| V ub | from QCD Sum Rules on the Light-Cone Patricia Ball IPPP - PowerPoint PPT Presentation

| V ub | from QCD Sum Rules on the Light-Cone Patricia Ball IPPP , Durham CKM06, 14 December 2006 Based on Ball/Zwicky, hep-ph/0406232; Ball, hep-ph/0611108.

Theory Input for Semileptonic Decays Form factors: � π ( p ) | ¯ uγ µ (1 − γ 5 ) b | B ( p + q ) � ( q + 2 p ) µ f + ( q 2 ) + m 2 B − m 2 π f 0 ( q 2 ) − f + ( q 2 ) � � = q µ q 2 1 0 ≤ q 2 ≤ ( m B − m π ) 2 m 2 B − m 2 � � ← → m π ≤ E π ≤ π 2 m B 0 ≤ q 2 ≤ 26 . 4 GeV 2 ← → 0 . 14 GeV ≤ E π ≤ 2 . 6 GeV Theoretical methods: lattice → J. Flynn’s talk dispersive constraints → I. Stewart’s talk QCD sum rules on the light-cone → this talk! – p.1

QCD Sum Rules on the Light-Cone Basic quantity: correlation function: � biγ 5 d ](0) | 0 � LCE T ( n ) uγ µ b ]( y )[ m b ¯ d 4 ye iqy � π ( p ) | T [¯ � ⊗ φ ( n ) i = π H n φ ( n ) π : π distribution amplitudes (DAs) T ( n ) H : perturbative amplitudes n : twist LCE: light-cone expansion m 2 � � B f B f + ( q 2 ) = 2 p µ + higher poles and cuts + . . . m 2 B − p 2 B B meson described by Euclidean current + plus analytical continuation – p.2

QCD Sum Rules on the Light-Cone Features of LCSRs: terms in LCE ordered in powers of 1 /m b → need to include higher-twist terms ( n > 2 ) � T ( n ) ⊗ φ ( n ) implies factorization – valid at higher twist? π H calculate O ( α s ) , known for T2 ( π (Khodjamirian et al. 97, Ball et al. 97) , ρ (Ball/Braun 98) ) T3 ( π (Ball/Zwicky 2001) ) → factorization OK use standard SR techniques: Borel-transformation, continuum model introduce irreducible systematic uncertainty ∼ 10% – p.3

QCD Sum Rules on the Light-Cone Ball/Zwicky 04: f + (0) = 0 . 258 ± 0 . 031 f � , f o , f T for Π 0.9 with theory input for 0.8 leading-twist π distribution 0.7 amplitude φ π ;2 0.6 0.5 Ball/Zwicky 05: constrain 0.4 φ π ;2 from experimental q 2 0.3 spectrum of B → πeν : 14 q 2 2 4 6 8 10 12 f + (0) ≈ 0 . 27 and BZ 04 | V ub | = (3 . 2 ± 0 . 4) · 10 − 3 Results for B → ρeν also available — but less experimental information. – p.4

Theory Assisted by Experiment 0.14 0.12 0.1 δ B / B 0.08 2006 BaBar results for q 2 0.06 spectrum in B → πeν in 0.04 12 bins (up from 5 bins in 0.02 2005) 0. 5. 10. 15. 20. q 2 [GeV 2 ] Strategy: Parametrise form factor, fit to data, extract | V ub | f + (0) . – p.5

Form Factor Parametrisations Becirevic/Kaidalov (BK) : f + (0) f + ( q 2 ) = � , 1 − q 2 /m 2 1 − α BK q 2 /m 2 � � � B ∗ B where α BK determines the shape of f + and f + (0) the normalisation; ▽ – p.6

Form Factor Parametrisations Becirevic/Kaidalov (BK) : f + (0) f + ( q 2 ) = � , 1 − q 2 /m 2 1 − α BK q 2 /m 2 � � � B ∗ B where α BK determines the shape of f + and f + (0) the normalisation; Ball/Zwicky (BZ): � � rq 2 /m 2 1 f + ( q 2 ) = f + (0) B ∗ + , 1 − q 2 /m 2 1 − q 2 /m 2 1 − α BZ q 2 /m 2 � � � � B ∗ B ∗ B with the two shape parameters α BZ , r and the normalisation f + (0) ; BK is a variant of BZ with α BK := α BZ = r . – p.6

Form Factor Parametrisations the AFHNV parametrisation (Flynn et al.), based on an ( n + 1) -subtracted Omnes representation of f + : n 1 � α i ( q 2 ) , f + ( q 2 ) n ≫ 1 � f + ( q i ) 2 ( s th − q 2 � = i ) s th − q 2 i =0 n s − s j s th = ( m B + m π ) 2 ; � with α i ( s ) = , s i − s j j =0 ,j � = i the shape parameters are f + ( q 2 i ) /f + ( q 2 0 ) with q 2 0 ,...n the subtraction points; the normalisation is given by f + (0) . – p.7

Form Factor Parametrisations the BGL parametrisation based on analyticity of f + : ∞ 1 0 )] k , � f + ( q 2 ) = a k ( q 2 0 )[ z ( q 2 , q 2 k a 2 � k ≤ 1 , P ( q 2 ) φ ( q 2 , q 2 0 ) k =0 0 ) = { ( m B + m π ) 2 − q 2 } 1 / 2 − { ( m B + m π ) 2 − q 2 0 } 1 / 2 z ( q 2 , q 2 { ( m B + m π ) 2 − q 2 } 1 / 2 + { ( m B + m π ) 2 − q 2 0 } 1 / 2 q 2 0 : free parameter, determines maximum | z | ; define 0 = 20 . 1 GeV 2 , | z | < 0 . 28 : q 2 BGLa : q 2 BGLb 0 = 0 , | z | < 0 . 52 systematic expansion in the small parameter z ; truncate at k max ; choose k max = 2 for BGLa and k max = 3 for BGLb . – p.8

| V ub | f + (0) from data Param. | V ub | f + (0) Remarks (9 . 3 ± 0 . 3 ± 0 . 3) × 10 − 4 χ 2 BK min = 8 . 74 / 11 dof α BK = 0 . 53 ± 0 . 06 (9 . 1 ± 0 . 5 ± 0 . 3) × 10 − 4 χ 2 BZ min = 8 . 66 / 10 dof α BZ = 0 . 40 +0 . 15 − 0 . 22 , r = 0 . 64 +0 . 14 − 0 . 13 (9 . 1 ± 0 . 6 ± 0 . 3) × 10 − 4 χ 2 BGLa min = 8 . 64 / 10 dof (9 . 1 ± 0 . 6 ± 0 . 3) × 10 − 4 χ 2 BGLb min = 8 . 64 / 9 dof (9 . 1 ± 0 . 3 ± 0 . 3) × 10 − 4 χ 2 AFHNV min = 8 . 64 / 8 dof from B − → π − π 0 (Arnesen et al.) (8 . 0 ± 0 . 4) × 10 − 4 SCET (tree-level, no 1 /m b corrections) All parametrisations agree – model-independent result! – p.9

Fitted Form Factor 1.04 f + ( q 2 ) /f BGLa ( q 2 ) f + ( q 2 ) + 6. 1.02 4. 1. 0.98 2. 0.96 0. 0. 5. 10. 15. 20. 25. 0. 5. 10. 15. 20. 25. q 2 [GeV 2 ] q 2 [GeV 2 ] Left panel: best-fit form factors f + as a function of q 2 . The line is an overlay of all five parametrisations. Right panel: best-fit form factors normalised to BGLa. Solid line: BK, long dashes: BZ, short dashes: BGLb, short dashes with long spaces: AFHNV. – p.10

Results for | V ub | Procedure 1: take FF from theory calculation, fit to BK and extract | V ub | from experimental partial branching ratio ( q 2 ≤ 16 GeV 2 for LCSR, q 2 ≥ 16 GeV 2 for lattice) α BK = 0 . 63 +0 . 18 LCSR f + (0) = 0 . 26 ± 0 . 03 , − 0 . 21 | V ub | = (3 . 5 ± 0 . 6(th) ± 0 . 1(exp)) × 10 − 3 | V ub | f + (0) = (9 . 0 +0 . 7 − 0 . 6 ± 0 . 4) × 10 − 4 α BK = 0 . 56 +0 . 08 HPQCD f + (0) = 0 . 21 ± 0 . 03 , − 0 . 11 | V ub | = (4 . 3 ± 0 . 7 ± 0 . 3) × 10 − 3 | V ub | f + (0) = (8 . 9 +1 . 2 − 0 . 9 ± 0 . 4) × 10 − 4 α BK = 0 . 63 +0 . 07 f + (0) = 0 . 23 ± 0 . 03 , FNAL − 0 . 10 | V ub | = (3 . 6 ± 0 . 6 ± 0 . 2) × 10 − 3 | V ub | f + (0) = (8 . 2 +1 . 0 − 0 . 8 ± 0 . 3) × 10 − 4 – p.11

Results for | V ub | Procedure 2: take FF from theory, fit to experimentally determined shape, BGLa , obtain f + (0) , extract | V ub | from full branching ratio. f + (0) = 0 . 26 ± 0 . 03 LCSR | V ub | = (3 . 5 ± 0 . 4(shape) ± 0 . 1( B )) × 10 − 3 HPQCD f + (0) = 0 . 21 ± 0 . 03 | V ub | = (4 . 3 ± 0 . 5 ± 0 . 1) × 10 − 3 FNAL f + (0) = 0 . 25 ± 0 . 03 | V ub | = (3 . 7 ± 0 . 4 ± 0 . 1) × 10 − 3 reduced theoretical uncertainty as shape of FF is fixed by experimental data reduced experimental uncertainty as total B ( B → πeν ) can be used – p.12

Summary form factor calculations from QCD sum rules on the light-cone in mature shape no scope for major improvement LCSR predictions for small and moderate q 2 < 16 GeV 2 → LQCD predictions for large q 2 > 16 GeV 2 ← reduce error of | V ub | determination by fixing shape of form factor from experiment instead of theory data both LCSR and FNAL prefer small | V ub | ∼ 3 . 6 × 10 − 3 HPQCD points at larger | V ub | ∼ 4 . 3 × 10 − 3 UTangles gives | V ub | = (3 . 50 ± 0 . 18) × 10 − 3 How sure are we about the inclusive result? (both th. and exp.) – p.13