1 Introduction


LMU-ASC 107/10




January 2011

Theory of decays at high :

OPE and quark-hadron duality

M. Beylich, G. Buchalla and Th. Feldmann444Address after January 2011:
IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK

CERN, Theory Division, CH–1211 Geneva 23, Switzerland

Ludwig-Maximilians-Universität München, Fakultät für Physik,

Arnold Sommerfeld Center for Theoretical Physics, D–80333 München, Germany

Physik Department, Technische Universität München,

James-Franck-Straße, D–85748 Garching, Germany

We develop a systematic framework for exclusive rare decays of the type at large dilepton invariant mass . It is based on an operator product expansion (OPE) for the required matrix elements of the nonleptonic weak Hamiltonian in this kinematic regime. Our treatment differs from previous work by 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

1 Introduction

The rare decays 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 -meson factories \@cite?? 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 colliders \@cite??.

The calculability of decay rates and distributions benefits from the fact that these processes are, to first approximation, semileptonic modes. Correspondingly, the hadronic physics is described by form factors, which multiply a perturbatively calculable amplitude. This simple picture is not exact because also the nonleptonic weak Hamiltonian at scale has matrix elements. The prominent example is given by hadronic interactions of the form , where the charm quarks annihilate into 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 completely 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 , for which the properties of charm loops are markedly different. For the presence of very narrow resonances leads to huge violations of quark hadron duality [12] and the hadronic backgrounds from , followed by , dominate the short distance rate by two orders of magnitude. This region in can be removed by experimental cuts.

For the kaon is very energetic and the charm loops can be computed systematically in the heavy-quark limit using QCD factorization for decays into light-like mesons [13, 14]. This approach was first employed for 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- region, has received comparatively little attention. In this case the kaon energy is around a or below, and (soft-collinear) QCD factorization is less justified, becoming invalid close to the endpoint of the spectrum at . On the other hand, the large value of defines a hard scale for the hadronic contribution to . Consequently an operator product expansion (OPE) can be constructed, which generates an expansion of the amplitude in powers of (or ). Charm loops, and other hadronic contributions, are thus approximated as effective interactions that are local on the soft scales set by and . This simplifies 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 discussed at lowest order in the OPE in [17], where it was applied to the endpoint region of and . In [18] the OPE was considered in some detail, including a discussion of power corrections.

In the present paper we formulate the OPE for the high- region of 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 -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. We also retain the kinematical dependence on in the coefficient functions, rather than expanding it around . We further discuss the issue of quark-hadron duality, which appears relevant because of the existence of resonance structure in the 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- region of . Further aspects and new results of our analysis will be summarized in sections 8 and 9. The main conclusion is that is under very good theoretical control also for . 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 reference. In section 3 our formulation of the OPE for at high is described and the power expansion is constructed explicitly, complete to second order in and with a discussion of weak annihilation as an example of a (small) third-order correction. 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 and transitions. Section 5 discusses the connection between the OPE for large and QCD factorization for energetic kaons, which are shown to give consistent results at intermediate . 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 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 transitions reads [19, 20, 21]




The operators are given by


Note that the numbering of is reversed with respect to the convention of [19]. Our coefficients correspond to in [19] and we include the factor of in the definition of . The sign conventions for the electromagnetic and strong couplings correspond to the covariant derivative . With these definitions the coefficients are negative in the Standard Model.

2.2 Dilepton-mass spectra and short-distance coefficients

We define the kinematic quantities (where is the dilepton invariant mass squared), , and


The differential branching fractions for can then be written as [22]


The coefficient contains the Wilson coefficient combined with the short-distance parts of the matrix elements of operators . The coefficient multiplies the matrix element of the local operator in the decay amplitude. The coefficient of the operator is determined by very short distances and is precisely known.

The corresponding formulas for can for instance be found in [23].

3 OPE for amplitudes at high

3.1 General structure

The amplitudes for the exclusive decays , where , , or a similar meson, are given by the matrix element of the effective Hamiltonian in (1) between the initial meson and the final state. The dominant contribution comes from the semileptonic operators . Their matrix elements are simple in the sense that all hadronic physics is described by a set of transition form factors. This is also true for the electromagnetic operator . The matrix elements of the hadronic operators are more complicated. They are induced by photon exchange and can be expressed through the matrix element of a correlator between the hadronic part of the effective Hamiltonian


and the electromagnetic current of the quarks


where is the electric charge quantum number of quark flavour and a summation over is understood. The decay amplitude may thus be written as


where and are the lepton spinors and


For transitions the contribution from is suppressed by the prefactor and can be neglected.

Exploiting the presence of the large scale , an operator product expansion (OPE) can be performed for the non-local term


which describes the contribution of 4-quark operators to the amplitude. Such an OPE corresponds to integrating out the hard quark loop, leading to a series of local effective interactions for the high- region. To leading order in the large- expansion this has been presented in [17]. A discussion of the OPE including higher-order contributions has been given in [18].

Before going into more detail we discuss the basic structure of the OPE for . The expansion may be written as


The operators are composed of quark and gluon fields and have the flavour quantum numbers of . They are ordered according to their dimension and carry an index labeling different operators with the same dimension. The are the corresponding Wilson coefficients, which can be computed in perturbation theory. The large scales justifying the expansion are and . They are counted as quantities of the same order. The coefficients then scale as in the heavy-quark limit. Since the matrix elements scale as , the matrix element of each term in (11) behaves as . Current conservation implies that all operators are transverse in ,


It is convenient to work with the -quark field in full QCD. This field could be further expanded within heavy-quark effective theory (HQET), in order to make the -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 -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 (), illustrated in Fig. 1,

Figure 1: OPE for the correlation function : Leading-power contributions (operators of dimension 3). The solid square and the virtual-photon attachment indicate the insertion 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 .

and in the chiral limit () there are two operators


Using the equations of motion for the external quarks it can be shown that all possible bilinears and arising from the correlator can be expressed in terms of (13) and (14). Consequently, no independent dimension-4 operators of the form can appear in the OPE. The complete proof is given in appendix A. As an example, the operator satisfies the equations-of-motion identity (for )


For any matrix element with momentum transfer this is equivalent to


Because of current conservation only the transverse part of such an operator 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 , two additional operators have to be considered


Since is small, and numerically similar to , we may formally count these as operators of dimension 4. Because they are absent at order , their impact will be suppressed to the level of , which is negligible. Note that these operators do in any case not introduce new hadronic form factors.

At (Fig. 2)

Figure 2: OPE for the correlation function : Second-order power corrections (operators of dimension 5). The crossed circles denote the places where the virtual photon can be attached.

we encounter operators with a factor of the gluon field strength , which have the form


where the denote Dirac and Lorentz structures. We will treat the OPE explicitly to the level of , that is including power corrections up to second order in .

Although we will not give a full treatment of dimension-6 corrections, we consider as an example the effect of weak annihilation (Fig. 3).

Figure 3: OPE for the correlation function : Weak annihilation as an example of third-order power corrections (operators of dimension 6). Crossed circles denote the various virtual photon attachments.

This contribution is characterized by the annihilation of the two valence quarks in the meson in the transition through the weak Hamiltonian. It is described by 4-quark operators, which read schematically


with Lorentz and Dirac structures indicated by , and the light quark field , in the case of non-strange mesons. Weak annihilation provides a mechanism to break isospin symmetry, directly at the level of the transition operator. In weak 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

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 corrections of . This order for is relevant in the next-to-leading logarithmic approximation to the amplitude. We may then write


The lower indices of the terms on the r.h.s. denote the dimension of the corresponding local operators, which come with a coefficient of order . The first term reads


The coefficient in (22) requires a UV renormalization, which has to be consistent with the definition of . The expression given here corresponds to the NDR scheme used in [19]. The function is (, , )


Next we have




with . The charm-loop contribution in (24), proportional to , can be inferred from [24]. Note that here we use the convention . In writing (24) we have neglected terms with the small QCD penguin coefficients .

Finally, weak-annihilation diagrams give the dimension-6 term


The terms in (26) only arise from QCD penguin operators, which have small coefficients.

We remark that all operators in (22), (24) and (26) vanish identically when contracted with , as required by gauge invariance.

3.3 corrections to the charm loop

The non-factorizable corrections to the charm loop arise from diagrams like the one shown on the r.h.s. of Fig. 1. The -dependence has been recently calculated in analytic form as a Taylor expansion in the small parameter [25]. Analytic results for had been presented in [26]. In the kinematical range relevant to 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 corrections to the charm loop lead to a 10-15% reduction of the real part of and contribute a negative imaginary part of again 10-15% relative to the short-distance contribution from (the precise value is scheme-dependent). This is in agreement with the effect found for the inclusive rate in the high- region, as discussed in [25], and is similar to the effect observed for the low- region in the exclusive decay modes, see Table 5 in [27].

It is to be stressed that these corrections almost compensate the factorizable charm-loop contribution (diagram on the l.h.s. in Fig. 1). The reason why the corrections 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 coefficients, the additional gluon exchange allows the -pair to be in a colour-octet state with no such suppression. At even higher orders in perturbation theory, with , on the other hand, the numerical effect on should really be small, as no new 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 amplitude at high . The cases of and transitions will be considered in turn.


The matrix element of the leading dimension-3 operator is given in terms of the familiar form factors , defined by ()


At the level of the dimension-5 correction in (24) one encounters operators of the form . Their matrix elements introduce, in general, new nonperturbative form factors. Using Lorentz invariance and the antisymmetry of and one can show that


In order to estimate the remaining term we assume for the kaon energy . In this limit the matrix element can be computed in QCD factorization. To leading order we then find


where , is the number of colours, and is the first inverse moment of the -meson light-cone distribution amplitude. This matrix element scales as relative to (27) in the heavy quark limit, and as for large .

In a similar way we can estimate the weak annihilation term


This contribution is power-suppressed as relative to (27). The suppression by the small Wilson coefficients is partly compensated by a large numerical factor of . To relative order there is no contribution from the term with and . Note that the result in (30) also corresponds to the matrix element obtained when naively factorizing the four-quark operators.

Normalized to the amplitude coefficient , the power corrections from and read


where , refers to the spectator quark in the meson.

Numerically, we find at for central values of the parameters. This number comes with a substantial uncertainty, in particular from and . Nevertheless, the correction to is very small, most likely below in magnitude. The correction in (31) diminishes further for larger , reaching at the endpoint. Towards the endpoint the kaon becomes soft and the result in (31), based on , can only be viewed as a rough model calculation. The conclusion that remains negligibly small should however still hold. The weak annihilation correction is at for and therefore entirely negligible, a consequence also of the small Wilson coefficients.


In the case of the decay into a vector meson the relevant form factors are defined as ()


It is convenient to treat the decay into longitudinally and transversely polarized vector mesons separately. Omitting terms proportional to


In the large energy limit , which we may use in the normalization of the power corrections, (4.2) and (36) simplify to [28, 29]


The case of a longitudinally polarized is very similar to the case of a pseudoscalar , discussed in section 4.1, and we find


where is the decay constant of .

For a with transverse polarization we obtain


where is the decay constant of .

Numerically we have and at for our standard set of parameters. These corrections are of similar size as for the pseudoscalar kaon and they are likewise negligible.

Finally, we quote the corrections from weak annihilation


where , refers to the spectator quark in the B meson. The corrections are tiny, at we have for , and , for . This is again a consequence of the suppression at large values of , see also [30].

5 Large vs. small recoil energy of the kaon

The OPE for the correlator (10), applied to , is valid as long as the energy of the kaon in the -meson rest frame


is small compared to . This condition is certainly fulfilled in the vicinity of the endpoint, but even for as low as , just above the narrow-resonance region, is still fairly small in comparison to the hard scale. On the other hand, such a value of is already larger than the QCD scale and one could consider using the factorization methods applicable to the case of energetic kaons. For