an average value h case further reduction of the uncertainty is u ed, we nevertheless quote an average 3 value, Vcb = (41.5 ± 0.6) 10 Vcb = (41.5 ± 0.6)

23 Vub

24 Exclusive Vub: B πl ν l Main ingredient: Form Factors π(p π ) V µ B(p B ) = [ ] f + (q 2 ) p µ B + pµ π m2 B m2 π q 2 q µ ] + f 0 (q 2 ) m2 B m2 π q 2 q µ Rate (vanishing lepton masses): dγ dq 2 = G2 F V ub 2 24π 3 p π 3 f + (q 2 ) 2.

25 Non-perturbative Method I: Light Cone Sum Rules F ( (p,q) = i (Ball, Braun, Zwicky, Khodjamirian, Melic,...) d 4 xe ipx ' + (q) T {ū$ ( b(x),m b bi$ 5 d(0)} 0 = Light-Cone Expansion, twist expansion Yields an estimate for f_b f_+(q²) Valid for small q² f_b is taken from a two-point sum rule Some theoretical uncertainties cancel

26 Uncertainties from Higher twists (>4) Results from LCSR b quark mass and renormalizations scale Condensate values Threshold and Borel parameter Pion distribution amplitude [ f + (0) = ± (5%) tw>4 ± (3%) mb,µ ± ± ±(3%) qq ± (3%) s B 0,M ± (8%) a π 2,4 ] f + (0) = ± V ub = (3.2 ± 0.1 ± 0.1 ± 0.3) 10 3 ; (Numbers from Ball, Zwicky)

27 Results from Lattice QCD 3 Unquenched estimates are available Results for large q² Extrapolation using Becirevic Kaidalov parametrization f 0 (q 2 ) HPQCD f + (q 2 ) HPQCD f 0 (q 2 ) Fermilab/MILC f + (q 2 ) Fermilab/MILC f + (q 2 ) = c B (1 α B ) (1 q 2 )(1 α B q 2 ) c (1 α ) Rate for q 2 > 16 GeV 2 V ub 2 (1.31 ± 0.33) ps 1, V ub 2 (1.80 ± 0.48) ps 1, q 2 in GeV 2 (HPQCD / Fermilab MILC)

28 Lattice Results for Vub V ub πlν q 2 >16 V ub πlν q 2 >16 = (3.87 ± 0.70 ± ) 10 3 (F NAL 04) = (4.73 ± 0.85 ± ) 10 3 (HP QCD)

29 Other Methods 1. Model-independent treatment of the long-distance effects in B K l + l at small recoil: OPE in 1/Q, Q q 2 m b 2. Improved heavy quark symmetry relations, including power corrections O(Λ/m b ) and hard gluon effects Complete systematic description of the exclusive rare radiative decays B K l + l at low recoil. Improved V ub determination including: NNLO hard gluon corrections power corrections O(Λ/m b )

30 dγ( B ρeν)/dq 2 dγ( B K l + l )/dq = V ub 2 2 V tb Vts 2 8π2 α 2 1 N eff (q 2 ) λ HB ρ λ (q 2 ) 2 λ HB K λ (q 2 ) 2 1. Compute the RG invariant factor N eff using the OPE N eff (q 2 ) = Ceff 9 ( 1 + 2m2 b q 2 C eff 7 C eff 9 ) 2 + C O( Λ m b ) 2. Use double ratios and D data to determine the SU(3) breaking. Define R B V (y) λ HB ρ λ (y) 2 R λ HB K λ (y) 2 D V (y) λ HD ρ λ (y) 2 λ HD K λ (y) 2 then Grinstein double ratios ( R B V (y) = R D V (y) 1 + O( m s m ) s ) m b m c

31 Perspectives in exclusive Vub Depends on progress in Lattice Gauge Theory Extend the range in q²: Moving NRQCD Perspectives in LCSR Talk by P. Ball

32 Inclusive Vub (Bigi, Uraltsev, Shifman, Bauer, Ligeti,... Bosch, Lange, Neubert, Paz (BLNP) Cut against charm background: Standard OPE fails * Standard Expansion of B Xu l ν: (y=2e/m) dγ dy = G2 F V ub 2 m 5 Q 192π 3 λ 1 + 3λ 2 3m 2 Q [( 2y 2 (3 2y) + 10y2 3 ] δ(1 y) λ 1 3m 2 δ (1 y) Q * Similar for the Photon Energy Spectrum in B Xs γ: λ 1 m 2 Q + 2y(6 + 5y) λ 2 m 2 Q ) Θ(1 y). (4.95)

33 Resummation: Shape function

34 SCET and shape functions * Inclusive rates in the endpoint region become dγ = H J S H: Hard Coefficient Function, Scales ~ m J: Jet Function, Scales ~ (m Λ), Contraction of Collinear Quarks (Sterman Korchemski) S: Shape Function, Scales ~ Λ, Matrix Element with soft quarks and Gluons

35 P + Example: P spectrum and B Xu l ν + P + = E H P H P = E H + P H m 2 π m B P + P m B (Recksiegel, M., P Bosch, Lange, Neubert, Paz)

36 Results for Vub S<:T6&J 9:,- 9&&&G&&HIF43&! KL Y,""&5<6B.=>5,6>#B.&>%ZZ#.Z 9:,- 9&&&G&&HIFLI&! 3F8M&! KL 9:,- 9&&&G&&HLFN3&! 2 U T&V 2 H"&*41 B01E4)0& * ) +- ' * ) +#, KL $M K'8+,-.< 9:'E/2/3# =243'M!;"# N@ON'7?@NA'7?@NP'7?@JP 8P@Q 7 O@Q''7 A@R<S N@PO'7?@A?'7?@AT'7?@UJ 8J?@N'7JJ@T'7 T@A<S A@AT'7?@AO'7?@A?'7?@UU 8JJ@J 7 O@?''7 A@O<S 9:,- 9&&&G&&HLFNO&! 3F84&! KL W [\"2/6" A@?Q'7?@TJ!"#$%&'#($%)*&+$,-"$%.%&/&01234*&2%567&8334 Belle BaBar V ub = (4.38 ± 0.19 ± 0.27) Current uncertainty mainly driven by the uncertainty in the b-quark mass

37 Perspectives in inclusive Vub Currently the best method for Vub Shape function insensitive methods Fantastic progress over the last few years... but uncertainties below 5% are futuristic

38 Summary V cb = (41.5 ± 0.6) V ub = (4.30 ± 0.30) 10 3 * Vcb is almost as well known as the Cabibbo angle! * Uncertainty in Vub is mainly driven by the b quark mass