QCD Effects in Exclusive Rare B-Meson Decays

Transcription

1 QCD Effects in Exclusive Rare B-Meson Decays Alexander Khodjamirian UNIVERSITÄT SIEGEN Continuous Advances in QCD, Minneapolis 2011 [A.K., Th. Mannel, A. Pivovarov and Yu-M. Wang, JHEP09 (2010) 089] Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 1 / 27

2 Introduction γ,z l + l B K ( ) l + l or B K γ, proceed via FCNC, BR b t W t s l + l b s benchmark channels for flavour physics, q 2 = (p l + + p l ) 2 or q 2 = 0 first data: BABAR, Belle, CDF, precise LHCb data expected B K ( ) Standard Model: t, W, Z and high-virtuality gluons absorbed in effective operators H eff, the rest - hadronic matrix elements Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 2 / 27

3 B K ( ) l + l in the Standard Model Computation of the decay amplitude: A(B K ( ) l + l ) = K ( ) l + l H eff B, H eff = 4G 10 F V tb Vts C i (µ)o i (µ), 2 i=1 dominant contributions from the b sll or b sγ operators: O 9(10) = αem 4π [ s Lγ µ b L ]lγ µ (γ 5 )l, C 9 (m b ) 4.4, C 10 (m b ) = 4.7, O 7 = em b 8π 2 [ sσ µν (1 + γ 5 )b]f µν, C 7 (m b ) 0.3 four-quark operators, combined with e.m. interaction: O (c) 1 = [ s L γ ρ c L ][ c L γ ρ b L ], C 1 (m b ) 1.1 O (c) 2 = [ c L γ ρ c L ][ s L γ ρ b L ], C 2 (m b ) 0.25 quark-penguin (O 3 6 ) and gluon-penguin (O 8 ) operators Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 3 / 27

4 Charm-loops a combination of the ( sc)( cb) weak interaction (O 1,2 ) and e.m.interaction ( cc)( ll) mimicking FCNC Charm-loop effect: l l l l l l b c c s b c similar effects: u, d, s, c, b-quark loops (quark-penguin operators O 3 6 ), u-loops from O1,2 u (CKM suppressed in b s ), A(B K ( ) l + l ) include specific hadronic matrix elements, not just form factors c s b c c x s Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 4 / 27

5 Charm-loop in B K ( ) l + l γ c b s B K K factorizable c-quark loop C 9 C 9 + (C 1 + 3C 2 )g(m 2 c, q 2 ) (a) (b) perturbative gluon corrections being factorized in O(α s ) and added to C 9 [M. Beneke,T. Feldmann,D. Seidel (2001)] (c) (d) how important are the soft gluons (low-virtuality, nonvanishing momenta) emitted from the c-quark loop? Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 5 / 27

6 Charm loop turns charmonium at q 2 mj/ψ 2,... an on-shell hadronic state: B J/ψK J/ψ l + l other ψ-resonances, open-charm states (e.g., B DDK K l + l ), M cc m B m ( ) K B ψk : the naive factorization fails, hinting at large nonfactorizable contributions to avoid huge backgr., J/ψ and ψ(2s) -regions are subtracted from the q 2 -distribution data in B K ( ) l + l the effect of intermediate/virtual cc states remains at q 2 m 2 J/ψ (nonperturbative at q2 4m 2 c ) Can we use the { loop corrections } ansatz? Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 6 / 27

7 Isolating the charm-loop in the decay amplitude the contribuiton of O 1,2 and e.m. interaction: A(B K ( ) l + l ) (O1,2) = (4πα em Q c ) 4G F lγ V tb Vts µ l 2 q 2 H(B K ( ) ) µ, the relevant hadronic matrix element: { H (B K ( ) ) µ (p, q) = i d 4 xe iq x K ( ) (p) T c(x)γ µ c(x), [ ]} C 1 O 1 (0) + C 2 O 2 (0) B(p + q), at small q 2 4mc 2 use the operator-product expansion (OPE) for the T -product: { } Cµ(q) a = d 4 xe iq x T c(x)γ µ c(x), c L (0)Γ a c L (0), Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 7 / 27

8 Expansion near the light-cone the dominant region in this T -product: x 2 1/(2m c q 2 ) 2 at q 2 4m 2 c: T - product of cc-operators can be expanded near the light-cone x 2 0, schematically, T { c(x)γ µ c(x), c L (0)γ ρ c L (0)} = C µρ 0 (x 2, m 2 c) + two-gluon term +... T { c(x)γ µ c(x), c L (0)γ ρ λ a 2 c L(0)} = 1 0 duc µραβ 1 (x 2, m 2 c, u)g a αβ(ux) +... after x-integration and taking hadronic matrix element: O(C 1 )/O(C 0 ) O(C n+1 )/O(C n ) Λ 2 QCD /(4m2 c q 2 ), but! no local expansion possible in each term of LC OPE: O(C 1 ) k=0 (qλ QCD) k /(4m 2 c q 2 ) k+1, q m b /2 and m b Λ QCD m 2 c. Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 8 / 27

9 The resulting effective operators LO reduced to simple cc -loop, no difference between local and LC, O µ (q) = (q µ q ρ q 2 9 g µρ ) 32π 2 g(m2 c, q 2 ) s L γ ρ b L. gluon emission: use c-quark propagator near the light-cone in the external gluon field [I. Balitsky, V. Braun (1999)] define LC kinematics (n ± ) in the rest-frame of B, q (m b /2)n + one-gluon emission yields a new nonlocal operator: Õ µ (q) = dω I µραβ (q, m c, ω) s L γ ρ δ[ω (in +D) ] G αβ b L, 2 Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 9 / 27

10 The local OPE limit ω 0 in the nonlocal operator, no derivatives of G µν Õ (0) µ (q) = I µραβ (q) s L γ ρ Gαβ b L, I µραβ (q, m c ) = (q µ q α g ρβ + q ρ q α g µβ q 2 g µα g ρβ ) 1 16π dt t(1 t) m 2 c q 2 t(1 t) At q 2 = 0, the quark-gluon operator obtained in B X s γ in [M.Voloshin (1997)] in B K γ [A.K.,G. Stoll,R. Rueckl,D. Wyler(1997)] the neccesity of resummation was discussed before [Z. Ligeti, L. Randall and M.B. Wise,(1997); A.K. Grant, A.G. Morgan, S. Nussinov and R.D. Peccei (1997); J. W. Chen, G. Rupak and M. J. Savage,(1997); G. Buchalla, G. Isidori and S.J. Rey (1997)] Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 10 / 27

11 Hadronic matrix elements for the charm-loop effect the LO: factorized cc loop [ ] H (B K ( ) ) µ (p, q) = fact ( C1 3 + C 2 reduced to B K ( ) form factors, nothing new ) K ( ) (p) O µ (q) B(p + q), The gluon emission yields: [ ] H (B K ( ) ) µ (p, q) = 2C 1 K ( ) (p) Õµ(q) B(p + q). nonfact new hadronic matrix element K ( ) (p) s L γ ρ δ[ω (in +D) ] G αβ b L B(p + q), 2 Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 11 / 27

12 Charm-loop effect in B K l + l The invariant amplitude: [ H µ (B K ) (p, q) ]fact.+nonfact. = [ (p q)q µ q 2 ] p µ [ (C1 ) ] 3 + C 2 A(q 2 ) + 2C 1 Ã(q 2 ) the factorizable part A(q 2 ) = 9 32π 2 g(m 2 c, q 2 )f + BK (q2 ) Wilson coefficients enhance the nonfact. part C 1 /3 + C 2 C 1 need nonperturbative QCD methods to calculate the form factor f + BK (q2 ) and the nonfactorizable amplitude Ã(q2 ) use one and the same LCSR approach Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 12 / 27

13 LCSR with B-meson distribution amplitudes [ A.K., T. Mannel, N.Offen,2005 ] p d B u b q (a) B B (b) (c) a similar approach: LCSR for B π in SCET [F.De Fazio, Th. Feldmann and T. Hurth, (2005)] Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 13 / 27

14 B-meson DA s 0 q 2α (x)[x, 0]h vβ (0) B v = if Bm B 4 0 dωe iωv x [(1 + /v) { } ] φ B +(ω) φb +(ω) φ B (ω) /x γ 5 2v x defined in HQET; key input parameter: the inverse moment 1 λ B (µ) = 0 dω φb +(ω, µ) ω QCD sum rules in HQET: λ B (1 GeV) = 460 ± 110 MeV [V.Braun, D.Ivanov, G.Korchemsky,2004 ] all B π, K ( ), ρ form factors calculated so far only tree-level calculations, 2,3-particle DA s model for 3-particle DA s obtained from 2-point QCD sum rules in HQET [A.K., T.Mannel, N.Offen (2007)] Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 14 / 27 βα

15 LCSR for the soft-gluon hadronic matrix element the correlation function: γ F (B K ) νµ (p, q) = i d 4 ye ip y 0 T {j K ν (y)õµ(q)} B(p + q), B b d s c hadronic dispersion relation in the kaon channel F νµ (B K ) (p, q) = if K p ν mk 2 [(p q)q p2 µ q 2 p µ ]Ã(q2 ) + ds ρ µν(s, q 2 ) s h s p 2 Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 15 / 27

16 Light-cone vs local OPE 4 2 A q 2 A A q 2 A q 2 GeV 2 charm-loop hadronic matrix elements in B K l + l : simple loop: dashed; soft-gluon emission from the loop: light-cone (local) OPE- solid (dash-dotted ) Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 16 / 27

17 Charm-loop effect in B K l + l in terms of C 9 the effective coefficient C 9 (µ = m b ) 4.4 a process-dependent correction to be added: C ( cc,b K ) 9 (q 2 ) = 32π2 H (B K ) (q 2 ) 3 f + BK (q2 ) = (C 1 + 3C 2 ) g(m 2 c, q 2 ) + 2C 1 32π 2 3 Ã(q 2 ) f + BK (q2 ) C 9 (0) = , (µ = m b = 4.2GeV) C9 c c, B K q 2 GeV 2 Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 17 / 27

18 Charm-loop effect for B K l + l factorizable part determined by the three B K form factors V BK (q 2 ), A BK 1 (q 2 ), A BK 2 (q 2 ), three kinematical structures for the nonfactorizable part: C ( cc,b K,V ) 9 (q 2 ) = (C 1 + 3C 2 ) g(m 2 c, q 2 ) 2C 1 32π 2 3 (m B + m K )ÃV (q 2 ) q 2 V BK (q 2 ) nonfactorizable part enhances the effect, 1/q 2 factor C ( cc,b K,V ) 9 (1.0GeV 2 ) = C ( cc,b K,A 1 ) 9 (1.0GeV 2 ) = C ( cc,b K,A 2 ) 9 (1.0GeV 2 ) = C9 c c, B K, M q 2 GeV 2, Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 18 / 27

19 Charm-loop effect in B K γ By-product of our calculation for B K l + l at q 2 = 0 factorizable part vanishes, nonfactorizable part yields a correction to C7 eff (m b) 0.3 in the two inv. amplitudes: C eff 7 C eff 7 + [ C ( cc,b K γ) 7 ] 1,2, [ C ( cc,b K γ) 7 ]1 [ C ( cc,b K γ) 7 ]2 = ( ) 10 2, the previous results in the local OPE limit, LCSR with K DA: [ C ( cc,b K γ) 7 ] BZ 1 = ( 0.39 ± 0.3) 10 2, [ C ( cc,b K γ) 7 ] BZ 2 = ( 0.65 ± 0.57) (1) [P.Ball, G. W. Jones and R. Zwicky (2007)] our result in the local limit is closer to 3-point sum rule estimate: [A.K.,G. Stoll,R. Rueckl,D. Wyler(1997)] Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 19 / 27

20 Accessing large q 2 with dispersion relation analyticity of the hadronic matrix element in q 2, dispersion relation: H (B K ) (q 2 ) = H (B K ) (0) + q 2[ ψ=j/ψ,ψ(2s) + f ψ A BψK mψ 2 (m2 ψ q2 im ψ Γ tot 4m 2 D ψ ) ρ(s) ] ds s(s q 2 iɛ) absolute values of the residues f ψ A BψK from exp. data the integral over ρ(s) fitted as an effective pole previous uses of dispersion relation: only factorizable part, positive residues, k-factors to accomodate data [F. Krüger, L. Sehgal (1997); A. Ali, P. Ball, L. T. Handoko, G. Hiller (2000)] Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 20 / 27

21 Charm-loop effect at large q 2 solid- central input, green-shaded - uncertainties the dispersion relation ansatz coinsides with OPE result at q 2 < 4.0 GeV 2 and is valid up to s = 4mD 2 (at q 2 < mj/ψ 2 largely independent of higher-states ansatz) C 9 c c, B K C9 c c, B K, M q 2 GeV q 2 GeV 2 Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 21 / 27

22 Summary soft-gluon emission from a c quark loop in B K ( ) l + l - a nonlocal operator, effective resummation of local operators, 1/(4m 2 c q 2 )-suppression LCSR with B meson DAs used to calculate the emerging hadronic matrix element accuracy can be improved by including O(α s ) effects analytical continuation using dispersion relation and data on B ψk allow to access q 2 4m 2 D q 2 (m B m K ) 2 (low recoil) region not accessible q 2 a local expansion in 1/q 2 possible, [B.Grinstein, D. Pirjol (2004); Th. Feldmann, G. Buchalla (2011)] but the problem of hadronic matrix elements remains to be solved (no access with LCSR, lattice QCD? ) still a lot of work ahead to get a reliable estimate of QCD effects in B K ( ) l + l, B K γ amplitudes Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 22 / 27

23 BACKUP Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 23 / 27

24 Form factors from LCSR with B-meson DA s form factor this work LCSR with light-meson DA s [P.Ball and R.Zwicky] ([Duplancic et al]) f + Bπ (0) 0.25± ±0.031, ( ) f + BK (0) 0.31± ±0.041±0.008 fbπ T (0) 0.21± ±0.028 fbk T (0) 0.27± ±0.037±0.009 V Bρ (0) 0.32± ±0.029 V BK (0) 0.39± ±0.033±0.031 A Bρ 1 (0) 0.24± ±0.024 A BK 1 (0) 0.30± ±0.028±0.023 A Bρ 2 (0) 0.21± ±0.023 A BK 2 (0) 0.26± ±0.027±0.022 T Bρ 1 (0) 0.28± ±0.021 T BK 1 (0) 0.33± ±0.028±0.024 Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 24 / 27

25 Observables for B K l + l differential distribution in q 2 with (solid) and without (dashed) charm-loop effect 3.0 N q 2 B0 K 0 Μ Μ q 2 GeV 2 forward-backward asymmetry : q 2 0 = GeV2 10% larger without nonfactorizable correction 0.4 dafb dq 2 B0 K 0 Μ Μ GeV q 2 GeV 2 Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 25 / 27

26 Influence on the observables for B K l + l adding δc 9 (q 2 ) to the decay amplitude differential distribution in q 2 with (solid) and without (dashed) charn-loop effect 10 7 dbr dq 2 B 0 K 0 Μ Μ GeV q 2 GeV 2 Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 26 / 27

27 Summary other similar loop effects accessible with the same method the role of soft gluons in "weak annihilation" to be studied B B b q O 1 6 s q K ( ) l l + Alexander Khodjamirian QCD Effects in Exclusive Rare B-Meson Decays 27 / 27