# QCD factorization for
^{1}^{1}1Supported in part by
National Natural Science Foundation of China and State Commission of
Science and Technology of China

###### Abstract

In this work, we give a detailed discussion for QCD factorization involved in the complete chirally enhanced power corrections in the heavy quark limit for decays to two light pseudoscalar mesons, and present some detailed calculations of radiative corrections at the order of . We point out that the infrared finiteness of the vertex corrections in the chirally enhanced power corrections requires twist-3 light-cone distribution amplitudes (LCDAs) of the light pseudoscalar symmetric. However, even in the symmetric condition, there is also a logarithmic divergence from the endpoints of the twist-3 LCDAs in the hard spectator scattering. We point out that the decay amplitudes of predicted by QCD factorization are really free of the renormalization scale dependence, at least at the order of . At last, we briefly compare the QCD factorization with the generalized factorization and PQCD method.

PACS numbers 13.25.Hw 12.38.Bx

## I introduction

The study of decays plays an important role in understanding the origin of violation and physics of heavy flavor. We expect that the parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the standard model, for instance, the three angles , and in the unitary triangle, can be well-determined from decays, especially from the charmless non-leptonic two-body B decays. Experimentally, many experiment projects have been running (CLEO, BaBar, Belle etc.), or will run in forthcoming years (BTeV, CERN LHCb, DESY HeraB etc.). With the accumulation of the data, the theorists will be urged to gain a deeper sight into decays, and to reduce the theoretical errors in determining the CKM parameters from the experimental data.

In the theoretical frame, the standard approach to deal with such decays is based on the low-energy effective Hamiltonian which is obtained by the Wilson operator product expansion method (OPE). In this effective Hamiltonian, the short-distance contributions from the scale above have been absorbed into the Wilson coefficients with the perturbative theory and renormalization group method. The Wilson coefficients have been evaluated to next-to-leading order. Then the main task in studying non-leptonic two-body decays is to calculate the hadronic matrix elements of the effective operators. However, we do not have a reliable approach to evaluate them from the first principles of QCD dynamics up to now.

Generally, we must resort to the factorization assumption to calculate the hadronic matrix elements for non-leptonic decays, in which the hadronic matrix element of the effective operator (in general, which is in the form of current-current four-quark operator) can be approximated as a product of two single current hadronic matrix elements; then it is parameterized into meson decay constant and meson-meson transition form factor. The most popular factorization model is the Bauer-Stech-Wirbel (BSW) model[1]. In many cases, BSW model achieves great success, which can predict the branching ratios of many modes of non-leptonic decays in correct order of magnitude. This factorization assumption does hold in the limit that the soft interactions in the initial and final states can be ignored. It seems that the argument of color-transparency can give reasonable support to the above limit. Because quark is heavy, the quarks from quark decay move so fast that a pair of quarks in a small color-singlet object decouple from the soft interactions. But the shortcomings of this simple model are obvious. First, the renormalization scheme and scale dependence in the hadronic matrix elements of the effective operators are apparently missed. Then the full decay amplitude predicted by BSW model remains dependent on the renormalization scheme and scale, which are mainly from Wilson coefficients. In past years, many researchers improved the simple factorization scheme and made many remarkable progresses, such as scheme and scale independent effective Wilson coefficients[2, 3], effective color number which is introduced to compensate the ‘non-factorizable’ contributions, etc. Furthermore, some progresses in nonperturbative methods, such as lattice QCD, QCD sum rule etc.[4, 5, 6], allow us to compute many non-perturbative parameters in decays, such as the meson decay constants and meson-meson transition form factors. Every improvement allows us to have a closer look at the nonleptonic decays.

Except for the factorization approximation, another important approach has been applied to study many exclusive hadronic decay channels, such as , , etc. This is PQCD method[7, 8, 9]. In this method, people assumes that exclusive hadronic decay is dominated by hard gluon exchange. It is analogous to the framework of perturbative factorization for exclusive processes in QCD at large momentum transfer, such as the calculation of the electromagnetic form factor of the pion [10]. The decay amplitude for decay can be written as a convolution of a hard-scattering kernel with light-cone wave functions of the participating mesons. Furthermore, in Ref.[8, 9] the Sudakov suppression has been taken into account.

Two year ago, Beneke, Buchalla, Neubert, and Sachrajda (BBNS) gave a QCD factorization formula in the heavy quark limit for the decays [11]. They pointed out that the radiative corrections from hard gluon exchange can be calculated by use of the perturbative QCD method if one neglects the power contributions of . This factorization formula can be justified in case that the ejected meson from the quark decay is a light meson or an onium, no matter whether the other recoiling meson which absorbs the spectator quark in meson is light or heavy. But for the case that the ejected meson is in an extremely asymmetric configuration, such as D meson, this factorization formula does not hold. The contributions from the hard scattering with the spectator quark in B meson are also involved in their formula. This kind of contribution cannot be contained in the naive factorization. But it appears in the order of . So they said that the naive factorization can be recovered if one neglects the radiative corrections and power suppressed contributions in the QCD factorization, and the ‘non-factorizable’ contributions in the naive factorization can be calculated perturbatively, then we do not need a phenomenological parameter to compensate the ‘non-factorizable’ effects any more[12, 13, 14].

This QCD factorization (BBNS approach) has been applied to study many meson decay modes, such as [15, 16], , [17, 18, 19] and other interesting channels[20, 21, 22, 23]. Some theoretical generalizations of BBNS approach have also been made, such as the chirally enhanced power corrections[18, 19, 24, 25] from the twist-3 light-cone distribution amplitudes of the light pseudoscalar mesons. In this work, we will take a closer look at this issue. This work is organized as follows: Sect. II is devoted to a sketch of the low energy effective Hamiltonian; in Sect.III, we will give a detailed overview of QCD factorization, in which some elaborate calculations are shown, especially for the chirally enhanced power corrections; Sect. IV is for some detailed discussions and comparison of BBNS approach to the generalized factorization and PQCD method; we conclude in Sect.V with a summary.

## Ii Effective Hamiltonian — First Step Factorization

decays involve three characteristic scales which are strongly ordered: . How to separate or factorize these three scales is the most essential question in B hadronic decays.

With the operator product expansion method (OPE), the relevant effective Hamiltonian is given by [26]:

(1) | |||||

where (for transition) or (for transition) and are the Wilson coefficients which have been evaluated to next-to-leading order approximation with the perturbative theory and renormalization group method.

In the Eq.(1), the four-quark operators are given by

(2) |

and

(3) |

with and being the tree operators, the QCD penguin operators, the electroweak penguin operators, and , the magnetic-penguin operators.

In this effective Hamiltonian for decays, the contributions from large virtual momenta of the loop corrections from scale to are attributed to the Wilson coefficients, and the low energy contributions are fully incorporated into the matrix elements of the operators[26]. So the derivation of the effective Hamiltonian can be called “the first step factorization”.

To evaluate the Wilson coefficients, we must extract them at a large renormalization scale (for example in the standard model) by matching the amplitude of the effective Hamiltonian () to that of the full theory (), then evolve them by the renormalization group equations from the scale to the scale . It should be noted that the extraction of the Wilson coefficients by matching does not depend on the choice of the external states, if we regularize the infrared (and mass) singularities properly[26]. All dependence on the choice of external states only appears in the matrix elements , and is not contained in . So only contains the short-distance contributions from the region where the perturbative theory can be applied. But for the matrix elements , the long-distance contributions appear, and are process-dependent.

Several years ago, the perturbative corrections to the Wilson coefficients in SM have been evaluated to next-to-leading order with renormalization group method[26]. As we know, the Wilson coefficients are generally renormalization scheme and scale dependent. So, in order to cancel such dependence, we must calculate the hadronic matrix elements of the effective operators to the corresponding perturbative order with the same renormalization scheme and at the same scale, then we can obtain a complete decay amplitude which is free from those unphysical dependences.

## Iii QCD Factorization For

After “the first step factorization”, the decay amplitude for can be written as

(4) |

in which, as mentioned in the previous section, the contributions from the large scale down to has been separated into the Wilson coefficients . The remaining task is to calculate the hadronic matrix elements of the effective operators. But for the complexity of QCD dynamics, it is difficult to calculate these matrix elements reliably from first principles. The most popular approximation is factorization hypothesis, in which the matrix element of the current-current operator is approximated to a product of two matrix elements of single current operator:

(5) |

Obviously, under this approximation, the original hadronic matrix element misses the dependence of the renormalization scheme and scale which should be used to cancel the corresponding dependence in the Wilson coefficients . A plausible solution to recover this scale and scheme dependence of is to calculate the radiative corrections. In one-loop level, they can be written as [13, 14, 27]:

(6) |

Here and represent the one loop corrections of QCD and QED respectively. Then one takes

(7) |

Therefore, the scheme and scale dependence of which are expressed in the form of and is recovered. But in quark level, and usually contain infrared divergences if we take the external quarks on-shell[28]. To remove or regularize the infrared divergence, the conventional treatment is to assume that external quarks are off-shell by . But this introduction of the infrared cutoff results in a gauge dependence of one-loop corrections. So how to factorize the infrared part of the matrix elements is a very subtle question. But maybe this question would get an important simplification in the case that the final states of B meson decay are two light mesons.

Two years ago, Beneke, Buchalla, Neubert and Sachrajda proposed a promising QCD factorization method for . They pointed out that in the heavy quark limit , the hadronic matrix elements for can be written in the form

(8) |

Obviously, the above formula reduces to the naive factorization if we neglect the power corrections in and the radiative corrections in . They find that the radiative corrections, which are dominated by hard gluon exchange, can be calculated systematically with the perturbative theory in the limit , in terms of the convolution of the hard scattering kernel and the light-cone distribution amplitudes of the mesons. This is also similar to the framework of perturbative factorization for exclusive processes in QCD at large momentum transfer, such as the calculation of the electromagnetic form factor of the pion [10]. Then a factorization formula for can be written as [11]:

(9) |

We call this factorization formalism as QCD factorization or the BBNS approach. In the above formula, and are the leading-twist wave functions of and pion mesons respectively, and the denote hard-scattering kernels which are calculable in perturbative theory. At the order of , the hard kernels can be depicted by Fig.1. Figures 1(a)-1(d) represent vertex corrections, Figs 1(e) and 1(f) penguin corrections, and Figs 1(g) and 1(h) hard spectator scattering.

In the heavy quark limit, both pions are energetic. The pion ejected from quark decay moves so fast that it can be described by its leading-twist light-cone distribution amplitude. The pair in the ejected pion is produced as a small-size color dipole. Consequently, the ejected pion decouples from the soft gluons at leading order of . Of course, only the cancellation of soft gluons is not enough to make the factorization hold, it is necessary that the pair also decouples from the collinear gluons. Both the cancellations of soft gluons and collinear gluons guarantee that the hard kernel is of infrared finiteness. Contrast to the pion ejected from b quark weak decay, the recoiling pion which picks up the spectator in B meson can not be described by its leading-twist light-cone distribution amplitude(LCDA), because the spectator is transferred to the recoiling pion as a soft quark. Here Beneke et al. take the point of view that the form factor cannot be calculated perturbatively. If we attempt to calculate the form factor within the perturbative framework, by the naive power counting, we find that the leading twist LCDA of pion does not fall fast enough to suppress the singularity at the endpoint where the quark from decay carries almost all momentum of the pion. It indicates that the contributions to form factor are dominated by the soft gluon exchange[16]. This point of view can be justified also from the calculation of the form factor by using light-cone sum rule (LCSR)[5, 6], in which the dominated contribution to comes from the region where the the spectator quark is transferred as a soft quark to the pion. So the transition form factor survives in the factorization formula as a nonperturbative parameter. However, when the spectator quark in meson interacts with a hard gluon from the ejected pion, the recoiling pion can be also described by its light-cone distribution amplitude. This hard spectator scattering is missed in the naive factorization, but calculable in the perturbative QCD at the leading power in . So with this factorization formula, the remaining hard part of the hadronic matrix element from the scale about has been factorized into the hard scattering kernel, and the long distance contributions are absorbed into the transition form factors and the light-cone wave functions of the participating mesons. Thus this is the “final factorization” for the two-body nonleptonic charmless decays.

An explicitly technical demonstration of the above argument has been presented in one-loop level in Refs.[11, 16]. For , this QCD factorization has been proved to two-loop order[16]. In the literature, the ejected pion is represented by its leading twist light-cone distribution amplitude(LCDA). However, since the mass of quark is not asymptotically large, in particular, some power corrections might be enhanced by certain factors, such as the scale of chiral symmetry breaking GeV, and have significant effects in studying two-body nonleptonic charmless decays. So, in this manner, the chirally enhanced power corrections must be taken into account. Accordingly, describing the ejected pion by its leading twist LCDA is not enough, the two-particle twist-3 LCDAs must be taken into account. Below, we will show the elaborate results of QCD factorization in these two cases. For illustration, we take as an example, but the result is easily generalized to the cases that the final states are the other light pseudoscalars.

### iii.1 Leading-twist Distribution Amplitude Insertion

When inserting leading-twist LCDA of the light pseudoscalar, in the heavy quark limit, the quark constituents of the ejected pion can be treated as a pair of collinear massless quark and antiquark with the momentum and respectively ( is the momentum of the ejected pion and we take as a hard light-cone momentum in calculation, ), because that the contributions from the transverse momenta of the quarks in ejected pion are power suppressed [16].

#### iii.1.1 Vertex Corrections

Now we move on to the explicit one-loop calculation of the diagram Figs. 1(a)-1(d) for . For illustration, we write down the one-gluon exchange contribution to the matrix element of the operator .

(10) | |||||

(11) | |||||

(12) | |||||

(13) | |||||

When we calculate the vertex corrections in the leading
power of , not only ultraviolet
divergence emerges but infrared divergence does also.
Infrared divergence arises from two regions where
the virtuality of the loop is soft or collinear to the momentum
of the pions. In Ref.[16], the authors gave an explicit
cancellation of soft and collinear divergence in vertex corrections
for in eikonal approximation. Figures 1(a),1(b)
and 1(c),1(d) cancel the soft divergence; 1(a),1(c) and 1(b),1(d)
cancel the collinear divergence. For
, the cancellation is similar except
that the collinear divergence also arises from the region where
is collinear to the momentum of the recoiling pion.
So Figs. 1(c),1(d) cancel not only part of soft divergence but also
part of collinear divergence.
Below, we give an explicit calculation of the Feynman
diagrams Figs. 1(a)-1(d) to show the cancellation of the
infrared divergences. In order to regularize the infrared
divergence, there are two choices for us. One is the
dimensional regularization (DR) scheme, in which the infrared divergence
can be regularized into the pole terms . In contrast to the
dimensional regularization of ultraviolet divergence, the infrared
divergence arises when , instead of
in the case of the ultraviolet divergence. So
the dimension in regularization for infrared divergence
must be set to be greater than 4. This is a subtle point,
but it will not cause any ambiguity in our calculation
because the infrared part and ultraviolet part can be safely
separated. The other method to
regularize the infrared divergence is the well-known massive gluon
(MG) scheme, in which the infrared divergence is handled by
replacing by in the gluon propagator.
Similar scheme has been applied in earlier computation of the radiative
corrections for , in which
the massless photon is replaced by a massive photon. In addition,
in our latter calculation, there are also several schemes in treating
, the most popular two are the naive dimensional regularization
(NDR) scheme and the ’t Hooft-Veltman renormalization (HV)
scheme. Both have been applied to calculate the Wilson coefficients
[26]. In this work, if there is no specification,
the NDR scheme is always applied in our calculations
for its simplicity
^{4}^{4}4Such choice does cause a scheme dependence
in the matrix elements. However,
when we choose the Wilson coefficients in the same
scheme as for the matrix elements, the final full decay amplitude is
free of scheme-dependence..

After a straightforward calculation in DR scheme and using the corresponding Feynman parameter integrals listed in Appendix C, we obtain

(14) | |||||

(15) | |||||

(16) | |||||

(17) | |||||

In above, we have set () in regularizing the infrared divergence. Then, after summing over all four diagrams, we find that all pole terms in are really cancelled before we integrate over the momentum fraction variable . So after modified minimal subtraction (), we get

(18) | |||||

Assuming that the light-cone distribution amplitude is symmetric, then the above equation can be simplified as follows:

(19) | |||||

It is easy to check that the above equations are consistent with the results in previous works. Actually, with the MG scheme, we get the same results as that by using the DR scheme.

With Eqs.(18,19), we can compute the vertex corrections no matter the LCDA is symmetric or asymmetric. This is very important in principle. For instance, when kaon is ejected from b quark decay, we must take the contributions from the asymmetric part of LCDA of kaon into account, although the contributions from the asymmetric part are very small numerically[19].

#### iii.1.2 Penguin Corrections

There are two kinds of penguin corrections. One is the four quark operators insertion [Fig. 1(e)]; the other is the magnetic penguin insertion [Fig. 1(f)]. The first kind is generally called BSS mechanism. In generalized factorization, BSS mechanism plays a very important role in violation because it is the unique source of strong phases. But in generalized factorization, the virtuality of the gluon or photon is ambiguous; usually one varies around . This variation does not have an important effect on the branching ratios, but it does for asymmetries. In QCD factorization, this ambiguity is rendered by taking the virtuality of the gluon as . When treating penguin contractions, one should be careful that Fig. 1(e) contains two kinds of topology, which is depicted in Fig. 2. They are equivalent in 4 dimensions according to Fierz rearrangement. However, since penguin corrections contain ultraviolet divergence, we must do calculations in d dimensions where these two kinds of topology are not equivalent [29]. Below we give an explicit calculation of or penguin insertions for which belong to the second topology, Fig. 2(b):

(20) | |||||

After subtraction and using the equations of motions, we get the finite result

(21) | |||||

where . The first topology, Fig. 2(a), for example, penguin insertion for , is similar to the results of the second topology, Fig.2(b), except that there is an extra factor :

(22) | |||||

For the magnetic penguin insertion, it is the easiest calculation of the radiative corrections. The result of insertion for is

(23) | |||||

#### iii.1.3 Hard scattering with the spectator

Hard spectator scattering [Fig.1(g) and (h)] is completely missing in the naive factorization. But in QCD factorization, it can be calculated in the perturbative QCD, and expressed by a convolution of the hard kernel and the LCDAs of mesons. At the leading power of , both of the light pseudoscalars from the meson decay can be represented by their leading twist LCDAs. So after a straightforward calculation, we obtain this contribution for from the operator insertion,

(24) | |||||

### iii.2 Chirally enhanced corrections — Twist-3 LCDAs insertion

It has been observed that QCD factorization is demonstrated only in the strict heavy quark limit. This means that any generalization of QCD factorization to include or partly include power corrections of should redemonstrate that factorization still holds. There are a variety of sources which may contribute to power corrections in ; examples are higher twist distribution amplitudes, transverse momenta of quarks in the light meson, annihilation diagrams, etc. Unfortunately, there is no known systematic way to evaluate these power corrections for exclusive decays. Moreover, factorization might break down when these power corrections, for instance, transverse momenta effects, are considered. This indicates that one might have to give up such an ambitious plan that all power corrections could be, at least in principle, incorporated into QCD factorization order by order. One might argue that power corrections in decays are numerically unimportant because these corrections are expanded in order of a small number . But this is not true. For instance, the contributions of operator to decay amplitudes would formally vanish in the strict heavy quark limit. However it is numerically very important in penguin-dominated rare decays, such as interesting channels , etc. This is because is always multiplied by a formally power suppressed but chirally enhanced factor , where and are current quark masses. So power suppression might probably fail at least in this case. Therefore phenomenological applicability of QCD factorization in rare decays requires at least a consistent inclusion of chirally enhanced corrections.

Chirally enhanced corrections arise from the two particle twist-3 light-cone distribution amplitudes, generally called and . So when chirally enhanced corrections are concerned, the final light mesons should be described by leading twist and twist-3 distribution amplitudes. Then it is crucial to show that factorization really holds when considering twist-3 distribution amplitudes. The most difficult part is to demonstrate the infrared finiteness of the hard scattering kernels . In addition, possible chirally enhanced power corrections can also appear in the hard spectator scattering. So, for consistency, we must involve these corrections.

#### iii.2.1 Vertex corrections

When we calculate the chirally enhanced power corrections, contrast to the leading-twist light-cone wave function insertion, the cancellation of the infrared divergences in the vertex corrections to operator (here it is or ) can not be shown simply by the eikonal approximation similar to what has been done at the leading power of , because the Dirac structure or spin structure of twist-3 light-cone wave functions of the light pseudoscalar makes the “on-shell” condition for the external quarks invalid. Thus, to justify the cancellation of the infrared divergence in vertex corrections, we must give the explicit calculation. As mentioned in the previous subsections, we have two choices to regularize the infrared divergence in one-loop calculation. One is the DR scheme; the other is MG scheme. Generally, these two schemes are equivalent, for instance, similar to what has been done in vertex corrections. However, in the DR scheme, it is difficult to extrapolate the twist-3 wave functions of the light pseudoscalar to d dimensions properly, although they are well-defined in 4 dimensions. Therefore, we prefer to use the MG scheme in our calculation for chirally enhanced corrections to avoid the above possible problems.

In addition, generally we calculate the Feynman diagrams in the momentum space, so the correct projection of the light-cone wave functions of the light pseudoscalar in the momentum space is necessary. From Appendix B, we find that it is easy to find the proper momentum space projection of the leading twist and type twist-3 wave function, but for , the projection is not very clear. Note that the coordinate in the definition of by the non-local matrix element must be transformed into a partial derivative of a certain momentum in the projection of momentum space. However, it is difficult to find the derivative which makes the projection only depend on the structure of the light pseudoscalar itself. Generally, the momentum which the partial derivative acts on is dependent on the hard kernel. Therefore, we prefer to compute the Feynman diagrams of the twist-3 wave functions insertion, especially insertion in the coordinate space. We think that such calculation can avoid the ambiguity about how to project the coordinate into the momentum space. We recalculate the leading twist insertion by using the same method, and obtain the same results as those in the previous sections. Below, we will show how to perform this trick in calculation of insertion. For example, let us consider Fig. 3. In coordinate space, we have

(25) | |||||