The Charm-loop effect in B K( )`+` ! ∗ − Yu-Ming Wang Theoretische Physik I , Universitat¨ Siegen A. Khodjamirian, Th. Mannel,A. Pivovarov and Y.M.W. arXiv:1006.4945 Workshop on QCD and hadron physics, Weihai, August, 2010 1 0 2 4 6 8 10 12 14 16 18 20 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 0 2 4 6 8 10 12 14 16 18 20 I. Motivation and introduction ( ) + B K ∗ ` `− induced by b s FCNC current: • prospectiv! e channels for the search! for new physics, a multitude of non-trival observables, Available measurements from BaBar, Belle, and CDF. Current averages (from HFAG, Sep., 2009): • BR(B Kl+l ) = (0:45 0:04) 10 6 ! − × − BR(B K l+l ) = (1:08+0:12) 10 6 : ! ∗ − 0:11 × − Invariant mass distribution and FBA (Belle,−2009): ) 2 1.2 /c 1 2 1 L / GeV 0.5 0.8 F -7 (10 0.6 2 0 0.4 1 0.2 dBF/dq 0.5 ) 0 FB 2 A /c 2 0.5 0 0.4 / GeV -7 0.3 (10 1 2 0.2 I A 0 0.1 dBF/dq 0 -1 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 0 2 4 6 8 10 12 14 16 18 20 q2(GeV2/c2) q2(GeV2/c2) Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 2 q2(GeV2/c2) q2(GeV2/c2) Decay amplitudes in SM: • A(B K( )`+` ) = K( )`+` H B ; ! ∗ − −h ∗ − j eff j i 4G 10 H = F V V C (µ)O (µ) ; eff p tb ts∗ i i − 2 iX=1 Leading contributions from O and O can be reduced to B K( ) 9;10 7γ ! ∗ form factors. Form factors are insufficient to describe the QCD dynamics of B K( )`+` • ! ∗ − decays due to nonfactorization corrections. Some important nonfactorizable effects: • a) Charm loop induced by the c-quark current-current operators and c-quark e.m. current. b) Quark loops generated by the penguin operators suppressed by the CKM factors and Wilson coefficients. c) Weak annihilation effect. Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 3 Charm loop in B K( )`+` ! ∗ − γ∗ c b s ¯ B¯ K K (a) (b) (c) (d) Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 4 The leading-order charm loop effect (fig. a) as well as the NLO corrections • (fig. c, d) are already included in the factorization formulae. (a) Factorizable charm loop: C C + (C + 3C )g(m2; q2) . 9 ! 9 1 2 c (b) Perturbative corrections (Beneke, Feldmann, Seidel, 2001) : QCD factorization, corrections to C9 . Two important questions concerning the charm loop effect remain: • (a) How important are the soft gluons emitted from the charm quark loop (fig. b) ? (b)How far in q2 can we go with the OPE for the estimation of the charm loop effect? Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 5 “Evolution” of the charm loop Charm quark pair can form the charmonium bound state at large q2: • P C a) Charmonium resonances with J = 1−−: J= , (2S), ... B) DD¯ continuum: B DD¯K Kl+l . ! ! − Failure of naive factorization for B J= K indicating sizeable nonfactoriz- • ! able corrections. The effect of virtual charmonium states remains at small q2 region. • Whether the ansatz loop gluon corrections is valid in all kinematical • f g region? L Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 6 Light-cone dominance of the charm loop The contribution of charm loop: • ¯ µ ( ) ( ) + (O ) 4GF `γ ` (B K ∗ ) A(B K ∗ ` `−) 1;2 = (4παemQc) V V ∗ µ ! (p; q) : ! − p2 tb ts q2 H The involved hadronic matrix element: • ( ) (B K ∗ ) 4 iq x ( ) µ ! (p; q) = i d xe · K ∗ (p) T ¯c(x)γµc(x) ; H Z h j f [C O (0) + C O (0)] B(p + q) : 1 1 2 2 gj i At small q2 region, OPE for the T-product • a 4 iq x a (q) = d xe · T ¯c(x)γµc(x); ¯c (0)Γ c (0) : Cµ Z f L L g The dominant contribution comes from the region: • 2 2 2 x 1=(2mc q ) : h i ∼ − q Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 7 Light-cone expansion and resummation Effective operator for the factorizable charm loop: • 2 9 2 2 ρ µ(q) = (qµqρ q gµρ) g(m ; q )¯s γ b : O − 32π2 c L L Local OPE and light-cone OPE are equivalent in this case. Effective operator for the nonfactorizable charm loop in light-cone OPE: • ρ (in+ ) µ(q) = d! I (q; !)¯s γ δ[! D ]G b : O Z µραβ L − 2 αβ L f e Failure of local OPE for the nonfactorizable charm loop: • m 1 (q k) O(C1) · : ∼ (4m2 q2)m mX=0 c − k—the momentum of soft gluon. Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 8 The local OPE limit The local OPE limit can be recovered by setting ! 0: • ! (0) (0) ρ µ (q) = I (q)¯s γ G b ; O µραβ L αβ L The non-local effectivf e operator in light-cone eexpansion is localized. Gauge invariant coefficient (Voloshin 1997; Khodjamirian, Ruckl, Stoll and • Wyler, 1997): (0) 2 I (q) = I (q; 0) = (qµqαg + qρqαg q gµαg ) µραβ µραβ ρβ µβ − ρβ 1 1 t(1 t) dt − ; ×16π2 Z0 m2 q2t(1 t) c − − Discussions on the resummation of local expansion: • Ligeti, Randall, and Wise, 1997; Grant, Morgan, Nussinov and Peccei, 1997; Chen, Rupak and Savage, 1997; Buchalla, Isidori and Rey, 1997. Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 9 Hadronic matrix element of the charm loop corrections Factorizable charm loop: • (B K) C1 [ µ ! (p; q)] = + C K(p) µ(q) B(p + q) : H jfac 3 2 h jO j i (a)Two-body local operator, (b) added to C9 in the quark level, process independent. Nonfactorizable charm loop: • (B K) [ µ ! (p; q)] = 2C K(p) µ(q) B(p + q) : H jnonf 1h jO j i (a) Three-body nonlocal operator, f (b) added to C9 in a process-dependent way. Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 10 Amplitude of charm loop in B K`+` ! − The invariant amplitude: • (B K) 2 C1 2 2 µ ! (q ) = + C A(q ) + 2C A~(q ): H 3 2 1 Factorizable charm loop: • 9 A(q2) = g(m2; q2)f + (q2): 32π2 c BK Small Wilson coefficient, sensitive to energy scale. Nonfactorizable charm loop: • 2 A~(q ) = K(p) µ(q) B(p + q) : h jO j i Large Wilson coefficient, insensitivefto energy scale. Nonperturbative method to compute f + (q2), A~(q2): B-meson LCSR • BK HQET B-meson state, Light-cone dominance of the coorrelator, B-meson DA, LO calculation. Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 11 Model for 3-particle DA of B-meson Decomposition of non-local operator matrix element ( Kawamura, Kodaira, • Qiao and Tanaka, 2001): (in+ ) 0 d¯α(y)δ[! D ]Gστ (0)b (0) B¯(v) h j − 2 β j i fBmB 1 iλy v = dλ e− · (1+ v) (vσγτ vτ γσ)[ΨA(λ, 2!) ΨV (λ, 2!)] 2 Z 6 − − 0 yσvτ yτ vσ yσγτ yτ γσ iσστ ΨV (λ, 2!) − XA(λ, 2!) + − YA(λ, 2!) γ5 : − − v y v y βα · · Model of B-meson DAs (Khodjamirian, Mannel, Offen, 2007): • λ2 E 2 (λ+!)=!0 ΨA(λ, !) = ΨV (λ, !) = 4 ! e− ; 6!0 λ2 E (λ+!)=!0 XA(λ, !) = 4 !(2λ !)e− ; 6!0 − λ2 E (λ+!)=!0 YA(λ, !) = 4 !(7!0 13λ + 3!)e− : −24!0 − 2 2 The normalization constant of the three-particle DA’s is λE = 3=2λB. Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 12 LCSR for the nonfactorizable charm loop amplitude The choice the correlation function: • (B K) 4 ip y K νµ ! (p; q) = i d ye · 0 T j (y) µ(q) B(p + q) ; F Z h j f ν O gj i f γ∗ c b B¯ s d Hadronic dispersion relation in the K channel: • 2 (B K) ifKpν 2 2 1 ρ~νµ(s; q ) νµ ! (p; q) = [(p q)qµ q pµ]A~(q ) + ds : F m2 p2 · − Zs s p2 K − h − Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 13 LCSR predictions of charm loop effect in B K`+` • ! − 4 L A q2 A 0 0.5 K 2 ® B H L H L , 0.0 c 0 c H 9 -0.5 C -2 ~ D 10*A q2 A 0 -1.0 -4 1 2 3 4 0 1 2 3 4 2 2 H L H L q2 GeV2 q GeV (a)The nonfactorizableH amplitudeL is about a few percent of the factorizable one and has the opposite sign. H L (b) The effective resummation the local operators in the framework of the light-cone OPE “softens” the gluon correction to the charm loop. + Charm loop effect in B K` `− in terms of ∆C9: • ! 2 2 (¯cc;B K) 2 2 2 32π A~(q ) ∆C ! (q ) = (C1 + 3C2) g(m ; q ) + 2C1 : 9 c 3 + 2 fBK(q ) (¯cc;B K) +0:09 ∆C9 ! (0) = 0:17 0:18 : − Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 14 Charm loop effect for B K `+` ! ∗ − Kinematical structures of charm loop amplitudes B K `+` : • ! ∗ − (B K∗) α β γ 2 µ ! (p; q) = µαβγ∗ q p 1(q ) H 2 2H 2 +i[(m m )∗ (∗ q)(2p + q)µ] 2(q ) B − K∗ µ − · H 2 q 2 +i(∗ q)[qµ (2p + q)µ] 3(q ) ; · − m2 m2 H B − K∗ Factorizable charm loop is described by three B K form factors: • ! ∗ BK∗ 2 BK∗ 2 BK∗ 2 V (q ), A1 (q ), A2 (q ).