Exclusive semileptonic B and Λ b decays at large hadronic recoil

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1 Exclusive semileptonic B and Λ b decays at large hadronic recoil Alexander Khodjamirian UNIVERSITÄT SIEGEN Workshop: Beautiful Mesons and Baryons on the Lattice, ECT* Trento, 2-6 April 2012 Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 1 / 37

2 Part 1: B K l + l at large hadronic recoil Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 2 / 37

3 b s flavour-changing neutral currents (FCNC) b sl + l, via virtual t, Z, W New Physics? effective local operators C(m b ) { s b l l} experiment (LHCb ) prefers exclusive channels, B K l + l, B K l + l, B s φl + l,... the accuracy improving : recently, LHCb observed the zero in the FB asymmetry in B K l + l can the theory provide a competetive accuracy? (in what follows: mainly B K l + l mode ) Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 3 / 37

4 The anatomy of the effective Hamiltonian H eff = 4G F 2 V tb V 10 ts i=1 C i (µ)o i (µ) µ mb local b sll 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, operators combined (nonlocally) with e.m. lepton-pair emission: O 7 = em b 8π 2 [ sσ µν (1 + γ 5 )b]f µν, C 7 (m b ) 0.3 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 O 8g = m b 8π 2 sσ µν(1 + γ 5 )bg µν, C 8 (m b ) 0.2 O quark-penguin operators, C 3,4,5,6 < 0.03 neglected the V ub V us part Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 4 / 37

5 B K l + l decay in Standard Model the decay amplitude: A(B K l + l ) = K l + l H eff B = 4G F 2 V tb V ts 10 i=1 C i K l + l O i B dominant contributions factorize: B K form factors l l l l b s b s B K B K O 9,10 O 7 K (p) sγ µ b B(p + q) f + BK (q2 ), K (p) sσ µν b B(p + q) f T BK (q 2 ) additonal nonlocal contributions: combining O c 1 2, O 3 6, O 8g with the emission of γ l + l Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 5 / 37

6 The nonlocal effects in B K l + l 4-quark operators: O1,2 c, Oq 3 6 (q = u, d, s, c, b) q b s B K LO loop NLO loops factorizable soft gluon emission spectator scatt. nonfactoriz. Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 6 / 37

7 The nonlocal effects in B K l + l gluon-penguin operator O 8g B B B factoriz. NLO nonfact.: soft gluon spectator scatt. weak annihilation: O q 3 6 q = u, d b b B K B K Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 7 / 37

8 Including nonlocal effects in the decay amplitude schematically: A(B K l + l ) = 4G [ F V tb V 2 + i=1,2,3..6,8 ts i=7,9,10 C i ( ll) H i BK C i ( ll) f i BK hadronic matrix elements of nonlocal contributions: { } HBK i (p, q) = K (p) d 4 x e iq x T j em (x), O i (0) B(p + q) in LO reducible to form factors loop factor (OPE) first estimates of NLO contributions to B K ( ) l + l in the QCD factorization scheme (m b ) at small q 2 ( large hadronic recoil) [Beneke, Feldmann, Seidel (2001)],... ] Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 8 / 37

9 QCD calculation of B K l + l B K form factors: lattice QCD, light-cone QCD sum rules (LCSR) (the same accuracy as LCSR for B π FF s including m s 0 effects) HBK i, the size of nonfactorizable nonperturbative contributions? ("soft" gluons, vacuum quark-antiquark pairs ) the dominant part : H 1,2 BK due to Oc 1,2 (charm loops) Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 9 / 37

10 Charm-loops in B K l + l Charm-loop effect: a combination of the ( sc)( cb) weak interaction (O 1,2 ) and e.m.interaction ( cc)( ll) l l l l l l b c c s b similar u, d, s, b-quark loops: suppressed by Wilson coeffs. (quark-penguin operators O 3 6 ) or by CKM (u-loops from O u 1,2 ) c c s b c c x s Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 10 / 37

11 Charm loop turns charmonium at q 2 mj/ψ 2, cc loop becomes a hadronic state: e.g., B K l + l = { B J/ψK J/ψ l + l } heavier ψ-levels (charmonia with J P = 1 ) at q 2 = m 2 ψ, cc states with the masses up to m B m ( ) K spectrum of ψ states as seen in e + e hadrons 4.8GeV( 4.4GeV) J/ψ Mark I Mark I + LGW Mark II PLUTO DASP Crystal Ball BES ψ(2s) ψ 3770 ψ 4040 ψ 4160 ψ 4415 c [PDG, V.V. Ezhela et al. hep-ph/ ] Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 11 / 37

12 The B ψk ( ) amplitudes naive factorization fails in B ψk ( ), (ψ = J/ψ, ψ(2s)), A(B ψk ) G F V 2 cb Vcs(c 1 (µ) + c 2 (µ)/3)f ψ f + BK (q2 = mψ 2 ) yields Γ(B ψk ) exp. indicating large nonfactorizable contribuitons ( which should effectively remove the µ-dependence) QCD factorization should not (and does not) work in these decays, due to heavy-mass final states Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 12 / 37

13 How to account for the charm-loop effect? cc loops ψ resonances - a double counting!! the experimentalists (BABAR,Belle,CDF,LHCb): subtract the bins of J/ψ and ψ(2s) 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 in the small q 2 4m 2 c region (large hadron recoil)? soft-gluon emission from c-quark loop from OPE on the light-cone [A.K., Th. Mannel, A. Pivovarov and Yu-M. Wang, [hep-ph]] Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 13 / 37

14 Charm-loop in B K l + l at q 2 4m 2 c the hadronic matrix element: { H µ BK (p, q) = i K (p) d 4 x e iq x T c(x)γ µ c(x), [ ]} C 1 [ s L (0)γ ρ c L (0) c L (0)γ ρ b L (0)] + C 2... B(p + q) at q 2 4m 2 c, contracting c-quark fields in propagators: c(0) c(x) and c(x) c(x), including gluon emission: Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 14 / 37

15 Expansion near the light-cone at q 2 4m 2 c, the dominant region: x 2 1/(2m c q 2 ) 2 T - product of cc-operators can be expanded near the light-cone x 2 0, diagrammatically: s 0 x the simple loop (unit operator LO NLO ) b s 0 b s G(ux) G(ux) x one-gluon emission (non-local, x 0) ( s(0)g(x)b(0)) b 0 G(vx) x two-gluon emission... Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 15 / 37

16 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 Exclusive semileptonic B and Λ b decays at large hadronic recoil 16 / 37

17 Hadronic matrix elements for the charm-loop effect the LO: factorized cc loop ] ( ) [H µ BK (p, q) = C1 fact 3 + C 2 K (p) O µ (q) B(p + q), reduced to f + BK (q2 ) form factors, ( loop function) The gluon emission yields: [H µ BK (p, q) ] nonfact = 2C 1 K (p) Õµ (q) B(p + q). new hadronic matrix element ( coeff.function) K (p) s L γ ρ δ[ω (in +D) ] G αβ b L B(p + q), 2 Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 17 / 37

18 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 K (p) Õµ (q) B(p + q) + ds ρ µν(s, q 2 ) p2 s h s p 2 B meson DA s used as nonperturbative input Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 18 / 37

19 Nonlocal effects in B K l + l in terms of C 9 the Wilson coefficient C 9 (µ = m b ) 4.4, a process-dependent correction to be added: C9 BK (q2 i ) Hi BK (q2 ) f + BK (q2 ) previously we investigated only the role of c-loop soft nonfact. effect vs LO and found BK ( cc,lo nf ) C9 (q 2 = 0) 0.2 Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 19 / 37

20 Nonlocal effects in B K l + l in terms of C 9 [A.K., Th.Mannel, Y.M. Wang, work in progress] preliminary results including nonlocal effects due to O1,2 c LO and soft nonfact. O 3 6 LO and soft nonfact. O 8 soft nonfact. (new LCSR estimate) O1,2 c NLO two-loop factorizable contributions using diagrams from [Asatrian,Greub,Walker(2002)], O1,2 c NLO spectator contributions [Beneke, Feldmann, Seidel (2001)],... we use OPE results at negative q 2 having in mind the presence of light-quark loops Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 20 / 37

21 Nonlocal effects for B K l + l in spacelike region C 9 f 0.0 C 9 nf C 9 G 0.2 Re C 9 NLO Re C 9 tot Im C 9 NLO 0.3 preliminary q 2 GeV 2 Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 21 / 37

22 Accessing the timelike q 2 region? analyticity of the hadronic matrix element in q 2, unitarity hadronic 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) the residues A BψK and f ψ determined by BR(B ψk ), BR(ψ l + l ) FSI phase attributed to A(B ψk ), (Im part in (p + q) 2 ) a certain ansatz ( z-parameterization) used for the integral dispersion relation fitted to the calculated H (B K ) at q 2 < 0 Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 22 / 37

23 The width of B K ll [preliminary] dbr B 0 K0Μ Μ dq q 2 GeV 2 exp. data on B K ll, [Belle 09 (violet), CDF 11(red)] solid ( green shaded-theor. uncertainties) and dotted - two different ansaetze for dispersion relation; dashed - without nonlocal effects Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 23 / 37

24 Part 2: Λ b p form factors from LCSR [ A.K., Ch.Klein, Th.Mannel, Y.-M. Wang arxiv: ] Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 24 / 37

25 The LCSR method N(P) u d u 0 z b P q q vacuum-to-nucleon correlation function: Π µ(5) (P, q) = i d 4 z e iq z 0 T { η Λb (0), b(z)γ µ (γ 5 )u(z) } N(P). q 2 m 2 b, (P q)2 m 2 b, P2 = m 2 N, Λ b interpolating 3-quark current, we use η (P) Λ b = (u C γ 5 d) b, η (A) Λ b = (u C γ 5 γ λ d) γ λ b. Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 25 / 37

26 Nucleon Distribution Amplitudes (DA s) [V.Braun, A.Lenz et al ( )], definition, schematically (z 2 0): 0 ɛ ijk u i α(0)u j β (z)d k γ (0) N(P) = t S t αβγ 3 dx 1 dx 2 dx 3 δ(1 x i )e ix2p z F t (x i, µ), i=1 twist expansion: 27 DA s of twist 3,4,5,6 coefficients and normalization parameters determined from 2-point sum rules proton e.m. form factors were calculated from LCSR Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 26 / 37

27 Accessing the Λ b p form factors hadronic dispersion relation, schematically Π µ(5) (P, q) = 0 η Λ b Λ b Λ b bγ µ (γ 5 )u N m 2 Λ b (P q) η Λ b Λ b Λ b bγ µ (γ 5 )u N mλ 2 + (P q) 2 b s h 0 ds ρ µ(5)(s, q 2 ) s (P q) 2 6 form factors, standard definitions (cf nucleon β decay): j Λ b (P q) b γ µ u N(P) = ū Λb (P q) f 1 (q 2 ) γ f 2(q 2 ) µ+i σ µνq ν + f 3(q 2 ff ) q µ u N (P), m Λb m Λb 0 q 2 (m 2 Λ b m 2 N ), γ µ γ µ γ 5, f i (q 2 ) g i (q 2 ) decay constant of Λ b from two-point sum rules Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 27 / 37

28 LCSR in detail specific problems for baryon QCD sum rules the contributions of Λ b, ( JP = 1/2 state, m Λ b m Λb MeV we used linear combinations of kinematical structures in the correlation function to eliminate Λ b baryon interpolating current: multiple choice we used pseudoscalar and axial currents replace b by c in LCSR Λ c N form factors (used to calculate strong couplings ) inputs: finite m b, a few universal parameters of nucleon DA s, two-point sum rules for η Λb currents: λ (A) Λ b = GeV 2, λ (P) Λ b = GeV 2, Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 28 / 37

29 Numerical results for the Λ b p vector form factors f 1 b p q f 2 b p q q 2 GeV q 2 GeV 2 q 2 11 GeV 2 direct calculation from LCSR, at larger q 2 z-parameterization and extrapolation reasonable agreement between sum rules with different baryon currents Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 29 / 37

30 Numerical results for the axial-vector Λ b p form factors g 1 b p q g 2 b p q q 2 GeV q 2 GeV 2 Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 30 / 37

31 The width of Λ b plν l decay can be used to extract V ub dγ dq 2 (Λ b plν l ) = G2 F m3 j ff Λb 192π 3 V ub 2 k 1 (q 2, m Λb, m N ) f 1 (q 2 ) normalized q 2 -spectrum d BR b p lν l d q 2 BR b p lν GeV 2 l q 2 GeV 2 Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 31 / 37

32 The width of Λ b plν l decay partially integrated width: pure prediction of LCSR ζ(0, qmax) 2 = 1 q 2 max V ub 2 0 dq 2 dγ dq 2 (Λ b plν l ) = ps 1 ( = ps 1) for axial-vector (pseudoscalar) interpolating current of Λ b improvements in the future possible: nucleon DA parameters, α s corrections Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 32 / 37

33 Conclusions with LC OPE and LCSR the effects of four-quark and penguin operators in b sl + l exclusive decays can be systematically accounted including soft-gluon effects in the region q 2 <few GeV 2, B K l + l allows for more accurate treatment than B K l + l, K γ both in FF s and nonlocal effects are (or will be ) the HBK i amplitudes accessible on the lattice? Λ b form factors can be calculated from LCSR both topics will be discussed further in the talk by Thorsten Feldmann Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 33 / 37

34 BACKUP SLIDES Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 34 / 37

35 The hierarchy of contributions in LC OPE after integrating over x and taking hadronic matrix element each extra gluon brings one power of Λ2 QCD 4m 2 c q 2 suppression perturbative gluon corrections are α s suppressed reexpanding the one-gluon nonlocal operator near x = 0 in derivatives of G µν (0) : term with k-th derivative q m b /2 and m b Λ QCD m 2 c. k=0 (qλ QCD ) k (4m 2 c q 2 ) k+1 the OPE near the light-cone works, but not the local OPE Alexander Khodjamirian Exclusive semileptonic B and Λ b decays at large hadronic recoil 35 / 37

36 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 Exclusive semileptonic B and Λ b decays at large hadronic recoil 36 / 37

37 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 Exclusive semileptonic B and Λ b decays at large hadronic recoil 37 / 37