PITHA 05/21
December 23, 2005
Spectator scattering at NLO in nonleptonic
decays: Tree amplitudes
M. Beneke and S. Jäger
Institut für Theoretische Physik E, RWTH Aachen
D–52056 Aachen, Germany
We compute the 1loop () correction to hard spectator scattering in nonleptonic decay tree amplitudes. This forms part of the NNLO contribution to the QCD factorization formula for hadronic decays, and introduces a new rescattering phase that corrects the leadingorder result for direct CP asymmetries. Among the technical issues, we discuss the cancellation of infrared divergences, and the treatment of evanescent fourquark operators. The infrared finiteness of our result establishes factorization of spectator scattering at the 1loop order. Depending on the values of hadronic input parameters, the new 1loop correction may have a significant impact on treedominated decays such as .
1 Introduction
The majority of observables at the factories is connected with branching fractions and CP asymmetries of hadronic decays to two charmless mesons, for which stronginteraction effects are essential. There is some control over these effects, since the decay amplitudes factorize in the heavyquark limit. In the QCD factorization framework [1] the matrix elements of the effective weak interaction operators take the (schematic) expression
(1) 
The longdistance stronginteraction effects are now confined to a form factor at , decay constants , and lightcone distribution amplitudes . The benefit is that information extraneous to twobody decays is available for these, and that the shortdistance kernels can be expanded in a perturbation series in the strong coupling . Both kernels are currently known from [1] at order . While for this includes a 1loop correction to “naive factorization”, in case of the order contribution is actually the leading term. It originates from the treelevel exchange of a hardcollinear gluon with the spectatorquark in the meson, as indicated in Figure 2 below. (The class of corrections from fermionloop insertions into the gluon propagator is also known [2]. In spectator scattering these terms are all connected with the hardcollinear scale and make no contribution to the hard 1loop correction, which we compute here.)
In this paper we shall compute the 1loop () correction to the spectatorscattering kernel for what is known as the (topological) “tree amplitudes” in twobody decays. There are several motivations for performing this calculation:

As in any perturbative QCD calculation the 1loop correction is necessary to eliminate scale ambiguities. In the present case of spectator scattering the characteristic scales are and . The latter being only about , a 1loop calculation is necessary to ascertain the validity of a perturbative treatment by showing that the expansion converges. The 1loop correction to spectator scattering forms part of the nexttonexttoleading order (NNLO) contribution to the decay amplitudes.

At order the strong interaction phases, and hence direct CP asymmetries, originate entirely from the imaginary part of the kernel in the first term on the righthand side of (1). The 1loop correction to introduces a new rescattering mechanism by spectator scattering. Its calculation represents an important, presumably dominant, part of the nexttoleading order (NLO) result for the CP asymmetries. The NLO result will be needed to resolve or understand potential discrepancies of the LO result with experimental data.
The organization of the paper is as follows. In Section 2 we set up the definitions and matching equations for the calculation of the hardscattering kernel , which is then described in Section 3. The expression for the kernel is given at the end of that section. In Section 4 we obtain the tree amplitudes in a convenient representation, where the lightcone distribution amplitudes are integrated in the Gegenbauer expansion. The numerical effect of the new correction on the tree amplitudes and the branching fractions is investigated in Section 5. We conclude in Section 6.
2 Setup and matching
2.1 Flavour and colour
We are concerned with the currentcurrent operators in the effective weak Hamiltonian for transitions given by
(2)  
with denoting color, and or . There are two possible flavour flows to the final state as illustrated in Figure 1 for . In case of the colourallowed tree amplitude (left), denoted following the notation^{1}^{1}1The normalization is such that at treelevel and with and . See [3], section 2.2, for the relation between decay amplitudes and the parameters. of [3], meson represented by the upgoing quark lines has the flavour quantum numbers of , and those of , where denotes the flavour of the spectator antiquark in the meson. In case of the coloursuppressed tree amplitude (right in Figure 1), the corresponding quantum numbers are and , respectively. In addition there exist “penguin contractions”, where the and fields from are contracted in the same fermion loop. Together with other operators from the effective Hamiltonian they contribute to the (topological) penguin amplitudes, which we do not consider in this paper. Thus, in the computation of the (topological) tree amplitudes there appear four shortdistance coefficients, two corresponding to the matrix element of as shown in Figure 1, and two corresponding to , which differ only by the colour labels at the operator vertex. It will be seen from the final result that only two of the four coefficients are different, because and are equivalent by a Fierz transformation, when the flavours and are not distinguished. However, since we use dimensional regularization, Fierz symmetry cannot be assumed to hold a priori.
We shall refer to the flavourflow diagrams, where the spinor indices are contracted along the quark lines of (left diagram of Figure 1), as the “right insertions” of ; the other contraction (right diagram) is the “wrong insertion”. Exactly the same diagrams contribute to the two right (wrong) insertions, only the colour factor is different for each diagram, since the two operators have different colourorderings. With colour and flavour thus understood, we will omit colour and flavour labels in the subsequent discussion of operator matching.
2.2 Matching onto SCET
The shortdistance kernels can be determined by extracting the hard and hardcollinear momentum regions from quark decay amplitudes according to the strategy of expanding Feynman diagrams by regions [4]. The calculation becomes more transparent, when it is organized as an operator matching calculation in softcollinear effective theory (SCET) [5]. The spectatorscattering kernel is obtained by the matching sequence QCD SCET SCET, by which hard fluctuations (, virtuality ) and hardcollinear fluctuations (, , , virtuality ) are integrated out in two steps. This method has by now been worked out completely for heavytolight form factors at large recoil energy of the light meson, both to all orders [6, 7], and by explicit 1loop calculations of the shortdistance coefficients [8, 9, 10]. For application to nonleptonic decays the effective theory has to be extended to include two sets of collinear fields corresponding to the (nearly) lightlike directions of the two finalstate mesons. As explained in [11] this is a relatively minor complication, because the collinear fields for different directions decouple already at the scale .
Our SCET conventions follow those of the formfactor calculations [6, 8, 10]. Meson , which picks up the spectator antiquark from the meson, moves into the direction of the lightlike vector . The collinear quark field for this direction is denoted by with , the corresponding collinear gluon field is . The second meson moves into the opposite direction , and the collinear fields for this direction are , satisfying , and . The heavy quark field is labeled by the timelike vector with .
In [6] a powercounting argument has been developed to identify the SCET operators that can appear at leading power in the expansion of heavytolight form factors. Applying this argument to the two collinear directions separately, we find that can match to only two operators in SCET with nonvanishing matrix elements for nonsinglet mesons .^{2}^{2}2Recall that we are not counting flavour degrees of freedom. Mesons with flavoursinglet components require additional twogluon operators, as well as a term that does not factorize in SCET [12]. The leading operator in the collinear2 sector is uniquely given by . The two operators are then constructed by multiplying this with an A0 and a B1type current for the transition [6]. Due to chirality conservation and the requirement that the operator be a Lorentz scalar, there is only one current of each type. The two SCET operators thus obtained can be arranged to reproduce the structure of the factorization formula (1) by defining
(3)  
The first operator includes the shortdistance coefficients , such that its matrix element is proportional to the form factor ( for vector mesons) in QCD (not SCET). The expressions for the coefficients to 1loop (more precisely, their momentum space Fourier transforms) can be found in [10], but they will not be needed here. In (3) fields without position argument are at , and the field products within the large brackets are coloursinglets. We do not consider colouroctet operators, since their matrix elements between meson states vanish. Although the second operator carries an apparent suppression, both operators are in fact leading, because the matrix element of is suppressed. Hence, at leading order in , the operators from (2) are represented in SCET by the equation
(4) 
with , , and () the momentum of (). Of the two matching coefficients is already known to the 1loop order () [1]. In this paper we compute the 1loop () correction to
(5) 
We recall that on accounting for flavour there are actually two copies of , with different flavour structure, and given the two operators in the effective Hamiltonian, there are four different coefficient functions , which we do not distinguish here to simplify the notation.
To see how (1) follows and to make the overall factors explicit, we evaluate the matrix element of (4) for the case that and are both pseudoscalar mesons. The SCET Lagrangian contains no leadingpower interactions between the collinear2 and collinear1 fields after decoupling soft gluons from the collinear2 sector by a field redefinition (second paper of [5]). The matrix elements of , fall apart into ([10], eqs. (18,81) with )
(6) 
such that
(7)  
A demonstration of factorization should provide an argument for the convergence of the various convolution integrals, an issue that is not solved to all orders in perturbation theory for the second term (spectator scattering) in the bracket. The convergence will be explicitly checked at 1loop in our calculation. At the 1loop order it is also easy to see by diagrammatic analysis that no operators other than , are needed to reproduce the hard momentum regions. In particular any diagrams that match directly onto sixquark operators already in SCET are powersuppressed.
2.3 Matching onto SCET
To complete the derivation of (1) the hardcollinear scale is integrated out by matching onto SCET. Hardcollinear momentum regions appear only in spectator scattering, since an external soft momentum is required. Thus the first term in the bracket of (7) is left unchanged, while the SCET form factor related to the matrix element (6) must be matched onto SCET. No new calculation is needed for this step, since we can use ([10], eq. (86))
(8) 
where the “jet function” has been calculated to 1loop in [9, 10], and is times the decay constant in the static limit ([10], eq. (83)). Inserting this into (7), we obtain
(9)  
which (up to a normalization factor ) is (1) with
(10) 
The jet function is unique, i.e. all four hardscattering functions are convoluted with the same .
The treelevel expressions for the hard coefficient functions (when not zero) and the jet function are
(11) 
where we introduced the QCD colour factors , , and the “bar notation”, in which for convolution variables . Only the two diagrams shown in Figure 2 have to be computed to obtain . The other two diagrams with attachments to the horizontal quark lines are included in the tree contribution to , and thus belong to the term in (4). Combining the tree coefficients, we obtain
(12) 
which reproduces the result from [1]. Note that denotes the quark pole mass, and the meson mass, but that factors of and have not been distinguished in [1], since the difference is a power correction.
3 1loop calculation
In this section we describe technical aspects of the computation of . We calculate the 5point amplitude
(13) 
and the corresponding SCET matrix elements of the righthand side of (4). With the exception of one class of diagrams to be discussed below, the parton momenta can be restricted to their leading components. Thus for the partons in the collinear2 direction we put , , for those in the collinear1 direction , , and for the heavy quark momentum . For such external momenta the SCET and HQET spinors coincide with the QCD ones.
We use dimensional regularization with and an anticommuting (NDR scheme). The amplitude (13) has ultraviolet (UV) and infrared (IR) singularities. The former must be subtracted in accordance with the definition of the operators in the effective Hamiltonian [13]; the latter in accordance with the definition of the jet function and lightcone distribution amplitudes. This is accomplished by using subtractions and a certain prescription for dealing with evanescent operators. We first discuss the “right insertion” of , in which the quark spinor indices are contracted according to . The “wrong insertion” leads to , which differs from the desired order (3) by a Fierz transformation.
3.1 Evanescent operators
The calculation in dimensional regularization is complicated by the presence of evanescent products of Dirac matrices (products that vanish in four dimensions). When such products multiply poles they need special treatment. In our calculation there are evanescent products that multiply UV singularities. Their definition is related to the renormalization convention for . The NDR scheme corresponds to setting
(14) 
whenever the lefthand side multiplies an UV pole. All other products multiplying UV singularities can be reduced to (14) by permutations.
The evanescent products that multiply IR poles are more complicated. Their treatment is related to the definition of evanescent operators of the type in SCET. To reduce the notation to the essentials, we strip off all the fields from and represent it only by its Dirac structure,
(15) 
where
(16) 
In our 1loop calculation we encounter the four operators
(17) 
In this notation equals . One easily checks that the other three operators are evanescent, i.e. vanish in four dimensions. These operators will disappear from the final result, since we shall renormalize them such that their IRfinite matrix elements vanish, but they must be kept in intermediate steps, hence the matching equation (4) has to be extended to include all four operators on the righthand side.
Evanescent operators appear already at tree level. In this approximation the matrix element (13) is given by
(18) 
(The “right insertion” of vanishes at tree level, because the colourtrace is zero.) The subscript “nf” (for “nonfactorizable”) means that the “factorizable” terms that belong to are omitted, and only the two diagrams in Figure 2 are included. While one can simply set here to recover (11), since no poles are present at tree level, the appearance of an evanescent operator at tree level implies that one must compute the mixing of into in the 1loop calculation.
3.2 UV renormalized 1loop amplitude
The calculation of the 1loop correction to involves the diagrams shown in Figure 3. The two lines directed upward represent the quark (antiquark) with collinear2 momentum proportional to . The horizontal lines describe an incoming bottom quark, and an outgoing collinear1 quark with momentum proportional to . The momentum of the external gluon is also in the direction. The calculation of diagrams with no gluon lines that connect the two upper lines to the horizontal lines is not necessary, since the definition of is chosen such that these diagrams contribute only to .
The calculation of the diagrams uses standard methods. The massive box integrals in dimensional regularization can be evaluated adapting the method of [14]. Alternatively, they can be reduced to vertex integrals, because all external momenta are linear combinations of only two vectors , . This observation also simplifies the tensor reduction, since one can use
(19) 
The classes A, B of 1particle reducible diagrams must be included in the amplitude calculation. The heavyquark propagator to the right of the external gluon line in class A is offshell by an amount of order , hence these diagrams contribute entirely to the shortdistance coefficient. Class B is more complicated, since the lightquark propagator with momentum has small virtuality, hence the diagram is not completely shortdistance. The nonlocal, longdistance contributions cancel in the matching relation against timeordered products of and the SCET interaction Lagrangian as discussed in [8]. The local contribution to the shortdistance coefficient can be extracted via the substitution
(20) 
A shortcut to this conclusion is obtained by observing that we can put , since we do not match operators with transverse derivatives, and keep only . For all relevant interaction terms from the SCET Lagrangian vanish, hence the class B diagrams are purely shortdistance. Indeed, since the term in the propagator does not contribute owing to the onshell spinor to the right, the substitution (20) becomes an identity for .
Ultraviolet renormalization of the amplitude involves standard counterterms from the QCD Lagrangian as well as the counterterms for . The UVrenormalized amplitude is written as
(21) 
where denotes the partonic treelevel matrix element of , equal to the Dirac matrix products (17) multiplied by the SCET quark spinors and gluon polarization vector. As already mentioned the tree matrix element of the “right insertion” of vanishes due to colour, so for . In this case the 1loop amplitudes are IRfinite and can be evaluated in (after UV renormalization is applied). Hence the evanescent terms vanish. The “right insertion” of has for [see (18)], and the 1loop amplitudes are IRdivergent. The 1loop terms have a singularity, proportional to the tree matrix element, as follows from the universality of soft singularities. has a pole, while turns out to be IRfinite. The IR divergences cancel when the QCD amplitude is related to the matching coefficient through (4) as explained in the following.
3.3 IR subtractions
We start from the matching equation (4) extended to include the evanescent operators
(22) 
Convolutions, which may involve one or two integrations, are now represented by an asterisk. Since we work with matrix elements in states with definite momentum it is convenient to use the momentumspace representation. Expanding all quantities to the 1loop order, making use of (21) and , we obtain
(23)  
The factorizable contribution on the lefthand side comes from 1loop diagrams with no gluon lines connecting the part of the diagram to the part. It is canceled by the term on the righthand side, since the 1loop matrix element of contains exactly these diagrams in the coefficient function in its definition (3). The UVrenormalized 1loop matrix elements of the are given by
(24) 
where is the bare matrix element, which depends on the IR regularization scheme , and the matrix kernel of ultraviolet renormalization factors. When dimensional regularization is used for UV and IR singularities as was done in the calculation of , the bare matrix elements vanish, since the 1loop diagrams are scaleless. Hence, inserting (24) into (23), using (3) and , we obtain
(25) 
by comparing the coefficient of . We also used that is zero for . The renormalization constants for the evanescent operators are determined by requiring that the IRfinite matrix elements () vanish [13, 15]. Here “IRfinite” means the matrix element computed with any IR regularization off other than dimensional and with dimensional regularization applied only to the UV singularities. According to (24) this fixes . Hence the 1loop shortdistance coefficient of the physical (nonevanescent) operator is given by
(26) 
Note that since the IRfinite matrix elements of the evanescent operators have been made to vanish, only the term survives in (22). It is therefore not necessary to determine the coefficient functions for . We also note that the renormalization constant is finite, and that is independent of the apparently arbitrary IR regulator. This is because the mixing of an evanescent operator into a physical operator arises through the multiplication of an ultraviolet pole with a term of order from the Dirac algebra, both of which are independent of the IR regularization. The poles do not contribute to operator mixing due to their universality.
Eq. (26) provides the final result for the two of the four matching coefficients associated with the right insertions. We briefly discuss the subtraction terms in (26). First note that for the right insertion of the tree amplitudes vanish, hence (26) is simply . This is consistent, since for this case is IRfinite as observed above. For the right insertion of , all three subtraction terms in (26) are present. Due to factorization in SCET, the renormalization of an operator falls apart into a renormalization factor for the collinear2 bracket and one for a B1type current. is therefore determined by the requirement that the lightcone distribution amplitude of and the jet function are defined in the scheme. This gives as the product of the BrodskyLepage kernel [16] and the renormalization kernel for the B1type current (first paper of [9], [10]). Subtracting from removes the IR singularities such that is finite as must be for a shortdistance coefficient. The third term on the righthand side involves the computation of the 1loop matrix element of the evanescent operator . We find that only a single diagram, shown in Figure 4, contributes to , such that is proportional to the spindependent part of the BrodskyLepage kernel. Explicitly,
(27)  
The fourth term follows from [10] and [1]
(28) 
with given in (40) below. Note that is a function of two variables, but like the tree contribution the two subtraction terms (27), (28) depend on , but not on the momentum fraction related to the collinear1 momenta.
3.4 “Wrong insertion”
The other two matching coefficients are related to the wrong insertions of as in the right diagram of Figure 1. We would like to express them as the coefficients of the same SCET operator , but the QCD calculation involves Dirac matrix products with a different contraction of spinor indices corresponding to
(29)  
In the second line we introduced again a shorthand notation that highlights the Dirac structure. The symbol means that the spinor indices are contracted as in . We deal with the required Fierz transformation and evanescent operators simultaneously by introducing the operators
(30) 
Here is the shorthand for . The basis is chosen such that and are Fierzequivalent and vanish in four dimensions. Hence we have one physical operator, , and four evanescent operators, , and . The tree matrix element is now given by
(31) 
and the 1loop amplitude reads
(32) 
This does not contain , since all diagrams have the “wrong” Fierzordering. Proceeding as before and requiring that the infraredfinite matrix elements of the four evanescent operators vanish, we find that (26) is replaced by
(33) 
with the bare 1loop mixing of into , and . The new term involves the difference of the mixing of and into themselves. This difference is finite and independent of the IR regulator for the same reason that is. There is one subtle aspect hidden in (33) that requires explanation. As in (23) we would like to cancel the factorizable QCD diagrams against the matrix element of , but the two terms appear in different Fierzorderings. The consequence of this is that there should be an extra term related to the factorizable diagrams on the righthand side of (33). Using that at treelevel only , , are nonzero, it is given by
(34) 
However, we find that this term vanishes, hence (33) is correct. The subtractions and are identical to the corresponding terms for the right insertion. The other two terms are once again related to an integral over the spindependent part of the BrodskyLepage kernel. Explicitly, they read
(35) 
Despite the different Dirac algebra and subtraction structure we find that the final result for the matching coefficient related to the wrong insertion of () is identical to the one for the right insertion of ().