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
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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 ∗ − | eff | i 4G 10 H = F V V C (µ)O (µ) , eff √ 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/ψ, ψ(2S), ... B) DD¯ continuum: B DD¯K Kl+l . → → −
Failure of naive factorization for B J/ψK 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 • { } 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) . → − √2 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 | { [C O (0) + C O (0)] B(p + q) . 1 1 2 2 }| 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 { L L }
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) = dω I (q, ω)¯s γ δ[ω D ]G b . O Z µραβ L − 2 αβ L f e
Failure of local OPE for the nonfactorizable charm loop: • m ∞ (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 ω 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 |fac 3 2 h |O | 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 |nonf 1h |O | 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 |O | 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[ω D ]Gστ (0)b (0) B¯(v) h | − 2 β | i fBmB ∞ iλy v = dλ e− · (1+ v) (vσγτ vτ γσ)[ΨA(λ, 2ω) ΨV (λ, 2ω)] 2 Z 6 − − 0 yσvτ yτ vσ yσγτ yτ γσ iσστ ΨV (λ, 2ω) − XA(λ, 2ω) + − YA(λ, 2ω) γ5 . − − v y v y βα · ·
Model of B-meson DAs (Khodjamirian, Mannel, Offen, 2007): • λ2 E 2 (λ+ω)/ω0 ΨA(λ, ω) = ΨV (λ, ω) = 4 ω e− , 6ω0 λ2 E (λ+ω)/ω0 XA(λ, ω) = 4 ω(2λ ω)e− , 6ω0 − λ2 E (λ+ω)/ω0 YA(λ, ω) = 4 ω(7ω0 13λ + 3ω)e− . −24ω0 − 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 | { ν O }| i f γ∗
c b B¯ s d
Hadronic dispersion relation in the K channel: • 2 (B K) ifKpν 2 2 ∞ 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 ). Charm loop corrections in B K `+` in terms of ∆C : • → ∗ − 9 (¯cc,B K∗, i) 2 2 2 (¯cc,B K∗, i) 2 ∆C9 → M (q ) = (C1 + 3C2) g(mc , q ) + 2C1g → M (q ) .
32π2 (m + m e )V (q2) (¯cc,B K∗, 1) 2 B K∗ 1 g → M (q ) = , − 3 2 BK∗ ( 2) q V qe e 64π2 (m m V (q2) (¯cc,B K∗, 2) 2 B K∗) 2 g → M (q ) = − , 3 2 BK∗ ( 2) q Ae1 q e Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 15 64π2 (m + m )V (q2) (¯cc,B K∗, 3) 2 B K∗ 2 g → M (q ) = 3 q2ABK∗ (q2) 2 e 2 e V3(q ) + . (m m )ABK∗ (q2) B − eK∗ 2 L 3 L 1 2.5 M M
, 3 * , *
2.0 K K 2 ®
® 1.5 B B , , c
c 1.0 1 c c H H 9
9 0.5
C 0 C D D 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 q2 GeV2 q2 GeV2 1/q2 enhancement of nonfactorizable charm loop correction. • H L H L (¯cc,B K∗,M1) 2 +0.57 ∆C9 → (1.0GeV ) = 0.72 0.37 , −
(¯cc,B K∗,M2) 2 +0.70 ∆C9 → (1.0GeV ) = 0.76 0.41 , −
(¯cc,B K∗,M3) 2 +1.14 ∆C9 → (1.0GeV ) = 1.11 0.70 . −
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 16 Charm loop effect for B K γ → ∗
A by-product of the calculation for B K `+` at q2 = 0. • → ∗ − Only the nonfactorizable soft-gluon contributes to the B K γ amplitude. • → ∗ The charm loop effect in terms of corrections to Ceff : • 7 eff eff (¯cc,B K γ) C C + [∆C → ∗ ] , 7 → 7 7 1,2 (¯cc,B K∗γ) (¯cc,B K∗γ) +0.9 2 [∆C7 → ]1 [∆C7 → ]2 = ( 1.2 1.6 ) 10− . ' − − × Within the errors, 8 % correction to C7eff . Local OPE predictions (Ball, Zwicky, 2007): • (¯cc,B K γ) BZ 2 [∆C → ∗ ] = ( 0.39 0.3) 10 , 7 1 − × − (¯cc,B K γ) BZ 2 [∆C → ∗ ] = ( 0.65 0.57) 10 . 7 2 − × − In the local limit, our prediction is consistent with 3-point QCDSR estimate • (Khodjamirian, Stoll, Ruckl, and Wyler, 1997).
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 17 Accessing large q2 with dispersion relation
Analyticity of the hadronic matrix element in q2 and dispersion relation: • (B K) 2 (B K) 2 fψABψK → (q ) = → (0) + q [ H H m2 (m2 q2 im Γtot) ψ=J/ψX,ψ(2S) ψ ψ − − ψ ψ ρ(s) + ∞ ds ]. Z4m2 s(s q2 i) D − − One substraction is performed due to the divergence of factorizable charm loop for s . → ∞ Experimental data on B ψK can fix the magnitude of the residues f A . • → | ψ BψK|
Nontrival phases of A from the final state interactions, which are ne- • BψK glected due to the absence of perturbative corrections. (a) Negative interference between J/ψ and ψ(2S) is favored. (b) The sign of J/ψ contribution is fixed by the higher derivatives of the dis- persion relation over q2 at small q2.
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 18 Parametrization of the integral over spectral density: • ρ(s) a(B K) ∞ ds → . Z 2 2 2 2 4mD s(s q ) ' m q − ∗ − (a)Small effect of continuum contribution:
(B K) 3 m = 4.06 GeV, a → = 0.06 10− . ∗ ×
fJ/ψABJ/ψK 3 fψ(2S)ABψ(2S)K 3 2 = 1.34 10− , 2 = 0.90 10− . mJ/ψ × mψ(2S) − × (b)The continuum contribution in terms of z-parameterization.
Comments on the previous dispersion relation (Kruger¨ and Sehgal, 1996): • (a) Ignorance of nonfactorizable charm loop. (b)Positive interference between different charmonium states’ contributions. (c)Artificial κ-factor to compensate the drawback of factorization assumption.
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 19 Charm-loop corrections below the DD¯ threshold
Solid line—-central values, green-shaded region—-errors. • L 4 1 4 M L , * K 2 2 K ® B ® ,
c 0 B 0 c , H c 9 c H C -2 -2 9 D C
-4 D -4 2 4 6 8 10 12 2 4 6 8 10 12 q2 GeV2 q2 GeV2
The dispersion relation isHconsistentL with the light-coneH OPEL for q2 < 4GeV2. • Our prediction at small q2 region (say, 5 6GeV2) is largely model indepen- dent. − For q2 > 4m2 , neither the approximation ¯cc-loop gluon corrections , • D { ⊕ } nor a simple sum over ψ resonances can provide an adequate description of this effect in B K( )`+` . → ∗ − Below the charmonium region the predicted ratio ∆C (q2)/C is 5% for • 9 9 ≤ B K`+` , but can reach as much as 20% for B K `+` . → − → ∗ −
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 20 Charm loop effect on the observables of B K`+` → −
Invariant mass distribution: dashed line—without charm loop effect, solid • line—with charm loop correction. L 2
- 1.00 GeV
H 0.70 L - Μ
+ 0.50 Μ 0 K ®
0 0.30 B H 2
dq 0.20 0.15 dBR * 7
10 0.10
2 4 6 8 10 12 q2 GeV2
H L The region between J/ψ and ψ(2S) is important to extract the information • on the interference of different charmonium states’ contributions.
The charm-loop effect becomes essential only if one approaches the J/ψ • resonance region.
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 21 Charm loop effect on the observables of B K `+` → ∗ −
Invariant mass distribution: • dashed line—without charm loop effect, solid line—with charm loop correc- tion. (a)Normalized at 1GeV2 to diminish the large uncertainties.
3.0
L 2.5 - Μ + Μ 0 *
K 2.0 ® 0 B H
L 1.5 2 q H N 1.0
0.5 2 4 6 8 10 12 q2 GeV2
H L
(b)1/q2 enhancement at small q2 region.
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 22 Forward-backward asymmetry: • 0.4 L 2 - 0.2 GeV H L - Μ + Μ
0 0.0 * K ® 0 B H
2 -0.2 dq FB
dA -0.4
2 4 6 8 q2 GeV2
(a) Zero-point of FBA: H L
m (m + m ) m T BK∗ (q2) m T BK∗ (q2) eff B b s 1 + K∗ 1 0 + 1 K∗ 2 0 C7 2 2 BK 2 q m V BK∗ (q ) − m A ∗ (q ) 0 B 0 B 1 0 1 (¯cc,B K∗, 1) 2 (¯cc,B K∗, 2) 2 +C9 + ∆C → M (q ) + ∆C → M (q ) = 0 . 2 9 0 9 0
(b)Charm loop effect on FBA: 10 % shift of zero point of FBA. 2 +0.2 2 2 2 q0 = 2.9 0.3 GeV , q0 without charm = 3.2GeV . − |
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 23 Summary
Nonfactorizable charm loop effect in B K( )`+` is estimated in terms • → ∗ − of OPE-controlled dispersion relation. Nonlocal three-body effective operator responsible for the nonfactorizable • charm loop is derived in light-cone OPE. Three-particle B meson DAs are used to describe the non-perturbative dy- • namics of K(p)( ) ¯s γρδ[ω in ]G˜ (0)b B(p + q) . h ∗ | L − + · D αβ L| i Below the charmonium region, the charm loop correction ∆C (q2)/C is • 9 9 5% for B K`+` , but can reach as much as 20% for B K `+` , ≤ → − → ∗ − the difference being mainly caused by the soft-gluon contribution. Our predictions can be further improved by including perturbative correc- • tions, light-quark loop and weak annihilation contribution. The method to estimate the charm loop effect presented here can be ex- • + + tended to the analysis of K πνν¯, Bs φl l−, Λb Λl l− and + → → → B Xsl l etc. → −
Yu-Ming Wang Workshop on QCD and hadron physics, Weihai, August, 2010 24