CERN-PH-TH/2010-305 LMU-ASC 107/10 TUM-HEP-788/10 IPPP/11/02 DCPT/11/04 January 2011 1 Theory of B → K(∗)l+l− decays at high q2: 1 0 OPE and quark-hadron duality 2 n a J M. Beylich2, G. Buchalla1,2 and Th. Feldmann3 6 § 2 ] h 1CERN, Theory Division, CH–1211 Geneva 23, Switzerland p - 2Ludwig-Maximilians-Universit¨at Mu¨nchen, Fakult¨at fu¨r Physik, p e Arnold Sommerfeld Center for Theoretical Physics, D–80333 Mu¨nchen, Germany h [ 3Physik Department, Technische Universit¨at Mu¨nchen, 1 James-Franck-Straße, D–85748 Garching, Germany v 8 1 1 5 . 1 Abstract 0 1 We develop a systematic framework for exclusive rare B decays of the type B 1 → : K( )l+l at large dilepton invariant mass q2. It is based on an operator product v ∗ − i expansion (OPE) for the required matrix elements of the nonleptonic weak Hamil- X tonian in this kinematic regime. Our treatment differs from previous work by a r a simplified operator basis, the explicit calculation of matrix elements of subleading operators, and by a quantitative estimate of duality violation. The latter point is discussed in detail, including the connection with the existence of an OPE and an illustration within a simple toy model. PACS: 12.15.Mm; 12.39.St; 13.20.He §Address after January 2011: IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK 1 Introduction The rare decays B K( )l+l are among the most important probes of flavour physics. ∗ − → They are potentially sensitive to dynamics beyond the Standard Model (SM) and have been intensely studied in the literature [1]. Measurements have been performed at the B-meson factories [2–6] and at the Fermilab Tevatron [7]. Excellent future prospects for detailed measurements are provided by the LHC experiments ATLAS, CMS, and LHCb at CERN [1], and, in the longer run, by Super Flavour Factories based on e+e colliders − [8–11]. Thecalculability ofB K( )l+l decayratesanddistributions benefitsfromthefact ∗ − → that these processes are, to first approximation, semileptonic modes. Correspondingly, the hadronic physics is described by B K( ) form factors, which multiply a perturba- ∗ → tively calculable amplitude. This simple pictureis notexact because also thenonleptonic weak Hamiltonian at scale m has B K( )l+l matrix elements. The prominent b ∗ − ∼ → example is given by hadronic interactions of the form (s¯b)(c¯c), where the charm quarks annihilate into l+l through a virtual photon. Such charm-loop contributions are more − complicated theoretically than the form-factor terms. Even though the charm loops are subdominant numerically in the kinematical regions of interest, they cannot be com- pletely neglected. In particular, the related uncertainty needs to be properly estimated in order to obtain accurate predictions. We need to distinguish three regions in the dilepton invariant mass q2, for which the properties of charm loops are markedly different. For 7GeV2 < q2 < 15GeV2 the presence of very narrow cc¯ resonances leads to huge violations of q∼uark ∼hadron duality [12] and the hadronic backgrounds from B K( )ψ, followed by ψ l+l , dominate ∗ − → → the short distance rate by two orders of magnitude. This region in q2 can be removed by experimental cuts. For q2 < 7GeV2 the kaon is very energetic and the charm loops can be computed systematica∼lly in the heavy-quark limit using QCD factorization for B decays into light- like mesons [13,14]. This approach was first employed for B K( )l+l in [15]. The ∗ − → results have many applications. A summary with detailed references can be found in [1] (see also [16] for a recent analysis). The high-q2 region, q2 > 15GeV2 has received comparatively little attention. In this case the kaon energy is aro∼und a GeV or below, and (soft-collinear) QCD factorization is less justified, becoming invalidclose totheendpoint ofthespectrumatq2 = (m m )2. B K − On the other hand, the large value of q2 defines a hard scale for the hadronic contri- bution to B K( )l+l . Consequently an operator product expansion (OPE) can be ∗ − → constructed, which generates an expansion of the amplitude in powers of E / q2 (or K Λ / q2). Charm loops, and other hadronic contributions, are thus approximated as QCD p effective interactions that are local on the soft scales set by E and Λ . This sim- p K QCD plifies the computation substantially. In fact, to leading order in the OPE the hadronic contribution reduces to a standard form-factor term. This picture has been first dis- cussed at lowest order in the OPE in [17], where it was applied to the endpoint region of B K( )l+l and B Kπl+l . In [18] the OPE was considered in some detail, ∗ − − → → 1 including a discussion of power corrections. In the present paper we formulate the OPE for the high-q2 region of B K( )l+l ∗ − → from the outset. Although our approach is similar in spirit to the analysis of [18], the concrete implementation is different. We will also go beyond the estimates presented in [18] in several ways. An important difference is that [18] combines the OPE with heavy-quark effective theory (HQET), whereas we prefer to work with b-quark fields in full QCD. The latter formulation has the advantage of a simplified operator basis, which makes the structure of power corrections and their evaluation considerably more transparent. Wealso retainthekinematical dependence onq2 inthecoefficient functions, rather than expanding it around q2 = m2. We further discuss the issue of quark-hadron b duality, which appears relevant because of the existence of cc¯ resonance structure in the q2 region of interest. Violations of duality are effects beyond any finite order in the OPE. Using a resonance model based on a proposal by Shifman, we quantify for the first time the size of duality violations in the high-q2 region of B K( )l+l . ∗ − → Further aspects and new results of our analysis will be summarized in sections 8 and 9. The main conclusion is that B K( )l+l is under very good theoretical control ∗ − → also for q2 > 15GeV2. Precise predictions can be obtained in terms of the standard form factors∼, with essentially negligible effects from the additional hadronic parameters related to power corrections and duality violation. The paper is organized as follows. Section 2 collects basic expressions for later refer- ence. In section 3 our formulation of the OPE for B K( )l+l at high q2 is described ∗ − → and the power expansion is constructed explicitly, complete to second order in 1/ q2 and with a discussion of weak annihilation as an example of a (small) third-order correc- p tion. In section 4 we present an estimate of the matrix elements of higher-dimensional operators and quantify their impact on the decay amplitudes for both B K and → B K transitions. Section 5 discusses the connection between the OPE for large q2 ∗ → and QCD factorization for energetic kaons, which are shown to give consistent results at intermediate q2 15GeV2. In section 6 we address the subject of duality violation ≈ in the context of a toy model analysis. The estimate is then adapted to the case of B Kl+l in section 7. In this section we also address conceptual aspects relevant for − → the existence of the OPE and the notion of quark-hadron duality. A comparison of our approach with the literature is given in section 8 before we conclude in section 9. Details on the basis of operators in the OPE are described in appendix A and some numerical input is collected in appendix B. 2 Basic formulas 2.1 Weak Hamiltonian The effective Hamiltonian for b sl+l transitions reads [19,20,21] − → G = F λ C Qp +C Qp + C Q (1) Heff √2 p" 1 1 2 2 i i# p=u,c i=3,...,10 X X 2 where λ = V V (2) p p∗s pb The operators are given by Qp = (p¯b) (s¯p) , Qp = (p¯b ) (s¯ p ) , 1 V A V A 2 i j V A j i V A − − − − Q = (s¯b) (q¯q) , Q = (s¯b ) (q¯q ) , 3 V A q V A 4 i j V A q j i V A − − − − X X Q = (s¯b) (q¯q) , Q = (s¯b ) (q¯q ) , 5 V A q V+A 6 i j V A q j i V+A (3) − − e X g X Q = m s¯σ (1+γ )Fµνb, Q = m s¯σ (1+γ )Gµνb, 7 8π2 b µν 5 8 8π2 b µν 5 α α ¯ ¯ Q = (s¯b) (ll) , Q = (s¯b) (ll) 9 V A V 10 V A A 2π − 2π − Note that the numbering of Qp is reversed with respect to the convention of [19]. Our 1,2 coefficients C correspond to C˜ in [19] and we include the factor of α/(2π) in the 9,10 9,10 definition of Q . The sign conventions for the electromagnetic and strong couplings 9,10 correspondtothecovariantderivativeD = ∂ +ieQ A +igTaAa. Withthesedefinitions µ µ f µ µ the coefficients C are negative in the Standard Model. 7,8 2.2 Dilepton-mass spectra and short-distance coefficients We define the kinematic quantities s = q2/m2 (where q2 is the dilepton invariant mass B squared), r = m2 /m2 , and K K B λ (s) = 1+r2 +s2 2r 2s 2r s (4) K K − K − − K The differential branching fractions for B¯ K¯l+l can then be written as [22] − → dB(B¯ K¯l+l ) G2α2m5 → − = τ F B V V 2 λ3/2(s)f2(s) a (Kll) 2 + a (Kll) 2 (5) ds B 1536π5 | ts tb| · K + | 9 | | 10 | (cid:0) (cid:1) The coefficient a (Kll) contains the Wilson coefficient C (µ) combined with the short- 9 9 distance parts of the B¯ K¯l+l matrix elements of operators Q ,...,Q . The co- − 1 8 → efficient a (Kll) multiplies the matrix element of the local operator Q in the decay 9 9 amplitude. The coefficient a (Kll) = C of the operator Q is determined by very 10 10 10 short distances 1/M and is precisely known. W The correspo∼nding formulas for B¯ K¯ l+l can for instance be found in [23]. ∗ − → 3 OPE for B → Ml+l− amplitudes at high q2 3.1 General structure The amplitudes for the exclusive decays B Ml+l , where M = K, K , or a similar − ∗ → meson, are given by the matrix element of the effective Hamiltonian in (1) between the 3 initial B meson and the Ml+l final state. The dominant contribution comes from the − semileptonic operators Q . Their matrix elements are simple in the sense that all 9,10 hadronic physics is described by a set of B M transition form factors. This is also → true for the electromagnetic operator Q . The matrix elements of the hadronic operators 7 Q ,...,Q ,Q are more complicated. They are induced by photon exchange and can be 1 6 8 expressed through the matrix element of a correlator between the hadronic part of the effective Hamiltonian 6,8 Hp C Qp +C Qp + C Q (6) ≡ 1 1 2 2 i i i=3 X and the electromagnetic current of the quarks jµ Q q¯γµq (7) q ≡ where Q is the electric charge quantum number of quark flavour q and a summation q over q is understood. The decay amplitude may thus be written as G α λ A(B¯ M¯l+l ) = F λ Aµ + uAµ u¯γ v +Aµ u¯γ γ v (8) → − −√22π t 9 λ cu µ 10 µ 5 (cid:20)(cid:18) t (cid:19) (cid:21) where u¯ and v are the lepton spinors and 8π2 Aµ = C M¯ s¯γµ(1 γ )b B¯ i d4xeiqx M¯ T jµ(x)Hc(0) B¯ 9 9h | − 5 | i− q2 · h | | i Z 2im +C bq M¯ s¯σλµ(1+γ )b B¯ 7 q2 λh | 5 | i 8π2 Aµ = i d4xeiqx M¯ T jµ(x)(Hu(0) Hc(0)) B¯ cu q2 · h | − | i Z Aµ = C M¯ s¯γµ(1 γ )b B¯ (9) 10 10h | − 5 | i For b s transitions the contribution from Aµ is suppressed by the prefactor λ /λ and → cu u t can be neglected. Exploiting the presence of the large scale q2 m2, an operator product expansion ∼ b (OPE) can be performed for the non-local term 8π2 µ(q) i d4xeiqxT jµ(x)Hc(0) (10) KH ≡ − q2 · Z which describes the contribution of 4-quark operators to the b sl+l amplitude. − → Such an OPE corresponds to integrating out the hard quark loop, leading to a series of local effective interactions for the high-q2 region. To leading order in the large-q2 expansion this has been presented in [17]. A discussion of the OPE including higher- order contributions has been given in [18]. 4 Figure 1: OPE for the correlation function µ: Leading-power contributions (operators KH of dimension 3). The solid square and the virtual-photon attachment indicate the in- sertion of the weak Hamiltonian and of the electromagnetic current, respectively. The diagram on the left shows the lowest-order term in QCD. On the right is a sample diagram for the next-to-leading QCD corrections of order α . s Before going into more detail we discuss the basic structure of the OPE for µ. The KH expansion may be written as µ(q) = C (q) µ (11) KH d,n Od,n d,n X The operators arecomposed ofquark andgluonfields andhave the flavour quantum d,n O numbers of (s¯b). They are ordered according to their dimension d and carry an index n labeling different operators with the same dimension. The C (q) are the corresponding d,n Wilson coefficients, which can be computed in perturbation theory. The large scales justifying the expansion are m2 and q2. They are counted as quantities of the same b order. The coefficients then scale as C m3 d in the heavy-quark limit. Since the matrix elements M¯ B¯ scale as √d,nm∼, thbe−matrix element of each term in (11) d,n b h |O | i 7/2 d behaves as m − . Current conservation implies that all operators are transverse in q, b q µ 0 (12) µOd,n ≡ It is convenient to work with the b-quark field in full QCD. This field could be further expanded within heavy-quark effective theory (HQET), in order to make the m - b dependence fully explicit. In such an approach many additional operators would arise whose hadronic matrix elements are not readily known. In contrast, the advantage of using the b-field in full QCD is that fewer operators appear and that the matrix elements of the leading ones are given by common form factors. In this method the OPE becomes particularly transparent and we will adopt it here. At leading order in the OPE (d = 3), illustrated in Fig. 1, and in the chiral limit (m = 0) there are two operators s qµqν µ = gµν s¯γ (1 γ )b (13) O3,1 − q2 ν − 5 (cid:18) (cid:19) im µ = bq s¯σλµ(1+γ )b (14) O3,2 q2 λ 5 5 Figure 2: OPE for the correlation function µ: Second-order power corrections (opera- KH tors of dimension 5). The crossed circles denote the places where the virtual photon can be attached. Using the equations of motion for the external quarks it can be shown that all possible bilinears s¯ Γb and s¯←D−Γb arising from the correlator µ can be expressed in terms of L KH (13) and (14). Consequently, no independent dimension-4 operators of the form s¯←D−Γb can appear in the OPE. The complete proof is given in appendix A. As an example, the operator s¯i←D−µ(1+γ )b satisfies the equations-of-motion identity (for m = 0) 5 s m 1 i s¯i←D−µ(1+γ )b bs¯γµ(1 γ )b+ ∂ (s¯σµν(1+γ )b)+ ∂µ(s¯(1+γ )b) (15) 5 5 ν 5 5 ≡ − 2 − 2 2 ¯ For any B X matrix element with momentum transfer q this is equivalent to s → m i 1 s¯i←D−µ(1+γ )b = bs¯γµ(1 γ )b q s¯σµν(1+γ )b+ qµs¯(1+γ )b (16) 5 5 ν 5 5 − 2 − − 2 2 Because of current conservation only the transverse part (g q q /q2)Oλ of such an µλ µ λ − operator Oµ can appear in the OPE. From (16) we see that this part can be reduced to a linear combination of (13) and (14). If we keep m = 0, two additional operators have to be considered s 6 qµqν µ = m gµν s¯γ (1+γ )b (17) O4,1 s − q2 ν 5 (cid:18) (cid:19) im m µ = s bq s¯σλµ(1 γ )b (18) O4,2 q2 λ − 5 Since m /m is small, and numerically similar to Λ/m , we may formally count these s b b as operators of dimension 4. Because they are absent at order α0, their impact will s be suppressed to the level of α m /m 0.5%, which is negligible. Note that these s s b ∼ operators do in any case not introduce new hadronic form factors. At d = 5 (Fig. 2) we encounter operators with a factor of the gluon field strength G , which have the form µν µ = s¯(gGΓ )µb (19) O5,n n where the Γ denote Dirac and Lorentz structures. We will treat the OPE explicitly to n the level of d = 5, that is including power corrections up to second order in Λ/m . b 6 Figure 3: OPE for the correlation function µ: Weak annihilation as an example of KH third-order power corrections (operators of dimension 6). Crossed circles denote the various virtual photon attachments. Although we will not give a full treatment of dimension-6 corrections, we consider as an example the effect of weak annihilation (Fig. 3). This contribution is characterized by the annihilation of the two valence quarks in the B¯ meson in the B¯ M¯ transi- → tion through the weak Hamiltonian. It is described by 4-quark operators, which read schematically µ = (r¯Γ bs¯Γ r)µ (20) O6ann,n 1 2 n with Lorentz and Dirac structures indicated by (Γ ) , and the light quark field r = u, 1,2 n d in the case of non-strange B¯ mesons. Weak annihilation provides a mechanism to break isospin symmetry, directly at the level of the transition operator. In µ weak KH annihilation, in addition to being a third order power correction, comes only from QCD penguin operators, which have small coefficients. The contribution to isospin breaking from this source will therefore be strongly suppressed. 3.2 OPE to leading order in α s In this section we give explicitly the first few terms in the OPE to leading order in renormalization-group improved perturbation theory, that is neglecting relative correc- tions of (α ). This order for is relevant in the next-to-leading logarithmic approx- s H imationOto the B¯ M¯l+l amKplitude. We may then write − → µ = µ + µ + µ + (α ,(Λ/m )3) (21) KH KH3 KH5 KH6a O s b The lower indices of the terms on the r.h.s. denote the dimension d of the corresponding local operators, which come with a coefficient of order 1/md 3. The first term reads b− qµqν µ = gµν s¯γ (1 γ )b KH3 − q2 ν − 5 · (cid:18) (cid:19) 1 h(z,sˆ)(C +3C +3C +C +3C +C ) h(1,sˆ)(4C +4C +3C +C ) 1 2 3 4 5 6 3 4 5 6 − 2 (cid:20) 1 2 h(0,sˆ)(C +3C )+ (3C +C +3C +C ) (22) 3 4 3 4 5 6 −2 9 (cid:21) 7 The coefficient in (22) requires a UV renormalization, which has to be consistent with the definition of C . The expression given here corresponds to the NDR scheme used in 9 [19]. The function h(z,sˆ) is (z m /m , sˆ q2/m2, x 4z2/sˆ= 4m2/q2) ≡ c b ≡ b ≡ c 8 m 8 8 4 2 1 √1 x h(z,sˆ) = ln b lnz + + x+ (2+x)√1 x ln − − +iπ (23) −9 µ − 9 27 9 9 − 1+√1 x (cid:18) − (cid:19) Next we have q qµ q qα C Q µ = εαβλρ β +εβµλρ β εαµλρ s¯γ (1 γ )gG b 1 cf(x) KH5 q2 q2 − λ − 5 αρ q2 (cid:20) (cid:21) q 4C Q λ s¯gG (gαλσβµ gαµσβλ)(1+γ )b 8 b (24) αβ 5 −m − q2 B Here x 1 √1 x f(x) = ln − − +iπ 2 (25) √1 x 1+√1 x − − (cid:18) − (cid:19) with x = 4m2/q2. The charm-loop contribution in (24), proportional to C , can be c 1 inferred from [24]. Note that here we use the convention ε0123 = 1. In writing (24) we − have neglected terms with the small QCD penguin coefficients C ,...,C . 3 6 Finally, weak-annihilation diagrams give the dimension-6 term 8π2 µ = q 2Q (r¯γµ(1 γ )b s¯ γλ(1 γ )r µ λ ) KH6a q4 λ r i − 5 j k − 5 l −{ ↔ } r=u,d(cid:20) X 2 iεµλβνr¯γ (1 γ )b s¯ γ (1 γ )r (δ δ C +δ δ C ) i β 5 j k ν 5 l ij kl 4 il kj 3 −3 − − (cid:21) 16π2i + q Q (r¯(1 γ )b s¯ σµλ(1+γ )r +r¯σµλ(1 γ )b s¯ (1+γ )r ) q4 λ r i − 5 j k 5 l i − 5 j k 5 l r=u,d(cid:20) X 1 (r¯(1 γ )b s¯ σµλ(1+γ )r r¯σµλ(1 γ )b s¯ (1+γ )r ) i 5 j k 5 l i 5 j k 5 l −3 − − − (cid:21) (δ δ C +δ δ C ) (26) ij kl 6 il kj 5 The terms in (26) only arise from QCD penguin operators, which have small coefficients. Weremarkthatalloperatorsin(22), (24)and(26)vanishidentically whencontracted with q , as required by gauge invariance. µ 3.3 O(α ) corrections to the charm loop s The non-factorizable (α ) corrections to the charm loop arise from diagrams like the s O one shown on the r.h.s. of Fig. 1. The q2-dependence has been recently calculated in analytic form as a Taylor expansion in the small parameter z = m2/m2 [25]. Analytic c b 8 results for m = 0 had been presented in [26]. In the kinematical range relevant to c our considerations, it has been shown that the convergence of the series is very good. We therefore use the mathematica input files provided by the authors of [25] in the online preprint publication for a numerical estimate. We find that the non-factorizable (α ) corrections to the charm loop lead to a 10-15% reduction of the real part of a s 9 O and contribute a negative imaginary part of again 10-15% relative to the short-distance contribution from C (the precise value is scheme-dependent). This is in agreement with 9 the effect found for the inclusive B X l+l rate in the high-q2 region, as discussed s − → in [25], and is similar to the effect observed for the low-q2 region in the exclusive decay modes, see Table 5 in [27]. It is to be stressed that these corrections almost compensate the factorizable charm- loopcontribution (diagramonthel.h.s. inFig.1). The reasonwhy the (α )corrections s O are not suppressed stems from the different colour structure of the diagrams. Whereas the factorizable charm loop comes with a colour-suppressed combination of Wilson co- efficients, the additional gluon exchange allows the cc¯-pair to be in a colour-octet state with no such suppression. At even higher orders in perturbation theory, (αn) with O s n 2, on the other hand, the numerical effect on a should really be small, as no new 9 ≥ additionally enhanced colour structures will arise. 4 Matrix elements and power corrections The computation of the amplitude from the OPE requires the evaluation of the matrix elements of the local operators. We estimate in particular the matrix elements of the higher-dimensional contributions. This will allow us to quantify power corrections to the B K( )l+l amplitude at high q2. The cases of B K and B K transitions will ∗ − ∗ → → → be considered in turn. 4.1 B → K The matrix element of the leading dimension-3 operator is given in terms of the familiar form factors f , defined by (p = k +q) ± K¯(k) s¯γµ(1 γ )b B¯(p) = 2f (q2)kµ +[f (q2)+f (q2)]qµ (27) 5 + + h | − | i − At the level of the dimension-5 correction in (24) one encounters operators of the form s¯G Γb. Their matrix elements introduce, in general, new nonperturbative form αρ factors. Using Lorentz invariance and the antisymmetry of G and σρτ one can show αβ that q K¯(k) s¯G (gαλσβµ gαµσβλ)(1+γ )b B¯(p) 0 (28) λ αβ 5 h | − | i ≡ In order to estimate the remaining term we assume Λ E m for the kaon energy K B ≪ ≪ E . In this limit the matrix element can be computed in QCD factorization. To leading K order we then find πα (E )C m f f k q K¯(k) µ B¯(p) = s K FC Q f(x) B B K kµ · qµ (29) h |KH5| i − N 1 c λ q2 − q2 B (cid:20) (cid:21) 9