Form Factors for Rare B Decays from Lattice QCD

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1 Form Factors for Rare B Decays from Lattice QCD Matthew Wingate DAMTP, Cambridge

2 Outline Motivation Form factors Rare (FCNC) decays Preliminary results Summary

3 Two views of a weak decay Phenomenologist: Illustration from I. Shipsey, Nature 427, 591 (2004) Experimentalist:

4 Two views of a weak decay Phenomenologist: Illustration from I. Shipsey, Nature 427, 591 (2004) Lattice theorist Experimentalist:

5 Two views of a weak decay Phenomenologist: Illustration from I. Shipsey, Nature 427, 591 (2004) Lattice theorist Experimentalist:

6 Full set of form factors Matrix element Form factor Relevant decay(s) P qγ µ b B f +, f 0 f T B πlν B Kl + l P qσ µν q ν b B B Kl + l V qγ µ b B V qγ µ γ 5 b B V A 0, A 1, A 2 { B (ρ/ω)lν B K l + l V qσ µν q ν b B V qσ µν γ 5 q ν b B T 1 T 2, T 3 { B K γ B K l + l

7 Full set of form factors Matrix element Form factor Relevant decay(s) P qγ µ b B f +, f 0 f T B πlν B Kl + l P qσ µν q ν b B B Kl + l V qγ µ b B V qγ µ γ 5 b B V A 0, A 1, A 2 { B (ρ/ω)lν B K l + l V qσ µν q ν b B V qσ µν γ 5 q ν b B T 1 T 2, T 3 { B K γ B K l + l... also make the spectator an s quark for Bs decays

8 B π l ν, reviewed by J. Laiho (CKM2010) BaBar result for V ub exclusive Exclusive B πlν and V ub Talk by Martin Simard. Two different analyses, π-η analysis and π-ρ analysis. Different fit and cut strategies. Fits to different numbers of modes, loose vs tight ν cut selection. Results for V ub consistent within the different approaches. Results from π-η analysis: New result from Belle (talk by Kevin Varvell) 2 ) / 2 GeV 2!B(q "10 Theory q 2 (GeV) 2 V ub (10 3 ) HPQCD > ± 0.13 ± FNAL > ± 0.12 ± LCSR < ± 0.07 ± Results for V ub ISGW2 HPQCD FNAL LCSR Data Unfolded q (GeV ) V ub status update from LCSR Warwick, S Using a BK parameterization for Belle experimental data V ub was extracted from the partial branching fraction for a number of different theories to give the normalization. Simultaneous fit to lattice (Fermilab/MILC) and Belle q 2 dependence (using the z parameterization) leads to a model independent result of V ub = (3.43 ± 0.33) The same procedure with the latest BaBar data leads to V ub = (3.14 ± 0.07 ± ) Talk by Patricia Ball. LCSR yields value for f B f + (q 2 ), and for consistency, calculation in progress to determine f B from QCD sum rules. Test sensitivity to radiative corrections by doing calculation to order α 2 s. Sum rule results for V ub have an 10% irreducible theory uncertainty, but are still useful at the moment, given the current precision of the lattice and the tension with the inclusive determination. Sum rule results (which use exclusive B πlν) tend to be less than 4 and in better agreement with lattice.

9 B π l ν, reviewed by J. Laiho (CKM2010) BaBar result for V ub exclusive Talk by Martin Simard. Two different analyses, π-η analysis and π-ρ analysis. Different fit and cut strategies. Fits to different numbers of modes, loose vs tight ν cut selection. Results for V ub consistent within the different approaches. Results from π-η analysis: Theory q 2 (GeV) 2 V ub (10 3 ) HPQCD > ± 0.13 ± FNAL > ± 0.12 ± LCSR < ± 0.07 ± Results for V ub Using a BK parameterization for Belle experimental data V ub was extracted from the partial branching fraction for a number of different theories to give the normalization. Simultaneous fit to lattice (Fermilab/MILC) and Belle q 2 dependence (using the z parameterization) leads to a model independent result of V ub = (3.43 ± 0.33) The same procedure with the latest BaBar data leads to V ub = (3.14 ± 0.07 ± ) Exclusive B πlν and V ub New result from Belle (talk by Kevin Varvell) 2 ) / 2 GeV 2!B(q "10-6 ISGW2 HPQCD FNAL LCSR Data Unfolded q (GeV ) V ub status update from LCSR Talk by Patricia Ball. Warwick, S LCSR yields value for f B f + (q 2 ), and for consistency, calculation in progress to determine f B from QCD sum rules. Test sensitivity to radiative corrections by doing calculation to order α 2 s.... [best] review talk... ever heard! -- la Repubblica Sum rule results for V ub have an 10% irreducible theory uncertainty, but are still useful at the moment, given the current precision of the lattice and the tension with the inclusive determination. Sum rule results (which use exclusive B πlν) tend to be less than 4 and in better agreement with lattice.

10 b s is rare in the SM b W s b t s t γ, Z W ν W H eff = G F 2 V tb V ts 10 i=1 C i (µ)q i (µ) l l b s b s γ l l

11 Dominant operators Decays B K γ B s φ γ B (ρ/ω)γ B K ( ) l + l B s φ l + l SM operators Q 7γ = e 8π 2 m b s i σ µν (1 + γ 5 )b i F µν Q 9V = e 8π 2 ( s b) V A ( l l) V Λ b Λ γ Λ b Λ l + l Q 2 = ( s c) V A ( c b) V A

12 Long distance effects Phenomenological calculations necessary Charmonium resonances b c c s Khodjamirian, et al, PLB 402 (1997) Grinstein & Pirjol, PRD 62 (2000), PRD 70 (2004) Khodjamirian, et al, arxiv: γ, Z Weak annihilation B b W s, d K ρ doubly Cabibbo-suppressed u u γ Ball, Jones, Zwicky, PRD 75 (2007)

13 Regions of applicability B X s l + l J/ψ ψ Short distance effects dominate at low q 2 Short distance effects thought to dominate at large q 2 q 2 (GeV 2 ) Plot from E Lunghiʼs CKM2008 talk

14 Form factors in B K (*) l + l - Differential branching fractions in q 2 regions Expt. in agreement with present SM estimates (Ali, Ball, Handoko, Hiller, PRD 61, 2000) Form factors needed as input for calculations of other observables: e.g. AFB (when nonzero), FL, AT (j) (Bobeth, Hiller, van Dyk, arxiv:1006:5013)

15 [ ] Form factor definitions q ν K (p, λ) sσ µν b B(p) =4T 1 (q 2 )ɛ µνρσ e ν λ pρ p σ, q ν K (p, λ) sσ µν γ 5 b B(p) =2iT 2 (q 2 ) [ eλµ (M2 B M2 K ) (eλ q)(p + ] p ) µ +2iT3 (q 2 )(eλ q) q [q 2 ] µ MB 2 (p + p ) µ M2 K K (p, λ) sγ µ b B(p) = 2iV (q2 ) M B + M K ɛ µνρσ e λν p ρp σ,. K (p, λ) sγ µ γ 5 b B(p) =2M K A 0 (q 2 ) [ +(M B + M K )A 1 (q 2 ) e µ λ A 2 (q 2 ) e λ q M B +M K ] e λ q q µ q 2 e λ q q 2 q µ [ p µ + p µ M2 B M2 K q 2 q µ ].

16 [ ] Form factor definitions q ν K (p, λ) sσ µν b B(p) =4T 1 (q 2 )ɛ µνρσ e ν λ pρ p σ, q ν K (p, λ) sσ µν γ 5 b B(p) =2iT 2 (q 2 ) [ e λµ (M2 B M2 K ) (e λ q)(p + p ) µ ] +2iT3 (q 2 )(e λ q) [q µ K (p, λ) sγ µ b B(p) = 2iV (q2 ) ɛ µνρσ eλν M B + M p ρp σ, K q 2 M 2 B M2 K (p + p ) µ Factor of 2 different from Becirevic, et al; Ball, et al. ]. K (p, λ) sγ µ γ 5 b B(p) =2M K A 0 (q 2 ) [ +(M B + M K )A 1 (q 2 ) e µ λ A 2 (q 2 ) e λ q M B +M K ] e λ q q µ q 2 e λ q q 2 q µ [ p µ + p µ M2 B M2 K q 2 q µ ].

17 Sum rule determinations Ball & Zwicky, Phys. Rev. D 71, (2005) q 2 /GeV 2 q 2 /GeV 2 ρ,ω final states in same paper; π, η, K final states in Ball & Zwicky, PRD 71, (2005)

18 Early LQCD work on B K* γ Bowler, et al. (UKQCD) (1994) Bernard, Hsieh, Soni (1994) Abada, et al. (APE) (1996) Bhattacharya and Gupta (1995) Del Debbio, et al. (UKQCD) (1998)

19 BLM quenched B K* Bećirević-Lubicz-Mescia, Nucl. Phys. B769, 31 (2007) Most recent study of form factor for B K γ Calculate with heavy quarks such that m H m D Allows calculation with q 2 = 0 Extrapolate using T H V (0) m 3/2 H s = c 0 + c 1 m 1 H s + c 2 m 2 H s Quenched result: T B K (q 2 = 0; µ = m b ) = 0.24 ± T B K (0)/T B ρ (0) = 1.2 ± 0.1

20 BLM quenched B K* Bećirević-Lubicz-Mescia, Nucl. Phys. B769, 31 (2007) B! K * " # 3 T 2 (q 2 ) lin. T 1 (q 2 ) lin. T 2 (q 2 ) quad. T 1 (q 2 ) quad. T 1,2 (q 2 ) q 2 [GeV 2 ]

21 UKQCD quenched B ρlν Bowler, Gill, Maynard, Flynn, JHEP 05 (2004) 035 Form factors A0, A1, A2, V Extrapolate up in mq Partially integrated decay rate For 12.7 GeV 2 <q 2 < 18.2 GeV 2 Γ PI / V ub = ps 1

22 UKQCD quenched B ρlν Bowler, Gill, Maynard, Flynn, JHEP 05 (2004) q 2 (GeV) A1 A2 q 2 (GeV) A0!=6.2!=6.0 V q 2 (GeV) 2 q 2 (GeV) 2 Figure 4: The form factors on both lattices. The vertical scale is different for each form factor. 0

23 Lattice NRQCD approach with Stefan Meinel, Zhaofeng Liu, Eike Müller, A. Hart, R. Horgan NRQCD formulation to calculate QCD dynamics of physically heavy b quark Matching to MSbar scheme in pert. th. (Hart, Horgan, Müller, in prep) Can work in lattice frame boosted relative to B (Horgan et al, PRD 80, 2009) Stat. and EFT errors mandate working at low recoil Nf = (MILC) configurations. No unquenched calculations of B V form factors published yet.

24 Lattice data Toward high statistics MILC lattices (2+1 asqtad staggered) a(fm) am sea Volume N conf N src am val coarse / / / /0.04 fine / /0.031 Light meson momenta (units of 2π/L) (p x, p y, p z ) = (0, 0, 0). ( q,0,0), (0, q,0), (0,0, q), where q=1 or 2. (1,1,0), (1,-1,0), (1,0,1), (1,0,-1), (0,1,1), (0,1,-1). (1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1). Many Source/Sink separations (16 coarse, 22 fine) So far, only v=0 NRQCD used (B at rest). Larger v (mnrqcd) next. Leading order (HQET) current presently used. 1/mb current matrix elements computed, matching calc. in progress

25 Preliminary results pf = Z. Liu, CKM2010

26 Preliminary results Extrapolation of T 1 and T 2 to q 2 =0 Pole dominance [Becirevic & Kaidalov (2000), Ball & Zwicky (2005), Becirevic et al. (2007)] T 1 (q 2 )= T (0) (1 q 2 )(1 α q 2 ), T 2(q 2 )= T (0) 1 q 2 /β, q2 = q 2 /MB 2 s. Z. Liu, CKM2010 T (0) = 0.161(45) if M B s T (0) = 0.164(38) if M B s is a free parameter (left graph). = GeV is fixed from PDG2010. with Stefan Meinel, Alistair Hart, Ron R. Horgan, Eike Zhaofeng Liu (DAMTP, University of Cambridge Lattice calculation of B K ( ) ll form factors University of Warwick 14 / 12

27 Preliminary results Extrapolation of T 1 and T 2 to q 2 =0 Pole dominance [Becirevic & Kaidalov (2000), Ball & Zwicky (2005), Becirevic et al. (2007)] T 1 (q 2 )= Multiply by 2 to compare T (0) (1 q 2 )(1 α q 2 ), T 2(q 2 )= T (0) 1 q 2 /β, q2 = q 2 /MB 2 s. Becirevic, et al; Ball, et al. Z. Liu, CKM2010 T (0) = 0.161(45) if M B s T (0) = 0.164(38) if M B s is a free parameter (left graph). = GeV is fixed from PDG2010. with Stefan Meinel, Alistair Hart, Ron R. Horgan, Eike Zhaofeng Liu (DAMTP, University of Cambridge Lattice calculation of B K ( ) ll form factors University of Warwick 14 / 12

28 Form factor shapes With limited data now, try Becirevic-Kaidalov, Ball- Zwicky formulae Much work done on (z) series expansion recently Boyd, Grinstein, Lebed, PRL (1995); Boyd, Savage, PRD (1997); Caprini, Lellouch, Neubert; Arneson et al, PRL (2005); Bharucha, Feldmann, Wick, arxiv: HPQCD D K paper: generalized expansion to simultaneously fit q 2, mq, & a dependences Na et al, arxiv:

29 Light quark mass extrapolation In present LQCD calculations, masses and kinematics are such that ρ and K* are stable Could there be large threshold effects in matrix elements? Simpler to study ρ and K* decay constants Internal consistency check: Vub from B πlν and B ρlν. Agreement would suggest extrapolations trustworthy within errors.

30 Summary Calculations in high gear now Lattice complications Worse statistics than pseudoscalar final states Light quark mass extrapolations through thresholds Phenomenology essential Estimate effects of long distance effects Check of sum rule results, esp. at large q 2 Being further from perfection is no excuse for giving up!

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